Marginal *Physical* Product of Capital Neither Necessary Nor Sufficient For Interest

Marginal *Physical* Product of Capital Neither Necessary Nor Sufficient For Interest
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The debate rages over Thomas Piketty’s runaway bestseller, Capital. As I explain in this post, it’s not just bohm-bawerk-eugenfree-market Austrian economists who are critical of Piketty’s framework; even “heterodox” economists hailing from the progressive camp are aghast at its treatment of capital and interest theory.

Here’s the basic problem: Piketty (and his mainstream defenders, such as Paul Krugman) adopt a very simple framework in which interest income is conceived as analogous to labor income. In standard competitive models, wages in equilibrium are equal to (“determined by”) the “marginal productivity of labor.” By analogy, the standard neoclassical approach says that in standard competitive models, the rate of return on capital (the interest rate, or what used to be called the “rate of profit” in classical days)  is equal to (“determined by”) the “marginal product of capital.”

This sounds straightforward enough. After all, in a competitive market the capitalist has to get paid the full value of what one more unit of his capital adds to total output, right?

No, that’s actually NOT right. In fact it’s totally muddled. The marginal product of capital explains the rental price per unit of time of a physical capital good. If using a tractor for an hour boosts the revenues of sharecroppers by $500, then (in a competitive equilibrium) they will pay the owner of that tractor $500 per hour to use it in their operation. That gives us absolutely no information, by itself, on whether the owner of the tractor is earning a high, low, zero, or even negative rate of return on the money he spent when buying the tractor.

The great Austrian economist Böhm-Bawerk spelled all this out in his famous critique of various interest theories in the late 1800s. For a more modern exposition, here is Ludwig von Mises:

A lengthening of the period of production can increase the quantity of output per unit of input or produce goods which cannot be produced at all within a shorter period of production. But it is not true that the imputation of the value of this additional wealth to the capital goods required for the lengthening of the period of production generates interest. If one were to assume this, one would relapse into the crassest errors of the productivity approach, irrefutably exploded by Böhm-Bawerk. The contribution of the complementary factors of production to the result of the process is the reason for their being considered as valuable; it explains the prices paid for them and is fully taken into account in the determination of these prices. No residuum is left that is not accounted for and could explain interest. [Human Action, Scholar’s Edition, pp. 526-527]

Capital & interest theory is admittedly a very difficult area; some have referred to it as “the black hole of economics.” One complication is that Mises himself differed from Böhm-Bawerk on some important points in the explanation of interest, as I spell out in the first chapter of my dissertation.

What is also tricky in this arena is that the mainstream neoclassical economists have formal models (which use either one good or make assumptions about a “steady state” that remove all of the problems) in which, mathematically, it seems as if the real interest rate really is directly pinned down by the marginal product of capital, i.e. the first derivative of the production function with respect to physical capital. (To see what’s going on here, refer to the technical appendix in my dissertation.)

Marginal Productivity of Capital Neither Necessary Nor Sufficient for Positive Interest Rate

The quickest way to demonstrate the poverty of the typical neoclassical theoretical framework is to show:

(A) Hypothetical economies where there is a positive interest rate–and savers earn a real return on their savings–even though capital goods do not physically contribute to output at all.


(B) Hypothetical economies where capital goods physically contribute to output, they do NOT physically depreciate, and yet their owners earn a zero rate of return on the money invested in them.


In this earlier Mises Canada blog post, I gave an example of (A). Specifically, I sketched out a simple world in which workers use their labor power in order to snatch low-flying birds, which are the only consumption good. Natural resources and physical capital goods play no role whatsoever in production. And yet, if the “catch” increases over time and the consumers have a desire to “smooth consumption” period to period, then someone who happens to catch many birds in period 1 can then lend them out at a positive real interest rate. This original “capitalist” can enjoy a perpetual stream of real bird consumption, period after period, expressed as the interest rate times his stockpile of accumulated birds. Thus I have shown an example of someone earning interest on his savings even though there is no physical investment in capital goods, and thus the “marginal product of capital” is zero.

Now in the present post, let me tweak the bird example to show case (B):

Suppose our hypothetical economy only last for 10 periods, and that everybody has perfect foresight of what’s coming. This time, however, the “bird catch” steadily declines each period, in order to perfectly offset the natural “time preference” in people’s subjective preferences. Thus, in equilibrium, a bird in period 1 trades for a bird in period 2, which trades for a bird in period 3, etc. Yes, *other things equal* people would prefer a present bird to a future bird, but other things aren’t equal: There is going to be very little bird consumption occurring in period 10, so a worker even way back in period 1 would be willing to trade away one of his present birds in order to obtain an airtight claim to someone else’s bird to be delivered in period 10.

Now then, the other wrinkle in our hypothetical economy is that a few workers start out in period 1 in possession of nets. A worker with a net can always catch 1 bird per hour more, than a worker who just uses his bare hands. (It’s true that over time it gets more difficult to catch birds per hour of labor, but workers also get more adept at handling the nets. The advantage of having a net in your possession is always 1 extra bird per hour, for periods 1, 2, …, 9, and 10.)

Notice that in this world, capital goods and labor contribute to total output. (When we explained that the real interest rate on birds would be exactly 0% because of the declining bird catch per period, we had already taken into account the fact that workers would be using the nets optimally.) We are assuming standard competitive markets, meaning that each worker earns the marginal product of his labor. The owners of the nets also earn the marginal product of their capital goods: Either they use the nets themselves and reap the extra output, or they rent the net to another person who then hands over the surplus above what that person would have caught with his bare hands. Either way, the earnings of the net owners are 1 bird/hour of usage.

Finally, to keep the math simple, suppose that all the workers just work for 10 hours each period. (If you want, you can suppose that there is only a 10-hour window each period when the birds fly low to the ground. Or, you can assume that the workers’ preferences for leisure just happen to yield the result that everybody always optimally choose to work exactly 10 hours each period, all things considered.) What can we say about the earnings of the capitalists, i.e. the owners of the nets?

Well, it’s clear that the owners of the nets will receive a flow of “real bird income” of 10 birds per period. If they want, the capitalists can thus eat 10 birds per period more than the workers who only have their raw labor power to generate an income. Now in Piketty’s preferred framework where we calculate a distribution of “national income” between “labor and capital” or between “the workers and the capitalists,” clearly we should say that the capitalists are getting some of the output, while the workers are getting the rest. According to Piketty, the share of income going to the capitalists is the real return to capital r multiplied by the stockpile of capital β. (Remember he uses the formula α = rXβ.) Since the result is positive–we know the owners of the nets are getting birds each period, while the workers aren’t retaining the full catch–it must be the case that the real rate of return on capital is positive.

Oops, except that it isn’t. If we calculate the market price of the nets in each period, we can see that their owners earn a zero return.

It’s easiest to work backwards. At the end of period 10, the market price of the nets will be 0 birds; nobody would trade even a single bird to buy the net from an owner, because the world ends in period 10.

At the end of period 9, people know that a net still has one period left to contribute to physical output. This contribution will be 10 birds (accruing one period in the future). But since (in this contrived example) we’ve adjusted things on purpose to get birds trading at par across time periods, that means a “bird-in-period-9” has the same market value–trades one-for-one against–claims to a “bird-in-period-10.” Thus, the market price of the net in period 9 is “10 present birds.” In other words, the owner of a net could sell it outright in period 9 to someone who would hand over 10 birds that could be eaten on the spot.

At the end of period 8, similar logic shows that the spot market price of the net must be 20 birds immediately available. At the end of period 7, the market price of the net will be 30 birds, etc. etc., and at the end of period 1 the market price of the net will be 90 birds. Also note that at the beginning of period 1–which is equivalent to the “end of period 0” before time begins–the net has the full value of 100 birds, which will be its lifetime contribution to physical output.

Now then, suppose the owner of a net asks his accountant to calculate the rate of return on his financial capital from period to period. What will the answer be?

The accountant will say, “At the beginning of period 1, your net had a market value of 100 birds. During that period, you rented it out to earn a gross return of 10 birds. However, even though the net was physically in perfect condition when the worker gave it back to you after his shift, by the end of period 1 the market price of the net had fallen to 90 birds. Thus your total wealth went from 100 birds at the start of the period, to 10+90 = 100 birds at the end of the period. You just treaded water financially speaking; you converted the form of your wealth from being 100% concentrated in a net to being 90% in a net and 10% in a stockpile of birds. But your total wealth–measured in market value–didn’t go up. Your rate of return was zero.”

The accountant could then give a similar explanation for time periods 2, 3, …, 9, and 10. The contribution of the net to physical output was fully accounted for in the very beginning, and cannot by itself generate a financial net return to the capitalist owner.


We have thus finished our task: In this earlier post I showed that we can have a positive interest rate in a world with zero marginal physical productivity of capital. Now in the present post I have shown that we can have positive marginal physical productivity of capital but a zero real interest rate.

This is what it looks like when someone like Piketty (and Krugman) uses a faulty economic framework to explain market phenomena. Yes, their defenders can come up with all sorts of rationalizations and rhetorical devices to get around my awkward counterexamples, but then again you could use the same tricks to “prove” that the labor theory of consumer goods value was actually right all along, too.

  • Jon Dorian

    Thank you kindly for your explanation. We believe that attacking the technicalities of a book that is technical is part of the "grand task" to debunk garbage science. But then, there is also the "other" side. The side of the common person that does not understand technicalities and must be shown "the big picture" (preferably in pictures). Here is where we came in. Our point is simple. All off Piketty's technical fireworks are directed at "demonstrating" that there will be a huge inequality in the future and this inequality is bad. The problem is, this is only true in a managed market, not in a free one. Hence our view Piketty Fences in the matter.
    Freedom and Power

    • David Eagen

      Piketty Fences from the link provided by Jon Dorian is GREAT. Piketty Fences has one of the easiest ways of understanding the debate of inequality.

      THE ONLY THING BETTER THAN Piketty Fences IS THE MAIN WEBSITE. There one can learn a great deal of the capitalist viewpoint with many easy to understand pictures and graphs. GO TO

      I am a new fan Jon

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Robert P. Murphy is the Senior Economist at the Institute for Energy Research, and a Senior Fellow with the Fraser Institute. He holds a PhD in economics from New York University. Murphy is the author of Choice: Cooperation, Enterprise, and Human Action (Independent Institute, 2015) as well as numerous other books and hundreds of articles.

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