The limits to (explosive) growth

What does AI mean for GDP?

In September, Ege Erdil and Tamay Besiroglu published a noteworthy paper reviewing the arguments for and against AI causing explosive growth in the not-so-distant future. It follows in the footsteps Tom Davidson’s 2021 report on explosive growth for Open Philanthropy.

I was recently asked for my take on these reports, and how likely I think explosive GDP growth from AI and automation is more generally. In short, I think both reports are useful as thought experiments but limited methodologically. I'll explain my perspective below.

But first, what do we mean by explosive growth in the first place? Both Davidson and Erdil & Besiroglu define explosive growth as an order of magnitude increase over historic rates, or a 30% annual growth rate in global GDP. This means the world economy would double every 2-3 years — a staggering and unprecedented rate of change, even for developing countries undergoing rapid catch-up growth.

As the historian Ian Morris points out on a recent 80,000 Hours podcast, merely extrapolating the historical average global GDP growth rate of 3.5% for another century already implies transformative economic and technical change:

You can’t just have what we’ve got now and pump it up with a bike pump and make it 30 times bigger. There aren’t enough resources in the world of the kind that we are currently using to produce 30 times as much GDP. Really, really profound things have to happen.

The miracle of compound growth is such that even small differences in sustainable growth rates can lead to enormous differences in wealth after a few decades. But by the same token, the herculean task of maintaining ~3% annual GDP growth in the context of a frontier economy should put some strain on the plausibility of growth rates running substantially higher.

Regardless, even if the explosive growth thesis is wrong, the future is still destined to look radically different from the present. Indeed, nothing I say here is meant to diminish the transformative potential of AI and related technologies. An intelligence explosion is coming that will bring with it an acceleration in scientific and technological progress and a step-change increase in the pace of change — assuming things don’t go terribly south in the transition. Relative to the man on the street, my growth expectations are thus directionally aligned with the theorists of explosive growth, even if our forecasts differ by orders of magnitude.

The limits to endogenous growth theory

Both Davidson and Erdil & Besiroglu arrive at their projections of explosive growth by drawing on endogenous growth theory, particularly the AK growth model:

The above equation is a standard Cobbs-Douglas production function, where Y is total output, K and L are the stocks of capital and labor inputs, and A is a constant representing how technology augments labor and capital. In the simplest AK model, α — the output elasticity of capital — is set to 1, representing constant returns. Output thus becomes linear in capital accumulation. That is, rather than suffering diminishing returns, accumulating more capital lets output grow indefinitely, allowing one to model a positive balanced growth rate in an economy’s steady-state.

Endogenous growth theory was developed in the 1980s to address the limitations of neoclassical growth models, which predict that an economy will stop growing in per capita terms once it reaches the production frontier set by the exogenous state of technology. In practice, far from converging to zero, frontier economies like the United States have continued to grow at a slower but still relatively consistent rate. This is because the factors influencing output growth are often endogenous, i.e. higher output creates the conditions for even higher output. When a researcher invents a new idea, the non-rivalry of ideas boosts the productivity other researchers, leading to more new ideas and technologies. Likewise, when a country gets richer, its people get healthier and better educated, and so the country gets richer still. Endogenous growth models are thus particularly popular for modeling human capital accumulation, learning-by-doing, and knowledge spillovers.

The endogeneity of growth is real and even under-rated, yet it is still not without diminishing returns — arguably the bed-rock principle in economics. Indeed, the AK growth model is notorious for its sensitivity to so-called “knife edge conditions.” If α is even a little bit less than 1, diminishing returns eventually set in and long-run output asymptotically converges to the neoclassical world of finite growth. Conversely, if α is greater than 1, accumulating more inputs has increasing returns, and output races off to infinity at a hyperbolic rate.

The explosive growth forecasted by Davidson and Erdil & Besiroglu is a trivial consequence of these knife-edge dynamics. In their models, the emergence of AIs that can fully automate human labor cause the L term to drop out of the production function. Output is thus solely a function of accumulable capital inputs, where capital now includes both physical machinery and an indefinite number of AI workers, researchers and scientists. They are aware of the fragile assumptions behind the AK model, but argue that accumulable inputs will have increasing returns to scale nevertheless.

But what if scientific research (whether done by AIs or humans) faces diminishing returns? In Are Ideas Getting Harder to Find?, Bloom et al. (2020) tackles this question and concludes that ideas are indeed getting hard to find, casting doubts on models that assume R&D has constant (much less increasing) returns to scale:

Our robust finding is that research productivity is falling sharply everywhere we look. Taking the US aggregate number as representative, research productivity falls in half every 13 years: ideas are getting harder and harder to find. Put differently, just to sustain constant growth in GDP per person, the United States must double the amount of research effort every 13 years to offset the increased difficulty of finding new ideas.

This analysis has implications for the growth models that economists use in our own research, like those cited in the introduction. The standard approach in recent years employs models that assume constant research productivity, in part because it is convenient and in part because the earlier literature has been interpreted as being inconclusive on the extent to which this is problematic. We believe the empirical work we have presented speaks clearly against this assumption. A first-order fact of growth empirics is that research productivity is falling sharply.

This would seem to severely undercut forecasts of explosive growth based on self-reinforcing R&D accumulation. Against this, Erdil & Besiroglu argue that α < 1 “still produces increasing returns to scale as long as the returns to idea-production diminish sufficiently slowly.” By incorporating Bloom et al.’s estimate of declining research productivity, they even find that hyperbolic growth occurs with values of α as low as 0.68. Forecasts of explosive growth are thus, in their words, “hard to avoid” even with “highly conservative assumptions.”

Yet arriving at this result requires making a few mammoth assumptions. I’ll walk through these below.

1) The limits to factor accumulation

First, the only reason α can go as low as 0.68 is because Erdil & Besiroglu argue that diminishing returns to accumulable inputs can be fully offset by the knowledge spillovers from new ideas, citing Bloom et al.’s estimate of the return on research investment of 0.32. That is, even if research productivity is falling, you can always offset that decline by throwing proportionally more AI researchers at the problem, as captured by this equation in Bloom et al.:

Technology or Total Factor Productivity thus becomes irrelevant, as the growth dynamics are fully driven by accumulable AI labor inputs, i.e. researchers. This is again exploiting a well-known feature (or bug?) or semi-endogenous growth theory: an economy can achieve explosive growth by simply growing its population. And while population growth is normally bounded by the limits of human reproduction, here we can simply let the AI researcher printing press go brrrrr. As E&B put it,

“this gives rise to a feedback mechanism where greater output gives rise to an increase in inputs that give rise to a greater-than-proportional increase in output. Hence, such models generically predict super-exponential growth conditional on AI that suitably substitutes for human labor.”

To call this analysis highly stylized is an understatement. Explosive growth follows trivially in an “assume a can-opener”-style fashion, as accumulation allows you to accumulate even more, ad infinitum. I could maybe see this working for research outputs, as scientific knowledge is in some sense an “endless frontier.” But what we care about is productive economic output, not academic findings. Thus even if doubling the stock of AI workers leads to rapid growth, it is still not without diminishing returns — hence why countries like Qatar treat their guest workers so poorly: given an effectively unlimited supply of migrants, their marginal product becomes exceedingly low, as reflected in their poor working conditions. I thus wouldn’t expect Bloom et al.’s estimated return on research investment of 0.32 to remain stable overtime, but to instead trend towards zero absent a major technological breakthrough. This is why, in neoclassical growth theory, the exogenous state of technology is the binding constraint on long-run growth, while for E&B, technology and growth are in some sense completely decoupled.

2) The limits to scale

AK models have been extensively critiqued for their implausible sensitivity to increasing returns to scale. Yet the increasing returns to scale assumption suffers from a more general problem: aggregation. That is, the appearance of increasing returns to scale in any given sector do not necessarily “aggregate up” to increasing returns to scale for the overall economy. Consider the scale effects from urban agglomeration. While letting smart people move to New York City might increase the city’s local returns to scale, there are also negative externalities in the regions suffering brain drain. Moreover, at some point congestion externalities set in, precluding a world where everyone simply moves to the one, giant metropole.

As Steven Bond-Smith explains,

But an approximation that knowledge has a single dimension leads to an implicit assumption that ideas production has returns to the aggregate scale, otherwise known as the “scale effect”. This is a well-known limitation of first generation endogenous growth theories most publicised by Jones (1995b). It implies that an increase in any rival factor endowment in the economy results in a higher growth rate without any microfoundation for such an increase. If any rival factor is growing, it implies an ever-increasing growth rate and explosive output in finite time. Time-series data on the growth of inputs to R&D for the United States is also not consistent with the functional form of ideas production in these first generation models of endogenous growth (Jones, 1995b). Returns to scale are indeed characteristic of innovation and endogenous growth, but the scale effect is widely recognised as an error in aggregation (Laincz and Peretto, 2006).

Indeed, so-called “strong scale effects” imply that merely increasingly an economy’s population leads to rising per-capita incomes, which is not empirically supported. As such, newer generations of endogenous growth theory seek to model long-run balanced growth with weak or non-existent scale effects. In these models, explosive growth is constrained by letting the increasing returns to ideas expands along two dimensions: quality improvements and new product varieties. Put differently, while individual firms (or cities and regions) may exhibit scale effects, these are ultimately tempered by new entrants, eliminating returns to scale in aggregate.

Imagine a pizzeria that exhibits increasing returns to scale. The more pizza they produce, the more revenue they can pour into pizza productivity, letting them produce even more pizza. At some point, however, we either enter a world of pizza super-abundance, or people start getting sick of pizza and shift their demand to hamburgers — a new product with its own scaling curve. Increasing returns at the firm-level thus fail to translate into increasing returns for the economy as a whole.

This simple example also illustrates the limits to the “nonrivalry of ideas,” as new ideas for making pizza are not necessarily new ideas for making hamburgers. In fact, as Charles Jones notes, “there is remarkably little work aimed at measuring the degree of increasing returns associated with the nonrivalry of ideas, despite the importance of this parameter.” Nor do we have any good reason to think estimated rates of returns to scale at current margins will persist far out of distribution. Economists thus usually treat AK theory as a toy model; not something to take too literally. Likewise, when economists calibrate growth models on historical data, it is often little more than an exercise in curve fitting, which is why there's still no agreement among economists over which theoretical structure best explains cross-national growth differences. The use of these models for predicting truly “long-run” growth is thus inherently dubious.

David Deming put it well in a recent post:

The problem with development accounting is that your results end up being incredibly dependent on measurement and on unverifiable assumptions about the structure of the aggregate production function. ... What does all this mean for the importance of human capital in development accounting? Jones (2014) and Caselli and Ciccone (2019) debate the credibility of different assumptions about the structure of the aggregate production function. Collectively they conclude that human capital explains somewhere between 0 and 100 percent of cross-country income differences. This is not very helpful.

Indeed, if human capital or some other accumulable factors explained cross-national differences in per-capita GDP, migrants to the US from poor countries wouldn't see their wages instantly close 40%+ of the gap with US natives. Instead, this suggests we're primarily rich because of our technology and institutions.

3) Consumption bottlenecks

Say's Law states that supply creates its own demand, as the production of a product necessarily creates demand for other products of like-value to be offered in exchanged. More generally, one person's production is by definition another person's consumption, just as one person's cost is another person's income. This basic "adding up" constraint is why, in general equilibrium, aggregate labor and capital income ratios remain highly constant across time, even in the face of substantial automation.

To arrive at explosive growth, E&B are forced to relax this assumption. This makes superficial sense, since if AI workers directly substitute for human labor, labor ceases to be a complementary factor of production, and the income share of the economy rapidly approaches subsistence. But if a tree falls in the forest and no one’s there to buy the lumber, does it really add to GDP? That is, even with AI workers that can perfectly emulate human workers in every respect, humans still play a distinctive and necessary role in growth models as the ultimate consumers of productive outputs.

This “perfect complementarity” between production and consumption is why automation doesn’t lead to technological unemployment. As C. Jones explains,

Suppose tasks are complementary in production, with an elasticity of substitution less than one. Then automation and capital accumulation push in opposite directions. As above, automation by itself tends to increase the capital share. However, because the elasticity of substitution is less than one, the input that becomes more scarce—labor here, since capital gets accumulated—sees its factor share rise. This is essentially a form of Baumol's (1967) cost disease. The increase in the fraction of the economy that is automated over time is just offset by a decline in the share of GDP associated with the automated sectors, such as manufacturing or agriculture. Economic growth is determined not by what we are good at but rather by what is essential and yet hard to improve. Labor gets concentrated on fewer and fewer tasks, but those tasks are essential, and therefore the labor share can remain high.

This is what we see in practice. Differential productivity growth is deflationary in the affected sectors, shifting nominal income to whatever remains labor intensive. Thus TVs have gotten cheap while health and child care have gotten more expensive, at least in terms of household budgets. My one quibble with Jones' description is with the word “essential.” The average wage for a Starbucks Barista in Washington, DC, is $17.47 / hour plus tips and benefits. Contrast that with the average wage in India of $2-3 / hour USD. Yet Baristas aren't an “essential” form of labor; coffee pouring is already fully automated. Barista wages are instead pulled-up by the broader productivity of the economy, which affords people with higher incomes to demand human coffee pourers. Baumol effects are thus only superficially about labor intensive tasks that resist automation, and more deeply a consequence of the “adding up” constraint that forces production to ultimately translate into incomes and thus human labor demand.

If the circular flow of spending ever de-linked from consumers (i.e. capital owners just spent the entirety of their income on building more capital), then production would rapidly outstrip consumption, profits would collapse, and we'd have malinvestment of the Chinese ghost city variety. This is essentially the Marxist theory of overproduction leading to a "declining rate of profit" and thus the inevitable crisis of capitalism. Fortunately, Keynes came along and showed that the Marxists were confusing overproduction for under-consumption. Apparent violations of Say’s Law are explained by unstable monetary and credit aggregates leading to temporary short-falls in demand. Aggregate demand is thus a short-run policy parameter set by monetary and fiscal authorities. Say's Law still holds that aggregate demand == aggregate supply in the long-run, as otherwise markets wouldn't clear. As Aghion et al put it:

When applied to a model in which A.I. automates the production of goods and services, Baumol’s insight generates sufficient conditions under which one can get overall balanced growth with a constant capital share that stays well below 100%, even with nearly complete automation. When applied to a model in which A.I. automates the production of ideas, these same considerations can prevent explosive growth.

Even consumption exhibits diminishing marginal returns. While Jay Leno may own more cars than I'll ever be able to afford, he doesn't truly “consume” those cars at a constant rate. They're more like a form of savings. After all, there are only 24 hours in a day and, on some margin, Jay Leno still values quality time with his friends and family at least a much as he does racing his ‘94 McLaren. This is a version of Baumol’s disease in micro, as the opportunity cost of Leno’s (or any other rich person’s) time is enormous relative to the median worker, and yet rich people make consumption trade-offs that aren’t that different from yours or mine.

The long-run is neoclassical

Semi-endogenous growth theory is incredibly useful because per-capita growth really is endogenous in many ways, just not without limit. Otherwise, we would expect to see divergence rather than convergence in national incomes, and transitory shocks would permanently reduce long-run output, when instead we observe rapid recoveries from real shocks.

This suggest per-capita GDP growth is often endogenous but bounded by exogenous growth factors like technology and institutions, which neoclassical growth models capture in the single parameter A, or Total Factor Productivity. TFP is a residual catch-all that doesn't distinguish new tech and ideas from better institutions or other x-factors that influence the efficiency of labor-capital input combinations. This can make it hard to interpret, but the conceptual point remains clear.

In the long-run, neoclassical growth models suggest input accumulation can drive growth in total output, but not per-capita, and this is what we see. Estimates of cross-state and cross-country variation in levels and growth rates in output per worker are mostly driven by variation in TFP, not accumulable inputs; growth divergence stems from lags in technology adoption; and TFP in previous periods is strongly correlated with future growth rates.

One can always find papers that put more emphasis on accumulable inputs, such as human capital, but there are good reasons to think these studies are underestimating the contribution of TFP, as TFP is by nature hard to decompose from labor and capital investment with embedded innovation. TFP also matters most for the long-run growth potential of frontier economies, which is difficult to isolate in the data, as even "within country" TFP growth is heterogeneous across firms.

Given these empirical and accounting challenges, I thus defer to the strong theoretical reasons for thinking of institutions and technology as the primary determinants of per-capita living standards in the long-run. This has major implications for explosive growth forecasts, as TFP is additive not multiplicative. This implies linear growth in per capita output in the long-run, balanced growth equilibrium, as incomes can continue to grow from a higher base even as the growth rate trends to zero. As Thomas Philippon explains, the main time series breaks in linear TFP growth thus stem from the introduction of general purpose technologies:¹

A symptom of the failure of the exponential model is that the estimated trend growth rates are unstable. By contrast, the additive TFP model displays very few breaks and, in most cases, these breaks have a plausible economic interpretation in terms of General Purpose Technologies (GPTs). For example, the process of US TFP increments has only one break over the past 130 years, around 1930, following the large-scale implementation of the electricity revolution (Gordon, 2016). I investigate growth before 1890 using UK GDP per capita and I find two breaks between 1600 and 1914. The first is somewhere between 1650 and 1700, when growth becomes positive. The second is around 1830. These breaks are consistent with historical research on the first and second industrial revolutions (Mokyr and Voth, 2010). These rare breaks represent the main source of convexity in the historical series and TFP growth appears to be linear between the breaks.

If this pattern holds, I expect AI — as a general purpose technology — to increase TFP's linear growth coefficient. Rather than being a single inflection point, the diffusion of AI across the economy will produce a series of sigmoids that aggregate into a faster but ultimately temporary growth rate speed-up before reaching a steady-state at a higher balanced growth rate, reflecting the AI-augmented productivity of labor and capital inputs. Given the genuinely endogenous role AI can play in science and R&D, this growth speed-up could be quite dramatic, but I still struggle to imagine annual GDP growth ever reaching 30% on a sustained basis, even with cheap fusion and other breakthroughs in the physical sciences. Nonetheless, the more dramatic the rate change, the shorter the interval of that higher rate, as the quicker we'd converge to the new production frontier, rather than fly off to infinity.

Throughout this process, it’s hard to imagine our institutions as they're presently designed surviving even a fraction of this pace of change. While E&B argue that “the experience of Chinese catch-up growth shows that sustained growth rates on the order of 10%/year and one-time growth rates on the order of 15%/year have precedent in economic history,” this is growth from a much lower base. A human can survive a car accelerating from 0 to 60 mph in 4 seconds, but accelerating from 60 to 600 mph in 4 seconds is enough to be deadly. Likewise, to the extent E&B argue growth will break through regulatory and other socio-technical bottlenecks, they're implicitly forecasting massive property rights violations and a broader institutional regime change, as our current institutions simply do not support the economic throughput they envision, even with all the provisos I've given here.

Like it or not, the long-run future is thus closer to the Robert Gordon world where we eventually pick all the fruit — low-hanging and otherwise — before settling into some absurdly high, but asymptotic, standard of living. Just as you can't discover Maxwell’s Equations or bring literacy rates up to 100% twice, AI can't cure cancer or automate the entire services industries twice. Even radical life extension has its limits, as memories fade. And while superintelligence may allow for all kinds of megaprojects and space colonization efforts, whether this adds to real living standards isn’t clear either. Certainly, it doesn't enhance my welfare if a tech billionaire sends robots off to terraform Mars. From the standpoint of per-capita consumption, space colonization is just an interstellar version of Jay Leno’s garage.

The (bounded) subjectivity of value

The economists’ claim that “human wants are infinite” is a fiction arising from the subjective theory of value. Subjectivism is a reasonable approximation, as beauty is in the eye of the beholder. Nonetheless, humans still have a finite sensory throughput, which puts a hard phenomenological bound on our potential quality of life.

Even at our relatively low level of “abundance,” wealthy countries exhibit a rise in post-material values. In rich countries, marginal consumption becomes increasingly conspicuous and zero-sum (Keeping up with the Joneses), before beginning to shift into non-material forms of consumption, like relative status, luxury beliefs, or experience goods like world travel. Competition over positional goods could carry measured GDP onward and upward, but the more that GDP goes into status games, the less well it proxies actual well-being, and the more we come to value regulations and incentives designed to forestall zero-sum competition.

The subjectivity of value further constrains explosive growth insofar as it reframes what growth is for. GDP is not well-being, just a proxy, as production is a price-valued quantity determined by the subjective value-added perceived by human buyers and sellers. Consider the planets in our galaxy made entirely of diamond. What GDP should we ascribe those planets? With no human to value it, I would argue their GDP is zero. But even if we could one day access those planets, the putative value of a quadrillion dollars worth of diamond would immediately collapse to the value of similarly abundant commodities like sand or water. Subjectivism thus implies that GDP is a social construction, not an objective quantity, as if the work performed by AI workers accumulated according to some Marxist labor theory of value.

Another way to see this is to imagine AI leading to breakthroughs in Brain Computer Interfaces that let us upload our minds and live in virtual worlds. In a virtual world, I can “own” 1000 cars and live in the Burj Khalifa with unlimited goods and services, all for the cost of electricity. “Actual” goods and services production would thus collapse, creating a world of subjective post-scarcity that decouples well-being from GDP. Something similar would happen if technology one day lets us directly modify our desires. With a few neuronal edits, I could make broccoli taste like chocolate, turn pain into pleasure, and otherwise transform my perceptions to whatever my meta-preferences desire. What would 30% GDP growth even mean in such a world? I posit it would be meaningless.

In practice, though, most people just want to live happy and healthy lives, to find a life project, and to contribute to their broader community before dying at a ripe old age surrounded by family. That is, human values aren't subjective in some completely arbitrary way, but are deeply rooted in our human nature and evolutionary psychology. My long-run expectation for AI (again, assuming humanity isn’t simply replaced) is thus a world that looks neo-traditional, a la Avatar's Pandora. With radical abundance, we can finally undo the malaise and alienation brought on by modernity — Freud’s Civilization and Its Discontents — in favor of a high-tech version of the hunter-gather modes of living that are most aligned, in an evolutionary mismatch sense, to our mental and physical flourishing.

That is, once our material wants are fully met, I suspect humanity’s focus shifts back to the spiritual. History thus ends where it began.

1. See Matt Clancy's commentary.