# The conflict between equity and efficiency

Equality is not the only social virtue to which we may aspire. Efficiency is another such virtue, a notion that is typically interpreted by economists in terms of the Pareto principle—namely, that if a state of affairs is better for someone without being worse for anyone, then it is a better state. If states of affairs are interpreted as income distributions, and if we assume that everybody prefers more income to less, then a “Paretian” will judge that x is a better distribution than y whenever at least one person has more income and no person has less income in x than in y.

How might one describe an egalitarian? One could say an egalitarian is a person who holds it to be bad that some persons should be worse off than others. Operationally, and in the context of income distributions, one way in which many economists and philosophers have interpreted egalitarianism is in terms of the requirement that x is a better distribution than y whenever x is a more equal distribution than y.

Given these interpretations of egalitarianism and efficiency, a very fine account of how the two principles could clash is available in the work of Albert Weale, now a professor of political theory at University College London. In much of what follows, I shall depend a great deal on Weale’s paper “The Impossibility of Liberal Egalitarianism” (*Analysis*, 1980).

To see clearly the nature of the conflict between equity and efficiency, assume a two-person world, comprising individuals 1 and 2. A typical income distribution is a two-component list in which the first component refers to person 1’s income, and the second to person 2’s income. One distribution will be said to be more equal than another if the income-share of the poorer person is larger in the first distribution than in the second.

Now consider two distributions x = (5,5) and y = (6,12). Clearly, x (in which the poorer person’s income share is 50%) is a more equal distribution than y (in which the poorer person’s income share is around 33%).

An egalitarian of the sort we have described earlier must hold x to be a better state than y. However, each person in y has a higher income than her corresponding income in x, so a Paretian must judge y to be a better state than x. In summary, x is better than y in egalitarian terms, and y is better than x in efficiency terms. Hence, the equity-efficiency conflict.

How can we ethically rank two income distributions x and y if the ranking is governed by two criteria of which one criterion ranks x over y and the other ranks y over x? The obvious way out, one may think, is to grant priority to one criterion.

The Pareto principle has long enjoyed the status of a sacred cow, so let us see what happens when we accord priority to this principle. In particular, given two distributions x and y, let us say that x will be ranked ethically superior to y if either (a) x is Pareto-superior to y, or, failing that, if (b) x is more equitable than y and y is not Pareto-superior to x. Call this the priority rule. The priority rule seems to be a natural way of affording priority to efficiency over equity. Does it resolve the conflict between the two principles?

It would seem so: going back to our original example involving the distributions x = (5,5) and y = (6,12), notice that by the priority rule we should declare the Pareto-superior distribution (if one exists) to be the ethically better distribution. As it happens, y is indeed Pareto-superior to x, so y should be ranked above x, and the conflict stands resolved.

Or does it? Consider a third distribution z = (7,4), in which the income-share of the poorer person is 4/11, or around 36%. Clearly, z is a more equitable distribution than y, and a less equitable distribution than x. By the priority rule, we should—as we have already seen—judge y to be ethically superior to x. Notice that since x is more equitable than z and z is not Pareto-superior to x, the priority rule will dictate that x should be ranked ethically superior to z.

Similarly, in the comparison between y and z, since z is more equitable than y and y is not Pareto-superior to z, by the Priority Rule, one must have: z is ethically superior to y. What then, in sum, do we have? The following: y ethically superior to x; x ethically superior to z; and z ethically superior to y. We have a complete cycle of ethical rankings, with each distribution chasing the other in an infinite regress (a cycle of this nature falls foul of a necessity of logic called the property of transitivity).

We could try and augment the role of the Pareto principle by according it even more priority, along lines which Weale has explored in his paper. But this only leads to additional contradictions and logical inconsistencies (the interested reader may wish to consult Weale’s paper for the finer details). In an extreme bid to avert conflict, we could assign a role only to efficiency, and none to equity, in the ethical ranking of income distributions.

But then, as Albert Camus has said, “If I attempt to solve a problem, at least I must not by that very solution conjure away one of the terms of the problem.”

Conflict is thus unavoidable if we insist on a view of egalitarianism such as has been presented in this article, and insist also on the ethical irreproachability of Paretianism. We shall see in a subsequent piece how this equity-efficiency conflict has tended to project egalitarianism in a poor light.

*S. Subramanian is an economist.*

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