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The contract curve is the locus of Pareto efficient allocations in an Edgeworth box—points where the indifference curves of the two consumers are tangent. Trade moves the economy along the contract curve. The set of allocations in the contract curve that both agents prefer to the initial endowment is the core, the set of outcomes that cannot be blocked by coalitions.
From the Edgeworth box, you know how to represent all possible allocations of two goods between two people in a single diagram, and from Pareto efficiency you know that an allocation is efficient when you cannot make one person better off without making the other worse off. The contract curve connects these two ideas: it is the line through the Edgeworth box that traces out every Pareto efficient allocation.
Geometrically, a point is on the contract curve when the two consumers' indifference curves are tangent to each other at that point. Tangency means their marginal rates of substitution (MRS) are equal — both consumers value the tradeoff between the two goods identically at the margin. Why does equal MRS imply efficiency? If the MRS values differ, there is room for a mutually beneficial trade. Suppose Alice values an extra unit of good X at 3 units of good Y, while Bob values it at only 1 unit of Y. Then Alice could give Bob 2 units of Y for 1 unit of X, and both would be happier. Only when their marginal valuations coincide have all gains from trade been exhausted — and that is exactly where the indifference curves are tangent.
The contract curve typically runs from one corner of the Edgeworth box to the opposite corner, passing through the interior. Its exact shape depends on the consumers' preferences. With identical homothetic preferences, the contract curve is the diagonal of the box. With very different preferences, it may bow strongly toward one side. Every point on the contract curve is efficient, but they differ dramatically in how the gains are distributed — at one end, Alice has nearly everything; at the other, Bob does. Efficiency alone says nothing about fairness.
Not every point on the contract curve is a plausible outcome of voluntary trade. Given an initial endowment (the starting allocation before trade), both consumers will only agree to move to allocations that make each of them at least as well off as they were initially. The subset of the contract curve where both consumers are on or above their initial indifference curves is called the core of the economy. The core narrows the efficient allocations to those that are individually rational — neither party would agree to a trade that leaves them worse off than their endowment. Competitive equilibrium, as you would expect from the First Welfare Theorem, lies within this core.
The contract curve is more than a geometric curiosity — it crystallizes the fundamental distinction between efficiency and distribution that runs through all of welfare economics. Choosing a point on the contract curve determines how the surplus is split between the two agents. Markets select one particular point (the competitive equilibrium); negotiation, bargaining power, or social choice mechanisms select others. The contract curve shows the full menu of efficient outcomes and makes visible the tradeoffs that any allocation mechanism must navigate.