Computing the competitive equilibrium given initial allocation

Question

Suppose that there are two agents, 1 and 2, and two goods, honey (h) and lemon (l), and that the agents' preferences over these goods are defined by the following utility functions:

$$u^1(x_h^1, x_l^1)=x_h^1+x_l^1$$ $$u^2(x_h^2, x_l^2)=min[x_h^2, x_l^2]$$

Furthermore, suppose that the agents are assigned the following initial allocations, of the form (h,l):

$$w^1=(1/2, 1/2)$$ $$w^2=(3/2, 1/2$$

Given these utility functions and initial allocations, one could draw an edgeworth box and compute the set of pareto efficient outcomes. Doing so, one obtains the set: $$x:=(x^1_h,x^1_l,x^2_h,x^2_l)|x^2_h=x^2_l , x_h^2\in[1/2,1].$$

In the above set of Pareto outcomes, I am assuming that markets clear, which is why I don't specifically mention the consumption of agent 2 (it's defined by what agent 1 does not consume). To calculate this set, I simply drew the edgeworth box and used the fact that Pareto efficiency is the set of points on the contract curve between the initial two indifference curves given by the initial allocation.

My ultimate goal is to find the competitive equilibrium which I know belongs to the set of Pareto efficient consumption levels, but I don't know how to do this. I know that CE is the pair of allocations and prices that cause the allocations to solve each agents consumption problem, but I'm honestly pretty lost in all of the different equations and constraints.

Could someone help clarify exactly what needs to be done?

Answer 1

The easiest way is to assume some prices $p_h, p_l$ and focus on only one of the goods, say lemons. Derive the demand for lemons for each agent given the assumed prices and set the sum of their demands equal to the total endowment (market clearing condition for lemons). This gives you an equation that has both prices as the only unknowns. Usually at this point is when I choose which good to be numeraire, i.e. to set its price to 1, in order to make the algebra easier. Now you have a single equation that depends on a single price and should be pretty easy to solve (given the utilities you are given).

Now you have the equilibrium prices for both goods, so you can plug them in the demand for lemons you found earlier and also figure out the demand for the other good given those prices. Since one market (lemons) is cleared, the other should be too.

This is a classic problem, so SPOILER ALERT, the price of both goods should be the same: 1. The logic is simple, the guy with perfect complement preferences has strong preferences for an intermediate basket (since he has positive endowment for both goods, CE cannot be in a corner), and the only way to incentivize the guy with perfect substitute preferences to take an intermediate basket is by choosing a price ratio equal to his MRS (in this example, 1). Of course I'm speaking loosely, but hopefully, that gives you some intuition.