This post illustrates why the non-substitution theorem includes an assumption of no joint production. I have previously gone a little into the the theory of joint production in an analysis of depreciation. I have also previously illustrated the non-substitution theorem with an example in which the theorem's assumptions are met (part 1, part 2, Kurz and Salvadori on the theorem).
2.0 The Technology
Consider three islands, Alpha, Beta, and Gamma. A competitive capitalist economy exists on each island. These islands are identical in some respects and differ in others. The point is to understand how differences in tastes can be related to other differences, particularly in prices.
All three islands have the same Constant Returns to Scale technology available. They also exhibit the same rate of profits, and have fully adapted production to their conditions. The technology consists of processes to produce rye and wheat, where workers use inputs of rye and wheat to produce rye and wheat available at the end of the time period associated with each process. This time is a year. That is, each production process requires a year to complete. Each process fully uses up their inputs in producing their outputs.
The processes here exhibit joint production. A process is an example of joint production when its output consists of more than one good. The production of wool and mutton is a well-known example. With joint production, there is room in the economy for processes producing the same set of outputs in different proportions, as in this example. Table 1 shows processes, some subset of which are chosen by the firms at the ruling prices in this example. I think a better example might fully specify a larger technology from which to chose processes.
|Labor||1 Person-Year||1 Person-Year|
|Rye||1/8 Bushel||3/8 Bushel|
|Wheat||1/16 Bushel||1/16 Bushel|
|Outputs||1 Bushel Rye|
& 1/2 Bushel Wheat
|1/2 Bushel Rye|
& 1 Bushel Wheat
3.0 Quantity Flows
The employed labor force grows at a rate of 100% on each island. Each island differs, however, in the mix of outputs that they produce.
The population on Alpha wants to eat only rye. They do not and will not consume wheat. Table 2 shows the quantity flows per employed laborer on Alpha. Notice that the commodity inputs purchased at the start of the year total 1/8 bushel rye and 1/16 bushel wheat. Since the rate of growth is 100%, 1/4 bushel rye and 1/8 bushel wheat will be needed for inputs into production in the following year. This leaves 3/4 bushels rye available for consumption at the end of the year per employed worker. There is also an excess output of 3/8 bushels wheat per worker, freely disposed of each year.
|Rye||1/8 Bushels Rye|
|Wheat||1/16 Bushels Wheat|
|Outputs||1 Bushel Rye|
& 1/2 Bushel Wheat
|GROSS OUTPUTS PER WORKER: (1 Bushel Rye, 1/2 Bushel Wheat)|
|CAPITAL PER WORKER: (1/8 Bushel Rye, 1/16 Bushel Wheat)|
|CONSUMPTION PER WORKER: 3/4 Bushel Rye|
The Beta population eats only wheat. Table 3 shows the quantity flows on Beta. Here the same sort of calculations reveal that Beta has 3/4 bushel wheat available for consumption at the end of the year, per employed worker.
|Inputs||Rye Process||Wheat Process|
|Labor||1/4 Person-Year||3/4 Person-Year|
|Rye||1/32 Bushels Rye||9/32 Bushels Rye|
|Wheat||1/64 Bushels Wheat||3/64 Bushels Wheat|
|Outputs||1/4 Bushel Rye|
& 1/8 Bushel Wheat
|3/8 Bushel Rye|
& 3/4 Bushel Wheat
|GROSS OUTPUTS PER WORKER: (5/8 Bushel Rye, 7/8 Bushel Wheat)|
|CAPITAL PER WORKER: (5/16 Bushel Rye, 1/16 Bushel Wheat)|
|CONSUMPTION PER WORKER: 3/4 Bushel Wheat|
|Inputs||Rye Process||Wheat Process|
|Labor||25/28 Person-Year||3/28 Person-Year|
|Rye||25/224 Bushels Rye||9/224 Bushels Rye|
|Wheat||25/448 Bushels Wheat||3/448 Bushels Wheat|
|Outputs||25/28 Bushel Rye|
& 25/56 Bushel Wheat
|3/56 Bushel Rye|
& 3/28 Bushel Wheat
|GROSS OUTPUTS PER WORKER: (53/56 Bushel Rye, 31/56 Bushel Wheat)|
|CAPITAL PER WORKER: (17/56 Bushel Rye, 1/8 Bushel Wheat)|
|CONSUMPTION PER WORKER: (9/14 Bushel Rye, 3/7 Bushel Wheat)|
4.0 Price System
Since these economies have adapted to their requirements for use, stationary prices prevail. Assume a rate of profits of 100%, identical across all three islands. Also assume the wage is paid at the end of the year.
4.1 Prices on Alpha
Recall that there is excess production of wheat on Alpha. "If there is excess production of [wheat], [wheat] becomes a free good" (J. Von Neumann, "A Model of General Economic Equilibrium," Review of Economic Studies, 1945-1946: 1-9). Asuming the wage is paid at the end of the year, the price system given by Equation 1 will be satisfied:
4.2 Prices on Beta and Gamma
The price system given by Equations 3 and 4 will be satisfied on Beta and Gamma:
Under the conditions satisfied by this example, in which the economies on different islands are fully adapted to tastes, the prices shown in Table 5 prevail. Differences in tastes between Beta and Gamma are associated with unchanged prices, even in this context. Different tastes on Alpha, however, are associated with a difference in which goods have positive prices and a consequent difference in the wage.
|Alpha||Beta & Gamma|
|Wheat (Bushel)||0 Bushel Rye||2 Bushel Rye|
|Labor (Person-Year)||3/4 Bushel Rye||3/2 Bushel Rye|
Note that if only goods with a positive price were shown in the techniques chosen on the respective islands, the input-output matrices would be square in all cases (1x1 on Alpha and 2x2 on the other two islands). I think that this property can arise in some cases where wages are not entirely consumed and profits not entirely invested. As I understand it, however, it is a theorem that the input-output matrices are square under this golden-rule condition.
If wages were the same across all three islands, then the rate of profits would vary between Alpha, on the one hand, and Beta and Gamma, on the other. Since the rate of growth is equal to the rate of profits, the rate of growth for a fully adjusted economy would be determined endogeneously. The different choices of the workers on how to consume their wages would result in a difference in the rate of growth between Alpha and the Beta and Gamma islands.
Even though differences in tastes can be associated with differences in prices, it is not clear that this example illustrates a model consistent with the neoclassical (scarcity) theory of value:
"..in a production context...it makes no sense to talk of 'endowments' of given physical quantities if these physical quantities, to be carried over from one period to another, are the unknowns to be determined. It makes no sense to talk of 'scarce' resources, if these resources can be produced in whatever quantities may be needed by the economic system...
When all inputs are themselves produced, a change in the composition of demand simply means that more of some inputs and less of other inputs will have to be produced, while the optimum technique remains the same. In other words, the process of adaptation to any given change in the composition of final demand is, in a production context, radically different from the one considered by traditional theory. Whereas, with given and fixed inputs (the traditional case), the only way to adapt is through a change of technique which may allow the substitution of some inputs for others, in a production context in which all inputs are themselves produced the obvious way to adapt is to produce the inputs which are needed and to cut down production of those which are no longer needed. There is no question of changing the technique. Input substitution, in a production context, has no role to play...
Another route which has been pursued to minimize the importance of the new results...consists in attributing the irrelevance of substitution to the 'very special' case of no joint production and constant coefficients [ = constant returns to scale -RLV ]. But the inconsistency of this contention is here brought into sharp relief by the very analysis of the previous pages...
As already pointed out...the joint production and nonconstant coefficients case is more complicated than, but not basically different from, the case concerning single products and constant coefficients. The complication arises from the fact that a change of the composition of demand may entail a change of the optimum technique and of the price structure. However, this does not enable us to say anything about the direction in which the input proportions will change.
...It is precisely the unambiguous direction in which relative prices and input proportions are related to each other that justifies talking of 'substitution.' But there is nothing of the sort in a production context. No general relation exists between the changes in the price structure and changes in the input proportions. More specifically, no monotonic inverse relation exists, in general, between the variation of any price, relative to another price, and the variation of the proportions among the two inputs to which these two prices refer. When this is so, to talk of 'substitution' among these inputs no longer makes any sense." -- Luigi L. Pasinetti (1977). Lectures on the Theory of Production, Columbia University Press: 186-188