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Thomas Piketty's best-selling book on inequality, "Capital in the Twenty-First Century" has attracted a lot of criticism on the right for its data analysis. Less well-known, however, is the criticism it's gotten from economists on the Left. Piketty uses an aggregate production function, where capital is represented by a single number K. This is a fairly standard thing to do in mainstream economics, but back in the 1960's there was a major debate in economics called the Cambridge Capital Controversy, revolving around the fact that capital is heterogenous and thus it may not be legitimate to represent it by a single number. So some economists argued that many standard conclusions in economists, like the notion that capital is paid its marginal product, are flawed.

Paul Krugman disagrees with this critique of mainstream economics, so in this blog post he defended Piketty from the charge that his conclusions depend on assuming a homogenous capital stock:

The thing to bear in mind, however, is that you really don’t need to reject standard economics either to explain high inequality or to consider it a bad thing.

There are a few economists on the left who seem to believe that:

  1. You need to believe in the existence of a perfectly well-defined aggregate measure of capital to believe in the marginal productivity theory of income distribution;
  2. If you believe in, or even use, marginal productivity theory, you are conceding that capitalists deserve their income.

Neither of these things are true. Nothing about marginal productivity theory depends on the exact truth of a simple aggregate production function with capital defined by a single number. And saying that capital gets its marginal product in no way says that the people who own that capital deserve what they get.

My question is about the statement in bold, where Krugman says that you can still get the conclusions of marginal productivity theory without assuming that capital can be described by a single number.

So my question is, what is the basis of Krugman's assertion? Can someone point me to a proof of the statement "capital is paid its marginal product" which doesn't depend on homogenous capital?

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Nothing about marginal productivity theory depends on the exact truth of a simple aggregate production function with capital defined by a single number

Different sorts of capital used as separate production technologies prevent clean aggregation to a representative form of capital but does not prevent capital from being paid its marginal product.

To make things simple, consider an economy with two kinds of capital $k_1$ and $k_2$ and no labor. There is only a single final good ($y$) but are two separate production technologies $y_1$ and $y_2$ where $y_1 = k_{1}^{\alpha_1}$ and $y_2 = k_{2}^{\alpha_2}$. Then aggregate production of $y$ is: $$y = k_{1}^{\alpha_1} + k_{2}^{\alpha_2}$$ If $0 < \alpha_1 < \alpha_2 < 1$ then for positive rental rates on capital $r_1$ and $r_2$ it will be efficient to choose a positive amount of both types of capital such that for each type $i$: $$ \alpha_i \cdot k_{i}^{\alpha_i - 1} = r_i\Rightarrow k_{i}^{*} = (r_i/\alpha_i)^{1/(\alpha_i - 1)} $$

There is no single $k$ that represents capital in this economy. Sure you could define $k=k_1 + k_2$, but that wouldn't mean exactly the same thing. On the margin the two sorts of capital don't have the same product and so aggregation doesn't make sense here. But in this setting, it is likely that the rental rate on capital would be equated ($r_1 = r_2$) because why would you buy one sort of capital when the other sort paid more?

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