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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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+In [[economics]], a consumer's '''indirect utility function'''
+<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]]

Acetic acid (data page): Difference between revisions

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