Indirect utility function

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 (p⁰, w⁰) and (∂v(p,w))/∂w≠0, then

-(∂v(p⁰,w⁰)/(∂p_i))/(∂v(p⁰,w⁰)/∂w)=x_i (p⁰,w⁰), i=1,…,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, Financial Times: Prentice Hall, pp. 28-33.