Question

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) = X^a Y^(1-a)

where 0<a<1.

Let 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 = X^a (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 demand curve for a price increase from p to q yielding a change in consumption of X from x_p to x_q is given by

                                                                ΔCS = [aBln(x_q) -px_q] - [aBln(x_p)-px_p] - (q-p)x_q

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.Calculate the compensating variation. 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. Following the same logic (but now for the equivalent variation concept), calculate the equivalent variation.

c. Compare ΔCS, as measured with the demand curve, to the compensating variation and equivalent variation.

Answer 1

The utility function is a Cobb Douglas one representing the equation U(X,Y) = X^a *Y*(1-a).

The person's budget is B. The feasible amount of consumption must satisfy the budget line equation, B = pX+Y. where p is the unit price of good X and the price of Y us set to 1.

The utility can be maximised by choosing X = a*B/p. When B= 100, a= 0.5, p=0.2,X=250 the utility is 111.8. If price is raised to l=0.3, then X falls to 204.12. To keep the utity constant to be 111.8 we need to increase the B. By putting the new values in the equation U = X^a (B-X)^(1-a)

Or, 111.80 = (204.12)^0.5 * (B - 204.12) ^ (1-a)

Or, 111.80 = 14.28 * (B-204.12)^0.5

Or, 7.82 = (B-204.12)^0.5

61.1524 = B - 204.12

B = 265.2724

The increase in B needed is , B = 265.2624 - 100 = 165.2624.

b) In case, of the rise in price and fall in X, a and B remaining the constant, the utility level would be,

U = X^a * (B-X)^(1-a)

U = 204.12^(.5) (100-204.12)^(1-0.5)

Or, U = 204.12 ^ (0.5) (-104.12)^0.5

Or, U = 14.28 * 10.20

Or, U = 145.656

To find EV we need to find the new utility at the old prices,

New utitly is 145.656.

So EV = 145.656 - 111.80 = 33.856

c)