Inverting systems of demand functions

This note gives a set of sufficient conditions under which a system of demand functions for “gross substitutes” can be inverted to obtain an equivalent system of inverse demand functions, and vice versa. These conditions also ensure that either system can be inverted to obtain a system of “mixed” demand functions, i.e., a mixture of demand and inverse demand functions. In addition, the resultant functions possess intuitive properties expected of gross substitutes.