Lecture 2: Efficiency and Equity in the Edgeworth Box

From Bundles to Allocations

Last time we talked about a single agent’s preferences over their own consumption. For such an individual, the 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

For this week and next, 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.”) In week 3 we will extend the model to include production.

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 between 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.

Preferences over Allocations

When we introduced the notion of indifference curves, we motivated our discussion by grounding it in the axiom of complete preferences: that is, given any two alternatives $X$ and $Y$, we assume that an agent will say that she either prefers $X$ to $Y$, prefers $Y$ to $X$, or is indifferent between the two. Therefore, an indifference curve through bundle $X$ divides the choice space into the “preferred set” (those bundles the agent prefers to $X$) and the “dispreferred set” (those bundles the agent disprefers to $X$).

The same logic may apply to the Edgeworth Box: if we presented Alison and Bob with a range of alternatives within the Edgeworth Box, they could each tell us which one they preferred or dispreferred. In particular, for any allocation $X$, they could tell us the set of bundles they prefer to $X$, the set of bundles they disprefer, and the set between which they are indifferent. In other words, for any bundle $X$, we can draw both Alison’s indifference curve and Bob’s indifference curve through it.

If we draw the indifference curves and shade in the preferred sets, something profound reveals itself:

If we look at the shaded regions, we can see that for any bundle X, the indifference curves partition the box into four regions:

• Bundles that make Alison better off and Bob worse off
• Bundles that make Alison worse off and Bob better off
• Bundles that make both Alison and Bob worse off
• Bundles that make both Alison and Bob better off

It’s this last category that’s most important to us as economists. Think about what this means: simply by reallocating goods from one person to another, we can make both of them better off. In other words: the world is improvable. There is cause for optimism.

To be more precise, we can say that if we have two bundles, $X$ and $Y$, then $Y$ Pareto dominates $X$ (or that moving from $X$ to $Y$ is a Pareto improvement) if $Y \succeq_i X$ for every agent $i$, and $Y \succ_i X$ for at least one agent $i$. That is, it makes at least one person strictly better off, and makes no agent worse off.

Generally speaking, at any allocation for which the agents’ marginal rates of substitution are not equal, there may be a set of allocations that represent potential Pareto improvements. For example, if Alison’s MRS is less than Bob’s, this would mean that Alison values her last unit of good 1 (in terms of good 2) less than Bob would value it; so there is the potential that by trading some of her good 1 for some of Bob’s good 2, they can each be made better off. Visually, there would be a “lens” of allocations which could potentially represent Pareto improvement above and to the left of the initial endowment, as shown in the diagram above. If Alison’s MRS is greater than Bob’s, on the other hand, the lens of potential Pareto improving allocations would lie below and to the right of the initial allocation. (Try dragging point $X$ to the upper-left part of the box and/or changing the parameters of the utility function to see how this works.)

Pareto Efficiency and the Contract Curve

We just showed that for an arbitrary allocation, there may be a set of alternative allocations which are Pareto improvements over that allocation: that is, it might be possible to make at least one person better off without making many anyone worse off.

However, there are some allocations for which there is no overlap between the set of bundles that Alison prefers, and the set that Bob prefers. Such an allocation is called Pareto efficient, meaning that you cannot make one person strictly better off without making at least one other person strictly worse off.

Try dragging the allocation in the following diagram around until the is no overlap in the set of allocations Alison prefers to X, and the set of allocations Bob does. Once you find such a point, try changing their preferences using the sliders, and try to figure out how the location of these kinds of points changes:

What makes a bundle Pareto efficient? For a case like, this, in which we have “well-behaved” preferences (smooth, monotonic, strictly convex, along with some other technical conditions), a Pareto efficient allocation is characterized by the condition that the marginal rates of substitution are equal: \(MRS^A = MRS^B\) In the diagrams in this section, we’ve been looking at two agents with Cobb-Douglas preferences. You might have noticed that you can change the coefficients on these preferences: Alison’s preferences have been given by \(u^A(x_1^A,x_2^A) = a \ln x_1^A + (1-a) \ln x_2^A \Rightarrow MRS^A = {a \over 1-a} \times {x_2^A \over x_1^A}\) while Bob’s preferences have been given by \(u^B(x_1^B,x_2^B) = b \ln x_1^B + (1-b) \ln x_2^B \Rightarrow MRS^B = {b \over 1-b} \times {x_2^B \over x_1^B}\) For a point $X$ to be Pareto efficient, therefore, we would need their MRS’s to be equal, or \({a \over 1-a} \times {x_2^A \over x_1^A} = {b \over 1-b} \times {x_2^B \over x_1^B}\) What can we glean from this? If both Alison and Bob have the same utility function (that is, if $a = b$), then we have \({x_2^A \over x_1^A} = {x_2^B \over x_1^B}\) That is, along the contract curve, they will each have the same ratio of good 2 to good 1, so the contract curve is a straight line connecting their two origins. However, if $a > b$, then the contract curve bends down and to the right: that is, Alison likes good 1 relatively more (and good 2 relatively less) than Bob does, so Pareto efficient allocations will generally give her more good 1 and Bob more good 2: \({x_2^A \over x_1^A} < {x_2^B \over x_1^B}\) On the other hand, if $a < b$, then the opposite is true, and the contract curve bends up and to the left, meaning \({x_2^A \over x_1^A} > {x_2^B \over x_1^B}\) Regardless, at every point along the contract (you can try dragging bundle $X$ left and right), there is no overlap Alison and Bob’s preferred sets:

Note that the locus of the contract curve is based solely on the dimensions of the Edgeworth Box and the preferences of the agents. In other words, it is a characteristic of every possible allocation to be Pareto efficient or not. It is not affected by whatever the initial allocation (“endowment”) might be.

The “Contract Curve” for non-well-behaved preferences

If preferences are not well-behaved, the set of Pareto efficient points may not be characterized by a point of tangency. For example, suppose both Alison and Bob viewed these goods as perfect substitutes, but Alison’s MRS was greater than Bob’s. For example, their utility functions might be \(u^A(x_1^A,x_2^A) = 2x_1^A + x_2^A\) \(u^B(x_1^B,x_2^B) = x_1^B + 2x_2^B\) The following diagram shows the set of bundles preferred to each allocation. Try to find the Pareto efficient points (i.e. the “contract curve”). Then check the box to reveal where it is:

What’s going on here? At every point in the Edgeworth Box, Alison likes good 1 relatively more than good 2, and Bob feels the opposite way. So there is always an opportunity for an improvement if Alison exchanges some of her good 2 for some of Bob’s good 1…as long as she has some good 2, and he has some good 1! Hence the set of points from which there is no potential improvement are the allocations in which Alison only has good 1 and/or Bob only has good 2.

Equity and Social Preferences

In the previous analysis, we restricted ourselves to looking at the preferences of the two agents in an Edgeworth Box model. We made no judgements about allocations, beyond categorizing some as Pareto efficient and others as not Pareto efficient. We also saw that there was a range of Pareto efficient allocations (i.e. the “contract curve”), some of which were quite unequal. Indeed, the two “origins” of the Edgeworth Box (in which one of the two agents has all of both goods, and the other has none) are generally Pareto efficient as long as both agents have strictly monotonic preferences: if one person has all the goods, then taking any away from them would make them worse off, so there is no possible Pareto improvement!

In this section we’ll think about different ways of evaluating allocations beyond just whether they are efficient or not. Just as we analyzed the “feasible set” for Chuck on a desert island in Part I, we’ll analyze the set of possible combinations of utility available within the Edgeworth Box. We’ll then talk about what preferences over those utility combinations might look like, and how a social observer who cares about both agents might determine the allocation they think is “best.”

The Utility Possibilities Frontier

At every point in the Edgeworth Box, each person has a certain amount of utility: that is, every allocation $[(x_1^A,x_2^A),(x_1^B,x_2^B)]$ is associated with a utility for person A $(u^A)$ and a utility for person B $(u^B)$.

Just as we can plot quantities of good 1 and good 2 in “good 1 - good 2 space,” we can plot the utilities obtained by each agent in “utility A - utility B space,”” or “utility space” for short. This is a graph showing A’s utility on the horizontal axis, and B’s utility on the vertical axis.

If you drag the allocation around the Edgeworth Box, you can see that not all utilities are possible. The utility functions in this example are \(u^A(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 4}\) \(u^B(x_1,x_2) = x_1^{1 \over 4}x_2^{1 \over 2}\) Therefore if agent A has all 20 units of good 1 and 10 units of good 2, her utility will be \(u^A(20,10) = 20^{1 \over 2} \times 10^{1 \over 4} \approx 7.85\) and Agent B’s utility would be 0; hence the point (7.95, 0) is the point in utility space that gives agent A the most utility.

Likewise, if agent B had all of both goods, his utility would be \(u^B(20,10) = 20^{1 \over 4} \times 10^{1 \over 2} \approx 6.69\) and Agent A’s utility would be 0; hence the point (0, 6.69) is the point in utility space that gives agent B the most utility.

Try starting from any non-Pareto-efficient allocation — that is, any point such as [(12,6), (8,4)] — and moving the allocation into the “lens” of Pareto improvements. Since a Pareto improvement makes both people better off, this results in the dot in utility space moving up and to the right.

What about when there is no possible Pareto improvement — that is, if the allocation lies along the contract curve? In that case making one person better off would mean making the other one worse off. Just as the Production Possibilities Frontier (PPF) is the boundary between combinations of goods which can be produced with available resources and those which cannot, of attainable production outputs, we can define the “Utility Possibilities Frontier” as the boundary between those utilities which are possible for some allocation within the Edgeworth Box, and those which are impossible. The utility combinations along the UPF, therefore, correspond to allocations along the contract curve: at such an allocation, you cannot make one person better off without making the other one worse off.

Social preferences

Analyzing the utility implications of allocations in the Edgeworth Box allows us to pursue some interesting philosophical thought experiments. In particular, we could ask ourselves what our own preferences over possible outcomes might be? Put another way: can a disinterested “social planner” have preferences over potential utility outcomes in utility space? Could we somehow define a “utility function over utilities” such as $W(u^A,u^B)$? And what would its properties be?

• Preference for efficiency. It’s clear that we probably feel that a Pareto improvement is a good thing: that is, moving up and to the right in utility space is something we should generally be in favor of. In other words, it would make sense that $W(u^A,u^B)$ should be monotonic: that is, it should be increasing in both $u^A$ and $u^B$.
• Preference for equity. As a matter of ethics, we might think that economic inequality is a bad thing. For example, we may prefer a point in the middle part of the contract curve (where each agent has a moderate amount of utility) to either origin (where one person has everything, and the other person has nothing). Indeed, we may even prefer a slightly inefficient allocation that gives each person a fair amount of utility to a highly unequal situation. In other words, our preferences over utilities may be convex.

Any social preferences will therefore have indifference curves in both the Edgeworth Box and utility space. For example, the diagram below shows the indifference curves for the simple Cobb-Douglas social welfare function $W(u^A, u^B) = u^Au^B$. Try experimenting with the following diagram by moving points X and Y around the space:

In the initial positions of X and Y, X is preferred to Y because although Y lies along the contract curve, X is a more equitable allocation. Of course, the optimal allocation (from this social welfare function, in any case) is along the contract curve, as it must be. (Why must that be the case?)

Hang on, we can’t use utils!

OK, I know what you’re thinking. After spending a whole chapter talking about how we shouldn’t take utility seriously, here we’re seeming to not only take it seriously for a single person, but think we can compare it across people? What gives?

One unsatisfactory answer is that we can use some method to “normalize” utility across people, so that we can get as close as we can to comparing apples to apples. While that might be possible, it’s hard to see how it’s rigorous.

A slightly more satisfactory answer is that some of the contexts we look at have the “goods” in question here be money. For example, we could think about an intertemporal consumption model in which “good 1” is present consumption and “good 2” is future consumption, and the utility functions are utility functions over money. In that case we might argue, with a bit more soundness, that we should assign similar utility values to the same amount of money enjoyed by different people: that is, the amount of “welfare” generated by distributed money among people shouldn’t depend on which person has which amount of money. (These are sometimes called “anonymous” social preferences.)

But the truth is, those are all pretty weak arguments. This framework is not meant to be solvable for a truly “ideal” allocation. Rather, it’s a useful tool for thinking about the tradeoffs between efficiency and equity in a simple, elegant model.

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