Indirect Utility Function

The indirect utility function relates $(\text{Prices, Income})$ directly to utility, assuming the person is utility-maximizing

$$\begin{array}{rl}\text{Utility Function: }& u:(x_1,x_2)\mapsto \text{utility}\\ \text{Indirect Utility Function: }& v:(p_1,p_2,I)\mapsto \text{utility}\end{array}$$

Derivation Process:

• Use Lagrangian Optimization to derive Demand Functions $x_1(p_1,p_2,I),x_1(p_1,p_2,I)$
• Substitute these demand functions into the original utility function $u(x_1,x_2)$ to get the indirect utility function.

Properties

• HD0 in $(\text{Prices},\text{Income})$ → Check after derivation!
  • In other words, inflation in prices and income doesn’t change utility
• Decreasing in prices ($\frac{\partial v}{\partial p_1},\frac{\partial v}{\partial p_2}<0$), Increasing in income ($\frac{\partial v}{\partial I}>0$)
  • → Check after derivation!
• Use Roy’s Identity to get back to the Marshallian Demand.