# Piketty’s Capital: III

In a recent post on *The Beacon* I argued that what Thomas Piketty called “the first fundamental law of capitalism” in his recent book, *Capital in the Twenty-First Century*, depicted the causal relationship between the value of capital and the return earned by capital backwards. Representing the return on capital as α, the rate of return as *r*, and the value of capital as β, Piketty argues that the value of capital, multiplied by its rate of return, determines the return on capital, or in mathematical notation, α=*r*xβ.

In fact, the value of capital is determined by the return it earns, so a mathematically equivalent but more economically correct way to restate Piketty’s fundamental law is β=α/*r*. See my earlier post for a more complete explanation.

In an accounting sense, Piketty’s equation is correct, but my restatement gives a more accurate representation of the causal relationships. I discussed the role of α in my earlier post, but this restatement also more accurately represents the role of *r*, the rate of return on capital.

A quick technical clarification: Piketty defines both the value of capital and the income from capital as a fraction of total income, so β is the value of capital divided by income and α is capital income as a fraction of total income. But for this discussion we can multiply both sides of the equation by income and just refer to β as the value of the capital stock and α as the income it earns without misrepresenting the concepts Piketty discusses, and simplifying the discussion.

In his discussion Piketty recognizes that more risky investments will earn a higher return, but that idea plays no role in his empirical analysis that follows, and in particular, the idea that capital owners make decisions to risk their capital and are not guaranteed any rate of return receives minimal consideration. As I discussed in my previous post, the income capital owners earn from their capital depends on their investing it effectively in projects that create value for the economy.

One can see that looking at my rearranged fundamental law, β=α/*r*, that for a given return on capital, if α rises, then the value of capital, β, will also rise. The value of capital grows in proportion to the income it produces, as opposed to the way Piketty states it, where the income it produces depends on its value.

What about the rate of return on capital? Piketty says it has been relatively constant over long periods of time, and I have no problem with this generalization as a stylized historical fact. One can see from looking at the equation that if the economic environment becomes more risky, so owners of capital receive a higher return (*r* rises), then the value of capital will go down even if the income it earns remains constant. The common sense of this is that if there is more risk involved in owning capital, the capital won’t be worth as much, confirming what the equation shows.

Likewise, if the economic environment becomes less risky, the owners of capital will receive a lower return, and the value of capital will rise.

Risk is not the only factor that affects the return on capital. Government policy can also have a substantial effect. Piketty shows that β has increased substantially in the twenty-first century, which supports his argument of growing inequality. But consider that monetary policy in most of the twenty-first century has been geared toward keeping interest rates low, first under the Greenspan Fed in response to the recession in the early 2000s, and then after 2007 in response to the growing financial crisis.

A lower interest rate lowers the rate of return on capital, so again looking at β=α/*r*, the Fed’s policy of lowering *r* has had the effect of increasing β, that is, of increasing Piketty’s measure of inequality. The bulk of the growing inequality Piketty sees in the twenty-first century is not the result of anything inherent in capitalism, as he claims, but rather is the result of a deliberate policy on the part of the Federal Reserve.

The conventional wisdom in financial markets is that the low rate of return on fixed interest investments has pushed money into the stock market, which has been responsible, at least in part, for the stock market’s rise since the crash after the recession. Piketty’s law supports this, because the Fed has forced a lower rate of return on investment, lowering *r* and causing β to be higher for a given α.

This being the case, twenty-first century data cannot be used to argue persuasively about rising inequality because β is rising, as Piketty argues. Interest rates can’t be pushed lower, and when interest rates rise, that will increase *r* which will reduce β for any given α. The effect Piketty documents in the twenty-first century is temporary.

I hope I haven’t taxed readers too much with a pair of fairly technical posts in Piketty’s book. (If you’ve read this far, you must have at least some interest.) I will post some additional thoughts on Piketty’s book in another post, lighter on the technical aspects and more reflecting on his overall message.