Edgeworth box

Edgeworth box

In economics, an Edgeworth box, named after Francis Ysidro Edgeworth, is a way of representing various distributions of resources. Edgeworth made his presentation in his famous book, "Mathematical Psychics: An essay on the application of mathematics to the moral sciences", 1881. Edgeworth's original two axis depiction was developed into the now familiar box diagram by Pareto in 1906 and was popularized in a later exposition by Bowley. The modern version of the diagram is commonly referred to as the Edgeworth-Bowley box.

The Edgeworth box is used frequently in general equilibrium theory, and can aid in finding the competitive equilibrium of a simple system.

Example

Imagine two people (Octavio and Abby) with a fixed amount of resources between the two of them &mdash; say, 10 liters of water and 20 hamburgers. If Octavio takes 5 hamburgers and 4 liters of water, then Abby is left with 15 hamburgers and 6 liters of water. The Edgeworth box is a rectangular diagram with Octavio's Origin on one corner (represented by the O) and Abby's origin on the opposite corner (represented by the A). The width of the box is the total amount of one good, and the height is the total amount of the other good. Thus, every possible division of the goods between the two people can be represented as a point in the box.

In theory, it is possible to draw among these points, indifference curves for both Abby and Octavio representing combinations of the goods that are of equal value, respectively, to Octavio and Abby. For example, Abby might value 1 liter of water and 13 hamburgers the same as 5 liters of water and 4 hamburgers, or 3 liters and 10 hamburgers. There is, of course, an infinity of such curves (assuming water and hamburgers to be infinitely divisible) that could be drawn among the combinations of goods for either consumer (Octavio or Abby). The indifference curve is often derived from each consumer's utility function.

Pareto set

Wherever one of these curves for Abby happens to just touch (but not cross) a curve of Octavio's, a unique combination of the two goods is identified that yields both consumers a maximum value (which consumer realizes the greater value cannot be known, even to the consumers). Such tangential contacts between the infinity of indifference-curve pairs, if plotted, will form a trace connecting Octavio's origin (O) to Abby's (A). The curve connecting points O and A is often called the Pareto set, since each point on the curve is Pareto optimal. The utilities which can be achieved by a given point on the Pareto set constitute what is sometimes called the utility possibility frontier. It is important to note that while it is often described as a curve, the Pareto set may be all points within some shape, or even the whole box.

The vocabulary used to describe different objects which are part of the Edgeworth box diverges. The entire Pareto set is sometimes called the contract curve, while Mas-Colell, Winston, and Green, in their famous Microeconomics text, "Microeconomic Theory" (1995), restrict the definition of the contract curve to only those points on the Pareto set which make both Abby and Octavio at least as well off as they are at their initial endowment. Other authors who have a more game theoretical bent, such as Martin Osborne and Ariel Rubinstein in their "A Course in Game Theory" (1994), uses the term core for the section of the Pareto set which is at least as good for each consumer as the initial endowment.

In order to calculate the Pareto set, the slope of the indifference curves for both consumers must be calculated. That slope is the negative of the marginal rate of substitution, so since the Pareto set is the set of points where both indifference curves intersect tangentially, it is also the set of points where each consumer's marginal rate of substitution is equal.

ee also

*Welfare Economics
*List of economics topics
*Utility

* [http://www.sscnet.ucla.edu/ssc/labs/cameron/e1f98/imapedge.html java applet UCLA]

References

* Mas-Colell, Andreu; Whinston, Michael D.; and Jerry R. Green. Microeconomic Theory. Oxford University Press, US: 1995, ISBN 0-19-507340-1
* Osborne, Martin J. and Ariel Rubinstein: A Course in Game Theory, MIT Press, 1994, ISBN 0-262-65040-1

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