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. Contents 1 Example 2 Pareto set 3 See also 4 References 5 External links Example

Imagine two people (Octavio and Abby) with a fixed amount of resources between the two of them say 10 liters of water and 20 hamburgers. If Abby takes 5 hamburgers and 4 liters of water then Octavio 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.

Indifference curves (derived from each consumer's utility function) can be drawn in the box for both Abby and Octavio representing combinations of the goods that are of equal value respectively to each party. Hence Abby is indifferent between any combination of goods on her indifference curve and the same is true for Octavio and his indifference curve. 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 infinite number of such curves (assuming water and hamburgers to be infinitely divisible) that could be drawn among the combinations of goods for each consumer (Octavio or Abby).

When an indifference curve for Abby crosses one of the indifference curves for Octavio at more than one point (i.e. the two curves are not tangent to each other) a space is created by the crossing of the two curves in which Pareto efficient trades can still take place. A contract curve can be drawn through the diagonal of the Edgeworth box which signifies the locus of all the Pareto efficient points. 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 necessarily 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. See also Efficiency between production and consumption List of economics topics Utility References Mas-Colell Andreu; Whinston Michael D.; and Jerry R. Green (1995). Microeconomic Theory. New York: Oxford University Press. ISBN 0195073401.  Osborne Martin J.; Rubinstein Ariel (1994). A Course in Game Theory. Cambridge: MIT Press. ISBN 0262650401.  External links java applet UCLA