The Edgeworth box is a visual tool for analyzing how two people can allocate two goods between themselves in a pure exchange economy. It captures everything you need to see: initial endowments, potential trades, and efficient outcomes. Understanding this diagram is essential for reasoning about general equilibrium and the welfare properties of exchange.

Structure and components of the Edgeworth box

The box is built by combining two individuals' indifference curve maps into a single diagram. Person A's origin sits at the bottom-left corner, and Person B's origin sits at the top-right corner. The width of the box equals the total amount of good 1 available, and the height equals the total amount of good 2.

Every point inside the box represents a complete allocation: how much of each good each person holds. If A has 3 apples and 5 bananas, B automatically has whatever remains of the total supply.
Indifference curves for A curve in the usual way (convex toward A's origin). Indifference curves for B curve toward B's origin, so they appear "flipped" relative to what you're used to.
The slope of each person's indifference curve at a given point is their marginal rate of substitution (MRS) between the two goods.

Analyzing allocations and trades

Start by marking the initial endowment point, the allocation each person begins with before any trade. Two indifference curves pass through that point, one for each person. The region enclosed between those two curves is called the lens.

Any allocation inside the lens makes at least one person better off without making the other worse off. These are the trades both parties would voluntarily accept.
Allocations outside the lens would leave at least one person on a lower indifference curve than at the endowment, so that person would refuse the trade. These allocations are not individually rational.
Trade narrows the lens. As the two parties move toward allocations where their indifference curves are tangent, the lens shrinks until no further mutually beneficial trades remain.

Contract curve and its significance

Definition and characteristics

The contract curve is the locus of all Pareto efficient allocations inside the Edgeworth box. At every point on this curve, the indifference curves of A and B are tangent to each other, which means:

$MRS_A = MRS_B$

Because the two MRS values are equal, there's no way to rearrange goods and make one person better off without making the other worse off.

The curve runs from A's origin (where A has nothing) to B's origin (where B has nothing), passing through the interior of the box.
Its exact shape depends on both individuals' preferences. If preferences are symmetric, the curve will be close to the diagonal. If one person has a much stronger preference for one good, the curve bows toward that person's origin.

Structure and components of the Edgeworth box, microeconomics - What's the role of initial endowments in an edgeworth box? - Economics Stack ...

Deriving the contract curve

To actually find the contract curve analytically, follow these steps:

Write down each person's MRS. For person $i$ with utility $U_i(x_1, x_2)$ , the MRS is $\frac{\partial U_i / \partial x_1}{\partial U_i / \partial x_2}$ .
Use the feasibility constraints to express B's consumption in terms of A's: $x_1^B = \bar{x}_1 - x_1^A$ and $x_2^B = \bar{x}_2 - x_2^A$ , where $\bar{x}_1$ and $\bar{x}_2$ are the total endowments.
Set $MRS_A = MRS_B$ and substitute the feasibility constraints in. This gives you one equation in two unknowns ( $x_1^A$ and $x_2^A$ ).
Solve for $x_2^A$ as a function of $x_1^A$ (or vice versa). That relationship traces out the contract curve.

For example, if both people have Cobb-Douglas utility $U_A = (x_1^A)^a (x_2^A)^{1-a}$ and $U_B = (x_1^B)^b (x_2^B)^{1-b}$ , the tangency condition becomes:

$\frac{a \cdot x_2^A}{(1-a) \cdot x_1^A} = \frac{b \cdot x_2^B}{(1-b) \cdot x_1^B}$

Substituting the feasibility constraints and solving gives you the contract curve as an explicit function. With identical exponents ( $a = b$ ), the curve is just the diagonal of the box.

Importance in economic analysis

The contract curve defines the entire set of efficient outcomes available in this two-person economy. That's useful for several reasons:

Bargaining analysis: When two people negotiate a trade, the final outcome should land somewhere on the contract curve (assuming they bargain to an efficient result). Where exactly it lands depends on bargaining power and the initial endowment.
Equity vs. efficiency trade-offs: Every point on the curve is efficient, but they differ dramatically in how well off each person ends up. A point near A's origin is efficient but terrible for A. This makes the contract curve a natural tool for thinking about distributional questions.
Moving along the curve always involves a trade-off: one person gains, the other loses. Moving toward the curve from an interior point can benefit both.

Efficiency of allocations in the Edgeworth box

Types of allocations

There are three categories to keep straight:

Efficient allocations lie on the contract curve. Indifference curves are tangent, $MRS_A = MRS_B$ , and no Pareto improvement is possible.
Inefficient allocations lie off the contract curve. The indifference curves cross rather than being tangent, so there exist trades that could make at least one person better off without harming the other.
Pareto improvements are movements from an inefficient point toward the contract curve. By definition, at least one person's utility rises and neither person's utility falls.

Structure and components of the Edgeworth box, Indifference Curves | OS Microeconomics 2e

Analyzing allocation efficiency

Given any inefficient allocation, you can identify the set of Pareto improvements by finding the lens between the two indifference curves passing through that point. The efficient allocations reachable from that starting point are the segment of the contract curve that lies inside the lens.

The core of the economy is the portion of the contract curve where both individuals are at least as well off as at the initial endowment. In other words, it's the segment of the contract curve inside the lens formed by the endowment point. No coalition of traders can block a core allocation, which is why competitive equilibria always lie in the core.

If a point is far from the contract curve, large potential gains from trade exist.
If a point is already on the contract curve, no further mutual improvement is possible.

A common mistake is thinking that any point on the contract curve is reachable from a given inefficient allocation. It's not. Only the segment inside the lens qualifies as a Pareto improvement from that specific starting point. Moving to a point on the contract curve outside the lens would make one person worse off than they were initially.

Contract curve and Pareto efficiency

Relationship between contract curve and Pareto efficiency

These two concepts are directly linked: the contract curve is the set of all Pareto efficient allocations in the Edgeworth box. Every point on the curve satisfies $MRS_A = MRS_B$ , and every point off the curve does not.

Moving from a point off the curve to a point on the curve (within the lens) is a Pareto improvement. Moving along the curve is not a Pareto improvement because one person necessarily loses.

Economic theorems and implications

Two foundational results connect the contract curve to competitive markets:

First Fundamental Theorem of Welfare Economics: Any competitive equilibrium allocation is Pareto efficient. In the Edgeworth box, this means the equilibrium price ratio leads to a point on the contract curve where both individuals' MRS equals the price ratio:

$MRS_A = MRS_B = \frac{p_1}{p_2}$

This result requires that preferences are locally nonsatiated (people always prefer a bit more of something). It does not require convex preferences.

Second Fundamental Theorem of Welfare Economics: Any Pareto efficient allocation on the contract curve can be achieved as a competitive equilibrium, provided you can redistribute initial endowments appropriately. This result does require convex preferences (diminishing MRS). It's powerful because it says efficiency and equity are, in principle, separable problems: pick the fair outcome on the contract curve, redistribute endowments via lump-sum transfers to support it, and then let markets reach that allocation through price-taking behavior.

Together, these theorems explain why the contract curve matters beyond the two-person case. The curve shows the full menu of efficient outcomes, and the welfare theorems tell you which ones markets can reach and how policy (through redistribution of endowments) can steer the economy toward different points on that curve.