# 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+ 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 $(p^{0},w^{0})$ and ${\frac {\partial v(p,w)}{\partial w}}\neq 0$ , then

$-{\frac {\partial v(p^{0},w^{0})/(\partial p_{i})}{\partial v(p^{0},w^{0})/\partial w}}=x_{i}(p^{0},w^{0}),\quad i=1,\dots ,n.$ ## 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 $p$ and utility level $u$ ::106

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