01 Aug 2017 29 Aug 2017

**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).

**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 , .

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.

__Scenario S2__: and . Under scenario S2, the marginal value product of labor is at least as high as its shadow cost but only for individual 1. Then, efficient labor is and and efficient production is . Under Scenario S2, individual 2 has strong preferences for leisure, implying that it is efficient for this individual to work less (compared to Scenario S1), thus lowering the quantity produced of the good . In the absence of transaction costs, an allocation that is both efficient and fair under Scenario S2 would satisfy for , implying that . This inequality can be satisfied only if . This generates two possible sub-scenarios: Scenario S2a where , and Scenario S2b where .

Under __Scenario S2a__, means that individual 2 (who chooses not to work) has a cost of time at least as high as individual 1 (who chooses to work). In this case, there exist allocations that are both fair and efficient. They are the allocations satisfying , , and . Given , note that this last inequality restricts the difference in consumption to be positive and bounded between and . It means that, under scenario S2a, in the absence of transaction costs and under both fairness and efficiency, individual 1 (who chooses to work) must consume more good than individual 2 (who chooses not to work). This shows that, being both fair and efficient, the allocations just discussed would correspond to utilities located on the Pareto utility frontier (see Figure 1). Thus, under zero transaction costs, Scenario S2a is an example where fairness can be introduced in the analysis without generating an efficiency loss. In addition, such fair allocations are clearly not egalitarian (as each individual consumes different consumption bundles). Thus, in contrast with Scenario S1, Scenario S2a is an example where egalitarian allocations would not be efficient (see Figure 1).

Under __Scenario S2b__, means that individual 2 (who chooses not to work) has a lower cost of time than individual 1 (who chooses to work). In this case, in the absence of transaction costs, fairness and efficiency are inconsistent with each other.[1] Note that this does not imply that fairness cannot be achieved. As noted above, egalitarian allocations exist; and they are always fair. But under Scenario S2b in the absence of transaction costs, such fair allocations cannot be efficient (see Figure 1). In such situations, in contrast with scenarios S1 and S2a, introducing fairness in the analysis cannot be done without generating an efficiency loss. Note that Scenario Sc2 corresponds to , implying that the individual who has a strong preference for leisure (individual 2) is also less productive (with . This shows that the conflict between efficiency and fairness arises from the heterogeneity in both time preferences and productivity across individuals. Under such a scenario, there is necessarily a tradeoff between efficiency and fairness. This raises the question: What is the cost of this tradeoff? We answer this question in section 3 below.

The above analysis was presented in the absence of transaction costs. What happens to the relationships between efficiency and fairness in the presence of transaction costs? To address this issue, we consider the case where labor exchange is costly and distinguish between "actual labor" and "effective labor" , the difference being the amount of labor "wasted" in labor exchange (e.g., due to commuting time). In this context, let , with , measuring the proportion of labor time wasted by the -th individual, . Then, the production function for becomes , where . In this case, transaction costs in the labor market imply a decline in labor productivity, with . But note that, conditional on the new parameters , the analysis of efficiency presented above remains valid.

Next, consider envy under transaction costs. Let be the cost accrued when the -th individual considers switching place with the -th individual, . The switching costs include information cost, relocation cost and retraining cost. Consider the case where prices are normalized such that the price of is 1. Then, the absence of envy under transaction costs means that no individual feels that they would be better off switching with others, implying that for all . In the above example, this gives the fairness constraints: . First, when the transaction costs are small (with ), then the fairness constraints would generate the results discussed above: fair allocations can be efficient under some situations (i.e., under Scenarios S1 and S2a); but fair allocations are inconsistent with efficiency under other situations (i.e., under Scenario S2b). Second, note the above fairness constraints are always satisfied when the transaction costs are sufficiently large. Thus, when transaction costs are large, fairness and efficiency would always be consistent with each other. This result holds under any leisure preference parameters and any labor productivity parameters . This shows that the presence of transaction costs can affect fairness in two significant ways: 1/ it increases the odds that any allocation would be seen as fair; and 2/ it makes it easier for efficiency and fairness to be fully consistent with each other. These insights are formally developed in the next section.

**Efficiency and Fairness**

This section presents an analysis of fairness under general conditions. Going beyond the simple case discussed in section 2, we consider an economy involving consumer goods and individuals under general preferences and technology. Each individual makes consumption and time allocation decisions. Let . As in section 2, denote the leisure time of the -th individual by , with corresponding labor time being , . Denoting consumer goods by , the -th individual has preferences represented by the utility function , where , . For welfare analysis, we define a reference bundle , where satisfies .

We make the following assumptions:

__As1__: The utility function is continuous and quasi-concave on , .

__As2__: The function is strictly increasing in for all , .

Assumption As1 is standard in consumer theory. And given , assumption As2 implies that preferences are non-satiated in the consumer goods .

For the -th individual, define the benefit function

if (1)

otherwise,

where , . The benefit function in (1) measures the number of units of the reference bundle that the -th individual facing utility is willing to give up to obtain the bundle (Luenberger, 1992b). When the reference bundle is chosen such that one unit of is worth , the benefit function is a monetary welfare measure of willingness-to-pay. Some key properties of the benefit functions will prove useful. First, the benefit function satisfies the translation property for any and any finite . Second, under As1 (i.e., under the quasi-concavity of ), is concave in (Luenberger, 1992b, p. 466).

In a production economy, labor is used to produce consumer goods. Denote by the goods produced. Denote labor inputs by , where is the quantity of the -th individual's labor used in the production process, . We use the netput notation (where outputs are positive and labor inputs are negative). Denoting netputs by , the production technology is represented by the feasible set be , where means that outputs can be produced using labor inputs . Note that having allows for heterogeneity in labor productivity across individuals.

__As3__: The set is closed and convex and satisfies .

Next, letting , we define a __feasible allocation__ as an allocation which satisfies , where . Accordingly, feasibility requires four conditions: 1/ household decisions are feasible: ; 2/ labor use cannot exceed labor supply: ; 3/ netputs are feasible as outputs can be produced using labor inputs : ; and 4/ aggregate consumption cannot exceed production: . Let . From As3, note that implies that .

We start our analysis with the characterization of efficiency. We rely on the standard Pareto efficiency criterion: an allocation is __Pareto efficient__ if it is feasible and there is no other feasible allocation that can make one individual better off without making someone else worse off. Consider the following optimization problem

, (2)

where .

__Definition 1__: An allocation is __maximal__ if it solves equation (2). And it is __zero-maximal__ if, in addition to being maximal, it satisfies .

Zero-maximal allocations are of interest given their close relationship with Pareto optimality. Luenberger (1992a, p. 230-232; 1995, p. 189-192) obtained two key results: 1/ under assumption As2, a Pareto efficient allocation is zero-maximal; and 2/ under assumptions As1-As3, a zero-maximal allocation is Pareto efficient with respect to all feasible allocations where for all. This has several useful implications. First, except for boundary points, a Pareto efficient allocation is equivalent to a zero-maximal allocation. Second, Pareto efficiency corresponds to maximizing aggregate benefit and then redistributing the maximized aggregate benefit among the individuals. Third, the set of points traces out the Pareto utility frontier.

Next, we introduce fairness considerations in the analysis. As motivated in section 2, we explore when are fairness and efficiency consistent with each other. In a way consistent with previous research, we identify fairness as the absence of envy. But we conduct the analysis of fairness allowing for the presence of transaction costs.

__Definition 2__: An allocation ( is __fair__ if it satisfies and for each , where is the number of unit of the reference bundle that would be dissipated by the -th individual if the -th individual were to switch place with the -th individual, .

In our definition of fairness, the condition } involves the term (. When the reference bundle is defined such that one unit of is worth , it follows that can be interpreted as the cost involved when the -th individual considers switching place with the -th individual, . In this context, the parameters represent transactions costs involved in the fairness evaluation made by individual . Note that the absence of switching costs is obtained as a special case when for all . This the case commonly examined in previous research (e.g., Feldman and Kirman, 1974). But having individuals switching places is expected to be costly: it would typically involve information cost, relocation cost and retraining cost. This suggests that neglecting the transaction costs of switching places is rather unrealistic in fairness analysis.

In Definition 2, fairness corresponds to the absence of envy under transaction costs. Indeed, the absence of envy means that no individual prefers someone else's consumption bundle taking into consideration the switching costs . Letting }, define . From assumptions As2 and As3, for any , note that is a feasible as well as a fair allocation. This gives the following result.

__Lemma 1__: For any , a fair allocation always exists: .

An allocation is defined to be __fair-efficient__ if it is fair and there is no other fair allocation that can make one individual better off without making someone else worse off. Letting be the utility level obtained by the -th individual, note that the fairness conditions (stating the absence of envy) } can be alternatively written as or under assumption As2 as . Using the translation property of the benefit function, this last condition can be written as .

Next, we consider evaluating fair-efficient allocations by introducing the fairness constraints into the optimization problem (2). For given and this gives

. (3)

where is the maximized aggregate benefit under fairness. Equation (3) indicates that fairness matters when some of the fairness constraints in are binding in (3).

__Definition 3__: For a given , an allocation is __fair-maximal__ if it solves equation (3). And it is __fair-zero-maximal__ if, in addition to being fair-maximal, it satisfies .

As discussed in Chavas (2008), there are close relationships between fair-zero maximal allocations and fair-efficient allocations. We say that a consumption bundle is in the -interior of if it satisfies for some . In the context of equation (3), Chavas (2008, p. 261-262) obtained the following results:

__Lemma 2__: Given defined in equation (3),

a/ Under assumption As2, a fair-efficient allocation is fair-zero-maximal.

b/ Under assumptions As1-As3, a fair-zero-maximal allocation is fair-efficient with respect to all feasible allocations where is in the -interior of for all.

Lemma 2 has several useful implications. First, except for boundary points, a fair-efficient allocation is equivalent to a fair-zero-maximal allocation. Second, a fair-efficient allocation is obtained by maximizing aggregate benefit subject to fairness constraints, and then redistributing the maximized aggregate benefit among the individuals. Third, the set of points traces out the fair utility frontier (Chavas, 2008).

Lemma 2 indicates that fair-maximal allocations satisfying (3) can be used to evaluate the efficiency of fair allocations. Two key issues are: Is there and efficiency cost to fairness? And there is, what is it? As written, equation (3) does not identify explicitly the cost of fairness. Next, we proceed to identify the cost of fairness explicitly. For that purpose, considering two individuals , define the function

if (4)

otherwise.

The function in (4) provides a welfare measure of the cost of envy (net of transaction costs) by the -th individual toward the -th individual, . Indeed, when the benefit function is finite and , then the -th individual does not envy the -th individual. In this case, in (4) and the associated cost of envy is zero. Alternatively, when is positive and finite, then in (4). This corresponds to situations where the -th individual envies the -th individual and thus where the allocation is unfair. The welfare interpretation of in (4) as the cost of envy (net of transaction costs) follows from being the number of units of the reference bundle the -th individual's is willing to give up to obtain the bundle consumed by the -th individual.

This willingness-to-pay interpretation of in (4) suggests that there is an alternative characterization of the optimization problem (3). This is stated next (see the proof in the Appendix).

__Proposition 1__: For a given and , let be as defined in equation (3). The maximized aggregate benefit under fairness can be alternatively written as

. (5)

Proposition 1 states that the equivalence of equation (3) and equation (5) in identifying the maximized aggregate benefit under fairness. But, compared to (3), equation (5) offers a significant advantage: it provides as an explicit measure of the cost of envy. This property will prove to be particularly useful in our investigation (see below).

From Proposition 1, the results stated in Lemma 2 also apply to the fair-maximal allocations identified in (5). In particular, a fair-maximal allocation is obtained by maximizing aggregate benefit net of envy cost as given in (5). And, except for boundary points, a fair-efficient allocation is a fair-zero-maximal allocation where aggregate net benefit is maximized (yielding in (5)) and the associated aggregate benefit is entirely redistributed, with .

The term in equation (5) reflects the aggregate cost of unfair allocations. Since from (4), this aggregate cost is always non-negative. More specially, this aggregate cost is zero under fair allocations (where there is no envy). But would be strictly positive for allocations that are unfair. Given and comparing the maximization problems in (2) and (5) implies the following result.

__Lemma 3__: For a given and ,

. (6)

The inequality in (6) reflects that introducing fairness considerations tends to reduce maximized aggregate benefit. In this context, provides a general measure of the cost of envy. And in the case where the reference bundle is chosen such that one unit of is worth , is a monetary measure of the cost of envy. In addition, equation (6) implies that the Pareto utility frontier is at least as high as the fair utility frontier . In other words, introducing fairness in efficiency analysis tends to shift the utility frontier inward. The implications for pricing are explored next.

**Pricing under Fairness**

This section develops a dual approach to efficiency analysis under fairness with the objective of evaluating pricing under fair efficiency. Let be the set of continuous, non-decreasing functions satisfying the translation property for any and . Consider the generalized Lagrangian functional

, (7)

where is a penalty function associated with the constraint and and (5). As discussed below, the functions provide a measure of the value of , expressed in number of units of the reference bundle . For a given and , consider the optimization problem

}. (8)

Equation (8) evaluates the saddle-point of the generalized Lagrangian (7). There is close relationship between the fair-maximal allocation given in (5) and problem (8). In general, in (8) is an upper bound to in (5): (Bertsekas, 1995; Rubinov et al. 2002). The difference has been called the "duality gap". Much interest has focused on situations of "zero duality gap" where . Under a zero duality gap, the primal problem (5) and the dual problem (8) become equivalent.

__As4__: For a given and , the function is upper-semi continuous in at , where

Rubinov et al. (2002) and Chavas and Briec (2012) established that Assumption As4 is a necessary and sufficient condition for a zero-duality gap where . This generates the following result.

__Proposition 2:__ For a given and , and under assumption As4, a fair-maximal allocation satisfies

(9)

}(10)

.(11)

And a fair-zero-maximal allocation satisfies (9), (10 and (11), where satisfies

. (12)

Equations (9) and (10) follow directly from the maximization problems in (8) with respect to and respectively. And the optimization in equation (11) is obtained from the minimization problem with respect to in (8). Equation (11) generating the maximized fair benefit follows from a zero-duality gap under assumption As4. Finally, equation (12) is obtained from the definition of a fair-zero-maximal allocation (in Definition 3).

The functions provide a measure of the value of labor and of the consumer goods , expressed in number of units of the reference bundle . Of special interest is the case where is chosen such that one unit of is worth $1. This amounts to imposing a price normalization rule under which has a monetary interpretation. Then, the slope of the function provides useful information about pricing for . When is linear, the slopes of with respect to are constant and identify uniform pricing supporting a fair-efficient allocation. But as discussed below, our analysis allows for to be nonlinear, in which case the slopes of are not constant, corresponding to nonlinear pricing.

Interpreting as the value of the netputs , equation (9) states that, conditional on the pricing scheme , fair-maximality implies aggregate profit maximization. This is an important result. While profit maximization is a well-known implication of Pareto optimality (e.g., Debreu, 1959), equation (9) shows that introducing fairness considerations in the analysis does not affect the validity of the profit maximization motive. What is new here is that, under fairness, the function is not restricted to be linear, implying the fair-efficient pricing solution to (11) can involve nonlinear pricing. See below.

Similarly, taking as a measure of value, the function in equation (10) can be interpreted as measuring aggregate consumer expenditure under fairness. To see that, consider the case where , i.e. where fairness considerations do not play any role. Then, equation (10) would reduce to

}. (13a)

Chavas and Briec (2012, p. 685) showed that equation (13a) can be alternatively written as

, (13b)

showing that is the minimized aggregate expenditure under the pricing scheme and utilities . This has two implications. First, in equation (13) is the minimized aggregate expenditure under fairness, conditional on the pricing scheme and utilities . Second, given , comparing (10) and (13) gives the following result.

__Lemma 4__: For a given and , and conditional on the pricing scheme , we have

. (14)

Equation (14) states that aggregate expenditure is as least as large under fair-efficient allocations than under Pareto efficiency. In other words, introducing fairness tends to increase aggregate consumer expenditure.

The solution to equation (11) identifies an optimal pricing scheme that supports a fair-zero-maximal allocation. fair-efficient allocation. As noted above, the pricing scheme can be nonlinear.

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__Figure 1__: Utility Frontiers under Efficiency and Fairness in the Absence of Transaction Cost

__Scenario S1__

__Scenario S2a__

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