Why Do Capitalists Earn Interest Income?
THE INTEREST PROBLEM
Eugen von Böhm-Bawerk's three-volume work, Capital and Interest, is a classic, both because of its brilliant analysis and its witty exposition. The first volume provides a history and critique of all preceding explanations of the "interest problem." For Böhm-Bawerk, the task of the interest theorist was to explain why a capitalist could regularly earn a net return on his financial assets, even though (unlike laborers) he apparently did nothing to "earn" this interest income.
BÖHM-BAWERK'S "AGIO THEORY"
Böhm-Bawerk's solution consisted of two steps. First, he framed the phenomenon of interest, not as a return to financial investments, but rather as a premium, or agio, in intertemporal exchanges. For example, take the case of a tractor. Typically, a capitalist who invests in a tractor (either directly or by lending funds to a farmer) can earn an interest return on his investment; that is, he will have more real wealth after the tractor has been used to harvest crops than before. What Böhm-Bawerk realized was that this phenomemon—the growth in real financial wealth through investment in the tractor—relies on an apparent undervaluation of the tractor.
To see this, suppose that the tractor is expected to yield an additional $1,000 worth of revenue every year, and that it will last ten years (before being junked). Böhm-Bawerk argued that the only reason a capitalist could earn money through ownership of the tractor is that its initial purchase price is less than $10,000. Only in that case could an investor use an initial amount of financial wealth and turn it into a greater subsequent amount (ten years later).
Thus, Böhm-Bawerk had transformed his original question. Rather than asking, "Why do capitalists earn an effortless flow of interest income?" he could instead wonder, "Why is it that the initial purchase prices of capital goods systematically fall short of the future profits their use is expected to yield?"
The second step in Böhm-Bawerk's solution was to make the claim that present goods are preferred to future goods. Generally speaking, a person values present apples, houses, etc. more than he values claims to such goods that cannot be redeemed until the future. In the case of our hypothetical tractor, its purchase price is denominated in present dollars, while it only offers the hope of a stream of future dividends (of $1,000 each year for ten years). Since no one would be willing to give $10,000 now in exchange for a promise of $1,000 payments for each of the next ten years, it naturally follows that no one would pay $10,000 for our hypothetical tractor. Because of this fact—that present goods are worth more than future goods—the tractor can be purchased for less than $10,000, and a capitalist can increase the market value of his wealth by investing in tractors (and waiting ten years).
THE "NAÏVE PRODUCTIVITY THEORY"
Of particular interest to modern Austrians is Böhm-Bawerk's refutation of a popular, rival explanation for the phenomenon of interest. Many economists would argue that, in the case of our tractor, the reason a capitalist earns a net return on his wealth is that the tractor is productive: After all, a farmer can harvest more crops, year after year, with a tractor than without one, and so naturally (these economists believe) someone who buys a tractor can earn an income over time. More generally, such economists argue that borrowers are willing and able to pay interest because of the "productivity of capital."
Böhm-Bawerk brilliantly refuted this line of reasoning, which he referred to as the "naïve productivity theory" of interest:
I grant without ado that capital actually possesses the physical productivity ascribed to it, that is to say, that more goods can actually be produced with its help than without. I will also grant…that the greater amount of goods produced with the help of capital has higher value than the smaller amount of goods produced without it. But there is not one single feature in the whole set of circumstances to indicate that this greater amount of goods must be worth more than the capital consumed in its production. And that is the feature of the phenomenon of excess value which has to be explained. (I, p. 93, italics original)
We can understand Böhm-Bawerk's argument in terms of our tractor example. The "naïve productivity" theorist claims that the owner of a tractor earns a net return on his investment because the tractor yields $1,000 in marginal revenue each year of its life. So this explains (so thinks the naïve productivity theorist) the annual percentage return reaped by the capitalist.
But Böhm-Bawerk points out that this is only looking at one side of the matter. Yes, the productivity of the tractor explains why its owner enjoys $1,000 per year in extra income; if he wished, the owner could rent out the tractor and charge up to $1,000 per year for its services.
However, this flow of income will only represent a net return on the original investment if the original purchase price is less than $10,000. For suppose that the tractor initially cost $10,000. In that case, its owner would still receive $1,000 per year for the ten years of the tractor's life, but at the end of the decade the capitalist would have recovered only his initial principal, $10,000. In other words, the depreciation of the tractor would exactly offset the flow of dividends, so that the net rate of interest on the investment would be zero. Note that this is perfectly consistent with the fact that the tractor is productive, and so the tractor's productivity as such cannot be the explanation for a positive rate of interest.
THE NEOCLASSICAL APPROACH
Modern mathematical economists, who explain economic phenomena through systems of simultaneous equations, are often bewildered by the Austrian stress on subjective intertemporal preferences—rather than capital productivity—when it comes to interest theory. Indeed, a standard condition in a typical mainstream model is
r = f '(k),
which denotes the fact that in equilibrium, the real rate of interest is equal to the marginal product of capital, i.e. the increment in output produced by an increment in the capital stock k.
On the face of it, the neoclassical approach seems to commit the very fallacy that Böhm-Bawerk pointed out over one hundred years ago: The mainstream economists seem to argue that the real rate of interest is directly proportional to (and in a sense "caused by") the extra output yielded by additional units of capital. So what's going on here? Do the mainstream models contain a logical error?
Actually, they do not. What has happened is that, because of their need for analytical simplicity, the mainstream models assume that the world has only one good. Consequently, capital goods and consumption goods are the same thing, and all of the difficulties in "Austrian" capital theory are assumed away.
We can see this most clearly by a simple example. In order to motivate their assumption of a single good serving as both capital and consumption, the neoclassicals might adopt a model in which sheep are the only good. In this fictitious world, people own stocks of sheep. They can choose to consume their sheep in the present, enjoying the current marginal utility of consumption, or they can postpone consumption (i.e. save their sheep) for a future period. If they choose the latter course, their stock of sheep will multiply (because of natural reproduction). If, say, the number of sheep doubles every year, then (the neoclassical would argue) the equilibrium real rate of interest in this fictitious world must be 100 percent. It is through reasoning such as this that the mainstream economist believes that the "marginal product of capital" is linked to the equilibrium real rate of interest.
However, as I claimed above, this type of model assumes away the thorny issues in capital theory, which only the more sophisticated Austrian analysis attempts to handle. Recall that in our tractor example, the fatal flaw in the naïve productivity explanation was that it did not explain the initial purchase price, or market valuation, of the tractor, in terms of dollars. The tractor represents a claim on future dollars, but we cannot know the implicit interest rate on the investment until we know the present market value of the tractor in terms of dollars.
In contrast, consider the sheep example. In a fictitious world where sheep are the only good, the only measure of a person's real financial wealth is the number of sheep that he owns. In this simplified scenario, yes, if someone's stock of sheep physically doubles every year, then the market-clearing (real) interest rate must be 100 percent.
To put it another way: One sheep now represents a claim on an endless stream of future sheep. But unlike the tractor example, we do not here run into the Böhm-Bawerkian problem: The current market value of one present sheep, in terms of sheep, is always one! In the tractor case, physical facts alone could not tell us how many dollars would exchange for the capital good; the tractor might cost $5,000, or $10,000, or $15,000. But in the case of the sheep, we can say what the real price of the capital good (sheep) in terms of its future consumption good (sheep) has to be: One sheep trades for one sheep. Thus, the incidental use of a one-good model has allowed the neoclassical to completely sidestep the "Austrian" problem of valuing the capital stock in terms of its eventual output of consumption goods.
AUSTRIAN ANALYSIS STILL RELEVANT
I would like to conclude with a personal anecdote that illustrates the relevance of Böhm-Bawerk's critique. After I had reconciled the verbal logic of Böhm-Bawerk with the mathematical models of the mainstream, I wrote a first draft of one of my dissertation essays in which I explained away the apparent conflict by pointing out the tremendous importance of the mainstream's assumption of a single-good world. I handed in my draft to a renowned mainstream economist, just to make sure that I hadn't misunderstood neoclassical theory.
When I got my draft back, I was quite surprised to find that the professor had clipped a single piece of paper to the front. On it he had written something like, "This is the only interest theory that I, and just about everyone else, understand." Below he had drawn a simple diagram, with C(t) (i.e. consumption in period t) on the x-axis, and C(t+1) on the y-axis. There was a semicircle connecting the two axes, which denoted the production possibilities frontier (PPF) for present and future consumption through tractors.
The professor had drawn two dots on the PPF. The dot that was higher on the circle represented the tradeoff that was available through saving: By moving to the left on the x-axis, a person reduced current consumption in order to invest in tractors. By moving up on the y-axis, a person increased future consumption because of the marginal output of the tractors.
And now the crucial step: Because of the shape of the PPF, and because he had chosen points on the right side of the curve, it turned out that the leftward shift in present consumption was smaller than the upward shift in future consumption. Therefore, my professor thought that this simple diagram had shown a technological cause of interest: Because of the productivity of tractors, my professor was claiming that a small reduction in present consumption would cause a great increase in future consumption, i.e. a positive rate of interest.
What was so frustrating about this diagram was not that it was wrong per se, but that it completely overlooked Böhm-Bawerk's critique! My professor had completely overlooked the problem of pricing the tractors! Yes, the technological facts allow us to say that a given increment in future consumption (i.e. the gap on the y-axis) will require the present investment in a definite number of tractors; this is an engineering problem that does not involve subjective preferences.
However, just because we know how many tractors we need to buy in the present, we do not know how much such an investment will reduce our present consumption. In order to know this, we need to know the market price of tractors in terms of present consumption. By drawing the gap on the x-axis, my professor had just assumed that the tractors would cost less in terms of present consumption than their future output. In other words, my professor had assumed a positive rate of interest.
After several minutes of discussion, I finally got the professor to realize that he had been assuming away this difficulty. But he still refused to concede that physical facts alone could not explain a positive interest rate. No, instead he proclaimed: "Assume we can turn tractors into bananas one-for-one."
In conclusion, Böhm-Bawerk's critique of the naïve productivity theory was a brilliant leap forward for subjectivist economics. Unfortunately, its lessons are as relevant today as they were in the 1880s.
Robert Murphy is a summer fellow of the Mises Institute. email@example.com
 Böhm-Bawerk, Eugen von. (1959 ) Capital and Interest (3 vols. in 1), South Holland, IL: Libertarian Press.
 Specifically, Böhm-Bawerk wondered, “Whence and why does the capitalist receive this endless and effortless flow of wealth?” (I, p. 1, italics removed).
 No one would lend out 10 sheep today in exchange for 15 sheep next year, because the owner could simply hold on to his 10 sheep and allow them to double into 20 sheep next year through reproduction.
 Actually, one does not need to use verbal logic to see the problem. In the mathematical appendix to my dissertation (available here http://homepages.nyu.edu/~rpm213/files/Dissertation.pdf ), I develop a few general equilibrium models with two goods to illustrate Böhm-Bawerk’s insight.