Question

Consider the general equilibrium of a pure exchange economy in which two agents may engage in trade. The utility functions for the two agents are identical: U₁=A₁ B₁ and U₂=A₂B₂, where Ai​ and Bi​ are consumption levels of the two goods, A and B, for agent i(i=1,2). The initial endowment vectors are given as follows:

Agent 1: (a₁, b₁)=(0,100); Agent 2: (a₂, b₂)=(100,0). Suppose that they both maximize utility and that each agent accepts as given the market price as quoted by a referee. Answer the following questions:
a. Calculate the demand functions of both agents.
b. Determine the equilibrium prices that will clear the markets.
c. Calculate the equilibrium allocation in consumption.
d. Construct the locus of equilibrium points (the offer curve) for each agent. Examine the shape of the two offer curves, using the geometrical construction of Edgeworth-Bowley consumption box.
e. Show that the results derived in (b) and (c) satisfy both offer curves.

Answer 1

To calculate the demand functions, find the marginal utility of each good for both agents and equate them to the market prices. Determine the equilibrium prices by setting the demand functions equal to the initial endowments. The equilibrium allocation is the point where the demand functions intersect. Construct the offer curves and check if they satisfy the equilibrium prices and allocation.

Step-by-step explanation:

To calculate the demand functions of both agents, we need to find the marginal utility of each good for both agents and equate them to the market prices. Let's start with agent 1:

1. Agent 1's marginal utility of good A (MU₁A) is equal to the derivative of the utility function U₁ with respect to A₁, which is B₁.
2. Agent 1's marginal utility of good B (MU₁B) is equal to the derivative of the utility function U₁ with respect to B₁, which is A₁.
3. To find agent 1's demand for good A, we equate MU₁A to the market price of good A.
4. To find agent 1's demand for good B, we equate MU₁B to the market price of good B.

We follow the same steps for agent 2 to find their demand functions. To determine the equilibrium prices, we set the demand functions of both agents equal to their respective initial endowments and solve for the prices. The equilibrium allocation in consumption is the point where the demand functions intersect. The offer curve can be constructed by plotting the allocation of goods for different price levels. Finally, we check if the equilibrium prices and allocation satisfy both offer curves.