Editing Edgeworth box (section)

==Market equilibrium==
[[File:Edgeworthprice1.svg|280px|thumb|Fig. 2. Edgeworth box market]]Since there are only two commodities the effective price is the exchange rate between them. Our aim is to find the price at which market equilibrium can be attained, which will be a point at which no further transactions are desired, starting from a given endowment. These quantities will be determined by the indifference curves of the two consumers as shown in Fig.&nbsp;2. 

We shall assume that every day Octavio and Abby go to market with endowments {{math|(ω<sub>''x''</sub>,ω<sub>''y'' </sub>)}} and {{math|(Ω<sub>''x''</sub> – ω<sub>''x''</sub> , Ω<sub>''y''</sub> – ω<sub>''y'' </sub>)}} of the two commodities, corresponding to the position '''ω''' in the diagram. The two consumers will exchange between themselves under competitive market behaviour. This assumption requires a certain suspension of disbelief since the conditions for [[perfect competition]] – which include an infinite number of consumers – aren't satisfied. 

If two ''X'''s exchange for a single ''Y'', then Octavio's and Abby's transaction will take them to some point along the solid grey line, which is known as a '''budget line'''. (To be more precise, a budget line may be defined as a straight line through the endowment point representing allocations obtainable by exchange at a certain price.) Budget lines for a couple of other prices are also shown as dashed and dotted lines in Fig.&nbsp;2.

[[File:Edgeworthequil.svg|280px|left|thumb|Fig. 3. Equilibrium in an Edgeworth box]]The equilibrium corresponding to a given endowment '''ω''' is determined by the pair of indifference curves which have a common tangent such that this tangent passes through '''ω'''. We will use the term 'price line' to denote a common tangent to two indifference curves. An equilibrium therefore corresponds to a budget line which is also a price line, and the price at equilibrium is the gradient of the line. In Fig.&nbsp;3 '''ω''' is the endowment and '''ω{{'}}''' is the equilibrium allocation.

The reasoning behind this is as follows.

[[File:Edgeworthx.svg|thumb|280px|Fig. 4. Division of a neighbourhood by crossing indifference curves]]Firstly, any point in the box must lie on exactly one of Abby's indifference curves and on exactly one of Octavio's. If the curves cross (as shown in Fig.&nbsp;4) then they divide the immediate neighbourhood into four regions, one of which (shown as pale green) is preferable for both consumers; therefore a point at which indifference curves cross cannot be an equilibrium, and an equilibrium must be a point of tangency.

Secondly, the only price which can hold in the market at the point of tangency is the one given by the gradient of the tangent, since at only this price will the consumers be willing to accept limitingly small exchanges.

And thirdly (the most difficult point) all exchanges taking the consumers on the path from '''ω''' to equilibrium must take place at the same price. If this is accepted, then that price must be the one operative at the point of tangency, and the result follows.

In a two-person economy there is no guarantee that all exchanges will take place at the same price. But the purpose of the Edgeworth box is not to illustrate the price fixing which can take place when there is no competition, but rather to illustrate a competitive economy in a minimal case. So we may imagine that instead of a single Abby and a single Octavio we have an infinite number of clones of each, all coming to market with identical endowments at different times and negotiating their way gradually to equilibrium. A newly arrived Octavio may exchange at market price with an Abby who is close to equilibrium, and so long as a newly arrived Abby exchanges with a nearly satisfied Octavio the numbers will balance out. For exchange to work in a large competitive economy, the same price must reign for everyone. Thus exchange must move the allocation along the price line as we have defined it.<ref>See Pareto, ''Manuale''/''Manuel'', Chap. III, §170. Notice that Pareto is careful ''not'' to say that constant prices are general, merely that they are the commonest and most important case.</ref>

The task of finding a competitive equilibrium accordingly reduces to the task of finding a point of tangency between two indifference curves for which the tangent passes through a given point. The use of [[Edgeworth box#Offer curves|offer curves]] (described below) provides a systematic procedure for doing this.