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In [[economics]], a consumer's '''indirect utility function'''
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<math>v(p, w)</math> gives the consumer's maximal [[utility]] when faced with a [[price level]] <math>p</math> and an amount of [[income]] <math>w</math>.  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 <math>v(p, w)</math> can be computed from its utility function <math>u(x)</math> by first computing the most preferred bundle <math>x(p, w)</math> by solving the [[utility maximization problem]]; and second, computing the utility <math>u(x(p, w))</math> the consumer derives from that bundle. The indirect utility function for consumers is analogous to the [[profit function]] for firms.
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Formally, the indirect utility function is:
 
*Continuous on R<sup>n</sup><sub>++</sub>+ R<sub>+</sub>;
*Decreasing in prices;
*Strictly increasing in income;
*[[homogeneous function|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.
* [[Quasiconvex function|quasi convex]] in (p,w);
Moreover,
*Roy's identity: If v(p,w) is differentiable at <math>(p^0, w^0)</math> and <math>\frac{\partial v(p,w)}{\partial w} \neq 0</math>, then
<math>
-\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.
</math>
 
==References==
* Andreu Mas-Colell, Michael D. Whinston, Jerry R. Green, 2007. ''Microeconomic Theory'', Indian Edition, pp.&nbsp;56–57: ''The Indirect Utility Function''.
*Jehle, G. A. and Reny, P. J. 2011. "Advanced Microeconomic Theory", Third Edition: Prentice Hall, pp.&nbsp;28–33.
 
{{DEFAULTSORT:Indirect Utility Function}}
[[Category:Utility]]

Latest revision as of 06:20, 22 April 2014

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