In microeconomics, the expenditure function represents the minimum amount of expenditure needed to achieve a given level of utility, given a utility function and the prices of goods.
Formally, if there is a utility function
that describes preferences over n goods, the expenditure function
is the desired utility level,
is the set of providing at least utility
Expressed equivalently, the individual minimizes expenditure
subject to the minimal utility constraint that
giving optimal quantities to consume of the various goods as
and the prices; then the expenditure function is Suppose
is a continuous utility function representing a locally non-satiated preference relation on
is (1) As in the above proposition, note that
(2) Continue on the domain
For the second statement, suppose to the contrary that for some
, which contradicts the "no excess utility" conclusion of the previous proposition (4) Let
The expenditure function is the inverse of the indirect utility function when the prices are kept constant.
I.e, for every price vector
and income level
:[1]: 106 There is a duality relationship between the expenditure function and the utility function.
If given a specific regular quasi-concave utility function, the corresponding price is homogeneous, and the utility is monotonically increasing expenditure function, conversely, the given price is homogeneous, and the utility is monotonically increasing expenditure function will generate the regular quasi-concave utility function.
In addition to the property that prices are once homogeneous and utility is monotonically increasing, the expenditure function usually assumes Expenditure function is an important theoretical method to study consumer behavior.
Expenditure function is very similar to cost function in production theory.
Dual to the utility maximization problem is the cost minimization problem [2][3] Suppose the utility function is the Cobb-Douglas function
which generates the demand functions[4] where
is the consumer's income.
One way to find the expenditure function is to first find the indirect utility function and then invert it.
The indirect utility function
is found by replacing the quantities in the utility function with the demand functions thus: where
when the consumer optimizes, we can invert the indirect utility function to find the expenditure function: Alternatively, the expenditure function can be found by solving the problem of minimizing
subject to the constraint
This yields conditional demand functions