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."

Value & Capital, Chapter VI, Section 5

This section, the final one in Chapter VI, returns to the case of perfect competition and spells out the conditions for equilibrium in the general case of a firm converting multiple input factors into multiple products.

As in the simple case of a single factor and a single output, we have a relation between the quantities of factors used as inputs and the quantities of products resulting from production.  In this case we can think of the relation as a surface in multiple dimensions.  It will be useful to think of the elevation of such a surface as reflecting a single quantity, so Hicks explains how, for example, "given all the quantities of factors, and all quantities of products but one, the maximum producible amount of the remaining product can be deduced.  Similarly, given all the quantities of products, and all quantities of factors save one, the minimum amount needed of the remaining factor can be deduced."  In a footnote Hicks points out that such a relation will not be defined everywhere, as there will be cases for which "no amount of a remaining factor will be sufficient to produce the given collection of products."

Starting from a set of factor quantities, and the quantities of products that result from using the factors in production, Hicks notes that variations in production can happen in many ways, but they can all be reduced to some combination of three types of variations:

(1) "One product may be increased at the expense of another, i.e. substituted for another at the margin."
(2) "One factor may be substituted for another."
(3) "One factor and one product may be simultaneously increased (or diminished)."

In a footnote, Hicks states that the first two types can actually be reduced to the third.

We naturally assume that the enterprise will seek to maximize its surplus (the value of products it produces minus the costs of the factor quantities required in producing those quantities of products).  This leads to three conditions of equilibrium corresponding to the condition that price must equal marginal cost:

(1) "The price-ratio between any two products must equal the marginal rate of substitution between the two products."  Hicks calls this a "technical rate of substitution" (as it reflects the technology of production rather than happening according to the preferences of a consumer).
(2) "The price-ratio between any two factors must equal their marginal rate of substitution."
(3) "The price-ratio between any factor and any product must equal the marginal rate of transformation between the factor and product (that is to say, the marginal product of the factor in terms of this particular product)."

Next, the conditions for an equilibrium to be stable are as follows.  For stability in the process of transforming a factor into a product, the condition is that of diminishing marginal rate of transformation (or diminishing marginal product);  this carries over directly from the one-factor one-product case.  For substituting one product for another the stability condition is that of increasing marginal rate of substitution, or as Hicks explains, "increasing marginal cost in terms of the other product (marginal opportunity cost)."  For stability in substituting one factor for another, the condition is diminishing marginal rate of substitution.  Hicks explains in a footnote the intuition behind the opposite direction of the product and factor substitution conditions.

Increasing marginal rate of substitution for products, because the total value of products secured has to be maximized;  diminishing marginal rate of substitution for factors, because the total value of factors used has to be minimized.  These conditions are easily verified graphically, if the amounts of other factors and products are assumed given, and the two products (or factors) in question are measured along two axes.

Hicks explains that the stability conditions must hold for a one-for-one substitution or transformation (one factor or product for one other factor or product) but also for group substitutions or transformations.  Also

The marginal rate of substitution between any pair of groups of products must increase, and between any pair of groups of factors must diminish; the marginal rate of transformation between any group of factors and group of products must diminish.

Finally, Hicks discusses the conditions related to the existence of positive surplus.  Instead of a single condition, there are now multiple conditions.  Namely, it must not pay to abandon production of any subset of the set of all products.

Therefore the average cost of producing each product must be rising, and the average cost of producing each group of products must be rising, including the whole group that includes all the products.

Having laid out the conditions for equilibrium in the general case, Hicks will proceed as in part I of the book.  He will assume that the stability conditions and the conditions for positive surplus hold in the neighborhood of an equilibrium point, and he will then derive laws of market conduct for the firm.

Value & Capital, Chapter VI, Section 4

In this section Hicks discusses some of "the above difficulties" -- apparently referring to difficulties of satisfying the conditions of equilibrium when there are economies of scale.  One way of proceeding in our analysis is by "sacrificing the assumption of perfect competition."  When a firm is to some extent a monopolist, it can set a price above its marginal cost.  This may be a necessary condition of profitability, because average cost could sometimes be greater than marginal cost; but the problem with extending the assumption of monopoly too far is that, "Under monopoly the stability conditions become indeterminate; and the basis on which economic laws can be constructed is shorn away."

His conclusion, essentially, is that the only way out of the situation ("this wreck" as he calls it) is to assume that most of the markets that we will analyze are not significantly different from perfectly competitive markets.  Thus if prices exceed marginal costs by some percentage, we will suppose that these percentages are "neither very large nor very variable."  We will also suppose that diminishing marginal costs are rare, and therefore that marginal costs generally increase with output at the equilibrium point.

Hicks acknowledges that this assumption is a "dangerous step," that may restrict "to a serious extent" the problems that our analysis will be able to address.  He is doubtful, though, that the problems thus excluded are even "capable of much useful analysis by the methods of economic theory."

Value & Capital, Chapter VI, Section 3

This section consists of a discussion of the validity of the assumption, made in the previous section, that production has decreasing returns to scale, namely increasing marginal and average cost, and diminishing marginal and average product, as the scale of production increases.

Hicks lists two considerations that sometimes lead to criticism of the assumed conditions.  One consideration is "the frequent conviction of entrepreneurs themselves" that they have decreasing average costs.  The other consideration is that of indivisibility of certain types of investment in factors of production.

Hicks explains that for short-run problems, the existence of "fixed equipment or plant of the firm, which has been built up in the past, and is likely to be to some extent unique" can cause a situation of a factor of production being combined with resources that the firm cannot purchase on the open market.  He argues that this kind of situation can tend to cause diminishing returns, or increasing costs.

For long-run problems, the argument for increasing marginal cost follows from "the increasing difficulty of controlling an enterprise, as its scale of production grows."

Hicks devotes the final paragraph of the section to discussing the implications of having conditions on both marginal cost and average cost.  As he notes, "Marginal costs must rise as the firm expands, in order to ensure that its expansion stops somewhere."  But this condition alone is not enough to specify where the expansion stops.  The firm can be expected to sell at a price equal to marginal cost, but this marginal cost must not be too close to its minimum;  otherwise marginal cost would be below average cost, and the firm would be selling at a loss.