Wikipedia / Expenditure function
Wikipedia / Expenditure function
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.
that describes preferences over n is defined as:
.
and the prices; then the expenditure function is
Properties
is a continuous utility function representing a locally non-satiated is
If the utility function is strictly quasi-concave, there is Shephard's lemma
Proofs
(1) As in the above proposition, note that
.
, which contradicts the "no excess utility" conclusion of the previous proposition
.
Expenditure and indirect utility
The expenditure function is the inverse of the indirect utility:
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
Example
Suppose the utility function is the Cobb-Douglas function which generates the demand functions
is the consumer's income. One way to find the expenditure function is to first find the indirect utility function is found by replacing the quantities in the utility function with the demand functions thus:
when the consumer optimizes, we can invert the indirect utility function to find the expenditure function:
and the expenditure function is then
See also
References
Further reading