Contract curve

In microeconomics, the contract curve or Pareto set[1] is the set of points representing final allocations of two goods between two people that could occur as a result of mutually beneficial trading between those people given their initial allocations of the goods.

The contract curve is the subset of the Pareto efficient points that could be reached by trading from the people's initial holdings of the two goods.

However, most authors[2][3][4][5][6][7][8][9] identify the contract curve as the entire Pareto efficient locus from one origin to the other.

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.

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.

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).

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.

refers to the level of utility that person 2 would receive from the initial allocation without trading at all, and

subject to: This optimization problem states that the goods are to be allocated between the two people in such a way that no more than the available amount of each good is allocated to the two people combined, and the first person's utility is to be as high as possible while making the second person's utility no lower than at the initial allocation (so the second person would not refuse to trade from the initial allocation to the point found); this formulation of the problem finds a Pareto efficient point on the lens, as far as possible from person 1's origin.

In order to trace out the entire contract curve, the above optimization problem can be modified as follows.

By varying the weighting parameter b, one can trace out the entire contract curve: If b = 1 the problem is the same as the previous problem, and it identifies an efficient point at one edge of the lens formed by the indifference curves of the initial endowment; if b = 0 all the weight is on person 2's utility instead of person 1's, and so the optimization identifies the efficient point on the other edge of the lens.

As b varies smoothly between these two extremes, all the in-between points on the contract curve are traced out.

Blue curve of Pareto efficient points, at points of tangency of indifference curves in an Edgeworth box . If the initial allocations of the two goods are at a point not on this locus, then the two people can trade to a point on the efficient locus within the lens formed by the indifference curves that they were originally on. The set of all these efficient points that could be traded to is the contract curve.
In the graph below, the initial endowments of the two people are at point X, on Kelvin's indifference curve K 1 and Jane's indifference curve J 1 . From there they could agree to a mutually beneficial trade to anywhere in the lens formed by these indifference curves. But the only points from which no mutually beneficial trade exists are the points of tangency between the two people's indifference curves, such as point E. The contract curve is the set of these indifference curve tangencies within the lens—it is a curve that slopes upward to the right and goes through point E.