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);

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.

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