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In economics, a consumer's indirect utility function v {\displaystyle v} gives the consumer's maximal attainable utility when faced with a vector p {\displaystyle p} of goods prices and an amount of income w {\displaystyle w}. 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 v {\displaystyle v} can be computed from his or her utility function u , {\displaystyle u,} defined over vectors x {\displaystyle x} of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector x {\displaystyle x} by solving the utility maximization problem, and second, computing the utility u ] {\displaystyle u]} 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 is differentiable at {\displaystyle } and ∂ v ∂ w ≠ 0 {\displaystyle {\frac {\partial v}{\partial w}}\neq 0} , then