imported>Scrooge Mcduc at 07:35, 24 June 2025

2025-06-24T07:35:50Z

← Previous revision		Revision as of 07:35, 24 June 2025
Line 17:		Line 17:
	Moreover, [[Roy's identity]] states that 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		Moreover, [[Roy's identity]] states that 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>		:<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),\quad		-\frac{\partial v(p^0,w^0)/\partial p_i}{\partial v(p^0,w^0)/\partial w}=x_i (p^0,w^0),\quad
	i=1, \dots, n.		i=1, \dots, n.
	</math>		</math>

2806:2F0:A600:9E82:C766:B52:772B:3CA2 at 19:06, 9 November 2024

2024-11-09T19:06:01Z

New page

{{More citations needed|date=May 2021}}
__NOTOC__
In [[economics]], a consumer's '''indirect utility function'''
<math>v(p, w)</math> gives the consumer's maximal attainable [[utility]] when faced with a vector <math>p</math> of goods prices and an amount of [[income]] <math>w</math>. It reflects both the consumer's preferences and 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 their utility function <math>u(x),</math> defined over vectors <math>x</math> of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector <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 resulting indirect utility function is

:<math>v(p,w)=u(x(p,w)).</math>

The indirect utility function is:

*Continuous on '''R'''''n''+ × '''R'''+ where ''n'' is the number of goods;
*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]] states that 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),\quad
i=1, \dots, n.
</math>

== Indirect utility and expenditure ==
The indirect utility function is the inverse of the [[expenditure function]] when the prices are kept constant. I.e, for every price vector <math>p</math> and utility level <math>u</math>:<ref>{{Cite Varian Microeconomic Analysis 3}}</ref>{{rp|106}}
:<math>v(p, e(p,u)) \equiv u</math>

==Example==

Let's say the utility function is the Cobb-Douglas function <math>u(x_1, x_2) = x_1^{0.6}x_2^{0.4},</math> which has the Marshallian demand functions<ref>{{cite book |last=Varian |first=H. |year=1992 |title=Microeconomic Analysis |url=https://archive.org/details/microeconomicana00vari_0 |url-access=registration |edition=3rd |location=New York |publisher=W. W. Norton }}, pp. 111, has the general formula. </ref>
::<math> x_1(p_1, p_2) = \frac{ 0.6w}{p_1} \;\;\;\; {\rm and}\;\;\; x_2(p_1, p_2) = \frac{ 0.4w}{p_2}, </math>
where <math>w </math> is the consumer's income. The indirect utility function <math>v(p_1, p_2, w) </math> is found by replacing the quantities in the utility function with the demand functions thus:

::<math> v(p_1, p_2, w) = u(x_1^*, x_2^*) = (x_1^*)^{0.6}(x_2^*)^{0.4} = \left( \frac{ 0.6w}{p_1}\right)^{0.6} \left( \frac{ 0.4w}{p_2}\right)^{0.4} = (0.6^{0.6}*.4^{.4})w^{0.6+0.4}p_1^{-0.6} p_2^{-0.4} = K p_1^{-0.6} p_2^{-0.4}w, </math>

where <math>K = (0.6^{0.6} * 0.4^{0.4}). </math> Note that the utility function shows the utility for whatever quantities its arguments hold, even if they are not optimal for the consumer and do not solve his utility maximization problem. The indirect utility function, in contrast, assumes that the consumer has derived his demand functions optimally for given prices and income.

== See also ==
* [[Gorman polar form]]
* [[Hicksian demand function]]
* [[Value function]]

== References ==
{{Reflist}}

== Further reading==
*{{cite book |last=Cornes |first=Richard |title=Duality and Modern Economics |location=New York |publisher=Cambridge University Press |year=1992 |isbn=0-521-33601-5 |chapter=Individual Consumer Behavior: Direct and Indirect Utility Functions |pages=31–62 }}
*{{cite book |last=Jehle |first=G. A. |author-link=Geoffrey A. Jehle |last2=Reny |first2=P. J. |author-link2=Philip J. Reny |year=2011 |title=Advanced Microeconomic Theory |edition=Third |location=Harlow |publisher=Prentice Hall |isbn=978-0-273-73191-7 |pages=28–33 }}
*{{cite book |first=David G. |last=Luenberger |author-link=David Luenberger |title=Microeconomic Theory |location=New York |publisher=McGraw-Hill |year=1995 |isbn=0-07-049313-8 |pages=103–107 }}
*{{cite book |first=Andreu |last=Mas-Colell |author-link=Andreu Mas-Colell |first2=Michael D. |last2=Whinston |author-link2=Michael Whinston |first3=Jerry R. |last3=Green|author3-link=Jerry Green (economist)|year=1995 |title=Microeconomic Theory |location=New York |publisher=Oxford University Press |pages=56–57 |isbn=0-19-507340-1 }}
*{{cite book |last=Nicholson |first=Walter |title=Microeconomic Theory: Basic Principles and Extensions |location=Hinsdale |publisher=Dryden Press |edition=Second |year=1978 |isbn=0-03-020831-9 |pages=57–59 }}

{{Microeconomics}}

{{DEFAULTSORT:Indirect Utility Function}}
[[Category:Utility function types]]

Indirect utility function - Revision history

imported>Scrooge Mcduc at 07:35, 24 June 2025

2806:2F0:A600:9E82:C766:B52:772B:3CA2 at 19:06, 9 November 2024