Solved problem · Intermediate

Competitive Equilibrium in an Exchange Economy

Find the competitive equilibrium of a two-person Cobb–Douglas exchange economy, and verify the First Welfare Theorem directly.

Problem

Two consumers trade two goods. Consumer 1 has utility $u_1(x_1, y_1) = x_1^{2/3} y_1^{1/3}$ and endowment $\omega_1 = (1, 0)$ . Consumer 2 has utility $u_2(x_2, y_2) = x_2^{1/3} y_2^{2/3}$ and endowment $\omega_2 = (0, 1)$ .

Find the competitive equilibrium prices and allocation.
Find the set of Pareto-efficient allocations, and verify that the equilibrium allocation belongs to it.

Solution

Step 1 — Demands. Normalise $p_y = 1$ and write $p_x$ for the price of good $x$ . Each consumer has Cobb–Douglas preferences, so spends fixed budget shares. Consumer 1’s income is $p_x \cdot 1 + 1 \cdot 0 = p_x$ , and consumer 2’s is $1$ :

x_1 = \frac{2}{3}\cdot\frac{p_x}{p_x} = \frac{2}{3}, \qquad y_1 = \frac{1}{3}\, p_x, \qquad x_2 = \frac{1}{3}\cdot\frac{1}{p_x}, \qquad y_2 = \frac{2}{3}.

Step 2 — Market clearing. Clear the market for good $x$ (good $y$ then clears by Walras’s law):

x_1 + x_2 = \frac{2}{3} + \frac{1}{3p_x} = 1 \;\;\Longrightarrow\;\; p_x = 1.

So the equilibrium price ratio is $p_x / p_y = 1$ , and the allocation is

(x_1^*, y_1^*) = \left(\tfrac{2}{3}, \tfrac{1}{3}\right), \qquad (x_2^*, y_2^*) = \left(\tfrac{1}{3}, \tfrac{2}{3}\right).

Step 3 — The Pareto set. Interior efficiency requires equal marginal rates of substitution. With $x_2 = 1 - x_1$ and $y_2 = 1 - y_1$ :

2\,\frac{y_1}{x_1} = \frac{1}{2}\cdot\frac{1 - y_1}{1 - x_1} \;\;\Longrightarrow\;\; y_1 = \frac{x_1}{4 - 3x_1}.

This is the contract curve. At $x_1 = \tfrac{2}{3}$ it gives $y_1 = \tfrac{2/3}{4 - 2} = \tfrac{1}{3}$ — exactly the equilibrium allocation. The equilibrium is Pareto efficient, as the First Welfare Theorem guarantees.

Check your understanding

Re-solve with endowments $\omega_1 = (1,1)$ , $\omega_2 = (0,0)$ . What happens, and what does it tell you about the role of endowments versus preferences?