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.