1.Cali and David both collect stamps (x) and fancy spoons (y). They have the following preferences and endowments.

U_c (x, y) = x_c.y_c²

_c = (10, 5.2)

U_D (x, y) = x_D. y_D

_D = (14, 6.8)

a.Write down the resource constraints and draw an Edgeworth box that shows the set of feasible allocations.

b.Label the current allocation inside the box. How much utility does each person have?

c.Show that the current allocation of stamps and fancy spoons is not efficient. (Hint: Define the contract curve, or the set of Pareto optimal allocations.)

d.Show that the following allocation would make both Cali and David better off compared to the original allocation. Is it possible to make any further pareto improvements?

_c = (6,8)

_D = (16, 6)

2.Suppose the economy consists of two individuals, Ariel and Brad, who consume two goods, and , with the following preferences and initial endowments.

U_A (x, y) = x_A. y_A

_A = (4, 2)

U_B (x, y) = x_B. y_B

_B = (2, 3)

a.In an Edgeworth Box label the initial endowment, draw an indifference curve through the initial endowment for each individual, and shade the region of allocations that would be a Pareto improvement to the initial endowment.

b.Derive an equation to describe the set of Pareto-optimal allocations, and draw it in the Edgeworth Box.

c.Define the competitive equilibrium allocation [(x_A, y_A), (X_B, y_B)] and price ratio P_X/P_Y. (Remember that without loss of generality, you may set P₁ 1.)

d.Find a set of transfers, and , where _A + = 0, such that the competitive equilibrium becomes [(1, 1), (5, 4)].

3.A firm's production function is just a function of labor (there is no capital).

() = 100 ².

a.What is the marginal productivity of labor at L=2?

b.What is the average productivity of labor at L=2?

4.A firm's production function is given by F (L, K) = L² + K.

a.What is the MRTS of the function?

b.Does this function exhibit diminishing MRTS?

c.Does this function exhibit increasing, decreasing, or constant returns to scale?

5.Suppose a production function is given by the equation = ^{1 /2}

a.Graph the isoquants corresponding to Q=10, Q=20, and Q=50

b.Find the marginal Rate of Technical Substitution (MRTS)

c.Do these isoquants exhibit diminishing MRTS?

d.Does this production function exhibit constant, increasing or decreasing returns to scale?

6.Suppose the production function for hamburgers is characterized by Q=LK. If the wage is $10 and rent is $1, find the cost minimizing combination of labor and capital that will produce 121,000 hamburgers.

7.Suppose there are two inputs in the production function - labor and capital - and that these two inputs are perfect substitutes. The existing technology permits one machine to do the work of three persons. The firm wants to produce 100 units of output. Suppose that the price of capital is $750 per machine per week and that the weekly salary of each worker is $300.

a.Which combination of inputs should the firm use?

b.Suppose that a worker's weekly salary is $225. Which combination of inputs should the firm use?

8.A firm produces a good with the production function Q=10LK. Rent is $4 and the wage is $2. The firm is currently using K = 16. And just enough L to produce Q=320. How much could the firm save if it were to adjust K and L to produce in the least costly way? Graph the original input ratio with an isoquant and isocost curve, then show the optimal point on the same graph.