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.