expenditure function

Quick Reference

The minimum cost for a consumer of achieving a given utility level. Consider a consumer choosing the quantities, x1 and x2, of two goods to minimize expenditure subject to a utility constraint. The cost minimization problem is

min{x1,x2}p1x1+p2x2subject to U (x1, x2)≥ U.

The function


The solution is described by the two compensated demand functions

x1 = h1(p1,p2,U)


x2 = h2(p1,p2,U).

Substituting the optimal choices back into the objective gives the minimized level of expenditure as


The function


is the expenditure function. Shephard's lemma states that

∂E/∂pi = hi(p1,p2,U),

a result that is useful for calculating the welfare consequences of a price change. See also indirect utility function.

Subjects: Economics.

Reference entries