Consider a competitive exchange economy with two agents(1&2)and two goods(X&Y). Agent 1's endowment of (X,Y) is (100,100)& agent 2's endowment of (X,Y) is (50,0).An allocation of agent i is denoted by (xi,yi)where xi is his allocation of X & yi is his allocation of Y.Agent1's objective is to choose(x1,y1) to maximise his utility min{x1,y1}.Agent2's objective is to choose (x2,y2)to maximise his utility x2+y2. An example of a competitive equilibrium allocation is
a)(x1,y1)= (100,50) and (x2,y2)= (50,50).
(b)(x1,y1)=(125,100)& (x2,y2)=(25,0).(c)(x1,y1)=(50,50)&(x2,y2)=(100,50). (d)(x1,y1)=(100,100)& (x2,y2)=(50,0).
An example of Pareto efficient allocation is
(a) (x1,y1)=(50,50)& (x2,y2)=(100,50).
(b) (x1,y1)=(125,100)& (x2,y2)=(25,0).
(c) (x1,y1)=(125,75)& (x2,y2)=(25,25).
(d) (x1,y1)=(50,100)& (x2,y2)=(100,0).
An example of a pair of competitive equilibrium prices (p1,p2) is
(a) (1,0)
(b) (0,1)
(c) (1/3,2/3)
(d) (2/3,1/3).
Take a shot, what do you think? Hint: As I understand, the initial endowments of persons 1 and 2 are both utility maximizing. For person 1, he will maximize when x1=y1. For person 2, he is indifferent between getting an extra x2 or a y2. Hint 2: On question 2, I don't believe any of the possible answers are pareto efficient.
To determine the competitive equilibrium allocation, we need to find the allocations of goods (X and Y) for each agent that satisfy their objectives and the market clearing condition.
For agent 1, the objective is to maximize the minimum of x1 and y1. Since agent 1's endowment is (100, 100), the maximum value that can be allocated to either good is 100. If agent 1 receives more than 100 of either good, it will violate the endowment constraint. Therefore, agent 1's allocation should be (x1, y1) = (100, 100).
For agent 2, the objective is to maximize x2 + y2. Since agent 2's endowment is (50, 0), the maximum value that can be allocated to good X is 50. However, since agent 2 does not have any endowment of good Y, the allocation of good Y should be 0. Therefore, agent 2's allocation should be (x2, y2) = (50, 0).
Now, to determine the competitive equilibrium prices (p1, p2), we need to find the prices that clear the market. The market clearing condition states that the total demand for each good should be equal to the total supply. In this case, the total demand for good X is x1 + x2, and the total supply is 100 + 50 = 150. Similarly, the total demand for good Y is y1 + y2, and the total supply is 100 + 0 = 100.
Since the allocations are (100, 100) for agent 1 and (50, 0) for agent 2, the total demand for good X is 100 + 50 = 150. Similarly, the total demand for good Y is 100 + 0 = 100. Therefore, to clear the market, the prices should be such that p1 * 150 = p2 * 100.
Examining the answer choices:
(a) (x1, y1) = (100, 50) and (x2, y2) = (50, 50) does not satisfy the market clearing condition, as 100 + 50 does not equal 150.
(b) (x1, y1) = (125, 100) and (x2, y2) = (25, 0) does not satisfy the market clearing condition, as 125 + 25 does not equal 150.
(c) (x1, y1) = (50, 50) and (x2, y2) = (100, 50) satisfies the market clearing condition, as 50 + 100 equals 150.
(d) (x1, y1) = (100, 100) and (x2, y2) = (50, 0) satisfies the market clearing condition, as 100 + 50 equals 150.
Therefore, the example of a competitive equilibrium allocation is (c) (x1, y1) = (50, 50) and (x2, y2) = (100, 50).
Next, let's determine the example of a Pareto efficient allocation. A Pareto efficient allocation is one where no agent can be made better off without making another agent worse off.
Looking at the answer choices:
(a) (x1, y1) = (50, 50) and (x2, y2) = (100, 50) is Pareto efficient because any reallocation of goods between agents would make one of them worse off.
(b) (x1, y1) = (125, 100) and (x2, y2) = (25, 0) is not Pareto efficient because agent 2 can be made better off by reallocating some goods from agent 1.
(c) (x1, y1) = (125, 75) and (x2, y2) = (25, 25) is not Pareto efficient because agent 2 can be made better off by reallocating some goods from agent 1.
(d) (x1, y1) = (50, 100) and (x2, y2) = (100, 0) is not Pareto efficient because agent 1 can be made better off by reallocating some goods from agent 2.
Therefore, the example of a Pareto efficient allocation is (a) (x1, y1) = (50, 50) and (x2, y2) = (100, 50).
Finally, let's determine the example of a pair of competitive equilibrium prices (p1, p2). The prices should be such that p1 * 150 = p2 * 100, which satisfy the market clearing condition.
Looking at the answer choices:
(a) (1, 0) does not satisfy the market clearing condition, as 1 * 150 does not equal 0 * 100.
(b) (0, 1) does not satisfy the market clearing condition, as 0 * 150 does not equal 1 * 100.
(c) (1/3, 2/3) satisfies the market clearing condition, as 1/3 * 150 equals 2/3 * 100.
(d) (2/3, 1/3) satisfies the market clearing condition, as 2/3 * 150 equals 1/3 * 100.
Therefore, the example of a pair of competitive equilibrium prices is (c) (1/3, 2/3).