22.4 Social Preferences and Equity

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 Possibilties 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.