Describe the core allocations of economy - characterize the

Question

PROBLEM SET:

1. Consider an exchange economy with two agents and two goods. Let x_ik denote agent i's consumption of good k. Consumption sets are the nonnegative orthant. Agent 1 has utility represented by Ax₁₁ + x₁₂ where A > 0 ; while agent 2's utility is x₂₁x₂₂. Endowments are (e, 0)
for agent 1, and (0, e) for agent 2, where e> 0 .

a) Characterize the core allocations for this economy.

b) Let A=1 and either argue directly that the core converges to the Walrasian equilibrium as the economy is replicated k > 1 times, or show that it does not.

2. Consider a two agent and two good exchange economy where agent 1 has preferences represented by u1 (x11, x12) = min(2.x₁₁, x₁₂) , and agent 2 has preferences represented by u₂ (x₂₁, x₂₂) = min(x₂₁, x₂₂) . Both agents have endowments (1,1) . The consumption sets of the agents are k₊² . Describe the core allocations of this economy.