Indirect utility function

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

${\displaystyle v(p,w)=u(x(p,w)).}$

The indirect utility function is:

• Continuous on Rⁿ₊ × 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 ${\displaystyle (p^{0},w^{0})}$ and ${\displaystyle {\frac {\partial v(p,w)}{\partial w}}\neq 0}$, then

${\displaystyle -{\frac {\partial v(p^{0},w^{0})/(\partial p_{i})}{\partial v(p^{0},w^{0})/\partial w}}=x_{i}(p^{0},w^{0}),\quad i=1,\dots ,n.}$

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 ${\displaystyle p}$ and utility level ${\displaystyle u}$:[1]^:106

${\displaystyle v(p,e(p,u))\equiv u}$

See also

• Hicksian demand function
• Gorman polar form
• Value function

References

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

Further reading

• 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.