Depreciation of the One Hoss Shay

1.0 Introduction
Some economists have mistakenly thought that depreciation is merely a matter of capital goods wearing out:

"One of the advantages of the von Neumann model is that it can handle capital goods without fuss and bother. A nondepreciating capital good simply enters both as input and as output in the corresponding process. If the capital good depreciates 3 per cent per unit of time, 1 unit of the good may appear as input, and 0.97 unit as output." --Robert Dorfman, Paul A. Samuelson, and Robert M. Solow (1958: 382-383)

The above is an incorrect characterization of the von Neumann model. Von Neumann got depreciation right, unlike Dorfman, Samuelson, and Solow in that quotation. (Luigi Pasinetti points out this error.)

2.0 A Three-Period Example
2.1 Production
As an example of the correct calculation of depreciation, consider a machine of constant efficiency that lasts three production cycles. Each cycle takes a year. Table 1 summarizes the Constant-Returns-to-Scale production processes assumed to be in use for this example. The left five columns show inputs applied over the course of the year. The three right-most columns show outputs available at the end of the year. Oil is representative of inputs used up in producing the output, which the machine is an example of fixed capital. Suppose this machine, at whatever age, is only used in producing wheat; it is not used as an input in any production processes other than those shown in Table 1.

Table 1: Three Production Processes

1 New Machine	&	a Barrels Oil	&	l Person Yrs	Produce	1 Quarter Wheat	&	1 One-Year Old Machine
1 One-Year Old Machine	&	a Barrels Oil	&	l Person Yrs	Produce	1 Quarter Wheat	&	1 Two-Year Old Machine
1 Two-Year Old Machine	&	a Barrels Oil	&	l Person Yrs	Produce	1 Quarter Wheat

This data illustrate the phenomenon of joint production. A process exhibits joint production when its output consists of more than one good. Because the von Neumann model is a model of general joint production, it "can handle capital goods without fuss or bother." Capital goods of different ages and history are distinguished as different goods; capital goods differing only in age are not treated as different quantities of identical goods. This is the point Dorfman, Samuelson, and Solow get wrong in the above quote.

A firm can be thought of as using all three of these production processes side by side. The managers of a firm will want to know how much to enter on thier books for old machines of different ages. They will also want to know the rate of (accounting) profits that they are making. The firm's book prices will be such that the firm is making the same rate of profits in all three process:

(1)

(2)

(3)

where:

• p0 is the price of a new machine,
• p1 is the price of a one-year old machine,
• p2 is the price of a two-year old machine,
• pa is the price of a barrel oil,
• pw is the price of a quarter wheat,
• w is the wage, and
• r is the rate of profits.

A little algebra is of interest here. Multiply each of Equations 1, 2, and 3 by the appropriate power of (1 + r) to obtain Equations 4, 5, and 6:

(4)

(5)

(6)

In summing, the prices of old machines drop out:

(7)

Or:

(8)

In the price equations for an input-output system, Equation 8 can replace the system of equations specified in Displays 1, 2, and 3. Suppose the original system can be solved for prices, given the numeraire and, say, the rate of profits. Then the new system can be solved, as well. And the price of old machines does not appear in the system.

One can think about Equation 8 as follows. A wheat-producing firm buys a new machine that lasts for three production cycles. It doesn't make sense in calculating profit to charge the cost of the machine totally at the time it is bought. The later two production cycles make use of the services of the machine. The cost of the inputs for all three cycles should include some part of the cost of the machine. Equation 8 shows how much the machine should contribute to costs of the inputs of each year's production process, where the costs are assessed at the end of the year.

2.2 An Annuity
Consider an annuity, where (equal) payments are made at the end of three consecutive years. If the interest rate is such that the present value of this stream of payments is equal to the price, p0, of the annuity at the start of the first year, Equation 9 must hold:

(9)

where x is the payment made at the end of the year. Summing the terms on the right, one obtains:

(10)

That is, an annuity of:

(11)

paid at the end of three years has a present value of p0.

The charge for the machine at the end of each production cycle, inclusive of profit and depreciation, is then the value of a fixed annuity paid at the end of each production cycle over the lifetime of the machine.

3.0 Generalizations and a Limit
The above analysis is easily generalized to a machine of constant efficiency that lasts for n years. The annual charge is:

(12)

One can also figure out the the price of the machine of constant efficiency at any age as a ratio of the price when new:

(13)

Figure 1 shows this ratio, for various rates of profits, for the above example of a machine that lasts three years. (Sraffa provides a graph for machine that lasts 50 years, and the rate of profits in his example ranges up to 20%.) A machine that lasts forever would have a charge of p0 r. This is a charge only for profit; there is no depreciation in the case of a machine with an infinite lifespan. This conclusion correctly suggests that joint production provides a framework in which to analyze the rent of land.

Figure 1: Dependence of Variation in Capital Value With Rate of Profits

The analysis can also be generalized to consider machines of varying technical efficiency throughout their age. An (industrial) plant might be analyzed as a collection of processes, where each process uses more than one kind of machine. These generalizations lead to more equations being appended to the system, as well as matching prices for machines of distinct ages and history. In principle, if the efficiency of the machines dependes on the history of the machines it is combined with as inputs in the processes, the number of equations can be countably infinite (Roncaglia 1978). The analysis must also be one of the choice of technique, where the technique includes the possibility of junking a machine while processes still remain that can use it. That is, the economic life of a machine may be shorter than its technical life.

It seems to me incredible that conventions for depreciation in practice could be the solutions of the general theory of joint production. One of the world’s foremost experts on the theory of joint production has this to say:

”Every economist knows that there is indeed much arbitrariness in accounting for depreciation or in the prices set for different derivatives from crude oil. Of course, competition always settles the matter somehow, but whereas it was easy – as de Quincey knew ... – to predict the value of shoes produced by a shoemaker in the nineteenth century, it is very difficult to predict on the basis of cost calculations how a modern refinery will recover its costs by setting administered prices for different refined products. Its rate of return for all its processes taken together can in general not deviate much from the rate of profit of the economy, but this observation does not explain how the relative prices of its individual products are set. Is this therefore another area where history must (as in the case of distribution) supplement theory because it is open-ended? The example of public goods as a form of joint social costs proves that there are cases of indeterminancy of possibly growing historical importance, but Sraffa has shown that there is a solution to the problem of the evaluation of joint products that does not require any explicit reference to the subjective elements of demand.” --Bertram Schefold (1989: 32)

The question becomes whether the number of cost-minimizing processes is equal to the number of products in the general case of joint production, except at flukes. I’m not convinced that the answer to this question is in the affirmative.

4.0 Conclusion
Suppose an economist claims that firms maximize profits. And that economist depicts depreciation as a technical datum, not dependent on prices and the rate of profits. Then, that economist has made a mistake in mathematics.

References

• Robert Dorfman, Paul A. Samuelson, and Robert M. Solow (1958). Linear Programming and Economic Analysis, Dover
• Luigi L. Pasinetti (1977). Lectures in the Theory of Production, Columbia University Press
• Alessandro Roncaglia (1978). Sraffa and the Theory of Prices (English translation), John Wiley & Sons
• Bertram Schefold (1989). Mr. Sraffa on Joint Production and Other Essays, Unwin-Hyman
• Piero Sraffa (1960). Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, Cambridge University Press