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."