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:

*Non-increasing in prices, because an increase in prices cannot open up an available bundle that would provide more utility;
*Non-decreasing in income, because when income rises, at worst you could consume the same bundle;
*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).