In economics, a consumer's indirect utility function gives the consumer's maximal attainable utility of goods prices and an amount of income. It reflects both the consumer's preferences and market conditions.

by solving the utility maximization problem the consumer derives from that bundle. The resulting indirect utility function is

The indirect utility function is:

• Continuous on Rn+ × R+ where n is the number of goods;

• Decreasing in prices;

• Strictly increasing in income;

• Homogenous with degree zero in prices and income; if prices and income are all multiplied by a given constant the same bundle of consumption represents a maximum, so optimal utility does not change;

• quasi-convex in (p,w).

Moreover, Roy's identity states that if v(p,w, then

Indirect utility and expenditure

The indirect utility function is the inverse of the expenditure function:

Example

which has the Marshallian demand functions

is found by replacing the quantities in the utility function with the demand functions thus:

Note that the utility function shows the utility for whatever quantities its arguments hold, even if they are not optimal for the consumer and do not solve his utility maximization problem. The indirect utility function, in contrast, assumes that the consumer has derived his demand functions optimally for given prices and income.

See also

• Gorman polar form

• Hicksian demand function

• Value function

References

Further reading