Overview

expenditure function


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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

E(p1,p2,U)

The solution is described by the two compensated demand functions

x1 = h1(p1,p2,U)

and

x2 = h2(p1,p2,U).

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

E(p1,p2,U)≡p1h1(p1,p2,U)+p2h2(p1,p2,U).

The function

E(p1,p2,U)

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.


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