22.1 From Bundles to Allocations

Up to now we have investigated how a single agent, endowed with a bundle of goods, would consider trading away from that endowment. In this part, we will conclude our study of the neoclassical microeconomic model by turning our attention to an economy with multiple people. We will assume that they each start with an endowment, and can trade with one another.

For an individual, our relevant choice space was the set of all combinations of good 1 and good 2. As far as any individual agent was concerned, a “bundle” $X$ was some combination of goods \(\text{bundle }X = (x_1,x_2)\) Let’s now consider the world’s simplest “economy,” of two agents, whom we’ll call Alison and Bob. Because we have two people, we’ll now need to identify now not only which good someone is consuming, but also who is consuming it. For that purpose, we’ll use a superscript with the person’s initials; so we’ll write Alison and Bob’s bundles as: \(\textcolor{#1f77b4}{\text{Alison's bundle} = X^A = (x_1^A, x_2^A)}\) \(\textcolor{#ff7f0e}{\text{Bob's bundle} =X^B = (x_1^B, x_2^B)}\) For good measure, as you can see, we’ll also color-code Alison using blue, and Bob using orange, just to keep them straight.

In analyzing the “economy” made of Alison and Bob, we can consider not only of how many goods each of them consumes in their own bundle, but in the allocation of goods across individuals. That is, we can define an allocation as a vector of individual bundles: \(\text{allocation }X = [\textcolor{#1f77b4}{(x_1^A, x_2^A)},\textcolor{#ff7f0e}{(x_1^B, x_2^B)}]\)

Visualizing an exchange economy: the Edgeworth Box

In this chapter, we’ll consider an exchange economy in which the total quantities of goods 1 and 2 are fixed: so an allocation is some division of the total quantity of goods among various agents. (In other words, there is no production in this economy; but we’ll assume that agents can trade with one another. Hence the term “exchange.”)

Suppose, for example, that between them, the “Alison and Bob” economy has 20 units of good 1 and 10 units of good 2. Any given division of this total is an allocation: for example, if Alison has $\textcolor{#1f77b4}{12}$ of the 20 units of good 1 and Bob has the other $\textcolor{#ff7f0e}{8}$, and Alison has $\textcolor{#1f77b4}{2}$ of the 10 units of good 2 and Bob has $\textcolor{#ff7f0e}{8}$, then this would correspond to the allocation \(X = [\textcolor{#1f77b4}{(12, 2)},\textcolor{#ff7f0e}{(8,8)}]\) We can illustrate all possible allocations in this economy in a diagram called an “Edgeworth Box.” This is my favorite diagram in all of economics, because it’s so unusual and packs a serious philosophical punch. It’s worth taking a good moment to ground ourselves in it.

The Edgeworth Box is, indeed, a box. The width of the box is the total amount of good 1 in the economy; the height of the box is the total amount of good 2. Any point in the box represents an allocation of goods 1 and 2 among Alison and Bob. That is, if Alison has $x_1^A$ units of good 1, then Bob must have the other $x_1^B = 20 - x_1^A$ units. Therefore, for any allocation $A$ in this box, $x_1^A$ is represented as the horizontal distance between the left-hand side of the graph and the point; and $x_1^B$ is represented as the horizontal distance between the point and the right-hand side of the box. Moving to the right means reallocating good 1 from Bob to Alison. In the same way, $x_2^A$ is the vertical distance bewteen the bottom of the graph and the point; and $x_2^B$ is the vertical distance from the point to the top of the box; so moving down means reallocating good 2 from Alison to Bob.

In other words, this is like a graph with two origins: from Alison’s perspective, the origin is the bottom left corner, so her consumption increases as you move up and to the right. Bob, on the other hand, views the world differently: his “origin” is in the top-right, so his consumption increases as you move down and to the left within the box.

The diagram below illustrates this point. The top rectangle is the Edgeworth Box; the bottom two graphs show how the allocation is viewed by each individual. Try dragging the allocation around the Edgeworth box to see how it affects the bundles consumed by Alison and Bob:

Note that neither Alison nor Bob necessarily “see” the box – they only see things from their own perspective, and may not know how many goods someone else has. But we, taking the economist’s (or “social planner’s”) view of the situation, both know how many of each good each agent has, and therefore know the set of all feasible reallocations of goods 1 and 2 between Alison and Bob. In that sense the Edgeworth Box represents the “choice space” for an outside policymaker considering the range of possible outcomes for this economy.