In economics, a consumer's indirect utility function
gives the consumer's maximal attainable utility when faced with a vector
of goods prices and an amount of income
It reflects both the consumer's preferences and market conditions.
This function is called indirect because consumers usually think about their preferences in terms of what they consume rather than prices.
A consumer's indirect utility
can be computed from their utility function
of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector
by solving the utility maximization problem, and second, computing the utility
the consumer derives from that bundle.
The resulting indirect utility function is The indirect utility function is: Moreover, Roy's identity states that if v(p,w) is differentiable at
, then The indirect utility function is the inverse of the expenditure function when the prices are kept constant.
which has the Marshallian demand functions[2] where
The indirect utility function
is found by replacing the quantities in the utility function with the demand functions thus: where
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.