Krugman is Wrong
Matthew Martin 12/28/2012 09:51:00 PM
"if it’s strongly enough capital-biased, they [workers] can actually be made worse off"
which is absolutely, unbelievably wrong.
That tells me that he has in mind something similar to the cobb-douglas production function:
BUT! Krugman is wrong in saying that wages fall as a result of an increase in alpha. Yes, he is right that the wage equals the slope of the curve in the graph. But the graph is deceptive--in reality the wage will rise!
I can't believe I'm the first economist to point this out. Fred Mosely and Nick Rowe both weighed in and, in my humble opinion, had nothing useful to say. But no one seems to have pointed out that Krugman's problem is that he is assuming that capital is fixed, when in reality we would expect the steady state equilibrium level of capital to rise in response to an increase in capital's share of income (alpha). Maybe that is what Rowe was hinting at here, though if so he's quite cryptic about it.
I'm relying on a series of well known results in macroeconomics, so for the non-math readers, here is the result: any capital-biased technological change will cause investors to invest in producing more capital, because the return to capital is now much higher than before (intuitively, they are more willing to give up current consumption to invest in capital now). This means that the capital stock rises to a new equilibrium value. Capital and labor are complements in production, so an increase in the capital stock also increases the returns to labor, causing wages to rise. While it is true that total output rises more than wages do, meaning that workers earn a smaller share of GDP, Krugman is absolutely wrong to suggest that workers are worse off than before--in fact, their incomes do rise, possibly quite substantially.
Anyway, here is the math (after the jump):
I'm relying on well-known results in macro, so for brevity I will skip the derivations--you may disagree with assumptions of the model, but I'm not wrong about the results those assumptions imply. So here are my assumptions: for simplicity, the labor supply is fixed (the same thing krugman assumes), capital and consumption are substitutes (meaning that I+C=Y, where I is the amount of new capital produced that period), the production function is cobb-douglas like the one above, and markets are perfectly competitive. It doesn't matter what units of measurement we use for labor, so lets just normalize it so that one unit of labor is supplied each period. Also, as is typical for all macro models, there are frictions in capital markets, so as my prof Dr Reynolds used to say "capital enters the production function with a one-period lag." For those of you familiar with the method, here is a concise statement of the model I'm considering (written out as a "planner's problem," also, my apologies for being lazy about writing expectations and time subscripts--that is supposed to be the expected value of summed utility, and alpha, A, K, and C are all supposed to have subscript t's.):
But here is the real point. Here is the formula for the steady state level of capital:
First of all, note that alpha does, in fact, enter into the formula for K, the steady-state level of capital. That means Krugman should not be assuming that capital is fixed! Now it is as easy as plugging in numbers. We will hold these parameters constant: A=1, beta=0.99, delta=0.1 and the only parameter we will change is alpha. Here if alpha is equal to 0.2, which implies that capital earns 20% of GDP, then we get K=2.1 and plugging this into the wage equation shows that total labor income in this economy is 0.93. If we plug in alpha=0.8, which represents a large capital-biased technological change exactly like what Krugman drew in his graph, then we get K=20253.2 and w=557.5. Just to be clear, I've copied and pasted the exact results from computing the formulae above:
So I find myself in an unusual place. Normally when I want to know what I think about something, I read Krugman. But in this case I find myself saying that Krugman is wrong. Even with "capital-biased" technological change, labor incomes still rise. And this allows us to apply results from the Ramsey optimal tax literature that also says Krugman is wrong: in particular, Krugman is incorrect that capital's rising share of income is a good reason to raise taxes on capital to pay for retirement programs like social security and Medicare. The owners of capital don't need their benefits to keep pace with their rising income from capital since they, by definition, make up for that in the fact that they have savings invested in capital to finance their own retirements. And workers' wages still rise by an ample amount in response to capital-biased change, so it is quite possible to finance the government by taxing labor income but not capital. And the mainstream Ramsey literature says that it is optimal (eventually) to cut taxes on capital to zero. There are a few caveats to this result, but these are not part of Krugman's argument so I won't address them here.