Editing Edgeworth box (section)

==Offer curves==
[[File:Edgeworthprice3.svg|280px|thumb|Fig. 8. Preferred points on different budget lines]]'''Offer curves''' provide a means of finding points of equilibrium, and are also useful for investigating their existence and uniqueness. 

Two such curves, one for each consumer and both depending on the endowment, can be drawn in the box. We pivot the budget line about '''ω''' and trace the two consumers' most favoured points along the line as shown by the coloured dots in Fig.&nbsp;8. These are points at which the line is tangential to their own indifference curves.

[[File:Edgeworthprice4.svg|left|280px|thumb|Fig. 9. Offer curves]]The locus of a consumer's most favoured points is his or her offer curve. Fig.&nbsp;9 shows Octavio's offer curve as dark blue and Abby's as brown. They meet at the point '''ω{{'}}''' and the equilibrium budget line (drawn in grey) is the one passing through this point. The indifference curves through '''ω{{'}}''' for the two consumers are shown in paler colours.

An offer curve necessarily passes through the endowment point '''ω'''. If we take Abby as an example, we note that one of her indifference curves must pass through '''ω''' and that a budget line can be chosen to have the same gradient as the indifference curve here, making '''ω''' a most favoured point for this line. 

In consequence the two consumers' offer curves necessarily intersect at '''ω'''; but the property which makes this happen is that '''ω''' is the only possible point of intersection consistent with budget lines of differing gradient,  and that therefore it doesn't necessarily constitute an equilibrium. 

Any intersection of offer curves at a point other than '''ω''' determines a stable equilibrium. If the two offer curves are tangential at the endowment point, then this point is indeed an equilibrium and their common tangent is the corresponding budget line.<ref>This account is based on Section 15.B of Mas-Colell et al. The illustration is their Example 15.B.1 with its Cobb-Douglas α set equal to 0·275.</ref>

===Terminology for offer curves===
Offer curves were first used by [[Vilfredo Pareto]] – see his ''Manuale''/''Manuel'' Chap. III, §97. He called them 'exchange curves' (''linee dei baratti''&nbsp;/&nbsp;''lignes des échanges''), and his name for Octavio's preferred allocation along a budget line was his 'equilibrium point'.

This preferred allocation is sometimes nowadays referred to as Octavio's 'demand', which constitutes an asymmetric description of a symmetric fact. An allocation determines Abby's holding as much as Octavio's, and is therefore as much a supply as a demand. 

''Offre'' is French for 'supply', so calling an offer curve a locus of demands amounts to calling a supply curve a locus of demands.

===Uniqueness of equilibria===
[[File:Edgeworthpathology1.svg|280px|thumb|left|Fig. 10. An Edgeworth box with multiple equilibria]]
[[File:Edgeworthpathology2.svg|280px|thumb|right|Fig. 11. An Edgeworth box with multiple equilibria (detail)]]
It might be supposed from economic considerations that if a shared tangent exists through a given endowment, and if the indifference curves are not pathological in their shape, then the point of tangency will be unique. This turns out not to be true. Conditions for uniqueness of equilibrium have been the subject of extensive research: see [[General equilibrium theory#Uniqueness|General equilibrium theory]].

Figs. 9 and 10 illustrate an example from Mas-Colell et al. in which three distinct equilibria correspond to the endowment point '''ω'''. The indifference curves are:

<math>\quad x - \tfrac{1}{8}y^{-8} = u</math> (Octavio)

<math>\quad y - \tfrac{1}{8}x^{-8} = u</math> (Abby).

The indifference curves fill the box but are only shown when tangential to some representative budget lines. 
The offer curves, drawn in Fig. 11, cross at three points shown by large grey dots and corresponding to exchange rates of {{frac|1|2}}, 1 and 2.