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.

Value & Capital, Chapter VI, Section 2

This section presents an analysis that derives the equilibrium conditions for a firm operating in a perfectly competitive market.  In the simplest case, the firm is able to take advantage of technical opportunities for converting a single factor A into a single output X.  When the firm faces prices for A and X that are given by the market, it will "embark upon production, so long as the total value of the product secured is greater than the total value of factor employed.  Further, it will be to its advantage to produce that quantity of product which will make the excess as large as possible."

Figure 18 illustrates this graphically.  The horizontal axis measures quantities of the input A, the vertical axis measures quantities of the output X, and the curved line illustrates the set of production possibilities (input-output combinations) available to the firm.

Suppose the distance ON corresponds to the quantity of factor being employed as input.  The curve shows that the distance PN gives the quantity of product that results.  Suppose the line PK has slope equal to the ratio of the unit price of factor to the unit price of product.  Then the distance MK represents the quantity of product having the same market value as ON, the quantity of factor consumed in production. (This interpretation of MK may be easier to grasp by keeping in mind the familiar definition of slope as "rise over run.")  Then the distance OK corresponds to the surplus product that the firm generates.  The market value of OK is the net value of receipts minus costs.  As Hicks notes, "The conditions of equilibrium are that OK should be a maximum, and should be positive."

As the text points out, OK is not maximized in Figure 18.  The line PK could be raised (thus increasing OK) until PK is tangent to the production curve, as in Figure 19.

The text goes on to set out the conditions of equilibrium as follows:

(1)  To maximize the excess value of production, the line PK must be tangent to the production curve.  Thus the slope of the production curve at equilibrium must equal the ratio of the price of the factor to the price of the product.  The slope of the production curve also represents the marginal product.  Therefore, as Hicks notes, this condition for equilibrium "can be put in either of two familiar forms:  the price of the factor equals the value of its marginal product, or the price of the product equals its marginal cost."  If this condition did not hold, the firm would find it more profitable to undertake the production process with some other level of input.

(2)  Because OK must be a maximum rather than a minimum, the production curve must be "convex upwards" at the point of tangency (or as one likely heard such a curve described in introductory calculus, "concave down").  This implies that the slope (and therefore the marginal product) must be decreasing at the equilibrium point.

Hicks then discusses the similarity between these two conditions and those derived earlier for the theory of subjective value.

The production curve, as we have drawn it, is remarkably similar in its properties to an indifference curve.  Where we had equality between a price-ratio and a marginal rate of substitution, we now have equality between a price-ratio and a marginal product -- which may be looked on, if we choose, as a marginal rate of transformation.  As for the stability condition, diminishing marginal rate of substitution is replaced by diminishing marginal product.  These two conditions are therefore substantially identical, and by their means we shall be able to construct a theory of the conduct of the firm closely similar to our theory of the conduct of the private individual.

Hicks then discusses a third condition of equilibrium, for which he notes that there is no correspondence in the theory of subjective value.

(3)  The surplus OK must be positive.  This can only be the case if the slope of OP is greater than that of PK.  Since the production curve lies at or below PK, this implies that the slope of OP must be diminishing as P moves to the right.  Since the slope of OP corresponds to the average product, this means that average product must be diminishing.  In a footnote, Hicks derives the equivalent expression of this condition in terms of average cost:

Alternatively, we may argue in the following way.  If there is a positive surplus, price must be greater than average cost.  But price equals marginal cost.  Therefore marginal cost must be greater than average cost.  Therefore the production of an additional unit must raise average cost.  Therefore average cost must be increasing.

Hicks concludes the section by summarizing the equilibrium conditions in the following lists:

1. Price of factor = value of marginal product.           1. Price of product = marginal cost.

2. Marginal product diminishing.                             2. Marginal cost increasing.

3. Average product diminishing.                              3. Average cost increasing.

Value & Capital, CHAPTER VI -- THE EQUILIBRIUM OF THE FIRM, Section 1

The author, Sir John Hicks, sets out in this opening section his goal for the chapter, which is "to bring out a certain parallelism which exists between the case of the firm and that of the private person."  By doing so, he intends "to extend the theory of exchange set out in the last chapter to take account of production as well."

In the previous chapter Hicks assumed that an individual consumer, having come into the market with some collection of commodities or services, could only obtain other ones by means of exchange.  Now he will allow the possibility that sometimes they can generate new commodities through production -- a process of technical transformation of a set of inputs into a set of outputs.  An opportunity to undertake such a transformation will be chosen only if the set of goods generated has a higher value than the set consumed in the transformation.  Changes in market conditions can result in changes in the set of production opportunities that are profitable.  In addition, the set of individuals who can take advantage of these opportunities may change.

In general, an entrepreneur will acquire services that are factors of production and will do so "not because he has any direct desire for them, but because he needs them for the full exploitation of his productive opportunities."  The production that is enabled by these factors will determine the amounts of factors employed.  The enterprise that converts the factors into products may be regarded as an economic unit, operating completely separately from the private account of the entrepreneur.  "It acquires factors and sells products;  its aim is to maximize the difference between their value."

Value & Capital, CHAPTER V, Section 8

In this brief section, the last one of the chapter, Hicks summarizes by saying that he doubts much more can be said about the theory of exchange, at least at a similar level of generality.  He states that it would be possible to proceed to dealing with applications -- especially given how much of the traditional theory of international trade was developed by studying the simple exchange of two goods -- but he is choosing not to proceed in that direction in this book.

He notes, however, that having spent so much time on the theory of exchange will prove useful in the upcoming chapters.  With production and with dynamic problems, "almost exactly the same questions" as those examined in this chapter will appear.  Although these questions will seem slightly more complicated at first, Hicks claims they can be cast in familiar forms, "and so it will turn out that we know the answers already."

This completes the discussion of the first five chapters; thank you for reading this far.

Value & Capital, CHAPTER V, Section 7

This section adds one final proposition to the discussion of the laws of exchange.  Hicks had previously shown that when demand for some good X rises, the price of X must rise.  In this section he addresses the question of what governs the extent of the rise in price.  His answer (again ignoring income effects) is the following:

It can be shown that a given rise in demand will affect the price of X less, the more substitutability or the less complementarity there is between any pair of commodities in the system.

It makes intuitive sense that if there are a large number of substitutes for X, then the increased demand for X will cause some consumers to shift to a substitute rather than pay the higher price for X.  Hicks's explanation is similar, but he adds that the substitutes themselves will also tend to rise in price, although the rise will be spread broadly over the whole set of substitutes and will therefore affect each one very little.

When there is less complementarity in the system, one could think of this as a situation in which it is relatively painless to shift demand from one good to another;  making do with less of a commodity does not tend to diminish the utility gained from consuming other commodities.  Hicks comes at the question from the other direction, assuming a large group of complements; then, an increase in the demand for X will lead suppliers of the increased volume of X to tend to dispose of goods complementary with X.  If the demand for these complementary goods had not increased, this would tend to drive down their prices, which would then lead to increased demand.  Since X is complementary with them, demand for X would increase as well, thus further increasing its price.

Hicks notes that we can apply these same concepts "at a second remove" to the substitutes and complements themselves.  If such a good (either a substitute or complement for X) has good substitutes, the effect on its price from the change in demand for X will be less pronounced.  If on the other hand it is a member of a set of complements, the effect on its price will be increased.

Hicks concludes the section by noting that

Complementarity, like imperfect substitutability, is therefore to be regarded as an element of rigidity in the system, which diminishes the elasticity of supply of any particular good.  Similarly, of course, if we had begun with an increase in supply of X, we should have found the same factors diminishing the elasticity of demand.