Piketty's book is like a giant attention-sucking vortex. I can't seem to escape it. This time I'm thinking about the criticism of Piketty's analysis that has to do with rates of return on capital. Piketty says that if ${r > g}$, where ${r}$ is the return to capital, and ${g}$ is the growth rate of aggregate GDP, then wealth will become more and more concentrated.

Critiques of Piketty have questioned the assumptions underlying this conclusion. The most recent one I've seen is in Larry Summers' review piece. Let's let him sum up the issues:

This rather fatalistic and certainly dismal view of capitalism can be challenged on two levels. It presumes, first, that the return to capital diminishes slowly, if at all, as wealth is accumulated and, second, that the returns to wealth are all reinvested. Whatever may have been the case historically, neither of these premises is likely correct as a guide to thinking about the American economy today.

With respect to the first assumption regarding the rate of return, here is what Summers says:

Economists universally believe in the law of diminishing returns. As capital accumulates, the incremental return on an additional unit of capital declines.

But Summers has fallen into what I think is a really common trap for economists. He presumes that his second statement ("As capital accumulates, the incremental return on an additional unit of capital declines") contradicts Piketty's assumption ("that the return to capital diminishes slowly, if at all, as wealth is accumulated"). These two statements are not mutually exlusive.

The issue is that Summers is confounding wealth and capital. This is not helped by Piketty, who uses "capital" in his title and in the book the way that normal people use it, as a synonym for "wealth". But from the perspective of an economist, these two concepts are *not* the same thing. The capital that Summers refers to in his critique (often denoted ${K}$) is a subset of the measure of national wealth (${W}$, as I'll call it) that Piketty documents.

Without going too deep into this, Piketty's measure of wealth consists of three parts: real estate, corporate capital, and financial assets. Only real estate and corporate capital are what economist have in mind when they say capital (${K}$). Wealth, however, consists of all three parts, so that Piketty's wealth is ${W = K + F}$, where ${F}$ is the value of financial assets. Asserting that the return to capital falls as the capital stock increases - as Summers does - does not imply that the return to *wealth* falls as the stock of wealth increases. Even if we assume that financial markets work so efficiently that the return to capital and the return to financial assets are identical, this does not mean that the return to wealth necessarily falls as wealth accumulates.

To see this, consider a really slimmed down version of the "bubble asset" model from Blanchard and Fischer (1989, p. 228). We have that the return on capital is ${r = f'(K)}$, where ${f'(K)}$ is the marginal product of capital. The ${f'(K)}$ is the derivative of the production function, and represents the marginal increase in output we'd get from adding one more unit of capital. Under our typical assumptions about diminishing returns, as ${K}$ goes up ${r}$ goes down. This is what Summers is using as his critique.

An efficient financial market would ensure that financial assets (F) would *also* have a return of ${r}$. If they did not, then people would buy/sell financial assets until the return was equal. (Yes, I'm ignoring risk entirely, but that doesn't change the main point here). So the return on all wealth is equal to ${r}$, and note that this is pinned down by the value of ${K}$ alone.

Now, we have assumed that ${r}$ falls as ${K}$ increases. Does this imply that ${r}$ falls as wealth (${W}$) increases? No. The relationship between ${r}$ and ${W}$ depends entirely on the composition of the change in ${W}$. If ${W}$ rises because ${K}$ rises (say ${F}$ stays constant), then the rate of return on wealth *falls* because the marginal product of capital has declined. This is what Summers and others have in mind.

However, it's perfectly plausible that ${W}$ rises even though ${K}$ falls, because the value of financial assets (${F}$) are increasing even more quickly. In this case, the marginal product of capital has increased, and the rate of return on wealth has *increased*. In this case, the rate of return rises with wealth.

Is it reasonable for an economy to experience falling capital but a rising value of financial assets? Sure. The point of Blanchard and Fisher's model of bubbles is that even though all individuals are acting rationally at all times, the economy can take off onto a weird path where the stock of capital (${K}$) gets run down while the value of financial assets (${F}$) rises. Eventually this is unsustainable, as we'd run out of capital, but there is no reason that a situation like this cannot persist for a while.

Will the return to wealth *necessarily* rise as wealth accumulates? No. There are other equally reasonable paths that the economy could take where wealth accumulation is driven mainly by capital accumulation and the rate of return falls as wealth accumulates, consistent with the Summers critique. The point I want to make is that there is no particular reason to believe in a fixed relationship between *wealth* and the return on *capital*. They can move completely independently of each other.

So Piketty can easily be right that we are currently in a world where both the wealth/income ratio is increasing *and* the rate of return on wealth is rising (or remaining roughly constant), and that this could persist for some indefinite period. On the other hand, it was not inevitable that this was going to happen, and it could just as easily end tomorrow as in 100 years.

I think the story that is milling around beneath the surface of Piketty's book is that recent wealth accumulation has been primarily of financial assets, not capital. Hence the return has stayed high and the concentration of wealth has continued. If the returns on that wealth are continually reinvested in financial assets as opposed to capital, then Piketty's death spiral of wealth concentration would likely be the outcome. To avoid that death spiral, you'd want to get the returns on wealth reinvested into real capital so that the return on capital (and hence wealth) gets pushed down.