Edgeworth box

Fig. 1) Edgeworth box.

Fig. 2) Some indifference curves for different utility levels of a household.

Fig. 3) Exchange process

Fig. 4) The optimization through exchange processes defines the contract curve .

The Edgeworth box , also known as the Edgeworth diagram (after Francis Ysidro Edgeworth ), is a graphic aid in microeconomics that is used to intuitively determine the general equilibrium of a pure-exchange economy made up of two individuals to investigate in the two-goods case. A Edgeworth box is made up of the amount of indifference curves of the two considered individuals together, respectively in the positive orthant associated sets quantities diagrams edged of them are diagonally composed mirror-reversed, so that the coordinate axes of a box (box) form.

construction

In Fig. 1, initially only the coordinate axes (and not the indifference curves) are drawn in to illustrate the construction. The coordinate system with the directional axes pointing upwards and to the right belongs to household 1, the other to household 2. Since a pure exchange economy is considered (no production activity), the dimensions of the Edgeworth box are always determined ex ante : The length corresponds exactly to that sum of the available (and potentially tradable) units of good 1, so the sum of the features of person 1 Good 1 , and furnish of person 2 Good 1 . The same applies to the height of the box, which corresponds to the sum of the equipment of both people with good 2 ,. ${\displaystyle e_{1}^{1}}$ ${\displaystyle e_{1}^{2}}$ ${\displaystyle e_{2}^{1}+e_{2}^{2}}$

Each point within an Edgeworth box has, to be precise, four coordinates, because it is determined for both households how much each of them has access to both goods. For this reason, each point within an Edgeworth box describes a complete allocation .

Let us denote the person's equipment vector ; it records the individual's equipment with good 1 and with good 2. In Fig. 1, a point has been drawn as an example that marks the initial equipment of household 1 and household 2. According to the purpose of an Edgeworth box, one can now imagine that exchange processes are carried out from this point, in the course of which the households gradually leave the orange-colored point and bring it into equilibrium. This process is described in the following section. ${\displaystyle \mathbf {e} ^{i}=\left(e_{1}^{i},e_{2}^{i}\right)}$ ${\displaystyle i=1,2}$ ${\displaystyle i}$ ${\displaystyle \left(\mathbf {e} ^{1},\mathbf {e} ^{2}\right)}$

Exchange process and example

Fig. 2 shows the lower part of an Edgeworth box and consists of some indifference curves from household 1. (The indifference curve is the geometric location of all combinations of quantities of goods - in this case of good 1 and good 2 - that create the same level of utility.)

Example: Fig. 2 assumes that the preferences of the household are represented by a utility function of the Cobb-Douglas type , specifically by the utility function. The indifference curves for the utility levelare then obtained viawith,as in Fig. 2. From the dimensions of the box, one can see that the sum of the facilities in both households for both goods is a total of 10. ${\displaystyle u(x_{1}^{1},x_{2}^{1})={\sqrt {x_{1}^{1}\cdot x_{2}^{1}}}}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle u(x_{1}^{1},x_{2}^{1})={\overline {u}}\Rightarrow x_{1}^{1}\cdot x_{2}^{1}={\overline {u}}^{2}\Rightarrow x_{2}^{1}={\overline {u}}^{2}/x_{1}^{1}}$ ${\displaystyle {\overline {u}}=3,4,5,\ldots }$

In this example, it is assumed that both households have identical preferences. If one then looks at the Edgeworth box in Fig. 3, the exchange process can be illustrated: The (red) point at the top left marks the initial allocation, i.e. the initial equipment of households 1 and 2. If one now looks at the indifference curve of household 1 (blue) that passes through this point; it reveals that the utility of household 1 from its initial configuration is 4. Every bundle of goods that is above this indifference curve (and thus on a higher indifference curve) is obviously preferred by household 1 (blue). The same applies to household 2 (green): He also prefers bundles of goods that are on higher indifference curves (note that the coordinate system is, as is known, shown in mirror image - higher indifference curves for household 2 are therefore below the bold green curve).

From this consideration it follows that each point within the lens-shaped area between the bold green and blue indifference curve in Fig. 1 puts both households in a better position than their respective initial equipment. Every point within the lens is, in the language of welfare economics, Pareto-superior (also: a Pareto improvement ) to the initial allocation , because without a household being worse off, at least one household is better off.

If the households now carry out such an exchange, they are now at some point within the "lens". One can then repeat the above consideration to see that there is generally a new, smaller lens-shaped area for each of these points, which is Pareto-superior with respect to the new allocation. However, this does not apply to every point. If you look at the (red) point in the middle of Fig. 3, there is no longer any Pareto improvement: If you want to improve household 1 (blue) and “raise” it to a higher indifference curve, this is only possible by household 2 (green) is placed worse.

Determination of the Pareto Optima - the contract curve

If in the previous section a certain point was only vaguely mentioned as an example of a Pareto optimum (for the designation see: Vilfredo Pareto ), the set of these efficiency points can also be precisely described. In fact, there are an infinite number of Pareto-efficient allocations. Its characteristic is that two indifference curves (namely one from household 1 and one from household 2) touch each other at these points. So if you want to find all Pareto optima within an Edgeworth box, you just have to go through all indifference curves of a household and mark the point at which it just touches any indifference curve of the other household. If you then connect these points as shown in Fig. 4, you get a curve, the so-called contract curve of the exchange economy. It contains all Pareto-efficient allocations of the economy. Fig. 4 illustrates this: starting from the lower right (red) point, all points on the contract curve are Pareto-Optima within the associated intersection, and that part of the contract curve that runs through the surface (indicated by the orange points) for each household is also a desirable improvement compared to the initial allocation.

The contract curve is generally not a straight line. In the example, this follows from the fact that the households have identical utility functions.

literature

• Friedrich Breyer: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2011, ISBN 978-3-642-22150-7 .
• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Hal Varian : Intermediate Microeconomics. A modern approach. 8th edition. WW Norton, New York and London 2010, ISBN 978-0-393-93424-3 .