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.