Critique of Austrian Economics
Part I: The Legacy of Friedrich Hayek
Section IV: The Average Period of Production
BöhmBawerk (1959, v. 2 p. 86) wrote of an average period of production and was roundly criticized for it:
Let us suppose, for instance, that the production of a consumption good costs a total of 100 working days. (Let us ignore the cooperating uses of land, just for the sake of simplicity.) Of these, one day was expended ten years ago, further, one day was expended in each of the following years from the ninth to the second. The remaining ninety days were expended in the year immediately preceding completion of the good.... The average recovery of the entire 100 days would then be in accordance with the following formula:
10+9+8+7+6+5+4+3+2+1

100
=
55  100

There is no necessity in giving this dead horse another kick. Skousen writes, “It was in part because of this average formulation that Menger referred to BöhmBawerk’s theory as ‘one of the greatest errors ever committed’” (1990, p. 24). He calls this “baffling,” though later he writes, “BöhmBawerk’s controversial solution [to the seemingly infinite period of production] is not only wrong, but it is unnecessary. It has been rightly criticized by economists, even by sympathizers. Rothbard calls the average period of production a ‘mythical concept.’ Mises states ‘[it] is an empty concept’” (1990, p. 151). Garrison writes, “[BöhmBawerk] obscured the essential subjectivist theme and needlessly exposed [Austrian economists] to criticism from a formalistic point of view. History records the dissatisfaction with these developments on the part of Menger and other members of the Austrian School. Mises rejected the arithmetic approach offered by BöhmBawerk and recast the arguments in a subjectivist mold” (1985, p. 163).
Menger’s criticism of his wayward student need not baffle us. BöhmBawerk made a name for himself in 1884 with his History and Critique of Interest Theories, which was really just shooting fish in a barrel. Five years later his Positive Theory of Capital was both good and original. Unfortunately, the good parts (value and price) were not original, being just a restatement of Menger (1981), and the original parts (the average period of production and roundaboutness) were no good. His defensive Further Essays did little to strengthen his position.^{5,6}
It was with the publication of BöhmBawerk’s 1889 Positive Theory of Capital (1959) that Austrian economists split into two branches. Menger did not find a worthy successor until Mises, two generations later, and together they laid the groundwork for the Axiomatic School founded by this author (1999). Meanwhile, BöhmBawerk was laying the groundwork for what would become the Hayekian School. Then the next generation divided both branches again. Mises and Hayek took up Menger’s and BöhmBawerk’s work, respectively. At the same time, BöhmBawerk’s student Strigl would work along the lines of Ricardo’s corn model^{7} and Morgenstern would team up with von Neumann to invent game theory.
But the focus on ethnicity obscured these essential distinctions. Keynes and Marshall are both English and Sraffa and Modigliani are both Italian, but nobody thinks to lump their theories together on the basis of race. Why do so with the Austrians?
Fortytwo years after BöhmBawerk’s attempt, Hayek tried and failed to measure the period of production. He defines “average” as half the time since the application of the original means of production:
As the average time interval between the application of the original means of production and the completion of the consumers’ goods increases, production becomes more capitalistic, and vice versa. In the case we are contemplating in which the original means of production are applied at a constant rate throughout the whole process of production, this average time is exactly half as long as the time which elapses between the application of the first unit of original means of production and the completion of the process (1967, p. 42).

Another dead horse. As everyone knows, the average and the midpoint of a distribution are not the same thing. 1930 was indeed a turning point for the Hayekians. Unfortunately however, and largely because of this failure, it was a turn towards oblivion. The following passage (1967, p. 43) is the turning point:
A perfectly continuous process of this sort is somewhat unwieldy for theoretical purposes; moreover such an assumption is not perhaps sufficiently realistic. It would be open to us to deal with the difficulties by the aid of higher mathematics. But I, personally, prefer to make it amenable to a simpler method by dividing the continuous process into distinct periods. 
The histograms did not help his exposition. Most of his third lecture (1967) is devoted to saying, basically, that the DWCS remains smooth and continuous even as its parameters change. In the context of a discreet histogram, that is a tough sell. Had he initially defined f(t) to be a smooth and continuous function, it would have been easy.^{8}
By substituting a histogram for his continuous function, f(t), Hayek was trying to make his simplistic conception of “average” seem more at home. Had he stayed with the continuous function he would have had to explain why it had such an abrupt tintersection and not, as would seem more intuitive, asymptotically approach the time axis. But histograms, being discrete, must have an abrupt beginning and hence a finite range. If one conveniently has an even number of bars each the same width, it seems natural to divide them into two groups with a faint dashed line.
Another fortyseven years later Garrison had a suggestion: “A third though not independent dimension of capital can be envisaged which represents a composite [product] of the two dimensions [time and money].... Much ambiguity can be avoided by using [this] concept of aggregate production time rather than average production time or average period of production” (1978, p. 170). And, indeed, the accompanying figure (p. 174), reprinted here as Figure 3, has the axes labeled $ and APT.
Quite frankly, this does not make any sense. A change of variables just makes the graph look different. It does not attach any meaning to its “average.” In any case, Figure 3 should be a parabola in the new coordinate system.
Hayek’s triangle in the mt plane is defined by t = \( \frac{c  m}{ r } \).
In the uv plane, with u = m and v = tm (Garrison’s APT dimension), it is v = \( \frac{u(c  m)}{ r } \). Here t = time, m = money, c = consumption expenditures and r = interest.
r = interest? Garrison claims that “the slope of the line [hypotenuse] is the (simple) rate of interest (profit) when the economy is in equilibrium” (1978, p. 173). No, it is not. Compound interest is exponential and interest is always compounded – there is no such thing as “simple” interest.^{9}
Readers familiar with calculus will recall changing variables to facilitate integration.
To integrate in the uv plane, however, Garrison needed to mention that the Jacobian of his transformation is \( \frac{l}{ u } \). But we already knew the formula for the area of a right triangle, so making this integral more tractable could not have motivated the change of variables. Whatever Garrison’s motivation was, very little of his paper makes any sense.
But there must be some temporal measure or the Hayekian’s incessant references to “lengthening the period of production” would not mean anything at all. Skousen asserts that the average “is not only wrong, but it is unnecessary” (1990, p. 151). He is half right. It is definitely wrong, but it (or a similar concept) is necessary. Thus, having banished BöhmBawerk’s average from the front door, he is forced to readmit it through the back door. Skousen’s explanation of business cycles depends on credit expansions lengthening the period of production and on the inevitable contraction shortening it. But it is impossible to talk about something being lengthened or shortened unless one knows how to measure it. If Hayekian business cycle theory is to be salvaged, it is necessary (but not sufficient) to attach a measuring rod to the period of production.
What to do? What to do?
First of all, Hayekians should stop using the word “average.” Words have meaning and the meaning of this one is the sum of a finite set of numbers divided by the number of elements in the set. Hayek’s statistic is actually the midpoint, half the range. But neither average nor range have any meaning in the context of a continuous distribution defined out to infinity.
Skousen and other modern Hayekians have stopped using this word, but not because they stopped using the concept. Basically, when confronted with BöhmBawerk’s inability to defend his average, Hayekians just substituted the phrase “lengthen the period of production” for the debunked phrase “lengthen the average period of production.^{10}
The next step is to attach some meaning to this oftused phrase. The continuous analog of the average is the mean. So does this imply that Hayekians can stop dancing around their now nameless concept and just slap this new word on it? No. They would have to prove that the mean meets the conditions that they have placed on the concept previously known as “average.” But what, exactly, are those conditions?
Rothbard writes, “the production structure is lengthened, and the prices of original factors specialized in the higher stages rises. The prices of capital goods change like a lever being pivoted on a fulcrum at its center; the prices of consumers’ goods fall most, those of firstorder capital goods fall less; those of highestorder capital goods rise most, and the others less” (1970, p. 855). He does not define “center.” But, since he is following Hayek, he probably means the midpoint. Hayek (1967, p. 75) discusses this pivoting action but does not specifically say that it occurs at the midpoint, or anywhere else, because he is using histograms. To describe a function that is pivoting around a moving point requires calculus, not bar charts.
Garrison (2001, p. 47) follows Hayek (1967, p. 39) to the letter by drawing his triangle with five stages: mining, refining, manufacturing, distributing and retailing. Yet he infers a great deal more from the same illustration. He explains “Five gives us just the appropriate degree of flexibility: a structural change that shifts consumable output into the future, for instance, would involve an expansion of the early stages (with the first stage expanding more than the second), a contraction of the late stages (with the fifth stage contracting more than the fourth), and neither expansion nor contraction of the (third) stage that separates the early and late stages” (2001, pp. 4647). Hence we see that Garrison explicitly places the pivoting action at the midpoint, whereas Hayek demurred.
O’Driscoll and Rizzo write, “A fall in interest rates, generated by monetary expansion, will not increase uniformly the value of all investment projects. The value of investment projects yielding consumption output in the more distant future rises relative to projects with more immediate payoffs. We call these projects and capital goods type 1 and type 2, respectively.... For type 1 goods the stream [of quasirents] tends to rise; for type 2 goods the stream tends to fall” (1985, pp. 205206). The terms “type 1” and “type 2” are then used freely without any effort being made to locate the boundary between them.
Similarly, Skousen writes, “At a certain point somewhere in the middle of the APS, the positive forces... equal the negative forces... and there is no change in output at that central point (C). Above point C, an expansion of higher order production takes place. Below point C, there will be a decline in lower order production” (1990, p. 237). No effort is made to locate exactly where this central point is in the middle of the APS.
None of these economists admit that their central/boundary point is the Hayekians’ old nemesis, the average period of production. Clearly, they still do not know how to measure it. Placing their pivot point “somewhere in the middle” is a little vague, to say the least.
But, if it is really true that a change in interest rates causes the DWCS to increase on one side of the boundary and decrease on the other, then the solution is simple: Just differentiate the DWCS with respect to r, set this derivative equal to zero and solve for t.
Before we can differentiate the DWCS we must define it mathematically. Skousen’s neverusedagain equations 1 and 2 are actually the best idea he had. Compound interest grows exponentially, so let us look at the exponential distribution, rert for 0 £ t < ¥. See Figure 4.
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.
\(\int_{0}^{\infty }re^{et}dt = l < \infty \) Eq. 3 Convergence
For continuous distributions, the definition of the mean is similar to the definition of the average except that integrals replace finite summations.
\(\int_{0}^{\infty }tre^{rt}dt = \frac{l}{r} \) Eq. 4 Mean
The DWCS is the exponential distribution scaled up by A, the wealth of the nation. That is, the DWCS is the function f(t) = Are^{rt} for 0 \(\leq \) t < \(\infty \).
Is the mean the central/boundary point of Rothbard, Garrison, O’Driscoll, Rizzo and Skousen? Is it the point where the function pivots like a lever on a fulcrum? This is an easy question to answer: Differentiate Are^{rt} with respect to r, set this derivative equal to zero and solve for t. Thus, \(\frac{\partial f}{\partial r} \) = Ae^{rt}  Atre^{rt}. The exponential function is always nonzero so, having set the derivative to zero, we can divide Ae^{rt} out of both sides to get 1 = tr. Hence, the central/boundary point is at t = \(\frac{1}{r} \), the mean. Q.E.D.^{11}
Considering the “roundabout” process that began with BöhmBawerk’s simplistic average, Hayek’s calling the midpoint the average, Garrison’s changing variables for no apparent reason and finally Skousen’s striking the aword from his vocabulary, this is a genuinely remarkable result. In spite of using the word “average” in its most colloquial sense, the Hayekians came surprisingly close to the solution to their problem. The continuous analog to the average does meet their conditions. They really can slap the word “mean” onto the concept previously known as “average.”
Thus, while we have not resurrected BöhmBawerk’s muchmaligned average period of production, we have entered a horse of the same color. That is as close as we can get to vindicating BöhmBawerk. His ghost can finally rest in peace and stop banging on the pipes at Auburn University. The Hayekians once again have a horse in the business cycle theory race.
^{5 } These three volumes are collected under the title Capital and Interest (BöhmBawerk 1959).
^{6 } BöhmBawerk was a better social philosopher than he was an economist. For instance, he writes, “We may define social capital as an aggregate of products which serve as a means of the acquisition of economic goods by society” (1959, v. 2 p. 32). He specifically excludes the means of subsistence of productive workers as a part of social capital. This focus on definitions may seem pedantic until we read that “There is but one basis for a contrary conclusion. That would be to classify workers, not as members of a civil society for the benefit of which the economy is conducted, but to regard them as objective labor machines. In that case – but only in that case – the workers’ maintenance would be in the same class as fodder for beasts of burden and fuel for machines; it would be a means of production; it would be capital” (1959, v. 2 p. 71). BöhmBawerk denounces “the tendency among English economists – often and quite justifiably censured – to regard workers as producing machines; that view made their wages a component part of production costs, and counted them a deduction from national wealth instead of a part thereof” (1959, v. 2 pp. 7273). By “English economists” he means Ricardo and his followers. Ironically, the first postMenger economist to raise the NeoRicardian flag was BöhmBawerk’s own student Richard Strigl. What Hoppe translates into “rations of the means of subsistence” should have been translated into “corn” to make it more clear to modern readers that Strigl is following Ricardo’s corn model (2000, p. 21).
^{7 } Lest there be any doubt about this, consider Strigl’s conception of his task: “The problem was formulated as such: what is the prerequisite for production’s taking advantage of the increased returns associated with choosing roundabout methods of production? We found that the existence of a subsistence fund was the prerequisite. While analyzing roundabout methods of production, we found further that there existed various specific provisions of goods whose production, on the one hand, was the result of choosing roundabout methods of production and whose expenditure, on the other hand, was necessary for the continuation of the roundabout process of production, and which had to be continuously reproduced in order to maintain it” (2000, p. 26). More concise is Sraffa’s title Production of Commodities by Means of Commodities (1960).
^{8 } Histograms do illustrate the proportion of the total movements of goods which is effected by exchange against money. Hayek writes, “If we divide the path traversed by the elements of any good from the first expenditure of original means of production until it gets in the hands of the final consumer into unit periods, and then measure the quantities of goods which pass each of these lines of division during a period of time, we secure a comparatively simple measure of the flow of goods without having recourse to higher mathematics. Thus, we may say that, in the instance we have been considering [in which the whole process of production is completed by a single firm], money has become more efficient in moving goods, in the sense that a given amount of exchanges against money has now become sufficient to make possible the movement of a greater volume of goods than before” (1967, p. 65). For example, early Fords were made almost entirely “in house,” that is, steel went in one end of the factory and Model Ts came out the other end. Today there are hundreds of companies each supplying Ford with some part and relying on other companies to supply them with even smaller parts. That there are many such industries so organized explains why the demand for money has increased far beyond what population figures would suggest. But this has nothing to do with business cycles. For that the structure of production must be defined as a distribution of wealth.
^{9 } Nobody accepts that the rate of interest can be represented by the slope of a straight line. That might have worked for a 1930 lecture, but today anyone with $20 can buy a calculator programmed to do timevalueofmoney calculations. They may not understand the math, but they know very well it is not linear.
^{10} See for example Cochran, Call & Glahe, “The credit expansion is the familiar MisesHayek business cycle theory.... Consumer preferences, augmented by an interestrateinduced overconsumption, are pulling resources into a shorter structure of production, while the credit expansion is attempting to attract resources to support a longer production structure. The resource base is ultimately not sufficient to allow completion of both structures simultaneously. This scenario is an updated version of the benchmark case used by Hayek in Prices and Production (1967)” (2003, p. 69, italics added). Shorter or longer what? Average?
^{11} Quite Easily Done. Or, as BöhmBawerk would say, as plain as a pikestaff.