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

Now, Judge, I Had Debts No Honest Man Could Pay/The Bank Was Holding My Mortage And They Were Gonna Take My House Away

1.0 Introduction
This example illustrates one aspect of how Sraffa analyzed natural resources. In this case, natural resources consist of land of various qualities or grades. The quantity of each grade is given; more land cannot be produced. Land is not destroyed either. Appropriate production processes yield as much land as a joint output as given as input. So this sort of model does not incorporate a natural resource like oil that is used up in production.

This example demonstrates that owners of less efficient (productive) natural resources can receive a greater rent.

By the way, Sraffa introduces a distinction between basic and non-basic commodities. Lands with positive rent are non-basic, and therefore their rent is a candidate for taxation.

2.0 Technology
This is a simple economy in which only corn is produced. Table 1 shows the available processes that have corn as an output. Each process requires the use of one grade of land, and no more than one process is known for each grade of land. (This is an example of a model of extensive rent.) Suppose this economy has available 150 acres of land of grade I, 162 acres of grade II land, and 210 acres of grade III land.

Table 1: Technology

Inputs	Process
	Alpha	Beta	Gamma
Labor (Person-Years)	2/5	1	2/7
Grade I Land (Acres)	1	0	0
Grade II Land (Acres)	0	3/2	0
Grade III Land (Acres)	0	0	1
Corn (Bushels)	2/5	1/6	4/7
Output (Bushels)	1	1	1

3.0 Prices
The question to be addressed is what prices and distribution of income are compatible with a long-period position, given the technology. The amount of corn required for net output is a parameter that must be known to answer this question.

3.1 When Only One Grade Of Land Is Cultivated
Suppose the requirements for use for corn in this economy can be satisfied by cultivating any one of the three grades of land. Two grades, and maybe some of the third grade, can lie fallow. So at most 90 bushels are produced each year, after replacinging up the seed corn.

And suppose that the wage is 1/2 bushels per person-year. The wage is paid out at the end of the year. These assumptions are enough to derive the factor-price curves shown in Figure 1. These curves are drawn under the assumption that all grades of land paid no rent.

Figure 2: Factor Price Curves

Since the beta factor-price curve is rightmost (on the outer frontier) at the given wage, all corn is produced on land of the second grade, and this land pays no rent. The wage and the rate of profits must satisfy the following equation:

(1/6)(1 + r) + w = 1

where a bushel corn is the numeraire. For a wage of 1/2 bushels per person-year, the rate of profits is 200%.

Under the given assumptions, the cost of producing a bushel corn on the first grade of land, even if that land were to pay no rent, is 7/5 bushels. Since this cost exceeds the revenues from selling a bushel of land, no capitalist would want to produce on the first grade of land. The reader can check that the cost of producing a bushel corn on the third grade of land also exceeds unity.

3.2 When Two Grades Of Land Are Cultivated
Now suppose the requirements for use are such that two grades of land must be cultivated. The net output of this economy is between 90 and 180 bushels corn. The wage remains 1/2 bushels per person-year. In this case, the first and second grades are cultivated, with the second grade paying rent. The price equations are:

(2/5)(1 + r) + (2/5)w + ρ₁ = 1

(1/6)(1 + r) + w + (3/2)ρ₂ = 1

ρ₁ ρ₂ = 0

ρ₁, ρ₂ ≥ 0

The equations specify that no land can have a negative rent and that at least one grade of land must have a rent of zero. The rate of profits is 100%, when the wage is 1/2 bushels per person-year. Land of grade I pays no rent, and the rent on land of grade II is 1/9 bushels per acre.

The cost of producing a bushel corn on the third grade of land, accounting just for outlays of seed corn and labor, is 9/7 bushels. Capitalists will not want to cultivate the third grade of land, even if it is free.

3.3 When Three Grades Of Land Are Cultivated
Finally, suppose the requirements for use for use require all three grades of land to be cultivated. The price equations are:

(2/5)(1 + r) + (2/5)w + ρ₁ = 1

(1/6)(1 + r) + w + (3/2)ρ₂ = 1

(4/7)(1 + r) + (2/7)w + ρ₃ = 1

ρ₁ ρ₂, ρ₃ = 0

ρ₁, ρ₂, ρ₃ ≥ 0

The rate of profits is 50%, when the wage is 1/2 bushels per person-year. The rent on land of grade I is 1/5 bushels per acre. The rent on grade II land is 1/6 bushels per acre

3.4 Orders Of Efficiency And Rentability
The above analysis has identified a definite order in which different grades of land will be cultivated as greater quantities of output are required to be produced. This is the order of efficiency. In this example, the order of efficiency, from most efficient to least efficient, is: Grade II, Grade I, Grade III.

One can also rank the grades of land from high rent to low rent, when all three grades of land are cultivated and the wage is 1/2 bushels per person-year. The order of rentability, from highest rent to zero rent, is: Grade I, Grade II, Grade III.

The orders of rentability and efficiency differ. It is possible for a less productive, that is, the less efficient, resource to provide its owner with a greater rent than the more efficient resource.

This example is not driven by the existence of switch points.

Reference

• Alberto Quadrio-Curzio, "Rent, Income Distribution, and Orders of Efficiency and Rentability", in Essays on the Theory of Joint Production (Edited by L. L. Pasinetti), Columbia University Press, 1980.