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."