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.