capital revisited
Dwarkesh and Trammell (D&T) wrote a post on "Capital in the 22nd Century" [1] that is interesting, but long and, in parts, unclear. I attempt a shorter version here that aims for clarity, and end with brief comments on the argument itself.
D&T's post is a riff on economist Thomas Piketty's "Capital in the 21st Century" [2], in which Piketty traces capital accumulation and its effects on societal inequality.
Since Piketty's model grounds D&T's argument, we'll first explain Piketty's workhorse model, and then work through it toward D&T's conclusion that AI could lead to runaway inequality.
Definitions
- Y is the income an economy produces each year.
- W is the privately owned wealth accumulated over time.
- β is wealth measured in years of income or W/Y.
If an economy produces Y = $100 each year and contains W = $500 of private wealth, then:
β = W/Y = $500/$100 = 5,
Or wealth is equivalent to 5 years' worth of income.
Now let r be the average annual return on that wealth (W). At a return of 5%, $500 of wealth produces $25 of income. Capital's share of income, which we call α, is therefore:
α = rβ = 5% * 5 = 25%
This is an accounting definition: capital's income share (α) equals the amount of wealth relative to income (β), multiplied by its return (r).
How Wealth's Share of Income Grows
How does wealth (β) grow? Let s be the fraction of annual income saved and reinvested, and g the annual growth rate of income. In a stable long-run relationship, wealth grows at the same rate as income, so Piketty argues:
gW ≈ sY, therefore β = W/Y ≈ s/g.
For example, if an economy saves s = 10% of its income while income grows by g = 2% a year, wealth (β) tends towards 5 years' worth of income. Higher saving (s) and/or slower growth (g) makes accumulated wealth (W) larger relative to what the economy currently produces (Y).
The question is what happens to return r as this wealth accumulates? If β doubles but r more than halves, capital's income share (α) falls since α = rβ. If r falls by less than half, capital's income share (α) rises.
This is separate from Piketty's famous inequality r > g. It says that wealth can compound faster than the economy grows, provided its owners save enough of their returns. But r > g does not by itself imply that capital's share of current income rises. For that, β must rise without r falling proportionately.
D&T want to know under what conditions capital's income share approaches 1, and in whose hands the resulting wealth concentrates.
Complements or substitutes?
To understand what happens with r, we need to know if capital and labor are complements or substitutes. Assume a production function:
Y = F(K, L)
- K is productive capital (e.g. computers).
- L is labor.
For this stripped-down model, assume that private wealth W consists mainly of claims on K. Let w be the wage paid for labor and r the return paid on capital.
The elasticity of substitution, σ, measures how readily firms change the ratio of capital to labor when their relative costs change:
σ = %Δ(K/L) / %Δ(w/r)
Suppose capital becomes 10% cheaper relative to labor and a firm uses only 5% more capital per worker, then σ = 0.5. If it uses 20% more, then σ = 2.
There are three cases to consider.
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If σ < 1, capital and labor are gross complements. Firms use more capital as it becomes cheaper, but not enough to offset the lower return on each unit. Capital's total income share falls.
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If σ = 1, as firms use more capital it exactly offsets the lower return. Capital and labor retain constant income shares.
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If σ > 1, capital and labor are gross substitutes. Firms use capital more than proportionally as it becomes cheaper, so capital's income share rises.
σ > 1 is What Matters
Capital changes according to:
K̇ = sY − δK
The dot means "change per year". New investment is the net saved fraction (s) of output (Y); δK is the depreciation.
When σ < 1, labor remains a bottleneck. As we add more capital per worker, each additional unit becomes less useful. Capital's return eventually falls so far that saving merely replaces depreciation. Capital accumulation converges towards a stable level.
If σ > 1 continues to hold as capital becomes extremely abundant, labor eventually ceases to be a binding constraint. Production then approaches:
Y = AK
where A is the amount of output produced per unit of capital. Substituting this into the accumulation equation gives:
K̇ = s(AK) − δK = (sA − δ)K
If sA > δ, capital grows at a positive percentage rate indefinitely. More capital produces more output; part of that output is invested to produce more capital (s); and the cycle repeats without requiring more labor. At the limit, capital's share (α) approaches 100% or 1.
D&T's Argument
This is the first part of D&T's argument. Piketty was probably wrong about the past because capital and labor have historically behaved as complements (σ < 1). But Piketty may be right about a future in which AI and robotics allow capital to reproduce without human labor being required to grow the economy.
The second part of their argument concerns ownership. Wealth is already much more concentrated than labor income. As income shifts from wages to returns on capital, income inequality therefore rises toward the existing inequality of wealth ownership. If wealthy households also save more or earn higher returns, wealth ownership itself can become progressively more concentrated.
The third part of their argument concerns who owns the capital most exposed to AI growth. Many young AI companies remain private during their fastest-growing phase. Their gains therefore accrue primarily to founders, employees, venture funds, private-equity investors, and institutions or wealthy individuals able to access those funds. Public companies and pension funds can obtain some indirect exposure, but ordinary investors may receive access only after much of the early appreciation has occurred. If private firms continue capturing a growing portion of corporate value, this "privatization of returns" could amplify the initial inequality in capital ownership, if AI takes us into a σ > 1 world.
D&T see two states of the future world: a Baumol or a Jevons world.
In a Baumol world, AI makes some outputs extremely cheap but cannot replace labor in every activity people value. Spending consequently shifts toward labor-intensive services such as elder care, live performance, sports, construction, and hospitality. Human labor remains a bottleneck, so its income share remains substantial and could even rise.
In a Jevons world, making AI services cheaper causes their use to increase by more than their unit price falls. D&T use this idea more broadly to describe a world in which capital remains useful no matter how abundant it becomes. Capital increasingly replaces labor, its income share rises, and accumulation can become self-sustaining.
Comments
Three main comments come to mind.
First, Piketty's model is obviously highly stylized, but it's a helpful aid to keep the logic tidy throughout. He's right to try to identify the underlying mechanism that allows for runaway capital accumulation and widening inequality. However, historical estimates generally place σ < 1 [3]. Rognlie finds that the postwar increase in the net capital share comes entirely from housing rather than productive capital accumulation [4]. And, Acemoglu and Robinson find no consistent correlation between r − g and top income shares across countries and time [5].
Second, there is not yet strong evidence for a substantial Jevons effect in AI inference [6]. Demirer, Fradkin, Tadelis, and Peng estimate a short-run price elasticity of approximately −1.11. Taken literally, a 10% decline in price produces roughly an 11% increase in use, leaving total expenditure nearly unchanged. The point estimate is therefore just on the Jevons side of the −1 threshold, but it is not statistically distinguishable from unit elasticity. Moreover, the estimate partly captures substitution among providers serving the same models, rather than the economy-wide response to cheaper intelligence. It consequently tells us little yet about D&T's much stronger claim that AI capital will become a substitute for labor throughout the economy.
Third, it would seem to me that embodied intelligence (e.g. robotics) would have to be sufficiently cheap, reliable, and performant for us to approach a Y = AK economy. Clearly this field is moving fast, but it isn't yet readily apparent that we can take highly capable LLMs and put them into autonomous robots to complete high-value economic activity (e.g. natural resource extraction, energy production, farming, construction, etc.) without a human in the loop. Having said that, I assume this is on the horizon, so this is more a comment on timing than whether this is a feasible outcome.
Taken together, the D&T scenario seems more plausible than not as a mechanism, but the current evidence runs the other way. The obstacles I can identify are about timing rather than possibility, which is a reason to hold the conclusion loosely rather than to discard it. In the meantime, two trends are worth watching: where consumer spending flows as AI-produced goods get cheaper, and how fast embodied intelligence arrives as AI diffuses further into the economy.
[1] https://philiptrammell.substack.com/p/capital-in-the-22nd-century
[2] https://en.wikipedia.org/wiki/Capital_in_the_Twenty-First_Century
[3] https://ezraoberfield.github.io/CESAggregation.pdf
[4] https://www.brookings.edu/wp-content/uploads/2015/03/RognlieText.pdf