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

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

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

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

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

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. 2-.25x –y = 0

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.

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

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 = .636294 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: (.636294 - .5 ) / .5 =.27 ~ 27%

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 = .193147 B +A = 0.123427 Error as a % of Compensating variation DW loss: = 0.06972 / 0.123427 = 0.564868 ~ 56% As Hausman points out, this error has actually gotten larger as P1-Po decreased.

REFERENCES Microeconomic Theory Andreu Mas-Colell; Jerry R. Green; Michael Dennis Whinston ISBN: 0195073401 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. 589-597. Exact Consumer's Surplus and Deadweight Loss Jerry A. Hausman The American Economic Review, Vol. 71, No. 4. (Sep., 1981), pp. 662-676.

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