(This question pertains to the Appendix; instructor-provided spreadsheet recommended) Imagine a person’s utility function over two goods, X and Y, where Y represents dollars. Specifically, assume a Cobb-Douglas utility function:
U(X,Y) = Xa Y(1-a)
where 0Let the person’s budget be B. The feasible amounts of consumption must satisfy the following equation:
B= pX+Y
where p is the unit price of X and the price of Y is set to 1.
Solving the budget constraint for Y and substituting into the utility function yields
U = Xa (B-pX)(1-a)
Using calculus, it can be shown that utility is maximized by choosing
X=aB/p
Also, it can be shown that the area under the Marshallian demand curve for a price increase from p to q yielding a change in consumption of X from xp to xq is given by
ΔCS = [aBln (xq)-p xq]- [aBln(xp)-p xp]-(q-p)xq
When B=100, a=0.5, and p=.2, X=250 maximizes utility, which equals 111.80. If price is raised to p=.3, X falls to 204.12.
a. Increase B until the utility raises to its initial level. The increase in B needed to return utility to its level before the price increase is the compensating variation for the price increase. (It can be found by guessing values until utility reaches its original level.)
b. Compare ΔCS, as measured with the Marshallian demand curve, to the compensating variation.