During droughts, cities often impose water use restrictions on consume

During droughts, cities often impose water use restrictions on consumers. Suppose a representative consumer has preferences for Water (W) and other goods (X) given by the utility function:

U(W,X) = WX.
Suppose the price of other goods is $1 and the price of water is initially 50¢. The consumer has a budget of $50/week.
a. How much water will the consumer purchase each week?
b. Suppose the government imposes a quota on water use of 50 units/week. Show that the quota reduces the representative consumer’s utility.
c. By how much does the quota harm the representative consumer? Specifically, compute the equivalent variation of the quota.

ANSWER

a. The consumer will choose 100 units of water per week.
b. Since the consumer can no longer consumer the optimal bundle from before, she will choose a bundle of 50 units of water (costing $25 ) and spend $75 on other goods. Her utility before was U(100,50 )=5000 and now is U(50,75 ) = 3750. She is indeed worse off from the quota.
c. We wish to find the amount of income which will lead to a utility of 3750 without a quota. The MRS = MRT condition is X/W = .5/1, or 2X = W. Plugging into the utility function, 2X2 = 3750 or X = 43.3. So Y = 86.6. The Expenditure for this bundle of goods is $86.6. Thus the EV is 86.6 – 100 = -13.4.