1. TEACHING HAUSMAN AND WILLIG USING MATHEMATICA BY MATT BOGARD, M.S.
CONSUMER WELFARE
TEACHING HAUSMAN AND WILLIG USING MATHEMATICA
BY MATT BOGARD, M.S.

2. Marshallian Surplus
Represented by the area between Po and P1 and to the left of the Marshallian demand curve.
Referred to as ‘area variation’
Observable from market data and consumer behavior

3. Equivalent Variation
Using the Hicksian Demand curve, the EV is the area to the left of h(u’) and between P1 and Po
Non-observable

4. Compensating Variation
CV = area to the left of h(u) and between P1 and Po
Non observable

5. Where does Marshallian Surplus fit in?
As stated previously, area variation = the area between Po and P1 and to the left of the Marshallian demand curve.
Willig points out:
CV < AV < EV
(for a price decrease)
For small price changes, these differences become smaller, and AV becomes a good approximation

6. Mathematical Representation
We can graph these demand functions as inverse demand functions
Specify dx by the inverse Marshallian demand function as Px = 1/X (blue)
and inverse Hicksian Demand as Px = 4/X^2 (red)
Note: these demand functions were derived by solving the utility maximization problem:
Max: U = X1/2Y1/2
s.t x –y = 0

7. Willig
The difference between area variation (Marshallian Surplus) and compensating variation ( Hicksian) is equal to D +B
As the difference between Po = .25 and P1 =1 decreases, so does the area D+B as a fraction of compensating variation.
Willig Marshallian Surplus is a good approximation for small price changes.

8. Hausman
Hausman- the approximation error might be quite large as a fraction of deadweight loss as measured by compensating variation
(A+C) = DW lossdx
(A + B) = DW loss  hx
The difference between (A+C) and (A+B) may not grow small as a fraction of true deadweight loss as P1-Po grows small
All points apply to equivalent variation as well

9. Calculating Welfare Changes
Suppose the market clearing price were .25 and a tax was imposed, raising the price to 1.0
The deadweight loss using the Marshallian demand curve would be C +A
By integration, this would be =
The deadweight loss according to the Hicksian demand curve would be A + B
By integration this is = .5
Error as a % of Compensating variation DW loss: ( ) / .5 =.27 ~ 27%

10. Change in DW loss II Lets assume that P1-Po gets smaller: P1 =.5 vs. 1
Integrating with Po = .25 and P1 = .5 will give us new values for the areas
C+A =
B +A =
Error as a % of Compensating variation DW loss: = / =
~ 56%
As Hausman points out, this error has actually gotten larger as P1-Po decreased.

11. REFERENCES Microeconomic Theory
Andreu Mas-Colell; Jerry R. Green; Michael Dennis Whinston
ISBN: Microeconomic Theory : Basic Principles and Extensions
Walter Nicholosn
Consumer's Surplus Without Apology
Robert D. Willig
The American Economic Review, Vol. 66, No. 4. (Sep., 1976), pp
Exact Consumer's Surplus and Deadweight Loss
Jerry A. Hausman
The American Economic Review, Vol. 71, No. 4. (Sep., 1981), pp

12. APPENDIX:

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