Expenditure function

The output function is a function used in microeconomics, which indicates how much a consumer must spend a minimum to achieve a given level of utility. Utility function and prices of the goods with which the benefits can be obtained, are given here.

Definition and meaning

The starting point for the derivation of the output function is the same as that used to derive the Hicksian demand. It consists in the output minimization problem

( Details can be found in the article Hicksian demand function. ) One solution to this releasable by means of the Kuhn -Tucker method optimization problem is known as Hicksian demand, with the vector of the requested quantities of goods, the corresponding price vector and an ex ante demanded ( minimum ) is use level. In words, it is in such demand that is to say that quantity of goods - depending on the commodity prices - which is necessary to the lowest possible cost to achieve a given level of utility. Substituting the Hicksian demand again in the minimized function, as one calls the resulting function as output function. It is therefore

While the Hicksian demand function gives the quantities of goods that are in demand in the output minimum, the output function returns the amount of expenditure which are necessary; In other words, the argument of the minimum, while the actual minimum, which is why you just could postulate that it is equal to the output function already in the above representation of the minimization problem.

Properties

It can be shown that under the usual conditions - continuous and strictly monotonically increasing - has among others the following properties:

  • Homogeneous of degree one in so and;
  • Monotonically increasing in;
  • Strictly increasing in for;
  • Concave.

Related to the indirect utility function

Derivation of the utility function

The output function was constructed above from the utility function out. Under certain conditions it is also possible to construct from the output function, the utility function.

Derivation of the utility function of the output function: Let the utility function u continuous, strictly increasing and quasi- concave in in. Be the output function. Then, the original, is used for the construction of the utility function is given as follows:

Convex function
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