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."