# Inequality of Returns and Reinvestment Opportunity

Larry Summers has one of the more careful reviews on Piketty’s Capital. His two central objections (in an otherwise glowing review) focus on Piketty’s assumptions that:

1. The elasticity of substitution between capital (K) and labor (L) is greater than 1. In other words, the decrease in capital income caused by an increase in the underlying stock (diminishing returns) is outweighed by the broader base, thereby decreasing labor share of income. (A point highlighted by Krugman, too).
2. All capital returns are reinvested. (That is, the s in the s/is at least constant and possibly growing).

I have nothing novel to say about the veracity of either claim. In fact I’d go a little bit farther than Summers in my skepticism of the former claim. As Summers notes:

But I think he misreads the literature by conflating gross and net returns to capital. It is plausible that as the capital stock grows, the increment of output produced declines slowly, but there can be no question that depreciation increases proportionally. And it is the return net of depreciation that is relevant for capital accumulation. I know of no study suggesting that measuring output in net terms, the elasticity of substitution is greater than 1, and I know of quite a few suggesting the contrary.

I would forget empirical studies for once and appeal to common sense. On top of Summers’ own skepticism regarding the elasticity of substitution accounting for depreciation (which is certainly lower than the gross figure) Piketty seems to glide over the challenge that this elasticity is also important in ensuring that as $g$ falls $r-g$ does not. (Naturally, they must both tend to zero together). Therefore, one must be convinced that the elasticity of substitution is substantially greater than one. Like Summers, I am not aware of many credible studies suggesting that this is the case (though, unlike Summers, my ignorance doesn’t mean much).

However – and I think this is a worry to which Summers is very sensitive – I am not sure that what we consider to be “accepted” elasticities today must hold in the near future. Here’s Summers:

Even where capital accumulation is concerned, I am not sure that Piketty’s theory emphasizes the right aspects. Looking to the future, my guess is that the main story connecting capital accumulation and inequality will not be Piketty’s tale of amassing fortunes. It will be the devastating consequences of robots, 3-D printing, artificial intelligence, and the like for those who perform routine tasks. Already there are more American men on disability insurance than doing production work in manufacturing. And the trends are all in the wrong direction, particularly for the less skilled, as the capacity of capital embodying artificial intelligence to replace white-collar as well as blue-collar work will increase rapidly in the years ahead.

What I read into this is that your good old $Y = F(K,L)$ production function will no longer capture the key themes of economic growth. The robotic future is better modeled with income as $Y = F(K, G(K_r, L))$ where $K_r$ is modern technology like “robots, 3-D printing, artificial intelligence, and the like” which is very easily substitutable with labor. If you take $F = K^{\alpha}G^{1-\alpha}$ and $G$ as another constant elasticity function what you notice is that as $K$ and $K_r$ tend to perfect substitutability, the $\sigma$ grows without bound. (Predicated on the assumption that robots and labor are theoretically substitutes which holds if, but not only if, that elasticity is greater than the elasticity between capital in general and the G). In any case, the elasticity that Summers doubts is very high is exponentially amplified by his own concerns of substitutable labor.

This analysis risks boiling town the tome of the decade into another trite debate about robots (though I think it is worthwhile to parse the internal contradictions of those making Summers’ argument something not uncommon in Piketty criticism). I would rather avoid that in arguing it really doesn’t matter much whether, qualitatively, returns to capital are diminishing quickly or whether savings are being reinvested. So let’s assume that neither of the two assertions Summers argues Piketty needs are true.

Summers says that returns are not reinvested since the most common form of return, imputed rent, is definitionally consumed. Yet this ignores the gaping wealth inequality that already exists (not one, granted, that I am very concerned about). While a quick read on Piketty suggests the central problem is “r>g”, the more dangerous problem is the inequality of returns itself. The distribution of stock ownership – the only real source of long-run returns – is far more skewed than that of wealth ownership. The bottom 95% own all of their wealth in two forms:

• Explicitly: Home ownership which forces saving over a period of thirty years.
• Implicitly: Social security which acts as a claim on America’s future GDP.

Each of these are invested in low-return (real estate and government bonds, respectively) securities. Diminishing marginal returns for the many are real. But if you’re a billionaire, you have access to the stock market, cleverly managed funds and, most importantly, the right information to make the right investments. One can reach the same conclusion as Piketty when wealth accessible to the broad middle-class is consumed and does face diminishing returns whereas that available to a percent reaching escape velocity does not.

While the wealth effect, suggesting that increases in wealth decrease savings, is certainly true for those who borrow on home equity loans and are liquidity-constrained (most of America), it certainly does not for those who are benefitting from wealth inequality, the top percent of a percent. Does anyone seriously think a multimillionaire is more likely to go on a fancy vacation to Europe – a minuscule fraction of his income and wealth – just because the S&P grew a little faster than he expected? I would imagine Summers does not think so, as his consistent calls for higher taxes on high income labor and capital gains would suggest otherwise. (And if it were the case that the superrich were inclined to consume more given higher capital returns, the case for progressive taxation of dividends and carried interest sours non-negligibly).

The Forbes 400 list cited in his review, while interesting, is hardly informative of the dynamics of our economy. The same list also suggests that equal opportunity has increased substantially over the past thirty years. (Not to mention that many on the list in the ’80s are not longer interested in chasing capital gains, but in finding meaningful ways to spend their amassed fortune – no mean task).

Ultimately, I think Summers’ analysis falls short in assuming that all wealth and all returns are the same. That there is one $r$, when there are many:

I am much less sure. At the simplest level, consider a family with current income of 100 and wealth of 100 as opposed to a family with current income of 100 and wealth of 500. One would expect the former family to have a considerably higher saving ratio. In other words, there is a self-correcting tendency Piketty abstracts from whereby rising wealth leads to declining saving […] The general conclusion of the research is that an increase of $1 in wealth leads to an additional$.05 in spending. This is just enough to offset the accumulation of returns that is central to Piketty’s analysis.

Of course, this argument would seem a lot less compelling if “a family” was replaced with “Bill Ackman”, even though that is precisely where wealth is increasing most readily and extraordinary returns are most ubiquitous. Unfortunately, sensible assumptions ensure that Piketty’s analysis is theoretically contestable until there are some massive changes in our social security and home ownership programs. And that’s assuming Google doesn’t triple the elasticity of substitution single-handedly in the next ten years.

1. Brett said:

Do you think the Social Security Fund should be allowed to use any income and surplus in the fund to invest in a broader set of securities than US Treasury Bonds, including higher yielding ones with the surplus returned as extra pay-outs to beneficiaries? I’ve been thinking about that lately, and I think it would be worthwhile to do so even if there’s a degree of moral hazard due to Social Security being backed by the Full Faith and Credit of the US.

I guess I still don’t see the robot factor that well. Even if automation is fully replaceable with human labor (which would imply some incredibly impressive artificial intelligence), would the ratio of using automation and humans simply vary depending on the relative cost of both? It would still be worthwhile to use human beings in various tasks where they might be cheaper than the comparable automation.

Or I guess we could end up in Mutual Fund Future, where everybody gets payouts and the capital share of the economy is something absurd like 90%, while robots do all the work.

• The short answer is yes. I’m not sure where the moral hazard is, exactly. I suppose you would want to incentivize the fund managers to make the best, most rewarding investments but without leverage I don’t really see this as a big problem. It’s an exercise in surplus optimization, something we’re quite good at doing.

With regard to robots, sure, perfect substitutability is unlikely. How you want to look at it is that if F is Cobb-Douglas and G is any CES function, than you have (by quick math, which may be wrong) that mu = 1 – 1/sigma where mu is the robot-labor substitutability and sigma is the elasticity of substitution that Piketty/Summers talk about. In this case, as robots tend to perfect substitution (mu = 1), sigma tends to infinity. It’s not hard to see sigma grows very quickly in mu.

And of course, some labor will not be substitutable. Chances are this labor is performed by the very affluent.

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