In an Edgeworth box, apples are measured along the horizontal axes, and oranges are measured along the vertical axes. Player 1 is in the lower left-hand corner; player 2 is in the upper right-hand corner. Player 1 likes apples but gets no utility from the consumption of oranges. Player 2 likes oranges but gets no utility from the consumption of apples. In the initial endowment, Player 1 has 5 apples and 4 oranges. Player 2 also has 5 apples and 4 oranges. What is the contract curve?

Question

BlackTom AI · Accepted Answer

The question asks for the contract curve in an Edgeworth box with two players having opposite, but non-overlapping, tastes: Player 1 cares only about apples and gets no utility from oranges, while Player 2 cares only about oranges and gets no utility from apples.

First, restating the setup: apples are on the x-axis, oranges on the y-axis. The total endowment is 10 apples and 8 oranges (since each player starts with 5 apples and 4 oranges).

To analyze Pareto efficiency, consider allocations described by (A1, O1) for Player 1, where A1 is Player 1’s apples and O1 is Player 1’s oranges. Then Player 2 has (10 − A1) apples and (8 − O1) oranges.

- Player 1’s utility: U1 = A1 (depends only on apples).
- Player 2’s utility: U2 = 8 − O1 (depends only on oranges, since he has 8 − O1 oranges).

Now, assess potential Pareto improvements along any feasible allocation:

- If A1 < 10, you can transfer one apple from Player 2 to Player 1 without changing Player 1’s oranges (or affecting Player 2’s utility via oranges, since apples don’t matter to Player 2). This would strictly increase U1 while leaving U2 unchanged, yielding a Pareto improvement. Therefore, any allocation with A1 < 10 is not Pareto efficient.

- If O1 < 8, you can transfer one orange from Player 1 to Player 2, increasing Player 2’s oranges (and hence U2) while reducing Player 1’s oranges but not affecting U1 (since U1 depends only on apples). This would strictly increase U2 without reducing U1, giving a Pareto improvement. Therefore, any allocation with O1 < 8 is not Pareto efficient.

Combining these, a Pareto-efficient allocation must satisfy A1 = 10 and O1 = 8 simultaneously. That is, Player 1 must hold all apples and all oranges, leaving Player 2 with nothing of either good. In other words, the contract curve is a single point: the allocation (A1, O1) = (10, 8) for Player 1, and (0, 0) for Player 2.

If a proposed option claimed a different allocation—such as Player 1 holding 10 apples but 0 oranges while Player 2 holds 8 oranges and 0 apples—that would not be Pareto efficient. In that alternative, since Player 2 still has oranges, one could transfer an orange from Player 1 to Player 2 (without affecting Player 1’s utility, which depends only on apples), improving Player 2’s utility while leaving Player 1’s objective unchanged. Hence that option would not lie on the contract curve.

In summary, given the preferences (P1: apples only; P2: oranges only) and the total endowment (10 apples, 8 oranges), the contract curve reduces to the corner where Player 1 has all goods (A1 = 10, O1 = 8) and Player 2 has none (A2 = 0, O2 = 0).

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