# Edgeworth box (Ofer Abarbanel online library)

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 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 his 1906 book “Manual of Political Economy” 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. It can aid in representing the competitive equilibrium of a simple system or a range of such outcomes that satisfy economic efficiency. It can also show the difficulty of moving to an efficient outcome in the presence of bilateral monopoly. In the latter case, it serves as a precursor to the bargaining problem of game theory that allows a unique numerical solution.

## 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 4 liters of water and 5 hamburgers, then Octavio is left with 6 liters of water and 15 hamburgers. 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. The points on each person’s indifference curve represent equally liked combinations of quantities of the two goods for that person. Hence Abby is indifferent between one combination of goods and another on any one of her indifference curves, and the same is true for Octavio. 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 are an infinite number of such curves that could be drawn among the combinations of goods for each consumer (Octavio or Abby).

With Octavio’s origin (the point representing zero of each good) at the lower left corner of the Edgeworth box and with Abby’s origin at the upper right corner, typically Octavio’s indifference curves would be convex to his origin and Abby’s would be convex to her origin.

When an indifference curve for Abby crosses one of the indifference curves for Octavio at more than one point (so the two curves are not tangent to each other), a space in the shape of a lens is created by the crossing of the two curves. Any point in the interior of this lens represents an allocation of the two goods between the two people such that both people would be better off than at the corners of the lens, since the interior point is on an indifference curve farther from both of their respective origins, and thus, each individual achieves a higher utility.

## Pareto set

Wherever one of these curves for Abby happens to be tangent to a curve of Octavio’s, a combination of the two goods is identified that yields both consumers a level of utility that could not be improved for one person by a reallocation without decreasing the utility of the other person. Such a combination of goods is said to be Pareto optimal. The set of tangential points of contact between pairs of indifference curves, if all traced out, will form a trace connecting Octavio’s origin (O) to Abby’s (A). This curve connecting points ‘O’ and ‘A’, which will not in general be a straight line, is called the Pareto set or the efficient locus, since each point on the curve is Pareto optimal.

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 (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 (1994), use 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 at each point. 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 are tangent, it is also the set of points where each consumer’s marginal rate of substitution is equal to that of the other person.

## References

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

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