1. Introduction

While the Pareto efficiency criterion can be used to rule out inefficient allocations, it allows for very unequal distribution of welfare among the  individuals. To the extent that extreme forms of inequality are seen as undesirable, this suggest a need to complement the Pareto efficiency criterion with some welfare distribution criterion. We do this below by introducing fairness in the analysis.

Note that fairness holds under ordinal preferences and does not involve any intertemporal comparison of utility (as envy by the -th consumer is evaluated using her own preferences).

2. Motivations

We want to answer two related questions: First, when is fairness consistent with Pareto efficiency? And when it is not, what is the welfare cost associated with introducing fairness in economic analysis? To motivate the answers to these questions, this section considers a simple case of an economy composed of two individuals making production and consumption decisions. Having  units of total time available, the -th individual chooses leisure time  and consumption . The -th individual has preferences represented by the utility function , where  denotes work time, and , .

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We first consider situations where there are no transaction costs. Labor times  are used in the production of output  according to the production function ) satisfying . This allows for heterogeneity in labor/leisure preferences as well as heterogeneity in labor productivity across individuals. The output  is a consumer good redistributed to the two individuals to satisfy .

As discussed in the introduction, fairness is defined as the absence of envy: under fair allocations, no individual prefers someone else’s bundle. In the absence of transaction costs, this implies that  for . Note that  and , , corresponds to an location that is both feasible and fair (although not efficient). Thus, a fair allocation always exists.

In this section, we illustrate the linkages between fairness and efficiency in the context of a particular example. The example is as follows: Let , , and , where the parameters  reflect the demand for leisure and the parameters reflect the productivity of labor for the two individuals. Normalizing prices so that the price of  is equal to 1, the shadow cost of time is  for the -th individual. In the absence of transaction costs, with  being the marginal value product of labor for the -th individual, the efficient labor supply is  when , . In this context, an allocation that is both efficient and fair would satisfy: , or  for . We consider several scenarios, all illustrated in Figure 1.

Scenario S1:  and . Under scenario S1, the marginal value product of labor is at least as high as its shadow cost for both individuals. Then, efficient labor is  and  and efficient production is . And under zero transaction costs, an allocation that is both efficient and fair under Scenario S1 satisfies  for . This implies that . In this case, there exists a unique allocation that is both fair and efficient: it is the egalitarian allocation where each individual receives the same bundle:  and , . This allocation generates utilities , , which is a unique point on the Pareto utility frontier. Figure 1 shows that, under scenario S1, fairness and efficiency can be consistent with each other. Figure 1 illustrates two results. First, under zero transaction costs, fairness excludes many “unfair” allocations located on the Pareto utility frontier. Second, under Scenario S1, fairness can be introduced in the analysis without generating an efficiency loss.