Robert Murphy: Mainstream Economists Don’t Even Get Their Dimensions Right

By: Robert P. Murphy

There are many differences between Austrian economics and the neoclassical mainstream, but one of the most critical involves the difficult field of “capital & interest theory.” (Here are three links of increasing difficulty to show the Austrian perspective on these issues: onetwo, and three.) This area has been dubbed the “black hole of economics” because it can devour researchers, but in the present post I can use a recent Paul Krugman blog entry to graphically illustrate the Austrian viewpoint. Specifically, Krugman’s diagram doesn’t even get the dimensions right!

Before diving in, I should acknowledge that this particular dispute has nothing to do with Keynesian policy recommendations. Rather, the problem in Krugman’s diagram is something that is taught in standard economics programs, whether Keynesian, Public Choice, or Chicago School.

In a debate over the impact of a cut in the corporate income tax rate, Krugman produced the following chart and commentary:

2017.10.31 Krugman Figure 1.png

[W]e can represent the economy with Figure 1, which has the stock of capital on the horizontal axis and the rate of return on capital on the vertical axis. The curve MPK is the marginal product of capital, diminishing in the quantity of capital because of the fixed labor force. The area under MPK – the integral of the marginal products of successive units of capital – is the economy’s real GDP, its total output. [Krugman, bold added.]

My point here is very simple: Look at the y-axis on Krugman’s Figure 1. As he says in his text, it refers to “the rate of return on capital.” In other words, the units of the y-axis are percentages.

Just to make sure you believe me, notice that the expression r*/(1-t) is showing the world rate of return on capital, adjusted by the tax rate on capital imposed by the U.S. federal government. Since tax rates are in percentages, clearly the units of the y-axis itself must be pure percentages.

But if that’s the case, consider the implications of Krugman’s commentary that I put in bold in the quotation above. If we’re calculating total GDP by summing up the contributions of the marginal contribution from each small dose of capital (which is shown along the ­x-axis), then it seems our answer is going to be that GDP is something like “304%.” That doesn’t make any sense.

To understand the problem, suppose that instead of capital on the x-axis, Krugman instead had been illustrating the labor market. In that case, instead of “r” on the y-axis, instead Krugman would have put “w,” to represent the real wage. Then we would see that the first unit of Labor would have a high contribution to output, where output was either measured in physical terms (such as apples or widgets) or in dollar terms. Either way, summing the area under the curve to get total output would have yielded something sensible, like “10 million widgets” or “$19 trillion.” We wouldn’t have gotten a nonsense answer like, “GDP is 304%.”

My observation here may seem elementary, but that’s sort of the point. Neoclassically trained economists launch into discussions of capital & interest, assuming that “interest equals the marginal product of capital” when in fact that is simply wrong. If we’re viewing particular capital goods, such as a tractor, then their marginal product doesn’t equal an interest rate—that doesn’t even have the right dimensions. No, the marginal product of a tractor is equal to the rental price of the tractor.

For more on this issue, you can consult the links I offered above, or if you really want to spend some time reading up on it, this collection of essays from Frank Fetter’s collection (and the introductory essay by Murray Rothbard) is a great resource.

A Note For Purists

Some readers—including presumably Krugman himself if he were to become aware of my post—might respond in the following way: “No Murphy, there isn’t a math problem here. Because there are heterogeneous capital goods in the economy, we aggregate them in terms of money-prices, the only sensible thing. (You can’t add up drill presses and hammers to get a single number.) So the units of the x-axis are actually dollars, meaning that when you multiply the vertical y-axis height by the horizontal x-axis width, the dimension of the product is actually a percentage-times-dollars, meaning dollars are what you end up with. The reason we express the rate of return on capital in percentage terms is simply for convenience and because of common usage. If you want to be pedantic, we could say the return to capital is $6 per $100 of a capital-year, just like we might say the wage rate is $30,000 per worker-year. But of course the former can be summed up as 6% per year.”

This is correct as far as it goes, though I suspect 95% of today’s PhD economists have never even thought about this subtlety, even though they were taught (and teach) graphs like Krugman’s Figure 1 all the time.

Yet it doesn’t really solve the problem, and here’s why: You can’t add up various types of capital goods according to their market prices, in an exercise in which you are going to independently vary the rate of return on capital. This is because the current spot market price of a particular machine (say) is a function of the rate of return on capital.

For example, suppose a particular tractor is expected to yield an increment in a farmer’s net income of $8,000 per year, for the next 10 years, after which time the tractor will be thrown out with no salvage value. How much “capital” in dollar terms does that tractor represent?

Well, if the market rate of interest is 0%, then the market price of that tractor is $80,000: This is just the non-discounted summation of the ten annual rents of $8,000.

However, if the market rate of interest is 5%, then the farmer must discount the future bursts of net income, in order to translate those future dollars into present dollars. In this case, the first increment of $8,000 (which for simplicity we assume comes into the farmer’s possession in one lump sum in a year’s time) is today only valued at about $7,619. (Note that $7,619 x 1.05 = $8,000.) The tenth and final burst of $8,000 in additional net income from the tractor is today only worth $4,911. Summing over all values, the farmer today would only be willing to pay about $61,774 for the tractor.

Note that we are talking here about the same physical tractor, and we are making the same assumptions about the additions it will make to net income over the next decade. But if we assume a prevailing market interest rate of 0%, then the “amount of capital” represented by the tractor is $80,000, while if we increase the annual interest rate to 5%, then all of a sudden the amount of “capital” in the tractor has shrunk to $61,774.

This isn’t at all analogous to what economists do with wages and labor. By varying the wage rate, the “quantity of labor hours” supplied and demanded will change, but the measurement of “an hour of labor” is itself independent of the wage rate. In contrast, the aggregate capital stock—even if we hold fixed the particular assortment of machines, tools, inventories, etc.—will apparently change, when measured in dollar terms, just by moving the rate of interest.

All of these thorny issues came up in the famous Cambridge Capital Controversy of the 1960s. It’s true, Krugman assured his readers that they didn’t need to worry about it, and that the orthodox mainstream economists had everything under control. But, as so often happens, Krugman was simply wrong. You don’t need to take my word for it — James Galbraith too was aghast at the lack of understanding of capital theory in Thomas Piketty’s bestselling book on capital.

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