Indirect utility function

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In economics, a consumer's indirect utility function v(p,w) gives the consumer's maximal utility when faced with a price level p and an amount of income w. It represents the consumer's preferences over 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(p,w) can be computed from its utility function u(x) by first computing the most preferred bundle x(p,w) by solving the utility maximization problem; and second, computing the utility u(x(p,w)) the consumer derives from that bundle. The indirect utility function for consumers is analogous to the profit function for firms.

Formally, the indirect utility function is:

  • Non-increasing in prices, because an increase in prices cannot open up an available bundle that would provide more utility;
  • Non-decreasing in income, because when income rises, at worst you could consume the same bundle;
  • 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.
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