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."

Value & Capital, CHAPTER V, Section 3

This section begins to address the case of exchange of more than two commodities.  For stability of an equilibrium in this context, it must be the case, as before, that a drop in the price of X will tend to make the demand for X greater than the supply.  Regarding stability, Hicks asks whether such an effect must be assumed to happen when the prices of all other commodities are given, or when these other prices are allowed to adjust so as to preserve equilibrium in the other markets.  Hicks argues that the answer is "that it is what happens when all other prices are adjusted that is really most important."  He notes that if a small rise in the price of X makes supply greater than demand, not by the working of the X-market alone, but rather through repercussions in the other markets, "the establishment of an equilibrium price system is going to be a more awkward business; but once equilibrium is reached it will still be a stable equilibrium, properly speaking.  A movement away from equilibrium will set up forces tending to restore equilibrium."

Hicks proposes to call a system in which all conditions of stability are satisfied perfectly stable.  He uses the term imperfectly stable to describe a system in which some conditions of stability are not satisfied, "but in which supply does become greater than demand when price rises if all repercussions are allowed for."

Hicks gives a brief preview of the fact that there are some problems where imperfect stability plays an important role.  As he notes, "Some of the most remarkable of them arise in connexion with the famous 'instability of credit.'"  He does not go into detail in the present section, but notes that "a pure system of multiple exchange, if it is stable at all, is likely to be perfectly stable."  He concludes the section by dismissing wholly unstable systems as "hardly interesting" and calling the derivation of their laws of change "a nonsense problem."

Value & Capital, CHAPTER V, Section 2

Hicks begins his examination of the stability of equilibrium with the case of simple exchange of two goods, X and Y.  In this case, the equilibrium condition is that supply equals demand for one of the goods (as this implies that supply equals demand for the other good as well).  The stability condition for equilibrium is that "a fall in the price of X in terms of Y will make the demand for X greater than the supply of X."  Hicks defines the excess demand as the difference between demand and supply at each price.  Given this definition, the equilibrium condition is that excess demand must equal zero, and the stability condition is that a fall in price should increase the excess demand (since it must become positive, after having been zero).  Hicks plots supply, demand, and excess demand curves together on Figure 14.

Hicks states that it is "obvious" from this diagram that "when the demand curve slopes downward to the right, and the supply curve upwards to the right, the excess demand curve must be downward sloping."  Just in case this doesn't seem obvious, note that on this diagram the demand curve is a line with negative slope, and the supply curve is a line with positive slope, therefore demand minus supply must be a line with negative slope.  (Note also that this figure, like most in economics, plots price on the vertical axis, so we read the quantity for a given price horizontally -- hence the price level at which the excess demand curve crosses the price axis is exactly the price level at which supply equals demand.)

Hicks then asks what can be said in general about the effect of a fall in price on excess demand.

As seen in Chapter II, both supply and demand effects can be analyzed in terms of income and substitution effects;  thus, Hicks argues, excess demand can be analyzed this way as well.  The substitution effect of a fall in price will work to increase demand and reduce supply;  thus the fall in price will tend to increase excess demand.  The income effect works by making buyers better off and sellers worse off.  If the good in question is not an inferior good, this will tend to increase both demand and supply.  Then, as Hicks concludes, "the direction of the income effect on excess demand depends on which of these two tendencies is the stronger."  These income effects could cancel each other out entirely, in which case the excess demand curve will slope downward, and the equilibrium will be stable.  In general, however, there will be some net increase or decrease in excess demand due to the redistribution of income between the buyers and sellers.  Because there will likely be some cancellation in effects, however, Hicks recommends that, "when dealing with problems of the stability of exchange, it is a reasonable method of approach to begin by assuming that income effects do cancel out, and then to inquire what difference it makes if there is a net income effect in one direction or the other."

Hicks points out that the equilibrium will still be stable if the net income effect goes in the same direction as the substitution effect.  He notes that the only possibility for instability comes "when there is a strong income effect in the opposite direction -- that is to say, the sellers of X will have to be much more anxious to consume more X when they become better off than the buyers of X are."   He illustrates such a case using Figure 15, in which point Q is unstable.  Even here, however, he notes that this excess demand curve "would still be able to turn round and produce stable positions (such as P or P')."

Hicks points out that the difficulty in a situation such as the one illustrated here is that there may be more than one stable equilibrium.  A change in the tastes of consumers may move the excess demand curve to the right, which in the case of starting from the equilibrium of P', could result in a sharp and discontinuous jump to the new equilibrium position of P.

Value & Capital, CHAPTER V -- THE WORKING OF THE GENERAL EQUILIBRIUM SYSTEM, Section 1

In this section, Hicks briefly outlines how the laws of change of the price system will be derived from stability conditions.  In the context of an exchange system, stability means that slight movements away from an equilibrium position will tend to cause reactions that push the system back toward equilibrium.  Under perfect competition, a rise in price tends to cause supply to exceed demand, which will tend to cause the price to fall.  Similarly, a fall in price tends to cause demand to exceed supply, which will cause the price to rise.  These relationships between supply and demand constitute the stability conditions for an equilibrium in an exchange system.

Since the theory of exchange is based on the theory of demand, Hicks proposes to check his investigations of the stability of exchange for consistency with the theory of demand as worked out in Chapters II and III.  From the stability conditions, Hicks will deduce laws of change, i.e. "rules about the way in which the price-system will react to changes in tastes and resources."  This investigation (still of a pure exchange economy) will occupy the eight sections of Chapter V.  In Chapter VI he will begin to examine markets with production.

Value & Capital, CHAPTER IV, Section 4

In this section Hicks briefly discusses the significance of what has been presented thus far about the systems of equations derived by the methods of Walras.  Hicks considers it "a great achievement to have shown, even so schematically, the interrelation of markets."  But he also acknowledges that many economists have felt that Walras's approach has a certain "sterility" about it.  He suggests that the reason for this sterility is that Walras did not proceed to work out the laws of change for these types of systems.  The theory presented up to this point tells us what conditions must be satisfied by a set of equilibrium prices established for a given system of resources and preferences, but it does not explain what will happen if these preferences or resources change.

Hicks will begin to address these questions in the next chapter.

Housekeeping note:  This will be the final post of 2015.  Thanks to everyone for reading; please continue to read the blog, and send me any feedback you have.  Ideally, I would like to finish the book in 2016, which would mean I would have to pick up the pace significantly:  there are about 240 pages remaining (excluding the mathematical appendix, which I do not intend to cover);  thus I would need to cover about 20 pages per month, instead of the 5 or so that I averaged in 2015.  This will be a challenge, but we'll see how it goes.

Best wishes to all for a happy new year.