On the Hicks-Allen Definition of Complements and Substitutes with Discrete Changes in Prices

For discrete price changes, compensated cross-price effects for two goods need not be equal if the household consumes three or more goods. In this paper I ask whether compensated cross-price effects must have the same sign even if they differ in magnitude. I show that with three or more goods, the answer to this question is "no". For discrete price changes, with more than two goods, the signs of the compensated cross-price effects for two goods can differ depending on which price changes. Hence, the Hicks-Allen definition of complements and substitutes can also differ depending on which price changes