Indirect utility function

In economics, a consumer's indirect utility function ${\displaystyle v(p,w)}$ gives the consumer's maximal attainable utility when faced with a vector ${\displaystyle p}$ of goods prices and an amount of income ${\displaystyle w}$. 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 ${\displaystyle v(p,w)}$ can be computed from his or her utility function ${\displaystyle u(x),}$ defined over vectors ${\displaystyle x}$ of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector ${\displaystyle x(p,w)}$ by solving the utility maximization problem, and second, computing the utility ${\displaystyle u(x(p,w))}$ the consumer derives from that bundle. The resulting indirect utility function is

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