Some Issues In Joint Production

1.0 Introduction
Sraffa's work on single product systems is sufficient for demonstrating the incorrectness of neoclassical economics, at least in applications in which equilibrium prices supposedly are ultimately, in some sense, scarcity indices. Sraffa's work on joint production is important in justifying a claim that Sraffa has rediscovered the logic behind the Classical theory of value. An interesting question is whether Sraffa's treatment of joint production holds up to a rigorous theoretical analysis. Christian Bidard, Heinz Kurz & Neri Salvadori, and Bertram Schefold are some economists who have gone into this question in some detail.

An example developed by J. E. Woods (1990: pp. 281-285) illustrates some questions raised by joint production. Unfortunately, I am not sure Woods is correct in his analysis; he constrains all goods to have positive prices in his example. I do not impose this constraint in my treatment of his example.

2.0 Technology
Consider an economy in which two goods, iron and coal, are produced. The managers of firms each know of three Constant-Returns-to-Scale processes for producing these goods (Figure 1). In each process, the inputs need to be available at the start of the year. The inputs are totally used up in these production processes, and the outputs become available at the end of the year. Since each process produces both iron and corn, this is an example of a model of joint production.

TABLE 1: Processes Exhibiting Joint Production

INPUTS	Process I	Process IIa	Process IIb
Labor	5 Person-Yrs	10 Person-Yrs	10 Person-Yrs
Iron	18 Tons	12 Tons
Coal			10 Cwt
OUTPUTS
Iron	48 Tons	12 Tons	12 Tons
Coal	10 Cwt	30 Cwt	30 Cwt

3.0 Quantity Flows
Suppose the final demand in this economy is for a composite good consisting of an equal amount of iron and coal. Four techniques can be formed from the technology to produce such a commodity. Two processes are used in each of the first two techniques. The Alpha technique consists of Process I and Process IIa operated in the proportions shown in Table 2. Notice that after the outputs are used to replace the inputs, the net output consists of one ton iron and one Cwt. coal, as required.

TABLE 2: Quantity Flows in Alpha Technique

INPUTS	Process I	Process IIa
Labor	1/6 Person-Yr	2/9 Person-Yr
Iron	9/15 Ton	4/15 Ton
Coal
OUTPUTS
Iron	1 9/15 Tons	4/15 Ton
Coal	1/3 Cwt	2/3 Cwt

The Beta technique consists of Process I and Process IIb used in the proportions shown in Table 2. Here too the net output is one ton iron and one Cwt. coal.

TABLE 3: Quantity Flows in Beta Technique

INPUTS	Process I	Process IIb
Labor	1/360 Person-Yr	1/108 Person-Yr
Iron	3/10 Ton
Coal		5/12 Cwt
OUTPUTS
Iron	4/5 Ton	1/2 Ton
Coal	1/6 Cwt	1 1/4 Cwt

If Process I is operated alone at unit level, the net output of the economy is 30 tons iron and 10 cwt. coal. The requirements for use would be satisfied if 20 tons of iron were thrown away - free disposal is assumed. If this technique is adopted, iron is a free good.

The last technique to be considered is the operation of process IIb alone. In this case more coal would be produced net than is needed. If this technique is cost minimizing, coal is a free good. (Notice than the combination of Process IIa and Process IIb would just produce more of coal, a free good. This is not economical.)

4.0 Prices
Since I want to consider cases where either iron or coal is a free good, neither can be chosen as the numeraire in an analysis of prices. Accordingly, suppose a person-year - in other words, labor commanded - is the numeraire.

4.1 The Alpha Technique
The price equations for the Alpha technique show the same rate of profits being obtained in the processes comprising the technique:

18 p_I(1 + r) + 5 = 48 p_I + 10 p_C

12 p_I(1 + r) + 10 = 12 p_I + 30 p_C

where p_I is the price of iron in units of person-years per ton, p_C is the price of coal in units of person-years per Cwt., and r is the rate of profits. By assumption, workers are paid at the end of the year. If the rate of profits as taken as given, the above system consists of two equations in two unknowns. The solution is:

p_I = 5/[6 (15 - 7 r)]

p_C = (5 - 2 r)/(15 - 7 r)

An economic restriction is that both prices be non-negative. Thus, the solution only obtains in the following interval for the rate of profits:

0 ≤ r ≤ 15/7

4.2 The Beta Technique
The price system for the Beta technique is:

18 p_I(1 + r) + 5 = 48 p_I + 10 p_C

10 p_C(1 + r) + 10 = 12 p_I + 30 p_C

Its solution is:

p_I = -5 r/[6 (r - 1)(3 r - 8)]

p_C = (4 - 3 r)/[(r - 1)(3 r - 8)]

Both prices are nonnegative if:

(4/3) ≤ r ≤ (8/3)

4.3 Choice of Technique
First, suppose Process I were operated alone. Since iron would be in excess supply, its price would be zero person-years per ton. Revenues would be equated to costs in Process I if p_C were 1/2 person-years per Cwt. But revenues would exceed costs in Process IIa by five person-years when operated at the unit level. Thus, firms would want to adopt Process IIa. Thus, operating Process I alone could not be cost-minimizing. (Since Process IIa alone cannot satisfy final demand, Process IIa could also not be operated alone.)

Second, suppose Process IIb were operated alone. In this case, coal would be a free good, and the price of iron would be 5/6 person-years per ton. For any non-negative rate of profits, revenues would never cover costs in Process IIa. On the other hand, for any rate of profits below approximately 133%, Process I would earn pure economic profits (Figure 1). That is, for rates of profits below this level, Process IIb would never be operated alone.

Figure 1: Profitability of Process I (Process IIb Prices)

Third, suppose prices corresponding to the Alpha technique were ruling. Figure 2 shows the difference in revenues and costs for Process IIb, the one process not in the Alpha technique. As usual in this analysis, costs include interest charges on the value of advanced capital. The Alpha technique is cost-minimizing only for rates of profits between zero and 200%, inclusive.

Figure 2: Profitability of Process IIb (Prices for Alpha Technique)

Last, suppose prices solved the price system for the Beta technique. Figure 3 shows the resulting difference in revenues and costs for process IIa, which lies outside the Beta technique. The Beta technique is cost-minimizing for rates of profits between 133 1/3 percent and 200%.

Figure 3: Profitability of Process IIa (Prices for Beta Technique)

5.0 Conclusions
The above analysis demonstrates that, for rates of profits between 0% and approximately 133%, the Alpha technique is cost-minimizing. For rates of profits above approximately 133%, Process IIb is operated alone and coal is free.

In a comparison of the Alpha and Beta techniques alone (without considering the possibility of operating a single process alone):

• The Alpha technique would be cost minimizing if the rate of profits were between 0% and 200% and the prices associated with the Alpha technique were ruling.
• The Beta technique would also be cost minimizing if the rate of profits were between approximately 133% and 200% and the prices associated with the Beta technique were ruling.
• The Beta technique would be cost minimizing if the rate of profits were between 200% and approximately 214% and the prices associated with the Alpha technique were ruling.
• The Alpha technique would be cost minimizing if the rate of profits were between 200% and approximately 267% and the prices associated with the Beta technique were ruling.

In short, the cost-minimizing technique would not be unique between the rate of profits of approximately 133% and 200%, if it were not for the possibility of operating Process IIb alone. The cost-minimizing technique would not exist between the rate of profits of 200% and approximately 267%, once again if it were not for the possibility of operating process IIb alone.

This example raises a question: Can examples arise with these sorts of non-uniqueness and non-existence problems, even allowing for the possibility of free goods? This is a theme in some of Christian Bidard's and Bertram Schefold's work. (Bidard amusingly names one of his articles "Is von Neumann Square?") I do not recall their conclusions. I think Bidard comes down negatively on Sraffa, based, I guess, partly on his analysis of joint production.

References

• Christian Bidard (2004) Prices, Reproduction, Scarcity, Cambridge University Press
• Heinz D. Kurz and Neri Salvadori (1995) Theory of Production: A Long-Period Analysis, Cambridge University Press
• Bertram Schefold (1989) Mr. Sraffa on Joint Production and Other Essays, Unwin-Hyman
• Bertram Schefold (1997) Normal Prices, Technical Change and Accumulation, Macmillan
• J. E. Woods (1990) The Production of Commodities: An Introduction to Sraffa, Humanities Press International