Value & Capital, CHAPTER VIII -- THE GENERAL EQUILIBRIUM OF PRODUCTION, Section 1

The opening paragraph of this section summarizes what has been covered in the book up to this point.  Chapters I-III explore "what determines the equilibrium of the private individual, and how he may be expected to react to changes in prices."  Chapters IV and V use the insights from the previous chapters "to elucidate the working of an economic system" consisting only of private individuals and in which the exchange of existing goods and services is the only economic activity. Chapters VI and VII introduce "a new kind of economic unit, the firm" and describe how a firm will conduct itself in the market.  At this point the stage is set "to examine the working of an economic system containing both kinds of units, private individuals and firms; so that the price-system does not only regulate exchange, but also regulates production."

Not surprisingly, the author, Sir John Hicks, calls the General Equilibrium of Production "an hypothesis of much wider applicability than the General Equilibrium of Exchange."  There are "quite a number" of economic problems where it can be applied safely, although it is possible to misuse it. In fact, Hicks claims that "the misuse of this system is one of the most fruitful sources of error in economic theory."  The reasons for this have to do with the areas of the economy that it abstracts away.

The section closes by enumerating the three "main deficiencies" of the system of the General Equilibrium of Production.  The first is that it leaves out the possibility of monopoly and imperfect competition.  The second is that "it abstracts from the economic activity of the State."  The third is that "it abstracts from capital and interest, saving and investment, and all that complex of activities ... earlier ... called 'speculation.'"  The book will treat this final deficiency in later chapters.

Value & Capital, CHAPTER VII, Section 6

Having treated some special cases in earlier sections, Chapter VII concludes in this section by summarizing what is known about the general case of a firm employing any number of factors to produce any number of products.  The summary still assumes that, "the factors ... co-operate with a fixed productive opportunity of limited capacity, so that the condition of increasing marginal cost is satisfied."  The discussion looks at what happens when a price of either a factor or a product changes (with all other prices being unchanged).

If there is a fall in price for some factor A, then the demand for A must increase.  If this happens, it must be balanced somehow -- either by the supply of some products increasing, or the demand for some other factors decreasing, or both.  Hicks goes on to explain:

The typical result of a fall in the price of a factor is then this:  that the supplies of products will expand, and the demand for other factors will expand too.  But to each of these general rules a limited amount of exception is possible, when the fixed resources are influential enough;  some factors may be substitutes for the first factor, some products may be regressive against it; the demands for substitute factors, and the supplies of regressive products, will decline.

Hicks explains in a footnote that regression seems to be more plausible in the multiple-product case than the single-product case.  If factor A plays an important role in the production of product X, then we may expect the output of X to increase when employment of A increases.  "But if the entrepreneur's fixed resources are devoted more to the production of X, they will be less available for the production of Y.  Thus A and Y may be regressive."

If there is a rise in the price of some product X, then the supply of X must increase.  Again, this effect must be balanced, and in this case it will be by an increase in employment of factors, a decreased output of some other products, or both.  Hicks also explains that

There are essentially the same reasons for expecting complementarity to be dominant among products as for expecting it to be dominant among factors (all the products must be complementary if the contribution to production of the entrepreneur's fixed resources is negligible).  Thus, though exceptions are possible, it is likely that the outputs of most of the other products will tend to rise.

Thus we would typically expect that an increased price of one product will cause an increased supply of other products and an increased demand for the factors.  "Substitute products and regressive factors will only be possible to a limited extent."

Although not stated explicitly, one can conclude that a price change for either a factor or product in the opposite direction of those assumed in the text will drive results in the opposite directions of those noted.

Hicks concludes by noting that these principles for the market conduct of a firm differ from those governing the behavior of an individual in two important ways:  (1) the income effect is absent, and (2) the general tendency will be for factors employed by the same firm to be complementary and for products jointly produced by a firm to be complementary.

This concludes Chapter VII.  Thank you for reading.

Noted in passing:  I am writing this post on the morning after Britain's vote to leave the European Union.  I wonder what Sir John Hicks would have made of this situation.  On the one hand, I tend to believe that he would generally support free trade and the economic integration facilitated by the EU.  On the other hand, some things I've read make me believe he had a tendency at times toward nationalist views, and I do think this vote was motivated largely by nationalism.

Value & Capital, CHAPTER VII, Section 5

This section further discusses the possible relationships among product and factors described in the previous section, where two factors A and B are used to produce one product X.  As the opening paragraph explains, "It is most likely that A and B will be complements, next most likely that no complementarity will be present and no regression, least likely of all that there will be regression.  The reasons for this all hang together."

The discussion then explores a limiting case in which complementarity must necessarily exist. This case assumes that there is no effect on marginal cost from the "fixed 'productive opportunity' of the enterprise" -- no economies of scale that result in cost savings for expanded production, and also no increase in marginal costs with output.  Hicks makes the following argument for A and B being complementary:

Since marginal cost is constant, the increase in product due to a simultaneous proportionate increase in both factors (the marginal product of the two factors taken together) must be constant.  But this joint marginal product is made up of four parts:

1. the marginal product of A with B constant;
2. the increment (or decrement) of this marginal product due to the simultaneous increase in B.  It will be an increment if A and B are complementary, a decrement if they are substitutes;
3. the marginal product of B with A constant;
4. the similar increment (or decrement) due to the increase in A.  To this the same rule applies.

He then claims that as the quantities of factors employed expand, the first and third of these parts must decline (this is because marginal product must be diminishing for an equilibrium to exist). But by assumption the whole does not decline.  Therefore the decline in 1 and 3 must be offset by increments in 2 and 4. Therefore A and B must be complementary.

So in this special case, in which the entrepreneur's "fixed opportunity" does not have a  limiting effect on the scale of production, the two factors must be complementary.  As soon as the fixed opportunity actually does something to limit expansion, the situation changes and the two factors are not necessarily complements.  (They still could be, if their join marginal product declines slowly.)

When might two factors employed to make a single product be substitutes?  For the case where output is variable, Hicks says this can only happen if
(1) "the fixed resources of the entrepreneur must make an appreciable contribution to production,"
(2) "the factors must be such that they would be close substitutes in the production of a given output."
He doesn't define "close" substitutes in the text, but I infer this to mean that the marginal rate of substitution is relatively high. (It's also worth noting that a footnote here says, "Thus in the case of constant costs and two factors, the two factors are necessarily complements in the production of a variable output, and necessarily substitutes in the production of a constant output.  This is a paradoxical situation, which may easily lead to misunderstandings unless we are careful about it."  He then states that it is more convenient not to regard the case of constant costs as the standard case, but as a limiting case in which the effect of the entreprenurial resources vanishes.  In the variable output case, a pair of factors employed by a single firm will ordinarily tend to be complementary.)

At this point, Hicks is able to provide an interpretation of the regression relationship in the current context:  If factor A and product X are regressive, then factors A and B must be substitutes.  From the preceding discussion, it follows that when A and X are regressive, the fixed resources of the entrepreneur must be playing a significant role in limiting production.  He then argues that this limiting effect, together with the regression relationship, imply that factor A must be especially suited to small-scale production, and factor B must be suited to production on a larger scale.  In this case, a decline in the price of A can lead to more employment of A, thereby leading to smaller-scale production and hence a decline in output.  Hicks then concludes, "Regression turns out to be a phenomenon of increasing returns;  one which is just consistent with perfect competition if the fixed entrepreneurial resources are important enough.  Still, it does not yet appear to be a possibility of which we need take much account."

Value & Capital, CHAPTER VII, Section 4

This section continues the "disentangling" of the possible substitution and complementarity relationships that might exist among products or factors of production.  It focuses on the case in which there are variations in the quantities of both factors and products.  

If the firm produces one product X, using two factors A and B, then, as before, a fall in the price of A will cause an increase in the demand for A.  But what happens with X and with B?  Section 1 and Section 2 of this chapter looked at each of these, respectively, in isolation.  Figure 20 indicated that the supply of X must increase, and Figure 21 indicated that the demand for B would decrease, but these arguments did not account for the possibility of complementarity.

When Hicks brings complementarity into the picture, he concludes that there would appear to be three ways in which to balance an increased demand for A:

(1) The supply of the product X may be increased, and the demand for the other factor B may be reduced (here no complementarity is present).
(2) The supply of X may be increased, but the demand for B may increase as well (here the factors A and B are complementary).
(3) The demand for the factor B may be reduced, but the supply of the product may be reduced too.  Here there is a queer sort of inverted complementarity between factor and product.

From figures 20 and 21 it is fairly clear that the typical relationship between factor and product -- in which more of the former will result in more of the latter -- is similar to the substitute relationship between two commodities, or between two factors, or between two products.  Given this similarity, it is natural to ask whether there is something that would be similar to complementarity, and Hicks identifies case (3) as that very thing.  He calls it "regression."  If factor A and product X are regressive, then substituting A for B will decrease the marginal product of B in terms of X.  This in turn will decrease the supply of X (given the prices of B and X).

Hicks closes this section with an amusing bit of sympathy for the reader:

I have a feeling that at this point the reader will rub his eyes, and declare that something must have gone wrong with the argument.  Regression is such a peculiar relation that it is hard to reconcile it with common sense.  Something, it would seem, must have been left out, which either excludes regression, or at least limits its possibility very drastically.  Let us see what that can be.

Hicks will address this question in the next section.

Value & Capital, CHAPTER VII, Section 3

This brief section begins the discussion of production in cases more complex than the simple cases treated in the previous two sections.  Those sections derived results about the necessary effects resulting from a factor or product price change in the one factor, one product case and in the fixed output, two factor case.

This section opens by discussing an analogy with utility theory, and how similar necessary results were obtained in simple cases. Thus the expectation is stated that we are getting these necessary results for the simple cases in production because we are working with only two variables -- one factor and one product, or two factors.  In more complex cases we may expect this "definiteness" to disappear.

This section considers the case of a firm producing a fixed output, using three factors A, B, and C.  Suppose the price of factor A falls; then, because the ratio of the prices of B and C stays the same, they can actually be considered as a single factor.  So we can conclude that the price drop for A will cause an increase in demand for A, and the demand for the combined factor of B and C must decrease.  As Hicks puts it, "There must be a substitution in favour of A at the expense of the other factors taken together."

Things change in the presence of complementarity. If B is complementary with A, the increased demand for A will cause an expansion in demand for B as well and therefore a substitution in favor of A and B, and against C.  Hicks explains that, as in utility theory, A and B are considered complementary when a substitution of A for C (B remaining unchanged) moves the marginal rate of substitution of B for C in favor of B.  Thus for a constant output, if we consider only substitutions among factors, the same rules emerge as for substitutions in a consumer's budget.

Practically the same thing would happen if the quantities of factors were kept constant and the firm varied its production of various products in response to changes in prices.  The only difference is that a rise in price of product X would lead to a substitution in favor of product X, as opposed to a price rise in a factor leading to a substitution against that factor.

Value & Capital, CHAPTER VII, Section 2

This section begins the "disentangling" (mentioned at the end of the previous section) of the possible substitution and complementarity relationships that might exist among commodities that could be products or be factors used in production of other products.  The first step in the analysis is to construct a simple case in which the firm will produce a fixed amount of output and, to do so, it will employ two factors, A and B.  The goal for the firm is to choose the quantities of the factors so as to minimize the cost of production.  Figure 21 illustrates the possible choices.

We assume the production curve is concave up.  This corresponds to the assumption of diminishing marginal rate of substitution between factors.  The line PK represents possible tradeoffs between quantities of the factors A and B, where each pair of quantities on the line has the same total cost, for the given factor prices.  The point P, where PK is tangent to the production curve, represents a position of equilibrium when the ratio of the prices of A and B is MK to PM.

Suppose the price of A were to fall.  Then, the amount of B having an equal value to the quantity ON of A would also fall, from MK to, say, MK1, and the total cost of production (valued in terms of factor B) falls from OK to OK1.  But since PK1 is not tangent to the production curve, the production costs can be reduced by moving to the point P' which is where the line PK2 (parallel to PK1) is tangent to the production curve.

At this new equilibrium, the production costs have been reduced to OK2; less of factor B is employed, and an additional quantity of factor A has been substituted for it.  These results follow just as necessarily as did the expansion of supply of the product when the factor price fell, in the case of one factor and one product.

Value & Capital, CHAPTER VII -- TECHNICAL COMPLEMENTARITY AND TECHNICAL SUBSTITUTION, Section 1

This chapter begins right where Chapter VI leaves off, by asking "what happens when a firm which has been at equilibrium at certain prices of products, and prices of factors, experiences a change in those prices."  How will those price changes affect the quantities of input factors it uses and the quantities of products it produces?  Hicks notes the similarity of the question to those addressed in Chapters II and III for the private individual.

Considering the simplest case, discussed in the last chapter, of an entrepreneur employing a single factor to produce a single product, the equilibrium is as shown in Figure 19 in the last chapter and can be seen in Figure 20 below -- in both figures denoted by P.  If the price of the factor falls, the most immediate effect (before any change is made in production) is that the entrepreneur's surplus increases from OK to OK1.  The reason for this is that the dashed line that represents the exchange of product for factor after the price change, will not decrease as much in moving from point P back to the vertical axis as did the prior exchange line PK;  this is because the quantity ON of factor consumed in production is not as costly in terms of product as before the price change.)  But the line PK1 is not tangent to the production curve, so OK1 is not the maximum surplus that the entrepreneur can achieve under the new conditions.  He will be better off at the new equilibrium P' on the production curve where the tangent P'K2 has the same slope as PK1.

We assume the production curve is concave down, so "the point P', where the tangent slopes upwards less steeply than at P, must lie to the right of P."  Therefore the fall in the price of the factor results in an increased use of the factor as an input to production and an increased output of the product.  As Hicks notes, a rise in the price of the product will also cause a decreased slope of the tangent, with the same effects.

In comparing these results with the earlier results for the private individual, Hicks notes that here the change in price leads to a new point where the tangent line touches the same (production) curve as before the price change, rather than a different curve.

Therefore, in the case of production, we do not have anything similar to the income effects which gave us so much trouble in utility theory. The only 'production effect' is something similar in character to the substitution effect; it is a movement along the curve (in this case a production curve, as in that case an indifference curve), the curve whose properties we know from the stability conditions.

Hicks notes another complexity within the production effect, however:  that of complementarity.  It turns out that complementarity is more complicated in production theory than in utility theory, because we have to consider the relations between two kinds of commodities -- the factors and the products.  Hicks closes with a brief glimpse of upcoming sections, saying "Their mutual relations and their cross-relations will take a little disentangling."