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 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 Then since 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 and and the expenditure function is then See also Expenditure minimization problem Hicksian demand function Slutsky equation Utility maximization problem Budget constraint Consumption set Shephard's lemma References Consumer theory Expenditure