Section I - You need to answer all questions in section I. Each part of the questions weighs.

1. Suppose an economy has two consumers, A and B, and two commodities X and Y. A's utility function and initial endowment are:

U_A(X_A, Y_A) = X_AY_A ω_A = (ω_AX, ω_AY) = (40, 80)

B's utility function and initial endowment are:

U_B(X_A, Y_B) = X²Y_B^B

ω_B = (ω_BX, ω_BY) = (60, 120)

Here X and Y are dividable.

(1) What is A's Marshallian demand function for X_A?

(2) What is A's Marshallian demand function for Y_A?

(3) What is B's Marshallian demand function for X_B?

2. Please find Y_A* in the general competitive equilibrium.

(8) Please find X_B* in the general competitive equilibrium.

(9) Please find Y_B* in the general competitive equilibrium.

There are 2,000 identical individuals in the market for commodity X, each with a demand function given by d(x) = 4-p, and 2,000 identical producers of commodity X, each with a function given by s(x) = 4p-2.

Suppose a sales tax of $0.4 per unit sold, from each of the 2,000 identical sellers of commodity X.

(1) Please find the market demand function with tax collection.

(2) Please find the market supply function with tax collection.

(3) Please calculate the market equilibrium price and quantity with tax collection.

(4) What is the total of taxes collected by the government?

(5) Please calculate the consumer surplus with tax collection.

(6) Please calculate the producer surplus with tax collection.

Suppose an economy has two consumers, Ellie and Kirstin, and two commodities X and Y. Ellie's utility function and initial endowment are:

U_E = min{X_E, Y_E}       ω_E = (ω_EX, ω_EY) = (10, 30)

Kirstin's utility function and initial endowment are:

U_K = min{X_K, Y_K}     ω_K = (ω_KX, ω_KY) = (30, 10)

(1) Please draw the Edgeworth box.

(2) Please label Ellie's and Kirstin's initial endowment points in the Edgeworth box.

(3) Please draw at least three Ellie's and Kirstin's indifference curves in the Edgeworth box.

(4) Please draw the contract curve in the Edgeworth box.

(5) Which point is the Pareto optimality? Why?

3. (4) What is B's Marshallian demand function for Y_B?

(5) In the general competitive equilibrium, what is P_X?

(6) Find X_A* in the general competitive equilibrium.

Section II - Please choose two questions to answer. Please do not answer all the questions. Each part of the questions weighs.

4. Suppose the production possibility frontier for guns (X) and butter (Y) is given by 2X² + Y² = 400.

(1) Graph this frontier.

(2) If individuals always prefer consumption bundles in which Y = X, how much X and Y will be preferred?

(3) Please indicate a point which is your solution from part (2) in your graph.

(4) At this point described in part (2), what price ratio will cause production to take place at that point?

5. Consider an economy with just one technique available for the production of each good, food and cloth. Suppose labor equals 90 and land equals 180, please answer the following questions.

(1) Please sketch the production possibility frontier.

(2) What are the intercepts of the production possibility frontier? Please indicate these two points in ?your graph.

(3) Please explain why the production possibility frontier is concave.

(4) Please find P_F, P_C.

6. Digging clams by hand in Sunset Bay requires both labor (L) and capital (K) inputs. The number of clams obtained per hour (q) is given by the production function such that

q = 2√2KL

Here we assume that K=32 to answer the following questions.

(1) Please graph the relationship between q and L. In your diagram, please put q on the vertical axis and L on the horizontal axis.

(2) What is the average productivity of labor in Sunset Bay? Please show it as a function of L.

(3) What is the marginal productivity of labor in Sunset Bay? Please show it as a function of L.

(4) According to your finding in part (3), please show the diminishing marginal productivities of labor.

Robinson Crusoe obtains utility from the quantity of fish he consumes in one day (F), the quantity of coconuts he consumes that day (C), and the hours of leisure time he has during the day (H) according to the utility function:

₁₁₁U = F₄C₄H₂

Robinson's production of fish is given by

F = √L_F

(where L_F is the hours he spends fishing), and his production of coconuts is determined by C = √L_C

(where L_C is the time he spends picking coconuts).?Here, assume Robinson decides to work 15 hours per day and, fish and coconut are dividable, to answer the following questions.

(1) Please graph his production possibility curve for fish and coconuts.

(2) Please show his optimal choices of these goods.