Welfare Measure with Price Changes

1. Welfare Measure with Price Changes • Compensating Variation (old utility, new prices) • Price Decrease • The income that could be taken away from the consumer to make them as happy as before • Price Increase • The income that would be given to make the consumer as happy as before the change • Equivalent Variation (new utility, old prices) • Price Decrease • The income that would have to be given to consumer to keep it at the higher price • Price Increase • The income that the consumer would pay to keep it at the lower price

2. Marshallian view of CS • If consumer surplus is to be useful construct, it must be capable of being observable in the real-world. • Marshallian CS is basically the area under the demand curve, above price paid, from zero to the amount purchased. • Does not take into the consideration the income effect resulting from charging the consumer the intramarginal values for each successive unit.

3. Two Different Demand Curves • Hicks Demand • Compensated • Utility held constant • Marshallian Demand • Uncompensated • Original demand

4. Expenditure Function • The dual problem is to allocate income so as to reach a given utility level with the least expenditure. • Reverse the problem. Dual Primal Solution gives us: Maximum utility (U*) with Income constrained to Solution gives us: Minimum Expenditure (I*) with Utility constrained to

5. Definition: Indirect Utility Function • Use utility maximization to derive optimal values of X as a function of price and income. • Now substitute X* into the utility function to get maximum utility. • U(X1*,X2*,…,Xn*) • V is the indirect utility function = maximum utility as a function of prices and income. • Can in some cases be estimated directly.

6. Definition: Expenditure Function • Minimize expenditure, subject to Ū=U(·). Derive compensated Demand curves • Optimal solution expressed as: • “E” is the expenditure function = minimum expenditure utility as a function of prices and utility.

7. Envelope Theorem (EMP) • The Hicksian demand functions are the first partials of the expenditure function. • We get the same demand “path” of price changes even if other prices change. • Area below the Hicksian demand functions are always compensating variation.

8. Envelope Theorem (UMP - Roy’s Identity) • The Marshallian demand functions are NOT in general partial derivatives of some integral function. • They are the first partials of the indirect utility function divided by the marginal utility of income. • This is the sum of changes in utility as price changes the 1/λM that converts change in utility to $$. The conversion factor changes with price. • If marginal utility of money (1/λM ) is constant then we get

9. Example • Derive compensated demand curves from dual problem (1) Set up Lagrangian and solve FOC’s:

10. (2) Use first two FOC’s to solve for X and Y (3) Derive compensated demand for Xc and Yc.

11. (4) Derive Expenditure Function (Minimum expenditure necessary to maintain constant utility, Ū) A (5) Indirect Utility function is (at A):

12. Compare to Ordinary (uncompensated) Demand

13. (1) Use first two FOC’s to solve for X* and Y* (2) Demand for X, X*, is found by substituting Y* into 3rd FOC and solving for X* (3) Substitute X* into Y*, and get Y*

14. Compare Ordinary Compensated Demand (uncomp.) Comp. Demand Expenditure Function Indirect Utility

15. Derive the Slutsky Equation Primal and dual give us same demand at the optimal bundle Differentiate both sides Re-arrange

16. Note that I* = PXX + PYY => Therefore Which is the Slutsky equation we discussed earlier….. Slope of Ordinary Demand Function = Slope of Compensated demand – (Slope of Engel Curve)(X*)

17. Calculate Substitution and Income Effects • What is the Substitution Effect? Xc(P1;U0) - Xu(P0;U0) • What is income Effect? Xu(P1; U1) -Xc(P1;U0) Y B I1 A C U0 U1 X P1 P0 S I

18. Summary: What’s the point? • Ordinary demand (Xu) • Derived from Utility Maximization • Compensated demand (Xc) • Derived from Expenditure Minimization • Relationship between Ordinary and compensated demand • Slutsky Equation • Slope of Ordinary Demand Function = Slope of Compensated demand – (Slope of Engel Curve)(X*) • Slope is same if income effect = 0 (dx/dI = 0) • Normal Good: (dX/dI >0) => • Inferior Good: (dX/dI <0) => • Giffen Good: (dX/dI <0) => slope of ordinary demand curve is “+”

19. Normal Good: For any change in price, you get a bigger shift along the ordinary demand curve than along the compensated demand curve. “Income effect reinforces substitution effect” PX For a price increase, EV<CS<CV. The intuition behind this is that for a normal good more income is required to compensate the individual for a rise in price to maintain utility than income to be taken away from an individual such that he lies on a same lower utility. Xu Xc x

20. Summary • Price changes have two effects: • Change the optimal bundle • Change income • When measuring change in CS, need to account for both effects • Marshallian CS = • Hicksian CS = • They will differ, depending on income effects • Difference is typically small for small (marginal) price changes