New Shallow Investigations: Telecommunications and Civil Conflict Reduction

We recently published two shallow investigations on potential focus areas to the Effective Altruism Forum. Shallow investigations, which are part of our cause selection process, are mainly intended as quick writeups for internal audiences and aren’t optimized for public consumption. However, we’re sharing these two publicly in case others find them useful.

The default outcome for shallow investigations is that we do not move forward to a deeper investigation or grantmaking, though we investigate further when results are particularly promising.

If you have thoughts or questions on either of these investigations, please use this feedback form or leave a comment on the EA Forum.


Telecommunications in Low and Middle-Income Countries (LMICs)

By Research Fellow Lauren Gilbert (EA Forum link)

  • Lauren finds that expanding cellular phone and internet access appears to cost-effectively increase incomes. Randomized trials and quasi-experimental studies in LMICs showed that gaining internet access led to substantial increases in income, with high social returns on investment.
    • We find these reported effects surprisingly large, and are continuing to dig into them more.
  • Lauren estimates that 3-9% of the world’s population do not have access to cellular service, and ~40% of the world’s population either have no access to mobile internet or do not use it. Lauren finds that the biggest barrier to usage is the cost of devices and coverage. These coverage gaps and costs are shrinking over time.
  • A large majority of spending on telecommunications is private/commercial, with a smaller amount of philanthropic spending. While the private investments are large, they aren’t as focused as a philanthropist might be on improving access for poor and rural communities.
  • Philanthropists could potentially help improve access by subsidizing investments in cell phone towers to improve coverage, and in internet cables to reduce the cost of internet. Lauren’s rough back-of-the-envelope calculation suggests that these investments may be cost-effective. A funder could also potentially lobby for policy changes to reduce costs — for example, reducing tariffs on imported electronics or changing the rules around how spectrum can be licensed.

Civil Conflict Reduction

Also by Lauren Gilbert (EA Forum link)

  • Civil conflict is a very important problem. Lauren estimates that civil wars directly and indirectly cause the loss of around 1/2 as many disability-adjusted life years as malaria and neglected tropical diseases combined. Civil wars also substantially impede economic growth, mostly in countries that are already very poor. 
  • While civil conflict is important and arguably neglected, it isn’t clear how tractable it is. However, some interventions have shown promise.
    • Lauren finds some evidence that UN peacekeeping missions are effective, and argues philanthropists could lobby for more funding.
    • Some micro-level interventions, such as mediation or cognitive behavioral therapy, also have suggestive empirical evidence behind them. Philanthropists could fund more research into these interventions.

Social returns to productivity growth

Intro

At Open Philanthropy we aim to do the most good possible with our grantmaking. Historically, economic growth has had huge social benefits, lifting billions out of poverty and improving health outcomes around the world. This leads some to argue that accelerating economic growth, or at least productivity growth,[1]If environmental constraints require that we reduce our use of various natural resources, productivity growth can allow us to maintain our standards of living while using fewer of these scarce inputs. should be a major philanthropic and social priority going forward.[2]For example, in Stubborn Attachments Tyler Cowen argues that the best way to improve the long-run future is to maximize the rate of sustainable economic growth. A similar view is held by many involved in Progress Studies, an intellectual movement that aims to understand and accelerate … Continue reading

In this report, I describe a model that helps assess this view and inform our Global Health and Wellbeing (GHW) grantmaking. Specifically, I focus on quantitatively estimating the social returns to directly funding research and development (R&D), in a relatively simple/tractable model. I focus on R&D spending because it seems like a particularly promising way to accelerate productivity growth, but I think broadly similar conclusions would apply to other innovative activities.

Having a reliable estimate of the social returns to innovation would allow us to quantitatively compare it to other potential philanthropic activities like cash transfers to the global poor or public health interventions that extend life, and then accordingly allocate our total funding in the optimal fashion. However, we’re not sure how much weight to put on this particular estimate given the simplifications and assumptions involved.

In brief, I find that:

  • In a highly stylized calculation, the social returns to marginal R&D are high, but typically not as high as the returns in some other areas we’re interested in (e.g. cash transfers to those in absolute poverty). Measured in our units of impact (where “1X” is giving cash to someone earning $50k/year) I estimate the cost effectiveness of funding R&D is 45X. This is 45% the ROI from giving cash to someone earning $500/year, and 4.5% the GHW bar for funding. More.
    • This is an estimate of the average returns to R&D, but the best R&D projects might have higher returns. In addition, leveraged opportunities to increase the amount of R&D — like advocating for a more liberal approach to high-skill immigration — could have significantly higher returns. More.
  • Returns to R&D were plausibly much higher in the past. This is because R&D was much more neglected, and because of feedback loops where R&D increased the amount of R&D occurring at later times. More.
  • The stylized estimate has many important limitations, and is not an all-things-considered estimate of the social returns to R&D. For example:
    • It omits potential downsides to R&D, e.g. increasing global catastrophic risks. For certain types of R&D, these downsides may significantly outweigh the benefits. This is a very significant limitation of the estimate. More.
    • It focuses on a specific scenario in which population growth stagnates and historical rates of returns to R&D continue to apply. In this scenario, productivity growth eventually stagnates. While this is arguably the most popular model of the future, it is not the only plausible one, and I discuss how placing weight on alternative scenarios would change the bottom line. More.
      • One such scenario is that R&D today brings forward the development of a future technology, like advanced AI, that accelerates R&D progress much more than past technologies. This could significantly increase the returns to R&D. But conditional on such a scenario Open Philanthropy sees a stronger case for reducing long-term risks from this future technology than accelerating its development. More.
  • Overall, the model gives us a new stylized value for the ROI of abstract marginal R&D spending that we may use in Global Health and Wellbeing cause prioritization work, though we have substantial uncertainty about how much weight to put on it given all of the assumptions and limitations. The stylized value we get out does make us think that some causes aimed at accelerating overall innovation, like science policy or high-skill immigration advocacy, would likely pencil out as above our GHW bar, but it also leaves us relatively skeptical of arguments that accelerating innovation should be the primary social priority going forward. More.

My estimate of the social returns to R&D

I draw heavily on the methodology of Jones and Summers (2020). I won’t explain their model in full, but the basic idea is to estimate two things:

  1. How much would a little bit of extra R&D today increase people’s incomes into the future, holding fixed the amount of R&D conducted at later times?[3]An example of an intervention causing a temporary boost in R&D activity would be to fund some researchers for a limited period of time. Another example would be to bring forward in time a policy change that permanently increases the number of researchers. 
  2. How much welfare is produced by this increase in income?

For part 1, I use economic growth models that connect R&D investments with subsequent productivity growth. For part 2, I use a simple log-utility model: welfare = k + m*log(income).[4]Three comments on the log-utility model. First, the results are the same whatever the values of the constants k and m. Second, I do a sensitivity analysis of the consequences of different utility functions; if the diminishing returns to income are steeper than log, this favours cash transfers more … Continue reading This log-utility model has two implications that I will use:

  • Increasing someone’s income by 10% has the same welfare effect whatever their initial income. 
  • Increasing one person’s income by 10% has roughly the same welfare effect as increasing ten people’s incomes by 1%, or as increasing 100 people’s incomes by 0.1%, or 1000 people’s incomes by 0.01%.[5]log(110) – log(100) ~= 10*[log(101) – log(100)] ~= 100*[log(100.1) – log(100)] ~= 1000*[log(100.01) – log(100)].

Toy Example

Here’s a toy example to roughly demonstrate how I calculate the social returns to R&D, and how this can be compared with cash transfers to people in global poverty. 

Let’s estimate the welfare benefits of spending $20 billion on R&D.

  • Total global R&D spend is $2 trillion per year. This produces frontier productivity growth of 1% per year.
  • $20 billion would increase global R&D spend by a fraction of 1/100 for 1 year. So in that year, rather than 1% productivity growth we’d expect to have 1.01% productivity growth.[6]More realistically, there will be a lag before productivity benefits are felt. Currently I don’t model this lag because it wouldn’t affect the results by much. I use a discount of 0.2%; so a 50 year lag would reduce the returns to R&D by ~10%. 
  • In subsequent years, everyone’s incomes will be 0.01% higher because of the extra money spent on R&D in that one year. 
  • The benefit in each year is equal to (number of people alive) * (value of raising someone’s income by 0.01% for one year).
  • Let’s ignore the benefits after 50 years, as a rough way to incorporate a discount rate. 
  • For simplicity, let’s assume the number of people alive is constant at 8 billion.

These assumptions imply that:

Social returns to $20 billion on R&D
= (number of people alive) * (value of raising someone’s income by 0.01% for one year) * (years of benefit)

= 8 billion * (value of raising someone’s income by 0.01% for one year) * 50

= 400 billion * (value of raising someone’s income by 0.01% for one year)

= 400 million * (value of raising someone’s income by 10% for one year)

This last line follows from the log-utility model: increasing 1000 people’s incomes by 0.01% has the same welfare effect as increasing one person’s income by 10%.

The conclusion is that spending $20 billion on R&D has the same welfare benefit as increasing the incomes of 400 million people by 10% each. (In fact it would increase many more people’s incomes by a much smaller amount; the log-utility model allows us to express the welfare benefit in this way.)

An alternative altruistic intervention is simply to transfer cash directly to the global poor.[7]GiveDirectly implements this intervention. Note, I use simplified numbers in this post that don’t exactly match GiveDirectly’s cost effectiveness, and I believe GiveDirectly is somewhat more impactful than the numbers I use imply. For someone living below the international poverty line on $500/year, $50 raises their income by 10% for one year. With $20 billion, you could do this 20 billion / 50 = 400 million times. The total benefit would equal 400 million * (value of raising someone’s income by 10% for one year).

In this toy example, the social returns to R&D exactly equals that from cash transfers. As we’ll see below, a more realistic calculation seems to favour cash transfers over R&D.

The main factor favouring R&D is its potential to help so many people as technological innovations spread across the world. The main factor favouring cash transfers is that even a small amount of money can significantly improve the lives of people in poverty.

A more realistic calculation

The actual calculation of the social returns to R&D differs from this toy example in a few important ways. This table summarises these differences, and their effects on the social returns to R&D. Some of the differences interact in complex ways, so there’s no simple way to describe their quantitative effect.

 

Difference from the toy example Effect on the social returns to R&D compared to the toy example
I count benefits over a longer period of time Increase
I recognise that ideas are getting harder to find Decrease
I use UN population projections Small increase
I think that some people might not benefit from frontier technological progress. Decrease, 0.7X
I only give R&D partial credit for productivity growth Decrease, 0.4X
I assume diminishing returns to adding more researchers within any given year Small decrease
I incorporate capital deepening: higher productivity → more machines → higher incomes.  Increase, 1.5X
Total, combining all the above differences  Decrease, 0.45X

Let’s discuss each difference in more detail.

I count benefits over a longer period of time. We should value improving someone’s life equally whether they live now or in 500 years time.[8]Note, rising incomes mean we don’t value adding an equal dollar amount to people’s incomes the same amount through time. We value a dollar more today because people today are poorer than they will be in the future. I use a small pure time discount of 0.2%, representing the possibility that a major disruption (e.g. extinction) prevents the welfare benefits of R&D from occurring at all. This small discount pushes towards placing a higher value on R&D, compared to the toy example.

I recognise that ideas are getting harder to find. Let’s say that R&D makes progress by coming up with new ideas, and define an ‘idea’ so that each new idea raises productivity by 1%. It turns out that there’s good evidence that it takes more research effort to find new ideas than it used to.[9] During the 20th century, the number of researchers grew exponentially, but productivity growth did not increase (in fact it decreased slightly). If R&D is responsible for the productivity growth, then more research effort is required to achieve each subsequent 1% gain in productivity. One plausible explanation is that the most obvious ideas are discovered first, so that over time increasingly difficult ones remain. Another explanation is that researchers must spend increasingly long studying before they’re able to contribute to their fields.

Importantly, ideas are getting harder to find despite researchers having better tools (e.g. the internet) to aid their research today than in the past. Even with these improved tools, it still takes more research effort to find new ideas than it used to.

How does this dynamic affect social returns to R&D? In the toy example, the extra R&D caused everyone’s incomes to be 0.01% higher forever. It turns out that once you incorporate ‘ideas getting harder to find’, this is no longer true. The % income benefit shrinks over time and approaches 0%.[10]Note: this does not mean that the absolute $ increase in incomes shrinks over time. It may decline, stay constant or increase, depending on the rate at which ideas are getting harder to find. Technically, if the “fishing out” parameter $$ \phi $$ > 0, then the absolute $ benefit increases … Continue reading

This is shown in the diagram below, which compares the total factor productivity (TFP) in a world without any intervention (orange) and a world where an intervention temporarily boosts R&D activity (blue). Note: the y-axis is log, so the gap between the lines represents the % difference in TFP, not the absolute difference.

The initial % increase declines towards 0% over time. This dynamic decreases the social returns to R&D, compared to the toy example.

Why does the % productivity increase decline in this way? Essentially, the initial extra R&D “steals” easier to find ideas from future years, making future research less productive (see footnote for more detail).[11]The key point is as follows: when ideas are getting harder to find, the number of new ideas found with a marginal researcher-year is roughly proportional to 1 / (total researcher-years so far). So if the 100th researcher-year finds 1/100 new ideas, the 200th researcher-year will find only 1/200 new … Continue reading

How quickly are ideas getting harder to find? I use an estimate from Bloom et al (2020), which looks at how research efforts translated into TFP growth in the US from 1930 to 2015. The implication is that each time TFP doubles, it becomes ~5X as hard to find a new idea.[12]Mathematically, in the semi-endogenous growth model the effort needed to find a new idea is proportional to TFP^($$ \phi $$ – 1), where $$ \phi $$ is the parameter controlling how quickly ideas are getting harder to find. I use $$ \phi $$ = -1.4, so every time TFP doubles the effort needed to … Continue reading

As a result of this dynamic, most of the benefits of today’s R&D occur in the first 100 years despite the small discount rate.

I use UN population projections. When estimating the number of future beneficiaries for today’s R&D, I used the UN population projections which forecast that the global population will rise to around 11 billion by 2100 and then remain at that level.[13]In the long run, there are reasons to think population will fall (fertility rates in developed countries), reasons to think it might increase (relating to biological and cultural evolution), and no compelling reason to think it will stay exactly the same. Still, this feels like a fair … Continue reading These projections also inform my estimate of the amount of R&D that will be done in each year in the future. This contrasts with formal economic models in which the population is typically assumed to be constant or increasing exponentially.

The toy example assumed that the population would remain at 8 billion. Compared to this, using UN population projections increases the number of beneficiaries and so increases the returns to R&D.

I think that some people may never benefit from frontier technological progress. The toy example assumed that everyone benefits from frontier technological progress. After all, people in all continents use technologies like smart phones, cars and solar panels.

However, I’m very uncertain about whether people in low-income countries will ever feel the full benefits of frontier TFP growth. For example, some agricultural R&D done in the US won’t ever be applicable in countries with different climates. Currently, I assume (arbitrarily) that if frontier TFP increases by 10% then TFP around the world will eventually increase by 7%.[14]The lag until productivity benefits are felt will probably be larger in low income countries than in high income countries. As mentioned above, I don’t model this lag because it wouldn’t affect the results by much. I use a discount of 0.2%, so a 50 year lag would reduce the returns to R&D … Continue reading This part of the calculation is particularly uncertain about and I’m interested in suggestions for how to think about this. This adjustment multiplies the social returns to R&D by a factor of 0.7.

I only give R&D partial credit for productivity growth. Activities with a potential claim to credit include misallocation reduction, business innovation (e.g. startups), learning by doing and capital accumulation. Ultimately, I credit R&D with 40% of productivity growth and explain my reasoning here. This straightforwardly multiplies the bottomline by 0.4. In an appendix, I briefly sense-check this assumption against studies using statistical techniques to tease out the causal impact of R&D on growth. Naively, this suggests I should give R&D more credit for growth, but there are a number of complications involved in the comparison.

I assume diminishing returns to adding more researchers within any given year. The toy example assumed that if you increase R&D funding by 1%, you’ll make 1% more R&D progress. That logic implies that doubling R&D funding would double the rate of R&D progress. However, there may in fact be diminishing returns to spending, e.g. because some research effort is duplicated. In line with this, the model assumes that the marginal $ spent on R&D causes only 75% as much R&D progress as the average $ spent.[15]In economic growth models, this corresponds to the “stepping on toes” parameter λ = 0.75. I’m not aware of data that pins down λ, and it seems like values between 0.4 and 1 could be correct. I use the estimate from Bloom et al. (2020) Appendix Table A1, where they set λ = 0.75 and then … Continue reading

I incorporate capital deepening: higher productivity → more machines → higher incomes. Suppose you invent a drug (e.g. caffeine) that makes everyone slightly better at their jobs. We can distinguish between two effects. A primary effect is that everything people buy is higher quality – e.g. better haircuts, tastier food, faster transport – because the people producing these goods and services are better at their jobs. A secondary effect is that when people and companies invest in buying tools to help them do their jobs (e.g. computers), they’ll get more tools. After all, the people producing these tools are better at their jobs. Having more tools makes people better at their jobs. In the economics literature, this secondary effect is called capital deepening.[16]The primary effect is recorded as a TFP increase because GDP went up holding constant the amount of labour and physical machinery. The secondary effect is recorded as capital deepening because each person has more physical capital (i.e. more or better machinery).

Both effects ultimately increase the quality or quantity of goods and services produced, and so raise incomes. My toy example included the primary effect, but not the secondary effect. Including both effects increases the benefit by a factor of 1.5.[17]Growth theory relates the size of these effects on income: (income increase from TFP and capital deepening) = (income increase from TFP alone) / (1 – capital share of GDP). The capital share is about 35%, so this multiplies the bottom line by 1 / (1 – 0.35) = 1.5.

Bottom line – social returns to R&D

Once we incorporate all these changes to the toy example, what are the social returns to R&D?

The toy example found that a marginal $20 billion to R&D has the same welfare benefit as increasing the incomes of 400 million people by 10% for one year. (In fact it would increase many more people’s incomes by a much smaller amount; the log-utility model allows us to express the welfare benefit in this way.)

With these changes, $20 billion to R&D has the same welfare benefit as increasing the incomes of 180 million people by 10% each for one year. (Again, it would actually increase many more people’s incomes by a much smaller amount; the log-utility assumption allows us to express the welfare benefit in this way.)

This is 45% of the benefit calculated in the toy example. We can break this decrease down into a 0.4 penalty from only giving R&D partial credit for productivity growth, a 0.7 penalty from uncertainty about whether frontier TFP progress really spills over to the whole world, and a 1.5 gain from capital deepening; combining these yields a 0.42 penalty. Then the other changes mostly cancel each other out. (If you want to know more, there’s a full description of the model in this appendix.)

To make this result more relatable, let’s consider smaller expenditures. Dividing both the costs and benefits by 180 million, ~$110 on R&D has the same welfare benefit as increasing one person’s income by 10% for one year.

These returns are high. However, they’re not as high as cash transfers to people in global poverty. $110 to someone living on $500/year increases their income by 22% for one year.

Intervention Welfare impact of $20 billion Welfare impact of $110 Cost effectiveness in Open Philanthropy’s units of impact
R&D (final calc) Increase the incomes of 180 million people by 10% for 1 year. Increase the income of one person by 10% for 1 year. 45X
In fact, R&D would increase many more people’s incomes by a much smaller amount and for a much longer time; the log-utility model allows us to state the benefit in this way.
Cash transfers to people on $500/year[18]As mentioned in a previous footnote, I think GiveDirectly is somewhat more impactful than the numbers in this row. Increase the incomes of 400 million people by 10% for 1 year. Increase the income of one person by 22% for 1 year. 100X
Cash transfers to people on $50,000/year Increase the incomes of 4 million people by 10% for 1 year. Increase the income of one person by 0.22% for 1 year. 1X

The GHW team at Open Philanthropy aims to only make grants whose expected impact is above a certain bar. Our current tentative bar is 1000X, as measured in units where “1X” is giving $1 to someone earning $50k/year. In these units R&D comes out as 45X, 4.5% of the bar.

There’s an important sense in which this comparison is biased in R&D’s favour. The calculated benefits from R&D include those that occur many decades into the future, while the income increase from a cash transfer is immediate. If we included long-run benefits from cash transfers, they would beat R&D by a wider margin.

The best pro-growth interventions are better than average R&D

One important caveat is that there may be leveraged ways to boost the amount of R&D. For example, lobbying for more high-skilled visas could increase the effective global number of skilled R&D workers, accelerating R&D progress more than paying for R&D directly.

A second caveat is that we’ve estimated the average impact of marginal R&D funding. Of course, the actual impact of any particular grant could be much larger or much smaller than this, depending on the project being funded. If a funder can consistently identify particularly promising projects, their impact could be larger than my estimate. One way to do this might be to focus on R&D projects that are specifically designed to help the global poor. Just as $1 goes further when transferred to the global poor, so too R&D might be more effective when targeted in this way.

Some of those involved with Progress Studies think accelerating innovation should be the world’s top priority. I discuss ways in which my outlook differs from theirs in this appendix.

 

Funding R&D was even better in the past

Today the returns to R&D are high. I think that they were even higher in the past for a couple of reasons.

R&D was more neglected. We’ve seen that ideas are getting harder to find over time as the easiest ones are discovered. In the past, much less R&D had been done in total and so ideas were significantly easier to find. Appendix F estimates that the fraction of the economy dedicated to research was 36 – 96X smaller in 1800 than today.

R&D increased the amount of R&D occurring at later times. I think that, historically, there were two mechanisms by which R&D caused more R&D to occur at later times.

  • Increasing the fraction of resources used for R&D. The first is providing evidence that R&D was a worthwhile activity. The fraction of people doing R&D has increased significantly over time.[19]For example data from Bloom et al. 2020 find the number of US researchers increasing by 4.3% per year on average since 1930. US population grew less than 1.5% per year on average in the same period, implying that the fraction of people doing research was growing.
    Earlier instances of R&D are probably an important reason, as their success fuelled the expansion of R&D efforts. The mechanism is: R&D → evidence of success → more R&D. Today this mechanism is probably less important, as there is already ample evidence of the fruitfulness of R&D (though funding specific neglected areas of R&D can have a similar effect today if you are better at predicting fruitful areas than the average funder). 
  • Increasing the future population. The second is that, going back hundreds and thousands of years, productivity improvements allowed a fixed supply of land to support larger populations, which meant more people to engage in innovative activity (though the fraction of people doing so was very low). The mechanism is: R&D → fixed supply of land can support larger populations → more people are alive to do R&D in the future. This dynamic is important in some prominent models of long-run growth.[20]See for example Lee (1988), Kremer (1993), Jones (2001) and Galor and Weil (2000). Today this mechanism is not important: population growth is determined by people’s fertility decisions rather than by how many people society is able to feed.

A back-of-the-envelope calculation suggests that the combined effects of these mechanisms could be very large, with the social impact of R&D hundreds of times greater in the past than today.

So it may be that historically R&D was the most promising philanthropic intervention, even if it isn’t quite as promising today.

 

Limitations of the model

The model ignores potential harms from R&D

Certain types of R&D might have large downside risks. For example, gain of function research can make pathogens more deadly and transmissible, potentially increasing global catastrophic risk from a pandemic.

This consideration might make funding certain types of R&D very harmful, reversing the conclusion of the model. I think it would be a mistake to act on the basis of this post without explicitly considering these downside risks.

Evaluating which types of R&D pose the largest risks is beyond the scope of this post. This is a significant limitation, and highlights that this post gives a stylized estimate of the returns to R&D but does not give an all-things-considered assessment.

The model implies growth will stagnate

My mainline scenario, used for the stylized calculation, implies that productivity growth will tend to 0% per year in the very long run.

Why does this happen? It’s the combination of two assumptions:

  • Ideas are getting harder to find. Each 1% increase in productivity requires more research effort over time, even accounting for the fact that researchers can use new technologies to aid their research effort.
  • Population stagnation. Population will rise to 11 billion and then remain roughly constant.

Together these two assumptions imply the pace of productivity growth will slow. Population stagnation implies that the number of researchers will eventually stagnate.[21]Though the fraction of people doing research can increase, this can only go on for so long. I discuss this possibility below. Ideas are getting harder to find then implies that it will then take increasingly long to find each new idea. This implication is explored in depth in Jones (2020).

The following graph shows my model’s prediction of productivity stagnation alongside a scenario where TFP grows exponentially at its recent historical rate forever.[22]There are good theoretical reasons to think TFP can’t grow exponentially at its recent rate for more than 10,000 years, but these don’t rule out exponential growth continuing for another 1000 years.

Productivity stagnation is a surprising implication of the model; people might wonder whether the model is overly pessimistic about productivity growth.

To address this, I explored four alternate scenarios in which productivity doesn’t stagnate:

  1. Maybe we’ll avoid productivity stagnation by increasing the fraction of people doing R&D (even more than in my model)? 
  2. Maybe ideas will not get harder to find in the future?
  3. Maybe the world’s population won’t stagnate?
  4. Maybe some trend-breaking future technology will allow us to avoid growth stagnation despite ideas getting harder to find? More.

I did very rough back-of-the-envelope calculations of how the social returns to R&D change in each scenario. Scenario 1 reduces the returns to R&D, multiplying the bottom line by ~0.4X. Scenarios 2 – 4 can multiply the bottom line by up to ~7X, or much more for some versions of scenario 4.[23]If R&D today expedites a future technology that massively accelerates future growth, the bottom line can increase by much more than 100X. More.

If I did a weighted average across these scenarios, it would probably increase my bottom line compared to simply using the mainline scenario. (The size of the increase would be very sensitive to the specific weights used, especially for the versions of scenario 4 with massively outsized returns.)

For now, the stylized estimate at the top of this post doesn’t put any weight on these scenarios. One reason for this is that the stylized estimate excludes scenarios where R&D causes large harm, so it feels fair to similarly exclude trend-breaking scenarios where R&D has large upsides. Another reason is that I want the stylized estimate to be comparable with Open Philanthropy’s impact estimates for other Global Health and Wellbeing cause areas, and we don’t consistently place weight on unlikely scenarios with large upsides.

In addition, I have specific reasons for excluding each of the four scenarios:

  1. My mainline scenario already involves significant increases in the fraction of people doing R&D. More.
  2. This scenario seems very implausible. More.
  3. The high returns in this scenario are driven by tiny benefits enjoyed by a massively expanded population 100s of years into the future. More.
  4. Conditional on the development of such a growth-boosting technology in the next century, Open Philanthropy currently prioritizes work reducing risks from this technology over work accelerating its arrival. More.

That said, I’m very uncertain about how much weight the GHW team should place on these scenarios and I think there is room for reasonable disagreement. It’s worth noting that even a 7X increase would leave unlevered R&D funding ~3X less effective than Open Philanthropy’s bar for funding within GHW (though it would likely imply that various levered advocacy or research spending opportunities should make up much more of our portfolio).[24]My mainline scenario found R&D to be 45% as impactful as giving cash to someone on $500/year. This implies R&D is 4.5% as impactful as our current bar for GHW grantmaking. A 7X increase would leave R&D 31.5% as impactful as the GHW bar.

I find scenario 4 the most plausible, and discuss it further in the next section. I discuss scenarios 1 – 3 at greater length in this appendix.

 

Maybe some trend-breaking future technology will allow us to avoid growth stagnation despite ideas getting harder to find?

Increasing research effort has been required to find new ideas, even though previous discoveries have made researchers more productive (e.g. calculators, coding tools, caffeine, the internet). Extrapolating this trend, we’d predict that future technological progress will make researchers somewhat more productive but that this won’t be enough to avoid productivity stagnation.

Perhaps, though, this trend won’t continue. Perhaps future technologies will enhance our research abilities more than those from the last 80 years. For example, if we develop advanced AI systems that can do independent research we might massively increase our research efforts.[25]Aghion et al. (2017) discuss the possibility that AI will accelerate productivity growth by automating research tasks. Another possibility is advanced bio-technology that radically enhances the productivity of human researchers. Call such technologies growth-enhancing technologies.[26]Note, a growth-enhancing technology might allow a constant population of human researchers to maintain ~2% productivity growth, or it might allow them to accelerate productivity growth. Open Philanthropy thinks the latter possibility is more likely than many actors seem to think, for reasons … Continue reading

Past technologies have enhanced research productivity somewhat, but we’ve still had to increase the number of human researchers to maintain constant productivity growth. Growth-enhancing technologies would (by definition) allow us to maintain constant productivity growth without increasing the number of human researchers.[27]Of course, growth-enhancing technologies might enable other trends to continue. E.g. the trend of ~2% annual growth in US GDP/capita over the past 150 years, or the trend of growth accelerating over the past 10,000 years.

If a growth-enhancing technology is developed sometime in the future, how would this alter the value of R&D today?

It turns out that this depends on whether R&D today affects the time at which the growth-enhancing technology is developed.

If R&D today doesn’t affect when a future growth-enhancing technology is developed, then its development reduces the value of R&D today. This is somewhat counter-intuitive: although more total R&D will happen, the marginal value of R&D today is lower. The reason is related to ideas getting harder to find. More R&D in later periods pushes us further up the diminishing returns curve for finding new ideas, so the additional R&D we funded makes less difference.[28]Let’s demonstrate this point with an example. Suppose an intervention causes an extra researcher-year to happen in 2021. Let’s consider its impact on TFP in 2100 if a growth-enhancing technology isn’t developed, and if it is developed. Suppose that if a growth-enhancing technology isn’t … Continue reading

If R&D today accelerates the development of a future growth-enhancing technology, its impact on future incomes could be much larger than my estimate. It would bring forward in time an income boost, raising people’s incomes for a long time into the future.[29]Appendix D does a very rough BOTEC on the returns to R&D for one possible growth-enhancing technology.

If this is the scenario anticipated by proponents of a “growth-first” worldview or the Progress Studies community, I think this would be worth being more explicit about; I take them to more typically be arguing from past trends rather than speculating about future trend-breaking technologies.

Open Philanthropy institutionally thinks the possibility of these sorts of trend-breaking future technologies is notably higher than many other actors in society seem to, and the Longtermist team of OP focuses explicitly on optimizing the expected impact of their spending in the world where such technologies are likely in the next century or so. However, conditional on placing high probabilities of the development of such technologies in the next century, the Longtermist side sees work on accelerating growth as lower impact than work reducing risks.[30]Eg. see this draft report by Joe Carlsmith on risk from power-seeking AI, or these two posts from the Cold Takes blog.

Absent more external consensus, we’re reluctant to have the prioritisation of our Global Health and Wellbeing team be driven by the possibility of trend-breaking future technologies. This is for reasons related to our views on worldview diversification.[31]Even if R&D isn’t competitive according to either worldview, might it look competitive according to a weighted sum of both? I think not. I estimate funding generic R&D to be ~10X worse than the GHW bar, and it looks significantly worse from a LTist perspective than alternative … Continue reading As such, I am currently not putting significant weight on this scenario in evaluating the social returns to R&D. We’d be interested to know if advocates for the primacy of growth or Progress Studies think that such trend-breaking future technologies are likely and/or crucial to their case for prioritizing growth going forward — that hasn’t been our impression from what we’ve read — and if so how they think about prioritizing growth relative to reducing longterm risks.[32]Appendix H discusses some potential differences between my perspective and that of Progress Studies advocates.

Other limitations of the model

Appendix K discusses two more debatable assumptions made by the model:

  • It assumes that increasing the amount of R&D in 2021 doesn’t affect the amount of R&D effort in future years.
  • It assumes that welfare increases with log(income).

In both cases, the assumptions of the model could be too aggressive or too conservative about the returns to R&D.

Appendix J lists additional ways in which the stylized calculation is arguably pessimistic or optimistic about the social returns to R&D.

How significant are these limitations?

I think these limitations are very significant. Including the downsides to R&D could make the returns to R&D substantially worse or even negative, while putting weight on scenarios with extreme upside could make the returns much better. 

 

Conclusion

The model discussed here gives Open Philanthropy a stylized value for marginal R&D spending that we may use in our GHW cause prioritization work. It suggests that the social returns to direct R&D spending are high, but not as high as some opportunities relating to poverty alleviation. Still, leveraged ways to boost R&D activity could be highly impactful, and Open Philanthropy may enter causes like high-skill immigration or science policy in part because of this modeling.

In addition, the research leaves me relatively skeptical of arguments that accelerating innovation is the primary social priority going forward. I estimated marginal R&D spending to be 4.5% of the GHW bar. Even if we ignore potential harms from R&D and consider an alternative scenario where R&D is 7X more valuable, directly funding R&D still doesn’t meet the GHW bar. The only scenario I considered where R&D returns are higher still – R&D today accelerates the development of a growth-enhancing technology – is one where Open Philanthropy currently, and in my view correctly, prioritizes reducing risks over accelerating timelines.

The stylized estimate in this post has huge limitations. Perhaps most important is that it excludes potential harms from R&D, and accounting for this factor could reverse the conclusion of the model. Based on this, and the model’s many other limitations, I see this post as opening a conversation about the returns to R&D rather than closing it.

 

Description of the appendices

The appendices, contained in a public google doc, dig into various aspects of this post in more detail. I recommend only reading appendices that are of particular interest.

  • Appendix A explains the model used in the stylized estimate of the returns to R&D, including all the assumptions needed to recover the result. 
  • Appendix B and appendix C do a deep dive into the model’s implications for how marginal R&D affects incomes in the short run and the long run.
  • Appendix D contains very rough estimates of the social returns to R&D in the alternative scenarios mentioned in the main post, allowing for a comparison with this post’s mainline scenario. Appendix E discusses some of these alternative scenarios qualitatively. The main takeaways from these two appendices were discussed above.
  • Appendix F contains a very rough estimate of the returns to R&D in 1800. The key takeaway is that incorporating the qualitative differences discussed above implies that R&D in 1800 could have been more than 100X more impactful than R&D today.
  • Appendix G sanity checks some of the assumptions of the model used in this post against economics papers that use statistical techniques to try and tease out the causal effect of R&D on growth.
  • Appendix H discusses how my current view differs from the views of those involved in the Progress Studies movement.
  • Appendix J lists ways in which the stylized calculation is arguably too optimistic or too pessimistic about R&D spending.
  • Appendix K briefly explores alternatives to two assumptions of the stylized calculation. The assumptions are:
    • The model assumes that increasing the amount of R&D in 2021 doesn’t affect the amount of R&D effort in future years; but it might increase or decrease it.
    • The model assumes that welfare increases with log(income).

Note that the calculations in the appendices are somewhat less vetted than those in the main text. The main text calculations have been reproduced using multiple methods, and checked by other researchers at Open Philanthropy.

 

Appendix A: Assumptions of the model

The calculation proceeds in two stages. First, I run a simple simulation to estimate the social impact of R&D. Second, I make a series of adjustments to make the bottom line more realistic.

Simulation assumptions

What model of R&D am I using?

I use the semi-endogenous growth model of Jones (1995).

TFP growth is given by:

$$$ g_A = constant * L^{\lambda}A^{\phi – 1} $$$

$$ L $$ is the number of researchers, A is the level of TFP, $$ g_A $$ is the growth rate of TFP.

$$ {\phi} $$ and $$ {\lambda} $$ are constants that control the diminishing marginal returns (DMR) to R&D. The lower these constants, the steeper the DMR.

  •  $$ \lambda $$ controls the DMR to more researchers at a given point in time (“stepping on toes”)
    • If  $$ \lambda $$=1, doubling researchers in a year doubles the TFP growth in that year
    • If $$ \lambda $$=0, doubling researchers in a year doesn’t change TFP growth that year.
    • If $$ \lambda $$=0.4, doubling researchers in a year increases TFP growth by 1.3X.
    • I think 0.4<<1 is reasonable.
      • Unfortunately, this is based on little but intuition. I think <1 because I think there are a few plausible mechanisms for giving at least a small stepping on toes effect. I tentatively think >0.4 because I’d be surprised if doubling the number of researchers speeds up R&D progress by <1.3X. I’m very interested to hear about further evidence on this point.
  • $$ \phi $$ controls whether progress today makes future progress harder
    • The smaller $$ \phi $$, the more progress today makes future progress harder (“fishing out”). 

What can the empirical data tell us about $$ \phi $$  and $$ \lambda $$?

The basic empirical trend is that (# researchers) has grown exponentially over the past 80 years, but TFP growth has stayed roughly constant.

Let’s assume that the R&D effort has been driving the TFP growth (or some constant fraction of it). Then there are then two explanations you could give for why TFP growth has stayed constant:

  1. Adding more researchers in a given year doesn’t actually increase the progress made very much. There’s a big “stepping on toes” effect. Small $$ \lambda $$.
  2. Each 1% increment in TFP is harder to achieve than the last. So more progress is required to achieve it. There’s a big “fishing out” effect. Small $$ \phi $$.

We can have (1) and (2) in various different combinations, as long as their combined effect is strong enough to explain the basic empirical fact. If you increase $$ \lambda $$, you’ll need to decrease $$ \phi $$ to compensate and make your predictions consistent with the historical data.

So the historical data don’t pin down both $$ \phi $$ and $$ \lambda $$ separately. Instead, I make an assumption about $$\lambda $$ and the data pins down $$ \phi $$. Two options mentioned in Bloom et al. (2020) are:

  • $$ \lambda $$=1, $$ \phi $$=-2.1
  • $$ \lambda=0.75 $$, $$ \phi $$=-1.4

I use the second; serial bottlenecks do place DMR on parallel research and marginal researchers are less talented (if our intervention increases researcher-concentration).

In appendix B I explain the consequences of this model for the social returns to R&D. The rest of this appendix continues to list the assumptions used.

 

How do I apply this model in the simulation?

The simulation compares utility in two worlds. In world 1 we slightly boost the amount of global R&D activity for 1 year.[33]The simulation assumes that the increase in R&D activity is proportional to the increase in funding. This may be optimistic: in reality you need both funding and researchers to do R&D. Essentially, the simulation assumes that more funding will bring with it more researchers, which may be … Continue reading In world 2 we do nothing. After this first year the R&D activity is the same in both worlds.

The simulation applies the R&D equation above in each year; making assumptions about the number of researchers L in each year and calculating the resultant TFP trajectory. It assumes R&D instantaneously boosts TFP around the whole world. It assumes incomes are proportional to TFP,[34]So it ignores the additional effect that capital deepening has on TFP increases in standard growth models. and calculates income trajectories for worlds 1 and 2.[35]It assumes everyone is on the world average income. Representing income inequality wouldn’t change the results. This is because we ultimately care about the percentage effect of R&D on income, and this is the same no matter what people’s starting incomes are. We care about the percentage … Continue reading It then converts these to utility trajectories, assuming utility = k + m*ln(income).[36]The specific values used for k and m do not affect the result as they cancel. In practice we use k=0 and m=1. It makes assumptions about the world population to calculate the total difference in utility between worlds 1 and 2 in each year.

To quantify the result, the simulation compares the utility from funding R&D to the utility that could have been achieved by increasing the consumption of someone on the average global income.[37]It quantifies the result in this way because this is a metric Open Philanthropy uses internally to compare the impacts from different kinds of intervention.

In particular, the simulation calculates the following quantity:

$$$ Impact \, multiplier = (utility \, from \, TFP \, gains \, of \, \$1 \, to \, R \&D) \, / \, (utility \, from \, \$1 \, to \, average \, world \, consumption) $$$

 

Other simulation assumptions

The simulation makes assumptions about the world population in each year and about the fraction of the world population who are researchers in each year.

  • Population 
    • The current population is 7.9 billion.
    • Population increases at 0.4% per year until 2100, when it reaches 11 billion.
    • Thereafter the population remains constant at 11 billion.
  • Fraction of population who are researchers
    • This eventually increases by a factor of 13 compared to today.
    • The fraction of the population doing research is proportional to ‘research intensity’, the fraction of GWP spent on R&D.
    • Current research intensity = 2.3%.
    • Research intensity increases linearly at an absolute rate of ~0.05% per year until it reaches 30%.[38]More precisely, the absolute size of the annual increment is 2% of current research intensity: 0.02 * 2.3% = 0.046%. So this assumption corresponds to thinking that research intensity has been growing exponentially at about 2% per year, but this exponential rate of increase will decline over time.

      • 2050: 3.7%
      • 2100: 6%
      • 2200: 10.6%
      • 2300: 15.2%
      • 2500: 24.4%
      • The research intensity increase can be understood as including the effects of catch-up growth. When countries develop they typically increase their R&D output; for example R&D from India and China is likely to significantly increase this century.

Note that the calculations in the appendices are somewhat less vetted than those in the main text. The main text calculations have been reproduced using multiple methods, and checked by other researchers at Open Philanthropy.

Adjustments made to simulation output

Expressing the benefit in terms of a % income increase

Most readers should skip this section. It describes a conceptual adjustment from the ‘impact multiplier’ metric used in the simulation to the ‘% income increase’ metric used in this blog.

The simulation calculates the following quantity:

$$$ Impact \, multiplier = (utility \, from \, \$1 \, to \, R \&D) \, / \, (utility \, from \, \$1 \, to \, average \, world \, consumption) $$$

It finds that this quantity = 21. Rearranging:

$$$ (utility \, from \, \$1 \, to \, R \&D) = 21 \times (utility \, from \, \$1 \, to \, average \, world \, consumption) $$$

Average world consumption (in nominal terms) is ~$10,000, so $1 to average world consumption would increase one person’s income by 0.01% for one year.

$$$ (utility \, from \, \$1 \, to \, R \&D)=21 \times \, (utility \, from \, increasing \, one \, person’s \, income \, by \, 0.01 \% \, for \, one \, year) $$$

$$$ (utility \, from \, \$1 \, to \, R \&D)=(utility \, from \, increasing \, 21 \, people’s \, income \, by \, 0.01 \% \, for \, one \, year) $$$

Multiplying the costs and benefits by 20 billion (in line with the discussion in the main text):
$$$ (utility \, from \, \$20b \, to \, R \&D)=(utility \, from \, increasing \, 420 \, billion \, people’s \, incomes \, by \, 0.01 \% \, for \, one \, year) $$$
$$$ (utility \, from \, \$20b \, to \, R \&D)=(utility \, from \, increasing \, 420 \, million \, people’s \, incomes \, by \, 10 \% \, for \, \, one \, year) $$$

 

Substantive adjustments

I multiply the above number by 0.42 due to three adjustments: 0.7 for the possibility that productivity benefits are not felt worldwide, 0.4 because I only credit R&D with 40% of TFP growth, and 1.5 because higher TFP leads to capital deepening. This implies:
$$$ (utility \, from \, \$20b \, to \, R \&D) = $$$
$$$ =(utility \, from \, increasing \, 420 \, million \, people’s \, incomes \, by \, 10% \, for \, one \, year)*0.42 $$$
$$$ =(utility \, from \, increasing \, 180 \, million \, people’s \, incomes \, by \, 10%  \, for \, one \, year) $$$

This gets us to the bottom line quoted in the main body. This sheet contains the above calcs for getting from the simulation output to the table in the main body.

The next section explains the 0.4X adjustment in more detail; the others are explained in the main post.

What proportion of productivity benefits should we credit to R&D?

I multiply by a factor of 0.4 because I credit measured R&D with 40% of productivity growth. Deciding how much credit to assign to different sources of income growth is a major source of uncertainty, and I don’t know of a principled and fully satisfactory approach for doing this. My reasoning was as follows:

  • Misallocation reduction gets 25% of the credit. This is based on Hsieh et al. (2013), which estimates that improvements in the allocation of talent explains 24% of growth in GDP/worker. A similar adjustment is made by Jones (2021).
  • Learning by doing gets 0% of the credit. 
  • Although learning by doing does lead to productivity improvements, this happens downstream of the introduction of new production processes. Without these new production processes, learning by doing would eventually dry up. I give the ultimate credit to the R&D (and other types of innovation) that develops these new production processes in the first place.
    • What we really care about is the counter-factual: if R&D progresses faster than it otherwise would have, what effect will this have on productivity growth? In this context, I am claiming that if R&D progressed faster, learning by doing would speed up in response (with a lag). The extra R&D would be counterfactually responsible for both more R&D progress and more learning by doing.
  • Measured R&D gets 55% of the remaining credit for innovation. 
    • In Eurostat’s survey of 28 countries, firms reported that R&D is 55% of total innovation costs.[39] See page 20 of Jones and Summers (2020). The other costs are linked to acquiring new equipment and software.
    • You could argue this is overly generous to measured R&D if R&D is systematically under-reported and this won’t be captured by survey responses. Some examples:
      • Wal-Mart sometimes doesn’t report R&D expenses but its logistics innovation has probably contributed to US TFP growth. 
      • Innovation related to improving services and introducing more product variety may also not be reported as R&D.
      • Startups are often focussed around highly innovative activities like taking new products to market, but may not bill much of this as R&D.

So I give R&D credit for 0.75*0.55 = ~40% of TFP growth.

Another method for arriving at ~40% is to assume that the combination of R&D spending and all net investment is responsible for 100% of growth. Applying this within the US, R&D spending is 2.7% GDP and net domestic investment is 4% of GDP. so the R&D spending is responsible for 2.7/6.7 = 40% of TFP growth.

In an appendix, I sense-check this assumption against studies using statistical techniques to tease out the causal impact of R&D on growth. Naively, this suggests I should give R&D more credit for growth, but there are a number of complications involved in the comparison.

Full list of assumptions

  • Utility = k + m * log(income)
  • R&D equation:
    • $$ g_A = constant * L^{\lambda}A^{\phi – 1} $$
      • Or equivalently, $$ dA = constant * L^{\lambda}A^{\phi} $$
    • $$ \lambda $$=0.75 
    • $$ \phi $$=-1.4
  • L proportional to $ spent
  • Initial value of $$ A $$ set by assuming a steady state with $$ L $$ growing at 3% per year – its average growth over the last 20 years in the Bloom et al. (2020) data set.
    • The formula for the steady state value of A is from the equation just below equation (19) on p.53 of Jones and Summers (2020).
    • If this initial value was lower, marginal R&D today would be more impactful.
  • Population 
    • Current population is 7.9 billion.
    • Population increases at 0.4% per year until 2100, when it reaches 11 billion.
    • Thereafter the population remains constant at 11 billion.
  • Fraction of population who are researchers
    • The fraction of the population doing research is proportional to ‘research intensity’, the fraction of GWP spent on R&D.
    • Current research intensity = 2.3%.
    • Research intensity increases linearly at an absolute rate of ~0.05% per year until it reaches 30%. Then it remains at 30%.
    • That’s a factor of 30/2.3 = 13 increase compared to today.
  • Discount rate on future utility: 0.2%
  • Adjustment for R&D benefits spreading all over the world: 0.7
  • Proportion of TFP growth credited to R&D: 0.5
  • TFP gains from R&D increase everyone’s incomes by the same % amount. 

You can see the full simulation code here, which provides sources for these assumptions. Alternatively, you can see a spreadsheet version of the calculation here.

 

Appendix B: understanding the implications of semi-endogenous growth for the social returns to R&D

I think about the impact of the intervention in terms of a ‘TFP wedge’: the % difference in TFP between the worlds with and without the intervention. First, I’ll discuss the initial size of the wedge; then how the wedge changes over time.

(In this section I quote various technical results; I derive these in appendix C.)

What’s the initial size of TFP wedge?

Suppose TFP growth in the world without the intervention is $$ g_0 $$, and the intervention increases the amount of R&D that year from L to L(1 + v). Then it turns out that the initial size of the wedge is:
$$$ wedge = {\lambda} g_0v $$$
$$$ wedge = {\lambda}* (growth \, without \, intervention)*(fractional \, increase \, in \, R \&D) $$$
For example, suppose $$ {\lambda} $$ = 0.5, TFP growth would be 2% without our intervention, and we boost total R&D spend one year by 1%. Then our intervention increases TFP by 0.5 * 2% * 1% = 0.01%.

 

How does the TFP wedge change over time?

Suppose the initial wedge is x% of total TFP. Over time x% falls towards 0%.

Why? Because ideas are getting harder to find. Suppose we counterfactually insert 1 extra researcher-year in 1800. Every year y after this, rather than R(y) researcher-years having occurred R(y)+1 have occured.

Initially, the level of technology might have been quite low, and this extra researcher-year might make a noticeable difference to TFP. Ideas are still easy to find.

Once the level of technology is high, however, this extra researcher-year makes little difference to TFP. Ideas are now very difficult to find.

To summarize: higher level of technology → ideas are harder to find → the extra researcher-year makes a smaller counterfactual difference to TFP → TFP wedge is smaller. Indeed, it turns out that the size of the wedge is inversely proportional to the level of technology (to some power).

The following diagram shows the technology level for two paths. The orange path is the one without an intervention; the number of researchers grows exponentially. The blue path has the same number of researchers each year as the orange path, except for the first year when it has 3X as many. You can see that the initial ‘ TFP wedge’ between the paths declines over time.

This means that the faster technology progresses, the faster the wedge declines. Faster tech progress → ideas become harder to find more quickly → TFP wedge declines more quickly.

It turns out that the wedge declines at an exponential rate of $$ (1-{\phi})g_A $$, where $$ g_A $$ is the growth rate of TFP. In an equilibrium where population grows at a constant exponential rate n, the wedge declines at an exponential rate of $$ {\lambda} $$n.

This has some interesting consequences.

  • If population growth is higher, the TFP wedge declines more quickly.
    • Faster population growth → faster tech progress → wedge declines faster
    • Naively, population growth would dramatically increase the returns to TFP boosts, but this is partly cancelled out by the faster-falling wedge.
    • Conversely: a stagnating population would naively significantly lower the returns to TFP boosts, but this is again partly cancelled by the slower-falling wedge. When population is constant, technological progress becomes slower and slower over time (ideas getting harder to find). So the wedge falls more and more slowly over time. If it takes 100 years to halve in size, it will take a further 200 years to halve again, and then 400 years, etc. 
  • If $$ \lambda $$ is smaller (more “stepping on toes”), the wedge declines more slowly.
    • Smaller $$ \lambda $$ → slower tech progress → wedge declines more slowly

We saw above that smaller $$ \lambda $$  leads to a small initial wedge. This effect is partly cancelled by the slower decline of the wedge. The “smaller initial wedge” effect dominates, except for very small discounts (<0.3%)[40]How long does it take for the “wedge declines more slowly” effect to dominate if we have no discount? Let’s assume we change our value of $$ \phi $$ to compensate when we change lambda. How long does it take for the total impact from $$ \lambda $$=0.75 to exceed the impact from $$ \lambda … Continue reading

 

Appendix C: deriving quantitative implications of semi-endogenous models for returns to R&D

This appendix derives some results discussed in appendix B and has some further discussion.

What’s the initial size of wedge?

TFP growth is given by:
$$$ g_A = constant * L^{\lambda}A^{\phi – 1} $$$

The initial size of this TFP ‘wedge’ is given by:[41]I get this expression by differentiating the expression for $$ g_A $$ with respect to $$ L $$: $$ wedge = d(g_A)/dL $$.
$$$ wedge_i = (constant * {\lambda}) / (A^{1- \phi}L^{1- \lambda}) $$$

What this means is:

  • The higher the current level of tech $$ A $$, the smaller the initial impact
  • The lower $$ {\phi} $$ (more “fishing out”), the smaller the initial impact
  • The larger the current research effort $$ L $$, the smaller the initial impact
    • Unless $$ {\lambda} $$ =1, in which case $$ L $$ makes no difference
  • The smaller $$ {\lambda} $$ (more “stepping on toes”), the smaller the initial impact

If TFP growth in the world without the intervention is $$ g_0 $$, and the intervention increases the amount of R&D that year from L to L(1 + v), we can simplify the above expression. The initial size of the wedge is:
$$$ wedge_i = {\lambda}g_0v $$$
$$$ wedge_i = {\lambda} * (growth \, without \, intervention) * (fractional \, increase \, in \, R \&D) $$$

How does the wedge change over time?

The wedge declines over time as the level of technology increases:
$$$ wedge(t) = wedge_i *[A_i /(t)]^{1- \phi} $$$
where $$ wedge(t) $$ gives the size of the wedge at time t, $$ A(t) $$ gives the level of technology at time t, and $$ A_i $$ gives the initial level of technology.

Therefore the wedge declines in size at the same rate at which $$ A(t)^{1- \phi} $$ grows. In other words, at an exponential rate of $$ (1- \phi)g_A $$, where $$ g_A $$ is the growth rate of TFP.

If the exponential growth of researchers n is constant, it turns out that the growth rate of $$ A(t)^{1- \phi} $$ equals $$ \lambda n $$. This makes sense. The more “stepping on toes” (small $$ \lambda $$), and the slower researcher growth (small $$ n $$), the slower the growth of technology.

So the wedge declines in size at an exponential rate of $$ \lambda n $$.

This can have counterintuitive consequences for how $$ \lambda $$ affects the intervention’s impact. The more “stepping on toes” (small $$ \lambda $$ ), the slower the wedge declines over time. In some circumstances, can mean that reducing $$ \lambda $$ actually increases the impact of the intervention over the very long run, despite the initial size of the wedge being smaller. The intuition is that tech progress is slower at later times, and this means that the intervention’s initial impact diminishes more slowly.

How does the annual utility from the intervention change over time? It decays at the exponential rate
$$$ (time \, discount) + (decline \, of \, wedge) – (population \, growth) = r + {\lambda}n – n. $$$
If this quantity is negative (if r  is very small and $$ \lambda<1 $$), the annual utility can grow over time. What’s going on here is the effect of the growing population outweighs that of the declining wedge.

 

Appendix D: very rough estimates of the social returns to R&D under different scenarios

These are very rough back-of-the-envelope calculations (BOTECs) of the social returns to R&D for a few different scenarios discussed in the blog. I haven’t made a special effort to make these easy to understand, but am including them for completeness.

The first scenario, the ‘mainline scenario’, is a simplified version of the model discussed in the main body of the blog. The others are variants on this model which avoid predicting productivity stagnation.

I calculated how much each scenario changes the returns to R&D compared to the mainline scenario.

Impact compared to mainline scenario
Scenario Ignore impacts after 100 years Ignore impacts after 500 years
Mainline scenario: Ideas getting harder to find and stagnating population drive productivity stagnation. 1X 1X
Maybe we’ll avoid productivity stagnation by increasing the fraction of people doing R&D? 0.7X 0.4X
Maybe ideas won’t get harder to find in the future? 2.3X, or less 6.6X, or less
Maybe some trend-breaking future technology will allow us to avoid growth stagnation despite ideas getting harder to find? ~2X, or much more ~6X, or much more
Maybe the world’s population won’t stagnate? 2.3X, or less 6.6X, or less

The sections below estimate the ‘impact multiplier’, defined as follows:

$$$ Impact \, multiplier = (utility \, from \, \$1 \, to \, R \&D) / (utility \, from \, \$1 \, to \, average \, consumption) $$$

This table summarises the results, and was used to construct the above table.

Scenario Impact multiplier (ignore impacts after 100 years) Impact multiplier (ignore impacts after 500 years)
Mainline scenario: Ideas getting harder to find and stagnating population drive productivity stagnation. 22X 38X
Maybe we’ll avoid productivity stagnation by increasing the fraction of people doing R&D? 15X 15X
Maybe ideas won’t get harder to find in the future? 50X, or less 250X, or less
Maybe some trend-breaking future technology will allow us to avoid growth stagnation despite ideas getting harder to find? 50X, or less 250X, or less
Maybe the world’s population won’t stagnate? 50X 250X

Mainline scenario: Ideas getting harder to find and stagnating population drive productivity stagnation.

  • R&D is 2% of US GDP, and currently produces 1% growth in incomes per year.
  • So 1% extra of GDP on R&D buys an initial income wedge of 0.5%. 
  • This wedge falls over time, as ideas get harder to find.[42]Why would this happen? We caused some extra counterfactual science to happen: R(t)+1 researcher-years rather than R(t) at each time t. But this extra science makes less % difference to income as ideas become harder to find. 100 vs 101 researcher-years makes a bigger % difference to income than 1000 … Continue reading More specifically, (it turns out that plausible parameters imply that[43]The wedge halves each time researcher population doubles (assuming the ‘stepping on toes’ parameter $$ \lambda $$=1). The relationship between TFP growth g and population growth n in steady state is given by g = $$ \lambda $$ * n / (1 – phi). Using $$ \lambda $$=1 and $$ \phi $$=-2 … Continue reading) each time the economy grows by 30% the wedge ~halves. 
    • (I’m assuming that the amount of R&D in subsequent years is unchanged by our intervention.)
  • So the wedge first halves after 15 years.
  • Population is constant but ideas are getting harder to find, so growth slows. More specifically, let’s assume each 30% of growth takes twice as long as the last. As a result, the wedge only halves a second time after 30 years, a third time after 60 years, and a fourth after 120 years.
  • So the effects of an extra 1% on GDP are roughly as follows:
    • 0.5% wedge for 15 years
    • 0.25% wedge for 30 years
    • 0.125% wedge for 60 years
  • The total effect in the first 100 years is: 0.5*15+0.25*30+0.125*55 = 22% income boost.
  • 22X direct consumption.
  • The total effect in the first 500 years is: 0.5*15+0.25*30+0.125*60 + (1/16)*120 + (1/32)*240 + (1/64)*35 = 38% income boost, 38X

Note: this is only a rough BOTEC and gives slightly different results to the full model discussed in the blog. The full model includes a pure discount rate and is more complicated in a number of other ways discussed in the main text post and appendix A.

Maybe we’ll avoid productivity stagnation by increasing the fraction of people doing R&D?

In a sentence “Yes in the last 80 years we’ve needed a growing population to sustain constant growth; but in the future increased R&D intensity will allow us to sustain constant growth with a constant population.”

A rough BOTEC:

  • R&D is 2% of US GDP, and currently produces 1% growth per year.
  • So 1% extra of GDP on R&D buys an initial income wedge of 0.5%. 
  • This wedge falls over time, as ideas get harder to find. As above, it halves each time technology increases 30%.
    • (I’m assuming that the amount of R&D in subsequent years is unchanged by our intervention.)
  • So the wedge halves every 15 years. 
  • So the effects of an extra 1% on GDP are roughly as follows:
    • 0.5% wedge for 15 years
    • 0.25% wedge for 15 years
    • 0.125% wedge for 15 years…
  • So the total effect in the first 100 years is: ~0.5*15*2 = ~15% income boost.
  • So spending on R&D is 15X direct consumption, with a 100-year horizon.
  • With a 500-year horizon, the effect is still ~15% income boost, 15X

The returns to extra R&D today are lower than in my baseline stagnation scenario. The extra R&D effort here improves technology faster, making ideas harder to find, so the extra counterfactual science we caused makes less difference.

Maybe ideas won’t get harder to find in the future?

Maybe the observed pattern over the last 80 years will stop, and constant researcher effort will be capable of sustaining a constant rate of rate. 

  • R&D is 2% of US GDP, and currently produces 1% growth per year.
  • So 1% extra of GDP on R&D buys an initial income wedge of 0.5%. 
  • This wedge is constant over time. It doesn’t diminish because ideas are not getting harder to find.
    • (I’m assuming that the amount of R&D in subsequent years is unchanged by our intervention.)
  • So the total effect in the first 100 years is: 0.5*100 = 50% income boost.
  • So spending on R&D is 50X direct consumption, with a 100-year horizon.
  • With a 500-year horizon, the effect is 0.5*500 = 250% income boost, 250X

This BOTEC assumes ideas stop getting harder to find just before our intervention. If they continue to get harder to find for a while, this would reduce the bottom line.

Maybe some trend-breaking future technology will allow us to avoid growth stagnation despite ideas getting harder to find?

For concreteness, suppose R&D today brings forward in time the tech level after which we can sustain 1% annual productivity growth with a constant population. If we bring forward that day by 1 year, we boost income in every year thereafter by 1%.

Rough BOTEC (this one is fiddly):

  • R&D is 2% of US GDP, and currently produces 1% growth per year.
  • Currently we need to increase the # researchers each year to sustain exponential growth, but once we reach tech level X a constant # researchers can sustain 1% exponential growth.
  • 1% extra of GDP moves the tech level forward by 0.5 years, and we reach tech level X 0.5 years earlier. Income at all later times is boosted by 0.5%.
  • Suppose we reach X immediately. Then the total effect in the first 100 years is: 0.5*100 = 50% income boost. That’s 50X.
    • The total effect in the first 500 years would be 0.5*500 = 250% income boost, 250X
  • Suppose we only reach X after 50 years. This would reduce the value of the intervention somewhat, for complicated reasons.
    • The benefits in the first 50 years are smaller than 0.5% as the 0.5% wedge shrinks over time.
    • If annual research effort has grown, we will reach tech level X less than 0.5 years earlier. E.g. if we do twice as much research by the time we reach tech level X, we’ll only reach it 0.25 years earlier, ~halving the impact each year thereafter.

This is an interesting and complicated case.

We could extend it to consider bringing forward periods of much faster growth, which would make the returns to R&D much higher. For example, if we bring forward an economic singularity (where growth is hyperbolic until it output approaches a very high ceiling), we could be bringing forward a time when all biological humans have lives that are unimaginably happy and fulfilled by today’s standard. In addition, if it’s possible for people to exist as simulations on a computer, that could expand the human population by a trillion-fold or much more.[44]See Bostrom (2003) for a slightly more detailed explanation of this point. If this happened, it would continue what I believe to be the long-run historical trend whereby R&D has had massive welfare returns by bringing a much richer, more populous and happier future forward in time (see appendix F for more about this model of historical R&D).

Maybe the world’s population won’t stagnate?

In a sentence: “yes ideas are getting harder to find, but the population growth will allow us to meet the demands for ever-more research for each 1% output gain”.

A rough BOTEC:

  • R&D is 2% of US GDP, and currently produces 1% growth per year.
  • So 1% extra of GDP on R&D buys an initial income wedge of 0.5%. 
  • This wedge falls over time, as ideas get harder to find. As above, it halves each time technology increases 30%. 
  • So the wedge halves every 15 years. 
  • But each time the wedge halves, the number of people alive ~doubles,[45]This is exactly true in the semi-endogenous framework when the “stepping on toes” parameter lambda=1. If lambda < 1 then the population more than doubles each time the wedge halves. so the total impact of the wedge is constant over time.
  • So the total effect in the first 100 years is: 0.5*100 = 50% GDP boost.
  • So spending on R&D is 50X direct consumption, with a 100-year horizon.
  • With a 500-year horizon, the benefits would be 250X.

This calc assumed that the population doesn’t stagnate at all. In practice, the realistic version of this scenario is one where population stagnates temporarily. If we incorporate a temporary population stagnation into this BOTEC, the estimated returns to R&D would decrease.

 

Appendix E: reasons growth might not stagnate, and how that would affect the bottom line

Appendix D contains very rough BOTECs calculating the returns to R&D in each of these scenarios.

Maybe we’ll avoid stagnating growth by increasing the fraction of people doing R&D?

As mentioned earlier, the fraction of people doing R&D (research intensity) has increased significantly over time. If it continues to increase, we can have a growing pool of researchers despite a stagnant population.

This mechanism can temporarily sustain steady productivity growth despite population stagnation. It cannot do so indefinitely as the fraction of people doing research cannot exceed 100% (and will likely cap out much earlier). Indeed, when I incorporate this mechanism into the model, the stagnation of productivity growth is only delayed temporarily.

How does an increasing research intensity in the future affect the impact of R&D today? Counterintuitively, it reduces the impact. The reason is related to ideas getting harder to find. More R&D in later periods pushes us further up the diminishing returns curve for finding new ideas, so the additional R&D we funded makes less difference.[46]Here’s another way to understand this effect. Because ideas are getting harder to find, the number of new ideas found with a marginal researcher-year is roughly proportional to 1 / (total years of research so far). Suppose we fund an extra year of research in 2021. As a result, in 2050 one more … Continue reading

In fact, the mainline estimate of the social return for R&D does assume that research intensity will continue to increase to some extent. If I put weight on a scenario where there are even greater increases in research intensity, I should also put weight on scenarios where there are smaller increases in research intensity. As it is, I’m happy to just use my central estimate in the stylized calculation of this post.

Maybe ideas won’t get harder to find in the future?

There is strong evidence that ideas became harder to find during the 20th century. But perhaps this is a temporary trend, and soon enough the average difficulty of finding a new idea will stay constant over time.

If this happens, a constant population would be able to sustain constant productivity growth. This would significantly raise the estimated value of today’s R&D. As mentioned above, the model implies that the % income increase due to R&D today will decline over time. But if ideas stop getting harder to find, this decline will stop. In other words, extra R&D today could raise people’s incomes by (e.g.) 0.5% forever into the future. Given my low discount rate, this would significantly raise the estimated social returns to R&D.

I don’t find this scenario plausible. The evidence suggests that ideas have been getting consistently harder to find since 1930, and there’s no reason to expect this trend to change.[47]See Bloom et al (2020) and https://mattsclancy.substack.com/p/innovation-gets-mostly-harder.

Another version of this claim would be that ideas have never been getting harder to find. Instead our R&D institutions have become exponentially worse over time, and this compounding inefficiency explains why an exponentially growing number of researchers has led to merely constant growth.

Concretely, the data implying that it takes 41X as much research effort to find a new idea than in 1930 is interpreted as implying that institutions have become 41X less efficient. The startling implication is that, if only institutions had remained at their 1930s level, productivity growth would be 41X faster today. That would involve TFP growing by ~40% every year! Again, I don’t find this scenario to be very plausible.

Maybe the world’s population won’t stagnate?

The argument for stagnating productivity growth assumes that the global population will level off at 11 billion. But if the world population grew exponentially in the long-run, growth would not stagnate. The number of researchers could continue to grow exponentially, maintaining constant productivity growth despite ideas getting harder to find.

Is this scenario plausible? It is the UN’s high-end projection out to 2300, and there are some reasons it could happen. Many subcultures and countries currently have very high population growth and, in the long-run, cultural and biological evolution will select for these groups. On the other hand, governments may make a concerted effort to avoid sustained population growth if it would have devastating environmental consequences. Overall, it’s hard to know whether this scenario is likely to happen.

How would sustained future population growth affect the value of R&D today?

Overall, it would significantly increase the value of R&D today. There are two effects, which point in opposite directions. The first effect is that there are more future beneficiaries of today’s R&D, with the number growing exponentially. The second effect is complex. There is more R&D at later times and so the counterfactual impact of today’s marginal R&D on TFP falls over time. In particular, if today’s R&D initially raised TFP by x% then over time x falls exponentially. Combining these two effects, marginal R&D causes a % TFP gain which falls off exponentially over time but is enjoyed by an exponentially growing population. Compared to a case with constant population, the first effect dominates and the impact of R&D is significantly higher.

However, a very large future population would raise the impact of many interventions with positive long-run effects. So it’s not clear that this scenario gives us reason to prefer R&D in particular. Partly on this basis, I’m not currently adjusting the stylized estimate based on this scenario.

 

Appendix F: back-of-the-envelope calculations of value of R&D in 1800

This appendix estimates the value of R&D in 1800 using a method that incorporates the two factors mentioned in the main text:

  • Historically, R&D was more neglected than it is today.
  • Historically, R&D increased the amount of R&D occurring at later times, both by increasing the fraction of resources used for R&D and by increasing future populations.

The calculation is very rough. Its purpose is to highlight just how much these two factors can increase the importance of historical R&D; it shouldn’t be interpreted as a precise estimate of R&D returns in 1800. The choice of 1800, rather than another year, is largely arbitrary.

The calculation uses a different model to one described in the main text to incorporate the second factor: R&D increasing the amount of R&D occurring at later times. The model implies that a researcher-year in 1800 was hundreds of times more impactful than in 2020.

Explaining the model

Suppose (as a toy example) that in 1800 there were exactly 100 researchers. What is the impact of funding an extra researcher-year in 1800? The assumption of this model is that the impact is to bring forward the technological level of the present world by 1/100 years. In other words, we start enjoying the level of wealth of the modern world 1/100 years earlier than we otherwise would have.

Why make this assumption? An extra researcher-year in 1800 means we make more technological progress in 1800. How much more? It increases the number of researchers by 1%, so by the end of 1800 technology is 1/100 years ahead of where it would have been.[48]101 researchers working for 1 year make the same amount of progress as 100 researchers working for (1 + 1/100) years. Based on the second factor (‘R&D increased the amount of R&D at later times’), this brings all subsequent R&D efforts forward in time by 1/100 years. I.e. the population passes each milestone 1/100 years earlier, and research concentration ramps up on a schedule that’s brought forward in time by 1/100 years. As a result, the whole future trajectory of R&D progress is brought forward by 1/100 years, and we reach modern levels of technology and wealth 1/100 years earlier.[49]Another way to think about this is that we assume that the total R&D effort at each time is determined by the level of technology. I.e. the level of technology determines both the population and the research concentration and so determines the total R&D effort. So if we reach a given level … Continue reading

So the effect of the extra researcher in 1800 is that we spend 1/100 years less time at 1800 levels of wealth and 1/100 years more time at modern levels of wealth.

How should we value reaching modern levels of wealth x years earlier (and spending x years fewer at 1800 levels of wealth)? The population in 1800 was 1 billion. So a conservative valuation is just the value of raising 1 billion people’s incomes from the average income in 1800 to the average income in 2020. Average global income today is $9600, compared with an estimate of $700 for 1800.[50] See data from Roodman (2020).

Assuming the log-utility model, this benefit is equivalent to increasing 27 billion people’s income by 10% for x years.[51]You need ~27 10% income increases to go from $700 to $9600. So you need 27 billion 10% income increases to raise 1 billion people’s income from $700 to $9600. Here I use the fact that all 10% income increases are valued equally, which comes from the log-utility model.
1.1^23 = 14 = 9600 / 700.

A less conservative valuation would also value the additional 7 billion people alive today, or take into account future increases in wealth that we will also reach x years earlier in time.

Applying the model

Currently we spend 2.3% of GWP on R&D.

Let’s say that percentage was 20-30X lower in 1800 — we suggest this is a conservative estimate below. Then R&D was only 0.1% of GWP. How many researchers would that buy? Let’s assume that with 0.1% of GWP you could pay for 0.1% of people to be researchers.[52]The assumption that researchers earn the average global wage, rather than a higher wage, will make no difference to the result. This is because I will make an equivalent assumption about the wages of researchers in 2020. The results would change somewhat if you think researchers today demand more … Continue reading The world population was 1 billion, so that would pay for 1 million researchers.

Funding an additional researcher-year in 1800 would bring forward the technological level of the present world by (1 / 1 million) years. This benefit is equivalent to increasing 27 billion people’s income by 10% for (1 / 1 million) years; or to increasing 27,000 people’s income by 10% for 1 year.

We can adjust this downwards based on some of the factors considered in the calculation of the value of R&D today:

  • Stepping on toes. Twice as many researchers make less than twice as much progress due to duplication. 0.75X
  • Proportion of tech progress credited to R&D. Only 50% of TFP growth comes from targeted R&D. 0.5X
  • Skepticism about global spillovers of R&D. 0.7X

This reduces the benefit to increasing 7100 people’s income by 10% for 1 year.

How does this compare to the benefit of funding a researcher-year today? The model described in the main body of this implies that the benefit is equivalent to raising 70 people’s income by 10% for a year.[53]For consistency with the 1800 calculation (see most recent fn), I assume that 1 researcher-year costs the same as the global average income. Today that is about $10,000. The model for R&D today implies that $100 to R&D has the same welfare effect as raising someone’s income by 7% for one … Continue reading

This calculation implies that R&D today has lower returns by a factor of 7100/70 = ~101.

One reason for this large factor is that the calculation in this section assumes an additional feedback loop between R&D and future R&D. I think that this captures a real difference between funding R&D now and in the past; but someone skeptical of that distinction might prefer a smaller factor.

On the other hand, the factor would be much larger if we made any of the following changes:

  • Used a time before 1800, when the fraction of resources used for R&D would be lower still.
  • Used a less conservative estimate of the value of arriving at the modern levels of wealth x years earlier (e.g. by placing value on the associated population increase, or by including the benefit of arriving at future levels of wealth x years earlier).
  • Compare the impact of $1 on R&D today vs in 1800, rather than comparing the impact of a researcher-year. $1 would buy much more research in 1800 than today, as salaries were much lower in 1800.

In light of this, my opinion is that historical R&D was more impactful than R&D today by even more than the factor of 101X estimated above.

We should be cautious with this comparison. As mentioned above, its purpose is to highlight just how much these two factors can increase the importance of historical R&D; it is not a precise estimate of R&D returns in 1800. However, the comparison does suggest that R&D was historically hundreds of times more impactful than R&D today.

Data on the research concentration in 1800

What was the research concentration (% of GWP was spent on R&D) in 1800? I’m not aware of high-quality data on this question. But various sources suggest that research-concentration in 2020 was probably at least 20-30X higher:

  • Data from Bloom et al. 2020 find the number of US researchers increasing by an average of 4.3% per year since 1930. US population grew less than an average of 1.5% per year in the same period, implying that the fraction of people doing research was growing by 2.8%. This implies a 12X increase in research intensity in 1930 – 2020.[54]1.028^90 = 12.
  • Bakker (2013) estimates growth in overall R&D using growth of specific types of R&D funding. One such estimate implies a 1.6% annual growth in research intensity between 1767 and 1904, corresponding to a 8X increase between 1800 and 1930.[55]1.016^130 = 8 Another such estimate implies a 0.8% annual increase between 1823 and 1941, corresponding to a 3X increase between 1800 and 1930.[56]1.008^130 = 3.
    • Combining these estimates with the Bloom et al. data for 1930-2020 implies a total increase in research-concentration of 36X – 96X.
  • Utility patents records show large increases between 1800 and 2020.
    • E.g. utility patent applications grew 300X since 1850. Accounting for a 15X population increase in the same period, that’s a 20X increase in patent application concentration.
    • Design patents similarly show a 30X increase in patent-concentration since 1850.
    • This is only weakly informative because patents are just proxies whose correlation with growth-enhancing research effort could chance over time.
  • Shuttleworth and Charnley (2016) claim science publications grew by 100X between 1800 and 1900. With a 1.6X increase in population in the same period, that’s a publication-concentration increase of 60X in that period alone.
    • Again, science publications are only a proxy for growth-enhancing research effort.

Based on the above numbers, a conservative estimate of the increase in research concentration from 1800 to 2020 is 20-30X.

 

Appendix G: top-down vs bottom-up calculations of the returns to R&D

Summary

This post uses a top-down approach to calculate the returns to R&D, using very high level empirical inputs and an assumption about what fraction of TFP growth is due to R&D. Many economics papers use bottom-up approaches, attempting to tease out the causal effect of R&D from micro-level data. We can use bottom-up approaches to sanity check this post’s assumption about the fraction of TFP growth that is due to R&D.

Bottom up approaches calculate a quantity called the social rate of return, and estimates vary widely between 30% and 130%. The social rate of return can also be calculated using inputs to my model; this calculation gives a value of only 13%. Naively, this implies I’m underestimating the returns to R&D. However, it’s possible that bottom-up approaches overestimate the social rate of return, or that the calculation using my inputs is not really comparable with the results from bottom-up approaches. So I only see this as weak evidence that I’m underestimating the returns to R&D.

Top-down vs bottom-up calculations of R&D returns

The methodology in this post, adjusted from Jones and Summers (2020), is a top-down approach to calculating the social returns to R&D. Its key empirical inputs are very high level quantities: frontier TFP growth and total global R&D expenditures. Based on these quantities, and a simple growth model relating them together, it estimates the welfare benefits from TFP growth then credits R&D with some portion of those benefits. A strength of this approach is its conceptual clarity. A weakness is its reliance on a bald assumption about what fraction of TFP growth is due to R&D.

Many economics papers estimate the social returns using bottom-up approaches. These look for correlations between R&D spending and subsequent TFP growth. Some papers focus on specific technologies or firms or industries. Some study entire countries or even groups of countries. A strength of these approaches is they can potentially identify the causal contribution of R&D to growth by applying statistical techniques to micro-level data and controlling for other causes of growth. Unsurprisingly, this causal identification faces many challenges. A weakness of these approaches is that they aren’t well suited to capturing spillovers that occur far away in space and time because these may not be included in the data used. Indeed, the coefficients estimated typically cannot be translated into all-things-considered estimates of the social returns to R&D without substantive further theoretical assumptions about how the benefit changes over time. In addition, spillovers to low-income countries are typically not included in calculations of the social returns in these papers. Appendix B of Jones and Summers (2020) contains a good overview of bottom-up approaches.

Using bottom-up approaches to calibrate the assumptions of the model in this post

Many bottom-up approaches estimate a quantity called the social rate of return,[57] See Bloom et al. (2013)Coe and Helpman (1995) and references within Jones and Williams (1998)  and Jones and Summers (2020). which means “if I invest a marginal $1 in R&D, how much will GDP increase?”. If the social rate of return is 50%, investing $1 in R&D raises GDP by $0.5. Formally, the social rate of return equals $$ dY/dR $$, where Y is output and R is the cumulative stock of R&D effort.

Jones and Williams (1998) relate empirical estimates of this quantity to the semi-endogenous model used in this post. They show that
$$$ r’ = {\lambda}{g_A}/s $$$
where r’ is an empirical estimate of the rate of return, $$ \lambda $$ is the stepping on toes parameter discussed in Appendix A, $$ g_A $$ is the growth rate of TFP due to R&D, and s is the fraction of GDP used for R&D.

We can use the model in this post to calculate the RHS of this equation, and compare it to empirical estimates of the LHS.

This post uses:

  • $$ \lambda $$=0.75
  • $$ g_A $$ = 1% * 0.4 = 0.4%
    • ~1% is the average value of TFP growth over the last 40 years in my data set. 
    • 0.4 is the fraction of TFP growth I’m crediting to R&D.
  • s = 2.3%

These imply a social rate of return of 13%.[58]0.75*0.4%/2.3% = 0.13. This is significantly lower than most estimates in the literature. Bottom-up approaches typically find the private rate of return — which only includes benefits to the innovator — is 20-30%. Estimates of the social rate of return are much higher and vary considerably, at 30 – 130%.[59]E.g. Bloom et al. (2013) estimate a private return of 21% and a social return of 55%. See multiple estimates in table 1 of Jones and Williams (1998) and in Appendix B of Jones and Summers (2020).

There are a few possible interpretations of this result:

1. This post underestimates the social benefits of R&D. 

A natural adjustment would be to increase fraction of TFP growth credited to R&D and to increase $$\lambda $$. Using $$ \lambda $$ =1 and crediting R&D with all frontier TFP growth would leave the RHS at 43%.

2. The empirical studies overestimate the social benefits of R&D. 

If other causes of TFP growth (e.g. capital expenditures) are correlated with measured R&D, these studies may overestimate the causal effect of measured R&D on TFP growth. Another possibility is that these studies focus on sectors in which R&D is particularly lucrative.

3. The LHS and RHS of the equation are not really comparable, for subtle theoretical reasons.

Here’s one such reason.

The model I use in this post assumes that there is no depreciation of the R&D stock. Once ideas are discovered, they are not forgotten. This means that the measured stock of R&D includes all historical R&D inputs without discount.

But some central empirical estimates of the social rate of return to R&D assume that the R&D stock depreciates at a very fast rate. Coe and Helpman (1995) use a depreciation rate of 5%; Bloom et al. (2013) use a rate of 15%. This means that the estimated stock of R&D only includes R&D inputs from recent years.

If instead these empirical estimates had assumed no depreciation of the R&D stock, they would estimate a larger stock of R&D and so a lower social benefit per $ of R&D stock. The estimated social rate of return would fall.[60]Conversely, if I’d used the depreciation assumptions used in the empirical estimates, I’d have expected the benefit from marginal R&D to diminish much more quickly over time and calculated a lower total return to R&D.

Fall by how much? A rough calculation suggests Bloom et al. (2013)’s estimate of the R&D stock would rise by a factor of 3, and so their estimate of the social rate of return would fall by a factor of 3 from 55% to 18%. Analogously, the estimate of Coe and Helpman (1995) would fall by a factor of 1.5 from 100% to 67%.

So if bottom-up empirical studies used the same “no depreciation” assumption as I use in this post, their estimates of the social rate of return would fall and the tension with my assumptions would reduce.

This depreciation issue is just one reason the LHS and RHS of the above equation are not straightforwardly compatible. I suspect digging into specific analyses in more detail would uncover further reasons why they are not straightforwardly comparable.

Which of these three interpretations is correct? Interpretations (2) and (3) both seem plausible to me; in combination I think they could explain the discrepancy. That said, I place some weight on interpretation (1) and doing this analysis updated me towards thinking that measured R&D accounted for a greater fraction of TFP growth.

This tension could be investigated further. An ambitious project would be to take the data used in these analyses and use them to directly fit the inputs to the semi-endogenous growth model used in this post.[61]The model here differs from the standard semi-endogenous growth model in two ways. First, the standard semi-endogenous growth model assumes that all TFP growth is due to R&D; this model relaxes this assumption. Second, this model translates income changes to welfare changes using a log-utility … Continue reading

 

Appendix H: Potential disagreements with Progress Studies

Progress Studies is an intellectual movement that aims to understand and accelerate civilisational progress. Some people involved in Progress Studies believe accelerating civilisational progress, for example by improving institutions for innovation, should be the world’s top priority.

My conclusion here seems to be somewhat in tension with this. My stylized estimate puts direct R&D spending at 45% the impact of cash transfers to the global poor, and 4.5% the impact of the GHW bar.

The apparent disagreement here may be smaller than it appears.

Firstly, the typical $ spent by governments of rich countries probably has many times less social impact than cash transfers to the global poor. This implies, in line with Progress Studies, that R&D spending has an unusually high social impact compared to typical government spending.

Secondly, leveraged ways to boost long-run innovation, like increasing high-skilled immigration or improving institutions, might be much more effective than directly funding R&D. I expect, partly based on unpublished work by Open Philanthropy, that some such opportunities do meet the GHW bar. In other words, I think that some interventions to boost innovation are among the best in the world for improving wellbeing. It’s not obvious to me that Progress Studies enthusiasts should be interpreted as making a stronger claim than this.

That said, I expect that I do have some substantive disagreements with Progress Studies enthusiasts. In particular:

  1. I expect that ideas will continue to become harder to find. Bloom et al. (2020) offers strong evidence that ideas (defined as insights that increase TFP by 1%) have been getting harder to find for 80 years. This means that even a permanent increase in the fraction of GDP used for R&D only temporarily increases the growth rate. This lowers the returns to R&D compared with if you could permanently increase the rate of economic growth, a possibility that Tyler Cowen highlights in his book Stubborn Attachments.

    Some people in Progress Studies have suggested that the evidence from Bloom et al. (2020) might be explained by institutions becoming worse over time.[62]E.g. see Patrick Collison here, 19:57-25:32. While this may be true to some extent, I don’t think it could be true to such a large extent that ideas haven’t been getting harder to find.
  2. I’m less inclined to place weight on trend-breaking scenarios where funding R&D has a very large upside. Stubborn Attachments acknowledges the possibility that any given intervention might only temporarily increase growth, but argues we should do an expected value calculation and put some weight on the possibility that we can permanently increase the economic growth rate. I feel reluctant for GHW grantmaking to place much weight on trend-breaking scenarios where the upside is very large; see here, here and here for further discussion. This is especially compelling when combined with the next point.
  3. I’m more wary of potential harms from R&D. Part of the reason I’m not inclined to put weight on scenarios with very large upside is because I’m not explicitly modeling possible harms. As mentioned above, this is not an all-things-considered analysis, but more like “How good does R&D look if we use our median estimate of R&D returns and future population growth, ignoring scenarios that could massively increase or massively decrease the returns?

Appendix J: ways in which the stylized estimate is too pessimistic vs too optimistic

Some assumptions of the report are listed as both “arguably too optimistic” and “arguably too pessimistic”. 

Arguably too optimistic

The stylized calculation:

  • Ignores potential harms from R&D, which could dominate the benefits in some cases.
  • Gives R&D substantial credit for long-run growth. 
    • An alternative view is that credit for growth should be distributed fairly evenly over economic activities rather than heavily concentrated on a few types of activity. For example, if R&D activity stopped tomorrow, growth would continue for decades due to learning by doing, business innovation, and diffusion of existing tech across the economy. 
    • A core reason I give R&D substantial credit is that over long timescales it seems that these sources of growth would dry up and R&D is necessary to sustain growth.
    • On the other hand, many reviewers of this post suggested that I credit R&D with 50% of the TFP growth or more, and this is supported by a naive comparison with bottom up calculations of the returns to R&D.
  • Assumes significant R&D spillover between countries.
    • In my stylized estimate, all global R&D goes into a common pot that boosts incomes around the world. 
    • In a model in which each country’s R&D was only relevant to their own growth, then the social returns to R&D would vary by country. The R&D in rich countries would be much less impactful than my stylized estimate because it would raise the incomes of fewer people. 
    • On the other hand, many reviewers of this post suggested that eventually 100% of frontier growth will spillover around the world, rather than the 70% that I assumed. 
  • Uses a utility function that may overestimate the benefits of increasing the incomes of people who are rich relative to those who are poor.
  • Assumes the marginal $ on R&D is 75% as impactful as the average $. You might think that the smartest people are already doing R&D, and the best projects are already funded, such that the marginal $ is worse than this.
  • Assumes R&D increases everyone’s incomes by the same %. If R&D increases the incomes of the rich by a larger % than those of the poor, the stylized calculation will overestimate the utility benefits from R&D. 
  • Includes benefits hundreds of years into the future. 
    • While I think this is right in principle, Open Philanthropy typically only includes near-term benefits in its analyses of philanthropic opportunities. So we should be very cautious when using the social impact estimate here to compare R&D with other opportunities; a naive comparison would give R&D an unfair advantage.
    • I use a discount of 0.2%, based on the possibility of a major disruption. Other sources of uncertainty about whether the benefits of R&D will really be felt might argue for a larger discount.

Arguably too pessimistic

The stylized calculation:

  • Ignores scenarios in which returns to R&D are much higher
    • e.g. expediting a large growth increase due to AI
    • Taking the expected value over those scenarios would probably significantly increase the estimated expected returns.
  • Doesn’t give current R&D credit for increasing the amount of research in future years, even though I think this was a significant dynamic historically and accelerating catch-up growth would have this effect.
  • Assumes ideas are getting harder to find rapidly (i.e. the diminishing returns are steep)
    • I use the estimate from Bloom et al. (2020), but my guess is that their data probably overestimate the growth in researchers and so overestimate the steepness of diminishing returns to finding new ideas.
    • Also, if innovation institutions have become worse over time (as people in the Progress Studies movement claim), then Bloom et al. (2020) will overestimate the steepness of diminishing returns to finding new ideas. 
  • Assumes that US R&D $ and non-US R&D $ are equally effective in boosting frontier growth, despite US $ being more focussed on frontier growth. An alternative assumption would be that US R&D does more to boost frontier growth per $ than other R&D.
  • Uses a harsh value for the DMR to R&D. It’s the value from “Are Ideas getting harder to find”, but I think their data probably overestimates the growth in researchers and so overestimates the steepness of returns. Adjusting for this would probably increase the bottom line by 1-1.5X.
  • Uses a utility function that may underestimate the benefits of technological progress by assuming the only effect is to raise incomes.
  • Assumes 30% of frontier innovation never spreads around the whole world. This would be fine if I assumed that only the US was driving productivity growth, but it seems kinda harsh when we’re giving all countries’ R&D equal credit for frontier productivity growth.

 

 

Appendix K: additional limitations of the model

The model assumes research today doesn’t change the amount of research tomorrow

My model assumes that funding additional research in 2021 doesn’t affect the amount of research effort in future years.

But you might think research today increases the amount of research tomorrow. I claimed this dynamic was important in the past, with successful R&D projects providing evidence that R&D was a fruitful activity. Today, I think this dynamic is especially plausible for small R&D sectors that have the potential to be significantly scaled up. For example, early stage solar panel R&D may have brought forward in time the point at which it was profitable for private entities to invest in R&D. In other words, early solar R&D increased the amount of solar R&D occurring at later times.

Another reason R&D today might increase R&D in later years is if it accelerates catch-up growth. Richer countries spend more on R&D, so faster catch-up growth implies more R&D spending. For example, India and China are contributing an increasing amount to global R&D, and faster catch-up growth would accelerate this process.

Conversely, more research today might remove a promising research project from the pool of possibilities, decreasing the amount of future research. For example, suppose that certain R&D funders only make grants if the project is sufficiently promising. If ideas are getting harder to find, projects will tend to become less promising over time. More research today would accelerate this process, reducing the appeal of future projects and so reducing the amount of future funding.

It seems plausible that R&D today could increase or decrease the amount of R&D that happens in the future. The current estimate doesn’t make an adjustment in either direction.

Uncertainty about the utility function

My calculation assumes that utility increases with the log of income. But there are plausible alternative assumptions that would change the bottom line in both directions.

On the one hand, you might think that utility diminishes more sharply than the log of income. This can be captured by using a CES function of income with $$ \eta >1 $$. This assumption increases the benefit of extra income for the very poorest people relative to richer people.[63]I.e. as income falls, the marginal utility of income increases by more.

This favours cash transfers targeted at the world’s poorest people) over R&D whose benefits are spread amongst people at all income levels.

But on the other hand, you might think that the welfare benefits of developing new technologies outstrip the income increases that they cause. For example, between 1990 and 2020 the average US real income increased from $40k to $60k.[64]https://ourworldindata.org/grapher/gdp-per-capita-worldbank?tab=chart&yScale=log&country=OWID_WRL~USA Imagine someone in 1990 who earns $40k being offered the following choice:

  1. Their income is raised to $60k, and they must spend all their income on 1990 goods and 1990 prices
  2. They receive the average income of someone in 2020, and must spend it on 2020 goods at 2020 prices

Plausibly, the second option is better because of the possibility of using entirely new products like the internet, smartphones, Amazon, etc.[65]To make the comparison fair, in option 2 you should be forced to buy all goods and services at 2020 prices. This means that some services will be more expensive than in 1990. Even so, I think option 2 is better because there are many new products that are cheap but useful. If this doesn’t seem … Continue reading The implication is that the increase in income underestimates the benefit of technological progress, or (relatedly) that inflation adjustments doesn’t adequately capture the value of new products and services.[66]How much does the income increase underestimate the welfare benefit of technological progress? You could roughly estimate this by increasing the cash payment in option 1. E.g. suppose you’re indifferent between option 2 and option 1’) A cash payment of $40k. This would imply that the true … Continue reading Using a utility function that adjusted for this would raise the returns to R&D.[67] Phil Trammell describes one such utility function here. Also see discussion by Matt Clancy in the “What about other benefits?” section of this post.

So some plausible changes to the utility function would reduce the social returns to R&D, while others would increase the returns.

Footnotes

Footnotes
1 If environmental constraints require that we reduce our use of various natural resources, productivity growth can allow us to maintain our standards of living while using fewer of these scarce inputs.
2 For example, in Stubborn Attachments Tyler Cowen argues that the best way to improve the long-run future is to maximize the rate of sustainable economic growth. A similar view is held by many involved in Progress Studies, an intellectual movement that aims to understand and accelerate civilisational progress.
3 An example of an intervention causing a temporary boost in R&D activity would be to fund some researchers for a limited period of time. Another example would be to bring forward in time a policy change that permanently increases the number of researchers.
4 Three comments on the log-utility model. First, the results are the same whatever the values of the constants k and m. Second, I do a sensitivity analysis of the consequences of different utility functions; if the diminishing returns to income are steeper than log, this favours cash transfers more strongly. Third, by expressing the benefits of R&D in terms of their welfare impact I differ from Jones and Summers (2020) who express the benefits in terms of $.
5 log(110) – log(100) ~= 10*[log(101) – log(100)] ~= 100*[log(100.1) – log(100)] ~= 1000*[log(100.01) – log(100)].
6 More realistically, there will be a lag before productivity benefits are felt. Currently I don’t model this lag because it wouldn’t affect the results by much. I use a discount of 0.2%; so a 50 year lag would reduce the returns to R&D by ~10%.
7 GiveDirectly implements this intervention. Note, I use simplified numbers in this post that don’t exactly match GiveDirectly’s cost effectiveness, and I believe GiveDirectly is somewhat more impactful than the numbers I use imply.
8 Note, rising incomes mean we don’t value adding an equal dollar amount to people’s incomes the same amount through time. We value a dollar more today because people today are poorer than they will be in the future.
9 During the 20th century, the number of researchers grew exponentially, but productivity growth did not increase (in fact it decreased slightly). If R&D is responsible for the productivity growth, then more research effort is required to achieve each subsequent 1% gain in productivity.
10 Note: this does not mean that the absolute $ increase in incomes shrinks over time. It may decline, stay constant or increase, depending on the rate at which ideas are getting harder to find. Technically, if the “fishing out” parameter $$ \phi $$ > 0, then the absolute $ benefit increases over time. If $$ \phi $$ < 0, it decreases over time. If $$ \phi $$ = 0 exactly, it stays constant. I use $$ \phi $$ = -2.1, estimated in Bloom et al (2020).)
11 The key point is as follows: when ideas are getting harder to find, the number of new ideas found with a marginal researcher-year is roughly proportional to 1 / (total researcher-years so far). So if the 100th researcher-year finds 1/100 new ideas, the 200th researcher-year will find only 1/200 new ideas and the 1000th researcher-year will only find 1/1000 new ideas.

Let’s work through the consequences of this point using an example. Suppose an intervention funds an extra researcher-year in 1900, and doesn’t change the amount of research happening in subsequent years. We’ll estimate the impact of the intervention on TFP in 1900 and in 2000.

What’s the impact of the intervention on TFP in 1900? Suppose that a total of 100 researcher-years have occurred by 1900. Then the intervention makes the difference between 100 researcher-years vs 101 researchers-years having happened, a difference of 1/100 new ideas. [Here I assume that the number of new ideas found with a marginal researcher-year = 1 / (total researcher-years so far).] 1/100 new ideas correspond to a 0.01% increase in TFP, because we’re defining “an idea” as a 1% TFP increase.

What’s the impact of the intervention on TFP in 2000? Suppose that a total of 1000 researcher-years had occurred by 2000. Then the intervention makes the difference between 1000 researcher-years vs 1001 researchers-years having happened, a difference of 1/1000 new ideas. [Here I again assume that the number of new ideas found with a marginal researcher-year = 1 / (total researcher-years so far).] 1/1000 new ideas correspond to a 0.001% increase in TFP.

So the intervention raises TFP by 0.01% in 1900, but only by 0.001% in 2000. Its impact on TFP falls towards 0% over time.

12 Mathematically, in the semi-endogenous growth model the effort needed to find a new idea is proportional to TFP^($$ \phi $$ – 1), where $$ \phi $$ is the parameter controlling how quickly ideas are getting harder to find. I use $$ \phi $$ = -1.4, so every time TFP doubles the effort needed to find a new idea increases by 2^2.4 = 5.3.
13 In the long run, there are reasons to think population will fall (fertility rates in developed countries), reasons to think it might increase (relating to biological and cultural evolution), and no compelling reason to think it will stay exactly the same. Still, this feels like a fair ‘default’ case to consider for calculating a stylised value of R&D for our Global Health and Wellbeing team. I discuss some alternative scenarios in appendix E, and list ways the model is optimistic and pessimistic in appendix J.
14 The lag until productivity benefits are felt will probably be larger in low income countries than in high income countries. As mentioned above, I don’t model this lag because it wouldn’t affect the results by much. I use a discount of 0.2%, so a 50 year lag would reduce the returns to R&D by ~10%. Note, the 70% average could arise from <<70% of benefits eventually spilling over to low income countries, and >70% of benefits eventually spilling over to everywhere else. E.g. the poorest half of the global population could get spillovers of 40% while the richest half get spillovers of 100%.
15 In economic growth models, this corresponds to the “stepping on toes” parameter λ = 0.75. I’m not aware of data that pins down λ, and it seems like values between 0.4 and 1 could be correct. I use the estimate from Bloom et al. (2020) Appendix Table A1, where they set λ = 0.75 and then estimate $$ \phi $$ = -1.4.
16 The primary effect is recorded as a TFP increase because GDP went up holding constant the amount of labour and physical machinery. The secondary effect is recorded as capital deepening because each person has more physical capital (i.e. more or better machinery).
17 Growth theory relates the size of these effects on income: (income increase from TFP and capital deepening) = (income increase from TFP alone) / (1 – capital share of GDP). The capital share is about 35%, so this multiplies the bottom line by 1 / (1 – 0.35) = 1.5.
18 As mentioned in a previous footnote, I think GiveDirectly is somewhat more impactful than the numbers in this row.
19 For example data from Bloom et al. 2020 find the number of US researchers increasing by 4.3% per year on average since 1930. US population grew less than 1.5% per year on average in the same period, implying that the fraction of people doing research was growing.
20 See for example Lee (1988), Kremer (1993), Jones (2001) and Galor and Weil (2000).
21 Though the fraction of people doing research can increase, this can only go on for so long. I discuss this possibility below.
22 There are good theoretical reasons to think TFP can’t grow exponentially at its recent rate for more than 10,000 years, but these don’t rule out exponential growth continuing for another 1000 years.
23 If R&D today expedites a future technology that massively accelerates future growth, the bottom line can increase by much more than 100X. More.
24 My mainline scenario found R&D to be 45% as impactful as giving cash to someone on $500/year. This implies R&D is 4.5% as impactful as our current bar for GHW grantmaking. A 7X increase would leave R&D 31.5% as impactful as the GHW bar.
25 Aghion et al. (2017) discuss the possibility that AI will accelerate productivity growth by automating research tasks.
26 Note, a growth-enhancing technology might allow a constant population of human researchers to maintain ~2% productivity growth, or it might allow them to accelerate productivity growth. Open Philanthropy thinks the latter possibility is more likely than many actors seem to think, for reasons discussed in this report.
27 Of course, growth-enhancing technologies might enable other trends to continue. E.g. the trend of ~2% annual growth in US GDP/capita over the past 150 years, or the trend of growth accelerating over the past 10,000 years.
28 Let’s demonstrate this point with an example. Suppose an intervention causes an extra researcher-year to happen in 2021. Let’s consider its impact on TFP in 2100 if a growth-enhancing technology isn’t developed, and if it is developed.

Suppose that if a growth-enhancing technology isn’t developed then a total of 1000 researcher-years will have happened by 2100. Then the intervention makes the difference between 1000 researcher-years vs 1001 researcher-years having happened, a difference of 1/1000 new ideas.  [Here I assume that the number of new ideas found with a marginal researcher-year = 1 / (total researcher-years so far). This is a consequence of ideas getting harder to find.]

If a growth-enhancing technology is developed, then a total of 9000 researcher-years will have happened by 2100. Then the intervention makes the difference between a 9000 researcher-years vs 9001 researchers-years having happened, a difference of 1/9000 new ideas.  [Here I assume that the number of new ideas found with a marginal researcher-year = 1 / (total researcher-years so far).]

So the intervention causes 1/1000 new ideas in 2100 if a growth-enhancing technology isn’t developed, but only 1/9000 new ideas if it is developed.

Note: I assume that the intervention doesn’t change the amount of research done in later years, so it always makes a difference of 1 researcher-year. But if the intervention brought the growth-enhancing technology forward in time, it would increase the amount of research in later years. This would significantly change the calculation.

29 Appendix D does a very rough BOTEC on the returns to R&D for one possible growth-enhancing technology.
30 Eg. see this draft report by Joe Carlsmith on risk from power-seeking AI, or these two posts from the Cold Takes blog.
31 Even if R&D isn’t competitive according to either worldview, might it look competitive according to a weighted sum of both? I think not. I estimate funding generic R&D to be ~10X worse than the GHW bar, and it looks significantly worse from a LTist perspective than alternative interventions.
32 Appendix H discusses some potential differences between my perspective and that of Progress Studies advocates.
33 The simulation assumes that the increase in R&D activity is proportional to the increase in funding. This may be optimistic: in reality you need both funding and researchers to do R&D. Essentially, the simulation assumes that more funding will bring with it more researchers, which may be optimistic.
34 So it ignores the additional effect that capital deepening has on TFP increases in standard growth models.
35 It assumes everyone is on the world average income. Representing income inequality wouldn’t change the results. This is because we ultimately care about the percentage effect of R&D on income, and this is the same no matter what people’s starting incomes are. We care about the percentage effect because we assume utility = k + m*ln(income).
36 The specific values used for k and m do not affect the result as they cancel. In practice we use k=0 and m=1.
37 It quantifies the result in this way because this is a metric Open Philanthropy uses internally to compare the impacts from different kinds of intervention.
38 More precisely, the absolute size of the annual increment is 2% of current research intensity: 0.02 * 2.3% = 0.046%. So this assumption corresponds to thinking that research intensity has been growing exponentially at about 2% per year, but this exponential rate of increase will decline over time.
39 See page 20 of Jones and Summers (2020).
40 How long does it take for the “wedge declines more slowly” effect to dominate if we have no discount? Let’s assume we change our value of $$ \phi $$ to compensate when we change lambda. How long does it take for the total impact from $$ \lambda $$=0.75 to exceed the impact from $$ \lambda $$=1? With constant exponential population growth of 1% it takes ~250 years. With population stagnating after 80 years, it takes ~5000 years. If we held $$ \phi $$ constant when changing lambda, or used a discount, it would take longer (perhaps never) for the impact from $$ \lambda $$=0.75 to exceed the impact from $$ \lambda $$=1.
41 I get this expression by differentiating the expression for $$ g_A $$ with respect to $$ L $$: $$ wedge = d(g_A)/dL $$.
42 Why would this happen? We caused some extra counterfactual science to happen: R(t)+1 researcher-years rather than R(t) at each time t. But this extra science makes less % difference to income as ideas become harder to find. 100 vs 101 researcher-years makes a bigger % difference to income than 1000 vs 1001.
43 The wedge halves each time researcher population doubles (assuming the ‘stepping on toes’ parameter $$ \lambda $$=1). The relationship between TFP growth g and population growth n in steady state is given by g = $$ \lambda $$ * n / (1 – phi). Using $$ \lambda $$=1 and $$ \phi $$=-2 (values from Bloom et al. 2020) implies g = n/3. Researcher population growth is 3X faster than TFP growth. By the time TFP has grown 30%, population has ~doubled and to the wedge has ~halved.
44 See Bostrom (2003) for a slightly more detailed explanation of this point.
45 This is exactly true in the semi-endogenous framework when the “stepping on toes” parameter lambda=1. If lambda < 1 then the population more than doubles each time the wedge halves.
46 Here’s another way to understand this effect. Because ideas are getting harder to find, the number of new ideas found with a marginal researcher-year is roughly proportional to 1 / (total years of research so far). Suppose we fund an extra year of research in 2021. As a result, in 2050 one more researcher-year has occurred. The number of ideas found with this extra researcher-year is proportional to 1 / (total years of research by 2050). If the fraction of people doing R&D increases, this quantity will be smaller.
47 See Bloom et al (2020) and https://mattsclancy.substack.com/p/innovation-gets-mostly-harder.
48 101 researchers working for 1 year make the same amount of progress as 100 researchers working for (1 + 1/100) years.
49 Another way to think about this is that we assume that the total R&D effort at each time is determined by the level of technology. I.e. the level of technology determines both the population and the research concentration and so determines the total R&D effort. So if we reach a given level of technology x years earlier, we also reach the corresponding level of R&D effort x years earlier. Again, this assumption captures the idea that R&D in 1800 increased the amount of R&D at later times.
50 See data from Roodman (2020).
51 You need ~27 10% income increases to go from $700 to $9600. So you need 27 billion 10% income increases to raise 1 billion people’s income from $700 to $9600. Here I use the fact that all 10% income increases are valued equally, which comes from the log-utility model.
1.1^23 = 14 = 9600 / 700.
52 The assumption that researchers earn the average global wage, rather than a higher wage, will make no difference to the result. This is because I will make an equivalent assumption about the wages of researchers in 2020. The results would change somewhat if you think researchers today demand more (or less) of the premium than researchers in 1800.
53 For consistency with the 1800 calculation (see most recent fn), I assume that 1 researcher-year costs the same as the global average income. Today that is about $10,000. The model for R&D today implies that $100 to R&D has the same welfare effect as raising someone’s income by 7% for one year. So $10,000 has the same effect as raising 100 people’s incomes by 7%. Given the log-utility model, this is roughly the same welfare effect as raising 70 people’s incomes by 10%.
54 1.028^90 = 12.
55 1.016^130 = 8
56 1.008^130 = 3.
57  See Bloom et al. (2013)Coe and Helpman (1995) and references within Jones and Williams (1998)  and Jones and Summers (2020).
58 0.75*0.4%/2.3% = 0.13.
59 E.g. Bloom et al. (2013) estimate a private return of 21% and a social return of 55%. See multiple estimates in table 1 of Jones and Williams (1998) and in Appendix B of Jones and Summers (2020).
60 Conversely, if I’d used the depreciation assumptions used in the empirical estimates, I’d have expected the benefit from marginal R&D to diminish much more quickly over time and calculated a lower total return to R&D.
61 The model here differs from the standard semi-endogenous growth model in two ways. First, the standard semi-endogenous growth model assumes that all TFP growth is due to R&D; this model relaxes this assumption. Second, this model translates income changes to welfare changes using a log-utility model.
62 E.g. see Patrick Collison here, 19:57-25:32.
63 I.e. as income falls, the marginal utility of income increases by more.
64 https://ourworldindata.org/grapher/gdp-per-capita-worldbank?tab=chart&yScale=log&country=OWID_WRL~USA Imagine someone in 1990 who earns $40k being offered the following choice:
65 To make the comparison fair, in option 2 you should be forced to buy all goods and services at 2020 prices. This means that some services will be more expensive than in 1990. Even so, I think option 2 is better because there are many new products that are cheap but useful.
If this doesn’t seem right, you might consider a more extreme version of the thought experiment. Instead of comparing 2020 with 1990, compare 2020 with 990. Imagine offering a medieval peasant enough medieval goods and services — food, land, servants — that their income is $60k a year. Would they prefer that, or being on $60k in 2020 with the conveniences of central heating, clean running water, varied food, modern medicine, etc? Here it seems clear to me that the latter option is better.
66 How much does the income increase underestimate the welfare benefit of technological progress? You could roughly estimate this by increasing the cash payment in option 1. E.g. suppose you’re indifferent between option 2 and option 1’) A cash payment of $40k. This would imply that the true welfare benefit is very roughly 2X the income increase. (This is very rough because the utility gains from receiving $40k are not twice those from $20k, due to diminishing returns to $.)
67 Phil Trammell describes one such utility function here. Also see discussion by Matt Clancy in the “What about other benefits?” section of this post.

History of Philanthropy: Work We’ve Commissioned

This page collects work we’ve commissioned on the history of philanthropy, as outlined on our focus area page.

 

Case studies we’ve commissioned

Case study on the Healthcare for the Homeless Program, by Ben Soskis:

Case study on the Pew Charitable Trusts’ drug safety legislation program, by Tamara Mann Tweel:

Case study on the role of multiple funders in the passage of the Affordable Care Act, by Ben Soskis:

Case study on the founding of the Center on Budget and Policy Priorities, by Suzanne Kahn:

Case study on the founding of the Center for Global Development, by Ben Soskis:

Case study on the role of the Center on Budget and Policy Priorities in state EITC programs, by Suzanne Kahn:

Case study on the Clinton Health Access Initiative’s role in global price drops for antiretroviral drugs, by Tamara Mann Tweel:

Case study on Philanthropy’s Role in The Fight for Marriage Equality, by Benjamin Soskis:

Case study on the role of philanthropy in promoting nuclear nonproliferation and threat reduction, by Paul Rubinson:

 

Case studies produced by the Center on Nonprofits and Philanthropy at the Urban Institute

We’ve also supported this work through a grant to the Urban Institute. Read more about this work at the Urban Institute’s website.

Case study on the Pugwash conferences on science and world affairs, by Paul Rubinson:

Philanthropy and US Student Movements: Four Cases, by Maoz Brown:

Literature Review: Conservative Philanthropy in Higher Education, by David Austin Walsh:

 

Case studies we’ve completed

Some case studies in early field growth, by Luke Muehlhauser:

 

Other work we’ve commissioned

Historian Ben Soskis reviewed the existing literature on the history of philanthropy and created the following resources:

  • Annotated bibliography: A list of books he identified as possibly informative. He briefly reviewed each book and summarized its contents.
  • Extended bibliography: A list of books he reviewed (covered in the list above); books he considered but did not review; and books he would review if he spent more time on this project.
  • Process and findings: The process he used to select books for consideration and his preliminary conclusions about the state of the literature.

We also support Ben’s work on HistPhil. This work is structured as part of Ben’s consulting for us rather than as a grant, which is why it does not appear in our grants database.

We have also commissioned reviews of the existing literature on several particular cases of philanthropic impact:

Could Advanced AI Drive Explosive Economic Growth?

This report evaluates the likelihood of ‘explosive growth’, meaning > 30% annual growth of gross world product (GWP), occurring by 2100. Although frontier GDP/capita growth has been constant for 150 years, over the last 10,000 years GWP growth has accelerated significantly. Endogenous growth theory, together with the empirical fact of the demographic transition, can explain both trends. Labor, capital and technology were accumulable over the last 10,000 years, meaning that their stocks all increased as a result of rising output. Increasing returns to these accumulable factors accelerated GWP growth. But in the late 19th century, the demographic transition broke the causal link from output to the quantity of labor. There were not increasing returns to capital and technology alone and so growth did not accelerate; instead frontier economies settled into an equilibrium growth path defined by a balance between a growing number of researchers and diminishing returns to research.

This theory implies that explosive growth could occur by 2100. If automation proceeded sufficiently rapidly (e.g. due to progress in AI) there would be increasing returns to capital and technology alone. I assess this theory and consider counter-arguments stemming from alternative theories; expert opinion; the fact that 30% annual growth is wholly unprecedented; evidence of diminishing returns to R&D; the possibility that a few non-automated tasks bottleneck growth; and others. Ultimately, I find that explosive growth by 2100 is plausible but far from certain.

1. How to read this report

Read the summary (~1 page). Then read the main report (~30 pages).

The rest of the report contains extended appendices to the main report. Each appendix expands upon specific parts of the main report. Read an appendix if you’re interested in exploring its contents in greater depth.

I describe the contents of each appendix here. The best appendix to read is probably the first, Objections to explosive growth. Readers may also be interested to read reviews of the report.

Though the report is intended to be accessible to non-economists, readers without an economics background may prefer to read the accompanying blog post.


2. Why we are interested in explosive growth

Open Philanthropy wants to understand how far away we are from developing transformative artificial intelligence(TAI). Difficult as it is, a working timeline for TAI helps us prioritize between our cause areas, including potential risks from advanced AI.

In her draft report, my colleague Ajeya Cotra uses TAI to mean ‘AI which drives Gross World Product (GWP) to grow at ~20-30% per year’ – roughly ten times faster than it is growing currently. She estimates a high probability of TAI by 2100 (~80%), and a substantial probability of TAI by 2050 (~50%). These probabilities are broadly consistent with the results from expert surveys,1 and with plausible priors for when TAI might be developed.2

Nonetheless, intuitively speaking these are high probabilities to assign to an ‘extraordinary claim’. Are there strong reasons to dismiss these estimates as too high? One possibility is economic forecasting. If economic extrapolations gave us strong reasons to think GWP will grow at ~3% a year until 2100, this would rule out explosive growth and so rule out TAI being developed this century.

I find that economic considerations don’t provide a good reason to dismiss the possibility of TAI being developed in this century. In fact, there is a plausible economic perspective from which sufficiently advanced AI systems are expected to cause explosive growth.


3. Summary

If you’re not familiar with growth economics, I recommend you start by reading this glossary or my blog post about the report.

Since 1900, frontier GDP/capita has grown at about 2% annually.3 There is no sign that growth is speeding up; if anything recent data suggests that growth is slowing down. So why think that > 30% annual growth of GWP (‘explosive growth’) is plausible this century?

I identify three arguments to think that sufficiently advanced AI could drive explosive growth:

  1. Idea-based models of very long-run growth imply AI could drive explosive growth.
    • Growth rates have significantly increased (super-exponential growth) over the past 10,000 years, and even over the past 300 years. This is true both for GWP growth, and frontier GDP/capita growth.
    • Idea-based models explain increasing growth with an ideas feedback loop: more ideas → more output → more people → more ideas… Idea-based models seem to have a good fit to the long-run GWP data, and offer a plausible explanation for increasing growth.
    • After the demographic transition in ~1880, more output did not lead to more people; instead people had fewer children as output increased. This broke the ideas feedback loop, and so idea-based theories expect growth to stop increasing shortly after the time. Indeed, this is what happened. Since ~1900 growth has not increased but has been roughly constant.
    • Suppose we develop AI systems that can substitute very effectively for human labor in producing output and in R&D. The following ideas feedback loop could occur: more ideas → more output → more AI systems → more ideas… Before 1880, the ideas feedback loop led to super-exponential growth. So our default expectation should be that this new ideas feedback loop will again lead to super-exponential growth.
  2. A wide range of growth models predict explosive growth if capital can substitute for labor. Here I draw on models designed to study the recent period of exponential growth. If you alter these models with the assumption that capital can substitute very effectively for labor, e.g. due to the development of advanced AI systems, they typically predict explosive growth. The mechanism is similar to that discussed above. Capital accumulation produces a powerful feedback loop that drives faster growth: more capital → more output → more capital …. These first two arguments both reflect an insight of endogenous growth theory: increasing returns to accumulable inputs can drive accelerating growth.
  3. An ignorance perspective assigns some probability to explosive growth. We may not trust highly-specific models that attempt to explain why growth has increased over the long-term, or why it has been roughly constant since 1900. But we do know that the pace of growth has increased significantly over the course of history. Absent deeper understanding of the mechanics driving growth, it would be strange to rule out growth increasing again. 120 years of steady growth is not enough evidence to rule out a future increase.

I discuss a number of objections to explosive growth:

  • 30% growth is very far out of the observed range.
  • Models predicting explosive growth have implausible implications – like output going to infinity in finite time.
  • There’s no evidence of explosive growth in any subsector of the economy.
  • Limits to automation are likely to prevent explosive growth.
  • Won’t diminishing marginal returns to R&D prevent explosive growth?
  • And many others.

Although some of these objections are partially convincing, I ultimately conclude that explosive growth driven by advanced AI is a plausible scenario.

In addition, the report covers themes relating to the possibility of stagnating growth; I find that it is a highly plausible scenario. Exponential growth in the number of researchers has been accompanied by merely constant GDP/capita growth over the last 80 years. This trend is well explained by semi-endogenous growth models in which ideas are getting harder to find.4 As population growth slows over the century, number of researchers will likely grow more slowly; semi-endogenous growth models predict that GDP/capita growth will slow as a result.

Thus I conclude that the possibilities for long-run growth are wide open. Both explosive growth and stagnation are plausible.

Acknowledgements: My thanks to Holden Karnofsky for prompting this investigation; to Ajeya Cotra for extensive guidance and support throughout; to Ben Jones, Dietrich Vollrath, Paul Gaggl, and Chad Jones for helpful comments on the report; to Anton Korinek, Jakub Growiec, Phil Trammel, Ben Garfinkel, David Roodman, and Carl Shulman for reviewing drafts of the report in depth; to Harry Mallinson for reviewing code I wrote for this report and helpful discussion; to Joseph Carlsmith, Nick Beckstead, Alexander Berger, Peter Favaloro, Jacob Trefethen, Zachary Robinson, Luke Muehlhauser, and Luisa Rodriguez for valuable comments and suggestions; and to Eli Nathan for extensive help with citations and the website.


4. Main report

If you’re not familiar with growth economics, I recommend you start by reading this glossary or my blog post about the report.

How might we assess the plausibility of explosive growth (>30% annual GWP) occurring by 2100? First, I consider the raw empirical data; then I address a number of additional considerations.

  • What do experts think (here)?
  • How does economic growth theory affect the case of explosive growth (here and here)?
  • How strong are the objections to explosive growth (here)?
  • Conclusion (here).

4.1 Empirical data without theoretical interpretation

When looking at the raw data, two conflicting trends jump out.

The first trend is the constancy of frontier GDP/capita growth over the last 150 years.5 The US is typically used to represent this frontier. The following graph from Our World in Data shows US GDP/capita since 1870.

The y-axis is logarithmic, so the straight line indicates that growth has happened at a constant exponential rate – ~2% per year on average.6 Extrapolating the trend, frontier GDP/capita will grow at ~2% per year until 2100. GWP growth will be slightly larger, also including a small boost from population growth and catch-up growth. Explosive growth would be a very large break from this trend.

I refer to forecasts along these lines as the standard story. Note, I intend the standard story to encompass a wide range of views, including the view that growth will slow down significantly by 2100 and the view that it will rise to (e.g.) 4% per year.

The second trend is the super-exponential growth of GWP over the last 10,000 years.7 (Super-exponential means the growth rate increases over time.) Another graph from Our World in Data shows GWP over the last 2,000 years:

Again, the y-axis is logarithmic, so the increasing steepness of the slope indicates that the growth rate has increased.

It’s not just GWP – there’s a similar super-exponential trend in long-run GDP/capita in many developed countries – see the graphs of US, English, and French GDP/capita in section 14.3.8 (Later I discuss whether we can trust these pre-modern data points.)

It turns out that a simple equation called a ‘power law’ is a good fit to GWP data going all the way back to 10,000 BCE. The following graph (from my colleague David Roodman) shows the fit of a power law (and of exponential growth) to the data. The axes of the graph are chosen so that the power law appears as a straight line.9

If you extrapolate this power law trend into the future, it implies that the growth rate will continue to increase into the future and that GWP will approach infinity by 2047!10

Many other simple curves fit to this data also predict explosive (>30%) growth will occur in the next few decades. Why is this? The core reason is that the data shows the growth rate increasing more and more quickly over time. It took thousands of years for growth to increase from 0.03% to 0.3%, but only a few hundred years for it to increase from 0.3% to 3%.11 If you naively extrapolate this trend, you predict that growth will increase again from 3% to 30% within a few decades.

We can see this pattern more clearly by looking at a graph of how GWP growth has changed over time.12

The graph shows that the time needed for the growth rate to double has fallen over time. (Later I discuss whether this data can be trusted.) Naively extrapolating the trend, you’d predict explosive growth within a few decades.

I refer to forecasts along these lines, that predict explosive growth by 2100, as the explosive growth story.

So we have two conflicting stories. The standard story points to the steady ~2% growth in frontier GDP/capita over the last 150 years, and expects growth to follow a similar pattern out to 2100. The explosive growth story points to the super-exponential growth in GWP over the last 10,000 years and expects growth to increase further to 30% per year by 2100.

Which story should we trust? Before taking into account further considerations, I think we should put some weight on both. For predictions about the near future I would put more weight on the standard story because its data is more recent and higher quality. But for predictions over longer timescales I would place increasing weight on the explosive growth story as it draws on a longer data series.

Based on the two empirical trends alone, I would neither confidently rule out explosive growth by 2100 nor confidently expect it to happen. My attitude would be something like: ‘Historically, there have been significant increases in growth. Absent a deeper understanding of the mechanisms driving these increases, I shouldn’t rule out growth increasing again in the future.’ I call this attitude the ignorance story.13 The rest of the main report raises considerations that can move us away from this attitude (either towards the standard story or towards the explosive growth story).

4.2 Expert opinion

In the most recent and comprehensive expert survey on growth out to 2100 that I could find, all the experts assigned low probabilities to explosive growth.

All experts thought it 90% likely that the average annual GDP/capita growth out to 2100 would be below 5%.14 Strictly speaking, the survey data is compatible with experts thinking there is a 9% probability of explosive growth this century, but this seems unlikely in practice. The experts’ quantiles, both individually and in aggregate, were a good fit for normal distributions which would assign ≪ 1% probability to explosive growth.15

Experts’ mean estimate of annual GWP/capita growth was 2.1%, with standard deviation 1.1%.16 So their views support the standard story and are in tension with the explosive growth story.

There are three important caveats:

  1. Lack of specialization. My impression is that long-run GWP forecasts are not a major area of specialization, and that the experts surveyed weren’t experts specifically in this activity. Consonant with this, survey participants did not consider themselves to be particularly expert, self-reporting their level of expertise as 6 out of 10 on average.17
  2. Lack of appropriate prompts. Experts were provided with the data about the growth rates for the period 1900-2000, and primed with a ‘warm up question’ about the recent growth of US GDP/capita. But no information was provided about the longer-run super-exponential trend, or about possible mechanisms for producing explosive growth (like advanced AI). The respondents may have assigned higher probabilities to explosive growth by 2100 if they’d been presented with this information.
  3. No focus on tail outcomes. Experts were not asked explicitly about explosive growth, and were not given an opportunity to comment on outcomes they thought were < 10% likely to occur.

4.3 Theoretical models used to extrapolate GWP out to 2100

Perhaps economic growth theory can shed light on whether to extrapolate the exponential trend (standard story) or the super-exponential trend (explosive growth story).

In this section I ask:

  • Do the growth models of the standard story give us reason beyond the empirical data to think 21st century growth will be exponential or sub-exponential?
    • They could do this if they point to a mechanism explaining recent exponential growth, and this mechanism will continue to operate in the future.
  • Do the growth models of the explosive growth story give us reason beyond the empirical data to think 21st century growth will be super-exponential?
    • They could do this if they point to a mechanism explaining the long-run super-exponential growth, and this mechanism will continue to operate in the future.

My starting point is the models actually used to extrapolate GWP to 2100, although I draw upon economic growth theory more widely in making my final assessment. First, I give a brief explanation of how growth models work.

4.3.1 How do growth models work?

In economic growth models, a number of inputs are combined to produce output. Output is interpreted as GDP (or GWP). Typical inputs include capital (e.g. equipment, factories), labor (human workers), human capital (e.g. skills, work experience), and the current level of technology.18

Some of these inputs are endogenous,19 meaning that the model explains how the input changes over time. Capital is typically endogenous; output is invested to sustain or increase the amount of capital.20 In the following diagram, capital and human capital are endogenous:

Other inputs may be exogenous. This means their values are determined using methods external to the growth model. For example, you might make labor exogenous and choose its future values using UN population projections. The growth model does not (attempt to) explain how the exogenous inputs change over time.

When a growth model makes more inputs endogenous, it models more of the world. It becomes more ambitious, and so more debatable, but it also gains the potential to have greater explanatory power.

4.3.2 Growth models extrapolating the exponential trend to 2100

I looked at a number of papers in line with the standard story that extrapolate GWP out to 2100. Most of them treated technology as exogenous, typically assuming that technology will advance at a constant exponential rate.21In addition, they all treated labor as exogenous, often using UN projections. These growth models can be represented as follows:

The blue ‘+’ signs represent that the increases to labor and technology each year are exogenous, determined outside of the model.

In these models, the positive feedback loop between output and capital is not strong enough to produce sustained growth. This is due to diminishing marginal returns to capital. This means that each new machine adds less and less value to the economy, holding the other inputs fixed.22 Even the feedback loop between output and (capital + human capital) is not strong enough to sustain growth in these models, again due to diminishing returns.

Instead, long-run growth is driven by the growth of the exogenous inputs, labor and technology. For this reason, these models are called exogenous growth models: the ultimate source of growth lies outside of the model. (This is contrasted with endogenous growth models, which try to explain the ultimate source of growth.)

It turns out that long run growth of GDP/capita is determined solely by the growth of technology.23These models do not (try to) explain the pattern of technology growth, and so they don’t ultimately explain the pattern of GDP/capita growth.

4.3.2.1 Evaluating models extrapolating the exponential trend

The key question of this section is: Do the growth models of the standard story give us reason beyond the empirical data to think 21st century growth of frontier GDP/capita will be exponential or sub-exponential?

My answer is ‘yes’. Although the exogenous models used to extrapolate GWP to 2100 don’t ultimately explain why GDP/capita has grown exponentially, there are endogenous growth models that address this issue. Plausible endogenous models explain this pattern and imply that 21st century growth will be sub-exponential. This is consistent with the standard story. Interestingly, I wasn’t convinced by models implying that 21st century growth will be exponential.

The rest of this section explains my reasoning in more detail.

Endogenous growth theorists have for many decades sought theories where long-run growth is robustly exponential. However, they have found it strikingly difficult. In endogenous growth models, long-run growth is typically only exponential if some knife-edge condition holds. A parameter of the model must be exactly equal to some specific value; the smallest disturbance in this parameter leads to completely different long-run behavior, with growth either approaching infinity or falling to 0. Further, these knife-edges are typically problematic: there’s no particular reason to expect the parameter to have the precise value needed for exponential growth. This problem is often called the ‘linearity critique’ of endogenous growth models.

Appendix B argues that many endogenous growth models contain problematic knife-edges, drawing on discussions in Jones (1999), Jones (2005), Cesaratto (2008), and Bond-Smith (2019).

Growiec (2007) proves that a wide class of endogenous growth models require a knife-edge condition to achieve constant exponential growth, generalizing the proof of Christiaans (2004). The proof doesn’t show that all such conditions are problematic, as there could be mechanisms explaining why knife-edges hold. However, combined with the observation that many popular models contain problematic knife-edges, the proof suggests that it may be generically difficult to explain exponential growth without invoking problematic knife-edge conditions.

Two attempts to address this problem stand out:

  1. Claim that exponential population growth has driven exponential GDP/capita growth. This is an implication of semi-endogenous growth models (Jones 1995). These models are consistent with 20th century data: exponentially growing R&D effort has been accompanied by exponential GDP/capita growth. Appendix B argues that semi-endogenous growth models offer the best framework for explaining the recent period of exponential growth.24However, I do not think their ‘knife-edge’ assumption that population will grow at a constant exponential rate is likely to be accurate until 2100. In fact, the UN projects that population growth will slow significantly over the 21st century. With this projection, semi-endogenous growth models imply that GDP/capita growth will slow.25 So these models imply 21st century growth will be sub-exponential rather than exponential.26
  2. Claim that market equilibrium leads to exponential growth without knife-edge conditions.
    • In a 2020 paper Robust Endogenous Growth, Peretto outlines a fully endogenous growth model that achieves constant exponential growth of GDP/capita without knife-edge conditions. The model displays increasing returns to R&D investment, which would normally lead to super-exponential growth. However, these increasing returns are ‘soaked up’ by the creation of new firms which dilute R&D investment. Market incentives ensure that new firms are created at exactly the rate needed to sustain exponential growth.
    • The model seems to have some implausible implications. Firstly, it implies that there should be a huge amount of market fragmentation, with the number of firms growing more quickly than the population. This contrasts with the striking pattern of market concentration we see in many areas.27 Secondly, it implies that if no new firms were introduced – e.g. because this was made illegal – then output would reach infinity in finite time. This seems to imply that there is a huge market failure: private incentives to create new firms massively reduce long-run social welfare.
    • Despite these problems, the model does raise the possibility that an apparent knife-edge holds in reality due to certain equilibrating pressures. Even if this model isn’t quite right, there may still be equilibrating pressures of some sort.28
    • Overall, this model slightly raises my expectation that long-run growth will be exponential.29

This research shifted my beliefs in a few ways:

  • I put more probability (~75%) on semi-endogenous growth models explaining the recent period of exponential growth.30
    • So I put more weight on 21st century growth being sub-exponential.
    • We’ll see later that these models imply that sufficiently advanced AI could drive explosive growth. So I put more weight on this possibility as well.
  • It was harder than I expected to for growth theories to adequately explain why income growth should be exponential in a steady state (rather than sub- or super-exponential). So I put more probability on the recent period of exponential growth being transitory, rather than part of a steady state.
    • For example, the recent period could be a transition between past super-exponential growth and future sub-exponential growth, or a temporary break in a longer pattern of super-exponential growth.
    • This widens the range of future trajectories that I regard as being plausible.

4.3.3 Growth models extrapolating the super-exponential trend

Some growth models extrapolate the long-run super-exponential trend to predict explosive growth in the future.31 Let’s call them long-run explosive models. The ones I’m aware of are ‘fully endogenous’, meaning all inputs are endogenous.32

Crucially, long-run explosive models claim that more output → more people. This makes sense (for example) when food is scarce: more output means more food, allowing the population to grow. This assumption is important, so it deserves a name. Let’s say these models make population accumulable. More generally, an input is accumulable just if more output → more input.33

The term ‘accumulable’ is from the growth literature; the intuition behind it is that the input can be accumulated by increasing output.

It’s significant for an input to be accumulable as it allows a feedback loop to occur: more output → more input → more output →… Population being accumulable is the most distinctive feature of long-run explosive models.

Long-run explosive models also make technology accumulable: more output → more people → more ideas (technological progress).

All growth models, even exogenous ones, imply that capital is accumulable: more output → more reinvestment → more capital.34 In this sense, long-run explosive models are a natural extension of the exogenous growth models discussed above: a similar mechanism typically used to explain capital accumulation is used to explain the accumulation of technology and labor.

We can roughly represent long-run explosive models as follows:35

The orange arrows show that all the inputs are accumulable: a marginal increase in output leads to an increase in the input. Fully endogenous growth models like these attempt to model more of the world than exogenous growth models, and so are more ambitious and debatable; but they potentially have greater explanatory power.

Why do these models predict super-exponential growth? The intuitive reason is that, with so many accumulable inputs, the feedback loop between the inputs and output is powerful enough that growth becomes faster and faster over time.

More precisely, the key is increasing returns to scale in accumulable inputs: when we double the level of every accumulable input, output more than doubles.36

Why are there increasing returns to scale? The key is the insight, from Romer (1990), that technology is non-rival. If you use a new solar panel design in your factory, that doesn’t prevent me from using that same design in my factory; whereas if you use a particular machine/worker, that does prevent me from using that same machine/worker.

Imagine doubling the quantity of labor and capital, holding technology fixed. You could literally replicate every factory and worker inside it, and make everything you currently make a second time. Output would double. Crucially, you wouldn’t need to double the level of technology because ideas are non-rival: twice as many factories could use the same stock of ideas without them ‘running out’.

Now imagine also doubling the level of technology. We’d still have twice as many factories and twice as many workers, but now each factory would now be more productive. Output would more than double. This is increasing returns to scale: double the inputs, more than double the output.37

Long-run explosive models assume that capital, labor and technology are all accumulable. Even if they include a fixed input like land, there are typically increasing returns to accumulable inputs. This leads to super-exponential growth as long unless the diminishing returns to technology R&D are very steep.38 For a wide of plausible parameter values, these models predict super-exponential growth.39

The key feedback loop driving increasing returns and super-exponential growth in these models can be summarized as more ideas (technological progress) → more output → more people → more ideas→…

These models seem to be a good fit to the long-run GWP data. The model in Roodman (2020) implies that GWP follows a ‘power-law’, which seems to fit the data well.

Long-run explosive models fitted to the long-run GWP data typically predict that explosive growth (>30% per year) is a few decades away. For example, you can ask the model in Roodman (2020) ‘When will be the first year of explosive growth?’ Its median prediction is 2043 and the 80% confidence range is [2034, 2065].

4.3.3.1 Evaluating models extrapolating the super-exponential trend

The key question of this section is: Do the growth models of the explosive growth story give us reason to think 21st century growth will be super-exponential? My answer in this section is ‘no’, because the models are not well suited to describing post-1900 growth. In addition, it’s unclear how much we should trust their description of pre-1900 growth. (However, the next section argues these models can be trusted if we develop sufficiently powerful AI systems.)

4.3.3.1.1.Problem 1: Long-run explosive models are not suitable for describing post-1900 growth

The central problem is that long-run explosive models assume population is accumulable.40 While it is plausible than in pre-modern times more output → more people, this hasn’t been true in developed countries over the last ~140 years. In particular, since ~1880 fertility rates have declined despite increasing GDP/capita.41 This is known as the demographic transition. Since then, more output has not led to more people, but to richer and better educated people: more output → more richer people. Population is no longer accumulable (in the sense that I’ve defined the term).42 The feedback loop driving super-exponential growth is broken: more ideas → more output → more richer people → more ideas.

How would this problem affect the models’ predictions? If population is not accumulable, then the returns to accumulable inputs are lower, and so growth is slower. We’d expect long-run explosive models to predict faster growth than we in fact observe after ~1880; in addition we wouldn’t expect to see super-exponential growth after ~1880.43

Indeed, this is what the data shows. Long-run explosive models are surprised at how slow GWP growth is since 1960 (more), and surprised at how slow frontier GDP/capita growth is since 1900 (more). It is not surprising that a structural change means a growth model is no longer predictively accurate: growth models are typically designed to work in bounded contexts, rather than being universal theories of growth.

A natural hypothesis is that the reason why long-run explosive models are a poor fit to the post-1900 data is that they make an assumption about population that has been inaccurate since ~1880. The recent data is not evidence against long-run explosive models per se, but confirmation that their predictions can only be trusted when population is accumulable.

This explanation is consistent with some prominent idea-based theories of very long-run growth.44 These theories use the same mechanism as long-run explosive models to explain pre-1900 super-exponential growth: labor and technology are accumulable, so there are increasing returns to accumulable inputs,45 so there’s super-exponential growth. They feature the same ideas feedback loop: more ideas → more output → more people → more ideas→…46

These idea-based theories are made consistent with recent exponential growth by adding an additional mechanism that makes the fertility rate drop once the economy reaches a mature stage of development,47 mimicking the effect of the demographic transition. After this point, population isn’t accumulable and the models predict exponential growth by approximating some standard endogenous or semi-endogenous model.48

These idea-based models provide a good explanation of very long-run growth and modern growth. They increase my confidence in the main claim of this section: long-run explosive models are a poor fit to the post-1900 data because they (unrealistically) assume population is accumulable. However, idea-based models are fairly complex and were designed to explain long-run patterns in GDP/capita and population; this should make us wary to trust them too much.49

4.3.3.1.2 Problem 2: It is unclear how much we should trust long-run explosive models’ explanation of pre-1900 growth

None of the problems discussed above dispute the explosive growth story’s explanation of pre-1900 growth. How much weight should we put on its account?

It emphasizes the non-rivalry of ideas and the mechanism of increasing returns to accumulable factors. This mechanism implies growth increased fairly smoothly over hundreds and thousands of years.50 We saw that the increasing-returns mechanism plays a central role in several prominent models of long-run growth.51

However, most papers on very long run growth emphasize a different explanation, where a structural transition occurs around the industrial revolution.52 Rather than a smooth increase, this suggests a single step-change in growth occurred around the industrial revolution, without growth increasing before or after the step-change.53

Though a ‘step-change’ view of long-run growth rates will have a lesser tendency to predict explosive growth by 2100, it would not rule it out. For this, you would have to explain why step change increases have occurred in the past, but no more will occur in the future.54

How much weight should we place in the increasing-returns mechanism versus the step-change view? The ancient data points are highly uncertain, making it difficult to adjudicate empirically.55 Though GWP growth seems to have increased across the whole period 1500 – 1900, this is compatible with there being one slow step-change.56

There is some informative evidence:

  • Kremer (1993) gives evidence for the increasing-returns mechanism. He looks at the development of 5 isolated regions and finds that the technology levels of the regions in 1500 are perfectly rank-correlated with their initial populations in 10,000 BCE. This is just what the increasing returns mechanism would predict.57
  • Roodman (2020) gives evidence for the step-change view. Roodman finds that his own model, which uses the increasing-returns mechanism, is surprised by the speed of growth around the industrial revolution (see more).

Overall, I think it’s likely that the increasing-returns mechanism plays an important role in explaining very long-run growth. As such I think we should take long-run explosive models seriously (if population is accumulable). That said, they are not the whole story; important structural changes happened around the industrial revolution.58

4.3.4 Summary of theoretical models used to extrapolate GWP out to 2100

I repeat the questions asked at the start of this section, now with their answers:

  • Do the growth models of the standard story give us reason beyond the empirical data to think 21st century growth will be exponential or sub-exponential?
    • Yes, plausible models imply that growth will be sub-exponential. Interestingly, I didn’t find convincing reasons to expect exponential growth.
  • Do the growth models of the explosive growth story give us reason beyond the empirical data to think 21st century growth will be super-exponential?
    • No, long-run explosive models assume population is accumulable, which isn’t accurate after ~1880.
    • However, the next section argues that advanced AI could make this assumption accurate once more. So I think these models do give us reason to expect explosive growth if sufficiently advanced AI is developed.
STANDARD STORY EXPLOSIVE GROWTH STORY
Preferred data set Frontier GDP/capita since 1900 GWP since 10,000 BCE
Predicted shape of long-run growth Exponential or sub-exponential Super-exponential (for a while, and then eventually sub-exponential)
Models used to extrapolate GWP to 2100 Exogenous growth models Endogenous growth model, where population and technology are accumulable.
Evaluation Semi-endogenous growth models are plausible and predict 21st century growth will be sub-exponential. Theories predicting exponential growth rely on problematic knife-edge conditions. Population is no longer accumulable, so we should not trust these models by default. However, advanced AI systems could make this assumption realistic again, in which case the prediction of super-exponential can be trusted.

4.4 Advanced AI could drive explosive growth

It is possible that significant advances in AI could allow capital to much more effectively substitute for labor.59 Capital is accumulable, so this could lead to increasing returns to accumulable inputs, and so to super-exponential growth.60 I’ll illustrate this point from two complementary perspectives.

4.4.1 AI robots as a form of labor

First, consider a toy scenario in which Google announces tomorrow that it’s developed AI robots that can perform anytask that a human laborer can do for a smaller cost. In this (extreme!) fiction, AI robots can perfectly substitute for all human labor. We can write (total labor) = (human labor) + (AI labor). We can invest output to build more AI robots,61 and so increase the labor supply: more output → more labor (AI robots). In other words, labor is accumulable again. When this last happened there was super-exponential growth, so our default expectation should be that this scenario will lead to super-exponential growth.

To look at it another way, AI robots would reverse the effect of the demographic transition. Before that transition, the following feedback loop drove increasing returns to accumulable inputs and super-exponential growth:

More ideas → more output → more labor (people) → more ideas →…

With AI robots there would be a closely analogous feedback loop:

More ideas → more output → more labor (AI robots) → more ideas →…

PERIOD FEEDBACK LOOP? IS TOTAL LABOR ACCUMULABLE? PATTERN OF GROWTH
Pre-1880 Yes: More ideas → more output → more people → more ideas →… Yes GWP grows at an increasing rate.
1880 – present No: More ideas → more output → more richer people → more ideas →… No GWP grows at a ~constant rate.
AI robot scenario Yes: More ideas → more output → more AI systems → more ideas →… Yes GWP grows at an increasing rate.

Indeed, plugging the AI robot scenario into a wide variety of growth models, including exogenous growth models, you find that increased returns to accumulable inputs drives super-exponential growth for plausible parameter values.62

This first perspective, analysing advanced AI as a form of labor, emphasizes the similarity of pre-1900 growth dynamics to those of a possible future world with advanced AI. If you think that the increasing-returns mechanism increased growth in the past, it’s natural to think that the AI robot scenario would increase growth again.63

4.4.2 AI as a form of capital

There are currently diminishing returns to accumulating more capital, holding the amount of labor fixed. For example, imagine creating more and more high-quality laptops and distributing them around the world. At first, economic output would plausibly increase as the laptops made people more productive at work. But eventually additional laptops would make no difference as there’d be no one to use them. The feedback loop ‘more output → more capital → more output →…’ peters out.

Advances in AI could potentially change this. By automating wide-ranging cognitive tasks, they could allow capital to substitute more effectively for labor. As a result, there may no longer be diminishing returns to capital accumulation. AI systems could replace both the laptops and the human workers, allowing capital accumulation to drive faster growth.64

Economic growth models used to explain growth since 1900 back up this point. In particular, if you adjust these models by assuming that capital substitutes more effectively for labor, they predict increases in growth.

The basic story is: capital substitutes more effectively for labor → capital’s share of output increases → larger returns to accumulable inputs → faster growth. In essence, the feedback loop ‘more output → more capital → more output → …’ becomes more powerful and drives faster growth.

What level of AI is required for explosive (>30%) growth in these models? The answer varies depending on the particular model:65

  • Often the crucial condition is that the elasticity of substitution between capital and labor rises above 1. This means that some (perhaps very large) amount of capital can completely replace any human worker, though it is a weaker condition than perfect substitutability.66
  • In the task-based model of Aghion et al. (2017), automating a fixed set of tasks leads to only a temporary boost in growth. A constant stream of automation (or full automation) is needed to maintain faster growth.67
  • Appendix C discusses the conditions for super-exponential growth in a variety of such models (see here).

Overall, what level of AI would be sufficient for explosive growth? Based on a number of models, I think that explosive growth would require AI that substantially accelerates the automation of a very wide range of tasks in the production of goods and services, R&D, and the implementation of new technologies. The more rapid the automation, and the wider the range of tasks, the faster growth could become.68

It is worth emphasizing that these models are simple extensions of standard growth models; the only change is to assume that capital can substitute more effectively for labor. With this assumption, semi-endogenous models with reasonable parameter values predict explosive growth, as do exogenous growth models with constant returns to labor and capital.69

A draft literature review on the possible growth effects of advanced AI includes many models in which AI increases growth via this mechanism (capital substituting more effectively for labor). In addition, it discusses several other mechanisms by which AI could increase growth, e.g. changing the mechanics of idea discovery and changing the savings rate.70

4.4.3 Combining the two perspectives

Both the ‘AI robots’ perspective and the ‘AI as a form of capital’ perspective make a similar point: if advanced AI can substitute very effectively for human workers, it could precipitate explosive growth by increasing the returns to accumulable inputs. In many growth models with plausible parameter values this scenario leads to explosive growth.

Previously, we said we should not trust long-run explosive models as they unrealistically assume population is accumulable. We can now qualify this claim. We should not trust these models unless AI systems are developed that can replace human workers.

4.4.4 Could sufficiently advanced AI be developed in time for explosive growth to occur this century?

This is not a focus of this report, but other evidence suggests that this scenario is plausible:

  • A survey of AI practitioners asked them about the probability of developing AI that would enable full automation.71 Averaging their responses, they assigned ~30% or ~60% probability to this possibility by 2080, depending on how the question is framed.72
  • My colleague Joe Carlsmith’s report estimates the computational power needed to match the human brain. Based on this and other evidence, my colleague Ajeya Cotra’s draft report estimates when we’ll develop human-level AI; she finds we’re ~70% likely to do so by 2080.
  • In a previous report I estimated the probability of developing human-level AI based on analogous historical developments. My framework finds a ~15% probability of human-level AI by 2080.

4.5 Objections to explosive growth

My responses are brief, and I encourage interested readers to read Appendix A, which discusses these and other objections in more detail.

4.5.1 What about diminishing returns to technological R&D?

Objection: There is good evidence that ideas are getting harder to find. In particular, it seems that exponential growth in the number of researchers is needed to sustain constant exponential growth in technology (TFP).

Response: The models I have been discussing take this dynamic into account. They find that, with realistic parameter values, increasing returns to accumulable inputs is powerful enough to overcome diminishing returns to technological progress if AI systems can replace human workers. This is because the feedback loop ‘more output → more labor (AI systems) → more output’ allows research effort to grow super-exponentially , leading to super-exponential TFP growth despite ideas becoming harder to find (see more).

Related objection: You claimed above that the demographic transition caused super-exponential growth to stop. This is why you think advanced AI could restart super-exponential growth. But perhaps the real cause was that we hit more sharply diminishing returns to R&D in the 20th century.

Response: This could be true. Even if true, though, this wouldn’t rule out explosive growth occurring this century: it would still be possible that returns to R&D will become less steep in the future and the historical pattern of super-exponential growth will resume.73

However, I investigated this possibility and came away thinking that diminishing returns probably didn’t explain the end of super-exponential growth.

  • Various endogenous growth models suggest that, had population remained accumulable throughout the 20th century, growth would have been super-exponential despite the sharply diminishing returns to R&D that we have observed.
  • Conversely, these models suggest that the demographic transition would have ended super-exponential growth even if diminishing returns to R&D had been much less steep.
  • This all suggests that the demographic transition, not diminishing returns, is the crucial factor in explaining the end of super-exponential growth (see more).

That said, I do think it’s reasonable to be uncertain about why super-exponential growth came to an end. The following diagram summarizes some possible explanations for the end of super-exponential growth in the 20th century, and their implications for the plausibility of explosive growth this century.

4.5.2 30% growth is very far out of the observed range

Objection: Explosive growth is so far out of the observed range! Even when China was charging through catch-up growth it never sustained more than 10% growth. So 30% is out of the question.

Response: Ultimately, this is not a convincing objection. If you had applied this reasoning in the past, you would have been repeatedly led into error. The 0.3% GWP growth of 1400 was higher than the previously observed range, and the 3% GWP growth of 1900 was higher than the previously observed range. There is historical precedent for growth increasing to levels far outside of the previously observed range (see more).

4.5.3 Models predicting explosive growth have implausible implications

Objection: Endogenous growth models imply output becomes infinite in a finite time. This is impossible and we shouldn’t trust such unrealistic models.

Response: First, models are always intended to apply only within bounded regimes; this doesn’t mean they are bad models. Clearly these endogenous growth models will stop applying before we reach infinite output (e.g. when we reach physical limits); they might still be informative before we reach this point. Secondly, not all models predicting explosive growth have this implication; some models imply that growth will rise without limit but never go to infinity (see more).

4.5.4 There’s no evidence of explosive growth in any economic sub-sector

Objection: If GWP growth rates were soon going to rise to 30%, we’d see signs of this in the current economy. But we don’t – Nordhaus (2021) looks for such signs and doesn’t find them.

Response: The absence of these signs in macroeconomic data is reason to doubt explosive growth will occur within the next couple of decades. Beyond this time frame, it is hard to draw conclusions. Further, it’s possible that the recent fast growth of machine learning is an early sign of explosive growth (see more).

4.5.5 Why think AI automation will be different to past automation?

Objection: We have been automating parts of our production processes and our R&D processes for many decades, without growth increasing. Why think AI automation will be different?

Response: To cause explosive growth, AI would have to drive much faster and widespread automation than we have seen over the previous century. If AI ultimately enabled full automation, models of automation suggest that the consequences for growth would be much more radical than those from the partial automation we have had in the past (see more).

4.5.6 Automation limits

Objection: Aghion et al. (2017) considers a model where growth is bottlenecked by tasks that are essential but hard to improve. If we’re unable to automate just one essential task, this would prevent explosive growth.

Response: This correctly highlights that AI may lead to very widespread automation without explosive growth occurring. One possibility is that an essential task isn’t automated because we care intrinsically about having a human perform the task, e.g. a carer.

I don’t think this provides a decisive reason to rule out explosive growth. Firstly, it’s possible that we will ultimately automate all essential tasks, or restructure work-flows to do without them. Secondly, there could be a significant boost in growth rates, at least temporarily, even without full automation (see more).74

4.5.7 Limits to how fast a human economy can grow

Objection: The economic models predicting explosive growth ignore many possible bottlenecks that might slow growth. Examples include regulation of the use of AI systems, extracting and transporting important materials, conducting physical experiments on the world needed to make social and technological progress, delays for humans to adjust to new technological and social innovations, fundamental limits to how advanced technology can become, fundamental limits of how quickly complex systems can grow, and other unanticipated bottlenecks.75

Response: I do think that there is some chance that one of these bottlenecks will prevent explosive growth. On the other hand, no individual bottleneck is certain to apply and there are some reasons to think we could grow at 30% per year:

  • There will be huge incentives to remove bottlenecks to growth, and if there’s just one country that does this it would be sufficient.
  • Large human economies have already grown at 10% per year (admittedly via catch up growth), explosive growth would only be 3X as fast.76
  • Humans oversee businesses growing at 30% per year, and individual humans can adjust to 30% annual increases in wealth and want more.
  • AI workers could run much faster than human workers.77
  • Biological populations can grow faster than 30% a year, suggesting that it is physically possible for complex systems to grow this quickly.78

The arguments on both sides are inconclusive and inevitably speculative. I feel deeply uncertain about how fast growth could become before some bottleneck comes into play, but personally place less than 50% probability on a bottleneck preventing 30% GWP growth. That said, I have spent very little time thinking about this issue, which would be a fascinating research project in its own right.

4.5.8 How strong are these objections overall?

I find some of the objections unconvincing:

  • Diminishing returns. The models implying that full automation would lead to explosive growth take diminishing returns into account.
  • 30% is far from the observed range. Ruling out 30% on this basis would have led us astray in the past by ruling out historical increases in growth.
  • Models predicting explosive growth have implausible implications. We need not literally believe that output will go to infinity to trust these models, and there are models that predict explosive growth without this implication.

I find other objections partially convincing:

  • No evidence of explosive growth in any economic sub-sector. Trends in macroeconomic variables suggest there won’t be explosive growth in the next 20 years.
  • Automation limits. A few essential but unautomated tasks might bottleneck growth, even if AI drives widespread automation.
  • Limits to how fast a human economy can grow. There are many possible bottlenecks on the growth of a human economy; we have limited evidence on whether any of these would prevent 30% growth in practice.

Personally, I assign substantial probability (> 1/3) that the AI robot scenario would lead to explosive growth despite these objections.

4.6 Conclusion

The standard story points to the constant exponential growth of frontier GDP/capita over the last 150 years. Theoretical considerations suggest 21st century growth is more likely to be sub-exponential than exponential, as slowing population growth leads to slowing technological progress. I find this version of the standard story highly plausible.

The explosive growth story points to the significant increases in GWP growth over the last 10,000 years. It identifies an important mechanism explaining super-exponential growth before 1900: increasing returns to accumulable inputs. If AI allows capital to substitute much more effectively for human labor, a wide variety of models predict that increasing returns to accumulable inputs will again drive super-exponential growth. On this basis, I think that ‘advanced AI drives explosive growth’ is a plausible scenario from the perspective of economics.

It is reasonable to be skeptical of all the growth models discussed in the report. It is hard to get high quality evidence for or against different growth models, and empirical efforts to adjudicate between them often give conflicting results.79  It is possible that we do not understand key drivers of growth. Someone with this view should probably adopt the ignorance story: growth has increased significantly in the past, we don’t understand why, and so we should not rule out significant increases in growth occurring in the future. If someone wishes to rule out explosive growth, they must positively reject any theory that implies it is plausible; this is hard to do from a position of ignorance.

Overall, I assign > 10% probability to explosive growth occurring this century. This is based on > 30% that we develop sufficiently advanced AI in time, and > 1/3 that explosive growth actually occurs conditional on this level of AI being developed.80 Barring this kind of progress in AI, I’m most inclined to expect sub-exponential growth. In this case, projecting GWP is closely entangled with forecasting the development of advanced AI.

4.6.1 Are we claiming ‘this time is different’?

If you extrapolate the returns from R&D efforts over the last century, you will not predict that sustaining these efforts might lead to explosive growth this century. Achieving 3% growth in GDP/capita, let alone 30%, seems like it will be very difficult. When we forecast non-trivial probability of explosive growth, are we essentially claiming ‘this time will be different because AI is special’?

In a certain sense, the answer is ‘yes’. We’re claiming that economic returns to AI R&D will ultimately be much greater than the average R&D returns over the past century.

In another sense, the answer is ‘no’. We’re suggesting that sufficiently powerful AI would, by allowing capital to replace human labor, lead to a return to a dynamic present throughout much of human history where labor was accumulable. With this dynamic reestablished, we’re saying that ‘this time will be the same’: this time, as before, the economic consequence of an accumulable labor force will be super-exponential growth.

4.7 Further research

  • Why do experts rule out explosive growth? This report argues that one should not confidently rule out explosive growth. In particular, I suggest assigning > 10% to explosive growth this century. Experts seem to assign much lower probabilities to explosive growth. Why is this? What do they make of the arguments of the report?
  • Investigate evidence on endogenous growth theory.
    • Assess Kremer’s rank-correlation argument. Does the ‘more people → more innovation’ story actually explain the rank correlation, or are there other better explanations?
    • Investigate theories of long-run growth. How important is the increasing returns mechanism compared to other mechanisms in explaining the increase in long-run growth?
    • Empirical evidence on different growth theories. What can 20th century empirical evidence tell us about the plausibility of various growth theories? I looked into this briefly and it seemed as if the evidence did not paint a clear picture.
  • Are we currently seeing the early signs of explosive GDP growth?
    • How long before explosive growth of GDP would we see signs of it in some sector of the economy?
    • What exactly would these signs look like? What can we learn from the economic signs present in the UK before the onset of the industrial revolution?
    • Does the fast growth of current machine learning resemble these signs?
  • Do returns to technological R&D change over time? How uneven has the technological landscape been in the past? Is it common to have long periods where R&D progress is difficult punctuated by periods where it is easier? More technically, how much does the ‘fishing out’ parameter change over time?
  • Are there plausible theories that predict exponential growth? Is there a satisfactory explanation for the constancy of frontier per capita growth in the 20th century that implies that this trend will continue even if population growth slows? Does this explanation avoid problematic knife-edge conditions?
  • Is there evidence of super-exponential growth before the industrial revolution? My sensitivity analysis suggested that there is, but Ben Garfinkel did a longer analysis and reached a different conclusion. Dig into this apparent disagreement.
    • Length of data series: How long must the data series be for there to be clear evidence of super-exponential growth?
    • Type of data: How much difference does it make if you use population vs GWP data?
  • How likely is a bottleneck to prevent an AI-driven growth explosion?

5. Structure of the rest of the report

The rest of the report is not designed to be read end to end. It consists of extended appendices that expand upon specific claims made in the main report. Each appendix is designed so that it can be read end to end.

The appendices are as follows:

  • Objections to explosive growth (see here).
    • This is a long section, which contains many of the novel contributions of this report.
    • It’s probably the most important section to read after the main report, expanding upon objections to explosive growth in detail.
  • Exponential growth is a knife-edge condition in many growth models (see here).
    • I investigate one reason to think long-run growth won’t be exponential: exponential growth is a knife-edge condition in many economic growth models.
    • This is not a core part of my argument for explosive growth.
    • The section has three key takeaways:
      1. Sub-exponential is more plausible than exponential growth, out to 2100.
      2. There don’t seem to be especially strong reasons to expect exponential growth, raising the theoretical plausibility of stagnation and of explosive growth.
      3. Semi-endogenous models offer the best explanation of the exponential trend. When you add to these models the assumption that capital and substitute effectively for human labor, they predict explosive growth. This raises my probability that advanced AI could drive explosive growth.
  • Conditions for super-exponential growth (see here).
    • I report the conditions for super-exponential growth (and thus for explosive growth) in a variety of economic models.
    • These include models of very long-run historical growth, and models designed to explain modern growth altered by the assumption that capital can substitute for labor.
    • I draw some tentative conclusions about what kinds of AI systems may be necessary for explosive growth to occur.
    • This section is math-heavy.
  • Ignorance story (see here).
    • I briefly explain what I call the ‘ignorance story’, how it might relate to the view that there was a step-change in growth around the industrial revolution, and how much weight I put on this story.
  • Standard story (see here).
    • I explain some of the models used to project long-run GWP by the standard story.
    • These models forecast GWP/capita to grow at about 1-2% annually out to 2100.
    • I find that the models typically only use post-1900 data and assume that technology will grow exponentially. However, the models provide no more support for this claim than is found in the uninterpreted empirical data.
      1. Other endogenous models do provide support for this claim. I explore such models in Appendix B.
    • I conclude that these models are suitable for projecting growth to 2100 on the assumption that 21st growth resembles 20th century growth. They are not well equipped to assess the probability of a structural break occurring, after which the pattern of 20th growth no longer applies.
  • Explosive growth before 2100 is robust to accounting for today’s slow GWP growth (see here)
    • Long-run explosive models predict explosive growth within a few decades. From an outside view perspective81, it is reasonable to put some weight on such models. But these models typically imply growth should already be at ~7%, which we know is false.
    • I adjust for this problem, developing a ‘growth multiplier’ model. It maintains the core mechanism driving increases in growth in the explosive growth story, but anchors its predictions to the fact that GWP growth over the last 20 years has been about 3.5%. As a result, its prediction of explosive growth is delayed by about 40 years.
    • From an outside view perspective, I personally put more weight on the ‘growth multiplier model’ than Roodman’s long-run explosive model.
    • In this section, I explain the growth multiplier model and conduct a sensitivity analysis on its results.
  • How I decide my probability of explosive growth (see here).
    • Currently I put ~30% on explosive growth occurring by 2100. This section explains my reasoning.
  • Links to reviews of the report (see here).
  • Technical appendices (see here).
    • These contain a number of short technical analyses that support specific claims in the report.
    • I only expect people to read these if they follow a link from another section.

6. Appendix A: Objections to explosive growth

Currently, I don’t find any of these objections entirely convincing. Nonetheless, taken together, the objections shift my confidence away from the explosive growth story and towards the ignorance story instead.

I initially discuss general objections to explosive growth, then objections targeted specifically at using long-run growth data to argue for explosive growth.

Here are the objections, in the order in which I address them:

General objections to explosive growth

Partially convincing objections

No evidence of explosive growth in any subsector of the economy
Growth models predicting explosive growth are unconfirmed
Why think AI automation will be different to past automation?
Automation limits
Diminishing returns to R&D (+ ‘search limits’)
Baumol tasks

Ultimately unconvincing objections

Explosive growth is so far out of the observed range
Models predicting explosive growth have unrealistic implications

Objections to using long-run growth to argue for explosive growth

Partially convincing objections

The ancient data points are unreliable
Recent data shows that super-exponential growth in GWP has come to an end
Frontier growth shows a clear slowdown

Slightly convincing objections

Long-run explosive models don’t anchor predictions to current growth levels
Long-run explosive models don’t discount pre-modern data
Long-run explosive models don’t seem to apply to time before the agricultural revolution; why expect them to apply to a new future growth regime?

6.1 General objections to explosive growth

6.1.1 No evidence of explosive growth in any sub sector of the economy

Summary of objection: If GWP growth rates were soon going to rise to 30%, we’d see signs of this in the current economy. We’d see 30% growth in sectors of the economy that have the potential to account for the majority of economic activity. For example, before the industrial revolution noticeably impacted GDP, the manufacturing sector was growing much faster than the rest of the economy. But no sector of the economy shows growth anywhere near 30%; so GWP won’t be growing at 30% any time soon.

Response: I think this objection might rule out explosive growth in the next few decades, but I’d need to see further investigation to be fully convinced of this.

I agree that there should be signs of explosive growth before it registers on any country’s GDP statistics. Currently, this makes me somewhat skeptical that there will be explosive growth in the next two decades. However, I’m very uncertain about this due to being ignorant about several key questions.

  • How long before explosive growth of GDP would we see signs of it in some sector of the economy?
  • What exactly would these signs look like?
  • Are there early signs of explosive growth in the economy?

I’m currently very unsure about all three questions above, and so am unsure how far into the future this objection rules out explosive growth. The next two sections say a little more about the third question.

6.1.1.1.Does the fast growth of machine learning resemble the early signs of explosive growth?

With regards the penultimate question, Open Philanthropy believes that there is a non-negligible chance (> 15%) of very powerful AI systems being developed in the next three decades. The economic impact of machine learning is already growing fast with use in Google’s search algorithm, targeted ads, product recommendations, translation, and voice recognition. One recent report forecasts an average of 42% annual growth of the deep learning market between 2017 and 2023.

Of course, many small sectors show fast growth for a time and do not end up affecting the overall rate of GWP growth! It is the further fact that machine learning seems to be a general purpose technology, whose progress could ultimately lead to the automation of large amounts of cognitive labor, that raises the possibility that its fast growth might be a precursor of explosive growth.

6.1.1.2 Are there signs of explosive growth in US macroeconomic variables?

Nordhaus (2021) considers the hypothesis that explosive growth will be driven by fast productivity growth in the IT sector. He proposes seven empirical tests of this hypothesis. The tests make predictions about patterns in macroeconomic variables like TFP, real wages, capital’s share of total income, and the price and total amount of capital. He runs these tests with US data. Five of the tests suggest that we’re not moving towards explosive growth; the other two suggest we’re moving towards it only very slowly, such that a naive extrapolation implies explosive growth will happen around 2100.82

Nordhaus runs three of his tests with data specific to the IT sector.83 This data is more fine-grained than macroeconomic variables, but it’s still much broader than machine learning as a whole. The IT data is slightly more optimistic about explosive growth, but still suggests that it won’t happen within the next few decades.84

These empirical tests suggest that, as of 2014, the patterns in US macroeconomic variables are not what you’d expect if explosive growth driven by AI R&D was happening soon. But how much warning should we expect these tests to give? I’m not sure. Nordhaus himself says that his ‘conclusion is tentative and is based on economic trends to date’. I would expect patterns in macroeconomic variables to give more warning than trends in GWP or GDP, but less warning than trends in the economic value of machine learning. Similarly, I’d expect IT-specific data to give more warning than macroeconomic variables, but less than data specific to machine learning.

Brynjolfsson (2017)85 suggests economic effects will lag decades behind the potential of the technology’s cutting edge, and that national statistics could underestimate the longer term economic impact of technologies. As a consequence, disappointing historical data should not preclude forward-looking technological optimism.86

Overall, Nordhaus’ analysis reduces my probability that we will see explosive growth by 2040 (three decades after his latest data point) but it doesn’t significantly change my probability that we see it in 2050 – 2100. His analysis leaves open the possibility that we are seeing the early signs of explosive growth in data relating to machine learning specifically.

6.1.2 The evidence for endogenous growth theories is weak

Summary of objection: Explosive growth from sufficiently advanced AI is predicted by certain endogenous growth models, both theories of very long-run growth and semi-endogenous growth models augmented with the assumption that capital can substitute for labor.

The mechanism posited by these models is increasing returns to accumulable inputs.

But these endogenous growth models, and the mechanisms behind them, have not been confirmed. So we shouldn’t pay particular attention to their predictions. In fact, these models falsely predict that larger economies should grow faster.

Response summary:

  • There is some evidence for endogenous growth models.
  • Endogenous growth models do not imply that larger economies should grow faster than smaller ones.
  • As well as endogenous growth models, some exogenous growth models predict that AI could bring about explosive growth by increasing the importance of capital accumulation: more output → more capital → more output →… (see more).

The rest of this section goes into the first two points in more detail.

6.1.2.1 Evidence for endogenous growth theories

6.1.2.1.1 Semi-endogenous growth models

These are simply standard semi-endogenous growth theories. Under realistic parameter values, they predict explosive growth when you add the assumption that capital can substitute for labor (elasticity of substitution > 1).

What evidence is there for these theories?

  • Semi-endogenous growth theories are inherently plausible. They extend standard exogenous theories with the claim that directed human effort can lead to technological progress.
  • Appendix B argues that semi-endogenous growth theories offer a good explanation of the recent period of exponential growth.
  • However, there have not been increasing returns to accumulable inputs in the recent period of exponential growth because labor has not been accumulable. This might make us doubt the predictions of semi-endogenous models in a situation in which there are increasing returns to accumulable inputs, and thus doubt their prediction of explosive growth.

6.1.2.1.2 Theories of very long-run growth featuring increasing returns

Some theories of very long-run growth feature increasing returns to accumulable inputs, as they make technology accumulable and labor accumulable (in the sense that more output → more people). If AI makes labor accumulable again, these theories predict there will be explosive growth under realistic parameter values.

What evidence is there for these theories?

  • These ‘increasing returns’ models seem to correctly describe the historical pattern of accelerating growth.87 However, the data is highly uncertain and it is possible that growth did not accelerate between 5000 BCE and 1500. If so, this would undermine the empirical evidence for these theories.
  • Other evidence comes from Kremer (1993). He looks at five regions – Flinders Island, Tasmania, Australia, the Americas and the Eurasian continent – that were isolated from one another 10,000 years ago and had significantly varying populations. Initially all regions contained hunter gathers, but by 1500 CE the technology levels of these regions had significantly diverged. Kremer shows that the 1500 technology levels of these regions were perfectly rank-correlated with their initial populations, as predicted by endogenous growth models.

6.1.2.2 Endogenous growth models are not falsified by the faster growth of smaller economies.

Different countries share their technological innovations. Smaller economies can grow using the innovations of larger economies, and so the story motivating endogenous growth models does not predict that countries with larger economies should grow faster. As explained by Jones (1997):

The Belgian economy does not grow solely or even primarily because of ideas invented by Belgians… this fact makes it difficult… to test the model with cross-section evidence [of different countries across the same period of time]. Ideally one needs a cross-section of economies that cannot share ideas.

In other words, the standard practice of separating technological progress into catch-up growth and frontier growth is fully consistent with applying endogenous growth theories to the world economy. Endogenous growth models are not falsified by the faster growth of smaller economies.

6.1.3 Why think AI automation will be different to past automation?

Objection: Automation is nothing new. Since 1900, there’s been massive automation in both production and R&D (e.g. no more calculations by hand). But growth rates haven’t increased. Why should future automation have a different effect?

Response: If AI merely continues the previous pace of automation, then indeed there’s no particular reason to think it would cause explosive growth. However, if AI allows us to approach full automation, then it may well do so.

A plausible explanation for why previous automation hasn’t caused explosive growth is that growth ends up being bottlenecked by non-automated tasks. For example, suppose there are three stages in the production process for making a cheese sandwich: make the bread, make the cheese, combine the two together. If the first two stages are automated and can proceed much more quickly, the third stage can still bottleneck the speed of sandwich production if it isn’t automated. Sandwich production as a whole ends up proceeding at the same pace as the third stage, despite the automation of the first two stages.

Note, whether this dynamic occurs depends on people’s preferences, as well as on the production possibilities. If people were happy to just consume bread by itself and cheese by itself, all the necessary steps would have been automated and output could have grown more quickly.

The same dynamic as with sandwich production can happen on the scale of the overall economy. For example, hundreds of years ago agriculture was a very large share of GDP. Total GDP growth was closely related to productivity growth in agriculture. But over the last few hundred years, the sector has been increasingly automated and its productivity has risen significantly. People in developed countries now generally have plenty of food. But as a result, GDP in developed countries is now more bottlenecked by things other than agriculture. Agriculture is now only a small share of GDP, and so productivity gains in agriculture have little effect on overall GDP growth.

Again this relates to people’s preferences. Once people have plenty of food, they value further food much less. This reduces the price of food, and reduces agriculture’s share of GDP. If people had wanted to consume more and more food without limit, agriculture’s share of the economy would not have fallen so much.88

So, on this account, the reason why automation doesn’t lead to growth increases is because the non-automated sectors bottleneck growth.

Clearly, this dynamic won’t apply if there is full automation, for example if we develop AI systems that can replace human workers in any task. There would be no non-automated sectors left to bottleneck growth. This insight is consistent with models of automation, for example Growiec (2020) and Aghion et al. (2017) – they find that the effect of full automation is qualitatively different from that of partial automation and leads to larger increases in growth.

The next section discusses whether full automation is plausible, and whether we could have explosive growth without it.

6.1.4 Automation limits

Objection: Aghion et al. (2017) considers a growth model that does a good job in explaining the past trends in automation and growth. In particular, their model is consistent with the above explanation for why automation has not increased growth in the past: growth ends up being bottlenecked by non-automated tasks.

In their model, output is produced by a large number of tasks that are gross complements. Intuitively, this means that each task is essential. More precisely, if we hold performance on one task fixed, there is a limit to how large output can be no matter how well we perform other tasks.89  As a result, ‘output and growth end up being determined not by what we are good at, but by what is essential but hard to improve’.

The model highlights that if there is one essential task that we cannot automate, this will ultimately bottleneck growth. Growth will proceed at the rate at which we can improve performance at this non-automated task.Response : There are two questions in assessing this objection:

  1.  Will there be an essential task that we cannot automate?
  2. If there is such a task, would this preclude explosive growth?

6.1.4.1 Will there be an essential task that we cannot automate?

The first question cannot be answered without speculation.

It does seem possible that we make very impressive progress in AI, automating wide-ranging cognitive abilities, but that there are some essential tasks that we still cannot automate. It is unclear how stable this situation would be: with many cognitive abilities automated, a huge cognitive effort could be made to automate the remaining tasks. Further, if we can restructure workflows to remove the necessity of an un-automated task, the bottleneck will disappear.

One reason to think full automation is plausible is that humans may ultimately have a finite set of capabilities (including the capability to learn certain types of new tasks quickly). Once we’ve developed machines with the same capabilities across the board, there will be nothing more to automate. When new tasks are created, machines will learn them just as quickly as humans.

One possibility is that tasks that will not be automated because we care intrinsically about having a biological human perform the task (e.g. carers, athletes, priests). I don’t expect this to be the sole factor preventing explosive growth:

  • In this scenario, if just one group didn’t have this intrinsic preference for human workers, it could grow explosively and ultimately drive explosive growth of GWP. So this scenario seems undermined by the heterogeneity of human preferences.
  • In this scenario the growth model of Aghion et al. (2017) implies that the percentage of GDP spent on tasks where we prefer human workers approaches 100%.90 But this seems unlikely to happen. Tasks crucial for gaining relative power in society, e.g. control of resources and military technology, can in principle be automated in this scenario. It seems unlikely that all actors would allow their spending on these tasks to approach 0%, essentially giving up relative power and influence.
    • If instead a constant fraction of output is spent on automated tasks, we could model this with a task-based Cobb Douglas production function. With this model, explosive growth then occurs if a sufficiently large fraction of output is spent on the automated tasks (see this model).

6.1.4.2 If there’s an essential task we cannot automate, does this preclude explosive growth?

Slightly more can be said about the second question.

Firstly, there can be super-exponential growth without full automation ever occurring. If we automate an increasing fraction of non-automated tasks each year, there can be super-exponential growth.

For example, the total fraction of automated tasks goes 0%, 50%, 80%, 95%,… We automate 1/2 the non-automated tasks in the first year, 2/3 in the second year, 3/4 in the third year, and so on. In this scenario, the economy is asymptotically automated, but never fully automated.

  • This situation implies that for any task i, that task is eventually automated. But this is also implied by the scenario favored in Aghion et al. (2017), in which a constant fraction of non-automated tasks are automated each year.
  • I am not claiming here that we will automate an increasing fraction of tasks each year, but just that such a situation is plausible (and perhaps similarly plausible to automating a constant fraction each year).
  • Note, super-exponential growth can only be sustained if there is some capital-augmenting technological progress happening in the background.91

What about if there’s some fixed fraction of tasks that we cannot automate?

This does rule out growth increasing without limit.92 However, it doesn’t rule out a significant but temporary increase in growth. There may be a long time before non-automated tasks become a bottleneck in practice, and growth may rise considerably during this time. For example, suppose that the number of human carers ultimately bottlenecks growth. In the long-run, most of GDP is spent on humans carers and productivity improvements elsewhere will make little difference to GDP growth. Nonetheless, there can be an interim period where human carers are still only a small share of GDP but the quantities of other goods and services are growing extremely rapidly, driving explosive growth of GDP. This explosive growth would end once spending on human carers is a large fraction of GDP.

Indeed, the authors of Aghion et al. (2017) acknowledge that even if there’s a limit to automation, ‘growth rates may still be larger with more automation and capital intensity’. Whether growth gets as high as 30% depends on how quickly the other tasks are automated,93 how quickly we increase the stock of capital,94how important the non-automated task is to the economy,95 and how well we initially perform the non-automated task.96

6.1.4.3 A drawback of the model

The model does not seem well suited for thinking about the introduction of new tasks. In their model, introducing a new task can only ever decrease output.97

6.1.4.4 Conclusion

This objection correctly highlights the possibility that very impressive progress in AI doesn’t lead to explosive growth due a few non-automatable tasks. This is a plausible scenario. Nonetheless, explosive growth could occur if we will eventually automate all tasks, or if we automate an increasing fraction of tasks each year, or if growth increases significantly before bottlenecks kick in.

6.1.5 Baumol tasks

Objection: Even if we automate both goods and ideas production, Aghion et al. (2017) raises the possibility that physical limits could constrain growth.98  In particular, they consider a model where each task has its own productivity. If there’s an absolute limit on the productivity of any essential task, then this ultimately limits overall TFP and can prevent explosive growth.

Response: This objection is correct: ultimately the growth process will come up against physical limits and TFP will reach an absolute ceiling. However, this doesn’t give us much reason to rule out explosive growth.

Firstly, even once TFP reaches its ceiling we could have fast exponential growth. If we automate all tasks Y = AmaxK; reinvestment is ΔK = sY – δK; Amax is the ceiling for TFP fixed by physical limits. The growth rate of the system is Amaxs – δ, which could be very high indeed.

Secondly, we may be a long way from achieving the maximum possible TFP. Before we reach this point, there could be super-exponential growth. The model raises the possibility that we may be closer to the ceiling than we think: if just one essential task hits a limit then this will limit total TFP. However, we should be wary of placing too much weight on this perspective. TFP has not yet been permanently limited by an essential but hard to improve task, despite the economy containing a huge array of tasks and experiencing lots of TFP growth. This is somewhat surprising to an advocate for Baumol tasks: surely just one of the many essential tasks should have hit a limit by now? The evidence to the contrary speaks to our ability to increase productivity in essential tasks despite physical limits, or to replace them with new tasks that avoid these limits.

6.1.6 What about diminishing returns to technological R&D?

Objection: There is good evidence that ideas are getting harder to find, at least when these ideas are weighted by their effects on economic growth.

Economists often understand ‘ideas’ in units such that a constant flow of ideas leads to constant exponential growth in A; each idea raises income by a constant percentage.

It is common to represent this effect using the parameter φ in the equation Ȧ = AφX, where X measures the amount of research effort (e.g. number of scientists) and A represents TFP. If ideas are getting harder to find, this means that φ < 1. This condition is important; it implies that X must increase exponentially to sustain exponential growth in A.

Bloom et al. (2020) observes steeply diminishing returns in 20th century R&D; they estimate φ = -2.1. Such steeply diminishing returns will surely prevent explosive growth. Perhaps they also explain the end of super-exponential growth in the 20th century.

Response: The feedback loop between output and inputs can be powerful enough to overcome these diminishing returns, especially if there are increasing returns to accumulable inputs. This is because the feedback loop can be strong enough for X to grow super-exponentially, leading to super-exponential growth in A.

This happens if increasing returns to accumulable inputs are powerful enough to overcome the diminishing returns to R&D.99 If labor is accumulable, or capital is substitutable with labor (elasticity of substitution > 1), models with plausible parameter values suggest there will be super-exponential growth despite the sharply diminishing to R&D observed by Bloom et al. More on the conditions for super-exponential growth in these models.

Consistent with this, various endogenous growth models suggest that the period of super-exponential growth did not end because the diminishing returns to R&D became too steep. Rather, they suggest that the demographic transition, which meant labor was no longer accumulable (in the sense that more output → more labor), was the key factor (see more).

Lastly, even if 20th century diminishing returns did rule out explosive growth, it is possible that returns will diminish less steeply in the future (the value of φ could increase).100 There could be an uneven technological landscape, where progress is slow for a time and then quicker again.

Further objection: Aghion et al. (2017) consider a model in which ideas production is fully automated, Ȧ = AφK, but growth still does not increase due to ‘search limits’. Importantly, in their model goods production is bottlenecked by labor, Y = AL.101 If φ > 0, the growth rate increases without limit, but if φ < 0, the growth rate decreases over time. φ < 0 is plausible. Theoretically, it could be explained by a fishing-out process, in which fewer and fewer good ideas remain to be discovered over time. Empirically, Bloom et al. (2020) estimates φ = -2.1 based on 80 years of US data.

Response: This correctly highlights the possibility that we fully automate R&D without seeing explosive growth. However, I still expect that full R&D automation would lead to explosive growth.

Firstly, in this model there would still be a temporary boost in growth while the ideas production was being automated. The automation process would cause research effort X to increase, perhaps very rapidly, leading to much faster growth temporarily.

Secondly, full automation of ideas production might facilitate full automation of the goods production (e.g. if it allows us to automate the process of automating tasks), Y = AK. Automating tasks is naturally thought of as a research activity. Full automation of goods production would lead to super-exponential growth, no matter what the value of φ.102This is the response I find most convincing.

Thirdly, even if φ < 0 in the economy on aggregate, it may be that >0 in certain important subsectors of the economy and this is sufficient for explosive growth. Of particular importance may be subsectors relating to how efficiently output can be reinvested to create more AI systems. If φ > 0 in these subsectors then, even if φ < 0 on aggregate, the number of AI systems can grow super-exponentially. This could in turn drive super-exponential growth of technology in all sectors, and thus drive explosive growth of output. I describe a toy model along these lines in this technical appendix.

Is φ > 0 in the relevant subsectors? The subsectors relating to how efficiently output can be reinvested to make AI systems are likely to be computer hardware and AI software. Bloom et al. (2020) find φ = 0.8 for a measure of computer hardware performance, and data from Besiroglu (2020) finds φ = 0.85 for a measure machine learning software performance. Of course this doesn’t show that this scenario is likely to happen, but reinforces the point that there is no easy inference from ‘φ < 0 in the aggregate’ to ‘AI automation of R&D wouldn’t drive explosive growth’.

Lastly, some papers find φ > 0. Even if it is currently below 0, it may change over time, and rise above 0.

6.1.7 Explosive growth is so far out of the observed range

Summary of objection: No country has ever grown at anywhere near 30%. Even when China was at its peak rate of catch-up growth, benefitting significantly from adopting advanced western technology, it grew at 8%. Never in history has a country grown faster than 10%. Explosive growth is so far out of the observed range that it should be regarded as highly improbable.

Response: This is a very natural objection, but ultimately I find it unconvincing.

The same kind of reasoning would have led people in 1750, when growth had never been higher than 0.3%, to rule out growth of 3%. And the same reasoning again would have led hypothetical economists alive in 5000 BCE, when the rate of growth had never been higher than 0.03%, to rule out growth of 0.3%. Growth rates have increased by two orders of magnitude throughout history, and so the reasoning ‘growth rates will stay within the historically observed ranges’ would have repeatedly led to false predictions.

It is true that a 30% growth by 2100 would involve a ten-fold increase in growth happening more quickly than any comparable increase in history. The increase from 0.3% to 3% took more than 150 years to occur and there are only 80 years left until 2100. But historically, increases in the growth rate have happened over progressively shorter time periods. For example, the increase from 0.03% to 0.3% took 6000 years. In 1700 it would have been a mistake to say ‘it took thousands of years for growth rates to increase ten-fold from 0.03% to 0.3%, so it will be thousands of years before growth increases ten-fold again to 3%’. This reasoning would ignore the historical pattern whereby growth increases more quickly over time. Similarly, it would be a mistake now to reason ‘it took hundreds of years for growth rates to increase from 0.3% to 3%, so it will be hundreds of years before growth could reach 30%’.103

So the fact that growth has never previously been anywhere near as high as 30% is not by itself a good reason to rule out explosive growth.

Relatedly, it would be unreasonable to assign an extremely low prior to 30% growth occurring.104Priors assigning tiny probabilities to GWP growth increasing well above its observed range would have been hugely surprised by the historical GWP trend. They should be updated to assign more probability to extreme outcomes.

6.1.8 Models predicting explosive growth have implausible implications

Summary of objection: The very same endogenous growth models that predict explosive growth by 2100 also predict that GWP will go to infinity in finite time. This prediction is absurd, and so the models shouldn’t be trusted.

This objection is in the spirit of a comment from economist Robert Solow:

It is one thing to say that a quantity will eventually exceed any bound. It is quite another to say that it will exceed any stated bound before Christmas.105

Response: Ultimately, I find this objection unconvincing.

Clearly, the economy cannot produce infinite output from a finite input of resources. And indeed this is exactly what certain endogenous growth models predict. But there are two ways to interpret this result.

  1. These models’ description of super-exponential growth is not realistic in any circumstances.
  2. Endogenous growth models’ description of super-exponential growth is only realistic up to a certain point, after which it ceases to be realistic.

I favor the second explanation for two reasons.

Firstly, it is very common for scientific theories to be accurate only in certain bounded regimes. This is true of both the hard sciences106 and the social sciences.107 As such, pointing out that a theory breaks down eventually only provides a very weak reason to think that it isn’t realistic in any circumstances. So the first explanation seems like an overreaction to the fact that theory breaks down eventually.

Secondly, it is independently plausible that the mechanism for super-exponential growth will break down eventually in the face of physical limits.

The mechanism is more output → more capital → better technology → more output →… But this cycle will eventually run up against physical limits. Eventually, we will be using the fixed input of physical resources in the best possible way to produce output, and further increases in output will be capped. At this stage, it won’t be possible to reinvest output in such a way as to significantly increase future output and the cycle will fizzle out.

In other words, we have a specific explanation for why we will never produce infinite output that leaves open the possibility that explosive growth occurs in the medium term.

So the fact that super-exponential growth must approach limits eventually – this particular objection – is itself only weak evidence that we have already reached those limits.

In addition to the above, many models predict explosive growth without implying output rises to infinity in finite time. For example, Nordhaus (2021) and Aghion et al. (2017) consider a model in which good production is fully automated but technological progress is still exogenous. This leads to a ‘type 1 singularity’ in which the growth rate increases without limit but never goes to infinity. Similarly, the models in Lee (1993) and Growiec (2020) both predict significant increases in growth but again the growth rate remains finite.

6.2 Objections to using the long-run growth to argue for explosive growth

6.2.1 The ancient data points used to estimate long-run explosive models are highly unreliable

Objection: We have terrible data on GWP before ~1500, so the results of models trained on this ‘data’ are meaningless.

Response: Data uncertainties don’t significantly affect the predictions of the long-run explosive models. However, they do undermine the empirical support for these models, and the degree of trust we should have in their conclusions.

6.2.1.1 Data uncertainties don’t significantly alter the predictions of long-run explosive models

Despite very large uncertainties in the long-run GWP data, it is clearly true that growth rates used to be much lower than they are today. This alone implies that, if you fit endogenous growth models to the data, you’ll predict super-exponential growth. Indeed, Roodman fit his model to several different data sets, and did a robustness test where he pushed all the data points to the tops and bottoms of their uncertainty ranges; in all cases the median predicted date of explosive growth was altered by < 5 years. This all suggests that data uncertainties, while significant, don’t drive significant variation in the predictions of long-run explosive models.

Using alternative data series, like GWP/capita and frontier GDP/capita, change the expected year of explosive growth by a few decades, but they still expect it before 2100.108

I did a sensitivity analysis, fitting Roodman’s univariate model to shortened GWP data sets starting in 10,000 BCE, 2000 BCE, 1 CE, 1000 CE, 1300 CE, 1600 CE, and 1800 CE. In every case, the fitted model expects explosive growth to happen eventually. (This is no surprise: as long as growth increases on average across the data set, long-run explosive models will predict explosive growth eventually.) The median predicted date for explosive growth is increasingly delayed for the shorter data sets;109 the model still assigns > 50% probability to explosive growth by 2100 if the data starts in 1300 CE or earlier. Sensitivity analysis on shortened data sets.

So the predictions of explosive growth can be significantly delayed by completely removing old data points; the obvious drawback is that by removing these old data points you lose information. Apart from this, the predictions of long-run explosive models do not seem to be sensitive to reasonable alterations in the data.

6.2.1.2 Data uncertainties undermine the empirical support for long-run explosive models

The long-run explosive models I’ve seen explain very long-run growth using the increasing returns mechanism. This mechanism implies growth should increase smoothly over hundreds and thousands of years.110

The data seems to show growth increasing fairly smoothly across the entire period 10,000 BCE to 1950 CE; this is a good fit for the increasing returns mechanism. However, I think the uncertainty of pre-modern data is great enough that the true data may show the growth in the period 5000 BCE to 1600 CE growth to be roughly constant. This would undermine the empirical support for the long-run explosive models, even if it wouldn’t substantially change their predictions.

Doubts about the goodness of fit are reinforced by the fact that alternative data series, like GWP/capita and frontier GDP/capita are less of a good fit to the increasing returns mechanism than the GWP series.

As an alternative to the increasing returns mechanism, you might instead place weight on a theory where there’s a single slow step-change in growth rates that happens between 1500 and 1900 (Ben Garfinkel proposes such a view here). Though a ‘slow step-change’ view of long-run growth rates will have a lesser tendency to predict explosive growth by 2100, it would not rule it out. For this, it would have to explain why step change increases in growth rate have occurred in the past, but more could not occur in the future.

Despite these concerns, it still seems likely to me that the increasing return mechanism offers an important role in explaining the long-run growth data. This suggests we should place weight on long-run explosive models, as long as population is accumulable.

6.2.2 Recent GWP growth shows that super-exponential growth has come to an end

Objection: Recently, GWP growth has been much lower than long-run explosive models have predicted. This shows that these models are no longer useful for extrapolating GWP

Response: Roodman (2020) does a careful analysis of how ‘surprised’ his model is by the recent data. His model is somewhat surprised at how slow GWP growth has been since 1970. But the data are not in very sharp conflict with the model and only provide a moderate reason to distrust the model going forward.

We can assess the size of the conflict between the model and the recent data in three ways: eyeballing the data, quantifying the conflict using Roodman’s model, and comparing the recent slowdown to historical slowdowns.

(Note, by ‘slowdown’ I mean ‘period where growth either remains at the same level or decreases’. This is a ‘slowdown’ compared to the possibility of super-exponential growth, even if growth remains constant.)

6.2.2.1 Eyeballing how much the recent data conflicts with Roodman’s model

First, here’s the graph we saw earlier of GWP against time. Though the recent points deviate slightly from Roodman’s trend, the difference is not significant. It looks smaller than previous historical deviations after which the trend resumed again.

A representation that highlights the deviation from the expected trend more clearly is to plot GWP against its average growth in the following period:

The last five data points indicate the growth after 1970 is surprisingly low. But again they do not seem to be in very sharp conflict with the trend.

6.2.2.2 Quantifying how much the recent data conflicts with Roodman’s model

It’s possible to quantify how surprised Roodman’s model is by a data point, given the previous data points (more). The results are that:

  • 1980 GWP is between the 40th and 50th percentiles, so isn’t surprising.
  • 1990, 2000, 2010, and 2019 GWP are between the 20th and 30th percentiles, so are surprising but not hugely surprising. If Roodman’s model incorporated serial correlation between random deviations from the underlying trend, the surprise would be smaller still.

6.2.2.3 The recent slowdown is large compared to other slowdowns in GWP growth

Growth in the period 1970 – 2020 has been slower than previously. During this time the economy has increased in size by a factor of 5.4. We can compare this to previous slowdowns after which the long-run super-exponential trend reasserted itself. If the recent growth slowdown is similar in size or smaller, this weakly suggests that the super-exponential trend will reassert itself once again, by analogy with previous slowdowns.

There are a couple of other slowdowns in GWP growth in the historical data:

  • Growth in the period 200 BCE – 1000 CE was consistently slower than in the previous thousand years. In this time the economy increased in size by a factor of 1.7.
  • Growth in the period 1200 CE – 1400 CE was slower than the previous period. In this time the economy did not increase in size.

So it seems the recent slowdown is shorter than previous slowdowns in terms of calendar years but longer when measured by the fractional increase of GWP.111 This weakly suggests the slowdown is not just random, but rather the result of some systematic factor. The return to super-exponential growth after past slowdowns is not a strong indicator that we’ll return to super-exponential growth after the current one.

The next section aims to strengthen this evidence further, by focusing on the growth of frontier economies (e.g. US, UK, France), rather than just merely GWP growth.

6.2.2.4 So what?

If we think the demographic transition explains the recent slowdown, we may not be moved by this objection. I argued in the main report that we can think of highly substitutable AI as reversing the demographic transition, after which we would expect super-exponential growth to resume. The report’s basic thesis that sufficiently advanced AI could lead to explosive growth is consistent with the recent data.

Alternatively, we might have a more agnostic approach to the causes of long-run growth and the recent slowdown (i.e. the ignorance story). In this case, the recent data provides a stronger reason to reduce the probability we assign to explosive growth. However, it doesn’t provide a decisive reason: the recent data is not hugely improbable according to Roodman’s model.

6.2.3 Frontier growth shows a clear slowdown

6.2.3.1 Summary of objection

The prolonged lack of super-exponential growth of GDP per capita in frontier countries is striking. US per capita income has grown steadily at 1.8% for 150 years (since 1870), and other frontier countries show similar trends. The only reason GWP data doesn’t show the same pattern is catch-up growth. The lack of super-exponential growth over such a long period is strong evidence against long-run explosive models.

Even the trend in frontier GDP/capita may be overly generous to long-run explosive models. Frontier GDP/capita has recently been boosted from a number of one-off changes: e.g. the reallocation of people of color from low wage professions to high wage professions, the entry of women into the workforce, and improved educational achievement. Hsieh et al. (2013) estimates that improvements in the allocation of talent may explain a significant part of U.S. economic growth over the last 60 years.112 If we adjusted for these factors, the trend in frontier GDP/capita would likely be even more at odds with the predictions of long-run explosive models.

This strengthens the objection of the previous section.

6.2.3.2 Elaboration of objection

This objection is hard to spell out in a conceptually clean way because endogenous growth models like Roodman’s are only meant to be applied to the global economy as a whole, and so don’t necessarily make explicit predictions about frontier growth. The reason for this is that the growth of any part of the global economy will be influenced by the other parts, and so modeling only a part will necessarily omit dynamics relevant to its growth. For example, if you only model the US you ignore R&D efforts in other countries that are relevant to US growth.

Nonetheless, I do feel that there is something to this objection. GWP cannot grow super-exponentially for long without the frontier growing super-exponentially.

In the rest of this section I:

  • Suggest the size of the ‘frontier growth slowdown’ is about twice as big as the already-discussed GWP slowdown.
  • Suggest that the most natural application of Roodman’s univariate model to frontier growth allows the objection to go through.

(Again, I use ‘slowdown’ to refer to a period of merely exponential growth, which is ‘slower’ than the alternative to super-exponential growth.)

6.2.3.2.1 How much bigger is the frontier growth slowdown than the GWP slowdown?

I have briefly investigated the timescales over which frontier growth has been exponential, rather than super-exponential, by eyeballing GDP and GDP/capita data for the US, England, and France. My current opinion is that the frontier shows clear super-exponential growth if you look at data from 1700, and still shows super-exponential growth in data from 1800. However data from about 1900 shows very little sign of super-exponential growth and looks exponential. So the slowdown in frontier growth is indeed more marked than that for GWP growth. Rather than just 50 years of slowdown during which GWP increased by a factor of 5.4, there’s more like 120 years of slowdown during which GDP increased by about 10-15X.113

My current view is that considering frontier GDP/capita data increases the size of the deviation from the super-exponential trend by a factor of 2-3 compared to just using GWP data. This is because the deviation’s length in calendar time is 2-3 times bigger (120 years rather than 50 years) and the GDP increase associated with the deviation is 2-3 times bigger (GDP increases 10-15X rather than 5X). Recent frontier growth poses a bigger challenge to the explosive growth theory than recent GWP growth.

This is consistent with the results Roodman got when fitting his model to French per capita GDP. Every observation after 1870 was below the model’s predicted median, and most lay between the 20th and 35th percentiles. The model was consistently surprised at the slow pace of progress.

6.2.3.2.2 The simplest way of extending Roodman’s model to frontier countries implies they should grow super-exponentially

Roodman’s model implies that GWP should grow super-exponentially but does not say how the extent to which this growth results from frontier vs catch-up growth should change over time.

The simplest answer seems to be that both frontier and catch-up growth is super-exponential. The same story that explains the possibility of super-exponential growth for the total world economy – namely increasing returns to endogenous factors including technology – could also be applied to those countries at the frontier. If frontier countries invested their resources in helping others catch up we might expect something different. But on the realistic assumption that they invest in their own growth, it seems to me like the story motivating Roodman’s model would predict super-exponential growth at the frontier.

The lack of frontier super-exponential growth is especially surprising given that frontier countries have been significantly increasing their proportional spend on R&D.114 Roodman’s model assumes that a constant fraction of resources are invested and predicts super-exponential growth. How much more surprising that we see only constant growth at the frontier when the fraction of resources spent on R&D is increasing! The expansion of the size of the frontier (e.g. to include Japan), increasing the resources spent on frontier R&D even further, strengthens this point.

Response: deny the frontier should experience smooth super-exponential growth

A natural response is to posit a more complex relationship between frontier and catch-up growth. You could suggest that while GWP as a whole grows at a fairly smooth super-exponential rate, progress at the frontier comes in spurts. The cause of GWP’s smooth increase alternates between spurts of progress at the frontier and catch-up growth. The cause of this uneven progress on the frontier might be an uneven technological landscape, where some advances unlock many others in quick succession but there are periods where progress temporarily slows.

I think that accepting this response should increase our skepticism about the precise predictions of Roodman’s model, moving us from the explosive-growth story towards the ignorance story. It would be a surprising coincidence if GWP follows a predictable super-exponential curve despite frontier growth being the result of a hard-to-anticipate and uneven technological landscape.115 So, for all we know, the next spurt of frontier progress may not happen for a long time, or perhaps ever.

6.2.3.3 So what?

Again, this objection may not move you much if you explain the slowdown via the demographic transition. The recent data would not undermine the belief that super-exponential growth will occur if we get sufficiently substitutable AI.

If you are more agnostic, this will provide a stronger reason to doubt whether explosive growth will occur. The length of the slowdown suggests a structural break has occurred, and the super-exponential trend has finished (at least temporarily). Still, without an explanation for why growth increased in the past, we should not rule out more increases in the future. 120 years of exponential growth, after centuries of increasing growth rates, suggests agnosticism about whether growth will increase again in the next 80 years.

6.2.4 Long-run explosive models don’t anchor predictions to current growth levels

Objection: The models predicting explosive growth within a few decades typically expect growth to already be very high. For example, the median prediction of Roodman’s model for 2020 growth is 7%. Its predictions aren’t anchored sufficiently closely to recent growth. I analyze this problem in more detail in an appendix.

Response: I developed a variant of Roodman’s model that is less theoretically principled but models a correlation between growth in adjacent periods. This ‘growth differences’ model anchors its predictions about future growth to the current GWP growth rate of 3%.

The model’s median predicted year for explosive growth is 2082 (Roodman: 2043), a delay of about 40 years; its 80% confidence interval is [2035, 2870] (Roodman: [2034, 2065]). This suggests that adjusting for this problem delays explosive growth but still leaves a significant probability of explosive growth by 2100. Explanation of the model I developed.

I find this model most useful as an ‘outside-view’ that projects GWP based solely off past data, without taking into account specific hypotheses like ‘the demographic transition ended the period of super-exponential growth’, or ‘we’d only expect to see super-exponential growth again once advanced AI is developed’. If we embrace specific inside-view stories like these, we’d want to make adjustments to the model’s predictions. (For the examples given, we’d want to further delay the predicted dates of explosive growth based on how far we are from AI that’s sufficiently advanced to boost the growth rate.)

How might we adjust the model’s predictions further based on our beliefs about AI timelines?

Suppose you think it will be (e.g.) three decades before we have AI systems that allow us to increase the rate of growth (systems before this point might have ‘level effects’ but not noticeably impact growth). You could make a further adjustment by assuming we’ll continue on our current growth trajectory for three decades, and then growth will change as shown in the graph. In other words, you’d delay your median predicted year for explosive growth by another 30 years to about 2110. However, you’ll still assign some probability to explosive growth occurring by the end of the century.

I plotted the 10th, 50th, and 90th percentiles over GWP from three methods:116

  • Surveying economists about GWP/capita and combining their answers with UN population projections to forecast GWP (‘ ’).
  • Fitting David Roodman’s growth model to long-run historical GWP data (‘ ’).
  • Fitting my variant on Roodman’s model to long-run GWP data (‘ ’).

I am currently inclined to trust the projections somewhere in between growth differences and Roodman’s model if we develop highly substitutable117 AI systems (though I don’t think any model is a reliable guide to growth in this scenario), and the projections of the standard story if we don’t.

See code producing these plots at the bottom of this notebook. (If the link doesn’t work, the colab file can be found in this folder.)

6.2.5 Long-run explosive models don’t discount pre-modern data

Objection: For example, Roodman’s model downweights ancient data points for their uncertainty, but does not additionally downweight them on the basis that they are less relevant to our current growth regime. But more recent data is more likely to be relevant because the underlying dynamics of growth may have changed.

Response: My ‘growth-differences’ model allows the user to specify the rate at which ancient data points are discounted.118 For my preferred discount rate,119 this delays explosive growth by another 15 years to ~2090; it still assigns a 10% chance of explosive growth by 2040. Adjusting for this problem delays explosive growth further but leaves a significant probability of explosive growth by 2100.120 Again, if you think AI won’t start to affect growth for several decades, you would need to delay your median projection further (see more).

I also perform a sensitivity analysis on the effects of removing pre-modern data points. I find that the prediction of explosive growth by 2100 is robust to removing data points before 1300, but not to removing data points before 1600 (see more).

6.2.6 Long-run explosive models don’t seem to apply to the time before the agricultural revolution; why expect them to apply to a growth regime in the future?

Summary of objection: Roodman (2020) does the most sophisticated analysis on the fit of his model to data before 10,000 BCE. He finds that if he fits his model to data from 1 million years ago to the modern day, the estimated model is not a good fit to the data series. It confidently predicts that civilization will collapse within the first few 100,000 years, with a 98% chance of eventual collapse. Given that Roodman’s model did not describe a previous era – that of hunter gatherers – we should not trust its predictions about a future era of supposed explosive growth.

Response: I think this objection might potentially justify agnosticism about explosive growth, but it doesn’t confidence that it will not occur.

Let’s distinguish between three attitudes towards explosive growth:

  1. Confidence that explosive growth will occur (explosive growth story).
  2. Ignorance about whether explosive growth will occur (ignorance story).
  3. Confidence that explosive growth won’t occur (standard story).

I think that, at most, this objection might move you from Attitude 1 towards Attitude 2. It’s not an argument for Attitude 3. The objection provides a reason to doubt the predictions of Roodman’s model, but doesn’t provide any specific reason to rule out explosive growth.

I personally regard this objection as only a weak argument against Attitude 1. This is because a key part of technological progress, the driver to super-exponential growth, is the ability for new ideas to spread throughout society. But human societies with natural language only developed 50,000 – 150,000 years ago.121 So we wouldn’t expect Roodman’s model to be accurate before this point. As Roodman points out:

Through language, humans could share ideas more efficiently and flexibly than any organism before. Arguably, it was then that technology took on its modern, alchemical character as a force in economic development. Before, hominins had developed important technologies such as handaxes. But it is not obvious that those intellectual mutations spread or evolved any faster than the descendants of those who wrought them. After, innovations could diffuse through natural language, the first new medium of arbitrary expressiveness on Earth since DNA.

In addition, humans couldn’t accumulate capital until we became sedentary. This happened around the neolithic era, giving another reason to think growth dynamics would be different before 10,000 BCE.


7. Appendix B: Constant exponential growth is a knife-edge condition in many growth models

The growth literature has found it very difficult to find a satisfactory theoretical explanation for why long-term growth would be exponential, despite decades of effort. In many endogenous growth models, long-run growth is only exponential under knife-edge conditions. This means that constant exponential growth only occurs when some parameter is exactly equal to some value; the smallest disturbance in this parameter leads to a completely different long-run behavior, with growth either going to infinity or to 0. Further, it seems that these knife-edge conditions are problematic: there’s no particular reason to expect the parameter to have the precise value that leads to constant exponential growth.

I argue the best candidates for addressing this problem are semi-endogenous models. Here the ‘knife-edge condition’ is merely that the population grows exponentially.122 For this and other reasons discussed in this section, I place more weight on semi-endogenous models (~75%) than on any other models in explaining the recent trend of exponential growth.

The UN forecast that population growth will slow over the 21st century. When you plug this assumption into semi-endogenous growth models, they predict that GDP/capita growth will slow. This raises my probability that 21st century growth will be sub-exponential. The difficulty of finding a non-knife edge explanation of exponential growth also raises my credence that the pattern of exponential growth is a transitional rather than the beginning of a steady state regime.123 Nonetheless, I still assign substantial probability (~20%) that there is some mechanism generating exponential growth that will continue to function until 2100, although I’m not sure what it would be.

The rest of this section is as follows:

  • I explain my intuitive understanding of the claim that constant exponential growth is an unmotivated knife-edge condition (here).
  • I review the knife-edge conditions in a number of endogenous growth models (here).
    • This section also makes some other objections to certain models, explaining my preference for semi-endogenous models.
  • I briefly review the sub literature that claims that a very large class of models have knife-edge conditions for exponential growth (here).
  • I discuss a recent model that claims to produce exponential growth without knife-edge condition (here).

7.1 An intuitive explanation for why exponential growth might be a knife-edge condition in endogenous growth models

Let’s focus on the endogenous factor of technology. Assume that we invest a constant fraction of output into technology R&D. This investment causes the level of technology to improve by a certain percentage each year. We’re interested in how this percentage changes over time, as technology advances. In other words, we’re interested in how the rate of technological progress changes over time, with this progress measured as a percentage.

As technology improves, there are (at least) two things that might affect the rate of future progress. Firstly, in the future there may be less low hanging fruit as we have made all the easy technological discoveries and only difficult ones remain. Call this the fishing out effect. Secondly, we can use the new technology in our future research, increasing the effectiveness of future R&D efforts (e.g. use of the internet). Call this the standing on shoulders effect.

These two effects point in opposite directions but there is no reason to expect them to cancel out exactly. The fishing out effect relates to the landscape of technological discoveries, and how quickly the easy discoveries dry up; the standing on shoulders effect relates to the extent to which we can harness new technologies to improve the process of R&D. The two effects relate to very different things. So by default, we should expect these factors not to cancel out exactly. And so we should expect the rate of technological progress to either speed up or to slow, depending on which effect is more powerful. But there’s no reason to think that the rate of progress should stay exactly constant over time. This would be like giving one tennis player a broken arm and their opponent a broken leg, and expecting the two effects to cancel out exactly.

More nuanced models add additional factors that influence the rate of technological progress (e.g. the ‘stepping on toes effect’). But these additional factors don’t make it any more plausible that everything should cancel out and growth should be exponential.

The conclusion of this line of thinking is that, theoretically speaking, we shouldn’t expect technology to grow exponentially.

A similar argument can be applied to output as a whole, rather than just technology. Consider a growth model where all inputs are endogenous. The intuition behind the argument is that some factors suggest growth should increase over time, other factors suggest growth should slow over time; further, there’s no particular reason to expect these factors to cancel out exactly. So we should expect growth to either slow down, or speed up over time.

More precisely, we’re interested in the percentage increase in the total output each year. We want to know how this percentage changes over time as total output increases. There are again (at least) two effects relevant to this question.

The first effect is that, as the endogenous inputs to production increase over time, they become harder to increase by a fixed percentage. This is true because i) a fixed percentage is an increasingly large absolute amount, ii) there may be diminishing marginal returns to efforts to improve the factor, and iii) because of other complex factors.124 If inputs are harder to increase by a fixed percentage, then output as a whole is also harder to increase by a fixed percentage. Let’s call this effect percentage improvements become harder; it roughly corresponds to the fishing out effect in the previous section.

The second effect is that, as the endogenous inputs increase, we have more resources to invest in increasing the inputs. This increased investment allows greater absolute increases to be made to the inputs, and so to output as a whole. Call this effect greater investment; it corresponds to the standing on shoulders effect from the previous section.

Again, these two effects point in opposite directions. The percentage improvements become harder effect suggests growth will slow over time, the greater investment effect suggests that growth will increase. Again, I know of no reason to think these effects should exactly cancel out.125 If they don’t cancel, growth won’t be exponential.

To be clear, I do not think that this intuitive argument is itself sufficient to establish that exponential growth is a knife-edge condition and highly surprising. I include because it generalizes the specific argument I make below in the context of specific models.

7.2 Knife-edges in popular endogenous growth models

Most endogenous growth models can be broadly divided into two camps: accumulation based models and idea-based models.126 In the former, the ultimate source of growth in GDP/capita is the accumulation of physical or human capital. In the latter, the ultimate source of growth is targeted R&D leading to technological progress; although there is capital accumulation, it isn’t the ultimate source of growth.127

I will discuss the knife-edge conditions in popular growth models of both types. I think the knife-edge conditions are more problematic in the idea-based models; although accumulation based models face further objections.

Very little of the content here is original; knife-edge critiques of endogenous models are discussed in Cesaratto (2008), Jones (1999), and Jones (2005). The problem is often discussed with different terminology, referring to the difficulty of avoiding ‘scale effects’ or the ‘linearity critique’ of endogenous growth models. I expect all economists familiar with endogenous growth models will be aware that knife-edge assumptions are typically needed for constant exponential growth. I expect most of them won’t draw my conclusion: that the best account that avoids knife-edge conditions implies that 21st century growth will be sub-exponential.

One strong objection to accumulation based models that I don’t discuss in this report on is their tension with growth accounting exercises, e.g. Fernald and Jones (2014). These empirical exercises decompose growth into its constituent parts, and typically find that TFP growth accounts for the majority of growth rather than the accumulation of physical or human capital. I think this gives us a good reason to prefer idea-based models.

7.2.1 Accumulation based models

Perhaps the most standard mechanism for growth here is the accumulation of physical capital. This is the strategy of the AK model, and variants thereof. I’ll start by discussing the model of Frankel (1962) and the variant proposed by Arrow (1962). Then I’ll briefly comment on some other capital accumulation models.

7.2.1.1 Frankel (1962)

The production function in Frankel (1962) starts out as:

 

$$$ Y=AK^α(BL)^{1−α} $$$

where B is labor augmenting technology. Technological progress is endogenous and happens as a by-product of capital accumulation. The equation for B is:

 

$$$ {B}= (\frac {K}{L})^γ $$$

Frankel assumes γ = 1. In other words, labor augmenting technology is the capital per worker. Twice as much capital per worker makes workers twice as productive. With this assumption production is simply:

 

$$$ Y=AK $$$

Here and in all other models in this section, I assume the standard reinvestment equation for capital: = sYδK. This implies that growth is exponential.

The knife-edge condition is γ = 1. To simplify the analysis, assume L is constant. If γ > 1, there are increasing returns to K and Y goes to infinity in finite time.128If γ < 1, there are diminishing returns to K and growth tends to 0.

Is this knife-edge condition problematic? I think so. It claims that doubling the amount of capital per worker exactly doubles the productivity per worker. But why not think it would increase productivity by a factor of 1.9, or 2.1?

The problem becomes more acute when we realize that there are two distinct mechanisms by which capital accumulation increases labor productivity. The first is that each worker has more machinery to work with, increasing their productivity.129 The second mechanism is that capital accumulation leads to new technologies via the process of ‘learning by doing’. These improvements have spillover effects as new technologies can be adopted by all firms. But it is mysterious why these two very different mechanisms should combine such that γ = 1 exactly. If the spillover effects were ever so slightly bigger or smaller, or if the benefits of having more machinery were ever so slightly bigger or smaller, growth would go to 0 or infinity rather than being constant.

Robert Solow comments, on this topic, ‘This version of the endogenous-growth model is very unrobust. It can not survive without exactly constant returns to capital. But you would have to believe in the tooth fairy to expect that kind of luck.’130 In support of this comment, I argue in this technical appendix that even the constancy of 20th century growth wouldn’t convince us that long-run growth would be constant if we believed that Frankel’s growth model was literally correct.

There is another problem with Frankel’s AK model. In order to get constant returns to capital accumulation, but avoid increasing returns to capital and labor in combination, the model removes the effect of labor on output entirely. The seemingly absurd implication is that adding more workers won’t increase output. A defense might be that the model is intended for a simplified setting where labor is constant. If so, then the model doesn’t seem to be appropriate for explaining the recent period of growth, during which there has been significant population growth.

One last thing to note about this AK model is that if there is any capital augmenting technological progress (e.g. an increase in A), this will increase growth.131

7.2.1.2 Arrow (1962)

Arrow (1962) develops a similar AK model.132 His definition of labor augmenting technology depends on the total capital accumulated rather than the capital accumulated per person.

$$$ B=K^γ $$$

with γ < 1.133 This leads to:

$$$ Y=AK^α(BL)^{1−α}=AK^μL^{1−α} $$$

with μ = γ(1 – α) < 1.

This model does not, in my view, have a problematic knife-edge. However, it does imply that growth will be sub-exponential over the 21st century.

The growth rate of y = Y/L turns out to be:

$$$ g_y=g_{L} \frac {γ}{(1−γ)} $$$

If the labor force doesn’t grow, then neither will GDP/capita. This prediction is not actually falsified by observation, as the population has grown continuously since the industrial revolution. In fact, I think that exponential population growth is the most plausible root explanation for the historical observed pattern of exponential growth.

This model is structurally very similar to the semi-endogenous model developed by Jones that I discuss later. In both models, the ultimate driver of exponential income growth is exponential growth in labor. Both models imply that growth over the 21st century will be sub-exponential, as population growth is expected to slow.

(A quick aside: if capital were perfectly substitutable with labor – the AI robot scenario – then this model predicts explosive growth. In this scenario, capital can play the same role as labor in production and so, if AI robots are cheaper than human labor, the model will ultimately approximate: Y = AK1 + γ(1-α). There are increasing returns to capital accumulation and so super-exponential growth. This is just to demonstrate that some accumulation models do imply that this scenario would lead to explosive growth.)

7.2.1.3 Other capital accumulation stories

Jones and Manuelli (1990) develop a model in returns to capital fall, but rather than falling to 0 as in most models falls to a constant and then stays at that constant. This means that capital accumulation is sufficient for sustained growth. Growth from capital accumulation will be sub-exponential as the returns to capital diminish towards the constant, and afterwards it will be exponential.

For this model to explain the recent period of exponential growth, then, it must claim that returns to capital have long ago diminished to their lowest possible value, and are now constant. Intuitively, this claim doesn’t seem plausible: returns to capital would diminish further if we equipped every worker with the highest quality equipment possible. Putting that aside though, the model in essence behaves the same way as AK in the regime where returns to capital are constant. So the same problems we saw above will apply.

Indeed, the knife-edge analogous to the one considered above applies. In the limit where returns to capital are constant we have:

$$$ \frac {dY}{dK}=K^ϕ $$$

with φ = 0. If φ > 0, growth from capital accumulation is super-exponential; if φ < 0, growth goes to 0. We can ask why φ = 0. The value of φ is again plausibly the product of two mechanisms: additional capital can be used directly to produce more output; accumulating capital involves some ‘learning by doing’ and produces new technologies that can be copied by others. I can see no reason for these two mechanisms to lead to exactly constant returns. Ultimately, I think Jones and Manuelli (1990) faces the same objections as the AK model; its main advantage is that it formally acknowledges diminishing returns to capital (though not during the regime where exponential growth is occurring).

Another way capital accumulation can lead to sustained growth is by using a CES production function where the elasticity of substitution between capital and labor is above 1. In this case, as with Jones and Manuelli, the returns to capital diminish initially and then approach some constant. While the returns are diminishing, growth from capital accumulation is sub-exponential; in the limit where these returns are constant, growth from capital accumulation is exponential. In the limit the model faces the same ‘knife-edge objection’ as Jones and Manuelli (1990): why would the direct and spillover effects of capital accumulation net out at exactly constant returns?134

There is another problem for the CES production function approach. In the limit where growth is exponential, the capital share is 1. The capital share has been around 0.3 for the last 50 years (although it has recently increased somewhat), so this model wouldn’t offer a good explanation of the recent period of exponential growth.

7.2.1.4 Human capital accumulation

Lucas (1988) suggests the ultimate driver of growth is not physical but human capital. The model is as follows:

$$$ Y=AK^α(lhL)^{1−α} $$$

$$$ \dot h=ϕh(1−l) $$$

where h is human capital per person, l is the proportion of time spent working, 1 – l is the proportion of time spent increasing h, φ is a constant, and A is a constant.

The knife-edge here is that = constant × hφ with φ = 1 exactly. If φ < 1, there would be diminishing returns to human capital accumulation and growth would fizzle out; if φ > 1 growth would go to infinity in finite time.

Is this knife-edge problematic? Again, I think so. There are two possible interpretations of h;135 I think the condition is problematic in both cases.

The first interpretation is that h is the knowledge and skills of an individual agent; 1 – l is the proportion of their time they spend studying.136 Here, the knife-edge φ = 1 means that if you know twice as much, you can learn and teach exactly twice as quickly. But why not think it allows me to learn only 1.9 times as quickly, or 2.1 times as quickly? Why is my learning speed exactly proportional to my knowledge? As with physical capital, there are both direct and spillover benefits of increasing h. The direct benefit is that I leverage my knowledge and skills to learn more effectively in the future. The spillover effect is that others may copy my discoveries and knowledge; this can help their future learning. It is again problematic that these two distinct effects combine to give φ = 1 exactly.

There’s another problem with this first interpretation. In addition, our minds and capabilities are limited by our finite minds and lifespans. Our knowledge and skills can’t grow exponentially without limit, but ultimately hit diminishing returns.

The second interpretation is that h represents all the accumulated technical and scientific knowledge of humanity; 1 – lis the proportion of people who are scientists.137φ = 0 would mean that each absolute increase in knowledge was equally difficult. φ = 1 means that if humanity knows twice as much, an absolute increase in our knowledge becomes exactly twice as easy to achieve. This is a very particular degree of increasing returns.138 There are (at least) two relevant effects. If we know more, then perhaps we’ve made all the easy discoveries and new ideas will be harder to find (‘fishing out’). Or perhaps our knowledge will make our future learning more effective (‘standing on shoulders’). I see no reason to think these forces should net out so that φ = 1 exactly.

The second interpretation faces another severe problem: the rate of knowledge discovery depends on the fraction of people who are scientists but not the absolute number. If we alter this so that increases with L, then (still assuming φ = 1), an exponentially growing population would lead to an exponentially increasing growth rate.

(A quick aside: if capital were perfectly substitutable with labor – the AI robot scenario – then this model would display constant returns to accumulable inputs L and K. If h – which would then be interpreted as ‘AI robot capital’ rather than ‘human capital’ – continues to increase, then output will grow super-exponentially. This is again to demonstrate that some accumulation models do imply that this scenario would lead to explosive growth. However, if the model was adjusted to include a fixed factor so that there were slightly diminishing returns to capital and labor, then AI robots would not lead to explosive growth. Instead it would lead to a one-off step-change in growth rates, assuming that continued to grow exponentially.)

7.2.2 Idea-based models

I’ve argued that some central capital accumulation and human accumulation models don’t provide compelling explanations of the observed pattern of exponential growth, partly because they make problematic knife-edge assumptions.

One general drawback of accumulation-based models is that they don’t directly engage with what seems to be an important part of the rise in living standards over the last 100 years: discovery of new ideas through targeted R&D. Private and public bodies spend trillions of dollars each year on developing and implementing new technologies and designs that are non-rival and can eventually be adopted by others.

Idea-based models represent this process explicitly, and see it as the ultimate source of growth. Whereas accumulation models emphasize that growth involves increasing the number of physical machines and gadgets per person (perhaps with technological progress as a side-effect), Idea-based models emphasize that it involves purposely developing new (non-rival) designs for machines, gadgets, and other technologies.

This section is heavily based on Jones (1999). I simply pull out of relevant points.

Jones groups idea-based models into three camps based on important structural similarities between them:

  1. R / GH / AH
  2. Y/P/AH/DT
  3. J / K / S
    • These are from Jones (1995), Kortum (1997) and Segerstrom (1998). These are known as semi-endogenous growth models.
    • The knife-edge condition is that there’s exactly exponential growth in the number of workers.

I think the knife-edge conditions for exponential growth for R / GH / AH and Y / P / AH / DT models are just as problematic, if not more problematic, than those for accumulation based models discussed above.

For semi-endogenous models (J / K / S), the knife-edge condition is much less problematic. Indeed we know empirically population growth has been roughly exponential over the last 100 years.139 However, this will not continue until 2100. The UN projects that population growth will slow; J / K / S semi-endogenous models imply GDP/capita growth will slow as a result.

7.2.2.1 R / GH / AH models

Output is given by:

$$$ Y=A^σK^α{L_Y}^{1−α} $$$

LY is the number of workers in goods production. There are constant returns to K and LY, and increasing returns to K, LYand A.

New ideas are produced via:

$$$ \dot A=δA^ϕL_A $$$

for some constant δ. LA is the number of workers in knowledge production. A constant fraction of people do research: LA = fL, LA + LY = L.

The knife-edge assumption is φ = 1. If φ ≠ 1, then growth over time either goes to 0 or infinity, as in the above examples. To repeat my comments on Lucas (1988): there are (at least) two relevant mechanisms affecting φ. If A is larger, then perhaps we’ve made all the easy discoveries and new ideas will be harder to find (‘fishing out’). This suggests lower value of φ. Conversely, perhaps we can leverage our knowledge to make our future learning more effective (‘standing on shoulders’). I see no reason to think these forces should net out so that φ = 1 exactly.

7.2.2.2 Y / GH / AH models

$$$ Y=NZ^σK^α{L_Y}^{1−α} $$$

where N is the number of product lines and Z is the average level of technology per product line.

The number of products increases with the size of the total population:

$$$ N=L^β $$$

The rate of technological progress depends on the number of researchers per product line:

$$$ \dot Z= \frac {δZ^ϕL}{N}=δZ^ϕ{L_A}^{1−β} $$$

It turns out that exponential growth relies on two knife-edge conditions in this model: β = 1 and φ = 1.

If φ ≠ 1 , then growth over time either goes to 0 or infinity, as above. And again, the assumption that φ = 1 involves a very specific degree of increasing returns to knowledge accumulation despite plausible mechanisms pointing in different directions (‘fishing out’ and ‘standing on shoulders’).

If β ≠ 1, the number of researchers per firm changes over time, and this changes the growth rate.

7.2.2.3 J / K / S models

We can represent these models as:

$$$ Y=A^σK^α{L_Y}^{1−α} $$$

$$$ \dot A=δ{L_A}^{λ}A^ϕ $$$

$$$ \dot L=nL $$$

with n > 1, φ < 1 and λ < 1. As before, we assume that a constant fraction of people do research: LA = fL, LA + LY = L.

The exponential growth in L drives exponential growth in A: φ < 1 implies each new % increase in A requires more effort than the last, but exponentially growing labor is able to meet this requirement. Exponential growth in A then drives exponential growth in Y and K and thus of GDP/capita.

Often L is made exogenous, but Jones (1997) makes it endogenous, using fertility equations such that population growth tends to a positive constant in the long-run.

The knife-edge condition here is the exponential growth of labor: = nLφ and φ = 1 exactly.

Jones (1997) justifies this by appealing to biology: ‘it is a biological fact of nature that people reproduce in proportion to their number’. Indeed, population growth was positive throughout the 20th century for the world as a whole or for the US.140 So it does seem that the model matches the rough pattern of 20th century growth.

Population growth fell over the 20th century.141  If there was no lag between research effort and productivity improvements, perhaps this theory implies we should have seen a more noticeable slowdown in frontier GDP/capita growth as a result. However, some lag is realistic, and there does seem to have been such a growth slowdown since 2000. In addition, numerous factors may have offset slowing population growth: increases in the fraction of people doing R&D, more countries on the economic frontier (and so a higher fraction of scientists doing R&D pushing forward that frontier), increased job access for women and people of color (reduced misallocation), increased educational attainment, and possibly random fluctuations in the economic returns to R&D (e.g. the IT boom).

Growth accounting exercises suggest that these other factors are significant. Fernald and Jones (2014) suggest that the growing fraction of people doing R&D accounts for 58% of the growth since 1950, and education improvements account for 20%. Hsieh et al. (2013) estimates that improvement in talent allocation can account for more than 20% of income increases since 1950.

Given these other factors, the juxtaposition of slowing population growth and steady income growth during the 20th century is only weak evidence against semi-endogenous growth theories. (Indeed, high quality empirical evidence on growth theories is very hard to come by.)

Overall, it seems that semi-endogenous growth theory does a good job of explaining the general pattern of 20th century growth and that it’s hard to adjudicate beyond this point due to the effects of numerous other important factors.142

What does semi-endogenous growth theory imply about 21st century growth? It’s The UN population projections – which have a fairly good track record – over the 21st century imply that population growth will slow significantly. In addition, the historical growth of the fraction of people doing R&D cannot be maintained indefinitely, as it is bounded below 1. Both these trends, the slowing of population growth and the slowing growth of the fraction of researchers, imply that the growth of the number of researchers will slow. When you plug this into semi-endogenous growth theory, it predicts that the GDP/capita growth rate will also slow.

Where does this prediction come from? Semi-endogenous models imply each % increase in GDP/capita requires more research than the last. If the number of researchers is constant, each % increase in GDP/capita will take longer to achieve and growth will slow. If the number of researchers does grow, but at an ever slower rate, the model still predicts that GDP/capita growth will slow.

Jones draws just this implication himself in Jones (2020); Fernald and Jones (2014) discuss how slowing growth in educational achievement and the fraction of workers doing R&D, as well as population, might slow future GDP/capita growth. Kruse-Andersen (2017) projects growth out to 2100 with a semi-endogenous model and predicts average GDP/capita growth of 0.45%, without even taking into account slowing population growth.

So J / K / S theories offer plausible explanations of 20th century exponential growth and ultimately suggest that 21st century growth will be sub-exponential.143

7.2.2.3.1 Additional knife-edges in J / K / S models?

J / K / S models make use of the ‘knife-edge’ claim that the number of researchers has grown exponentially. I argued that this is not problematic for explaining the past as the empirical evidence shows that the assumption is approximately true.

But it could be argued that the power-law structure of J / K / S models is an additional knife-edge. Consider the knowledge production:

$$$ \dot A=δ{L_A}^{λ}A^ϕ $$$

The model assumes that φ is constant over time. If φ rose as A increased, then exponential growth in researchers would lead to super-exponential growth. If φ fell as A increased, then exponential growth in researchers would lead to sub-exponential growth.

To explain sustained exponential growth, J / K / S must assume that φ is constant over time, or at least asymptotes towards some value.

In my mind, this knife-edge is considerably less problematic than those of other models considered.

Firstly, a small deviation from the assumption does not cause growth to tend to 0 or infinity. If φ changes slightly over time, the rate of exponential growth will vary but it will not tend to 0 or infinity. For this to happen, φ would have to increase enough to exceed 1 (growth then tends to infinity) or decrease without bound (growth then tends to 0). But both these trajectories for φ are extreme, and so there is a vast region of possibilities where growth remains positive but bounded. I.e. a less idealized model might claim that φ varies over time but typically stays within some region (e.g. -3 < φ < 1). This broad assumption avoids extreme growth outcomes.

Secondly, all the endogenous models considered in this section use some sort of power-law structure like the J / K / Smodel. They are all guilty of some ‘knife-edge’ assumption equivalent to assuming that φ is constant over time. However, the other models in the section additionally assume that the power takes a particular value. In addition to assuming that φ is constant over time, they assume that φ takes a particular value. And I’ve argued that the particular value chosen is without good justification, and that changing that value ever so slightly would cause growth to go to 0 or infinity.

7.3 An economic sub-literature claims constant exponential growth is a knife-edge condition in a wide class of growth models

Growiec (2007) proves144 that:

Steady-state growth… necessarily requires some knife-edge condition which is not satisfied by typical parameter values. Hence, balanced growth paths are fragile and sensitive to the smallest disturbances in parameter values. Adding higher order differential/difference equations to a model does not change the knife-edge character of steady-state growth.

It generalizes the proof of Christiaans (2004), which applies to a more restricted setting.

My own view is that these proofs suggest that knife-edge problems are generic and hard to avoid, but do not establish that the knife-edge conditions of all models are problematic. Growiec has agreed with me on this point in private discussions, and in fact helped me understand why.

The reason is that not all knife-edges are problematic. Here are a few examples:

  • It’s plausible that there are constant returns to labor, capital, and land taken together, holding technology constant. This is supported by a thought experiment. Double the number of factories, the equipment inside them, and the workers in them; this should double output as you can make twice as much of each item. If this was the same knife-edge that was required for exponential growth, it would be less problematic than the knife-edges considered above (which roughly speaking requires constant returns to capital and technology holding labor constant).
  • Galor and Weil (2000) use a negative feedback loop to explain exponential growth. The more people there are, the more R&D effort there is and the faster the economy grows. In addition, when growth is faster people have fewer kids, instead focusing on education. This leads to the following dynamic: higher growth → lower fertility → lower growth. And conversely: lower growth → higher fertility → higher growth. This negative feedback loop stabilizes growth. It doesn’t involve any problematic knife-edge conditions, even though the theory satisfies the axioms of Growiec (2007). I don’t find this particular story convincing, as I trust the UN forecast that fertility will indeed fall over the century. Nonetheless, it is an existence proof of a theory without a problematic knife-edge condition.
  • There may be an alternative framework in which the ‘knife-edge’ case occurs for a thick set of parameter values. Indeed I discuss an attempt to do this for Y / GH / AH models in the next section, though I know of no other explicit attempts to do this.
  • The knife-edge may not be problematic at all if it involves the introduction of a completed new unwarranted term to the equations.145 Some of the knife-edges discussed above involved introduced a new exponent φ that was implicitly set to 1 in the original model. How problematic the knife-edge is depends on whether the new class of theories introduced is a natural extension of the original. In other words, are other values of φ plausible, or is φ = 1 a privileged case that we can expect to hold exactly? I argued that other values are plausible on a case by case basis above. But this is a matter of judgement; more of an art than a science.146

7.4 Might market dynamics eliminate the need for a knife-edge condition?

In the ambitious 2020 paper Robust Endogenous Growth, Peretto outlines a fully endogenous growth model that (he claims) achieves constant growth in equilibrium without knife-edge conditions. I consider the paper to be a significant technical contribution, and a very impressive attempt to meet the knife-edge challenge. However, I doubt that is ultimately successful.

The mechanism for achieving stable growth is somewhat complex – indeed the model as a whole is extremely complex (though well-explained). Very briefly, the economy is split into N firms, and the average quality of technology at a firm is denoted by Z. N increases when individuals decide to invest in creating new firms, Z increases when individuals decide to invest in improving their firm’s technological level. These decisions are all made to maximize individual profit.

There are increasing returns to investment in Z. This means that if N were held fixed and a constant share of output were invested in increasing Z then growth would explode (going to infinity in finite time). In this sense, the system has explosive potential.

However, this explosive potential is curbed by the creation of new firms. Once new firms are created, subsequent investment in Z is diluted, spread out over a greater number of firms, and Z grows more slowly. Creating new firms raises output in the short-term but actually reduces the growth of the economy in the long run.147

There are diminishing returns to N, so creation of new firms does not lead to explosive growth. We can think of the diminishing returns of N as ‘soaking up’ the excess produced from the increasing returns to Z.

I believe that if the growth rate of N was slightly faster or slower then long-run growth would diverge (either be explosive or tend to 0). If so, there should be a robust explanation for why N grows at exactly the rate that it does.

So the key question from the perspective of the knife-edge critique is:

Why does N grow just fast enough to curb the explosive growth potential of Z, but not fast enough to make long-run growth sub-exponential (tending to 0 in the long run)?

Despite studying the paper fairly closely, and it being well explained, I don’t have a fully satisfactory answer to this question. I discuss my best answer in this appendix.

Does the model fulfill its promise of avoiding knife-edge conditions? A recent review article answers with an emphatic ‘yes’, and I couldn’t see any papers disputing this result. However, the paper was only published in 2020, so there has not been much time for scrutiny. Although there seem to be no knife-edge conditions in the production function, it is possible that they are located elsewhere, e.g. in the equations governing firms’ profits. Indeed, in private correspondence Growiec has indicated that he believes there must be a knife-edge condition somewhere that Peretto does not explicitly discuss and may not even be aware of.

My own guess is that a knife-edge is present in the expression for the fixed cost a firm must pay to produce goods. This fixed cost is assumed to be proportional to Z. I believe that if it were proportional to Zφ with φ ≠ 1, then growth would either tend to infinity or to 0. If so, φ = 1 would be a knife-edge condition. Indeed, Peretto confirmed in private correspondence that if instead the fixed cost were proportional to Z0.9, the model would not produce exponential growth, and he thought the same was likely true if they were proportional to Z1.1. Growiec also thought this seemed like a plausible candidate for such a knife-edge condition. However, no-one has worked through the maths to confirm this hypothesis with a high degree of confidence. Further, this ‘knife-edge’ may not be problematic: φ = 1 may be the only assumption that prevents fixed costs from tending to 0% or 100% of the total costs of production.

Putting the knife-edge issue aside, the model seems to have two implausible problems:

  1. Problem 1. Though the model avoids knife-edge conditions, it has a perplexing implication. In particular, like all Schumpeterian growth models,148 it implies that if no new products were introduced – e.g. because this was made illegal – and we invested a constant fraction of output in improving technology then there would be explosive growth and output would approach infinity in finite time. This means that there is a huge market failure: private incentives to create new companies massively reduce long-run social welfare.
  2. Problem 2. In addition, it is not clear that market fragmentation happens as much as the model implies. A small number of organizations have large market shares of industries like mass media, pharmaceuticals, meat packing, search engines, chip production, AI research, and social networks.149 Indeed, in some areas market concentration has been increasing,150 and market concentration is one of the stylized facts of the digital era.151

Overall, this impressive paper seems to offer a fully endogenous growth model in which constant growth is not knife-edged. Though I doubt it is ultimately successful, it does identify a mechanism (individual incentives) which can cause an apparent knife-edge to hold in practice. The paper slightly raises my expectation that long-run growth is exponential.

7.5 Conclusion

It seems that many, and perhaps all, endogenous growth models only display constant exponential growth only under problematic knife-edge conditions that we have little reason to suppose hold exactly. The main exception is semi-endogenous growth models J / K / S, but these imply that 21st century growth will be sub-exponential given the projected slowing population growth.

There are a few important takeaways from the perspective of this report:

  • Theoretical considerations, combined with the empirical prediction that population growth will slow, implies 21st century growth will not be exponential, but rather sub-exponential.
  • The semi-endogenous models that I argue give better explanations for 20th century growth also imply that full automation of goods and knowledge production would lead to explosive growth. In particular, when you add to these models the assumption that capital can substitute for labor, they predict explosive growth. (See the endogenous models discussed here.)
  • It’s surprisingly hard to find a robust theoretical explanation of the empirical trend of exponential growth that implies it will continue until 2100. This suggests that exponential may be transitory, rather than a steady state. This in turn should raise our probability that future growth is sub- or super-exponential.

There are three caveats to these conclusions.

Firstly, a very recent endogenous growth model seems to allow for constant growth that does not depend on knife-edge conditions. Although I’m not convinced by the model, it highlights possible mechanisms that could justify a seemingly problematic knife-edge condition in practice.

Secondly, I have not done a review of all growth models. Perhaps an existing endogenous growth model avoids problematic knife-edge conditions and delivers exponential growth. I would be surprised if this is the case as there is a sub-literature on this topic that I’ve read many papers from (linked during this section), and they don’t mention any such model. For example, this review article on knife-edge problems doesn’t mention any such model, and argues that only Peretto’s 2020 paper solves the knife-edge problem.

Thirdly, perhaps there is a mechanism producing exponential growth that growth theorists aren’t aware of. The process of economic growth is extremely complex, and it’s hard to develop and test growth theories. If there is such a mechanism, it may well continue to produce exponential growth until 2100.

Based on these caveats, I still assign ~25% probability to ~2% exponential growth in frontier GDP/capita continuing until 2100, even if there’s sub-exponential growth in population.


8. Appendix C: Conditions for super-exponential growth

This section lays out the equation for various growth models, and the conditions under which super-exponential growth occurs. I don’t make derivations or explain the results. Its purpose is to support some key claims made in the main report.

There are two high-level sections, each of which support a key claim in the main report:

  • Long-run explosive models:
    • Key claim: Long-run explosive models assume that capital, labor and technology are all accumulable. Even if they include a fixed factor like land, there are increasing returns to accumulable inputs. This leads to super-exponential growth as long unless the diminishing returns to technology R&D are very steep. For a wide range of plausible parameter values, these models predict super-exponential growth.
  • Standard growth models adjusted to study the effects of AI:
    • Key claim: The basic story is: capital substitutes more effectively for labor → capital becomes more important → larger returns to accumulable inputs → faster growth. In essence, the feedback loop ‘more output → more capital → more output → …’ becomes more powerful and drives faster growth.

I also use this section to evidence my claims about the AI robot scenario, in which AI substitutes perfectly for human labor (the AI robot scenario):

Indeed, plugging this [AI robot] scenario into a range of growth models, you find that super-exponential growth occurs for plausible parameter values, driven by the increased returns to accumulable inputs.

This third claim is evidenced at the bottom of both the high-level sections, here and here.

Lastly, I use this section to evidence one further claim:

This suggests that the demographic transition, not diminishing returns, explains the end of super-exponential growth.

I evidence this final claim here.

8.1 Long-run explosive models

Long-run explosive models are endogenous growth models fit to long-run GWP that predict explosive growth will occur in a few decades.

In the main report I claim:

Long-run explosive models assume that capital, labor and technology are all accumulable. Even if they include a fixed factor like land, there are increasing returns to accumulable inputs. This leads to super-exponential growth as long unless the diminishing returns to technology R&D are very steep. For a wide range of plausible parameter values, these models predict super-exponential growth.

I support these claims by analysing some long-run explosive models.

8.1.1 Roodman (2020)

I analyze a simplified version of the model.152

The equations for the model:

$$$ Y=AK^αL^βW^{1−α−β} $$$

$$$ \dot K={s_K}Y−{δ_K}K $$$

$$$ \dot L={s_L}Y−{δ_L}L $$$

$$$ \dot A={s_A}A^{ϕA}Y−{δ_A}A $$$

A is technology, K is capital, L is labor; all three of these inputs are accumulable. W is the constant stock of land (fixed factor), φA controls the diminishing return to technology R&D, and δi controls the depreciation of the inputs.153

There are increasing returns to accumulable inputs. If you double A, K and L then Y more than doubles. (In Cobb Douglas models like this, there are increasing returns to some inputs just when the sum of the exponents of those inputs exceeds 1. In this case 1 + α + β > 1.)

A sufficient condition for super-exponential growth (deduced here) is:154

$$$ α+β> \frac {−ϕA}{1−ϕA} $$$

This inequality reflects the claim ‘there’s super-exponential growth if the increasing returns to accumulable factors [α + β] is strong enough to overcome diminishing returns to technological R&D’.

If α + β = 0.9 (the fixed factor has exponent 0.1) then the condition on φA is φA > -9. Even the cautionary data of Bloom et al. (2020) suggests φA = -3.155 So there is super-exponential growth for a wide range of plausible parameter values.

8.1.2 Kremer (1993)

I analyze the version of the model in Section 2:156

$$$ Y=Ap^{α}W^{1−α} $$$

$$$ \dot A=δA^{ϕ}p^{λ} $$$

A is technology, p is population, W is the fixed factor land. δ is constant, φ and λ control the diminishing return to technology R&D.

Kremer assumes GDP/capita is fixed at some Malthusian level ȳ:

$$$ p= \frac {Y}{ \bar y} $$$

So larger Y → larger p: population is accumulable. Further, larger Y → larger p → larger Ȧ: technology is also accumulable. There are increasing returns to accumulable factors: 1 + α > 1.

A sufficient condition for super-exponential growth (deduced here):

$$$ α> \frac {−λ}{1−ϕ}+1 $$$

Again it depends on whether increasing returns to accumulable factors can overcome diminishing returns to technology R&D.

Bloom et al. (2020) derive φ = -2.1, on the assumption that λ = 1. This estimate of φ is conservative compared to others. The condition then reduces to α > 2/3. This is plausible given that 1- α is the exponent on the fixed factor land.157 (To look at it another way, if we added capital to the model – Y = ApαKβW1-α-β – the condition would become something like α + β > 2/3.)

8.1.3 Lee (1988)

$$$ Y=Ap^{α}W^{1−α} $$$

$$$ \frac {\dot A}{A}=δlog(p), A_0 \, given $$$

$$$ \frac {\dot p}{p}=[log ( \frac {Y}{p})−log(\bar y)]×constant, p_0 \, given $$$

Constants have the same meaning as in Kremer (1993). Both population and technology are accumulable, and there are increasing returns to both in combination (1 + α > 1). The system grows super-exponentially.158 There is no parameter describing diminishing returns to R&D efforts, so no inequality.

8.1.4 Jones (2001)

$$$ Y=A^{σ}{L_Y}^{α}W^{1−α} $$$

$$$ \dot A=δA^ϕ{L_A}^λ $$$

LY is the amount of labor spent on producing output, LA is the amount of labor spent on research.159 Other symbols are as in Kremer (1993).

Changes in total labor L depend on GDP/capita, Y/L. The exact relationship is complex, but /L is an upside-down U-shaped function of income Y/L. (Initially /L increases with income, then it decreases.) In the initial period, L is output bottlenecked: higher Y → higher income → higher /L.

A is also accumulable: higher Y → higher L → higher Ȧ.

The system cannot be solved analytically, but the system grows super-exponentially if the following condition holds (as explained here):160

$$$ α> \frac {−λσ}{1−ϕ}+1 $$$

This is very similar to the condition in Kremer. Again, we have super-exponential growth as long as increasing returns to accumulable factors (α, σ) are sufficiently powerful enough to overcome diminishing returns.

Bloom et al. (2020) derive φ = -2.1, on the assumption that λ = 1 and σ = 1. The condition then reduces to α > 2/3. This is plausible given that 1 – α is the exponent on the fixed factor land.

8.1.5 How does the case of perfect substitution (‘AI robots’) relate to these models?

AI is naturally thought of as a form of capital, and most of the above models do not contain capital. However, I suggest above that we can also think of AI as making the labor accumulable (the ‘AI robot’ scenario). With this assumption, all the above models predict super-exponential growth under a range of plausible parameter values.

8.1.6 Can diminishing returns to innovative effort explain the end of super-exponential growth?

Perhaps the diminishing returns to innovative effort have become steeper over time. Jones (2001) estimates φ = 0.5 from population and GDP/capita from the last 10,000 years.161 Bloom et al. (2020) estimate φ = -2 from 20th century data on US R&D efforts and TFP growth. Could increasingly steep returns to innovative effort explain the end of super-exponential growth?

Summary

The models considered above suggest that the answer is ‘no’. When labor is accumulable, they predict super-exponential growth even with the conservative estimate of φ from Bloom et al. (2020). By contrast, when labor is notaccumulable (it grows exponentially) they predict exponential growth for a wide range of φ values. In other words, changing φ from 0.5 to -2 doesn’t change whether growth is super-exponential; for any φ in this range (and indeed a larger range), growth is super-exponential just if labor is accumulable.

In these models, the key factor determining whether growth is super-exponential is not the value of φ, but whether labor is accumulable. While diminishing returns to innovative effort may be part of the story, it does not seem to be the key factor.

Analysis

We’ve seen above that when labor is accumulable, these models comfortably predict super-exponential growth even with the conservative estimate of φ = -2 from Bloom et al. (2020); they also predict super-exponential growth higher larger values of φ. Growth is super-exponential under a wide range of values for φ.

By contrast, when labor is not accumulable, but instead grows exponentially regardless of output, these models predict exponential growth for a wide range of φ values.

  • Jones (2001) and Kremer (1993) Part 3 make exactly this assumption. They specify fertility dynamics leading to exponential population growth, and GDP/capita growth is exponential as long as φ < 1. Growth is exponential for a wide range of φ.
  • We can also see this in the case of Roodman (2020). When labor grows exogenously, there’s exponential growth if:
    $$$ α< \frac {−ϕA}{1−ϕA} $$$ where α is the exponent on capital. The capital share suggests α = 0.4, This implies there’s exponential growth as long as φA < -0.67. (This threshold is much higher than the estimate φA = -3 derived from Bloom et al. (2020) data.) Again, for a wide range of φA values, growth is exponential when labor isn’t accumulable. You can get a similar result for the endogenous growth model inspired by Aghion et al. (2017) discussed below.

    • Roodman (2020) estimates φA = 0.2 based on data going back to 10,000 BCE. This implies super-exponential growth, even with exogenous labor.
    • However, the absence of super-exponential growth over the last 120 seems like strong evidence against such high values of φA being accurate in the modern regime. Indeed, if you restrict the data set to start in 1000 AD, Roodman’s methodology implies φA = -1.3. With this value we again predict exponential growth when exogenous labor.
    • It is possible Roodman’s estimate unintentionally includes the effect of one-off changes like improved institutions for R&D and business innovation, rather than just estimating the diminishing returns to R&D.

8.2 Standard growth models adjusted to study the effects of AI

This section looks at standard growth models adjusted to study the possible growth effects of AI. These models treat AI as a form of capital. Some have their roots in the automation literature.

In the main report I claim that in many such models:

The basic story is: capital substitutes more effectively for labor → capital becomes more important → larger returns to accumulable inputs → faster growth. In essence, the feedback loop ‘more output → more capital → more output → …’ becomes more powerful and drives faster growth.

Here I look at a series of models. First I consider endogenous growth models, then exogenous ones, then a task-based model. Within each class, I consider a few different possible models.

8.2.1 Endogenous growth models

Explosive growth with partial automation, Cobb-Douglas

First consider a Cobb-Douglas model where both goods production and knowledge production are produced by a mixture of capital and labor:

$$$ Y=A^ηK^α{L_Y}^{γ}W^{1−α−γ} $$$

$$$ \dot A =A^{ϕ}K^{β}{L_A}^λW^{1−β−λ} $$$

$$$ \dot K=sY−δK $$$

A is technology and K is capital – both of these factors are accumulable. LA and LY are the human labor assigned to goods and knowledge production respectively – they are either constant or growing exponentially (it doesn’t affect the result either way). W is a fixed factor that can be interpreted as land or natural resources (e.g. a constant annual supply of energy from the sun).

The model is from Aghion et al. (2017), but I have added the fixed factor of land to make the model more conservative.

It is essentially a simple extension of the standard semi-endogenous model from Jones (1995), recognizing the roles of capital and natural resources as well as labor.

There is super-exponential growth, with growth rising without bound, if:

$$$ \frac {ηβ}{1−α}>1−ϕ $$$

(This claim is proved in this technical appendix.)

Intuitively, the condition holds if the increasing returns to accumulable factors (represented by α, β , η) are stronger than the diminishing returns to technology R&D (represented by 1 – φ).

How far is this condition from being satisfied? Bloom et al. (2020) estimates φ = -2 on the assumption that η = 1 (which can be seen as a choice about the definition of A).162 This estimate of φ is more conservative than other estimates. The condition becomes:

$$$ \frac {β}{1−α}>3 $$$

Recent data puts the capital share at 40%, suggesting α = β = 0.4:

$$$ \frac {0.4}{0.6}>3 $$$

The condition is not satisfied. It would be satisfied, however, if the capital share rose above 0.75 in both goods and knowledge production.163 At current trends, this is unlikely to happen in the next couple of decades,164 but could happen by the end of the century. (This condition can be thought of as an empirical for whether explosive growth is near, like these discussed in Nordhaus (2021). It lowers my probability that TAI will happen in the next 20 years, but not far beyond that.)

(Note: Arrow (1962) is another Cobb Douglas endogenous model which implies advanced AI can drive explosive growth – see here.)

Explosive growth with full automation, CES production function.

Cobbs Douglas models assume that the elasticity of substitution = 1. Constant Elasticity of Substitution (CES) production functions provide a more general setting in which the elasticity of substitution can take on any value. The expression KαL1-α is replaced by:

$$$ F_σ(K,L)=(αK^ρ+(1−α)L^ρ)^{\frac {1}{ρ}}, with \, ρ=\frac {σ−1}{σ} $$$

We can use this to generalize the above model as follows:

$$$ Y=A^ηF_{σY}(K,L)^αW^{1−α} $$$

$$$ \dot A=A^ϕF_{σA}(K,L)^βW^{1−β} $$$

$$$ \dot K=sY−δK $$$

where σY and σA are the elasticities of substitution between capital and labor in goods and knowledge production respectively. When σY = σA = 1, this reduces to the Cobb-Douglas system above. α and β now represent the returns to doubling both labor and capital. It is standard to assume α = β = 1 but I continue to include W so that the model is conservative.

(Again this model is a generalization of the endogenous growth model in Aghion et al. (2017). A similar model is analyzed very carefully in Trammell and Korinek (2021) Section 5.2.)

In this setting, σY and σA are the crucial determinants of whether there is explosive growth. The tipping point is when these parameters rise above 1. This has an intuitive explanation. When σ < 1, Fσ(K, L) is bottlenecked by its smallest argument. If L is held fixed, there is limit to how large Fσ(K, L) can be, no matter how large K becomes. But when σ > 1, there is no such bottleneck: capital accumulation alone can cause Fσ(K, L) to rise without limit, even with fixed L.

The conditions for sustained super-exponential growth depend on whether and are above or below 0. I discuss four possibilities.

When σY < 1, σA < 1, there is not super-exponential growth unless φ > 1, as shown here.

When σY > 1, σA < 1 the condition is φ > 1 or α ≥ 1, as shown here. In other words, if there are constant returns to labor and capital in combination, a standard assumption, then increasing σY above 1 leads to super-exponential growth. (Note: even if α < 1, there may be an increase in growth when σY rises above 1. I discuss this dynamic more in the task-based model below.)

When σY < 1, σA > 1, a sufficient condition is (as deduced here):

$$$ ηβ>1−ϕ $$$

Super-exponential growth occurs if increasing returns are sufficient to overpower diminishing returns to technology R&D. Aghion et al. (2017) analyze the standard case where η = 1 and β = 1. The condition becomes:

$$$ ϕ>0 $$$

(I discuss the related ‘search limits’ objection to explosive growth in a previous section.)

When σY > 1, σA > 1, a sufficient condition is (as deduced here):

$$$ \frac {ηβ}{1−α}>1−ϕ $$$

Remember, α and β now represent the returns to doubling both labor and capital, so values close to 1 are reasonable. Let’s take α + β =0.9. Bloom et al. (2020) estimate φ = -2 on the assumption that η = 1; let’s use these values. The condition is satisfied:

$$$ 9>3 $$$

(The latter two conditions can be derived from the from Cobb-Douglas condition using the following substitutions:

$$$ F_{σ<1}(K,L)→L $$$

$$$ F_{σ>1}(K,L)→K $$$

These substitutions can also be used to derive super-exponential growth conditions when σA = 1, σY ≠ 1, or when σA ≠ 1, σY = 1.)

The takeaway is that if AI increases the substitutability between labor and capital in either goods or knowledge production, this could lead to super-exponential growth. Reasonable parameter values suggest doing it in both would lead to super-exponential growth, but doing so in just one may not be sufficient.

Trammell and Korinek (2021) Section 5.1. discusses an endogenous ‘learning by doing’ model where a similar mechanism can lead to super-exponential growth.

8.2.2 Exogenous growth models

No fixed factor

Nordhaus (2021) considers the following model:

$$$ Y=F_σ(AK,L) $$$

$$$ \dot K=sY−δK $$$

$$$ A=A_0e^{gt} $$$

The key differences with the endogenous growth model considered above are:

  • No ideas production function: technology is exogenous.
  • No fixed factor W in the goods production. We add this later.
  • Technology only augments capital. This doesn’t affect the result.

If σ > 1 then the capital share rises to unity and the model approximates the following:

$$$ Y=AK $$$

$$$ \dot K=sAK−δK $$$

$$$ A=A_0e^{gAt} $$$

(This approximation, as well as the case σ < 1, is discussed in detail in this technical appendix.)

Now the growth rate of capital itself grows exponentially:

$$$ gK=sA−δ=sA_0e^{gAt}−δ≃sA_0e^{gAt} $$$

The growth rate of output follows suit:

$$$ gY=gK+gA=sA_0e^{gt}−δ+gA≃sA_0e^{gAt} $$$

Growth is super-exponential. (Note: although growth increases without bound, output does not go to infinity in finite time.) Again the pattern of explanation is: capital becomes more substitutable with labor → capital becomes more important → growth increases.

Even if technological progress halts altogether, growth is still:

$$$ gY=gK=sA_f−δ $$$

where Af is the final level of technology. This growth could be very fast.

How robust is this result to our initial assumptions?

  • We would have the same result if the model had been Y = AFσ(K, L) rather than Y = Fσ(AK, L). If the model was Y = Fσ(AK, L), we would not have unbounded growth.165
  • You get the same result in the human-capital accumulation model of Lucas (1988) – see here.
  • The result really depends on constant returns to K and L, combined with some form of capital augmenting technological progress.
  • The next section relaxes the assumption of constant returns to K and L.

With a fixed factor

Let’s consider a more conservative case, where there are diminishing returns to labor and capital in combination due to some fixed factor and where full automation doesn’t occur. This model is inspired by Hanson (2001):166

$$$ Y=(AK)^αL^βW^{1−α−β} $$$

The equations for A and K are as above. We assume L is constant.

The steady state growth rate is (proof here):

$$$ g_y= \frac {αg_A}{1−α} $$$

If there is an increase in the capital share due to AI, growth will increase.

Suppose AI increases the capital share from α to α + fβ. (In a task-based model this corresponds to automating fraction f of tasks.) Production becomes:

$$$ Y=(AK)^{α+fβ}L^{(1−f)β}W^{1−α−β} $$$

Growth increases to (proof here):

$$$ g_Y= \frac {(α+fβ)g_A}{1−α−fβ} $$$

Again, the basic story is that the importance of (accumulable) capital increases, and growth increases as a result.167

If α + β is close to 1, and f = 1 (full automation) the new growth rate could be very high. If α + β = 0.9 then:

$$$ g_Y=9g_A $$$

Hanson uses a more realistic model of AI automation. He separates out standard capital from computer capital, and assumes the productivity of computer capital doubles every two years, in line with Moore’s law. He finds that fully automating labor with computer capital can cause growth to rise from 4.3% a year to 45%.

Trammell and Korinek (2021) Section 3.4 discusses other exogenous growth models where a similar mechanism causes growth to increase.

8.2.3 Task-based models

So far all the models have treated the economy as a homogenous mass, and talked about how well AI substitutes for human labor in general. Really though, there are many distinct tasks in the economy, and AI might substitute better in some tasks than others. Aghion et al. (2017) develops a model along these lines. In the model tasks are gross complements. Technically, this means that the elasticity of substitution between tasks is below one. Intuitively, it means that each task is essential: total output is bottlenecked by the task we perform least well.

I will not describe the mathematics of the model (interested readers can read the paper), but rather its implications for growth.

Firstly, it no longer makes sense to talk about the substitutability of capital and labor in general. Rather the substitutability varies between tasks. This is sensible.

Secondly, we can permanently increase growth in the model by automating a constant fraction of non-automated tasks each year. Automating a task requires the elasticity of substitution to exceed 1 for that task. Presumably we are already automating some tasks each year and this is contributing to growth. But if advanced AI unleashes a process by which the rate of task automation itself increases, this would increase growth.168 The quicker the pace of automation, the higher the growth rate. If we automate an increasing fraction of tasks each year we can maintain super-exponential growth.169 However, this prediction assumes we can seamlessly reallocate human labor to the remaining tasks. If this isn’t possible (which seems likely!), then the actual boost to growth would be lower than that predicted by the model.

This path to higher growth is consistent with the basic story discussed in the main report: AI increases the substitutability of capital → capital is increasingly important (it performs an increasingly large fraction of tasks) → super-exponential growth.

Thirdly, if some fixed set of essential tasks remain unautomated, they will eventually bottleneck growth. Growth will fall back down to the background growth rate (that doesn’t depend on automation). I discuss whether this undermines the prospect of explosive growth here.

8.2.4 How does the case perfect substitution (‘AI robots’) relate to these models?

The case of perfect substitution corresponds to σ = ∞. So it corresponds to σ > 1 in the CES models. In the Cobb-Douglas models it corresponds to the share of capital rising to what was previously the joint share of capital and labor. This case leads to faster growth in all the above models with plausible parameter values, and to super-exponential growth in all the models except the conservative exogenous model.

8.3 What level of AI would be sufficient for explosive growth?

Given all of the above growth models, what’s our best guess about the level of AI that would likely be sufficient for explosive growth? Here I ignore the possibility that growth is bottlenecked by a factor ignored in these models, e.g. regulation. A better statement of the question is: if any level of AI would drive explosive growth, would level would be sufficient?

Answering this question inevitably involves a large amount of speculation. I will list the main possibilities suggested by the models above, and comment on how plausible I see them. It goes without saying that these predictions are all highly speculative; they may be ‘the best we have to go on’ but they’re not very ‘good’ in an absolute sense.

Here are three main answers to the question: ‘What level of AI is sufficient for explosive growth?’:

1. AI that allows us to pass a ‘tipping point’ in the capital share. The Cobb-Douglas models typically suggest that as the capital share in goods production and knowledge production rises, growth will be exponential until a ‘tipping point’ is passed. (We imagine holding the diminishing returns to R&D fixed.) After this point, growth is super-exponential and there will be explosive growth within a few decades.

I put limited weight on this view as the ‘tipping points’ are not reproduced in the CES setting, which generalizes Cobb-Douglas. Nonetheless, Cobb-Douglas provides a fairly accurate description of the last 100 years of growth and shouldn’t be dismissed.

2. AI that raises the elasticity of substitution σ between capital and labor above 1. When σ < 1 there is a limit to how large output can be, no matter how much capital is accumulated. Intuitively, in this regime capital is only useful when there’s labor to combine it with. But when σ > 1, capital accumulation alone can cause output to rise without limit, even with a fixed labor supply. Intuitively, in this regime capital doesn’t have to be combined with labor to be useful (although labor may still be very helpful). When this condition is satisfied in goods or knowledge production, explosive growth is plausible. When it’s satisfied in both, explosive growth looks likely to happen.

I put more weight on this view. However, these models have their limits. They assume that the degree of substitutability between labor and capital is homogenous across the economy, rather than depending on the task being performed.

3. AI that allows us to automate tasks very quickly. (This could either be because an AI system itself replaces humans in many tasks, or because the AI quickly finds ways to automate un-automated tasks.) In the task-based model of Aghion et al. (2017), automating a task provides a temporary boost to growth (a ‘level effect’). If we automate a constant fraction of un-automated tasks each year, this provides a constant boost to growth. If we automate a large enough fraction of non-automated tasks sufficiently quickly, growth could be boosted all the way to 30%.170 A special case of this story is of course full-automation.

I put the most weight on this third view. Nonetheless, it has some drawbacks.

  • It doesn’t address the process by which tasks are automated and how this might feed back into the growth process.
  • It doesn’t seem to be well-positioned to consider the possible introduction of novel tasks. In their model, introducing a new task can only ever decrease output.
  • Like any model, it makes unrealistic assumptions. Most striking is the assumption that human workers are seamlessly reallocated from automated tasks to un-automated tasks. Friction in this process could slow down growth if we haven’t achieved full automation.
  • It emphasizes the possibility of growth being bottlenecked by tasks that are hard to automate but essential. But it may be possible to restructure workflows to remove tasks that cannot be automated. This should reduce the weight we place on the model.

One common theme that I’m inclined to accept is that explosive growth would not require perfectly substitutable AI. Some weaker condition is likely sufficient if explosive growth is possible at all. Overall, my view is that explosive growth would require AI that substantially accelerates the automation of a very wide range of tasks in production, R&D, and the implementation of new technologies.


9. Appendix D: Ignorance story

According to the ignorance story, we’re simply not in a position to know what growth will look like over the long-term. Both the standard story (predicting roughly exponential growth) and the explosive growth stories are suspect, and we shouldn’t be confident in either. Rather we should place some weight in both, and also some weight in the possibility that the pattern of long-run growth will be different to the predictions of either story.

The ignorance story is primarily motivated by distrusting the standard story and the explosive growth story. This leaves us in a position where we don’t have a good explanation for the historical pattern of growth. We don’t know why growth has increased so much over the last 10,000 years, so we don’t know if growth will increase again. And we don’t know why frontier per-capita growth has been exponential for the last 150 years, so we don’t know how long this trend will continue for.

We shouldn’t confidently expect explosive growth – this would require us to trust the explosive growth story. But nor can we confidently rule it out – we’d either have to rule out sufficient AI progress happening by the end of the century, or rule out all of the growth models that predict explosive growth under the assumption that capital substitutes for labor. I discuss some of these here, and Trammell and Korinek (2021) discusses many more.

9.1 The step-change story of growth

This report focuses on the possibility that GWP grew super-exponentially from 10,000 BCE to 1950, with some random fluctuations. The increasing returns mechanism, important in some other prominent theories of long-run growth, provides a plausible explanation for historical increases in growth..

However, the pre-modern GWP data is poor quality and it is possible that GWP followed a different trajectory. More precisely, GWP may have grown at a slow exponential rate from 10,000 BCE to 1500, and then there may have been a one-off transition to a faster rate of exponential growth. If this transition is allowed to last many centuries, from 1500 to 1900, this ‘step change’ story is consistent with the data.

Let a ‘step-change’ model be any that doesn’t use the mechanism of increasing returns to explain very long-run growth, but instead focuses on a one-off structural transition around the industrial revolution.171

Step-change models are typically complex, using many parameters to describe the different regimes and the transition between them. This isn’t necessarily a drawback: perhaps we should not expect economic history to be simple. Further, the step-change model is more consistent with the academic consensus that the industrial revolution was a pivotal period, breaking from previous trends.

9.2 The step-change story of growth lends itself to the ignorance story

What should you think about explosive growth, if you accept the step-change story?

Hanson (2000) is an example of the step-change story. Hanson models historical GWP as a sequence of exponential growth modes. The Neolithic revolution in 10,000 BCE was the first step-change, increasing growth from hunter-gatherer levels to agricultural society levels. Then the industrial revolution in 1700, the second step-change, increased growth from agricultural levels to modern levels. (In some of Hanson’s models, there are two step-changes around the industrial revolution.)

If we were in the final growth mode, Hanson’s model would predict a constant rate of exponential growth going forward.172 However, Hanson uses the pattern of past step-changes to make predictions about the next one. He tentatively predicts that the next step-change will occur by 2100 and lead to GWP doubling every two weeks or less (growth of ≫ 100%).173

But we should not be confident in our ability to predict the timing of future step-changes in growth from past examples. Plausibly there is no pattern in such structural breaks, and it seems unlikely any pattern could be discerned from the limited examples we have seen. Someone embracing Hanson’s view of long-run GWP should see his predictions about future step-changes as highly uncertain. They may be correct, but may not be. In other words, they should accept the ignorance story of long-run GWP.

Could you accept the step-change theory and rule out explosive growth? You would need to believe that no more step changes will occur, despite some having occurred in the past. What could justify having confidence in this view? A natural answer is ‘I just cannot see how there could be another significant increase in growth’. However, this answer has two problems. Firstly, it may not be possible to anticipate what the step-changes will be before they happen. People in 1600 may not have been able to imagine the industrial processes that allowed growth to increase so significantly, but they’d have been wrong to rule out step-changes on this basis. Secondly, mechanisms for a faster growth regime have been suggested. Hanson (2016) describes a digital economy that doubles every month and various economic models suggest that significant automation could lead to super-exponential growth (more).

9.3 How ignorant are we?

I think the ignorance story is a reasonable view, and put some weight on it.

Ultimately though, I put more weight on a specific view of the long-run growth. This is the view offered by models of very long run growth like Jones (2001): increasing returns (to accumulable inputs) led to super-exponential growth of population and technology from ancient times until about 1900. Then, as a result of the demographic transition, population grew exponentially, driving exponential growth of technology and GDP/capita.

Of course, this view omits many details and specific factors affecting growth. But I think it highlights some crucial dynamics driving long-run growth.

This view implies that 21st century growth will be sub-exponential by default: population growth is expected to fall, and so GDP/capita growth should also fall. However, if we develop AI that is highly substitutable with labor, then models of this sort suggest that increasing returns (to accumulable inputs will once again lead to super-exponential growth (more).


10. Appendix E: Standard story

This is not one story, but a collection of the methods used by contemporary economists to make long-run projections of GWP, along with the justifications for these methodologies.

In this section I:

  • Briefly describe three methods that economists use to project GWP, with a focus on why they judge explosive growth to be highly unlikely (here).
  • Show a probability distribution over future GWP that, from my very brief survey, is representative of the views of contemporary economists (here).
  • Summarize the strengths and potential limitations of this collection of methods (here).

Note: this section focuses solely on the papers I found projecting GWP out to 2100. It does not cover the endogenous growth literature which contains various explanations of the recent period of exponential growth. I discuss these explanations in Appendix B.

10.1 Methods used to project GWP

I have only done an extremely brief review of the literature on long-term GWP extrapolations. I have come across three methods for extrapolating GWP:

  1. Low frequency forecasts – use econometric methods to extrapolate trends in GDP per capita, usually starting 1900 or later (more).
  2. Growth models – calculate future growth from projected inputs of labor, capital and total factor productivity (more).
  3. Expert elicitation – experts report their subjective probabilities of various levels of growth (more).

I’m primarily concerned with what these views say about the prospect of explosive growth. In summary, all three methods assign very low probabilities to explosive growth by 2100. My understanding is that the primary reason for this is that they use relatively modern data, typically from after 1900, and this data shows no evidence of accelerating growth – during this time the rate of frontier GDP per capita growth has remained remarkably constant (source, graphs).

10.1.1 Low frequency forecasts of GDP per capita data since 1900

10.1.1.1 How does it work?

Low-frequency forecasting174 is a econometrics method designed to filter out short-horizon fluctuations caused by things like business cycles and pick up on longer-term trends.

I’ve seen two applications of low-frequency forecasting to project GWP until 2100175. The first176 simply takes a single data series, historical GWP per capita since 1900, and projects it forward in time. The second177 fits a more complex model to multiple data series, the historical GDP per capita of various countries. It can model complex relationships between these series, for example the tendency for certain groups of countries to cluster together and for low-income countries to approach frontier countries over time. Both models essentially project low-frequency trends in GDP per capita forward in time, without much reference to inside-view considerations.178

Econometric models of this kind have the benefit of providing explicit probability distributions.. E.g. these projections of US and Chinese GDP/capita from Muller (2019).

10.1.1.2 Relation to the possibility of explosive growth

The structure of the model leads it to assign very low probabilities to the growth rate increasing significantly. So it assigns very low probabilities to explosive growth. In particular, the model assumes that the long-run growth rate oscillates around some constant.

More precisely, the models I’ve studied assume that per capita GWP growth179 is given by:

$$$ g_t=μ+u_t $$$

μ is a constant and ut is a (possibly random) component whose expected long-run average is 0. gt either follows a random walk centered on μ180, or oscillates around μ deterministically. Either way, μ is the long-run average growth rate. Growth in successive periods is correlated and can differ from μ for some time, but in the long run average growth will definitely tend towards μ. These models assume that long-run growth rate is constant; they assume that long-run growth is exponential.

The only way that these models represent the possibility of explosive growth is through the hypothesis that the long-run growth rate μ is very large but, by a large coincidence, the random component ut has always canceled this out and caused us to observe low growth. The resultant probability of explosive growth is extremely small. In both the papers the estimate of average GWP growth until 2100 was about 2% with a standard deviation of 1%. Explosive growth would be > 25 standard deviations from the mean!

Models with this structure essentially rule out the possibility of an increasing growth rate a priori.181 This could be a valid modeling decision given that post-1900 GWP data, and certainly the frontier GDP data, shows no pattern of increasing per capita growth, and it is in general reasonable for a model’s assumptions to foreclose possibilities that have no support in the data. The problem, as we shall discuss later, is that pre-1900 data does show a pattern of super-exponential growth. Either way, it is fair to say that the low-frequency models are not designed to assess the probability of explosive growth, but rather to model the probability of hypotheses that are plausible given post-1900 data.

Could we use the low-frequency methodology to get a more accurate idea of the probability of explosive growth? It should in principle be possible to fit a low-frequency model that, like Roodman’s, contains a parameter that controls whether long-run growth is sub- or super-exponential.182 The possibility of explosive growth would then be represented by our uncertainty over the value of this parameter (as in Roodman’s model). I suspect that this model, trained on post-1900 data, would conclude that growth was very probably sub-exponential, but assign some small probability to it being slightly super-exponential. Explosive growth would eventually follow if growth were super-exponential. So I suspect that this methodology would conclude that the probability of explosive growth was small, but not as small as in the low-frequency models I have seen.

10.1.2 Growth models

10.1.2.1 How do they work?

Growth models describe how inputs like labor, capital and total factor productivity (TFP) combine together to make output (GDP). They also describe how these inputs change over time.

Here I’ll just describe how an extremely simple growth model could be used to generate GWP projections. Then I’ll list some ways in which it could be made more realistic.

Output Y in a year is given by the following Cobb-Douglas equation:

$$$Y=AK^αL^β $$$

where

  • A is TFP.
  • K is the capital, a measure of all the equipment, buildings and other assets.
  • L is labor, a measure of the person-hours worked during the timestep.
  • α and β give the degree of diminishing returns to capital and labor; it’s often assumed that α + β = 1, meaning that a doubling the number of workers, buildings and equipment would double the amount of output.

The inputs change over time as follows:

  • A grows at a constant exponential rate – the average rate observed in the post-1900 data.
  • L in each year is given by UN projections of population growth.
  • The change in K between successive years is ΔK = sY – δK, where s is the constant rate of capital investment and δ is the constant rate of capital depreciation.
    1. The value of K in year n can be calculated from the values of K and Y in the year n – 1

You generate GWP projections as follows:

  • Identify Y with GWP.
  • Get starting values of Y, A, K and L from data.
  • Project A and L for future years as described above.
  • Project K and Y for future years as follows:
    1. Predict next year’s K using the current values of K and Y.
    2. Predict next year’s Y using your projections for A, K, and L next year. Now you have K and Y for next year.
    3. Repeat the above two steps for later and later years.

The above model is very basic; there are many ways of making it more sophisticated. Perhaps the most common is to project each country’s growth separately and model catch-up effects.183 You could also use a different production function from Cobb-Douglas, introduce additional input factors like human capital and natural resources184, use sophisticated theory and econometrics to inform the values for the factors185 and constants186 at each timestep, control for outlier events like the financial crisis, and model additional factors like changing exchange rates. These choices can significantly affect the predictions, and may embody significant disagreements between economists. Nonetheless, many long-run extrapolations of GWP that I’ve seen use a growth model that is, at its core, similar to my simple example.

My impression is that these models are regarded as being the most respected. They can incorporate wide-ranging relevant data sources and theoretical insights.

One down-side of these models is that the ones I’ve seen only provide point estimates of GWP in each year, not probability distributions. Uncertainty is typically represented by considering multiple scenarios with different input assumptions, and looking at how the projections differ between the scenarios. For example, scenarios might differ about the rate at which the TFP of lower-income countries approaches the global frontier.187 The point estimates from such models typically find that average per capita GWP growth will be in the range 1 – 3%.

10.1.2.2 Relation to the possibility of explosive growth

Most of the long-run growth models I’ve seen set frontier TFP exogenously, stipulating that it grows at a constant rate similar to its recent historical average.188 While individual countries can temporarily grow somewhat faster than this due to catch-up growth, the long-run GDP growth of all countries is capped by this exogenous frontier TFP growth (source).

The structure of most of these models, in particular their assumption of constant frontier TFP growth, rules out explosive growth a priori.189 This is supported by the relative constancy of frontier TFP growth since 1900, but is undermined by earlier data points.

A few models do allow TFP to vary in principle, but still do not predict explosive growth because they only use post-1900 data. For example, Foure (2012) allows frontier TFP growth to depend on the amount of tertiary education and finds only moderate and bounded increases of TFP growth with tertiary education.190

The more fundamental reason these models don’t predict explosive growth is not their structure but their exclusive use of post-1900 data, which shows remarkable constancy in growth in frontier countries. This data typically motivates a choice of model that rules out explosive growth and ensures that more flexible models won’t predict explosive growth either.

10.1.3 Expert elicitation

10.1.3.1 How does it work?

GWP forecasts are made by a collection of experts and then aggregated. These experts can draw upon the formal methods discussed above and also incorporate further sources of information and the possibility of trend-breaking events. This seems particularly appropriate to the present study, as explosive growth would break trends going back to 1900.

I focus exclusively on Christensen (2018), the most systematic application of this methodology to long-run GWP forecasts I have seen.

In this study, experts were chosen by ‘a process of nomination by a panel of peers’ and the resultant experts varied in both ‘field and methodological orientation’.191 Experts gave their median and other percentile estimates (10th, 25th, 50th, 75th, 90th percentiles) of the average annual per-capita growth of GWP until 2100. For each percentile, the trimmed mean192 was calculated and then these means were used as the corresponding percentile of the aggregated distribution.

As well as providing aggregated quantile estimates, Christensen (2018) fits these estimates to a normal distribution. The mean per capita growth rate is 2.06% with a standard deviation of 1.12%. This provides a full probability distribution over GWP per capita for each year.

10.1.3.2 Relation to the possibility of explosive growth

If any expert believed there was > 10% chance of explosive growth, this would have shown up on the survey results in their 90th percentile estimate. However, Figure 7 of their appendix shows that no expert’s 90th percentile exceeds 6%. Strictly speaking, this is compatible with the possibility that some experts think there is a ~9% probability of explosive growth this century, but practically speaking this seems unlikely. The experts’ quantiles, both individually and in aggregate, were a good fit for a normal distribution (see Figure 7) which would assign ≪ 1% probability to explosive growth.

Nonetheless, there are some reasons to think that the extremely high and extremely low growth is somewhat more likely than these the surveys suggest:

  • There is a large literature on biases in probabilistic reasoning in expert judgement. It suggests that people’s 10 – 90% confidence intervals are typically much too narrow, containing the true value much less than 80% of the time. Further, people tend to anchor their uncertainty estimates to an initial point estimate. These effects are especially pronounced for highly uncertain questions. The survey tried to adjust for these effects193, but the same literature suggests that these biases are very hard to eliminate.
  • The experts self-reported their level of expertise as 6 out of 10, where 5 indicates having studied the topic but not being an expert and 10 indicates being a leading expert.194 The authors ‘take this as suggestive that experts do not express a high level of confidence in their ability to forecast long-run growth outcomes’. It also seems to suggest that there is no clear body of experts that specialize in answering this question and has thought deeply about it. This increases the chance that there are legitimate ways of approaching the problem that the experts have not fully considered.

10.2 Probability distribution over GWP

I want an all-things-considered probability distribution over GWP that is representative of the different views and methodologies of the standard story. This is so I can compare it with distributions from the other big pictures stories, and (at a later time) compare it to the economic growth that we think would result from TAI. If you’re not interested in this, skip to the next section.

I’ve decided to use the probability distribution constructed from above-discussed expert elicitation in Christensen (2018). It has a mean of 2.06% and a standard deviation of 1.12%. I chose it for a few reasons:

  • The experts can use the results of the other two methods I’ve discussed (econometric modeling and growth models) to inform their projections.
  • Experts can take into account the possibility of trend-breaking events and other factors that are hard to incorporate into a formal model.
  • The experts in Christensen (2018) were selected to represent a wide-range of fields and methodologies.
  • The central aim of Christensen’s paper was to get accurate estimates of our uncertainty, and its methodology and survey structure was designed to achieve this goal.
  • The expert elicitation distribution is consistent with point estimates from growth models.195 This is important because I believe these growth models incorporate the most data and theoretical insight and are consequently held in the highest regard.
  • One possible drawback of this choice is that the distribution may overestimate uncertainty about future growth and assign more probability to > 3% than is representative.
    • The 90th percentile of the distribution is higher than any point estimates I’ve seen.196
    • The 10 – 90th percentile range is wider than the equivalent range from econometric methods.
    • This may be because the expert elicitation methodology can incorporate more sources of uncertainty than the other models.

The expert elicitation probability distribution is over GWP per capita. To get a distribution over GWP I used the UN’s median population projections (which have been accurate to date).197

10.3 Strengths and limitations

Advocates of standard story use a range of statistical techniques and theoretical models to extrapolate GWP, that are able to incorporate wide-ranging relevant data sources. If we were confident that the 21st century would resemble the 20th, these methods would plausibly be adequate for forecasting GWP until 2100.

However, I do believe that the methodologies of the standard story are ill-equipped to estimate the probability of a regime-change leading to explosive growth. This is due to a couple of features:

  • The papers I’ve seen exclusively use post-1900 data, and often only post-1950 data.198 While reasonable for short-term growth forecasts, this becomes more questionable when you forecast over longer horizons. The post-1900 data is silent on the question of whether 21st century growth will follow a similar pattern to 20th century growth and of what it might look like if it does not.
  • Its models typically foreclose the possibility of explosive growth by assuming that the long-run frontier growth rate is constant. This assumption is supported by the post-1900 data but not, we shall see, by endogenous growth theory or by data sets that go back further in time. As a result of this assumption, its models do not assess the probability that 21st century GWP growth is super-exponential, a critical question when assessing the plausibility of explosive growth.
  • An important caveat is that expert elicitation does seem well placed to anticipate a regime-change, but experts assign < 10% to explosive growth and probably < 1%. I find this the most compelling evidence against explosive growth from the standard story. It is hard to fully assess the strength of this evidence without knowing the reasons for experts’ projections. If they have relied heavily on the other methods I’ve discussed, their projections will suffer from drawbacks discussed in the last two bullet points.

These limitations are not particularly surprising. The methods I’ve surveyed in this section were originally developed for the purposes of making forecasts over a few decades, and we saw above that even the most expert people in this area do not consider themselves to have deep expertise.


11. Appendix F: Significant probability of explosive growth by 2100 seems robust to modeling serial correlation and discounting early data points

The model in Roodman (2020) assigns 50% probability to explosive growth happening by 2044, 10% by 2033, and 90% by 2063. However, there are reasons to think that Roodman’s model may predict explosive growth too soon, and its confidence intervals may be too narrow.

An appendix discusses two such reasons:

  • The growth rates in nearby periods are correlated, but Roodman’s model implies that they are independent.
  • Recent data is more relevant to predicting 21st century growth than ancient data points, but Roodman’s model doesn’t take this into account.

(Note: there are other reasons to think explosive growth will happen later than Roodman predicts. In particular, population is no longer accumulable, where accumulable means more output → more people. This section does not adjust Roodman’s model for this objection, but only for the two reasons listed.)

How much would accounting for these two factors change the predictions of Roodman’s model? Would they delay explosive growth by a decade, a century, or even longer? To get a rough sense of the quantitative size of these adjustments, I built a simple model for projecting GWP forward in time. I call it the growth multiplier model. (At other places in the report I call it the ‘growth differences’ model.)

The growth multiplier model retains some key features of Roodman’s univariate endogenous growth model. In particular, it retains the property of Roodman’s model that leads it to predict sub- or super-exponential growth, depending on the data it is fit to. The justification for these features is the same as that for Roodman’s model: long-run GDP data displays super-exponential growth and endogenous growth models predict such growth.

At the same time, the growth multiplier model aims to address some of the drawbacks of Roodman’s model. Most significantly, it incorporates serial correlation between growth at nearby periods into its core. In addition, the user can flexibly specify how much extra weight to give to more recent data points. The model also incorporates randomness in a simple and transparent way. The cost of these advantages is that the model is considerably less theoretically principled than the endogenous growth models.

With my preferred parameters, the model assigns a 50% chance of explosive growth by 2093 and a 70% chance by 2200.199 There is still a 10% chance of explosive growth by 2036, but also a 15% chance that explosion never happens200. While I don’t take these precise numbers seriously at all, I do find the general lesson instructive: when we adjust for serial correlation and the increased relevance of more recent data points we find that i) the median date by which we expect explosion is delayed by several decades, ii) there’s a non-negligible chance that explosive growth will not have occurred within the next century, and iii) there is a non-negligible chance that explosive growth will occur by 2050. In my sensitivity analysis, I find that these three results are resilient to wide-ranging inputs.

The rest of this section explains how the model works (here), discusses how it represents serial correlation (here), compares its predictions to the other big-picture stories about GWP (here), does a sensitivity analysis on how its predictions change for different inputs (here), and discusses its strengths and weaknesses (here).

The code behind the growth multiplier model, Roodman’s model, and this expert survey is here. (If the link doesn’t work, the colab file can be found in this folder.)

11.1 How does the growth multiplier model work?

Put simply, the model asks the question ‘How will the growth rate change by the time GWP has doubled?’, and answers it by saying ‘Let’s look at how it’s changed historically when GWP has doubled, and sample randomly from these historically observed changes’. Historically, when GWP has doubled the growth rate has increased by about 40% on average, and so the model’s median prediction is that the growth rate will increase by another 40% in the future each time GWP doubles.

The model divides time into periods and assumes that the growth rate within each period is constant. The length of each period is the time for GWP to increase by a factor r – this choice is inspired by the properties of Roodman’s univariate model.

So we divide the historical GWP data into periods of this kind and calculate the average growth rate within each period. Then we calculate the change in average growth rate between successive periods. Again inspired by Roodman’s univariate model, we measure this change as the ratio between successive growth rates: new_growth_rate / old_growth_rate.201 Call these ratios growth multipliers. The growth multiplier of a period tells you how much the average growth rate increases (or decreases) in the following period. For example, if 1800-1850 had 2% growth and 1850-1900 had 3% growth, then the growth multiplier for the period 1800-1850 would be 1.5.

Here’s an example with dummy data, in which r = 2.

To extrapolate GWP forward in time, we must calculate the growth rate g of the period starting in 2025. We do this in two steps:

  • Randomly sample a value for the previous period’s growth multiplier. In this example, gm is the growth multiplier of the period finishing in 2025. gm is randomly sampled from the list [2, 2, 1.5, 0.5].202 All items on the list need not be equally likely; we can specify a discount rate to favor the sampling of more recent growth multipliers. This discount rate crudely models the extra weight given to more recent data points.
  • Multiply together the growth rate and growth multiplier from the previous period. In this example, g = 1.5 × gm.
  • Calculate the duration of the next period from its growth rate. In this example, we calculate YYYY from g.203 Notice that we already know the GWP at the end of the next period (in this example $25,600b) as we defined periods as the time taken for GWP to increase by a factor of r.

We’ve now calculated the growth rate and end date of the next period. We can repeat this process indefinitely to extrapolate GWP for further periods.

The two seemingly arbitrary assumptions of this model – defining each period as the time for GWP to increase by a factor of r, and calculating the next growth rate by multiplying the previous growth rate by some growth multiplier – are both justified by comparison to Roodman’s univariate model. The former assumption in particular corresponds to a core element of Roodman’s model that drives its prediction of super-exponential growth. I discuss this in greater detail in this appendix.

11.2 How does the growth multiplier model represent serial correlation?

In Roodman’s model, the median predicted growth for 2020-40 is higher than the observed growth in 2000-20 for two reasons:

  1. The model believes, based on historical data, that when GWP increases growth tends to increase.
  2. Growth in 2000-20 was below the model’s median prediction; it treats this as a random and temporary fluctuation, uncorrelated with that of 2020-40; it expects growth to return to the median in 2020-40.

It is Factor 2 that causes the model to go astray, failing to capture the serial correlation between growth in the two periods. Factor 2 alone raises the model’s median prediction for 2019 growth to 7.1%.204

The growth multiplier model addresses this problem by predicting growth increases solely on the basis of Factor 1; Factor 2 has no role. Unlike Roodman’s model, it does not track a ‘median’ growth rate as distinct from the actual growth rate; rather, it interprets the current growth rate (whatever it is) as ‘the new normal’ and predicts future growth by adjusting this ‘new normal’ for increases in GWP (Factor 1).

As a result, the growth multiplier model builds in serial correlation between the growth in different periods. If the current growth rate is ‘surprisingly low’ (from the perspective of Roodman’s model) then this will directly affect the next period’s growth rate via the formula new_growth_rate = old_growth_rate × growth_multiplier.205 In this formula, the role of ‘× growth_multiplier’ is to adjust the growth rate for the increase in GWP (Factor 1). The role of old_growth_rate is to link the next period’s growth directly to that of the previous period, encoding serial correlation. A single period of low growth affects all subsequent periods of growth in this way. A single period of low growth affects all subsequent periods of growth in this way. Further, this effect does not diminish over time, as the growth of period i + n is proportional to the growth of period i for all n.

There are possible models that display degrees of serial correlation intermediate between Roodman’s model and the growth multiplier model. I think such models would be more realistic than either extreme, but I have not attempted to construct one. I discuss this possibility more in this appendix. So while I regard Roodman’s predictions as overly aggressive, I regard those of the growth multiplier model as adjusting too much for serial correlation and in this sense being overly conservative. We should expect some return to the longer-run trend.

11.3 What are the model’s predictions for my preferred parameters?

The following table describes the two inputs to the growth difference model and what my preferred values for these inputs are:

 

INPUT MEANING PREFERRED VALUE CONSIDERATIONS THAT INFORMED MY CHOICE OF R
r r controls the lengths of the periods that the model divides GWP into. A smaller value for rmeans we look at how growth has changed over shorter periods of time, and extrapolate smaller changes into the future.

Its value is fairly arbitrary; the division into discrete periods is done to make the model analytically tractable. My sensitivity analysis suggests the results are not very sensitive to the value of r – predicted dates for explosive growth change by < 10 years.

1.6 If r is too small, the GWP data is too coarse-grained to contain successive data points where GWP only differs by a factor of r. For example, GWP increases by a factor of 1.5 between some successive ancient data points.

If r is too large the assumption that growth is constant within each period is less plausible, and we lose information about how growth changes over shorter periods. For example, if r > 1.6 we lose the information that growth was slower from 2010-19 than from 2000 to 2010.

Discount rate How much we discount older data points. A discount of 0.9 means that when GWP was half as big we discount observations by a factor of 0.9, when GWP was 1/4 the size the discount is 0.92, when it was 1/8 the size the discount is 0.93, and so on. 0.9 This discount means that, compared to a 2000 observation, the 1940 observation has 73% of the weight, the 1820 observation has 53% of the weight, and the 3000 BCE observation has 23% of the weight.

With these inputs the model’s percentile estimates of the first year of explosive growth (sustained > 30% growth) are as follows:

These probabilistic GWP projections can be shown alongside those of Roodman’s model and the standard story.

See code producing this plot at the bottom of this notebook. (If the link doesn’t work, the colab file can be found in this folder.)

I believe the probabilities from the growth multiplier model are closer than Roodman’s to what it’s reasonable to believe, from an outside-view perspective, conditional on the basic ideas of the explosive growth story being correct.206

If we trust the standard story’s view that growth will continue at roughly its current level (1 – 3%) over the next decade or so, then we should decrease the probability of explosive growth by 2100 relative to these plots.

11.4 Sensitivity analysis: how do the growth difference model’s predictions change for different inputs?

I investigated how changing both inputs affects the model’s projections. Full details are in this appendix, but I summarize the key takeaways in this section. For reference, Roodman’s percentile predictions about the first year of explosive growth are as follows:

PERCENTILE EXPLOSIVE GROWTH DATE
10 2034
30 2039
50 2043
70 2050
90 2065

When I used my preferred inputs, the growth multiplier model differs from Roodman’s in two ways:

  • It models serial correlation. This is implicit in the model’s structure.
  • It places a larger discount on older data points. This is via my choice of discount rate.

We’ll now investigate the effect of each factor in turn, including how sensitive these are to the choice of r.

11.4.1 Serial correlation alone could delay explosive growth by 30-50 years

We can isolate the impact of the first factor by choosing not to discount older data points (discount rate = 1). In this case, still using r = 1.6, the percentiles of the growth multiplier model are as follows:

A further sensitivity analysis on r shows that using different values of r between 1.05 and 3 could change the median date by ± 10 years in either direction, change the 10th percentile by ± 5 years in either direction, and change the 90th percentile by ± 100s of years.

11.4.2 A reasonable discount can delay explosive growth by 20 years

The following table shows information about different discount rates. It shows how severely each discount downweights older data points, and how many years it delays the median predicted date of explosive growth.

WEIGHT OF OLDER DATA POINTS (A 2000 DATA POINT HAS WEIGHT 100%) DELAY TO MEDIAN DATE OF EXPLOSIVE GROWTH (YEARS)
DISCOUNT RATE 1940 1820 3000 BCE R = 1.6 R = 2
0.95 86% 74% 49% 4 1
0.9 73% 53% 23% 10 4
0.85 61% 38% 10% 21 10
0.8 51% 26% 4% 46 19
0.75 34% 12% 0.6% 89 29
0.7 22% 5% 0.1% 190 34

I consider values of discount rate equal or lower than 0.8 to be unreasonable. They place overwhelming importance on the last 50 years of data when forecasting GWP over much longer periods of time than this. For long-range forecasts like in this report, I favor 0.9 or 0.95. For reasonable discounts, explosive growth is delayed by up to 20 years.

The effect on the 10th percentile is much smaller (< 10 years), and the effect on the 70th and 90th percentiles is much larger. See this appendix for more details.

Even with very steep discounts, long term growth is still super-exponential. The recent data, even when significantly upweighted, don’t show a strong enough trend of slowing GWP growth to overwhelm the longer term trend of super-exponential growth.

Smaller values of r are slightly more affected by introducing a discount rate. I believe that this is because with smaller values of r the can model is fine-grained enough to detect the slowdown of GWP growth in the last ~10 years and a discount heightens the effect of this slowdown on the predictions. See more details about the interaction between r and the discount rate in this appendix.

11.5 Strengths and limitations of the growth multiplier model

The growth multiplier model is really just an adjustment to Roodman’s model. Its key strength is that it addresses limitations of Roodman’s model while keeping the core elements that drive its prediction of super-exponential growth.

Its prediction of explosive growth invites many criticisms which I address elsewhere. Beyond these, its key limitation is that its modeling choices, considered in isolation, seem arbitrary and unprincipled. They are only justified via comparison to the increasing returns of endogenous growth models. A further limitation is that its description of the evolution of GWP is both inelegant and in certain ways unrealistic.207 Lastly, a somewhat arbitrary choice about the value of r must be made, and results are sensitive to this assumption within a couple of decades.


12. Appendix G: How I decide my overall probability of explosive growth by 2100

The process involves vague concepts and difficult judgement calls; others may not find it useful for deciding their own probabilities. I do not intend for the reasoning to be water-tight, but rather a pragmatic guide to forming probabilities.

Here are my current tentative probabilities for the annual growth of GWP/capita g over the rest of this century:

  • Explosive growth, g > 30%: There’s a period, lasting > 10 years and beginning before 2100, in which g > 30%: ~30%.
  • Significant growth increase, 5% < g < 30%: There’s no explosive growth but there’s a period, lasting > 20 years and beginning before 2100, in which g > 5%: ~8%.
  • Exponential growth, 1.5% < g < 5%: There’s no significant growth increase and average growth stays within its recent range of values: ~25%.
  • Sub-exponential growth, g < 1.5%: We never have a significant growth increase, and average annual growth is near the bottom or below its recent range: ~40%.208

I’ve rounded probabilities to 1 significant figure, or to the nearest 5%, to avoid any pretence at precision. As a result, the probabilities do not add up to 100%.

Note, the specific probabilities are not at all robust. On a different day my probability of explosive growth by 2100 might be as low as 15% or as high as 60%. What is robust is that I assign non-negligible probability (>10%) to explosive growth, exponential growth, and sub-exponential growth.

The diagram below summarizes the process I used to determine my probabilities. I use the toy scenario of ‘AI robots’discussed in the main report to help me develop my probabilities. Each AI robot can replace one human worker, and do the work more cheaply than a human worker. I use this scenario because it is concrete and easy to represent in economic models: AI robots allow capital to substitute perfectly for labour in goods production and knowledge production.

The following sections go through the diagram, explaining my decisions at each node. I recommend readers keep the diagram open in a tab to help them follow the logic. At several points, I feel I’ve been somewhat conservative about the probability of explosive growth; I indicate these as I go.

12.1 Will we develop AI robots (or AIs with a similar impact on growth) in time for explosive growth to occur by 2100?

I split this into two sub-questions:

  1. What level of AI is sufficient for explosive growth (assuming AI robots would drive explosive growth)?
  2. Will we develop this level of AI in time for explosive growth to occur by 2100?

12.1.1 What level of AI is sufficient for explosive growth (assuming AI robots would drive explosive growth)?

What’s the lowest level of AI that would be sufficient for explosive growth, assuming AI robots would be sufficient?

My view on this question is mostly informed by studying the growth models that imply AI robots would drive explosive growth. I analyze models one by one here, and draw my conclusions here. My (rough) conclusion is that ‘explosive growth would require AI that substantially accelerates the automation of a very wide range of tasks in production, R&D, and the implementation of new technologies.’ This would require very rapid progress in both disembodied AI and in robotics.

Consider a ‘virtual worker’ – AI that can do any task a top quality human worker could do working remotely (it could be one AI system, or multiple working together). I believe, for reasons not discussed in this report, that a virtual worker would probably enable us to quickly develop the level of robotics required for explosive growth.

I use a ‘virtual worker’ as my extremely rough-and-ready answer to ‘what’s the lowest level of AI that would drive explosive growth?’. Of course, it is possible that a virtual worker wouldn’t be sufficient, and also possible that a lower level of AI would be sufficient for explosive growth.

12.1.2 Will we develop a ‘virtual worker’ in time for explosive growth to occur by 2100?

There are two sub-questions here.

  1. By when must we develop a virtual worker for there to be explosive growth by 2100?
  2. How likely are we to develop a virtual worker by this time?

I have not investigated the first sub-question in depth. In the growth models I’ve studied for this report, it seems that even in the ‘AI robot’ scenario it could take a few decades for growth to increase to 30%.209 So I provisionally treat 2080 as the answer to the first sub-question. For reasons not discussed in this report, I believe this is conservative and that developing a virtual worker would drive explosive growth within years rather than decades.

The second sub-question is then ‘How likely are we to develop a virtual worker by 2080?’.

My view on this is informed by evidence external to this report:

  • Expert forecasts about when high-level machine intelligence will be developed.210
    • If this was my only source of evidence I would assign ~45% by 2080.211
  • A framework by my colleague Ajeya Cotra analyzing when the computation required to develop TAI will be affordable.
    • Her high-end estimate assigns ~90% probability by 2080.
    • Her best-guess estimate assigns ~70% probability by 2080.
    • Her low-end estimate assigns ~40% probability by 2080.
  • My own report on what prior we should have about when Artificial General Intelligence is developed.212
    • My high-end estimate assigns ~30% probability by 2080.
    • My best-guess estimate assigns ~15% probability by 2080.
    • My low-end estimate assigns ~4% probability by 2080.

Personally, I put most weight on Ajeya’s framework (0.7), and roughly similar weight to the other two sources of evidence (~0.15 each). Conditional on Ajeya’s framework, I am closer to her low-end estimate than her best guess, at around 50% probability by 2080.213 Overall, I’m currently at around ~45% that we will develop a virtual worker by 2080.214

This explains my reasoning about the top-level node of the diagram. The next section looks at the nodes on the left hand side of the diagram, assuming we do develop a ‘virtual worker’, the section after looks at the right hand side of the diagram.

12.2 Assuming we do develop AI with a similar impact on growth to AI robots (left fork)

12.2.1 Would AI robots drive explosive growth, absent any unintended bottlenecks?

Another way to understand this question is: Do AI robots have a strong tendency to drive explosive growth?

My opinion here is influenced by the history of economic growth and the choice between different growth models:

  • There are broadly speaking two classes of theories: accumulation models and idea-based models. In accumulation models, the ultimate source of growth in GDP/capita is the accumulation of physical or human capital. In idea-based models, the ultimate source of growth is targeted R&D leading to technological progress.
  • Idea-based models imply that AI robots would lead to explosive growth, when you use realistic parameter values.
    • These models have increasing returns to inputs as a central feature, but do not predict super-exponential growth as labour is not accumulable. With AI robots there are increasing returns to accumulable inputs which can drive super-exponential growth.
    • I analyze many of idea-based models in Appendix C,215 subbing in the AI robot scenario. I find that the increasing returns to accumulable inputs drives super-exponential growth when you use realistic parameter values.216
    • Idea-based models offer a simple and plausible account of very long-run growth, according to which increasing returns to accumulable inputs has caused growth to increase over time.
      • They are compatible with the importance of one-off structural transitions occurring around the industrial revolution.
    • Appendix B argues that some idea-based theories (semi-endogenous growth models) offer the best explanation of the recent period of exponential growth.
  • For accumulation-based models, the link between AI and growth is less clear but it’s still plausible that AI robots would drive explosive growth conditional on these models.
    • Many of these models imply that the AI robot scenario would lead to explosive growth.
      • For example, the learning by doing model of Arrow (1962) (more) or the human capital accumulation model of Lucas (1988) (more).217
      • It’s possible to dismiss this prediction as an unintended artifact of the model, as the primary mechanism generating sustained growth in these models (capital accumulation) has no strong intuitive link with AI. This is in contrast to idea-based models, where there is an obvious intuitive way in which human-level AI would speed up technological progress.
    • Some accumulation theories don’t imply that the AI robot scenario would cause explosive growth.
      • For example, see Frankel (1962) (more), or simply a CES production function with the elasticity of substitution between labour and capital greater than 1 (more).
      • I suggest these models face serious problems.
    • Appendix B argues that accumulation theories require problematic knife-edge conditions for exponential growth.
    • Growth accounting exercises, e.g. Fernald and Jones (2014), find that TFP growth accounts for the majority of growth rather than the accumulation of physical or human capital. This gives us reason to prefer idea-based models.
  • Overall, I put ~80% weight on idea-based theories.
  • Exogenous growth models can be understood as expressing uncertainty about the ultimate driver of growth. Even in a conservative exogenous growth model, where a fixed factor places diminishing returns on labour and capital in combination, capital substituting for labour in goods production can cause a significant one-time increase in growth (although this may not be sufficient for > 30% annual growth).

So, overall, would AI robots this century drive explosive growth, assuming there are no unanticipated bottlenecks? My starting point is the 80% weight I put on idea-based models, based on their explanation of very long-run growth and the recent period of constant growth. I bump this up to 90% as various exogenous models and accumulation-based models also imply that AI robots would drive explosive growth. Lastly, I cut this back to 80% based on the possibility that we can’t trust the predictions of these models in the new regime where capital can entirely replace human labour.

Most of the 20% where AI robots don’t have a tendency to drive explosive growth corresponds to none of our theories being well suited for describing this situation, rather than to any particular alternative model.

So I put ~80% on AI robots driving explosive growth, absent unanticipated bottlenecks.

12.2.2 Will there be unanticipated bottlenecks?

I have done very little research on this question. Above, I briefly listed some possible bottlenecks along with reasons to think none of them are likely to prevent explosive growth. I put ~25% on a bottleneck of this kind preventing explosive growth.

This means my pr(explosive growth | AI robots this century) = 0.8 × 0.75 = ~60%. If I had chosen this probability directly, rather than decomposing it as above, I’d have picked a higher number, more like 75%. So the ‘60%’ may be too low.

12.2.3 If there is an unanticipated bottleneck, when will it apply?

This corresponds to the node ‘Does the bottleneck apply before g>5%?’.

Suppose we develop AI that has a strong tendency to drive explosive growth, but it doesn’t due to some bottleneck. How fast is the economy growing when the bottleneck kicks in?

Large countries have grown much faster than 5% before,218 suggesting the bottleneck probably kicks in when g > 5%. In addition, there’s a smaller gap between the current frontier growth (~2%) and 5% than between 5% and 30%.

On the other hand, it’s possible that the unknown bottleneck is already slowing down frontier growth, suggesting it would limit growth to below 5%.

Somewhat arbitrarily, I assign 80% to the bottleneck kicking in when g > 5%, and 20% to it kicking in when g < 5%.

12.2.4 If we develop a ‘virtual worker’ but it has no tendency to drive explosive growth, will growth slow down?

This corresponds to the left-hand node ‘Will growth slow down?’.

My first pass is to fall back on the scenario where we don’t make impressive advances in AI at all (I discuss this scenario below). This implies ~65% to sub-exponential growth and ~35% to exponential growth.219 I give 50% to each because highly advanced AI might help us to sustain exponential growth even if it has no tendency to produce explosive growth.

12.3 Assuming we don’t develop AI robots, or AI with similar impacts on growth (right fork)

12.3.1 Is there explosive growth anyway?

If we are skeptical of the explanations of why growth increased in the past, and why it has recently grown exponentially, we may be open to growth increasing significantly without this increase being driven by AI. Growth has increased in the past, perhaps it will increase again.

Even if we can’t imagine what could cause such an increase, this is not decisive evidence against there being some unknown cause. After all, hypothetical economists in 1600 would have been unlikely to imagine that the events surrounding the industrial revolution would increase growth so significantly. Perhaps we are just as much in the dark as they would have been.

Further, brain emulation technology could have similar effects on growth to advanced AI, allowing us to run human minds on a computer and thus making population accumulable. Perhaps radical biotechnology could also boost the stock of human capital and thus the rate of biotechnological progress.

I currently assign 2% to this possibility, though this feels more unstable than my other probabilities. It’s low because I put quite a lot of weight in the specific growth theories that imply that super-exponential growth was fueled by super-exponential growth in the human population (or the research population) and so wouldn’t be possible again without advanced AI or some tech that expanded the number or capability of minds in an analogous way; I’m conservatively assigning low probabilities to these other technologies. I think values as high as 5-10% could be reasonable here.

12.3.2 If there isn’t explosive growth anyway, does growth slow down?

This corresponds to the right-hand node ‘Will growth slow down?’.

I put ~75% weight in semi-endogenous growth theories, which is my first-pass estimate for the probability of sub-exponential growth in this scenario.

You could try to account for further considerations. Even if semi-endogenous growth theory is correct, g could still exceed 1.5% if the fraction of people working in R&D increases fast enough, or if other factors boost growth. On the other hand, even if semi-endogenous growth theory is wrong, growth could slow for some reason other than slowing population growth (e.g. resource limitations). I assume these considerations are a wash.

I do make one more adjustment for the effect of AI. Even if we don’t develop AIs with comparable growth effects to AI robots, AI might still increase the pace of economic growth. Aghion et al. (2017) focus on scenarios in which AI automation boosts the exponential growth rate. I assign 10% to this possibility, and so give 65% to sub-exponential growth in this scenario.


13. Appendix H: Reviews of the report

We had numerous people with relevant expertise review earlier drafts of the report. Here we link to the reviews of those who give us permission to do so. Note: the report has been updated significantly since some of these reviews were written.


14. Technical appendices

14.1 Glossary

GDP

  • Total stuff produced within a region, with each thing weighted by its price.

GWP

  • Total amount of stuff produced in the whole world, with each thing weighted by its price.

GDP per capita

  • GDP of a region divided by the region’s total population.
  • So GWP/capita is GWP divided by the world population.

Frontier GDP

  • GDP of developed countries on the frontier of technological development. These countries have the highest levels of technology and largest GDP/capita.
  • Often operationalized as OECD countries, or just the USA.

Physical capital

  • Machinery, computers, buildings, intellectual property, branding – any durable asset that helps you produce output.
  • I often refer to this as merely ‘capital’.
  • Doesn’t include land or natural resources.

Human capital

  • Human skills, knowledge and experience, viewed in terms of its tendency to make workers more productive.

Total factor productivity (TFP) growth

  • Increase in output that can’t be explained by increases in inputs like labor and capital.
  • If TFP doubles, but all inputs remain the same, output doubles.
  • TFP increases correspond to better ways of combining inputs to produce output, including technological progress, improvements in workflows, and any other unmeasured effects.
  • In the report I often don’t distinguish between TFP growth and technological progress.

Exponential growth

  • Example 1: the number of cells doubling every hour.
  • Example 2: the number people infected by Covid doubling every month.
  • Example 3: GWP doubling every 20 years (as it does in some projections).
  • Definition 1: when ‘doubling time’ stays constant.
  • Definition 2: when a quantity increases by a constant fraction each time period.
  • yt+1 = yt(1 + g), where g is the constant growth rate.
    • US GDP / capita has grown exponentially with g = 1.8% for the last 150 years. The doubling time is ~40 years.

Super-exponential growth

  • When the growth rate of a quantity increases without bound (e.g 1% one year, 2% the next year, 3% the next year…).
  • One example would be yt+1 = yt(1 +kyt).
  • The time taken for the quantity to double falls over time.
  • Examples:
    • In ancient times it took 1000s of years for GWP to double, but today GWP doubles much faster. GWP doubled between 2000 and 2019.
    • Some solutions to endogenous growth models imply GWP will increase super-exponentially.
    • When univariate endogenous growth models are fit to historical GWP data from 10,000 BCE, they typically imply growth is super-exponential and that GWP will go to infinity in finite time.

Sub-exponential growth

  • When the growth rate of a quantity decreases over time (e.g 1% one year, 0.5% the next year, 0.2% the next year…).
  • One example would be yt+1 = yt(1 +k/yt).
  • Another example is simply linear growth yt+1 = yt + k.
  • The time taken for the quantity to double increases over time.
  • Examples:
    • The world’s population has doubled since 1973, but UN projections imply it will not double again this century.
    • Some solutions to endogenous growth models that imply GWP will increase sub-exponentially. In these models growth ultimately plateaus.
    • When univariate endogenous growth models are fit to historical GWP data from 1950, they typically imply growth is sub-exponential and that GWP will plateau.

Constant returns to scale

  • If the inputs to production all double, the output doubles.
  • For example, suppose output is created by labor and capital. Mathematically, we write this as Y = F(L, K). Constant returns to scale means that F(2L, 2K) = 2Y.

Increasing returns to scale

  • If the inputs to production all double, the output more than doubles.
  • For example, suppose output is created by labor, capital and technology. Mathematically, we write this as Y = F(L, K, A). Increasing returns to scale means that F(2L, 2K, 2A) > 2Y.

Exogenous growth model

  • Growth model where the ultimate driver of growth lies outside of the model.
  • E.g. in the Solow-Swan model growth is ultimately driven by the growth of inputs that are assumed to grow exponentially. The growth of these inputs is the ultimate source of growth, but it isn’t explained by the model.
  • Technological progress is not explained by exogenous growth models.

Endogenous growth model

  • Growth model that explains the ultimate driver of growth.
  • E.g. Jones (2001) describes dynamics governing the increase in population and of technology, and the growth of these inputs is the ultimate source of growth.
  • Typically endogenous growth models explain the growth of technology.

14.1.1 Classifications of growth models

I introduce some of my own terminology to describe different types of growth models.

Long-run explosive models predict explosive growth by extrapolating the super-exponential trend in very long-run growth. I argue they should only be trusted if population is accumulable (in the sense that more output → more people).

Idea-based models explain very long-run super-exponential growth by increasing returns to accumulable inputs, including non-rival technology. They include long-run explosive models and models that have a demographic transition dynamic such as Jones (2001) and Galor and Weil (2000).

Step-change models. These models of very long-run growth emphasize a structural transition occurring around the industrial revolution that increases growth. They stand in contrast to models, like long-run explosive models, that emphasize the increasing return mechanism and predict growth to increase more smoothly over hundreds and thousands of years.

Explosive growth models predict that perfect substitution between labor and capital would lead to explosive growth.

14.2 Models of very long-run growth that involve increasing returns

The purpose of the literature on very long run growth is to understand both the long period of slow growth before the industrial revolution and the subsequent take-off from stagnation and increase in growth.

I focus on two models on very long-run growth – Jones (2001) and Galor and Weil (2000). They are both characterized by increasing returns to accumulable inputs until a demographic transition occurs.Both these models predict super-exponential growth before the demographic transition, and exponential growth after it.

For both models I:

  • Discuss the mechanisms by which they initially produce super-exponential growth, comparing them to the mechanisms of long-run explosive models.
  • Explain how these models later produce exponential growth.
  • Analyze the mechanisms why these models preclude explosive growth, and suggest that highly substitutable AI could prevent these mechanisms from applying.

14.2.1 Jones (2001)

14.2.1.1 The model

There are two accumulable factors in this model: technology A and labor L. There is also a fixed supply of land, T. They are combined together to create output in the following equation:

$$$ Y=A^σ{L_Y}^βT^{1−β} $$$

where LY is the amount of labor spent on producing output (people choose to divide their time between three activities: producing output, doing research, and having children).

Improvements in technology are determined by:

$$$ \dot A=δA^ϕ{L_A}^λ $$$

where LA is the amount of labor spent on doing research, and δ > 0 is a constant. φ describes whether the productivity of research increases (φ > 0) or decreases (φ < 0) with the level of technology; Jones assumes φ < 1. λ allows for diminishing returns to additional researchers: 0 < λ < 1. In equilibrium, the growth rate of A is proportional to the growth rate of L:

$$$ g_A=constant×g_L $$$

Increases in L depend on income per capita, via its effects on the death rate and the birth rate. For a very low level of income per capita, gL = 0. As income rises above this level, gL increases, mostly because as the death rate decreases; as income rises further gL starts decreasing again as the demographic transition reduces the birth rate. So gL as a function of income per capita is an upside-down U.

The general pattern of growth is then as follows:

  • Initially per capita incomes are just high enough for the population to increase very slowly. The rate of technological innovation is very slow at this stage.
  • Eventually, the population increases to a stage where technological innovation is happening somewhat quickly. There is then a powerful positive feedback loop: faster technological progress → larger per capita income → larger population → even faster technological progress →…
  • This feedback loop leads to fast growth of population, technology, and of per capita income.
  • Once per capita income is high enough, the demographic transition sets in, reducing population growth. This stabilizes the growth of technology and per capita incomes, and there is steady exponential growth.

14.2.1.2 Generating super-exponential growth

Jones places a restriction on λ, φ, β, and σ so that the model is characterized by increasing returns to accumulable factors (see p. 9). For example, suppose that φ + λ = 1, so that there are constant returns in the technology production function to accumulable factors. Then Jones’ restriction simplifies to σ + β > 1 – increasing returns in production to the accumulable factors A and L.

These increasing returns allow the model to generate super-exponential growth. In this sense, the model’s mechanism for generating super-exponential growth is very similar to that of Roodman’s model. Both models produce super-exponential growth via increasing returns to accumulable factors.

However, the details of exactly how labor is accumulated differs between Jones’ and Roodman’s models. In Roodman’s model, a constant fraction of output is reinvested to increase the labor supply:220

$$$ \dot L=sY $$$

This implies that the growth rate of labor is proportional to per capita income.

$$$ g_L≡ \frac {\dot L}{L}= \frac {sY}{L} $$$

By contrast, in Jones’ model labor accumulation is more complex. gL is the birth rate minus the death rate. The death rate falls with per capita income. The birth rate initially rises with per capita income because people can achieve subsistence with less work and so have more time to raise children. These combined effects mean that gL initially increases with per capita income.

Although Jones does not have direct proportionality between gL and per capita income, the initial behavior is similar to that of Roodman’s model. In both cases, higher per capita income drives a higher gL which drives increases to gA and gY. The following super-exponential feedback loop is present in both models:

Higher per capita income → higher gL → higher gA and gY → even higher per capita income…

14.2.1.3 Generating exponential growth

In Roodman’s model, the above dynamic continues without limit, and growth becomes ever faster. In Jones’s model, by contrast, gL only increases up to a point (see Figure 2 on p. 12). As per capita income increases further, the birth rate falls because wages are high enough that people choose to work over having children. Moreover, the birth rate falls faster than the death rate, which by now is fairly low. This means that gL decreases. This fall in gL is the model’s way of representing the demographic transition.

As a result of the fall in gL, gA and gY also fall. The model exogenously specifies a minimum for gL (via specifying a minimum death rate and birth rate), which determines the long-run values of gA and gY. The ultimate source of the exponential growth is this exogenous assumption that in the long-run gL tends to a constant.

14.2.1.4 Precluding explosive growth

Explosive growth in this system requires gL to keep rising until it pushes up gA and gY to the point where gL > 30%. The model has two mechanisms that prevent this from happening.

The first we’ve already seen. gL only increases with per capita income up to a point; beyond this point gL falls. This represents the demographic transition.

However, even without this mechanism, there is a limit to how long super-exponential growth could proceed in this model. This limit is the maximum number of children a person can have. People have a finite supply of time, and in the model they must use a fixed amount of time on each of their children. This limits birth rate, and so places another (higher) cap on gL.

It seems unlikely that either of these limits would apply if AI systems were developed that were perfectly substitutable with human labor. In this case, we could increase the effective size of the labor force by creating more AI systems. Roodman’s equation for the increase in the labor supply ( = sY), in which the increase in the stock of generalized labor (generalized labor = human labor + AI labor) is proportional to output, then seems more reasonable. For this is the reinvestment equation commonly used for capital, and AI systems would be a form of capital.

14.2.1.5 Institutions

Jones gets a better fit to the long-run historical data on GDP/capita and population when he models institutions that encourage innovation, like property rights and patents. He crudely represents these with a parameter π that controls the proportional of income paid to researchers. π influences how much effort is made to improve technology. He finds that adding shocks that boost π allows the model to better imitate the sudden rise in living standards around the time of the industrial revolution. Indeed, Roodman’s model is surprised at the speed of growth at this time.

14.2.2 Galor and Weil (2000)

The general model here is very similar to Jones (2001) in several respects.

  • It has accumulable factors technology and labor, and land as a fixed factor.
  • Improvements in technology are caused by people having new ideas. As a consequence, larger populations lead to faster technological progress.
  • Increases in labor are determined by people’s decisions about how to split their time between work and having children.221

One significant addition is that parents can invest in educating their children. The more education, the more human capital per person and the faster the pace of technological progress. There are diminishing returns to education on the pace of technological progress.

The general pattern of growth is as follows:

  • Initially per capita income is low. People must spend most of their time working to achieve subsistence income, so have few children. The supply of labor, level of technology, and per capita income all grow slowly.
  • As per capita income rises, parents can achieve subsistence with less time working, and spend more time having children. The population rises more quickly. This leads to faster technological growth, which in turns leads to faster growth in per capita income.
  • There is a positive feedback loop: higher per capita income → more people → faster technological progress → even higher per capita income →…
  • Once technological progress is fast enough, parents are incentivized to have fewer children. This is because they’re instead incentivized to invest time in their children’s education.
    • This causes growth to increase more quickly for a while with the following feedback loop: faster technological progress income → better educated people → faster technological progress →…
    • Eventually this leads population growth to decline: the demographic transition. When population growth declines to 0, the amount of human capital and rate of technological progress are also constant.222

14.2.2.1 Generating super-exponential growth

Galor and Weil (2000) generates super-exponential growth in an analogous fashion to Jones (2001). As in Jones (2001), the equations by which the factors are accumulated are characterized by increasing returns. Once both endogenous inputs (technology and population) have doubled, the growth rates of both these inputs increase.

The super-exponential feedback loop is roughly as follows:

Higher per capita income → higher L → higher gA and gY → even higher per capita income…

Around the industrial revolution, educational investment raises human capital per person and leads to faster increases in growth:

Higher gA → more educational investment → higher human capital per person → higher gA

14.2.2.2 Generating exponential growth

We’ve touched upon the mechanism that generates constant exponential growth. There is a negative feedback loop that returns the growth rate of technology to a fixed point. In brief, the feedback loop is:

Faster growth → smaller population → lower growth

Slower growth → larger population → faster growth

Why the link from growth to population? Parents have to decide whether to spend time on having more children or on educating them; between having fewer better-educated children versus more worse-educated children. They make this choice to maximize the total income of their children (but not their children’s children). A higher growth rate of technology increases the value of education in the market, and so shifts incentives towards fewer better-educated children. Fewer children then reduces the rate of technological growth. The same negative feedback loop happens in reverse if technological growth is too low.

Faster growth → incentive to have fewer children → population falls → slower growth

Slower growth → incentive to have more children → population rises → faster growth

In equilibrium we have:

Equilibrium growth → fixed incentives to have children → population constant → constant growth

This negative feedback loop returns the growth rate of technology to a fixed point and then keeps it there.223

14.2.2.3 Precluding explosive growth

The same negative feedback loop as discussed in the last section explains why super-exponential growth is avoided. If growth ever became too high, the population would decrease until growth settled back down again.

As with Jones (2001), it doesn’t seem like this mechanism would apply if AI systems were developed that were perfectly substitutable with human labor. In this case, we could increase the effective size of the labor force by creating more AI systems, and then increases in labor wouldn’t be limited by the finite amount of time that parents’ can give to childbearing.

Again, Roodman’s equation for the increase in the labor supply ( = sY), in which the increase in the stock of generalized labor (= human labor + AI labor) is proportional to output, seems more reasonable in this hypothetical.

14.3 Graphs showing frontier GDP growth

14.3.1 Summary

There isn’t good quality long-run data on the economic frontier because the frontier changes over time, and old data points are highly uncertain. Here I eyeball data for the USA, England, and France.

The data looks as if growth is super-exponential if you look at data going back to 1700 or earlier. However, when you remove data before 1900 the trend looks roughly exponential.

14.3.2 Graphs of super-exponential growth in frontier GDP/capita

14.3.2.1 United States (source)

14.3.2.2 England (source)

 

14.3.2.3 France224

 

EconomicGrowthV.png

 

The next two sections analyze the US and English data in a little more detail.

14.3.3 US per capita GDP growth

US per capita growth from 1650 looks super-exponential (source). Constant exponential growth would look like a straight line as the y-axis is log.

EconomicGrowthQ.png

 

Even from 1800, there is a trend of super-exponential growth. The red line shows the average growth rate from 1800 – 1890; the blue line shows the average growth rate since then. It looks like growth sped up throughout the 19th century and then maintained at a constant rate.

 

EconomicGrowth12.png

 

However, if you restrict the data set to data from 1870 it looks exponential. You can see the slowing growth from 2000.

 

EconomicGrowthL.png

 

This pattern of slowing growth since 2000 is confirmed by the data in Vollrath’s recent book, Fully Grown, and by the following data from the world bank (source).

EconomicGrowthE.png

 

14.3.4 England total GDP growth – sensitivity analysis

Long-run English GDP since 1300 looks super-exponential (source):

EconomicGrowth14.png

 

It’s still super-exponential if you exclude data before 1700:

EconomicGrowth1.png

 

However, if you restrict the data to post-1800, the super-exponential trend disappears. The trend is well approximated by exponential growth – see the red line. Notice, though, that the average growth rate after WW1 (blue line) is faster than that before it (red line):

EconomicGrowthC.png

 

14.4 Graph of GWP per capita

Data is from Roodman (2020) – see Table 2.

14.5 Graphs of population growth

14.5.1 Frontier population

US and UK data shows a slight slowing of population growth.225 This may be offset by more countries joining the economic frontier.

US and UK separately:

EconomicGrowth13.png

 

US and UK combined:

 

EconomicGrowthP.png

 

14.6 Uncertainty about global population vs uncertainty about per capita GWP

This section argues that, for the standard story, uncertainty about GWP/capita is a much bigger source of uncertainty about 2100 GWP than uncertainty about population.

The following plot shows the GWP projections for various assumptions about GWP per capita growth and future population. In particular, I compare projections for the 10th and 90th percentiles of GWP per capita growth and the 5th and 95th percentiles of population. Uncertainty about population affects the GWP projections much less than uncertainty about per capita GWP growth.

 

EconomicGrowth2.png

 

14.7 Endogenous and exogenous growth models

14.7.1 What’s the difference between endogenous and exogenous growth models?

In brief, exogenous growth models stipulate the rate at which technology A changes over time and the growth of technology is the ultimate source of growth in these models. By contrast, endogenous growth explains the ultimate source of growth, often by explaining the increase of technology.

I’ll illustrate this difference between endogenous and exogenous growth models by comparing the standard exogenous Solow-Swan model with the endogenous ‘learning by doing’ model of Frankel (1962).

Both models use the same Cobb-Douglas production function (though with different parameter values):

$$$ Y=A^σK^αL^{1−α} \, (0) $$$

where:

  • Y is the total output.
  • A is technology.
  • L is the labor input.
  • K is the capital input.

Both models treat K as endogenous: production Y is invested into increasing K:

$$$ \dot K=s_KY−δ_KK \, (1) $$$

where sK jointly represents both the proportion of Y that is invested into capital and how much capital that amount of investment is able to produce, and δK is the rate at which capital loses value due to depreciation. There is a feedback loop between Y and K where Y is invested to increase K (Equation 1), which in turn increases Y (equation 0), which further increases investment in K, and so on.

Let’s consider the two growth models in turn.

In the exogenous Solow-Swan model, σ = 1 – α so the production function is:

$$$ Y=K^α(AL)^{1−α} $$$

Both A and L are assumed to grow at a constant exponential rate:

$$$ \dot L=L_0e^{nt} $$$

$$$ \dot A=A_0e^{gt} $$$

The feedback loop between Y and the endogenous factors – in this case just K – fizzles out due to the diminishing returns to Y from increases in the endogenous factors. In this model, these diminishing returns correspond mathematically to.226

As a result, the long-run growth rate of Y is n + g and the long-run growth rate of per capita income Y / L is g. Long-run growth is constant because n and g are constant. The constancy of the long-run growth rate is not explained by exogenous growth models, but is rather assumed via their stipulations about A and L.

Endogenous growth models allow the rate of technological progress to be determined within the model, e.g. by investment in R&D. There are many endogenous growth models, but I’ll use just one example to demonstrate.

In the endogenous growth model of Frankel (1962)227, σ = 1 and so the production function is:

$$$ Y=AK^αL^{1−α} $$$

Crucially, technological progress is endogenous and happens through a process of ‘learning by doing’. As capital is accumulated, technology improves as a by-product. The current level of technology A depends on the total amount of capital K that has been accumulated:

$$$ A=A_0K^η $$$

Technological progress happens (indirectly) as a result of the investment of output, rather than being exogenous to the model. The constant η controls the marginal returns to technology from accumulation of capital. If η > 1, each successive increment to K increases A by a larger and larger amount. If η < 1, there’s diminishing returns. Subbing in the expression for A into the production function, we get:

$$$ Y=A_0K^{α+η}L^{1−α} $$$

Labor is treated as exogenous. It turns out that in this model, in contrast to the Solow-Swan model, the long-run growth rate can depend on the actions on the rate of investment in capital sK228

14.7.1 .1 The conditions for super-exponential growth in Frankel’s simple endogenous growth model

We saw that in Frankel’s model the production function is

$$$ Y=A_0K^{α+η}L^{1−α} \, (3) $$$

where α < 1 gives the diminishing marginal returns of K to output, and η control the marginal returns of K to technology, A = A0Kη. K can increase due to the investment of output:

$$$ \dot K={s_K}Y−{δ_K}^K \, (4) $$$

It turns out that Y and K grow super-exponentially if α + η > 1. To understand why, substitute (3) into (4) and then divide both sides of (4) by K to get K’s growth rate, /K:

$$$ \dot K/K=A_0K^{α−η−1}L^{1−α}−δ_k $$$

$$$ \dot K/K=A’K^{α−η−1}−δ_k \, (5) $$$

Here I have defined A = A0L1-α to make the dependence on K clearer.

From (5), we see that if α + η > 1, K’s growth rate increases whenever K increases. This is what is meant by super-exponential growth. Intuitively, we have a strong feedback loop between K and Y. Y is invested to increase K, which in turn increases Y , which in turn increases investment into K, and so on. This feedback loop doesn’t peter out but gets more and more powerful as there are increasing returns to Y from increments in K. This is due to K’s dual role as a direct input to production and in increasing the level of technology.

14.8 If we believed Frankel’s model, the striking constancy of 20th century growth wouldn’t convince us that long-run growth was constant

In Frankel’s model we have:

$$$ Y=AK^α(BL)^{1−α} $$$

Technology progress from ‘learning by doing’ is given by:

$$$ {B}= (\frac {K}{L})^γ $$$

For simplicity, let’s assume L = 1. This implies:

$$$ Y=AK^{α+γ} $$$

Exponential growth requires the knife-edge condition α + γ = 1.

If we truly believed in Frankel’s endogenous growth model and fitted it to data on 20th century US growth, we would conclude that the equality was very nearly satisfied. But we couldn’t use the data to distinguish the possibilities that i) growth is exactly exponential, ii) growth is slightly super-exponential, iii) growth is slightly sub-exponential. With any natural prior over the values of α and η our posterior would assign much less probability to option (i) than to (ii) or (iii).229 If we then extrapolated growth into the future then our probability in (ii) and (iii) would lead us to attach significant weight to explosive growth eventually occurring and significant weight to growth eventually reaching a plateau.

14.9 Roodman’s endogenous growth model

14.9.1 Description of Roodman’s univariate stochastic model

The starting point for Roodman’s univariate endogenous growth model is:

$$$\dot Y=sY^{1+B}+δY \, (6) $$$

where:

  • Y is output, in this case GWP.
  • s jointly describes the proportion of output invested into increasing future output and the effectiveness of this investment.
  • δ is the depreciation rate of output.
  • B controls whether growth is sub- or super-exponential. Growth is super-exponential is B > 0, sub-exponential if B< 0 and exponential if B = 0.

Roodman augments (6) through the use of stochastic calculus, which models the randomness in the change of Y. This introduces an additional term W(t), a random walk whose cumulative variance at t units of time equals t (see Roodman’s paper for more details):

$$$ \dot Y=sY^{1+B}+δ Y+σ \sqrt {YY^{1+B}} \dot W \, (7) $$$

Notice that if B = 0, the amount of randomness in Y’s evolution is proportional to Y.230

So there is an element of randomness in determining . This randomness has a persistent effect on the subsequent trajectory. For example, if GWP is boosted by chance in 1800, we would not expect it to regress back to its previous trendline in 1850, but rather to continue to rise at the normal rate from its 1800 value. To put this another way, GWP is modeled as a Markov process where the next GWP value depends only on the current value.

14.9.2 How to quantify how surprised the model is by each data point

One of the advantages of Roodman’s model is that we can quantify how surprised the model is by each data point, conditional on the previous data points. Suppose we wanted to test how surprised the model is by GWP in 1700. First, we estimate the model parameters using only previous data points, up to 1600. Second, we calculate the probability distribution over GWP in 1700 conditional on the observed GWP in 1600. This probability distribution represents the model’s prediction for GWP in 1700 given all the previous data points. Lastly, we compare the actual GWP in 1700 to this probability distribution. If the actual GWP is higher than the 90th percentile of the probability distribution, the model is surprised at how high GWP was in 1700. If it’s lower than the 10th percentile, the model is surprised at how low it is. If the actual GWP is close to the distribution’s median, it isn’t surprisingly high or surprisingly low.

14.9.3 Graph of how surprised Roodman’s model is by French GDP/capita data

EconomicGrowthK.png

 

The model is surprised by how slow growth is in all the data points after 1870.

14.10 Some reasons Roodman’s model’s may underestimate the time until explosive growth occurs

In this section I discuss some technical features of Roodman’s model that lead it to predict explosive growth in just a few decades. These objections motivate the analysis above that explores the robustness of this prediction to changing these features.

14.10.1 Roodman’s model is overly surprised by the industrial revolution and by slow modern day growth because it assumes that random influences on growth at nearby points in time are uncorrelated

One of the advantages of Roodman’s model is that we can quantify how surprised the model is by each data point.231 Suppose we wanted to quantify how surprised the model is by the value of GWP observed in 1700. We would use the model to generate a probability distribution over 1700 GWP, conditional on all the previous data points. If the actual GWP in 1700 is higher than the 90th percentile of this probability distribution, the model is surprised at how high GWP was in 1700. If actual GWP is lower than the 10th percentile, the model is surprised at how low it is. If the actual GWP is close to the distribution’s median, it isn’t surprisingly high or surprisingly low.

Figure 13 of Roodman’s paper shows the percentile of each observation from 1600, each conditioning on the previous ones:

 

EconomicGrowthF.png

 

The observations in 1820, 1870 and 1913 are all above the 90th percentile, so the model is consistently surprised by how large growth is in this period. The observations in 1990, 2000, 2010 and 2019 are all below the 30th percentile, indicating that the model is consistently surprised by how low growth is in this period. The correlation between the model’s surprise in successive data points in time is striking.

Part of the reason for this surprise is that the random component of the model does not account for the serial correlation in the random fluctuations affecting successive data points. For example, after the model sees surprisingly high growth in 1820 and 1870 it does not think ‘whatever caused this recent surprisingly high growth might affect the next observation in the same way’; instead it recalculates the random component for the next observation from scratch. This leads it to be consistently surprised by successive observations, rather than adjusting its expectations. The low-frequency econometric methods discussed above can model this kind of serially-correlated deviation from an underlying trend.

Not modeling serial correlation has an impact on the model’s projections of GWP into the future. There are two effects on its GWP projections.

Firstly, the model will not infer from the surprising slowdown in growth observed in the last 50 years that ‘whatever caused this recent slowdown232 might affect the next observation in the same way’. Rather, the model will treat these recent deviations from its predictions as unrelated to future values of GWP. As a result, it will predict explosive growth sooner than if it had taken serial correlation into account in some way. This problem is highlighted by the model’s median prediction for the 2020 growth rate: 7.1%.

One way to think about this problem is that Roodman’s model expects growth to increase for two reasons:

  1. Recent growth is surprisingly low, as judged by the other data points to which the model has been fitted.
  2. Growth tends to increase as GWP increases.

Factor 1 causes the median projected growth to jump immediately up to 7.1%, Factor 2 then causes it to increase to 30% by 2044. The next section explores another model in which only Factor 2 plays a role.

The second consequence of not modeling serial correlation is that the model’s uncertainty intervals around future values of GWP are too narrow. Assuming that randomness is uncorrelated between two successive periods reduces your probability that the same extreme outcome will occur in both periods. As a result, you’ll underestimate the probability of these extreme outcomes. To correct for this mistake, you should widen your uncertainty intervals so that they include more extreme outcomes in both directions. Indeed, the model’s confidence intervals do seem too narrow. Its 80% confidence interval for the first year of explosive growth is [2033, 2063].

14.10.2 Roodman’s model doesn’t account for the fact that recent data points are more relevant to future growth than ancient data points

When Roodman estimates his model parameters, he downweights ancient data points to account for our uncertainty about their true values. However, he does not further discount them on the basis that patterns of growth in ancient times are less likely to be relevant to 21st century growth than patterns of growth in modern times. But this additional downweighting seems reasonable. For example, it is possible that in ancient times growth was super-exponential but that we’ve recently moved into a region of sub-exponential growth. To correct for this we could weigh more modern data more heavily when estimating the model parameters, or find some other way for the model to put more weight on recent data points.

14.11 Growth multiplier model

14.11.1 Detailed explanation of how the growth multiplier model works

The model takes as its starting point an insight from Roodman’s univariate endogenous growth models: each time GWP increases by a factor of r, the growth rate should be multiplied by some number.

To see this, consider Roodman’s univariate endogenous growth (before he adds in randomness):

$$$ \dot Y=sY^{1+B}+δY $$$

Rearrange this to find the growth rate /Y as a function of GWP Y:

$$$ \dot Y/Y=sY^B+δ \, (8) $$$

When Roodman estimates the parameters for this model he finds that B > 0 > 0 and the value of δ is extremely small233. As a result, the contribution of δ to the growth rate for modern day values of Y is negligible (see previous footnote). We can simplify the equation:

$$$ \dot Y/Y=sY^B \, (9) $$$

If Y increases by a factor of r, the growth rate is multiplied by rB. Using Roodman’s estimated parameters, and letting r= 2, growth increases by a factor of 20.55 = 1.46. So Roodman’s model predicts that when GWP doubles the growth rate will on average increase by about 46%234, although the randomness in his model means the exact amount will vary. In the terminology introduced above, the average value of the growth multiplier is 1.46. This is the basic driver of super-exponential growth in Roodman’s model.

This is the basis for the growth multiplier model splitting up time into periods in which GWP changes235 by a factor r, and calculating the difference of growth rates between successive periods as the ratio between their growth rates. Roodman’s model suggests that the growth rates should increase by some multiple between periods so-defined.236

The growth multiplier model departs from Roodman’s model in how it models randomness. Rather than calculating new and independent random contributions each infinitesimal timestep, it models randomness crudely by sampling growth multipliers from the historical data. In essence it asks the question ‘What will the ratio be between this period’s growth rate and that of the next period?’ and answers ‘Let’s sample randomly from the values of that ratio for analogous periods throughout history’.

Here is a step by step description of how to implement this model for the case of r = 2.

  • Create a shortened version of Roodman’s full GWP dataset where each data point’s GWP is twice that of the preceding data point, and the last data point is 2019.
    • Here is a picture of the final three rows of the resultant dataset:
      EconomicGrowthO.png

       

    • The full dataset has 16 rows and goes back until 5000 BCE; GWP halves each time until its 5000 BCE value of 2.02.
  • Calculate the average annualized growth rate between each pair of rows:
    EconomicGrowth8.png

     

    • The last row reads ‘NaN’ because we don’t know what the average growth rate will be between 2019 and when GWP is twice its 2019 value.
  • Calculate the ratio between each pair of successive growth rates. Each ratio is essentially a sampled value for the multiplier on growth rates 2B when GWP doubles:
    EconomicGrowth10.png

     

    • The average growth rate for 2000-2019 was 1.19 times higher than for 1977-2000. So the growth multiplier for the period starting in 1997 is 1.19.
    • The growth rate and growth multiplier of each row combine to give the growth rate of the next row.
  • Extrapolate GWP for the period starting in 2019.
    • First calculate the growth rate of the period starting in 2019. Take the previous period’s growth rate (0.04), randomly select a growth multiplier (using your discount rate to increase the probability of selecting more recently observed multipliers), and multiply them together. Suppose we selected a growth multiplier of 1.25, then the new growth rate is 0.04 × 1.25 = 0.05.
    • Then calculate the length of the period starting in 2019. This can be calculated using the formula237 ln(r) / ln(1 + g), in our case ln(2) / ln(1 + 0.05).
    • The GWP at the end of the next period is 2 × (GWP in 2019).
  • Repeat the above bullet for the next period (the one following the period starting in 2019). Repeat for as many periods as you wish.

14.11.2 Models that display a degree of serial correlation intermediate between Roodman’s model and the growth multiplier model

While in Roodman’s model the growth of successive periods are completely unrelated (leaving aside the effect of changing GWP238), in the growth multiplier model a period’s growth affects growth in all subsequent periods in equal measure. These two models can be seen as occupying opposite ends of a spectrum; an intermediate case would be that a period’s growth has a diminishing influence on the growth of future periods.

One concrete example of such an intermediate model would be a version of Roodman’s model with a different random component. We can think of Roodman’s model as sampling the random contribution to the growth rate from a Normal distribution in each timestep (this is how the randomness is implemented in practice). So Rt = N(0, σ). Instead, the random contribution in time step t could in part depend in part on the random contribution of the previous timestep: Rt = εRt – 1 + N(0, σ), with 0 < ε < 1. The constant ε can be adjusted to control the size of serial correlation, the larger it is the larger the degree of serial correlation. The techniques of low frequency forecasting, discussed in the main body, might be appropriate for constructing a model of this kind.

In this terminology, the growth multiplier model is conceptually similar to having Rt = Rt – 1 + N(0, σ). This is because in the growth multiplier model, and in this equation, the following hold:

  • The growth rate in period i has a persistent effect on the growth of all subsequent periods that does not decay over time.
  • As a result, the expected size of the deviation of the non-random element of Roodman’s model increases over time. This deviation behaves as a random walk with no tendency to return to 0.

14.11.3 Adding sampled growth multipliers to the list of historically observed growth multipliers

One interesting, and I suspect controversial, feature of the model is that each time a growth multiplier is sampled it is added to the list of historically observed growth multipliers. It is then more likely to be resampled when calculating future periods’ growth rates. In the example in the text, if we sampled gm = 1.5, then the next time we sampled it would be from the list [2, 2, 1.5, 0.5, 1.5].

The intuitive reason for this is that if we observe (for example) slowing growth during the next period, this should increase our probability that the period afterwards will again contain slowing growth. And if we observe slowing growth for the next five periods, our confidence in growth continuing to slow in the sixth period should be higher still. And if we ask the model now ‘what is it reasonable to believe about GWP, conditional upon five periods of slow growth’ its predictions for the sixth period should take into account those five periods even though they have not actually happened. By adding observed growth multipliers to the list, I ensure that the five periods of slowing growth are taken into account in this scenario.

More formally, I want the model to satisfy the following desideratum. The model’s current prediction, conditionalized on growth of X in the next period, is the same as what the model would predict if X growth actually happened and the model was retrained with the extra data from X. I’ll motivate this desideratum with the same example as above. Suppose we observed five periods of slowing growth and tried to extrapolate GWP with this model. We would of course include the data of the most recent five periods, and this would influence the model’s predictions. If we then ask our current model ‘What should we believe about future GWP conditional on the next five periods containing slowing growth’ our model should give the same answer.

The strongest case for the desideratum comes from an extreme case. Roodman’s model assigns a very tiny probability to GWP staying roughly constant over the next million years. But if you condition on this extreme case and then extrapolate GWP further into the future, the model predicts that GWP would almost certainly start growing quickly again afterwards. In fact, its predictions are identical to if you had simply conditioned on the historical data. Its conditional predictions are only based on historically observed data, not on the data we’ve asked the model to conditionalize on. This makes the model’s conditional predictions unreasonable in this extreme case. But even in non-extreme cases I think it makes the model’s uncertainty intervals too narrow. I believe my desideratum prevents the growth multiplier model from making unreasonable conditional predictions and appropriately increases the uncertainty of its predictions.

If I remove this controversial feature, and only sample from the actual historical data, the main result is to narrow the model’s confidence intervals. With my other preferred inputs, the date by which there’s a 10% chance of explosive growth goes back two years from 2036 to 2038; the date by which there’s a 70% chance of explosive growth moves forward 30 years from 2200 to 2147; the probability that explosion never happens goes from ~15% to 0%. The median date of explosive growth comes forward by only three years from 2093 to 2090.

14.11.4 Sensitivity analysis

14.11.4.1 How does r affect predictions without a discount rate?

EconomicGrowth15.png

 

14.11.4.2 How does r affect predictions with a discount rate of 0.95?

 

EconomicGrowth7.png

14.11.4.3 How does r affect predictions with a discount rate of 0.9?

EconomicGrowthU.png

14.11.4.4 How does r affect predictions with a discount rate of 0.85?

 

EconomicGrowth4.png

 

14.11.4.5 Median singularity date by r and discount rate

I’ve highlighted my preferred inputs and preferred output in bold.

DISCOUNT RATE R = 2 R = 1.6 R = 1.3 R = 1.1
1 2071 2084 2087 2073
0.95 2072 2088 2096 2084
0.9 2075 2093 2108 2098
0.85 2081 2105 2130 2116
0.8 2090 2130 2170 2141
0.75 2100 2173 2282 2182
0.7 2105 2274 3756 2228

I don’t trust values of r < 1.5 for two reasons.

  1. We would need to know how the average growth rate changed when GWP increased by a factor of (e.g.) 1.3. But the historical GWP data is often too coarse-grained to contain data about this for some periods. For example, within somd periods each GWP data point is 1.5X as large as the previous data point; within such periods there’s no information on how growth changed when r increased by less than a factor of 1.5. We have to interpolate the GWP data, assuming that the growth rate didn’t change in these periods.
  2. As r becomes smaller, we’re more likely to pick up the effects of business cycles that aren’t relevant to the economy’s potential for long-term growth. Such cycles involve growth increasing and then immediately decreasing again; these changes are negatively correlated such that they cancel out of the medium term. But the growth difference model will treat these changes as uncorrelated (it samples randomly from the growth multipliers), and will subsequently overestimate the propensity for the growth rate to significantly change.

 

14.12 Elaborations on objections to long-run explosive models

14.12.1 How do the predictions of the explosive growth story change if we omit old data points?

I did a sensitivity analysis on the effect of removing ancient data points. The following table summarizes the predictions of Roodman’s model and the growth multiplier model for data sets that begin at various different times.

EARLIEST DATA POINT
-10,000 BCE -2000 BCE 1 CE 1000 CE 1300 CE 1600 CE 1800 CE
Roodman (2020) 50% 2043 2046 2051 2091 2098 2171 2213
Growth differences model 10% 2036 2035 2038 2037 2037 2043 2059
50% 2093 2090 2092 2082 2089 2117 2302
90% Never Never Never Never Never Never Never

[Note: Roodman’s model is not a good fit to the data sets starting in 1300, 1600 and 1800239; small changes in the model, data or optimization methods might change its predictions by several decades (see most recent footnote for an example). Similarly, using a different value of r in the growth multiplier model can change the result by several decades.]

The prediction of super-exponential growth is robust to removing data points until 1800; the prediction of explosive growth by 2100 is robust to removing the data points until 1300. Again, if you made further adjustments based on thinking AI won’t increase growth for several decades, this would cause a further delay to the predicted date of explosive growth.

14.12.2 You’re only predicting explosive growth because of the industrial revolution

The step-change hypothesis vs the increasing-returns mechanism

Summary of objection: Yes, in ancient times growth rates were very low. And yes, in modern times growth rates are much higher. But this is just because the industrial revolution caused a step-change in growth rates. There’s no persistent trend of super-exponential growth beyond the one-off change caused by the industrial revolution. So the explosive growth story is wrong.

Response: I find this objection plausible but indecisive.

We can compare two hypotheses.

  1. Growth increased gradually – ‘smooth increase’ hypothesis.
    • This is implied by the increasing-returns mechanism in long-run explosive models.
  2. There was a one-off discrete increase around the industrial revolution – ‘step change’ hypothesis.

If you eyeball the long-run GWP data, it doesn’t look like there is a step change in growth. I calculated the annualized growth rate between successive GWP data points, and looked at how this changed over time.240

EconomicGrowthR.png

 

It looks as if growth increased fairly steadily until the second half of the 20th century. That said, the data is highly uncertain, and it’s definitely possible that the true data would show a step-change pattern. Especially if the step-change is understood as having occurred over a few centuries.

A similar story seems to be true of GWP per capita and frontier GDP per capita, although this is much harder to discern due to the lack of long-run data. Here’s GWP per capita.

EconomicGrowthN.png

 

This data is even more uncertain, and so again the true data could show a clearer step-change pattern.

Here’s French GDP per capita – a (flawed) proxy for GDP per capita on the economic frontier.241

EconomicGrowthV.png

 

This data shows something like a step-change around 1800.

Overall, while it looks to me that the data fits better with the ‘smooth increase’ hypothesis, the data are highly uncertain and provide very little evidence between the two hypotheses.

Note, there may be other reasons to prefer the ‘smooth change’ theory. It’s implied by the plausible-seeming increasing-returns mechanism that features in idea-based theories. Further, even if you accept the step-change hypothesis, I suggest you should still not rule out explosive growth (see more).

The following section provides a little further evidence that the data supports the smooth change theory. Even excluding data before/after the industrial revolution, there seems to be evidence of increasing growth.

14.12.2.1 Investigation: Is super-exponential growth only due to the industrial revolution?

I investigated whether the prediction of explosive growth was robust to omitting data before and after the industrial revolution. In particular, I fit Roodman’s model and the growth multiplier model on pre-1600 data and post-1800 data. The following table summarizes the predictions of both models about these two data sets.

WHEN IS THERE AN X% CHANGE OF EXPLOSIVE GROWTH?
DATA SET
10,000 BCE – 2020 10,000 BCE – 1600242 1800 – 2020
Roodman (2020) 50% 2043 2951 2213
B243 0.55 0.18 0.03
Growth differences model 10% 2036 2054 2059
50% 2093 2166 2302
90% Never Never Never

The data shows growth increasing either side of the industrial revolution, but not fast enough for Roodman’s model to predict explosive growth by 2100.

(Note: it is not surprising that Roodman’s model predicts explosive growth happening eventually. As long as growth has increased on average across the data set, the model will find super-exponential growth and predict an eventual singularity (growth going to infinity).)

The plot below shows pre-1600 GWP vs growth data. It does not look like an exponential model would be a good fit to this data (it would be a horizontal line) and indeed Roodman’s model assigns virtually no weight to values of B close to 0.244 To me, it seems that a super-exponential curve is a natural fit. If we treat the cluster at the bottom right as anomalous, it looks as if Roodman’s model might be a better visual fit when it’s estimated with the full dataset (orange line) than with just pre-1600 data (blue line).

EconomicGrowthG.png

 

If you do not trust the GWP data (see earlier objection) you should be particularly hesitant to accept the suggestion that super-exponential growth is present in the pre-1600 data. See this write up by Ben Garfinkel for a more detailed investigation into whether the pre-industrial era contains evidence of super-exponential growth. He finds very little evidence of smooth super-exponential growth and he views the pre-industrial data he used as highly unreliable.

14.13 Possible explanation for exponential growth in Perretto’s model

The key question about this model from the perspective of this report is:

Why does N grow just fast enough to curb the explosive growth potential of Z, but not fast enough to make long-run growth sub-exponential (tending to 0 in the long run)?

My best attempt is the following argument that by the time Z has doubled, the technology investment per firm also doubles, and so Ż doubles. This implies exponential growth.

Here’s the argument:

In the model the cost of creating a new firm must always be equal to the firm’s total revenue (more precisely, the discounted revenue stream that a firm provides). If the costs are lower than this, more firms will be created, lowering the per-firm revenue. (Although having more firms increases total output, it decreases output per firm and so decreases revenue per firm.) So in market equilibrium, the costs equal the revenue.

The cost of creating a firm is assumed to be proportional to Z. So we can argue as follows: Z doubles → the cost of creating a firm doubles → the revenue from each firm doubles.

Now, further assume that each firm invests a constant fraction of its revenue in technology investment (this only happens if this investment maximizes their profits; for now we just assume it). Then we can argue: the revenue from each firm doubles → per firm investment in technology doubles → Ż doubles.

Putting these arguments together we get: Z doubles → the cost of creating a firm doubles → the revenue from each firm doubles → per firm investment in technology doubles → Ż doubles. In other words Zdoubles → Ż doubles. This implies that Z grows exponentially.245

14.14 Toy model where favourable R&D returns in an important subsector drives explosive growth in GWP

There are two types of technology, standard technology A and investment technology Ainv. A plays the normal role in goods production, while Ainv governs the efficiency of capital investment. Capital is divided equally between developing both types of technology.

$$$ Y=AL $$$

$$$ \dot A=A^ϕ (\frac {K}{2}) $$$

$$$ \dot A_{inv}=A^{ϕinv} (\frac {K}{2}) $$$

$$$ \dot K=sA^{ϕinv}L $$$

If φinv > 0, the latter two equations are sufficient to generate super-exponential growth in Ainv and K. This then drives super-exponential growth in A and Y via the first two equations, no matter what the value of φ.

Informally, we can think of K as representing the number of AI systems and Ainv as the efficiency with which these systems can be created. Concretely, Ainv might relate to the level of hardware (‘how much memory and how many calculations per computer chip?’) and the level of software (‘how much memory and how many calculations do you need to run your AI?’). Then the story behind these equations is as follows. Investment in hardware and software (Ainv) causes explosive growth in the number of AIs (K). This drives explosive growth in all areas of technology (A) and so in GWP (Y).

14.15 Mathematical derivations of conditions for super-exponential growth

Section written by Guilhermo Costa.

Note that the derivations below specify the conditions for some of the models discussed in Appendix C to present sustained super-exponential growth; a couple of these models might possibly have short periods of super-exponential growth if the conditions below are not met. Throughout we also assume sufficient saving.

Additionally, some of the following proofs only hold if the ‘fishing out’ effect dominates the ‘standing on shoulders’ effect, so that, as the stock of technology increases, technological progress gets harder on the margin. Mathematically, this is represented by the parameter φ being smaller than 1.

14.15.1 Kremer (1993)

$$$ Y=AL^αW^{1−α} $$$

$$$ \dot A=δA^ϕL^λ $$$

$$$ \bar y=\frac {Y}{L}=constant $$$

Let us analyze the growth rate of output gY:

$$$ g_Y=g_A+αg_L= \frac {δL^λ}{A^{1−ϕ}}+αg_L $$$

In order for output per capita to remain constant, gY = gL and thus:

$$$ (1−α)g_Y= \frac {δL^λ}{(Y/L^αW^{1−α})^{1−ϕ}} $$$

Since ȳ is constant, YL , and so gYYλ – (1 – α)(1 – φ).

Therefore, the condition for super-exponential growth is:

$$$ λ>(1−α)(1−ϕ) $$$

In the case in which we are usually interested, φ < 1, we can rewrite the above condition as:

$$$ α> \frac {1−λ}{1−ϕ} $$$

14.15.2 Roodman (2020)

$$$ Y=AK^αL^βW^{1−α−β} $$$

$$$ \dot K=s_KY−δ_KK $$$

$$$ \dot L=s_LY−δ_LL $$$

$$$ \dot A=s_AA^{ϕA}Y−δ_AA $$$

We set W = 1 for simplicity. Roodman (2020) expresses this model using vectors v=(A,K,L),v=(1,α,β) and φ=(φA,0,0) and finds the following sufficient condition for growth to be super-exponential:246

$$$ \overrightarrow α \cdot \overrightarrow ϕ+(1−ϕ_0)(\overrightarrow α \cdot \overrightarrow u−1)>0 $$$

where u=(1,1,1) and the vectors are zero-indexed. Evaluating the dot products, we obtain:

$$$ ϕA+(1−ϕA)(α+β)>0 $$$

Taking φA < 1, we can rewrite the above condition as:

$$$ α+β> \frac {−ϕ_A}{(1−ϕ_A)} $$$

14.15.3 Hanson (2001)

$$$ Y=(AK)^αL^βW^{1−α−β} $$$

$$$ \dot K=sY−δK $$$

$$$ A=A_0e^{gAt} $$$

This model eventually settles into a balanced growth path, so we analyze how that path changes as the level of automation increases.247

Balanced growth requires:

$$$ g_K= \frac {sK}{Y}−δ=constant⇒g_Y=g_K $$$

Substituting the above into the expression for the growth rate of output:

$$$ g_Y=α(g_A+g_K)+βg_L⇒g_Y= \frac {αg_A+βg_L}{1−α} $$$

We now take gL = 0 for simplicity. The paper models automation as increasing the capital share and reducing the labor share correspondingly:

$$$ α→α’=α+fβ,β→β’=(1−f)β $$$

The long-run growth rate increases as follows:

$$$ g_K→{g_K}’= \frac {α’g_A}{1−α’}= \frac {(α+fβ)g_A}{1−α−fβ} $$$

14.15.4 Nordhaus (2021)

$$$ Y=F_ρ(AK,L)=[(AK)^ρ+L^ρ]^ \frac {1}{ρ} $$$

$$$ \dot K=sY−δK $$$

$$$ A=A_0e^{gAt} $$$

In the case ρ = 0, this model reduces to that found in Hanson (2001), so we focus on the cases ρ < 0 and ρ > 0.

14.15.4.1 Case #1 — ρ < 0

Let ν = |ρ| = – ρ. Writing the production function in terms of ν, we obtain:

$$$ Y= \frac {1}{[( \frac {1}{AK})^ν+ ( \frac {1}{L})^ν]^{ \frac {1}{ν}}} $$$

As labor is fixed but technology grows exponentially, eventually AKL and thus:

$$$ Y≈ \frac {1}{ [\frac {1}{L^ν}]^ {\frac {1}{ν}}}=L $$$

Therefore, growth is sub-exponential and output stagnates.

14.15.4.2 Case #2 — ρ > 0

Once again using the fact that AK &Gt L eventually, we obtain:

$$$ Y≈[(AK)^ρ]^{ \frac {1}{ρ}}=AK $$$

The growth rate of capital is given by:

$$$ g_K= \frac {sY}{K}−δ=sA−δ=sA_0e^{gAt}−δ $$$

and thus growth is super-exponential in the long-run, as the growth rate itself grows exponentially.

14.15.5 Aghion et al. (2017)

 14.15.5.1 Cobb-Douglas model

$$$ Y=A^ηK^α{L_Y}^γW^{1−α−γ} $$$

$$$ \dot A=A^ϕK^β{L_A}^λW^{1−β−λ} $$$

$$$ \dot K=sY−δK=sA^ηK^α{L_Y}^γW^{1−α−γ}−δK $$$

Writing the model in terms of gK and gA:

$$$ g_K= \frac {sA^η{L_Y}^γW^{1−α−γ}}{K^{1−α}}−δ $$$

$$$ g_A= \frac {K^β{L_A}^λW^{1−β−λ}}{A^{1−ϕ}} $$$

Since the production function is Cobb-Douglas, growth is super-exponential if and only if either ġK > 0 or ġA > 0 or both.248 But, if one of these growth rates is increasing, then so is the other, as both η and β are positive. Therefore, the following inequalities hold just if growth is super-exponential:

$$$ \dot g_K=g_K[ηg_A+γg_{L_Y}−(1−α)g_K]>0 $$$

$$$ \dot g_A=g_A[βg_K+λg_{L_A}−(1−ϕ)g_A]>0 $$$

Following the paper, we introduce the parameter ξ = βη / [(1 – α)(1 – φ)] We consider gLY = gLA = 0 for simplicity. We show that ξ > 1 if and only if growth in the long-run is super-exponential.

First, we prove that ξ > 1 is necessary for super-exponential growth. To do this, we prove the contrapositive; that is, we assume growth is super-exponential and deduce ξ > 1. Rewriting the inequalities above, we obtain:

$$$ \dot g_K>0⇔ηg_A>(1−α)g_K $$$

$$$ \dot g_A>0⇔βg_K>(1−ϕ)g_A $$$

Notice that gA > (1 – α)gK / η and gK > (1 – φ)gA / β together imply gA > gA / ξ. Remembering that gA > 0, we obtain ξ > 1, as desired.

Now we show that ξ > 1 is sufficient for super-exponential growth. To do this, we’ll once more prove the contrapositive claim that, if growth is not super-exponential, then ξ ≤ 1. As mentioned above, output grows super-exponentially if either capital or technology do the same, and, if one of these factors grows super-exponentially, so does the other. Therefore, if growth is not super-exponential, then:

$$$ \dot g_K≤0⇔ηg_A≤(1−α)g_K $$$

$$$ \dot gA≤0⇔βg_K≤(1−ϕ)g_A $$$

But these inequalities yield:

$$$ g_A≥ \frac {βg_K}{1−ϕ}≥ \frac {βηg_A}{(1−α)(1−ϕ)}=ξg_A⇒ξ≤1 $$$

as we wished to show, completing the proof.

Under the assumption φ < 1, the condition ξ > 1 can be written as:

$$$ \frac {ηβ}{1−α}>1−ϕ $$$

as we do in the body of the report.

14.15.5.2 CES model

$$$ Y=A^η[F_Y(K,L)]^αW^{1−α} $$$

$$$ \dot A=A^ϕ[F_A(K,L)]^βW^{1−β} $$$

We assume α, β, η > 0, φ < 1, and we take labor to be fixed. We assume that = sY – δK ≥ 0, so that capital accumulates. Writing the model in terms of growth rates:

$$$ g_K= \frac {A^η(F_Y)^αW^{1−α}}{K}−δ $$$

$$$ g_A= \frac {(FA)^βW^{1−β}}{A^{1−ϕ}} $$$

The general condition for super-exponential growth is that:

$$$ \dot g_Y=η \dot g_A+α \dot g_{F_Y}>0 $$$

Note that the case φ > 1 always leads to super-exponential growth, as, in this case, the growth rate of technology goes towards infinity even holding all other factors constant, and hence the growth rate of output also explodes.

14.15.5.2.1 Case #1 — ρY < 0, ρA < 0

Notice that, as L/K → 0, FYFAL. Consequently, eventually gALβW1 – β / A1 – φ is decreasing with A (using the fact that φ ≤ 1) and, therefore, gA → 0 and A grows sub-exponentially (in particular, A roughly grows as t1/(1 – φ)). It turns out that this implies that both capital and output grow sub-exponentially.

14.15.5.2.2 Case #2 — ρY > 0, ρA < 0

Notice that, as L/K → 0, FYK and FAL. The growth rates become:

$$$ g_K≈sA^ηK^{α−1}W^{1−α}−δ $$$

$$$ g_A≈ \frac {L^βW^{1−β}}{A^{1−ϕ}} $$$

The dynamics of A can be solved independently from that of K. Solving the differential equation for A, we conclude that it grows sub-exponentially (in particular, A roughly grows as t1/(1 – φ)), just as in Case #1.

Capital has more complex dynamics. If α > 1, then capital grows super-exponentially and, therefore, output also grows in that fashion. The case α = 1 implies super-exponential growth, as, in this case, gKAη – δ is an increasing function of time. The case α ∈ (0, 1) leads to capital and output growing sub-exponentially. To see this, consider that, if capital was growing exponentially or faster, then the (super-)exponential increase in capital would outweigh the power-law increase in technology, and the growth rate of capital would decrease.

14.15.5.2.3 Case #3 — ρY < 0, ρA > 0

Notice that, as L/K → 0, FYL and FAK. The growth rates become:

$$$ g_K≈ \frac {A^ηL^αW^{1−α}}{K}−δ $$$

$$$ g_A≈ \frac {K^βW^{1−β}}{A^{1−ϕ}} $$$

The condition for super-exponential growth becomes:

$$$ \dot g_Y=η \dot g_A+α \dot g_L>0⇒ \dot g_A>0 $$$

as labor is fixed and η is positive. But this occurs just if ġK is also positive, as evaluating these derivatives yields:

$$$ \dot g_K≈g_K(ηg_A−g_K) $$$

$$$ \dot g_A≈g_A[βg_K−(1−ϕ)g_A] $$$

and the above implies that sustained super-exponential growth of one of the variables implies in the sustained super-exponential growth of the other.

Therefore, growth is super-exponential if and only if ġA > 0, or, equivalently, if:

 

$$$ g_A> \frac {g_K}{η} $$$

$$$ g_K> \frac {(1−ϕ)g_A}{β} $$$

as these inequalities hold just if ġA > 0 and ġK > 0. Combining these inequalities, we conclude that growth is super-exponential if and only if:

 

$$$ g_A> \frac {g_K}{η}> \frac {(1−ϕ)g_A}{βη}⇒βη>1−ϕ $$$

14.15.5.2.4 Case #4 — ρY > 0, ρA > 0

Notice that, as L/K → 0, FYFAK. The growth rates become:

$$$ g_K≈A^ηK^{α−1}W^{1−α}−δ $$$

$$$ g_A≈ \frac {K^βW^{1−β}}{A^{1−ϕ}} $$$

The condition for super-exponential growth is:

$$$ \dot g_Y=η \dot g_A+α \dot g_K>0 $$$

Evaluating the derivatives of the growth rates, we obtain:

$$$ \dot g_K≈g_K[ηg_A−(1−α)g_K] $$$

$$$ \dot g_A≈g_A[βg_K−(1−ϕ)g_A] $$$

Once more, we notice that, if one of the variables exhibits sustained super-exponential growth, so must the other. Similarly, if one of the variables exhibits sustained sub-exponential growth, so must the other. Therefore, the derivatives of the growth rates always have the same sign and super-exponential growth occurs if and only if both ġK > 0 and ġA > 0.

A similar argument to that used in Case #3 yields:

$$$ g_A> \frac {(1−α)g_K}{η}> \frac {(1−ϕ)(1−α)g_A}{βη}⇒βη>(1−ϕ)(1−α) $$$

14.15.6 Jones (2001)

This model is somewhat more complicated than the others presented above, as it endogenizes labor, so it warrants some additional explanation.

People can choose to devote their time to labor or to having children. In the model, each person devotes a fraction l of their time to work and a fraction 1 – l having children. The number b of births per capita is given by:

 

$$$ b=α(1−l) $$$

where α is a constant. The mortality rate is given by:

 

$$$ d=f(c)+ \bar d $$$

where f is a decreasing function and c = Y / N is consumption per capita. Population growth is described by:

 

$$$ g_N=b−d $$$

as expected.

Labor can be devoted to research or to final good production; the resource constraint for labor is:

 

$$$ L_Y+L_A=L=l \cdot N $$$

The production function for the final good is Y = AσLYβW1 – β, while the path of technological growth is given by Ȧ = δAφLAλ.

Following the paper, we define a parameter:

 

$$$ θ= \frac {λσ}{1−ϕ}−(1−β) $$$

where, as usual, we assume φ < 1. We show that growth is super-exponential if and only if θ > 0.

The condition for super-exponential growth is:

 

$$$ \dot g_Y=σ \dot g_A+β \dot {g_L}_Y=σ \dot g_A+β \dot g_N>0 $$$

We first show that ġA > 0 if and only if ġN > 0; therefore, both growth rates have the same sign and hence super-exponential growth requires both to hold. Then, we show that ġA > 0 and ġN > 0 both hold just if the desired inequality holds.

First, we show that ġN > 0 ⇒ ġA > 0. Observe that:

 

$$$ g_N=b−d=α(1−l)= \bar d – f(c) $$$

and hence ġN > 0 just if c = Y / N is increasing. But Y / NY / LYAσLYβ – 1, so ġN > 0 just if:

 

$$$ σg_A>(1−β)g_N $$$

We suppose that β < 1, to ensure decreasing returns to labor. Therefore, sustained super-exponential growth in population requires sustained super-exponential growth in technology, as otherwise the inequality obtained above would be violated eventually.

Now we show that ġA > 0 ⇒ ġN > 0. The growth rate of technology is:

 

$$$ g_A= \frac {δ{L_A}^λ}{A^{1−ϕ}} $$$

and thus its derivative is:

 

$$$ \dot g_A=g_A[λ{g_L}_A−(1−ϕ)g_A]=g_A[λg_N−(1−ϕ)g_A] $$$

Hence:

 

$$$ \dot g_A>0⇔λg_N>(1−ϕ)g_A $$$

and thus sustained super-exponential growth in technology requires sustained super-exponential growth in population, as desired. Therefore, growth is super-exponential if and only if both ġA > 0 and ġN > 0.

Finally, we show that the conjunction of ġA > 0 and ġN > 0 holds if and only if θ > 0 For the first of these inequalities holds just if ġN > (1 – φ)gA / λ, while the second holds just if ġA > (1 – β)gN / σ. Therefore both ġA > 0 and ġN > 0 if and only if:

 

$$ g_A> \frac (1−β)gNσ>(1−β)(1−ϕ)gAλσ $$

$$ ⇔1>(1−β)(1−ϕ)λσ $$

$$ ⇔λσ1−ϕ>1−β $$

$$ ⇔θ>0 $$

where we use ġA > 0. This completes the proof.

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South Asian Air Quality

This is a writeup of a medium investigation, a brief look at an area that we use to decide how to prioritize further research.


In a nutshell

  • What is the problem? South Asia experiences some of the world’s highest levels of population-weighted PM2.5 air pollution. Our understanding is that poor air quality contributes significantly to negative health outcomes for the more than 1.8 billion people in the region, and that reducing the levels of particulate matter present in the air could save millions of lives.
  • What are possible interventions? Possible interventions, many of which require coordinated state action, include improving air quality monitoring programs and crafting and implementing sector-specific policies to reduce emissions. A philanthropist interested in reducing South Asia’s air pollution levels could support efforts to inform the design, implementation, and enforcement of more effective air pollution abatement policies, such as funding monitoring programs, research, and technical assistance.
  • Who else is working on this? Philanthropic interest in South Asian air quality appears to be limited but growing rapidly, although many major philanthropic actors seem to address air pollution as a climate concern rather than as a health issue. Outside of the philanthropic sector, we think it’s likely that governments are the biggest spenders on improving air quality. We remain substantially uncertain about the exact levels of funding South Asian governments are directing toward the issue.

1. The problem

South Asia – and in particular the Indo-Gangetic Plain that covers parts or all of India, Pakistan, Bangladesh, and Nepal – experiences some of the world’s highest population-weighted air pollution levels.1 Our understanding is that poor air quality contributes significantly to negative health outcomes for the more than 1.8 billion people in this area.2 Of the pollutants present in South Asia’s air, we focus on PM2.5 – particulate matter no larger than 2.5 micrometers in diameter – which we understand to be associated with the greatest health costs.3 In total, the State of Global Air report – a collaboration between the Health Effects Institute and the Institute for Health Metrics and Evaluation’s Global Burden of Disease project – attributes approximately 71.4 million disability-adjusted life years (DALYs) across South Asia annually to air pollution.4 According to the Institute for Health Metrics and Evaluation, air pollution in South Asia accounts for nearly 3% of all DALYs worldwide – i.e. eliminating dangerous levels of air pollution in South Asia alone would reduce the number of prematurely lost years of healthy life by 3% each year.5

Exposure to PM2.5 air pollution can occur outdoors or within households, with the two settings associated with different concentrations, health outcomes, and interventions. Sources of outdoor, or ambient, air pollution in South Asia include brick kilns, vehicles, coal power plants, and crop burning.6 According to the State of Global Air report, South Asia’s average experienced ambient air pollution in 2019 was 78.2 µg/m3, a concentration higher than both the World Health Organization’s recommended standard of 10 µg/m3 and its intermediate standard of 35 µg/m3.7

While we have not investigated the overall evidence base thoroughly, we have encountered widespread agreement that long-term exposure to ambient PM2.5 pollution can result in significant negative health effects, such as chronic respiratory and cardiovascular diseases, that reduce life expectancy. The State of Global Air report, for example, estimates that, in 2019, almost 40 million DALYs in South Asia were attributable to ambient PM2.5 air pollution.8 This number appears to be stable in some South Asian countries and increasing in others.9 While we have not independently vetted this or other mortality and morbidity estimates, it seems reasonably plausible to us based on South Asia’s population-weighted air pollution levels and what the literature we’ve reviewed says about air pollution’s role in chronic illnesses.10

Concentrations of household (as opposed to ambient) air pollution in South Asia appear to be far more difficult to measure, with estimates we found ranging from 35 µg/m3 to over 2,000 µg/m3.11 We found more certainty, however, that household air pollution is widespread: one source estimates that roughly 60% of people in South Asia use solid cooking fuels, the primary source of household air pollution.12 This percentage is apparently decreasing as people switch to cleaner energy sources.13

The lack of reliable household PM2.5 concentration data makes it difficult to confidently discern health effects. The available evidence indicates that negative health outcomes of household air pollution in South Asia may include low birth weight, preterm birth, and other conditions that are correlated with an increased risk of infant death.14 The State of Global Air report, for example, attributed approximately 95,000 infant deaths within the first month of life to household air pollution in South Asia in 2019, estimating an overall impact of approximately 30 million DALYs within the region for that year.15

Of the nations comprising South Asia, India appears to experience among the highest average annual population-weighted ambient air pollution levels – 83.2 µg/m3 – and to contain the greatest number of DALYs attributable to both ambient and household air pollution – 31.1 million and 20.9 million, respectively.16 South Asia’s growing and aging population means that the burden from air quality – all else equal – is rising. In the case of household air pollution, this appears to be more than offset by people switching to cleaner cooking fuels, reducing the burden over time.17 However, the number of DALYs attributable to ambient air pollution appears to be increasing as outdoor air quality is worsening, accentuating demographic trends.18 The outsize impact of air pollution in India relative to other South Asian nations suggests to us that improving India’s air quality could greatly reduce South Asia’s population-weighted annual PM2.5 concentrations and DALYs resulting from air pollution.19

1.1 Why believe these estimated harms

We often have concerns about the quality and reliability of non-experimental social scientific evidence, and prefer to be able to replicate key inputs to our calculations ourselves. That is not possible with the State of Global Air report, which does not have open data and code. So we start with some skepticism that these large DALY estimates should be taken at face value. However, we did a review of the underlying literature and – while, as with all social science literatures, we think there could be room for improvement – we came away thinking that we would probably not want to adjust the State of Global Air burden estimates downward by more than a factor of two.

More specifically, biological mechanisms appear to support the conclusion that exposure to air pollution results in significant negative health effects, including mortality, in humans. Both the American Heart Association and the Lancet Commission on pollution and health, as well as epidemiologists we have spoken with, state that breathing particulate matter generates inflammation and vascular damage. These effects in turn are linked to conditions such as atherosclerosis and high blood pressure, which are known to cause life-threatening diseases such as ischemic heart disease and ischemic stroke.20 In infants, the proposed pathway seems to be that particulate pollution causes lower transmission of nutrients to fetuses, resulting in lower birth weight and nutrient deficiencies associated with higher infant mortality and lifelong health complications.21

There are various animal and human RCTs and studies on these biological mechanisms. The studies generally find that particulate pollution causes vascular inflammation, atherosclerosis, and low birth weight.22 More recent animal studies, however, do not seem to use mortality as an outcome of interest, and some much older studies found null effects of air pollution exposure on mortality.23 According to Open Philanthropy’s scientific research team, the null mortality results in animal models in the older studies are not necessarily evidence against mortality effects in humans, largely due to innate differences in biology and lifespans, though we do take them to be a mild negative update.

Outside of studies on the relevant biological mechanisms, we have found various natural experiments conducted by economists that attempt to isolate the causal effect of particulate pollution on mortality. Ebenstein et al., 2017 in particular examines the health effects of air pollution in conditions similar to those in South Asia, although we’re skeptical of this paper’s headline mortality effects.24 Other quasi-experimental papers, many of which focus on short-term exposure to particulates, generally find meaningful effects on mortality on both infants and adults.25 These papers have reassured us that the non-experimental social science literature we have found is likely not detecting the mortality effects of a confounding variable.

We have not found any meta-analyses that look for publication bias in the quasi-experimental evidence mentioned above. There is an epidemiological literature, however, that contains funnel plots that aim to identify publication bias. In a literature with no publication bias, one would expect to see a symmetric, triangle-shaped pattern of dots in the scatter, with the lower-powered analyses equally likely to fall on the right or left of the high-powered analyses. The plots within Pope et al., 2020 (specifically Figure 4), which examines epidemiological papers on the causal effect of air pollution on mortality in cohort studies, appear to have some asymmetry in the middle of the funnels.26 We very tentatively believe that a publication-bias adjustment based on these charts would reduce the mortality effect size to a number slightly to moderately below the consensus in the epidemiological literature.27


2. Possible Interventions

2.1 Government action

Many potential air quality improvements require coordinated state action. The following abatement policies are some of the ones that we thought had a mix of potentially addressing a large portion of the pollution problem and were likely administratively feasible.28

2.1.1 Retrofitting and building efficient brick kilns

20% of clay bricks are produced in South Asia, although PM2.5 emissions attributable to the sector seem to vary by country and be concentrated in urban areas.29 A report by the World Bank estimates that the brick sector is the second-largest PM2.5 contributor in Bangladesh and Nepal, responsible for 11% and 3% of PM2.5 emissions, respectively.30 In India, meanwhile, the share of PM2.5 emissions attributable to brick kilns appears to be comparatively lower, although we have encountered substantial uncertainty around this point. The Health Effects Institute offers one of the lower estimates we found, tracing approximately 2% of India’s PM2.5 pollution and 2 to 3% of its PM2.5-related deaths to brick kilns.31 The World Bank’s report has the highest estimate of the sources we gathered, attributing 8% of India’s PM2.5 emissions to the brick sector.32 The World Bank estimates that retrofitting existing kilns could reduce PM2.5 by 30-50%, as well as improving energy efficiency.33

Despite the uncertainties around emission levels, we think it is plausible (but by no means decisive) that a government-championed effort (e.g., regulations and/or subsidies) to retrofit and build efficient brick kilns would be administratively feasible and could meaningfully reduce PM2.5 pollution from the brick sector.34

2.1.2 Implementing and enforcing a ban on older vehicles

At least since 2015, government bodies in India have indicated an interest in limiting the use of older vehicles.35 There are some regional bans, but it’s unclear to us to what extent they have been implemented or enforced.36 This existing – if inconsistent – interest in banning older vehicles, along with what appears to be a low number of vehicles over 10 years old (meaning political/economic costs of a ban are smaller), suggests that this is a potentially promising area for further government action.37

We’re uncertain about the percentage of the population-weighted PM2.5 pollution in South Asia that is vehicular, although it appears fairly significant. A report by India’s Ministry of Environment, Forest and Climate Change estimates that vehicles contribute approximately 28% of population-weighted PM2.5 emissions in Delhi during the winter and 4% nationally when accounting for all modes of transportation.38 The Energy and Resources Institute attributes 50% of Bangalore’s PM2.5 load to automobile emissions.39 A source apportionment study of Mangalore attributed 70% of particulate pollution to vehicles.40 Older vehicles in particular appear to be a significant contributor to vehicle emissions, with one estimate we found claiming that vehicles older than 15 years account for 15% of total vehicular pollution, and tend to pollute 10 to 25 times as much as newer vehicles.41 Based on these numbers, we think it is likely that a ban on older vehicles could reduce total PM2.5 pollution, although we’re very uncertain about the total reduction we could reasonably expect and how enforceable (and beneficial) a ban would realistically be.

2.1.3 Mandating and enforcing coal scrubbers

Most of the estimates we found attribute approximately 15% of India’s PM2.5 emissions to coal power generation.42 It seems plausible to us that coal is a significant source of PM2.5 emissions, given the prominence of coal in India’s electricity generation and CO2 emissions.43

One report we saw claims that installing wet coal scrubbers in power plants could reduce PM2.5 emissions by as much as 98% and newer fabric filters can reach efficiencies as high as 99.7%.44 While we have not independently vetted this estimate, if accurate, it indicates to us that coal scrubbers could significantly improve India’s air quality.45

The Indian government has already mandated that plants install coal scrubbers to limit emissions, although compliance appears to be limited.46 Given the apparent magnitude of coal power emissions and the government’s existing interest in pursuing mitigation measures, additional efforts to install coal scrubbers might be a promising intervention.

Below, we share our rough back-of-the-envelope calculations (BOTECs) on the potential cost-effectiveness of philanthropic support for the installation of coal scrubbers.

2.1.4 Reducing crop burning with better targeted tractor subsidies

Our impression is that crop burning is a relatively minor source of emissions in India; one article claims that it constitutes an average of 5% of annual PM2.5 pollution in Delhi, although it reaches up to 40% at certain points in the year.47 The vast majority of farmers appear to burn their crops, with only an estimated 20% using tractors to till their fields.48 It seems plausible that better targeting tractor subsidies to increase the percentage of farmers using tractors, while decreasing the percentage of those who burn stubble, could moderately improve Delhi’s air quality.49 We are unsure of the potential impact tractor subsidies might have on air quality across the broader South Asian region.

2.1.5 Better targeting liquified petroleum gas subsidies

From what we have found, solid cooking fuels – still used by approximately 60% of households – account for roughly 40% of the health burden from PM2.5 pollution in South Asia.50 We tentatively assume that substantial reductions in solid cooking fuel use could lead to large reductions in health impacts. The main replacement for solid cooking fuel (e.g. wood, agricultural refuse, charcoal, etc.) is liquified petroleum gas (LPG).

The Indian government already subsidizes LPG use, currently entitling each household to 12 LPG cylinders per year.51 The subsidies, however, do not provide significant discounts to the market price, suggesting that LPG cylinder prices may remain too high for many poor households to afford.52 As a means of increasing the subsidies available to the poor, the government has unsuccessfully attempted to convince wealthier households to voluntarily pay for unsubsidized LPG.53 Better targeting the subsidies by increasing availability and subsidy size for poorer households could plausibly help reduce the number of households that use solid cooking fuels.

2.2 What could a philanthropist support?

We have encountered widespread uncertainty around the share total and population-weighted PM2.5 that is attributable to different sources in India and across South Asia. Addressing this information deficiency seems to be crucial to appropriately targeting abatement strategies. As such, it seems likely that philanthropic efforts may be able to productively focus on 1) improving the information ecosystem for local decision makers and other stakeholders and 2) increasing the technical capacity of key governmental agencies to address air quality. A philanthropist interested in supporting either of these two outcomes might pursue any of a variety of activities, some of which we’ve listed below.

2.2.1 Source apportionment studies

As we mentioned previously, we have found that the deficiency in pollution source apportionment data has made it difficult to gauge the potential impacts of available interventions. Source apportionment studies are scientific studies that attempt to measure what share of the total PM2.5 concentration in a given city or region can be attributed to different sources, e.g. transportation, power generation, other industrial sources, etc.54

Source apportionment studies could be conducted in partnership with interested cities needing technical assistance.55 Such localized studies, in providing governments with rough estimates of their cities’ largest sources of air pollution, could potentially improve governments’ (and philanthropists’) abatement strategies.

Below, we share our rough back-of-the-envelope calculations on the potential cost-effectiveness of philanthropist support for source apportionment studies.

2.2.2 Air quality monitoring

A philanthropist might fund either low-cost sensors or advanced monitoring stations. Low-cost sensors, which we have already supported, can be installed locally and could contribute data to real-time air quality maps that report shifts in pollution amounts. We assume that these maps could help improve public awareness of local pollution levels and precipitate minor behavior change, as well as enable governments and other entities to track the impacts of abatement methods. The limited accuracy of low-cost sensors may impede pollution measurements, however, as individual sensors may not be able to detect small changes in concentrations.56

Advanced monitoring stations are much more accurate – also significantly more expensive – and could be installed in each of India’s airsheds. Potentially combining the stations with sun photometers to measure the atmospheric column could allow for significantly more accurate and frequent satellite measurements of air pollution sources and concentrations.57 These measurements could, in turn, provide governments with more precise pollution targets, enable the effects of abatement policies to be tracked, and contribute to general air quality reporting.

We think that air quality monitoring could be a fairly large source of philanthropic spending in the short term, with smaller ongoing costs after the initial implementation. From our conversations, we also received the impression that Indian air monitoring is comparatively well-funded, and that supporting monitoring in other South Asian regions might generate more impact on the margin. We have not independently vetted these claims.

2.2.3 Research on the abatement curve and air pollution health effects in India

We perceive research on the abatement curve (the graph describing the financial costs and volumes of PM2.5 reductions by intervention pursued) and air pollution health effects in India as having a variety of benefits. A better defined abatement curve data could serve as a menu of options for interested philanthropists or policymakers. Research on health effects could provide more targeted data on the health effects of PM2.5 pollution, including potentially distinguishing between the health effects of different types of pollutants.

In addition, such research could help drive awareness among governments and the public of the extent of the problem and accordingly encourage the adoption of targeted abatement measures (particularly if this research identifies a more narrow set of lower cost policy changes that could solve a large share of the total problem). We have heard that this may be more effective if the research is based at national institutions that also provide expertise to local governments or non-governmental organizations working on this issue.

2.2.4 Technical assistance

Providing technical assistance to government entities could improve the outcomes of pollution abatement measures by increasing governmental capacity to implement, enforce, and monitor air pollution abatement measures. A funder interested in this outcome could, for example, work with outside consultants to provide technical assistance to India’s pollution control boards, which, for a number of reasons, have struggled to enforce air quality regulations.58 We have found conflicting estimates of the pollution control boards’ current spending, although it seems to be between $100 million to $300 million a year, split between air pollution, water pollution, noise pollution, and waste management.59

2.2.5 Policy outreach

The interventions outlined in the section above are largely under the purview of the government. As such, philanthropic efforts might focus on providing decision makers with data and resources to craft effective air pollution abatement policies. Potential funding areas could include source apportionment and concentration research, real-time air quality maps, and reporting on air quality in local news outlets. Other means of increasing the salience of air quality might include funding programs like the Clean Air Fund’s Doctors for Clean Air, which raises awareness of the health impacts of air pollution, or supporting air quality programs at universities.

2.3 How cost effective could spending in this area be?

If air pollution costs 71.4 million DALYs annually in South Asia, and we were spending $20 million per year, we would need to be pulling forward solutions to approximately 0.06% of the problem by 10 years for every year of our spending in order to clear our 1,000x bar.60 It is difficult to reason about small numbers like that but given the relatively limited scale of other philanthropy in this space, we do not think that would be an unreasonable bar for us to clear.

We do not have a specific plan for how to spend money cost-effectively on this problem at that level, but we’ve done a few back-of-the-envelope calculations on promising-seeming potential projects, described in more detail below, that also make us think they could clear the 1,000x bar.

2.3.1 Air quality monitoring

We have already recommended funding, totalling $3 million, to install a network of low-cost air quality sensors in India. We have removed our current BOTEC since it’s related to our hiring process for a South Asian air quality program officer.

2.3.2 Source apportionment studies

By our rough calculations, a source apportionment study would need to accelerate a reduction of 0.8 µg/m3 in pollution by 10 years for a city of 5 million to reach our 1,000x bar.61 This calculation assumes that:

  • The cost of a source apportionment study would scale as a function of population size. We roughly estimate that a study in a city of 5 million would cost $500,000.
  • Source apportionment studies would only measure, and thus impact, ambient air pollution levels.
  • The PM2.5 concentration in cities is proportional to the national concentration. The average annual population-weighted concentration of ambient air pollution in India is 83.2 µg/m3. Approximately 31,140,452 DALYs in India are due to ambient air pollution, and every DALY is valued at $50,000.62
  • India’s population is 1,366,000,000.63

2.3.3 Coal scrubbers

According to a report by the Disease Control Priorities Network, installing coal scrubbers in all power plants would cost approximately $1.7 billion.64 The same report estimates that to retrofit the plants with the lowest cost per life saved would cost $615 million, although other sources we’ve encountered estimate costs that are more than an order of magnitude higher.65 If the $615 million figure were correct, paying to install coal scrubbers could reach and perhaps surpass our 1,000x bar, assuming the following conditions are true:

  • As we discussed above, coal power generation contributes approximately 15% of India’s PM2.5 emissions.
  • Installing scrubbers reduces PM2.5 emissions by at least 80%.66
  • The selected coal power plants are responsible for 75% of the sector’s DALY costs.67
  • The health effects of air pollution in India cost approximately $2.68 trillion/year.68
  • Given that the government is already mandating the installation of coal scrubbers, our funding speeds installation up by five years.

Under these conditions, we would estimate an ROI of $2.68 trillion (total cost of ambient air pollution) × .15 (power sector share of total PM2.5) × .75 (selected plants’ share of power sector DALYs) × .8 (reduction in PM2.5 from scrubbers) × 5 (years of speed-up) / $615 million (cost of scrubbers) = ~1,960x, though again we do not know these assumptions to be correct and have seen much higher cost estimates in the literature.

2.4 What scale could a program in this area possibly reach?

Based on our understanding of the available funding opportunities, we think there is a high likelihood that a program in this area could spend at least $25 million per year on activities such as air quality monitoring, abatement and source apportionment studies, technical assistance, scaling existing organizations working on air quality, and policy outreach, at a cost-effectiveness level comparable to other funding opportunities we pursue. We think there is a lower likelihood of significantly more than $25M/year of capacity in opportunities we would consider quite cost-effective.


3. Who else is working on this?

3.1 Philanthropic organizations

Philanthropic interest in South Asian air quality appears to be limited but growing rapidly: an estimate by the Clean Air Fund, which was cited to us in multiple conversations, puts the philanthropic spending in this area at roughly $7 million in 2019, up from $1 million in 2015.69 We have not vetted the report’s estimates and would guess there are structural underestimates because the report is based on self-reported data from foundations, some of whom may not participate in data sharing, but the estimates are broadly consistent with what we heard in conversations.

The international philanthropic actors working on South Asian air quality that we have heard mentioned most frequently are Bloomberg Philanthropies, the Children’s Investment Fund Foundation, ClimateWorks, the IKEA Foundation, the MacArthur Foundation, the Oak Foundation, Pisces Foundation, and the William and Flora Hewlett Foundation. Some major Indian funders, such as Ashish Dhawan, have also come up in our conversations with experts and funders in the field. We do not believe that this is an exhaustive list: we would guess that we have accounted for the largest philanthropic funders working in this area, but we are certainly missing smaller investments from non-profits and activists.

Many of these major philanthropic actors appear to address air pollution as a contributor to climate change rather than in terms of direct negative health effects from particulates. Climate-focused philanthropic spending on air quality is part of a broader effort to mitigate emissions in India, with philanthropic annual spending on emissions reductions that we think is on the order of $100M-$350M.70

It is unclear to us to what extent treating air quality as a climate concern versus as a health issue would result in significantly different funding strategies. There is definite potential for overlap between climate and air quality spending, as many interventions that reduce greenhouse gasses tend to reduce PM2.5 emissions as well (e.g., limiting reliance on coal for electricity generation). But the two goals can also come apart (e.g., flue-gas desulphurization units on coal plants help improve air quality for health, but as far as we know do not mitigate climate impacts). Overall, we do not think that the presence of significant climate funders mitigates the need for more focused work to improve air quality from a health perspective.71

3.2 Government

We found it difficult to find reliable estimates of governmental spending on air quality. According to one source we found, in the 2019-2020 budget cycle, the Indian government created and allocated a fund of Rs 44 billion (approximately $609 million at the time of conversion) to address air pollution in large cities.72 Additionally, a 2020 report released by the Council on Energy, Environment and Water and UrbanEmissions notes that the National Clean Air Plan, which directs cities to create action plans to reduce particulate matter concentrations by 20-30% by 2024, receives Rs 4.6 billion (approximately $63 million at the time of conversion). However, the report also noted that there are no penalties for non-achievement or “legal mandate for reviewing and updating plans.”73 In fact, only nine cities seem to have noted the costs of execution, which ranged from Rs 890 million to Rs 160 billion (approximately $11.9 million to $2 billion at the time of conversion, respectively).74 We remain substantially uncertain of the accuracy of these estimates and recognize that it’s plausible that there may be additional state funds we do not know about. Overall, we think it’s likely that the government is the biggest spender on improving air quality, but that the current spending is substantially lower than the amount required to adequately reduce air pollution.


4. What have we done so far?

Air quality monitoring stood out to us as a particularly tractable abatement strategy that has the capacity to absorb immediate funding. We have accordingly recommended grants totalling $3 million to support a three-year collaboration between Professor Joshua Apte of UC Berkeley, the Indian Institute of Technology Delhi (IIT Delhi), and the Council on Energy, Environment, and Water (CEEW) to install a network of low-cost air quality sensors in South Asia.

The aim of the collaboration is that the data from the sensors will inform the design, implementation, and enforcement of more effective air pollution abatement policies. Additionally, we also see this project as an early learning opportunity for the testing and deployment of low-cost sensors across South Asia; if successful, we predict that the sensors could have spillover effects on the speed at which other low-cost sensors are deployed, although we have not consulted experts on this point. Both of these outcomes could plausibly result in significant reductions to South Asian air pollution levels.

For our back-of-the-envelope calculations on the potential cost-effectiveness of these grants, see above.


5. Potential risks and downsides

We have identified a number of potential risks and downsides to funding air quality improvements efforts in South Asia, including:

  • Immediate spending capacity appears to be limited. We have identified a few abatement activities – such as air quality monitoring and certain forms of technical assistance – that could benefit from immediate funding. We have otherwise struggled, however, to identify areas that have the capacity for large-scale recurring support, and we expect that a philanthropist hoping to make South Asian air quality a long-term funding area may need to consistently find novel grantmaking opportunities.75
  • Air quality interventions largely depend on government regulation and enforcement. Accordingly, most philanthropic efforts in South Asian air quality would be limited to activities that inform government policies but that may not directly impact air quality (for example, funding air quality monitoring stations that in turn provide decision makers with data to craft effective abatement measures). It will likely be difficult to predict what the likely impact of these efforts might be.
  • There are risks and restrictions specific to funding work in India. The Indian government has historically regulated foreign grantmaking within India and recently implemented additional restrictions on foreign funding to Indian NGOs.76 It is our impression that all of the potential philanthropic efforts listed in this writeup are permissible under current laws, but there is a risk that the Indian government could implement additional restrictions that would change that.
  • Philanthropic efforts could lead to overly restrictive policies in some domains, which in turn could create space for corruption or potentially slow economic growth. While this writeup does not take into account risks from air pollution beyond mortality, we recognize that some abatement strategies may not be economically feasible or desirable after accounting for their costs.
  • Philanthropic interest in South Asian air quality appears to be growing, so additional funding now could risk crowding out other funders who would otherwise enter.

6. Our process and next steps

We talked to a number of experts and major funders in the field in the process of researching South Asian air quality. The following individuals agreed to being named as sources for this report, though this should not be interpreted to mean that any experts named here endorse our conclusions in part or in total:

  • Aaron Van Donkelaar
  • Ambuj Sagar
  • Amita Ramachandran
  • Arden Pope
  • Avijit Michael
  • Brikesh Singh
  • Dan Kass
  • Ishwar Gawande
  • Jarnail Singh
  • Josh Apte
  • Kanchi Gupta
  • Matt Whitney
  • Melanie Hammer
  • Michael Greenstone
  • Pallavi Pant
  • Randall Martin
  • Reecha Upadhyay
  • Rohini Pande
  • Sam Ori
  • Sangita Vyas
  • Santosh Harish
  • Siddarthan Balasubramania
  • Vinuta Gopal

We continue to be open to learning about more opportunities in this space and may make additional grants in the future.


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Scale-up economics for cultured meat

Open Philanthropy commissioned a report from David Humbird on the potential for cultured meat production to scale up (to the point where it would be sufficiently available and affordable to replace a substantial portion of global meat consumption).

Abstract

This analysis examines the potential of cultured meatproducts made from edible animal cell culture to measurably displace the global consumption of conventional meat. Recognizing that the scalability of such products must in turn depend on the scale and process intensity of animal cell production, this study draws on tech-no economic analysis perspectives in industrial fermentation and upstrea