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.

Value & Capital, CHAPTER V, Section 6

In this section Hicks begins to lay out his explanation of how economic laws can be derived from the stability conditions for a system of exchange.  He starts by supposing that some of the persons trading experience an increased desire for one of the commodities.  He explains that they are prepared to satisfy this desire by "increasing their supply (or diminishing their demand) for the standard commodity" while leaving their demands and supplies for all other commodities unaffected.  (To be clear, he's talking about increasing their supply of the standard commodity to the market, not increasing their own reserves of the commodity.)  He then asks what changes in prices should result.  To get at this answer he notes that the changes must cause an increase in supply from the other traders that would be sufficient to match the increase in demand from the first group.  The stability conditions have already specified what changes in prices will lead to an excess supply:  to increase the supply of a good X, its price must be raised.

What about the effects on other prices?  If we ignore income effects, and if we can assume that market reactions only take place for one other good Y, then the effect of the increased demand for X on the price of Y follows from the same analysis as shown in section 4:  the price will increase if X and Y are substitutes and fall if they are complementary.  Only this kind of change will maintain equilibrium in the Y-market.

Things get more interesting if more than one other price is affected.  Hicks runs through several possibilities, including the following:  "If Z is a substitute for X, the price of Z will be raised [by the increased demand for X]; and if Y is also a substitute for Z, this in its turn will raise the price of Y."  Also, "if Z is complementary with X and a substitute for Y, the effect through the Z-market will be to lower the price of Y."  He summarizes these cases as follows:

Indirect effects through third markets thus obey the rule that an increased demand for X will raise the prices of those goods which are substitutes of substitutes, or complements of complements, for X; it will lower the prices of those goods which are complements of substitutes, or substitutes of complements.

In cases where there are more commodities and prices involved, multiple indirect effects may be significant.  "Sometimes, perhaps often, they will all go in the same direction," Hicks states, giving the example where X and Y are part of a group of goods that are all substitutes for each other.  When X and Y are members of a group of goods that are all complements, however, things are more complicated;  the direct effect would be to lower the price of Y, but due to indirect effects, the net effect on the price of Y could go in either direction.

According to Hicks, "A system of multiple exchange in which no complementarity was present at all would obey a simple rule.  However many indirect effects were allowed for, they would all go in the same direction."  In addition, an increase in the demand for X would raise the price of X;  it would also raise the prices of all the other goods, but proportionately less than the price of X.  Hicks states that "Complete absence of complementarity, in this manner, is of course not at all a probable condition," but he places an interesting footnote here, in which he points out that the case of international currency exchange is an example where this property may be realized approximately.  "To the foreign-exchange dealers, bills in various currencies are probably all substitutes for one another."

Hicks spends the next-to-last paragraph of this section arguing that many actual situations can be expected to approximate the situation of complete absence of complementarity.  He states that in taking two goods at random, we would more likely expect them to be substitutes than complements.  In addition, indirect effects of complementarity tend to neutralize the direct effects.

Hicks's conclusion to this section's discussion is that "it does appear that an increase in demand for a particular good (or group of goods) is most likely to have an upward effect upon prices in general."  Due to indirect effects, it's possible that some goods that are directly or indirectly complementary with the one whose demand increased could have their prices fall, but these would be the exception.  In addition, the general upward effect on prices will not be widespread unless the good or goods whose price increased were "of considerable importance."

Value & Capital, CHAPTER V, Section 5

Hicks opens this section by stating his conclusion about the first of the two questions raised in the previous section.  His "tentatively negative answer" is as follows:  "If the market for a commodity X is stable, taken by itself, it is not likely to be rendered unstable by reactions through other markets."  He turns next to his second question -- that of whether a market for X that is unstable when considered by itself can likely be made stable by the reactions that happen through other markets.

In order for the market for X to be unstable when considered by itself, a rise in the price of X (with other prices given) will raise the excess demand for X.  (Looking back at the excess demand curve given in section 2 of this chapter, we can see that a rising excess demand curve already puts us in very strange territory.  Demand would have to outpace supply -- and increasingly so -- as the price for X rises.)  Reactions in other markets can only stabilize the market for X if they cause a lower excess demand for X, and Hicks argues that this is very unlikely.  Consider some other commodity Y, and assume for now that there are no income effects.  If Y is a substitute for X, a rise in the price of X should increase the excess demand for Y, thereby raising the price of Y;  this in turn should increase the excess demand for X.  If Y is complementary with X, a rise in the price of X should lower the excess demand for Y, thereby lowering the price of Y;  but because of the complementarity, this should increase the excess demand for X.  So in both cases the indirect reactions should increase the excess demand for X.  Thus, Hicks concludes, a market for X that is unstable, when taken alone, must be even more unstable when indirect effects are considered.

Hicks then goes on to acknowledge that this argument is not conclusive, because of potential complications when more than one other market is considered as well as the potential of income effects.  Hicks notes that the X-market will only be unstable to begin with, taken by itself, when income effects are large.  And if income effects tended to increase the demand for X when the price of X goes up, a similar effect could be possible when the price of Y changes.  As a result, the Y-market could exercise a stabilizing influence on an X-market that is unstable when taken by itself.  Hicks downplays the importance of this possibility but notes it as a possible exception to the rules he intends to set out in the next section of this chapter.

Hicks summarizes his conclusions about stability as follows:

There is no doubt that the existence of stable systems of multiple exchange is entirely consistent with the laws of demand.  It cannot, indeed, be proved a priori that a system of multiple exchange is necessarily stable. But the conditions of stability are quite easy conditions, so that it is quite reasonable to assume that they will be satisfied in almost any system with which we are likely to be concerned.  The only possible ultimate source of instability is strong asymmetry in the income effects. A moderate degree of substitutability among the bulk of commodities will be sufficient to prevent this cause being effective.

Finally, noting that if a system of exchange is stable at all it is likely to be perfectly stable, Hicks considers it "quite justifiable" to proceed to investigate how a perfectly stable system of multiple exchange reacts to changes in prices.

Value & Capital, CHAPTER V, Section 4

This somewhat lengthy section begins by raising two questions about the stability of a system of exchange involving more than two goods:

(i) Granted that the market for X is stable, taken by itself (that is to say, a fall in the price of X will raise the excess demand for X, all other prices being given), can it be rendered unstable by reactions through the markets for other commodities?  (ii) Supposing that the market for X is unstable, taken by itself, can it be made stable by reactions through other markets?

 The remainder of this section explores the first of these questions.

Hicks uses diagrams like those shown in Figure 16 to study the effect on the market for X of market reactions for another commodity Y (assuming given prices for all the other commodities). Starting with a diagram where the prices of X and Y correspond to the axes, he describes the construction of a curve XX' as follows:

Corresponding to any arbitrary price of Y, we can determine the price of X which will equate the supply and demand for X, and thus bring the X-market into equilibrium. ... Plotting this as a point on the diagram, let us then construct a series of similar points, by starting with other arbitrary prices of Y.  These points will form a curve, which we shall call XX'.

Hicks then begins to explore what can be said about such curves.

If the price of Y were to change, this would affect the levels of supply and demand of X at various prices of X.  These effects could be observed as changes in the excess demand curve for X.  If a rise in the price of Y raises the excess demand curve for X, the equilibrium price of X will be raised, and thus the curve XX' will be positively inclined.  Conversely, if the price rise for Y lowers the excess demand curve for X, the curve XX' will be negatively inclined.

But how is the excess demand curve for X affected by a rise in the price of Y?  As Hicks notes, this happens through an income effect and a substitution effect.  As mentioned in Section 2 of this chapter, the income effect will often be small (because it consists of two parts that likely work in opposite directions).  As an approximation,  Hicks supposes that we can neglect the income effect and concludes that "XX' will slope upwards when X and Y are substitutes and downwards when they are complementary."

In perhaps the most complicated passage in this section, Hicks devotes a paragraph to examining the case in which prices of X and Y both rise in the same proportion, leaving the ratio of their prices unchanged.  He notes that this has exactly the same effect  as "an equal proportionate fall in the prices of all other goods than X and Y (including the standard commodity), which can thus be lumped together and treated as a single commodity T."  If we ignore income effects, we expect a fall in the price of T to lower the excess demand for X unless X and T are complementary.  This means that the price of X would have to fall in order for equilibrium in the market for X to be restored.  Hicks therefore concludes that, "excepting when X is complementary with T, the rise in the price of X needed to maintain equilibrium in the market for X must be less than proportional to the rise in the price of Y.  The XX' curve must be inelastic."

Thus Hicks draws the following conclusions about the XX' curve, when no income effects are considered. When X is a substitute both for Y and for T (the composite good mentioned above), the curve XX' must slope upwards, and its elasticity must be less than one.  This is the case illustrated in the upper left diagram of Figure 16.  If X and Y are complementary, XX' slopes downwards;  this is the case shown in the upper right diagram of Figure 16.  If X and T are complementary, XX' slopes upward with elasticity greater than unity;  this is the case illustrated in the lower diagram of Figure 16. 

Similar properties hold when it comes to constructing the curve YY', representing the prices that bring the market for Y into equilibrium, given prices for X.  The only complication comes when considering complementarity between Y and T.  With the price of X measured along the horizontal axis, we will have YY' inelastic when Y and T are complementary, elastic when Y is a substitute for X and T.

With these properties of the XX' and YY' curves established, we can now analyze the stability of the multi-commodity system.  If the XX' and YY' curves intersect at some point P, then at the prices represented by that point, both the X-market and the Y-market will be in equilibrium.  The equilibrium will be stable if a small rise in the price of X causes a reaction in the price of Y, that in turn causes the price of X to decrease.  For this to happen, XX' must slope upwards more steeply than YY', as illustrated in Hicks's Figure 17.

Consider a price of X that is greater than P.  The Y-market would be brought to equilibrium at a point to the right of P on the curve YY' (suppose it is the point labelled Q).  The vertical coordinate of Q represents the price of Y for this equilibrium in the Y-market.  But the X-market is now out of equilibrium;  to restore equilibrium in the X-market for the given price of Y will require moving to the point R on the XX' curve.  Hicks's explanation simply notes that the price of X at point R "is nearer to the equilibrium position than that from which we started."  This is sufficient to establish his stability condition.  But it is also interesting to consider how further adjustments might proceed.  At point R the Y-market is out of equilibrium, so if the next step in the adjustment process involved moving to a new equilibrium in the Y-market for the given price of X, it would be at the point on the YY' curve that is directly below R.  One can visualize how such an adjustment process would continue to work itself out between the XX' and YY' curves, in a stair-step pattern, back toward the equilibrium point P.

Hicks notes that the stability condition above implies that "if there is no complementarity in the system, so that X, Y, and T are all substitutes for one another, then the system must be stable."  Furthermore, the second diagram in Figure 16 indicates that the presence of complementarity does not automatically imply instability. Hicks concludes this section by arguing that even in the case of the maximal degree of complementarity -- namely the case in which the XX' and YY' curves coincide -- the complementarity is not sufficient to cause instability; therefore "in our case of three-way exchange it is not possible for complementarity to be a source of instability."  And, he notes, this result "can be proved to hold mathematically for any number of goods."