Indirect utility function
In economics, a consumer's indirect utility function formula_1 gives the consumer's maximal attainable utility when faced with a vector formula_2 of goods prices and an amount of income formula_3. 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 formula_1 can be computed from his or her utility function formula_5 defined over vectors formula_6 of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by  the vector formula_7 by solving the utility maximization problem, and second, computing the utility formula_8 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 formula_10 and formula_11, then Indirect utility and expenditure. The indirect utility function is the inverse of the expenditure function when the prices are kept constant. I.e, for every price vector formula_2 and utility level formula_14: Example. Suppose the utility function is the Cobb-Douglas function formula_16 which has the Marshallian demand functions where formula_18 is the consumer 's income. The indirect utility function formula_19 is found by replacing the quantities in the utility function with the demand functions thus: where formula_21 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.