# Finance:Indirect utility function

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 his or her 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;
- 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 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[edit]

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]:^{[1]}^{:106}

- [math]v(p, e(p,u)) \equiv u[/math]

## See also[edit]

- Hicksian demand function
- Gorman polar form
- Value function

## References[edit]

- ↑ Varian, Hal (1992).
*Microeconomic Analysis*(Third ed.). New York: Norton. ISBN 0-393-95735-7.

## Further reading[edit]

- Cornes, Richard (1992). "Individual Consumer Behavior: Direct and Indirect Utility Functions".
*Duality and Modern Economics*. New York: Cambridge University Press. pp. 31–62. ISBN 0-521-33601-5. - Jehle, G. A.; Reny, P. J. (2011).
*Advanced Microeconomic Theory*(Third ed.). Harlow: Prentice Hall. pp. 28–33. ISBN 978-0-273-73191-7. - Luenberger, David G. (1995).
*Microeconomic Theory*. New York: McGraw-Hill. pp. 103–107. ISBN 0-07-049313-8. - Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995).
*Microeconomic Theory*. New York: Oxford University Press. pp. 56–57. ISBN 0-19-507340-1. - Nicholson, Walter (1978).
*Microeconomic Theory: Basic Principles and Extensions*(Second ed.). Hinsdale: Dryden Press. pp. 57–59. ISBN 0-03-020831-9.

*https://en.wikipedia.org/wiki/Indirect utility function was the original source. Read more*.