A Reswitching Example With Fixed Capital

1.0 Introduction
This post presents an example of reswitching with fixed capital. This is Example 5 in Chapter 9 of J. E. Woods, The Production of Commodities: An Introduction To Sraffa (Humanities Press, 1990). Fixed capital is a special case of joint production. I often analyze the choice of technique in circulating capital models by constructing the so-called factor-price frontier as the outer envelope of factor-price curves for each technique. This method of analysis does not generalize to general models of joint production. I confine myself in this post to methods of analysis that apply generally to joint production. This example illustrates the reswitching of techniques. A manifestation of reswitching in this example is the non-monotonic dependence of the economic life of machinery on the rate of profits.

2.0 Technology
The example is of a vertically-integrated industry producing a net output of corn with inputs of corn, machines, and labor. Machines can last for two yearly cycles of production, while corn inputs are entirely used up each year in production. Firms have available the three Constant-Returns-to-Scale (CRS) processes shown in Figure 1. New machines are produced by the first process, while corn is produced by both of the two remaining processes. The third process uses a one-year old machine as an input. The one-year old machine is produced jointly with corn by the second process.

Table 1: The Technology

	Machine Industry	Corn Industry
Inputs
Labor	1/10 Person-Year	43/40 Person-Year	1 Person-Year
Corn	1/16 Bushel	1/16 Bushel	1/4 Bushel
New Machines		1	0
Old Machines		0	1
Outputs
Corn	0 Bushels	1 Bushel	1 Bushel
New Machines	1	0	0
Old Machines	0	1	0

3.0 Quantity Flows
A choice of technique arises over the economic life of the machine. A firm might choose to discard the machine after the first year and never use the third process, whether in conjunction with the other processes or not. (I assume that machines can be freely disposed of after both one and two years of operation.)

The proportions in which processes are operated varies, depending on how many years the machine is used. Table 2 shows the first two process being used to produced a net output of one bushel corn. Notice that when these processes are operated in parallel new machines are simultaneously produced by the first process and used up by the second process. (I suppose I could scale up the processes in Tables 2 and 3 by 91 so that an integral number of machines is used in each technique and the net output of both is the same quantity of bushels of corn.)

Table 2: Use Of Machine For One Year

	Machine Industry	Corn Industry
Inputs
Labor	4/35 Person-Year	43/35 Person-Year
Corn	1/14 Bushel	1/14 Bushel
New Machines	0	8/7
Outputs
Corn	0 Bushels	8/7 Bushel
New Machines	8/7	0

Table 3 shows how all three processes must be scaled when machines are used for two years and the net output is one bushel corn. Tables 2 and 3 can be used to calculate the flow of labor inputs and capital goods required to produce a net output of one bushel corn for each technique. Table 4 exhibits the result of this calculation. If machines are junked after one year, 2/455 person-years more labor is hired per bushel corn net, as compared to machines being junked after two years.

Table 3: Use Of Machine For Two Years

	Machine Industry	Corn Industry
Inputs
Labor	4/65 Person-Year	43/65 Person-Year	8/13 Person-Year
Corn	1/26 Bushel	1/26 Bushel	2/13 Bushel
New Machines	0	8/13	0
Old Machines	0	0	8/13
Outputs
Corn	0 Bushels	8/13 Bushel	8/13 Bushel
New Machines	8/13	0	0
Old Machines	0	8/13	0

Table 4: Inputs Per Net Output of 1 Bushel Corn

	Years Machine Is Operated
	1 Year	2 Years
Labor	47/35 Person-Years	87/65 Person-Years
Capital	1/7 Bushel, 8/7 New Machines	3/13 Bushel, 8/13 New Machines, 8/13 Old Machines

4.0 Prices and the Choice of Technique
Firms choose the technique - that is how long machines are used - based on profitability. Hence, an analysis of the choice of technique requires an analysis of prices.

4.1 Prices When Machine is Operated One Year
Suppose, to begin with, that the machine is operated for one year only. Prices are such that no pure economic profits can be earned in producing new machines or in operating machines for the first year. Furthermore, costs (inclusive of a charge on capital goods advanced) do not exceed revenues in either of these two processes. These assumptions imply two equalities are met:

(1/16) (1 + r) + (1/10) w_α = p_{0, α}

[(1/16) + p_{0, α}](1 + r) + (43/40) w_α = 1,

where a bushel corn is the numeraire, p_{0, α} is the price of a new machine, w_α is the wage, and r is the rate of profits. These equations embody the implicit assumption that wages are paid after the laborers have been working for a year. The price of a year-old machine is zero.

This is a system of two equations in three unknowns. If the rate of profits is taken as given outside this system, the other two variables can be found as a function of the rate of profits:

w_α(r) = (5/2)(14 - 3 r - r²)/(4 r + 47)

p_{0, α}(r) = (1/16)(103 + 39 r)/(4 r + 47)

The larger is the wage, the smaller is the rate of profits. The maximum rate of profits in this system arises for a wage of zero. That maximum is approximately 253.11%.

The above solution can be used to analyze the profitability of operating the machine for another year. Figure 1 shows the costs and revenues (in bushels corn) for producing a bushel of corn by operating the third process in Table 1. Inputs are valued at the prices given by the above solution. Note that revenues would exceed costs if the rate of profits were below approximately 33%. Likewise, revenues would exceed costs if the rate of profits were above 50% and below the maximum. In these regions, firms would choose to operate the machines for a second year, and the prices would not be equilibrium prices.

Figure 1: Costs and Revenues for Operating Machine for Second Year

On the other hand, the above solution gives equilibrium prices if the rate of profits lies between approximately 33% and 50%. In this region, the costs of operating the machine for a second year exceed the revenues. Firms would only operate the machine for one year, consistently with the hypothesis for this case.

4.2 Prices When Machine is Operated Two Years
The other case arises when prices are consistent with the machine being operated for two years. By assumption, pure economic profits cannot be earned in any of the three processes in the technology. Likewise, costs do not exceed revenues for any of the three processes. These assumptions yield a system of three equations:

(1/16) (1 + r) + (1/10) w_β = p_{0, β}

[(1/16) + p_{0, β}](1 + r) + (43/40) w_β = 1 + p_{1, β}

[(1/4) + p_{1, β}](1 + r) + w_β = 1

where p_{0, β} is the price of a new machine, p_{1, β} is the price of a machine after operating for one year, w_β is the wage, and r is once again the exogeneously specified rate of profits.

This system has one more equation than the system in Section 4.1. But it contains one more variable also. Thus, the wage and the prices of the new and one year-old machine can be found in terms of the rate of profits:

w_β(r) = (5/2)(26 + 7r - 4r² - r³)/(87 + 51r + 4 r²)

p_{0, β}(r) = (1/80)(135 - 166 w_β - 86 w_β r + 50 r - 5 r²)/(1 + r)²

p_{1, β}(r) = (1/4)(3 - 4 w_β - r)/(1 + r)

In this case also, the maximum rate of profits arises when the wage is zero. The maximum rate of profits is approximately 258.77%

The analysis of the choice of technique in this case is based on examining the price of the year-old machine (Figure 2). Only non-negative prices are consistent with equilibria. The price of an used machine becoming negative is a sign that it is cheaper to truncate the use of a machine after it is used for one year. That is, for a rate of profits between approximately 33% and 50%, firms will junk the machine after one year of operation.

Figure 2: Price of One-Year Old Machine

4.3 Choice of Years of Operation
The analysis of the choice of technique is consistent, whether one begins with the price system for machines being used for one or two years. When the rate of profits is below approximately 33%, but non-negative, machines are used for two years. When the rate of profits exceeds 50% and below the maximum, machines are also used for two years. When the rate of profits exceed approximately 33% and are below 50%, machines are discarded after one year of operation.

5.0 Conclusion
Suppose one incorrectly accepts pre-Sraffian neoclassical or Austrian intuition. Then one would believe that a low rate of profits reflects capital, in some sense, being relatively less scarce than labor. And one would expect firms to adopt a more capital-intensive technique at a lower rate of profits. In this context, one would expect the economic lives of machines to be longer at a lower rate of profits. But, as shown by Figure 3, such a belief is incorrect, in general.

Figure 3: Economic Life of Machine versus Rate of Profit

If one had such incorrect pre-Sraffian neoclassical or Austrian intuition, one would likewise expect firms to hire more workers at lower wages, also under conditions of perfect competition. Figure 4 demonstrates this belief is false as well. So much for the idea that wages and employment are determined by the interaction of well-behaved supply and demand functions.

Figure 4: Labor Intensity versus Wage