Economics Problem Set-

QUESTION 1: Sometimes goods are rationed, so people cannot buy as much as they want at the announced price. A good example is Super bowl tickets, which are sold at below market price. Suppose that a consumer has the following Cobb-Douglas utility function

U(X₁, X₂) = X₁^0.2X₂^0.8

(a) Find the marshallian demand for the 2 goods given p1 = 2, p2 = 2 and I = 200.

(b) Now assume that the consumer can only buy 10 units of X1. What is the demand for good 2 is the supply for good 1 can be rationed at 10 units per customer?

(c) What is the MRS at the new rationed equilibrium?

(d) Supposed that a black market opens up in good 1. How much would the consumer pay for one more unit of good 1?

QUESTION 2: Consider the optimal choice problem of labor and rcreation. Suppose a consumer works the first 8 hours of the day at a wage of $10, but received an overtime wage of $20 for additional time worked.

Assume there is zero non-labor income, the price of the consumption good is 1. Since we are looking at how many hours this worker works in one day, it is safe to assume that T = 24.

(a) Draw the budget constraint on an optimal choice diagram.

(b) Draw the set of indifference curves that would make it optimal for this individual to work 4 house of overtime each day.

QUESTION 3: The labor supply problem can also be approached from an expenditure minimization perspective. Suppose a person's utility function takes the CobbDouglas form

U(C, R) = C^0.4R^0.6

C is consumption, R is recreation, Assume that the price of the consumption good is 1 and that the consumer has zero non-labor income (M=0). We want to find the number of units of recreation and consumption chosen per day. Therefore, T=24.

Let M = 0

Derive the expenditure function for these w, M, p and T (Hint: You will be deriving the compensated demand functions for consumption and recreation first.)

QUESTION 4: Similar to question 4, but with general terms.

An individual has the following utility function

U(C, R) = C^αR^{1- α}

Assume that the consumer has no endowment of income. T = 24.

(a) Use the expenditure minimization problem to derive the expenditure function.

(b) Use the envelope theorem to derive the compensated demand functions for consumption and recreation. The envelope theorem fives you a short cur to recover the compensated demand functions.

(c) Derive the compensated labor supply function. Show that (∂L^c/ ∂w) > 0 where Lc is the compensated labor supply function.