Expenditure Function

def. Expenditure Function. ($\approx$Income Function) Relates prices and a certain utility to the required income to achieve that level of utility.

$$E:(p_1,p_2,\bar{u})\mapsto I$$

Derivation. Substitute Hicksian Demand functions into the Budget Constraint.

Properties.

• HD1 in prices ($E(tp1,t{p}_2,\bar{u})=tE(p_1,p_2,\bar{u})$)
• Increasing in prices ($\frac{\partial E}{\partial p_1},\frac{\partial E}{\partial p_2}\ge 0$)
  • equal when you’re not buying that good.
  • Higher prices require more income
• Increasing in utility ($\frac{\partial E}{\partial u}>0$)
  • More utility requires more expenditure.
• Use Shepard’s Lemma in order to get back to Hicksian Demand Curve