Wikipedia

# 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

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