When he’s not botching basic historical facts about tax rates and minimum wages–in ways that conveniently serve his political narrative–Thomas Piketty, author of the celebrated *Capital in the 21st Century*, is botching the basic theory of capital and interest. Here at Mises Canada I’ve already pointed out a fundamental theoretical problem with Piketty’s whole structure. But in Larry Summers’ recent book review, he confirms another enormous problem with Piketty’s whole approach. Here’s Summers:

Economists universally believe in the law of diminishing returns. As capital accumulates, the incremental return on an additional unit of capital declines. The crucial question goes to what is technically referred to as the elasticity of substitution. With 1 percent more capital and the same amount of everything else, does the return to a unit of capital relative to a unit of labor decline by more or less than 1 percent? If, as Piketty assumes, it declines by less than 1 percent, the share of income going to capital rises. If, on the other hand, it declines by more than 1 percent, the share of capital falls.

Economists have tried forever to estimate elasticities of substitution with many types of data, but there are many statistical problems. Piketty argues that the economic literature supports his assumption that returns diminish slowly (in technical parlance, that the elasticity of substitution is greater than 1), and so capital’s share rises with capital accumulation. 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.

This is an absolutely devastating observation; it topples Piketty’s whole argument. Incidentally, I first saw a PhD student at MIT raise this point (in the comments of a Tyler Cowen blog post). Brad DeLong didn’t object to the student’s point, and now Larry Summers too is explicitly confirming it. So let me walk people through what’s going on here, to show why it is so critical. (Note that for the rest of this post, I am going to take the standard neoclassical approach to capital and interest at face value; Piketty is wrong even in the narrow confines of his own chosen paradigm.)

With his now famous claim that “r > g” for the indefinite future, Piketty is saying that the real return to capital will remain higher than the growth rate in total output. Therefore, Piketty argues, the size of the “capital stock” measured in terms of “years’ worth of output” will continue to rise in the coming decades. In other words, “capital/income” will increase.

Now if the size of the capital stock rises, what can we say about the share of income going to the capitalists each year, compared to the laborers? Well there are two forces. On the one hand, the larger the capital stock, the more units of capital the owners possess, earning returns out there in the market. So that would tend to make the total earnings of the capitalists go up. On the other hand, the larger the capital stock, the lower we’d expect “r” which is the real return on capital. (This is because of diminishing returns; it’s just as we’d expect wages to fall if the labor stock grows.) So that would tend to make the total earnings of the capitalists go down. The actual outcome depends on which of these two forces is stronger: As the number of units of capital goes up by x%, does the individual earnings per unit fall by more or less than x%? (In the special case of a “Cobb-Douglas production function,” the two forces always exactly balance each other, meaning that capital always earns the same percentage of output each period, with labor getting the rest.)

Piketty wants to scare his readers into believing that the percentage of income each period going to the capitalists will increase over time, meaning that “the workers” will earn a lower portion of total output in, say, 2100 than they do right now. So he needs to argue that the parameters on his assumed production function are such that an increase in the capital stock of x% will lower the return to capital “r” by less than x%. In the jargon of economists, Piketty needs to argue that empirically, the “elasticity of substitution” is greater than 1.

Piketty does indeed try to show this, by pointing to some of the high-end estimates in the literature putting the elasticity at 1.25. Yet as Rognlie the MIT grad student points out, this is confusing gross with net returns. If you are a capitalist who owns a bunch of machines, your *net* income each year is equal to

==> your gross rental income from your machines

==> minus the drop in the market value of your machines, either due to a change in prices or physical depreciation.

So as Rognlie and now Larry Summers confirm, there are no empirical estimates of an elasticity in substitution being so much above 1 that the share of *net* income (after accounting for physical depreciation) going to capitalists would be expected to indefinitely increase over the coming decades, as the capital stock accumulates.

Furthermore, beyond this problem with Piketty’s reading of the literature, Summers points out in his book review that Piketty’s thesis doesn’t work at all when looking at the richest people in America from 1982 – 2012, as Bryan Caplan highlights.

Now you might think that since Summers himself blew up the entire foundation of Piketty’s argument, he (Summers) might chide his fellow progressive economists in their over-the-top praise for the book. Nope… Summers says the book “richly deserves all the attention it is receiving.”

How to explain this apparent imbalance between the coherence of the book’s case, and the praise it is receiving? No need for us to cynically speculate; we can let Larry Summers himself explain what’s going on here:

Piketty, in collaboration with others, has spent more than a decade mining huge quantities of data spanning centuries and many countries to document, absolutely conclusively, that the share of income and wealth going to those at the very top—the top 1 percent, .1 percent, and .01 percent of the population—has risen sharply over the last generation, marking a return to a pattern that prevailed before World War I. There can now be no doubt that the phenomenon of inequality is not dominantly about the inadequacy of the skills of lagging workers. Even in terms of income ratios, the gaps that have opened up between, say, the top .1 percent and the remainder of the top 10 percent are far larger than those that have opened up between the top 10 percent and average income earners. Even if none of Piketty’s theories stands up, the establishment of this fact has transformed political discourse and is a Nobel Prize-worthy contribution.

And there you have it. “Even if none of Piketty’s theories stands up”–and Summers has already shown us that they are prima facie wrong–Piketty should get a Nobel Prize because he’s documented how much richer the super rich have gotten over the last generation.

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