The relationship between the direct and indirect utility functions may be described by a set of duality theorems. The following illustrative theorems are provided without proof.
Theorem 1: Let f be the finite regular strictly quasi-concave increasing function which obeys the interior assumption (the utility for a commodity combination in which one or more quantities is zero is lower than the utility for any combination in which all quantities are positive). The g determined by equation (c) is a finite regular strictly quassi-convex decreasing function for positive prices.
Theorem 2: Let g be a finite regular strictly quassi-convex decreasing function in positive prices. The h determined by equation (g) is a finite regular strictly quassi-concave increasing function.
Theorem 3: Under the above assumptions
h(q1,...,qn) = g[V1(q1,...,qn),..., Vn(q1,...,qn)] and
g(q1,..., qn) = h[D1(q1,..., qn),..., D1(q1,..., qn)]
The direct utility function determined by the indirect is the same as the direct utility function that determined the indirect.
Duality in consumption forges a much closer link between demand and utility functions for the purposes of empirical demand studies. It is sometimes possible to go from demand functions to the indirect utility function by using Roy's identity, and then to the corresponding direct utility function. Duality is also useful in comparative statics analysis. Homotheticity, separability, and additivity each have counterparts for the indirect utility function. Consequently, many theoretical analyses can be conducted in terms of either the direct or indirect utility function, whichever is more convenient.