Model of an economic market
In economics, an Edgeworth box, sometimes referred to as an Edgeworth-Bowley box, is a graphical representation of a market with just two commodities, "X" and "Y", and two consumers. The dimensions of the box are the total quantities Ω"x" and Ω"y" of the two goods.
Let the consumers be Octavio and Abby. The top right-hand corner of the box represents the allocation in which Octavio holds all the goods, while the bottom left corresponds to complete ownership by Abby. Points within the box represent ways of allocating the goods between the two consumers.
Market behaviour will be determined by the consumers' indifference curves. The blue curves in the diagram represent indifference curves for Octavio, and are shown as convex from his viewpoint (i.e. seen from the bottom left). The orange curves apply to Abby, and are convex as seen from the top right. Moving up and to the right increases Octavio's allocation and puts him onto a more desirable indifference curve while placing Abby onto a less desirable one. 
Convex indifference curves are considered to be the usual case. They correspond to diminishing returns for each good relative to the other.
Exchange within the market starts from an initial allocation known as an "endowment".
The main use of the Edgeworth box is to introduce topics in general equilibrium theory in a form in which properties can be visualised graphically. 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.
History.
Development of the Edgeworth box.
The Edgeworth box is named after Francis Ysidro Edgeworth, who presented it 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 "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 mathematical theory of economic equilibrium.
The conceptual framework of equilibrium in a market economy was developed by Léon Walras and further extended by Vilfredo Pareto. It was examined with close attention to generality and rigour by twentieth century mathematical economists including Abraham Wald, Paul Samuelson, Kenneth Arrow and Gérard Debreu. This was part of a wider movement in which Wald also sought to bring greater rigour to decision theory and many mathematicians concentrated on minimising dependence on the axiom of choice.
The theory of Walrasian markets has taken pains to find the most general premises from which a given conclusion can be obtained. Areas in which premises can be strengthened or weakened include:
Assumptions are also made of a more technical nature, e.g. non-reversibility, saturation, etc. 
The pursuit of rigour is not always conducive to intelligibility. In this article indifference curves will be treated as primitive. At first we will view them as convex and differentiable and concentrate on interior equilibria, but we will subsequently relax these assumptions.
Market equilibrium.
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. 2. 
We shall assume that every day Octavio and Abby go to market with endowments (ω"x",ω"y" ) and (Ω"x" – ω"x" , Ω"y" – ω"y" ) 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 "Xs 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. 2.
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. 3 ω is the endowment and ω' is the equilibrium allocation.
The reasoning behind this is as follows.
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. 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.
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 offer curves (described below) provides a systematic procedure for doing this.
Pareto set.
An allocation of goods is said to 'Pareto dominate' another if it is preferable for one consumer and no worse for the other. An allocation is said to be 'Pareto optimal' (or 'Pareto efficient') if no other allocation Pareto dominates it. The set of Pareto optimal allocations is known as the Pareto set (or 'efficient locus').
Consider a pair of tangential curves, one for each consumer as illustrated in Fig. 5, where the point of tangency is shown by the purple dot. Then convexity guarantees that the curves cannot intercept other than at the point of tangency, and the box is accordingly divided into 3 regions. The pale blue area is preferable to the point of tangency for Octavio but worse for Abby; the pale orange area is preferable for Abby but worse for Octavio; and the white area is worse for both. Similar considerations apply to the boundaries. It follows that the point of tangency is Pareto optimal.
Thus the Pareto set is the locus of points of tangency of the curves. This is a line connecting Octavio's origin (O) to Abby's (A). An example is shown in Fig. 6, where the purple line is the Pareto set corresponding to the indifference curves for the two consumers.
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 et al. 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, use the term core for the section of the Pareto set which is at least as good for each consumer as the initial endowment.
Since the Pareto set is the set of points where the consumers' indifference curves are tangential, it is also the set of points where each consumer's marginal rate of substitution is equal to that of the other person.
The first fundamental theorem of welfare economics.
We have seen that the points of tangency of indifference curves are the Pareto optima, but we also saw previously that the economic equilibria are those points at which indifference curves are tangential to a common price line. It follows that the equilibria are precisely the Pareto optima. 
This argument applies with one restriction even if the curves are undifferentiable or if the equilibrium is on the boundary. The condition for equilibrium is that no further exchange will take place, and the condition for no further exchange to take place is that there is no direction of motion which benefits one consumer without harming the other; and this is equivalent to the definition of a Pareto optimum.
The restriction is that equilibrium implies that no "local" improvement can be made – in other words, that the point is 'locally' Pareto optimal. But Pareto optimality is nowadays considered global by definition. Thus if the nature of the indifference curves allows non-global optima to arise (as cannot happen if they are convex), then it is possible for equilibria not to be Pareto optimal.
Perfect competition is not a precondition for the theorem. So long as the consumers are free to exchange, and will continue to do so until no mutually acceptable exchange is available, equilibrium will be reached and will be (at least 'locally') Pareto optimal.
The second fundamental theorem of welfare economics.
Now consider an economy in which the consumers have endowments ω as shown in Fig. 7. Left to itself a free market will take them to ω'. But suppose that some other position in the box – say α' – is considered socially preferable. We can assume that the socially desired position is Pareto optimal. 
We may think of the price lines (shown as dashed in the diagram) as corresponding to different distributions of real income, and movement along them as reallocation of resources while incomes remain fixed. 
Then in order to reposition society at the desired point α' it is not necessary for the government to redistribute resources in such a way that Octavio holds (α'"x",α'"y") and Abby holds the complement: it is sufficient to reallocate resources to take the economy to "any" point (say α) on the price line through α', and then leave the market to find its own equilibrium. Indeed, so long as the government recognises a desirable distribution of income it does not need to have any idea of the optimal allocation of resources.
In a statement for a more general economy, the theorem would be taken as saying that α' can be reached by a monetary transfer followed by the free play of market exchange; but money is absent from the Edgeworth box.
The second fundamental theorem does not provide a blueprint for righting society's ills. The government may decide to reallocate resources between Octavio and Abby, moving them from ω to α in advance of the day's trading; and in consequence whoever loses out may decide to take less to market the next day. The second fundamental theorem takes no account of the distortions introduced by the reallocation.
Offer curves.
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