On the “optimality” of inequality

Suppose there exists a one period society consisting of n identical individuals, each of whom is capable of producing output worth $1/n. Suppose that this society’s central planner imposes the following sharing rule: all n individuals in this society receive equal shares of total social output.

How much wealth will be created by this society? We already know that the maximum possible value for social wealth is $1. This outcome is only possible if we impose some highly restrictive behavioral assumptions. One possibility is that all n individuals’ objectives are to maximize total social wealth, and that no individual in this society suffers any “disutility” associated with putting forth the effort required in order to produce his or her maximum output. This outcome is also theoretically possible in a society consisting of self-interested individuals who suffer disutility from expending effort, so long as the central planner can not only perfectly and costlessly monitor effort, but also perfectly and costlessly enforce a “maximum effort” mandate for everyone in this society. In either case, total social wealth is $1, and everyone’s share is $1/n.

Now, suppose that there are costs to monitoring effort, and that self-interested individuals suffer disutility from expending effort. What happens then? The obvious answer is that shirking occurs, which reduces total social wealth as well as everyone’s per capita shares. Consider this problem from the perspective of one such individual who decides to shirk entirely; i.e., produce no output whatsoever. However, assume that the remaining n-1 individuals do not shirk. Then social wealth falls by $1/n, going from $1 to $(n – 1)/n; yet the shirker’s personal wealth falls by only $1/n2, going from $1/n to $(n – 1)/n2.  Thus the net benefit of shirking is substantial; the shirker only has to forgo the utility of $1/nin order to avoid the disutility (or, equivalently, utility “cost”) associated with expending effort. This is commonly referred to as the “free rider” problem. Therefore, it is not possible to have a free and prosperous society without some level of inequality in the distribution of income.

I became interested in this topic from listening to George Mason University economist Russ Roberts’ podcast featuring Cornell University economist Robert Frank. Although Professor Frank “…argues for a steeply rising tax rate on consumption that would reduce disparities in consumption” (cf. http://www.econtalk.org/archives/2010/11/robert_frank_on_1.html), he concedes that inequality is inevitable in a free society, based upon a similar thought experiment to the one provided here.