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."
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