Contract curve

The contract curve is a term used in economics and describes the curve in an Edgeworth box , which connects all Pareto-optimal solutions for the mutually beneficial exchange of two quantities of goods between two households, given the household's initial equipment. In the production theory , in contrast to the household theory , two companies exchange two sets of input factors between themselves.

The economic subjects A and B try to change their initial equipment with the goods X and Y in such a way that their utility is maximized. Both swap until a market equilibrium is reached. A Pareto-optimal market equilibrium exists when neither can increase its benefit without reducing the benefit of the other.

Geometrically, the curve is the connection of all points at which the indifference curves of the two economic subjects touch each other (i.e. they have identical marginal rates of substitution ).

Contract curve

The black line represents the contract curve, the blue curves are exemplary indifference curves for economic subject A, the orange curves the corresponding indifference curves for B.

Theoretically, the economic agents can get all points on the contract curve through exchange . As soon as a point on the curve is reached, there is no more change, since one of the two individuals would definitely be worse off by swapping. Which point is actually reached depends on what initial endowments the individuals have of both goods and what utility functions they have. The contract curve does not necessarily have to pass through the two corners of the Edgeworth box.

Points on the contract curve satisfy both the first welfare theorem and the second welfare theorem.

It is important to note that the contract curve only highlights optimal distributions from the point of view of efficiency . Problems of distributive justice are not taken into account. This restriction is particularly important if a model result is to be transferred to real situations - an inefficient point can definitely be preferred by all participants because other aspects play a role.

example

Robinson and Friday live on a lonely island cut off from the rest of the world. Robinson has an initial supply of 40 coconuts and 10 fish, while Friday is equipped with 10 coconuts and 40 fish. Other goods are of no concern.

Both islanders have a utility function of , where represents the number of coconuts consumed and the number of fish consumed. This leads to the fact that both aim to own coconuts and fish in identical numbers. Obviously, the initial configuration does not meet this requirement. If trade is allowed, both will try to improve their bundles of goods. ${\displaystyle U(K,F)={\sqrt {K\cdot F}}}$ ${\displaystyle K}$ ${\displaystyle F}$

If, on the other hand, a well-meaning dictator orders a redistribution, which for example results in the following bundle

• Robinson: 30 fish, 30 coconuts and Friday: 20 fish, 20 coconuts
• Robinson: 25 fish, 25 coconuts and Friday: 25 fish, 25 coconuts
• Robinson: 20 fish, 20 coconuts and Friday: 30 fish, 30 coconuts

so no further trade will be sought. Both have received an optimal bundle of benefits (according to their possibilities) and cannot improve further without stealing fish or coconuts from the other and thus reducing their benefits.

Therefore, Pareto improvements are no longer possible, the given distributions are Pareto-optimal .

The contract curve represents the set of all Pareto-optimal distributions according to which each participant is at least as good as according to the initial configuration. In this case, we are talking about all distributions where both Robinson and Friday have an identical number of coconuts and fish, and each has at least 20 coconuts and fish.

In the Edgeworth box (see drawing above) the contract curve in this example would take the form of a straight line from point (20.20) (Robinson has 20 fish and coconuts, Friday has 30 fish and coconuts) to point (30.30 ) (Robinson has 30 fish and coconuts, Friday 20 fish and coconuts) adopt. The three example distributions mentioned then lie on this line.

literature

• Hal Varian : Intermediate Microeconomics. Chapter 30 (pp. 540 ff.), 6th Edition, WW Norton & Company, New York / London 2003, ISBN 0-393-97830-3 .
• Microeconomics Chapter 16 (p. 563 ff.), Pindyck, Robert S., Rubinfeld Daniel L., 6th Edition, Prentice-Hall Series in Economics, 2004, ISBN 0-130-08461-1 .