# Indirect utility function

(Redirected from Value function)

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);

Moreover,

• 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

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

## References

• 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.