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:

  • Continuous on Rn+++ R+;
  • Decreasing in prices;
  • Strictly increasing in income;
  • 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.
  • quasi convex in (p,w);


  • Roy's identity: If v(p,w) is differentiable at (p^0, w^0) and \frac{\partial v(p,w)}{\partial w} \neq 0, then

-\frac{\partial v(p^0,w^0)/(\partial p_i)}{\partial v(p^0,w^0)/\partial w}=x_i (p^0,w^0),
i=1, \dots, n.


  • Andreu Mas-Colell, Michael D. Whinston, Jerry R. Green, 2007. Microeconomic Theory, Indian Edition, pp. 56–57: The Indirect Utility Function.
  • Jehle, G. A. and Reny, P. J. 2011. "Advanced Microeconomic Theory", Third Edition: Prentice Hall, pp. 28–33.