Three Ways of Looking at alpha = r k

Piketty's "first law of capitalism" is the accounting identity

α = r k

where α is the share of capital income in total output, r is the average return on capital, and k is the aggregate capital-output ratio.

As accounting, this is true by definition. As economics, what kind of economic behavior does it describe? There are three ways of looking at it. 

In the standard version, the profit share is determined by a production function, which is given by technology. The profit rate r* required by capital owners is fixed by technology in combination with time preferences. In this closure, k is the endogenous, or adjusting, variable.  Investment rises or falls whenever the realized profit rate differs from the required rate, thus keeping k at the level that satisfies the equation for r  = r*. 

In Piketty’s version, r is fixed (somehow; the mechanism is not clear) and k is determined by savings behavior and (exogenous) growth according to his "second law of capitalism": 

k = s/g

That leaves α to passively accommodate r and k. Capitalists get whatever the current capital stock and fixed profit rate entitle them to, and workers get whatever is left over; in effect, workers are the residual claimants in Piketty's system. (This is the opposite of the classical view, in which wages are fixed and capitalists get the residual.)

In a third interpretation, we could say that α and r are set institutionally -- α through some kind of bargaining process, or by the degree of monopoly, r perhaps by the interest rate set in the financial system. The value of the capital stock is then given by capitalizing the flow of profits α Y at the discount rate r. (Y is total output.) This interpretation is the natural one if we think of “capital” as a claim to a share of the surplus as opposed to physical means of production. 

This interpretation clearly applies to pure land, or to the market value of a particular firm. What if it applied to capital in general? Since claims on the surplus -- including claims exercised through nonproduced assets like land -- are not created by reserving output from consumption, aggregate savings would be a meaningless accounting construct in this case. (Or we could adopt a Hicksian view of saving in which it equals the change in net wealth by definition.) Looking at things this way also puts r > g in a different light. Suppose we think of the capital stock as a whole as something like the stock of a firm, which entitles the owners to the flow of profits from that firm. If the profits today are α Y and output is expected to grow at a rate g, what is the value of the stock today? If we discount future profits at r, then it is the sum from t=0 to t=infinity of α Y (1 + g)^t / (1 + r)^t, which works out to α Y / (r - g). So if we can take the rate of return on capital as the discount rate on future profits, then r > g is implied by a finite value of the capital stock.

We shouldn't ask what capital "really" is. It really is a quantity of money in a process of self-expansion, and it really is a mass of means of production, and it really is authority over the production process. But the particular historical questions Piketty is interested in may be better suited to thinking of capital as a claim on the social surplus than as a physical quantity of means of production. Seth Ackerman has some very interesting thoughts along these lines in his contribution to the Jacobin symposium on the book.