Indirect utility function

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 their 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 ${\displaystyle \frac{\mathrm{\partial }v(p,w)}{\mathrm{\partial }w}\ne 0}$, then

${\displaystyle -\frac{\mathrm{\partial }v(p^0,w^0)/\mathrm{\partial }{p}_i}{\mathrm{\partial }v(p^0,w^0)/\mathrm{\partial }w}=x_i(p^0,w^0),i=1,\dots ,n.}$

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 ${\displaystyle u}$:

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