Contract Curve - Example

Example

Assume the existence of an economy with two agents, Octavio and Abby, who consume two goods X and Y of which there are fixed supplies, as illustrated in the above Edgeworth box diagram. Further, assume an initial distribution (endowment) of the goods between Octavio and Abby and let each have normally structured (convex) preferences represented by indifference curves that are convex toward the people's respective origins. If the initial allocation is not at a point of tangency between an indifference curve of Octavio and one of Abby, then that initial allocation must be at a point where an indifference curve of Octavio crosses one of Abby. These two indifference curves form a lens shape, with the initial allocation at one of the two corners of the lens. Octavio and Abby will choose to make mutually beneficial trades — that is, they will trade to a point that is on a better (farther from the origin) indifference curve for both. Such a point will be in the interior of the lens, and the rate at which one good will be traded for the other will be between the marginal rate of substitution of Octavio and that of Abby. Since the trades will always provide each person with more of one good and less of the other, trading results in movement upward and to the left, or downward and to the right, in the diagram.

The two people will continue to trade so long as each one’s marginal rate of substitution (the absolute value of the slope of the person's indifference curve at that point) differs from that of the other person at the current allocation (in which case there will be a mutually acceptable trading ratio of one good for the other, between the different marginal rates of substitution). At a point where Octavios’s marginal rate of substitution equals Abby’s marginal rate of substitution, no more mutually beneficial exchange is possible. This point is called a Pareto efficient equilibrium. In the Edgeworth box, it is a point at which Octavio’s indifference curve is tangent to Abby’s indifference curve, and it is inside the lens formed by their initial allocations.

Thus the contract curve, the set of points Octavio and Abby could end up at, is the section of the Pareto efficient locus that is in the interior of the lens formed by the initial allocations. The analysis cannot say which particular point along the contract curve they will end up at — this depends on the two people's bargaining skills.