Indirect utility function

In economics, a consumer's indirect utility function $v(p, w)$ gives the consumer's maximal attainable utility when faced with a vector $p$ of goods prices and an amount of income $w$. 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 $v(p, w)$ can be computed from his or her utility function $u(x),$ defined over vectors $x$ of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector $x(p, w)$ by solving the utility maximization problem, and second, computing the utility $u(x(p, w))$ the consumer derives from that bundle. The resulting indirect utility function is

$v(p,w)=u(x(p,w)).$

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 states that if v(p,w) is differentiable at $(p^0, w^0)$ and $\frac{\partial v(p,w)}{\partial w} \neq 0$, then

$-\frac{\partial v(p^0,w^0)/(\partial p_i)}{\partial v(p^0,w^0)/\partial w}=x_i (p^0,w^0), i=1, \dots, n.$

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 $p$ and utility level $u$:^[1]:106

$v(p, e(p,u)) \equiv u$

See also

• Hicksian demand function
• Gorman polar form

References

1. Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0393957357.

• Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (2007). Microeconomic Theory (Indian ed.). pp. 56–57.
• Jehle, G. A.; Reny, P. J. (2011). Advanced Microeconomic Theory (Third ed.). Prentice Hall. pp. 28–33.
• Nicholson, Walter (1978). Microeconomic Theory: Basic Principles and Extensions (Second ed.). Hinsdale: Dryden Press. pp. 57–59. ISBN 0-03-020831-9.