Question

Consider a 2x2 pure exchange Edgeworth box economy. Each consumer is endowed with two units of x and one unit of y. Consumer A has (strictly) monotonic preferences over good x and is otherwise indifferent between any levels of good y. The preferences for consumer B are given by Ub(x, y) = x + 2y. What ratio of prices Px/Py will clear both markets?

Answer 1

Solution:

Let's assume that price of good x is $Px per unit, and that of good y is $Py per unit. Then, for consumer A, the endowment of goods x and y are 2 and 1, respectively. Then A's income = Px*2 + Py*1 = 2*Px + Py

Similarly, for consumer B as well, with endowment of 2 units and 1 unit for goods x and y respectively, total income becomes 2*Px + Py

As A has a strictly monotonic preference over x, anything greater in x gives him a higher utility, while any unit of good y has no impact on the utility of A. So, with current endowment of A, we can say that utility of A is some factor or power of units of x. For simplicity, let's say it is same as units of x he has. So current utility of A is 2. For B, with U(x, y) = x + 2y = 2 + 2*1 = 4.

As A generates utility only via good x, we can make such trade that A has all units of good x, and all y goes to B, such that no one's utility is lower that the current situation. So, with all x to A, utility of A = 4 (as total of 4 units of x exists in the economy). Similarly, all y to B (with total y's availability in the economy = 1 from A + 1 from B = 2 units), B's utility = 0 + 2*2 = 4. Notice how A's utility has increased while B's is same as before. This is the optimal allocation for the economy.

Now, with budget line Px*x + Py*y = M, M is income, x is units of good x, y is units of good y demanded. Then,

For A, as x = 4, y = 0, using the income he had, we derive: Px*4 + Py*0 = 2*Px + Py

So, 4*Px - 2*Px = Py

2*Px = Py

Px/Py = 1/2

We can verify this using B's budget line: x = 0, y = 2 for B, so Px*0 + Py*2 = 2*Px + Py

2*Py - Py = 2*Px

Py = 2*Px

Px/Py = 1/2 (hence, verified)

Thus, the ratio of price that clears both the markets is Px/Py = 1/2.