A Rejoinder to Mr. Murphy
Victor Aguilar
Robert Murphy's Criticisms of the Form of the DWCS
I said earlier that "the principle question that divides Garrison and myself is not the form of the graph (though that is important too), but what it represents: The Aggregate Production Structure, APS, represents income and the Distribution of Wealth over the Capital Structure, DWCS, represents wealth." However, having addressed this basic issue, let us now turn to Murphy's criticisms of the form of the DWCS, particularly its smoothness, its being defined out to infinity and the meaning of its mean (no pun intended).
Some of Murphy's criticisms of the DWCS could be leveled against any distribution. For instance, he asks, "what if we wish to depict an economy that does not have the smoothness inherent in this function?... It seems that once we want the DWCS to more accurately represent an actual capital structure, we would be forced to abandon the elegant mathematical construction and end up drawing a histogram" (Murphy, p. 16). Actually, as anyone who has worked with confidence intervals knows, if one has at least thirty samples, there are really no problems with using the normal distribution which, like the exponential distribution that we are using, is smooth and continuous. There are a lot more than thirty people in America and they each own thousands of distinct capital-goods items, so I think we are safe to define the DWCS to be smooth and continuous.11
Also, Murphy notes that the DWCS is defined out to infinity yet, in reality, people do not look much more than a few decades into the future, if that. Murphy writes, "Aguilar might respond that this is a harmless simplification, since the height of his graph goes to zero as we go out to infinity on the time axis" (Murphy, pp. 15-16), but Murphy clearly does not think it harmless himself. Actually, all continuous distributions have a tail that trails off into infinity. For instance, the normal distribution (a.k.a. the bell-shaped curve) is defined from -
to +, but there is only a 0.04% chance of observing phenomena more than three and a half standard deviations from the mean. Just because nobody has ever seen such unusual phenomena does not mean it is wrong to grant them a smidgen of probability. Anyway, regarding the DWCS, people do sometimes look to eternity - the designers of the Great Pyramid certainly did.
Murphy writes, "Although [Aguilar] may be right that the Hayekian approach is bankrupt, even so Aguilar's suggested fix doesn't fit the bill. Again, Hayek is trying to describe the entire capital structure that must be maintained if one is to yield a constant stream of consumption goods. This capital structure cannot be defined from the initial time to ’infinity,' for the simple reason that people can't wait forever to eat (or drive cars or watch TV)" (Murphy, p. 9).
I agree: The Hayekian approach is bankrupt. However, the capital structure can be defined from the initial time to infinity. As I said when I defined the DWCS, "Durable goods are spread out over time according to their depreciation function, weighted by time-preference, so the area under it is the item's current value. Inventory items that do not depreciate are discounted for time-preference on the expected time until they make their contribution to final consumption. Thus, the height of the DWCS graph at each point on the time axis is the present value of all the capital goods that are contributing to consumption at that future date" (Aguilar, p. 5). Note that, when I say "initial time," I mean "right now, at time zero" (Aguilar, p. 8). It is possible that Murphy is looking "back to axes carved by prehistoric men" (Murphy, p. 6) when he says "initial time." That analysis is of no value to us. We should be looking forward to future consumption, not back to caveman days.
Murphy did not read my paper very thoroughly. Of course people cannot wait forever to eat. That is why I weight the value of goods that will be consumed in the future by time preference. Mine would not be much of a theory if it could be dismissed simply by pointing out that people value a hamburger today more than they value a hamburger next week.
It is possible that Murphy is unaware of functions like 1/t which asymptotically approach the time axis, similar to the way Are-rt does, but do not converge, that is, the area under 1/t out to infinity is infinite. I write, "Convergence must be our first result. The area out to infinity represents wealth so, clearly, it cannot be infinite. There should be no need for the initial cutoff as in Skousen's Figure 2 or Garrison's Figure 3" (Aguilar, p. 14). Why does Murphy think I had to prove convergence, and do it first before proving any other results? It was to meet exactly the sort of criticism that he is leveling against me here.
Like "convergence," "average" is another difficult word for the Austrians. Murphy writes, "I have heard plenty of people use average as a general term, which could include the mean, median, and mode under its umbrella." Yes, plenty of people, but not mathematicians. "In any event," Murphy continues, "Aguilar thinks that what the Hayekians really mean to say is ’the midpoint, half the range.'" (Murphy, p. 8). Actually, that is not what Hayekians mean, it is what Hayek himself meant (1967, p. 42). Forty-two years earlier Böhm-Bawerk had his own definition (1959, v. 2 p. 86) and forty-seven years later, Garrison had a suggestion (1978, p. 170). The objective of this passage is not to identify what the Hayekians really mean, but to point out that they do not actually know what they mean when they use this ambiguous word.
Murphy writes, "[Böhm-Bawerk's] notorious concept of the average period of production was designed to quantitatively assess how long a unit of factor inputs was ’tied up' in the pipeline. Whatever Aguilar's other objections, he can't condemn Hayek for relying on something that is finite in scope. The original factors are necessarily tied up for only a finite time" (Murphy, p. 9). Are they? What about those copper axes that Murphy was so concerned about in the previous section? How long have they been tied up? Again, not to belabor the point, but Murphy's constant tendency to look back into the past to the original factors of production betrays his Marxist leanings.
11 Readers who wish to learn more about the exponential function can find a chapter on it in any introductory statistics textbook. As a probability density function, it describes the probability of having to wait t time-units for the first change in a Poisson process. For instance, the number of customers to enter a retail store in an hour has a Poisson distribution and the time, in minutes, before the next customer arrives has an exponential distribution, assuming that customers are rare enough that there is a negligible chance of getting two at a time. In industry, the number of defects to appear in the stream of manufactured products rolling off an assembly line is described as a Poisson process, again assuming that there is a negligible chance of a single item having more than one defect.
