Section 4

Pareto Efficiency and the Contract Curve

Definition

Pareto efficiency provides the standard welfare benchmark in exchange economies. The definitions are precise.

Pareto improvement: An allocation $x'$ is a Pareto improvement over $x$ if $u^i(x'^i) \ge u^i(x^i)$ for all $i$ , with strict inequality for at least one agent — someone is strictly better off and nobody is worse off.
Pareto efficient: An allocation $x$ is Pareto efficient if no feasible Pareto improvement exists — there is no reallocation of the fixed total endowment that makes someone better off without making anyone worse off.

Pareto efficiency is a weak normative requirement. It says nothing about fairness or the distribution of welfare between agents — an allocation that gives everything to agent A and nothing to agent B may well be Pareto efficient. The criterion only asks whether free welfare gains remain on the table. Any allocation that fails the test is difficult to defend: there exists a change that helps someone and hurts no one, yet the economy is not at that change.

Characterising efficiency: MRS equality

At an interior Pareto-efficient allocation, the two agents' indifference curves must be tangent to each other. If they were not tangent — if they crossed — the lens-shaped region between them would contain a Pareto-improving reallocation of the same total endowment. That would contradict efficiency. Tangency requires equal slopes, which in the language of the MRS derived in §3 is:

\mathrm{MRS}^A(x^A) = \mathrm{MRS}^B(x^B)

This tangency condition is both necessary and sufficient for interior efficiency under strictly convex preferences. Corner solutions — where an agent consumes zero of one good — can also be efficient, but they require the MRS inequality to be satisfied weakly rather than exactly. For the Cobb-Douglas preferences used here, the optimum is always interior, so the equality is the right condition throughout.

Deriving the contract curve

The contract curve is the set of all Pareto-efficient allocations in the Edgeworth box. We derive it by imposing the tangency condition together with feasibility. Let $\Omega_1$ and $\Omega_2$ denote the total endowments of goods 1 and 2: $\Omega_j = \omega^A_j + \omega^B_j$ .

Step 1
$\frac{\alpha_A}{1-\alpha_A} \cdot \frac{x^A_2}{x^A_1} = \frac{\alpha_B}{1-\alpha_B} \cdot \frac{x^B_2}{x^B_1}$
Equate the two MRS expressions from §3.
Step 2
$x^B_1 = \Omega_1 - x^A_1, \quad x^B_2 = \Omega_2 - x^A_2$
Substitute feasibility: B consumes whatever A leaves.
Step 3
$\frac{\alpha_A}{1-\alpha_A} \cdot \frac{x^A_2}{x^A_1} = \frac{\alpha_B}{1-\alpha_B} \cdot \frac{\Omega_2 - x^A_2}{\Omega_1 - x^A_1}$
Solve for $x^A_2$ as a function of $x^A_1$ .
Step 4
$x^A_2(x^A_1) = \frac{\alpha_A (1-\alpha_B) \, \Omega_2 \, x^A_1}{(1-\alpha_A)\, \alpha_B \, (\Omega_1 - x^A_1) + \alpha_A (1-\alpha_B) \, x^A_1}$
The contract curve in A's coordinates. Plot this and you get the locus of all Pareto-efficient allocations — every other point in the box is dominated.

Efficiency narrows the set of candidate allocations from the entire interior of the Edgeworth box — a two-dimensional region — down to a one-dimensional curve. Individual rationality narrows further to the segment of the contract curve that lies inside the lens from §3. That segment is the core of the exchange economy: every allocation on it is both efficient and individually rational, and every allocation off it can be improved upon by at least one agent or by both. The next section asks which point on the contract curve a competitive market — one price ratio clearing both goods simultaneously — actually selects.