The Edgeworth box and the contract curve

A pure exchange economy has no production. There are two people, 1 and 2, and two goods, x and y, in fixed totals: say 10 units of x and 6 of y. Whatever one person holds, the other holds the rest.

That makes every allocation a single point. Draw a rectangle 10 wide and 6 tall. Measure person 1’s bundle from the bottom-left corner, O₁: a point at (4, 2) means she holds 4 of x and 2 of y. Measure person 2’s bundle from the top-right corner, O₂, with the axes pointing the other way. The same point then tells you person 2 holds 10 − 4 = 6 of x and 6 − 2 = 4 of y. One dot, two bundles, and nothing is wasted or created. This is the Edgeworth box.

Each person has her own indifference curves: the bundles that give her the same utility. Person 1’s curves are drawn as usual, bowing toward O₁. Person 2’s are drawn from her own origin, so in the box they look turned round: they bow toward O₂. Through any dot there is one curve of each person.

The slope of a curve is the marginal rate of substitution, the rate at which she is willing to swap one good for the other and stay just as well off:

A steep curve means she values x highly relative to y: she would give up a lot of y for a little x. A flat one means the reverse. For a Cobb–Douglas person with u = x^ay^1−a the MRS is $\frac{a}{1-a}\cdot \frac{y}{x}$: the more y she holds relative to x, the more she is willing to give up.

Start at any dot. A change is a Pareto improvement if it makes at least one person better off and nobody worse off (CUET 2022). An allocation is Pareto efficient (or Pareto optimal) if no Pareto improvement exists: you cannot make someone better off without making someone else worse off (CUET 2019, UOH 2021).

In the box this is visible. Person 1 prefers every dot above-right of her curve (more of both goods); person 2 prefers every dot below-left of hers (that is above-right from her corner). Where the two preferred regions overlap there is a lens, and every allocation in the lens makes both people better off. If the two curves cut through each other, the lens is not empty, so trade can help both. If they just touch, the lens is empty and the dot is efficient.

Take the numbers: person 1 would give up 4 units of y for one more x (MRS₁ = 4), person 2 would give up only 1 (MRS₂ = 1). If person 2 hands over one x and takes 2 units of y back, person 1 is better off (she paid 2, was willing to pay 4) and so is person 2 (she got 2, wanted 1). Both gain, so the starting point was not efficient.

Drag the dot. Watch the two curves through it and the lens between them. Then find a point where the lens vanishes: that is on the contract curve. Switch economies to see how tastes shape the efficient set, and tick “Highlight the core” once you have set a starting point.

Join up all the Pareto efficient allocations and you get the contract curve: the curve of tangencies, the locus where the MRSs are equalised (JNU 2017). Edgeworth, who set out the idea in his 1881 book Mathematical Psychics, called the curve of possible final bargains the contract curve. The phrase now covers any pair, whether two consumers or two inputs: CUET 2026 asks which curve shows the efficient allocations of goods between two consumers, or of two inputs between two production functions. In a pure exchange economy, with fixed endowments and no production, it is the set of efficient allocations of goods, and nothing about production belongs to it (UOH 2019).

To find it for Cobb–Douglas people, equate the two MRSs. With person 1 holding (x, y) of a box W wide and H tall:

If their tastes are identical (a = b), the weights cancel and y(W − x) = x(H − y), so y = (H/W)·x: the diagonal of the box. If person 1 puts more weight on x than person 2 (a > b), the curve bows below the diagonal. Person 1 holds relatively more x, person 2 more y: each ends up with a lot of the good she likes best.

Exam questions rarely stop at Cobb–Douglas. Use the lab’s presets, and keep the principle: efficient means no swap both would accept. Where indifference curves have kinks or corners, MRS equality may not hold and you reason from the shape.

People do not start at an arbitrary point: they start with what they own, the endowment, a dot in the box. Nobody agrees to a trade that leaves her worse off than the endowment. So only the allocations in the lens through the endowment are on offer. Those that are also on the contract curve form the core: efficient allocations that neither person would block.

The core is a stretch of the contract curve, not a point, so exchange alone does not say where bargaining ends. The name goes back to Edgeworth’s “range of final settlements”, which Martin Shubik revived in 1959 as the core. Lesson 2 adds prices, and with them a single prediction.

Efficiency says nothing about who gets what. Giving the whole box to person 1 is Pareto efficient: nobody can be made better off without making her worse off. So is giving everything to person 2. Both are on the contract curve (they are its end points) yet one person has nothing.

Examiners test this. A cake split 0.45 and 0.45 wastes 0.1, so it is inefficient (JNU 2018): someone could have the spare tenth. Split 0.45 and 0.55, or give A the lot, and it is efficient (JNU 2018). The assertion-and-reason question CUET 2023 gives, as its reason, that “Pareto efficiency takes into account distributive justice”, which is false; the assertion, that an allocation is inefficient when someone can gain without harming anyone, is true.