Find the set of pareto optimal allocations in this economy

Question

Problem 1: Exchange economy

Description of the economy:
Consider a 2x2 Exchange economy with two goods X and Y, and two consumers A and B. Consumer As preferences are described by the utility function U_A : R²₊ → R, with

u_A (x_A, y_A) = x^α_Ay^1-α_A

where x_A is the quantity of good X consumed by A, y_A is the quantity of good Y consumed by A, and α ∈ [0, 1] is a preference parameter.

Consumer B's preferences are described by the utility function U_B : R²₊ → R, with

u_A (x_B, y_B) = x^α_By^1-α_B

where x_B is the quantity of good X consumed by B, y_B is the quantity of good Y consumed by B, and a is a preference parameter that has the same value as for consumer A.

Consumer A's endowments of good X and Y are (x¯_A, y¯_A) = (3, 1), and consumer B's endowments are (x¯_B, y¯_B) = (1, 3).

(1) Find the set of Pareto optimal allocations in this economy and illustrate them using the Edgeworth box diagram.

(2) Find the competitive equilibrium in this economy and illustrate it. in the Edgeworth box diagram.

Problem 2: Production economy

Description of the economy:

Consider a. 2.x2 Production economy with two factors L. (labor) and K (capital), and two firms (one that. produces good X and one that produces good Y).

The technology of the firm that. produces good X is described by the production function fx :R²₊ → R, with

f_x(l_x, k_x) = λl_x + (1 - λ)k_x if l_x ≥ 2; 0   if l_x < 2

where l_x is the quantity of factor L employed by the firm, k_x is the quantity of factor K employed by the firm, and λ ∈ (0, 1) is a technology parameter. Note that firm X is only able to produce a strictly positive quantity if it employs at least 2 units of labor.

The technology of the firm that, produces good Y is described by the production function fy :R²₊ → R, with

f_y(l_y, k_y) = (1 - λ)l_y + λk_y if k_y ≥ 2; 0   if k_y < 2

where l_y is the quantity of factor L employed by the firm, k_y is the quantity of factor K employed by the firm, and A is the same technology parameter as for firm X. Note that firm Y is only able to produce a strictly positive quantity if it employs at least 2 units of capital.

There is a total endowment of factors L and K of (L¯, K¯) = (8, 8) available in the economy.

(1) In an appropriate diagram, illustrate the map of isoquants for firm X

(2) For a fixed wage rate ω > 0, rental rate r > 0, and fixed target quantity x > 0, formulate and solve the cost minimization problem for firm X, and find its conditional input demand functions and cost function.

(3) Find the set of production efficient allocations in the economy and illustrate it. in the Bowley box diagram when (i) λ = 1/3 and (ii) λ = 2/3.

(4) Find the production possibility frontier of the economy and illustrate it in the appropriate diagram when (i) λ = 1/3, and (ii) λ = 2/3

(5) Suppose there are given output prices (p x, py) >> 0. Find the competitive equilibrium wage rate and rental rate when λ = 2/3

Problem 3: Production-exchange economy Description of the economy:

Consider a 2x2x2x2 Production-exchange economy with two factors L and K, two goods X and 1.7, two firms (one that produces good X and one that produces good Y), and two consumers A and B.

The consumers have preferences represented by the utility functions in Problem 1.

The firms have technologies represented by the production functions in Problem 2.

Consumer A owns L‾_A = 8 units of labor, no capital, all the shares of firm X, and no share of firm Y.
Consumer B owns L‾_B = 8 units of capital, no labor, no share of firm X, and all the shares of firm Y.

(1) Find the set of Pareto optimal allocations in this economy and illustrate it. in a diagram using the production possibility frontier and the Edgeworth box when (i) λ = 1/3, or (ii) λ = 2/3

(2) Find the competitive equilibrium in this economy when α ∈ [1/12, 11,12] and λ = 2/3

(3) Suppose that a planner is trying to maximize the welfare of the consumers. Would the planner prefer the technologies where λ = 1/3 or where λ = 2/3? Discuss your answer.