Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 5

In this section Hicks sums up his conclusions about the effect that a change in price of a commodity has on a consumer's expenditure.  Noting that the fall in price of some good X affects both the demand for X and the
demand for other commodities through an income effect and a substitution effect, Hicks discusses four cases in detail:

(1) A good Y may be highly complementary with X.  In this case the substitution effect will likely be large enough to drown out any income effect, so the demand for Y will definitely increase.

(2) Y may be mildly complementary with X.  In this case the income effect becomes important.  This will usually mean that both effects cause an increase in demand for Y, unless Y is an inferior good, in which case the strength of the income effect will determine whether demand for Y increases, decreases, or remains unchanged.

(3) A good Y may be mildly substitutable for X.  As Hicks notes, the income and substitution effects work in opposite directions in this (very common) case.  Thus the net effect on the demand for Y would tend to be small and could go either way.  (If Y is an inferior good, however, its demand will definitely decrease in this case.)

(4) A good Y may be highly substitutable for X.  In this case, Hicks notes, "the substitution effect will be decidedly dominant, and the demand for Y must diminish."

Finally Hicks asks which cases can have a fall in the price of X that results in no influence on the demand of Y.  This can happen if both the income and substitution effects are negligible or else if they are not negligible but tend to cancel each other out.  Hicks concludes this section by speculating that a fair number of commodities that economists have usually treated as 'independent' of a particular commodity are actually cases of non-negligible effects cancelling each other out.  In other words, "one feels that a good deal of mild substitutability must be present which is prevented from showing itself by being offset by income effects."

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 4

In this section Hicks elaborates on a few details about substitution effects, noting, to begin with, that the substitution being discussed in the context of complementary and competitive goods is exactly the same thing as the substitution discussed in earlier chapters.

When the consumer is choosing consumption amounts of two (and only two) goods, then the goods must necessarily be substitutes.  It is only when there are more than two goods involved that other kinds of relations among them become possible.  Hicks notes that this explains why complementarity cannot be represented on an indifference diagram for two goods, "for X and Y can only be complementary if there is some third thing at whose expense substitution in favor of both X and Y can take place."  A complementary group of commodities requires something outside the group for them to be substituted against.

So with multiple goods it is theoretically possible, in an extreme case, that all but one good could form a complementary group, with each good in the group being a substitute for the one good outside the group.  At the other extreme, there may be no complementary goods at all.  Hicks notes that it will usually be the case that a good will have a relatively small "knot" of other goods that are complementary with it, but "its most probable relation with any other good taken at random will be one of (doubtless mild) substitutability."

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 3

This section examines the operation of the income and substitution effects on complementary and substitute goods.  Hicks begins by noting that indifference diagrams are of little use in this context;  the problem is that the two-dimensional indifference diagrams cannot easily represent the relevant interactions of quantities of the two related goods along with money.  Hicks refers the reader to an algebraic version of the theory in the book's Appendix.  Here he describes the theory in words.

The case of the income effect is relatively straightforward.  As Hicks puts it, "A fall in the price of X acts like a rise in income, and thus tends to increase the demand for every good consumed, excepting inferior goods."  Hicks also notes that these effects will tend to be small if the consumer's spending on X is a small proportion of income.

The substitution effect is somewhat more complicated.  Substitution effects, as Hicks put it, "must involve a substitution in favor of X; and therefore against something other than X."  If we were to lump all other goods into a single composite commodity, then the substitution effect would cause the demand for this "commodity" to decrease with a fall in the price of X.  But it need not be the case that the demand decreases for every one of the commodities making up the composite one.  If Y is one of these commodities and if it is complementary with X, then the increased demand for X will tend to lead to an increased demand for Y.  Hicks gives a detailed explanation of this in terms of marginal rate of substitution for money.  To spell it out in slightly different terms, let me note that the definition of complementary goods (given in the previous section) states that when X is substituted for money, the marginal rate of substitution of a complementary good Y for money is increased.  But we have not assumed the price of Y to have changed, so there is now a mismatch between the price of Y and its marginal rate of substitution for money, which we know from Chapter I Section 6 means the individual cannot be in equilibrium.  The marginal rate of substitution of Y for money would have to decrease to restore equilibrium, which by the principle of Diminishing Marginal Rate of Substitution discussed in Chapter I Section 7, means the substitution of Y for money (i.e. the demand for Y) would have to increase.

By a similar process, a fall in the price of X would encourage a substitution of money against the good Y if Y were a substitute for X.  As Hicks states, "It is our definition of complementarity which draws the exact line between these two situations."

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 2

In this section Hicks explains how to overcome the difficulties described in the previous section regarding the definitions of complementary and competitive (i.e. substitute) goods.  The key step is to replace the use of marginal utility in the definitions with "marginal rate of substitution for money."  The definition of a substitute good then becomes:

Y is a substitute for X if the marginal rate of substitution of Y for money is diminished when X is substituted for money in such a way as to leave the consumer no better off than before.

Similarly, Y is complementary with X if the above substitution of X for money results in an increase in the marginal rate of substitution of Y for money.  Hicks motivates the specific nature of the reduction of money in the substitution of X by noting that the definition of a substitute good should make it "absolutely certain that an extra unit of the same physical commodity is a substitute for preceding units."  And we can only be certain of this when the extra unit of X is substituted for money in a way that leaves the consumer no better off than before; then the result is guaranteed by the principle of diminishing marginal rate of substitution.

As Hicks notes, the resulting definition is free from any dependence on a quantitative measure of utility.  In addition, the symmetry properties described in the previous section hold (namely, if Y is a substitute for X, then X is a substitute for Y, and similarly for complements).  Also this definition reduces to the Edgeworth-Pareto definition if the marginal utility of money is assumed constant, while being directly applicable in cases where the assumption does not hold.

Value & Capital, CHAPTER III -- COMPLEMENTARITY

This section begins with the definition of complementary and competitive goods as used by the economists Francis Ysidro Edgeworth and Vilfredo Pareto.

Y is complementary with X in the consumer's budget if an increase in the supply of X (Y constant) raises the marginal utility of Y;  Y is competitive with X (or is a substitute for X) if an increase in the supply of X (Y constant) lowers the marginal utility of Y.

To put this in familiar terms, one could think of hotdog franks and buns as being an example of a pair of complementary goods.  Conversely, one might think of french fries and onion rings as being substitutes.

With the above definition, the complementary-competitive relationship is symmetric or reversible:  If Y is complementary with X, then X is complementary with Y, and similarly for competitive goods.  Also if the marginal utility of money is constant, this definition implies that, for complementary goods, a fall in the price of X, increasing the demand for X, will raise the marginal utility of Y, which will lead to an increase in demand for Y.  Similarly, if X and Y are substitutes, a fall in the price of X will lower the demand for Y.

Hicks then goes on to describe Pareto's difficulties in trying to translate the definitions of complementary and competitive goods into the terms of indifference curves.  Pareto was able to find a connection between the case of complementary goods (according to the definition above) and the case of indifference curves that are highly bent, as in Figure 12.

Similarly, Pareto found a parallelism between the case of X and Y being substitutes and the case of indifference curves that are very flat, as in Figure 13.

But as Hicks explains, Pareto was not able to discover what degree of curvature corresponds to the distinction between complementary and substitute goods.  In addition, Hicks notes that the definition above violates Pareto's principle of not assuming utility to be measurable.  Hicks will show in the next section how these difficulties can be overcome.

Note to Chapter II -- CONSUMER'S SURPLUS

In this section, Hicks uses some of the results reached in this chapter to examine the doctrine of consumer's surplus.  He refers to Alfred Marshall's work on the topic, as well as an earlier paper by Jules Dupuit (that lacks an important qualification supplied by Marshall).  According to Hicks, Dupuit illustrated consumer's surplus using a price-quantity demand diagram, as shown below.

Dupuit claimed that the utility secured by being able to purchase 0n units of a commodity at the price pn is given by the area dpk on the diagram.  According to Hicks, Marshall uses the same diagram and arrives at the same result, but with an important qualification that the marginal utility of money is assumed to be constant.

Hicks recasts the analysis of consumer's surplus using indifference diagrams, as shown in Figure 11.

The consumer's income is given by OM, and the price of good X is given by the slope of the line ML, which touches an indifference curve at P.  Then ON will be the amount of X purchased, and PF will be the amount of money paid for it. (It may be easy to get confused here, as Figures 10 and 11 have slightly different interpretations.  It happens that the quantity pn in Figure 10 is the price paid by the consumer, whereas in Figure 11 PN is the quantity of money retained (not spent on X) by the consumer.)  The point P lies on a higher indifference curve than the point M does.  The consumer, starting with income OM, would be willing to pay RF to consume quantity ON of good X (since he'd be on the same indifference curve, at point R, as when he started).  Because he only has to pay PF instead of RF, consumer's surplus is given by the length of the line RP.

Hicks explains the derivation of Marshall's conclusion as follows:

If the marginal utility of money is constant, the slope of the indifference curve at R must be the same as the slope of the indifference curve at P, that is to say, the same as the slope of the line MP.  A slight movement to the right along the indifference curve MR will therefore increase RF by the same amount as a slight movement along MP will increase PF.  But the increment in PF is the additional amount paid for a small increment in the amount purchased  at the price given by MP, an amount measured by the area pnn'z' in Fig. 10.  The length RF is built up out of a series of such increments, and must therefore be represented on Fig. 10 by the area built up out of increments such as pnn'z'.  This is nothing else than dpno.

RP will therefore be represented on Fig. 10 by dpk -- Marshall's consumer's surplus.

Hicks then goes on to discuss the basis for Marshall's assumption that marginal utility of money is constant.  This assumption neglects the difference between the slopes of the indifference curves at P and R in Figure 11.  This difference will be important if the commodity under consideration is important in the consumer's budget.  Even if this isn't the case, the difference will still be important, according to Hicks, "if RP is large, if the consumer's surplus is large, so that the loss of the opportunity of buying the commodity is equivalent to a large loss of income."

Hicks goes on to argue that this weakness in Marshall's argument need not be retained, as the notion of consumer's surplus "is not wanted for its own sake; it is wanted as a means of demonstrating a very important proposition, which was supposed to depend upon it."  Although it isn't clear at this point just what "important proposition" Hicks is talking about, he states a page later that

This is all that is necessary in order to establish the important consequences in the theory of taxation which follow from the consumer's surplus principle.  It shows, for example, why (apart from distributional effects) a tax on commodities lays a greater burden on consumers than an income tax.

So, this is where Hicks is headed.  How does he get there?

He states that consumer's surplus is "the compensating variation in income, whose loss would just offset the fall in price, and leave the consumer no better off than before."  He goes on to show a lower bound on this compensating income, which is all that is needed for his argument.  He illustrates the bound on compensating income by means of the following example:

Suppose the price of oranges is 2d. each, and at this price a person buys 6 oranges.  Now suppose that the price falls to 1d., and at the lower price he buys 10 oranges.  What is the compensating variation in income?  We cannot say exactly, but we can say that it cannot be less than 6d.  For suppose again that, at the same time as the price of oranges fell, his income had been reduced by 6d.  Then, in the new circumstances, he can, if he chooses, buy the same amount of oranges as before, and the same amounts of all other commodities;  what had previously been his most preferred position is still open to him;  so he cannot be worse off.

In a footnote, Hicks says that the "compensating variation can thus be proved to be greater than the area kpzk' on Figure 10."  To see that this is the case, think of Hicks's oranges example as being depicted on Figure 10. Buying 6 oranges at the price 2d. corresponds to buying the quantity 0n at the price pn, and buying 10 at the price 1d. corresponds to buying 0n' at the price p'n' (distances are not to scale).  The area kpzk' equals 6d.  In the footnote, Hicks examines whether the compensating variation can be proved to be less than the area kz'p'k'.  In discussing this question, he explains that

At first sight, one might think so;  but in fact it is not possible to give an equally rigorous proof on this side.  This comes out clearly if we use the indifference diagram (Fig. 11).  The line exhibiting opportunities of purchase, when the price of oranges falls by 1d. and income is reduced by 10d., no longer passes through the original point of equilibrium P.  Thus we have no reliable information about the indifference curve it touches.

And without this indifference curve, we cannot compute the compensating variation for the price change.

This completes Hicks's demonstration that a tax on commodities is more burdensome on consumers than an income tax.  He states that other deductions that have been drawn using the concept of consumer's surplus could be similarly analyzed, and in a footnote he points to a then-recent paper by Harold Hotelling, published in Econometrica in July of 1938, as making a similar argument.

CHAPTER II -- Section 7

We're nearing the end of Chapter II of Value and Capital -- after this section, the only remaining material is a technical note connecting the results of this chapter with the topic of consumer's surplus.  In Section 7, Hicks extends his analysis to consider the case of a consumer who comes to the market as both a buyer and a seller of some commodity X.

If we assume the price of X remains fixed, then the previous conclusions of this chapter are unaffected.  The consumer can be assumed to sell at the given price whatever stock of commodity X he brought to the market, then use the proceeds (and whatever other income he had) to purchase a bundle of commodities to maximize his utility.

If the price of X can vary, the situation changes slightly.  The substitution effect works the same as before;  a fall in the price of X will increase the demand for X through consumers substituting X for some of the other purchases they would have made.  The income effect is different, though.  A seller of X is made worse off by a fall in the price of X, so he will decrease his own purchases of X (unless X is for him an inferior good).  Thus, for a seller, income and substitution effects work in opposite directions (except in the unusual case of inferior goods), whereas for buyers the two effects work in the same direction.

Hicks notes that sellers often derive large parts of their income from one particular thing they sell and that in such a case "the income effect is just as powerful as the substitution effect, or is dominant.  We must conclude that a fall in the price of X may either diminish its supply or increase it."  He goes on to argue that this phenomenon is most pronounced in the case of the factors of production.

Thus a fall in wages may sometimes make the wage-earner work less hard, sometimes harder; for, on the one hand, reduced piece-rates make the effort needed for a marginal unit of output seem less worth while, or would do so, if incomes were unchanged; but on the other, his income is reduced, and the urge to work harder in order to make up for the loss in income may counterbalance the first tendency.

Hicks notes that this asymmetry between supply and demand had long been known.  But he regards the explanation of its cause in terms of income and substitution effects "as one of the first-fruits of our new technique."