Nonlinear pricing under inequity aversion

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Nonlinear pricing under inequity aversionR Cavit Görkem Destan a, Murat Yılmaz b,∗ a b

Bonn Graduate School of Economics, University of Bonn, Germany Department of Economics, Bog˘ aziçi University, Turkey

a r t i c l e

i n f o

Article history: Received 16 November 2018 Revised 1 November 2019 Accepted 9 November 2019 Available online xxx JEL classification: D03 D42 L11 Keywords: Price discrimination Nonlinear pricing Fairness Inequity aversion

a b s t r a c t We consider a monopoly which practices nonlinear pricing, where buyers may have inequity-averse preferences. Each buyer has a valuation for the good, drawn from a distribution. The monopoly knows the distribution but not the realizations. We introduce the possibility that any buyer can be inequity-averse (fair types) or not (neutral types), in addition to different demand types. Fair types get a disutility from inequity captured through a utility function in the spirit of the one introduced by Fehr and Schmidt (1999). We characterize the optimal nonlinear pricing and show that the degree of suboptimality increases, and the monopoly profit decreases, as the degree of the inequity aversion increases. We also show that some or all of the neutral types with low demand are strictly better off, when there are inequity-averse types in the environment. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Pricing strategies of a monopoly facing different demand groups have been widely studied. Depending on how much a monopoly can tell different buyers apart in terms of their values of willingness to pay, the monopoly can adopt different degrees of price discrimination. When the monopoly knows that there are different groups of buyers with different demand structures, but is unable to tell which buyer is in which group, the monopoly can practice second degree price discrimination. This could be done through two-part tariffs, which involve a fixed fee and a unit price, or more generally through a nonlinear pricing scheme. There is a large number of real-life examples in which firms engage in price discrimination practices even when they do not directly observe which demand group a particular buyer belongs to. For instance, different memberships for gyms or clubs (some with a higher fee and better facility access, some with a lower fee and limited facility access), bulk discounts (for instance, in electricity consumption), tying and bundling (conditioning the sale of one good on the purchase of the other good from the same seller, for instance, tying printers and cartridges) and different pricing by airline companies (charging different prices for the same flight depending on the type of seat and/or service, also framed as quality discrimination). The purpose of these practices is to get different buyers to buy the good at prices closer to their own individual willingness to pay values, resulting in higher profit for the firm.

R We would like to thank Andrew F. Newman, Patrick Legros, Georg Kirchsteiger, Johannes Johnen, Øyvind Aas, Hamid Aghadadashli, Orhan Torul and Deniz Selman for their useful comments. We would also like to thank both the co-editors and the two anonymous referees for their suggestions and comments. ∗ Corresponding author. E-mail addresses: [email protected] (C.G. Destan), [email protected] (M. Yılmaz).

https://doi.org/10.1016/j.jebo.2019.11.013 0167-2681/© 2019 Elsevier B.V. All rights reserved.

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When a monopoly applies nonlinear pricing, it offers a set of pairs of total quantity and total payment, and lets each buyer choose whichever pair she wants. Each quantity-payment pair is targeted at a specific demand group consisting of buyers who find that pair optimal.1 In a simple setting with two types of buyers (low-demand and high-demand), the monopoly leaves no surplus to the low demand buyers, and leaves positive surplus to the high demand buyers, as information rents. Moreover, the low demand buyers ends up consuming a quantity that is socially suboptimal; more precisely, the marginal benefit from its consumption is higher than its marginal cost.2 One example of nonlinear pricing is that airlines offer their passengers a variety of seats and services: by paying more, one can get a first-class seat instead of an economy seat, or can choose to get priority boarding. DeCelles and Norton (2016) argue that the presence of a first-class cabin exposes economy passengers to inequality (compared with when there is no first-class cabin), and that “air rage”, a form of antisocial behavior by airplane passengers becoming abusive or unruly, will be almost four times more likely on flights with a first-class cabin relative to those without one.3 This observation indicates that it is plausible to consider possible fairness concerns of the buyers in the context of nonlinear pricing. In an environment with different types of buyers in which the firm cannot tell which buyer is of which type, buyers are likely to be allocated different quantity-payment pairs, resulting in different net surpluses. For instance, a low demand type may end up with no information rent (zero net surplus), whereas a high demand type may receive a positive information rent (positive net surplus). This allocation may make the low demand type (and/or the high demand type) unhappy because of the inequity in terms of different net surpluses. A buyer may easily be able to justify receiving a lower quantity (or quality) level when her corresponding total payment and/or demand level are also lower. However, the buyer may not be able to justify net surplus differences between hers and other demand type buyers, which may arise under a price discriminating monopoly. Thus, it is plausible to be concerned about buyers’ potential aversion to inequity which may result from the differences in the information rents/net surpluses each buyer gets. It is, therefore, important to consider the effect of buyers’ inequity aversion on the incentive scheme designed by the monopoly. Milgrom and Roberts (1992) pointed out that when providing incentives, preferences that take into account others’ well-being do matter: “A given level of pay may be viewed as good or bad, acceptable or unacceptable, depending on the compensation of others in the reference group, and as such may result in different behavior.” Also, Boyle (20 0 0) indicated that consumer reaction to price discrimination usually has a sense of unfairness: “...people often react to differential pricing for the same good with a sense of unfairness. No matter how many times they are lectured by the economists that it is actually to the benefit of all that producers be able to charge different prices to groups with different ability and willingness to pay, the popular reaction is normally ‘that’s not fair’.” There have been some influential studies, both theoretical and experimental, that explore preferences for fairness and inequity aversion.4 In their seminal paper, Fehr and Schmidt (1999) studied inequity-averse preferences and provided a simple model of inequity aversion together with a utility form that represents preferences for fairness. In this paper, we incorporate fairness concerns into a model of a nonlinear pricing monopoly, adopting a version of the utility form provided by Fehr and Schmidt (1999) to account for inequity aversion. We consider inequity aversion in the sense that if a buyer may derive disutility from the differences between her net surplus and those of other buyers. We consider two environments, one with discrete and one with a continuum of demand types in which a monopoly uses a nonlinear pricing scheme facing different types of buyers: different both in terms of their valuations for the good and their preferences for fairness. In the discrete case we have four types: neutral low, neutral high, fair low and fair high types, where a fair high type is inequityaverse and has a high demand parameter, a neutral low type is not inequity-averse and has a low demand parameter, etc. The monopoly knows the distribution of types but cannot tell which buyer is which type. We first solve for the optimal nonlinear pricing assuming all types are served, and show that the degree of suboptimality strictly increases as the degree of inequity aversion increases, where suboptimality of a quantity level is roughly the difference between its social marginal benefit and its social marginal cost.5 This is because the stronger the inequity aversion, the larger the neutral low type’s net surplus, which creates the need for stronger incentives for the high demand type. The monopoly provides these incentives by offering a smaller quantity for the low demand type in order to make that bundle less attractive for the high demand type. This lower quantity increases the difference between the marginal benefit and the marginal cost of the quantity targeted at the low demand type. Second, we show that, in the discrete case with four types, the neutral low type is strictly better off, and in the continuum case, a positive mass of neutral low types are strictly better off when there are inequity-averse types, relative to the case when there are no inequity-averse types in the environment. The intuition behind this result is that leaving zero net surplus for the neutral low type is not compatible with the participation constraint of the fair low type, forcing the monopoly to leave positive net surplus for the neutral low type. We also show that the monopoly profit decreases as the degree of inequity aversion increases in both the discrete and continuum cases. This is because inequity aversion makes it more costly to provide incentives to different types of buyers to self select, since the individual rationality

1

See Goldman et al. (1984), Maskin and Riley (1984) and Oren et al. (1984) for relatively early studies which have focused on this problem. Tirole (1988) has depicted these main findings. 3 For some media attention see https://www.economist.com/gulliver/2016/05/11/resentment- of- first- class- passengers- can- be- a- cause- of- air- rage and https://www.ft.com/content/8d915e44- 169f- 11e6- 9d98- 00386a18e39d. 4 Prasnikar and Roth (1992) provided experimental results that point towards fairness concerns in various game experiments, while Rabin (1993) studied the notion of fairness from a game theoretical perspective. 5 This is formally defined in Section 3.4. 2

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of the inequity-averse type is harder to satisfy. We also analyze various extensions and discuss some issues, including the case in which the monopoly is allowed to exclude some of the buyers from the market by not selling to them. An alternative individual rationality condition, as well as an alternative evaluation of the disutility from inequity, are considered. The scope of this paper falls into the intersection of the literature on nonlinear pricing by a monopoly and the literature on inequity aversion.6 In the nonlinear pricing literature, the optimal nonlinear pricing of monopolies involves marginal cost pricing for the last unit and higher marginal prices for all lower quantities (Mussa and Rosen, 1978).7 As we discussed above, the monopoly leaves some information rent for the high demand type and no surplus for the low demand type, and the quantity for the low demand type is socially suboptimal (Tirole, 1988).8 There is a growing number of studies on behavioral aspects and their implications in the nonlinear pricing context, such as biased beliefs about future preferences (Eliaz and Spiegler, 2008), time-inconsistency (DellaVigna and Malmendier, 2004), information ambiguity aversion (Zheng et al., 2015) and loss aversion (Carbajal and Ely, 2016; Hahn et al., 2018). In their seminal paper, Fehr and Schmidt (1999) provide a simple representation for inequity-averse preferences and solve the following puzzle: when players are given the opportunity to punish free riders, stable cooperation is maintained, although punishment is costly for those who punish. If some players care about equity, the puzzle is solved. Since their study, a large number of both theoretical and empirical studies emerged on other-regarding preferences, including inequity aversion and its implications.9 In the contracting context, there is experimental evidence which shows that when fairness concerns are present, principals prefer less complete contracts over more complete ones (Fehr and Schmidt, 20 0 0). In a moral hazard setting, there is experimental evidence showing that bonus contracts are chosen over incentive contracts implying that principals may have fairness concerns (Fehr et al., 2007).10 In a principal-agent setup in which the agent has inequity aversion, the optimal contract involves linear sharing rules (Englmaier and Wambach, 2010). There is experimental evidence for buyers setting prices sufficiently above the market clearing price and sellers responding with high quality levels in the presence of fairness concerns (Fehr et al., 1993). While there are many studies both on nonlinear pricing in various behavioral contexts and on inequity aversion in wellstudied economics problems, none has explored the implications of inequity aversion in the context of a nonlinear pricing monopoly, and our work fills this gap. To provide a better understanding of our contribution, we discuss two closely related papers: one from the nonlinear pricing literature focusing on loss-averse buyers, and the other from the fairness/reciprocity literature focusing on third degree price discrimination. Hahn et al. (2018), focusing on loss aversion in the context of nonlinear pricing, show that a monopoly finds it optimal to offer the same bundle (full pooling menu) if the likelihood of low demand buyer is sufficiently large and the loss aversion is in an intermediate range. Their study is the most closely related to ours, as the inequity-averse buyers in our model suffer from a disutility similar to the loss experienced by the loss-averse buyers in their model. However, there are crucial differences. The first difference is in terms of timing. In their model, the monopoly offers a menu of bundles, the buyers see the menu and form a reference point before they learn their demand type; then they learn their demand type, after which the choice of bundle is made. In addition to consumption utility, gain/loss is realized relative to the reference point. Overall, there are two choices for the buyer: the reference point (chosen ex-ante) and the consumption bundle (chosen ex-post). In our model, there is only one choice: the ex-post choice of the consumption bundle, made after demand type is realized. The gain/loss in Hahn et al. (2018) depends on the reference point chosen by the buyer, whereas the disutility from inequity in our model is based on the choices of other buyers. In their model, the two choices (reference point and consumption bundle) must be made in accordance with each other, requiring the use of the personal equilibrium concept. In contrast, we do not need to employ this concept, since given the menu, a buyer makes only one choice (the consumption bundle). Secondly, the heterogeneity in Hahn et al. (2018) is only through the demand types, as every buyer is loss-averse with the same loss aversion parameter. In contrast, our model allows for some buyers to be inequity-neutral and some inequityaverse. Thirdly, while they focus more on the pooling nature of the optimal menu and its conditions, we focus more on the effect of inequity aversion on the welfare of neutral buyers, monopoly profit, and social optimality. Finally, in their model loss aversion applies to consumption utility and payment disutility separately. In our model inequity aversion is calculated through the differences in the the sum of these two utilities, that is, consumption utility net of payment, or total net utility. However, they also show that if they introduce loss aversion through comparing total net utilities, the optimal menu is the same as in the standard model without loss aversion.11 This is not the case in our model: when inequity aversion is introduced through the differences in total net utilities, the optimal nonlinear pricing differs from the one in the standard model without inequity aversion. The reason for this nontrivial effect of inequity aversion in our model in contrast to theirs

6 Varian (1989) provides an extensive discussion of the different types of price discrimination, including nonlinear pricing, and their applications in the absence of any inequity aversion. 7 When buyer entry is introduced through learning own preference type after incurring privately known entry costs, the optimal nonlinear pricing contract changes significantly: distortion, exclusion and bunching are reduced (Ye and Zhang, 2017). 8 See Maskin and Riley (1984) and Wilson (1996) for a mechanism design approach for a monopoly’s nonlinear pricing problem. See Hendel et al. (2014) for a model of nonlinear pricing of storable goods, where they show that storability restricts monopolist’s ability to extract surplus. 9 Also, see Fehr and Schmidt (2006) for empirical foundations and theoretical approaches of other-regarding preferences. 10 However, trust contracts are not chosen as much as incentive contracts implying that the agents are not necessarily inequity-averse. 11 See Section S.5.3 in the Supplementary Material of their paper.

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is two-fold.12 First, when we set the outside option to include disutility from utility differences in case of not buying any of the bundles, then inequity aversion does not have an effect on the optimal nonlinear pricing.13 When the outside option entails no disutility from utility differences, then inequity aversion does have a nontrivial effect. Second, an inequity-averse buyer does not receive a “gain utility” when her total net utility is higher than others’ total net utilities, whereas a lossaverse buyer would receive a “gain utility” when her total net utility is higher than that of the reference point.14 In another related study, Englmaier et al. (2012), consider third degree price discrimination when consumers have reciprocity, and show that the difference in prices is smaller relative to the case with no reciprocity. Our study is similar to theirs in the sense that we both analyze the effects of inequity aversion on price discrimination. However, we focus on second degree price discrimination instead of third degree price discrimination. Also, the way fairness enters the buyer’s utility is different: they follow the reciprocity model provided by Falk and Fischbacher (2006), and we adopt the inequity aversion model provided by Fehr and Schmidt (1999). In our model, for a buyer inequity aversion emerges from the differences between her own net surplus and other buyers’ net surpluses, whereas in their model a buyer regards price discrimination as fair or unfair depending on the price she is charged compared to the prices other buyers are charged. Also, a buyer reciprocates toward the seller, by raising or lowering her quantity demanded, as a reaction to perceived fairness or unfairness. Although this reciprocity is not present in our model, a buyer’s participation constraint is affected by her inequity aversion, which in turn affects the quantities offered by the monopoly. Given the volume of experimental studies that provide evidence for fairness concerns of agents and inequity-averse preferences, it is important to analyze implications of inequity aversion in various economic models and environments. In the nonlinear pricing literature, the buyers are assumed to have only self-regarding preferences and no fairness concerns. In the light of the experimental evidence that suggests otherwise, it becomes crucial to adjust existing theoretical models to incorporate such preferences. This paper contributes to the literature in this sense, by studying the implications of inequity aversion in the context of a monopoly’s nonlinear pricing, where buyers may get disutility when their net surplus resulting from the nonlinear pricing adopted by the monopoly is smaller than the net surpluses of other buyers.15 We depict the model in Section 2. We consider a discrete type space and solve the optimal nonlinear pricing in Section 3. In Section 4, we solve the model with a continuum of types and characterize the optimal nonlinear pricing and its efficiency properties. In Section 5, we provide an analysis for the case in which the monopoly is allowed to exclude some buyers, and also discuss a few other issues. Section 6 concludes. 2. Model There is a monopoly which produces and sells a single product at a constant marginal cost c > 0 to a set of buyers. The monopoly engages in nonlinear pricing by offering a set of quantity-total payment bundles. The monopoly faces total demand from different types of buyers which the monopoly cannot tell apart.The buyers differ from one another in two aspects. (1) Each buyer has a demand parameter θ , which is distributed over the set  with a distribution function f(θ ). (2) Each buyer is either fair (has a utility function which depends on personal consumption as well as consumption of others) with probability γ or neutral (has a utility function which only depends on personal consumption) with probability (1 − γ ). Thus, the type space is given by  × {f, n}, where f denotes a fair type and n denotes a neutral type. The type of the buyer is private information, that is, only the buyer knows her type (θ , t), where θ ∈  and t ∈ {f, n}. The intrinsic utility of a neutral buyer with demand parameter θ , from a bundle b = (q, T ) is given by the function V (b, θ ) = θ u(q ) − T , where q is the quantity consumed, u(·) is a strictly concave and increasing function with u(0 ) = 0, and T is the payment for the quantity q, made by the buyer to the monopoly. The type space and the distribution function are common knowledge. The fair buyers have a special case of Fehr and Schmidt (1999) type utility functions. A fair type with demand parameter θ , who gets the bundle b = (q, T ) with q > 0, receives the following net utility:

  W (b, θ ) = V (b, θ ) − α γ max{0, V (b f (θˆ ), θˆ ) − V (b, θ )} f (θˆ )dθˆ    + (1 − γ ) max{0, V (bn (θˆ ), θˆ ) − V (b, θ )} f (θˆ )dθˆ 

where α > 0 is the parameter that captures fairness of the buyer, and b f (θˆ ) = (q f (θˆ ), T f (θˆ )) is the bundle (quantity and payment) for a fair type with demand parameter θˆ , and bn (θˆ ) = (qn (θˆ ), Tn (θˆ )) is the bundle (quantity and payment) for a 12 Heterogeneity in terms of inequity aversion in our model does not account for the nontrivial effect on the optimal nonlinear pricing, because we show that the ratio of inequity-averse buyers does not enter the optimal nonlinear pricing. Thus, our result does not change even if all buyers are inequity-averse. See the end of Section 3.4. 13 We discuss this in Section 5.3. 14 Under inequity aversion, there may even be a disutility when own total net utility is higher than others’ net utilities. This is discussed later in footnote17 . 15 The scope of our study is limited to this kind of other-regarding preferences and we do not focus on other-regarding preferences that represent trust, reciprocity, altruism, and spitefulness. For instance, the preferences depicted by Bolton and Ockenfels (20 0 0), where each agent compares own payoff to the average payoff in a reference group, and those depicted by Levine (1998) who studies altruistic and spiteful preferences, are beyond the scope of this study. Also see McCabe et al. (2003) who provide an experimental analysis regarding trust and reciprocity.

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neutral type with demand parameter θˆ .16 The larger the α , the larger the fairness concern of a fair type. When a fair type buyer’s benefit from consumption net of payment is less than the benefit from consumption net of payment of other buyers, her net utility decreases.17 That is, a buyer of type θˆ evaluates her net utility and also the net utility of the other types, V(b(θ ), θ ), in order to calculate the disutility she gets from inequity. This evaluation is feasible since she knows the type space and the distribution function, and from the nonlinear pricing offered she can calculate which quantity-total payment bundle is targeted at which type. There are two reasons why the disutility from inequity stems from the differences in net utilities/net surpluses. First, although buyers’ demand parameters are different, it is still the monopoly’s set of offers that determines the final utility levels. For instance, it is up to the monopoly to extract the entire surplus of the low type; the monopoly does not constrain itself by making sure that all or some types end up with the same net utility. Thus, we think that it is plausible to consider buyers who are averse to ending up with a lower surplus level than others, even if they know that others have different demand parameters. The second reason is that for a certain demand type buyer who chooses not to get the bundle another demand type buyer gets, there is no particular reason to get disutility from the fact that the bundles are different, since she could have chosen the other bundle if it provided her with a higher net surplus. Thus, calculating disutility in that way effectively assumes away any inequity aversion that buyers may have.18 The monopoly offers a set of bundles, {bi }i ∈  × {f,n} , where bi = (qi , Ti ) and each bi is targeted at the corresponding type i. Henceforth, we use {qf (θ ), Tf (θ ), qn (θ ), Tn (θ )}, instead of {(qi , Ti )}i ∈  × {f,n} , to ease notation. The monopoly solves the following profit maximizing problem to find the optimal nonlinear pricing, that is, the optimal menu of contracts {qf (θ ), Tf (θ ), qn (θ ), Tn (θ )}



max{qt (θ ),Tt (θ )}θ ∈,t∈{ f,n}



[γ [T f (θ ) − cq f (θ )] + (1 − γ )[Tn (θ ) − cqn (θ )]] f (θ )dθ

subject to individual rationality and incentive compatibility constraints of each type.19 First, we analyze the case in which there is a discrete set of types in Section 3 below, with the smallest possible type space. The purpose of Section 3 is to see the implications of fairness in the simplest possible setting. Then, we consider the continuum set of types in Section 4, where we generalize the insights from Section 3. 3. Discrete types There are two demand types, low demand types, with proportion λ, and high demand types, with proportion 1 − λ. From a bundle, b = (q, T ), low demand types receive a net intrinsic benefit of θ1 u(q ) − T and high demand types receive a net intrinsic benefit of θ2 u(q ) − T , where θ 2 > θ 1 > 0, and u(·) is a strictly increasing and strictly concave function, that is, u > 0 and u < 0, with u(0 ) = 0. Each demand type buyer has fairness concerns with probability γ and has no fairness concerns with probability 1 − γ . Thus, we have effectively four types of buyers: a proportion λγ are low demand types with fairness concerns (fair low type buyers), a proportion λ(1 − γ ) are low demand types with no fairness concern (neutral low type buyers), a proportion (1 − λ )γ are high demand types with fairness concern (fair high type buyers), and a proportion (1 − λ )(1 − γ ) are high demand types with no fairness concern (neutral high type buyers). A fair type buyer with θ i , i = 1, 2, receives a net benefit from a bundle b = (q, T ), given by

Wi (q, T ) = V − α [λγ max{V1 f − V, 0} + λ(1 − γ ) max{V1 − V, 0} + (1 − λ )γ max{V2 f − V, 0} + (1 − λ )(1 − γ ) max{V2 − V, 0}] We abuse notation by writing W(b, θ ), as W depends on the entire menu of bundles which determine other buyers’ surpluses. This version of fairness is one-sided, in the sense that an inequity-averse buyer cares about only when her payoff is less than other buyers’ payoffs, and she does not care about the case where her payoff is larger than other buyers’ payoff. Fehr and Schmidt (1999) provide a two sided fairness concern, where the agent cares about both types of inequities. With two sided fairness we would have 16

17

    W (b, θ ) = V (b, θ ) − α γ max{0, V (b f (θˆ ), θˆ ) − V (b, θ )} f (θˆ )dθˆ + (1 − γ ) max{0, V (bn (θˆ ), θˆ ) − V (b, θ )} f (θˆ )dθˆ       −β γ max{0, V (b, θ ) − V (b f (θˆ ), θˆ )} f (θˆ )dθˆ + (1 − γ ) max{0, V (b, θ ) − V (bn (θˆ ), θˆ )} f (θˆ )dθˆ 



where α captures fairness concern when the buyer receives less relative to others, and β captures fairness concern when the buyer receives more relative to others, where α ≥ β . In this paper, we assume α > β = 0 and focus on fairness concern stemming from when one receives less from others. When β > 0 is allowed, we believe that the distortion we get in our setting, with α > β = 0, would be even more salient. The monopoly would be forced to provide a more balanced set of utilities, relative to the case we study here. Thus, we believe that our results will not change qualitatively if we include a positive β , and it may not be worth including this type of aversion, as it makes the analysis even more cumbersome. 18 We could think of type θˆ calculating the utility she would get from the other types’ bundles, V (q(θ ), θˆ ), and then calculating the disutility from possible inequities, resulting from the difference V (q(θˆ ), θˆ ) − V (q(θ ), θˆ ). However, comparing own net surplus at different bundles and looking at disutility from those differences do not have any bite, because this type of inequity aversion would simplify to the disutility generated from incentive compatibility constraints that do not hold. This is never going to happen under optimal nonlinear pricing. We discuss this alternative way of calculating the disutility from inequity further in Section 5.4. 19 These constraints will be given in detail in Sections 3.4 and 4.1 below for two different type spaces.

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where V = θi u(q ) − T , V1 f = θ1 u(q1 f ) − T1 f , V1 = θ1 u(q1 ) − T1 , V2 f = θ2 u(q2 f ) − T2 f , and V2 = θ2 u(q2 ) − T2 .20 Vif , qif and Tif are for the fair type buyer with θ i demand parameter, and Vi , qi and Ti are for the neutral type buyer with θ i . Here, α > 0 summarizes the degree of fairness concern: the larger α , the larger the disutility from inequity to which a buyer is exposed. Monopoly chooses {(qi f , Ti f ), ((qi , Ti ))}i=1,2 to maximize its expected profit level:

max{(qi f ,Ti f ),((qi ,Ti ))}i=1,2

λγ [T1 f − cq1 f ] + λ(1 − γ )[T1 − cq1 ] + (1 − λ )γ [T2 f − cq2 f ] + (1 − λ )(1 − γ )[T2 − cq2 ]

subject to the individual rationality constraints and incentive compatibility constraints, to be specified below. We assume that the outside option, that is, not buying at all, or getting the bundle (q = 0, T = 0 ), is zero. There are two reasons for this. First, in the flights with first-class cabin example, passengers are more exposed to inequality when there is a first-class cabin on their flight. There is no particular reason for a passenger who takes the train or another flight with no first-class cabin to feel inequality due to the first-class cabin on the flight she did not take. Thus, the outside option would not account for the differences in utilities of those who actually take the flight that has two classes of seats. Second, consider a demand type with a very small demand parameter θ , who would not be willing to get any bundle offered even if the total price is very small. If not buying at all entails disutility emerging from those who are buying, then this buyer, who doesn’t want the good even at very low prices would feel disadvantaged, which would not be sensible. We, however, note that under this assumption (that not buying at all does not generate additional disutility), when a buyer buys an q =  amount at an arbitrarily small total price, where  is arbitrarily small, then the disutility from inequity is realized. This change gives rise to a discontinuity in the overall utility at the outside option level zero. Thus, although we provide reasons for this assumption, it is not entirely innocuous.21 Therefore, the individual rationality constraint for a fair low type buyer is given by IR1f : W1 (q1f , T1f ) ≥ 0, and for a fair high type buyer it is given by IR2f : W2 (q2f , T2f ) ≥ 0, where

W1 (q1 f , T1 f ) = V1 f − α [(1 − λ )γ max{V2 f − V1 f , 0} + λ(1 − γ ) max{V1 − V1 f , 0} + (1 − λ )(1 − γ ) max{V2 − V1 f , 0}] W2 (q2 f , T2 f ) = V2 f − α [λγ max{V1 f − V2 f , 0} + λ(1 − γ ) max{V1 − V2 f , 0} + (1 − λ )(1 − γ ) max{V2 − V2 f , 0}] And the individual rationality constraints for the two neutral types are simply IR1 : V1 ≥ 0 and IR2 : V2 ≥ 0. The monopoly offers a bundle and makes a take-it-or-leave-it offer to each type: bundle (q1f , T1f ) to the fair low type, bundle (q2f , T2f ) to the fair high type, bundle (q1 , T1 ) to the neutral low type, and bundle (q2 , T2 ) to the neutral high type buyers. 3.1. Observable types: first best When each of the four types is observable, there is no incentive compatibility constraint. Also, the monopoly offers bundles that leave no surplus/rent to any of the types, and all four individual rationality constraints bind. Thus, V1 = V2 = 0 and W1 = W2 = 0.22 Then, we have W1 (q1 f , T1 f ) = V1 f − α (1 − λ )γ max{V2 f − V1 f , 0} = 0, and W2 (q2 f , T2 f ) = V2 f − αλγ max{V1 f − V2 f , 0} = 0, which together imply V1 f = V2 f = 0.23 Therefore, we get Ti = θi qi for each i = 1, 1 f, 2, 2 f . Plugging these values into the profit function and taking the derivatives, we get θi u (qFi B ) = c for each i = 1, 1 f, 2, 2 f . These quantity levels are the first-best, efficient levels. 3.2. Observable Demand type, unobservable fairness type: second best I When the type regarding fairness concern is not observed, fair or neutral, but the demand type is observed, the monopoly offers two bundles, (q1 , T1 ) and (q1f , T1f ), targeted at the neutral low type and fair low type, respectively, and only these bundles are available for these two types. Likewise, the monopoly offers two bundles, (q2 , T2 ) and (q2f , T2f ), targeted at neutral high type and fair high type buyers, respectively, and only these bundles are available for these two types. The relevant incentive compatibility constraints are the ones that make sure the neutral low type does not get the bundle targeted at the fair low type, and vice versa, and the ones that make sure the neutral high type does not get the bundle targeted at the fair high type, and vice versa. Since the demand type is observed, there is no incentive problem regarding a high (low) demand type mimicking a low (high) demand type. The relevant incentive compatibility constraints clearly hold (with equality) at the first best levels, since each type receives zero rent at the first best levels and the demand types are observable. Thus, the first-best offers, (qFi B , TiF B ) with i = 1, 1 f , 2, 2 f , constitute the optimal set of offers. 3.3. Observable fairness type, unobservable demand type: second best II When the type regarding fairness concern is observed, fair or neutral, but the demand type is not, the monopoly offers two bundles, (q1f , T1f ) and (q2f , T2f ), targeted at the fair low type and fair high type, respectively, and only these bundles We drop the subscript n for the neutral type. So qi is the quantity for the neutral type with θ i and qif is the quantity for the fair type with θ i . We discuss nonzero outside option case, which would arise from the disutility from inequity when not buying any amount, in Section 5.3. 22 Note that leaving no surplus for the neutral type makes it easier to satisfy the fair type’s individual rationality constraint. 23 Otherwise, say if V2f > V1f , then W1 = V1 f − α (1 − λ )γ (V2 f − V1 f ) = 0 and W2 = V2 f = 0. Then, V1f < 0, which implies W1 < 0 contradicting W1 = 0. So, it cannot be V2f > V1f . The case V2f < V1f is ruled out similarly. Thus, we must have V2 f = V1 f , which in turn implies V2 f = V1 f = 0. 20

21

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are available for these two types. Likewise, the monopoly offers two bundles, (q1 , T1 ) and (q2 , T2 ), targeted at neutral low type and neutral high type buyers, respectively, and only these bundles are available for these two types. The individual rationality constraints are the ones described above: (IR1f ): W1 (q1f , T1f ) ≥ 0, (IR2f ): W2 (q2f , T2f ) ≥ 0, (IR1 ): V1 ≥ 0, and (IR2 ): V2 ≥ 0. The relevant incentive compatibility constraints are the ones that make sure the neutral low type does not get the bundle targeted at the neutral high type, and vice versa, and the ones that make sure the fair low type does not get the bundle targeted at the fair high type, and vice versa. Since the fairness type is observed, there is no incentive problem regarding a fair (neutral) type mimicking a neutral (fair) type. Thus, we only have the following four incentive constraints:

(IC12 ff ) : V1 f − α [(1 − λ )γ max{V2 f − V1 f , 0} + λ(1 − γ ) max{V1 − V1 f , 0} + (1 − λ )(1 − γ ) max{V2 − V1 f , 0}] ≥ V12 f − α [(1 − λ )γ max{V2 f − V12 f , 0} + λ(1 − γ ) max{V1 − V12 f , 0} + (1 − λ )(1 − γ ) max{V2 − V12 f , 0}]

(IC21 ff ) : V2 f − α [λγ max{V1 f − V2 f , 0} + λ(1 − γ ) max{V1 − V2 f , 0} + (1 − λ )(1 − γ ) max{V2 − V2 f , 0}] ≥ V21 f − α [λγ max{V1 f − V21 f , 0} + λ(1 − γ ) max{V1 − V21 f , 0} + (1 − λ )(1 − γ ) max{V2 − V21 f , 0}]

(IC12 ) : V1 ≥ V12 (IC21 ) : V2 ≥ V21 where V1 = θ1 u(q2 f ) − T2 f is the net benefit fair low type would get if she gets the bundle targeted at the fair high type, 2f

V2 = θ2 u(q1 f ) − T1 f is the net benefit fair high type would get if she gets the bundle targeted at the fair low type, V12 = θ1 u(q2 ) − T2 and V21 = θ2 u(q1 ) − T1 . 1f

Lemma 1. The profit maximizing {(qi , Ti )}i=1 f ,1,2 f ,2 is given by †

†

λθ1 c = c˜ λθ2 − (θ2 − θ1 ) λθ1 (1 − b) θ1 u (q†1 f ) = c = cˆ λθ2 (1 − b) − (θ2 − θ1 ) θ2 u (q†2 ) = θ2 u (q†2 f ) = c θ1 u (q†1 ) =

α (1−λ )γ † † † † α (1−λ ) where b is either Kγ or L, with K = 1+ α (1−λ ) and L = 1+α (1−λ )γ , and c˜ < cˆ. Moreover, q2 f = q2 > q1 > q1 f .

Proof. See Appendix B online.



Lemma 1 shows that the optimal quantity for the high type is equivalent to the socially optimal levels, and fair low type gets a quantity lower than the one neutral low type gets. When fairness type is observed, there is no concern for the neutral type to mimic the fair type and vice versa. Thus, the neutral type receives the same quantity as in the standard incomplete information case with no inequity aversion. However, the quantity (and the payment) for the fair low type is (are) lower than the one for the neutral low type. The reason is as follows. The neutral type receives the standard bundles where the neutral low type gets zero surplus (V1 = 0). The fair low type is exposed to inequity aversion only from the high types, and needs to be compensated in terms of a better V1f to meet her IR constraint. Also, since the fairness type is observed, the relevant incentive problem is across demand types given an aversion type, and not across aversion types. For the fair high type not to mimic the fair low type, a better bundle (higher V1f ) must be constructed carefully: if both quantity and payment are increased (and a higher V1f is achieved), then it is easier for the fair high type to mimic the fair low type since at lower q levels the extra marginal utility from increased quantity is much larger than the extra payment (through concavity of u(·)). Thus, both quantity and payment must instead be reduced (so that V1f is higher) in order to also make it undesirable for the high type. 3.4. Optimal nonlinear pricing with unobservable types: third best Before we describe the nonlinear pricing problem of the monopoly, we first invoke revelation and taxation principles in our setting. The revelation principle allows one to restrict attention to direct mechanisms where the seller asks the buyers to report their types, and the profit maximizing mechanism specifies quantity and total payment as functions of the reported types, q(·) and T(·), so that each type finds it optimal to report own type truthfully.24 The taxation principle, on the other hand, allows the seller to offer a menu of contracts, T(q), while each buyer, knowing her type, picks one of the the utility maximizing contracts. Here, T(q) is designed by the seller to maximize the expected profits. These two approaches are equivalent in our model with inequity aversion, as they are in the standard model with no inequity aversion.25 24 The revelation principle may fail when the principal (seller in this case) cannot commit and renegotiation is possible. It may also fail in some settings where the agents (the buyers) contract with several principals simultaneously. Neither of these is the case in our model: there is full commitment and there is only one seller. It may also fail when there is bounded rationality. In our model, however, it holds and we verify this in online Appendix A. 25 See online Appendix A for details of this equivalence.

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We take the taxation principle approach and let the monopoly offer a menu of contracts. The monopoly offers four bundles, (q1f , T1f ), (q1 , T1 ), (q2f , T2f ) and (q2 , T2 ), targeted at fair low type buyers, neutral low type buyers, fair high type buyers and neutral high type buyers, respectively. The monopoly solves the following profit maximizing problem to find the optimal nonlinear pricing, {(qi f , Ti f ), ((qi , Ti ))}i=1,2

max{(qi f ,Ti f ),((qi ,Ti ))}i=1,2 λγ [T1 f − cq1 f ] + λ(1 − γ )[T1 − cq1 ] + (1 − λ )γ [T2 f − cq2 f ] + (1 − λ )(1 − γ )[T2 − cq2 ] subject to individual rationality and incentive compatibility constraints of each type. Through the incentive compatibility constraints the monopoly makes sure that each type chooses the bundle targeted at her. The individual rationality constraints are the ones described in Section 3.1: (IR1f ) : W1 (q1f , T1f ) ≥ 0, (IR2f ) : W2 (q2f , T2f ) ≥ 0, (IR1 ) : V1 ≥ 0, and (IR2 ) : V2 ≥ 0. The incentive compatibility constraints for a fair low type buyer are as follows.

(IC11 f ) : V1 f − α [(1 − λ )γ max{V2 f − V1 f , 0} + λ(1 − γ ) max{V1 − V1 f , 0} + (1 − λ )(1 − γ ) max{V2 − V1 f , 0}] ≥ V1 − α [(1 − λ )γ max{V2 f − V1 , 0} + λ(1 − γ ) max{V1 − V1 , 0} + (1 − λ )(1 − γ ) max{V2 − V1 , 0}]

(

IC12 f

) : V1 f − α [(1 − λ )γ max{V2 f − V1 f , 0} + λ(1 − γ ) max{V1 − V1 f , 0} + (1 − λ )(1 − γ ) max{V2 − V1 f , 0}] ≥ V12 − α [(1 − λ )γ max{V2 f − V12 , 0} + λ(1 − γ ) max{V1 − V12 , 0} + (1 − λ )(1 − γ ) max{V2 − V12 , 0}]

(IC12 ff ) : V1 f − α [(1 − λ )γ max{V2 f − V1 f , 0} + λ(1 − γ ) max{V1 − V1 f , 0} + (1 − λ )(1 − γ ) max{V2 − V1 f , 0}] ≥ V12 f − α [(1 − λ )γ max{V2 f − V12 f , 0} + λ(1 − γ ) max{V1 − V12 f , 0} + (1 − λ )(1 − γ ) max{V2 − V12 f , 0}] where V1 = θ1 u(q1 ) − T1 is the net benefit the fair low type would get if she gets the bundle targeted at the neutral low type, V12 = θ1 u(q2 ) − T2 is the net benefit the fair low type would get if she gets the bundle targeted at the neutral high

type, and V1 = θ1 u(q2 f ) − T2 f is the net benefit the fair low type would get if she gets the bundle targeted at the fair high 2f

type.

IC11 f ,

2f

IC12 f and IC1 f make sure that the fair low type buyer prefers the bundle targeted at her over the bundles targeted

at the neutral low type, neutral high type and fair high type, respectively. The incentive compatibility constraints for a fair high type buyer are as follows.

(IC21 f ) : V2 f − α [λγ max{V1 f − V2 f , 0} + λ(1 − γ ) max{V1 − V2 f , 0} + (1 − λ )(1 − γ ) max{V2 − V2 f , 0}] ≥ V21 − α [λγ max{V1 f − V21 , 0} + λ(1 − γ ) max{V1 − V21 , 0} + (1 − λ )(1 − γ ) max{V2 − V21 , 0}]

(

IC22 f

) : V2 f − α [λγ max{V1 f − V2 f , 0} + λ(1 − γ ) max{V1 − V2 f , 0} + (1 − λ )(1 − γ ) max{V2 − V2 f , 0}] ≥ V2 − α [λγ max{V1 f − V2 , 0} + λ(1 − γ ) max{V1 − V2 , 0} + (1 − λ )(1 − γ ) max{V2 − V2 , 0}]

(

IC21 ff

) : V2 f − α [λγ max{V1 f − V2 f , 0} + λ(1 − γ ) max{V1 − V2 f , 0} + (1 − λ )(1 − γ ) max{V2 − V2 f , 0}] ≥ V21 f − α [λγ max{V1 f − V21 f , 0} + λ(1 − γ ) max{V1 − V21 f , 0} + (1 − λ )(1 − γ ) max{V2 − V21 f , 0}]

where V2 = θ2 u(q2 ) − T2 is the net benefit the fair high type would get if she gets the bundle targeted at the neutral high type, V21 = θ2 u(q1 ) − T1 is the net benefit the fair high type would get if she gets the bundle targeted at the neutral low

type, and V2 = θ2 u(q1 f ) − T1 f is the net benefit the fair high type would get if she gets the bundle targeted at the fair low 1f

1f

type. IC21 f , IC22 f and IC2 f make sure that the fair high type buyer prefers the bundle targeted at her over the bundles targeted at the neutral low type, neutral high type and fair low type, respectively. 1f 2f The incentive compatibility constraints for the neutral low type are as follows. (IC12 ): V1 ≥ V12 , (IC1 ): V1 ≥ V1f and (IC1 ): 2f

1f

2f

V1 ≥ V1 . IC12 , IC1 and IC1 ensure sure that the neutral low type buyer prefers the bundle targeted at her over the bundles targeted at the neutral high type, fair low type and fair high type, respectively. 1f 1f The incentive compatibility constraints for the neutral high type are as follows. (IC21 ): V2 ≥ V21 , (IC2 ): V2 ≥ V2 and 2f

1f

2f

Assumption 1. θ1 > (1 −

1+α (1−λ )

(IC2 ): V2 ≥ V2f . IC21 , IC2 and IC2 ensure sure that the neutral high type buyer prefers the bundle targeted at her over the bundles targeted at the neutral low type, fair low type and fair high type, respectively. λ

) θ2 .

Assumption 1 is a necessary condition when the monopoly serves all types. We focus on the case where the monopoly serves all types, characterize the optimal nonlinear pricing and show a result regarding the efficiency of this optimal nonlinear pricing.26 First, we identify the binding and non-binding constraints through Lemma 2 and Lemma 3, and then reduce the monopoly’s problem.

26 When Assumption 1 fails, some exclusion may be optimal. In Section 5.1, we discuss the case where the monopoly excludes some of the types. We provide sufficient conditions for exclusion of certain type(s) to be optimal.

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1f

Lemma 2. (i) V1 = V1 f and both IC1 and IC11 f bind. (ii) V2 = V2 f and both

2f IC2

and IC22 f bind.

2f

(iii) IC12 f and IC1 f are redundant. Proof. See online Appendix B.



Since the demand parameters are the same for the fair low type and neutral low type, if the utility of the bundle is different for the two, the one with the lower utility will switch to the other bundle. Thus, we need V1 = V1 f . A similar argument for the high types implies V2 = V2 f as well. Also, if the neutral low type does not deviate to the bundle of the neutral high type (V1 ≥ V12 ), then for the fair low type, deviation to the high type’s bundle brings more disutility, because now, the difference V2 − V12 is more than the difference V2 − V1 . Thus, neutral low type’s incentive compatibility implies that of the fair low type. At this point, we assume V2 ≥ V1 , which will be verified at the end of the proof of Proposition 1. Lemma 3. When V2 ≥ V1 , we have (i) IR1 , IR2 and IR2f hold with strict inequality. 1f

(ii) IC21 f and IC2 f are redundant. Proof. See online Appendix B.



The fact that IR2 and IR2f are not binding is not surprising as these are individual constraints for the high types. Also, since the net benefits for the two high types are the same, if one of the high types does not deviate to a low type’s bundle, 1f 1f the other high types also does not deviate to a low type’s bundle: IC21 f and IC2 f are implied by IC21 and IC2 . The fact that IR1 is not binding means that the neutral low type gets some positive rent, unlike the standard model. We will discuss this in more detail below in Proposition 1 and what follows that. Now, we derive a reduced version of the monopoly’s problem. Using V1 f = V1 and V2 f = V2 , IR2f simplifies to V2 − α [λγ + λ(1 − γ )] max{V1 − V2 , 0} ) ≥ 0. Then, with V2 ≥ V1 , IR2f becomes V2 ≥ 0 which does not bind since V2 ≥ V1 > 0 (V1 > 0 since IR1 does not bind). And, IR1f becomes V1 − α (1 − λ )(V2 − V1 ) ≥ 0, which can be written as [1 + α (1 − λ )]V1 ≥ α (1 − λ )V2 , or

V1 ≥

α (1 − λ ) V2 1 + α (1 − λ )

α (1−λ ) Denoting K = 1+ α (1−λ ) , we get V1 ≥ KV2 , which is equivalent to IR1f . Thus, the problem is reduced to

max{(qi ,Ti )}i=1 f ,2 f ,1,2 λγ [T1 f − cq1 f ] + λ(1 − γ )[T1 − cq1 ] + (1 − λ )γ [T2 f − cq2 f ] + (1 − λ )(1 − γ )[T2 − cq2 ] sub ject to V1 = V1 f ≥ KV2 = KV2 f V1 ≥ V12 V1 ≥ V12 f V2 ≥ V21 V2 ≥ V21 f V2 = V2 f implies θ2 u(q2 ) − T2 = θ2 u(q2 f ) − T2 f , that is, θ2 [u(q2 ) − u(q2 f )] = T2 − T2 f . If q2 > q2f , then θ1 [u(q2 ) − u(q2 f )] < T2 − T2 f , that is, θ1 u(q2 ) − T2 < θ1 u(q2 f ) − T2 f . Then, we have θ1 u(q1 ) − T1 ≥ θ1 u(q2 f ) − T2 f > θ1 u(q2 ) − T2 = V12 . Thus, when 2f

2f

q2 > q2f , IC12 is redundant. Likewise, if q2 < q2f , IC1 is redundant. If q2 = q2 f , then both IC12 and IC1 are the same. IC12

2f IC1

We ignore both and for now, and will characterize the optimal nonlinear pricing in the reduced problem without these two constraints. Then, after characterizing the optimal nonlinear pricing, we will show that both ICs will hold. Therefore, we focus on the reduced problem below.

max{(qi ,Ti )}i=1 f ,2 f ,1,2 λγ [T1 f − cq1 f ] + λ(1 − γ )[T1 − cq1 ] + (1 − λ )γ [T2 f − cq2 f ] + (1 − λ )(1 − γ )[T2 − cq2 ] sub ject to V1 = V1 f ≥ KV2 = KV2 f V2 ≥ V21 V2 ≥ V21 f . Proposition 1. Under Assumption 1, the profit maximizing {(q∗i , Ti∗ )}i=1 f ,1,2 f ,2 is given by

θ1 u (q∗1 ) = θ1 u (q∗1 f ) = c > c.

T1∗ = T1∗f =

θ2 u (q∗2 ) = θ2 u (q∗2 f ) = c

T2∗ = T2∗f

θ1 − K θ2

u(q∗1 ) 1−K = θ2 (u(q∗2 ) − u(q∗1 )) + T1∗

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λ(1−K )θ1 α (1−λ )  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ where K = 1+ α (1−λ ) ∈ (0, 1 ) and c = λ(1−K )θ2 −(θ2 −θ1 ) c. Moreover, q2 f = q2 > q1 = q1 f and T2 f = T2 > T1 = T1 f . The fair low 27 type buyers get zero surplus, while neutral low type buyers and high type buyers get positive surplus.

Proof. See online Appendix B.



In the standard model, the high type receives some information rent and the monopoly makes sure that the high type does not deviate to the low type’s bundle, and the low type is left with zero surplus. When there is inequity aversion, however, it is harder to satisfy the participation constraint of the fair low type buyer, since she gets additional disutility when other buyers receive a higher utility from the bundle they receive. Thus, the monopoly has motivation to provide a relatively more balanced set of utilities, and at the same time it needs to provide incentives for each type not to deviate, especially for the high type. If the bundles for the fair low type and the neutral low type are different, yielding different utilities, then the one with the low utility will deviate to the other bundle. Same is true for the fair high and neutral high types.28 Thus, it is not possible to have four different bundles and separate all the types. Instead, the monopoly chooses to give the fair and neutral low types the same bundle, different than the bundle for the high types, which is same across these two high types. This makes the neutral low type strictly better off relative to the case with no inequity aversion; the reason is that the existence of fair low type buyers forces the monopoly to leave some information rents to the neutral low type as well, in the sense that the neutral low type is not the “worst type” anymore. Now, the monopoly is also forced to change the total payment of the high type to induce her not to deviate. However, there are two forces affecting her payment, an upward pressure from the lower quantity of the low type and a downward pressure from the lower payment of the low type. The balance of this tradeoff depends on the model parameters and the degree of the concavity of the utility function u(·). We provide a resolution to this tension in Corollary 4 below, where we show that the neutral high type is adversely affected from inequity aversion of the fair types, under some mild conditions. Next, we provide a result that shows how suboptimality is affected by the degree of inequity aversion. Before we provide this result below in Proposition 2, we first explain what we mean by suboptimality in this context. In the nonlinear pricing context, a quantity level is socially optimal if its social marginal benefit is equal to its social marginal cost and socially suboptimal otherwise. For a neutral type i, if ∂∂qV = θi u (qi ) = kc with k > 0, then qi is socially optimal if k = 1 and suboptimal i

if k = 1. For a fair type buyer, we need to take into account her disutility from other buyers’ higher surpluses as well. The net utility of a fair type buyer is summarized by W instead of V. Thus, if ∂∂ W = kc with k > 0, then qf is socially optimal if q f

k = 1 and suboptimal if k = 1. Finally, we define |k − 1| to be the degree of suboptimality, that is, the distance between k and the socially optimal case, which is k = 1. Proposition 2. The degree of suboptimality increases as the degree of inequity aversion, α , increases. Also, the quantity level for the low type (both fair and neutral) is smaller than the socially optimal level and gets smaller as the degree of inequity aversion, α , increases. Proof. We know by Proposition 1, the quantity for the high type is already socially optimal, as its social marginal benefit equals to the marginal cost: θ2 u (q∗2 ) = c.29 Proposition 1 also shows that the suboptimality involved in the quantity for the neutral low type, q∗1 , is summarized in the difference between c and c , where c > c with c = kc, that is, k > 1. Note that λ(1−K )θ α (1−λ ) k = λ(1−K )θ −(θ1 −θ ) , where K = 1+ α (1−λ ) ∈ (0, 1 ). We need to show that k increases as α increases. 2 2 1

 dK λ ( 1 − K ) θ1 c λ ( 1 − K ) θ2 − ( θ2 − θ1 ) d α   (1 − λ )(1 + α (1 − λ )) − (1 − λ )α (1 − λ ) 1−K d = λθ1 c dK λ(1 − K )θ2 − (θ2 − θ1 ) (1 + α (1 − λ ))2 −λθ2 (1 − K ) + (θ2 − θ1 ) + λθ2 (1 − K ) 1−λ = λθ1 c (λ(1 − K )θ2 − (θ2 − θ1 ))2 (1 + α (1 − λ ))2 θ2 − θ1 1−λ = λθ1 c >0 (λ(1 − K )θ2 − (θ2 − θ1 ))2 (1 + α (1 − λ ))2

dk d = dα dK



since 0 < λ < 1, θ 2 > θ 1 > 0 and c > 0. Thus, as α increases, the suboptimality of q1 also increases. 27

1 c, which is equivalent to the condition in the standard case: θ1 u (q∗1 ) = Note that when α = 0 (thus, K = 0), we have θ1 u (q∗1 ) = λθ −λθ 2 ( θ2 −θ1 )

c θ −θ 1− 1−λλ 2θ 1 1

.

See Tirole (1988), page 154 for this condition. Also, note that Assumption 1 implies λθ2 > θ2 − θ1 , which ensures that the denominator is positive, assuming all types are served when there is no inequity aversion. 28 Of course, the monopoly can offer two different bundles to the fair low and neutral low types, both achieving the same utility, one with higher quantity and higher payment levels than the other. But then the incentive compatibility for, say, the neutral high type must generate indifference between the bundle targeted at her and the bundle targeted at the low type, the one with the higher quantity and payment levels. But this makes it harder to satisfy the incentive constraint for this high type, as opposed to the case where the two bundles for the fair low and neutral low types are the same. 29 We have already taken into account the extra social cost that arises in the form of disutility the fair type gets when measuring the suboptimality involved in the quantity for the fair type.

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Now, we turn to the suboptimality involved in, q∗1 f , the quantity for the fair low type buyer. Since the bundles for the fair low type and neutral low type buyers are the same, we have V1 f = V1 . This implies that W = V1 f − α (1 − λ )(V2 − V1 f ) = [1 +

α (1 − λ )]V1 f − α (1 − λ )V2 . For the fair low type, the marginal benefit of q∗1 f is then given by ∂∂qW = [1 + α (1 − λ )]θ1 u (q∗1 f ). 1f From Proposition 1, we know θ1 u (q∗1 f ) = kc. Thus, the marginal benefit of q∗1 f is given by [1 + α (1 − λ )]kc. We just showed that k increases in α . Clearly, 1 + α (1 − λ ) also increases in α . Thus, as α increases, [1 + α (1 − λ )]k also increases, hence the suboptimality increases.30 Overall, since both suboptimalities are increasing in α , the result follows. From Proposition 1, we have θ1 u (q∗1 ) = θ1 u (q∗1 f ) = c > c, where c = θ1 u (qF1B ) = θ1 u (qF1Bf ). By concavity of u, we have q∗1 f < qF1Bf and q∗1 < qF1B , that is, the quantity level for both low types is smaller than the socially optimal level. We know that as α increases k also increases. Thus, with a higher α , both q∗1 f and q∗1 decrease since u is concave. Hence, the quantity level for both low types gets smaller as the degree of inequity aversion, α increases. 

The optimal quantity level for the high type buyer (q∗2 ) is socially optimal, that is, marginal utility of consumption is the same as its marginal cost. However, the quantity level for the fair low type buyer (q∗1 f ) and the one for the neutral low type buyer (q∗1 ) are socially suboptimal. This result is parallel to the standard case, where each type is neutral and the low type gets the quantity level which is socially suboptimal. However, in the standard case, it also happens that the type of buyer who gets a socially suboptimal quantity level is left with zero surplus, which is not the case in the current model: The neutral low type buyer gets a socially suboptimal quantity level, yet she ends up with some positive surplus since her individual rationality holds with strict inequality. Proposition 2 implies that the stronger the fairness concern, the larger the social inefficiency of the optimal nonlinear pricing.31 The reason for this result is parallel to our intuition for Proposition 1. When there is inequity aversion, the neutral low type is no longer the “worst type” from the monopoly’s perspective. Once the monopoly leaves no surplus for the fair f low type (through binding IR1 ), then the neutral low type receives some positive surplus as some information rent. This positive surplus is more pronounced as the degree of inequity aversion increases since now monopoly has to leave a larger consumption benefit (net of payment) to the fair low type to satisfy her individual rationality, which in turn increases the neutral low type’s surplus. This increase in surplus for the neutral low type causes her indifference curve to shift right (on a payment-quantity space), and thus it puts an extra pressure on the incentives to be provided for the high type. In order to 1f keep satisfying the high type’s incentive compatibility (IC21 and IC2 ), that is, for her not to deviate, the quantity for the low   type must decrease (q1 to q1 ), which can be seen from the partial analysis depicted in Fig. 1 below.32 This further increases the social suboptimality, as the quantity level without inequity aversion is already below the socially optimal level. We also observe that when inequity aversion increases, the difference between the quantities for the low type and the one for the high type increases, which is a direct corollary to Proposition 2. Corollary 1. As the degree of inequity aversion, α increases, the variations in optimal quantities, q∗2 − q∗1 and q∗2 f − q∗1 f , both increase. Proof. From the proof of Proposition 2, we know that as α increases both q∗1 f = q∗1 decrease, whereas q∗2 f = q∗2 both stay the same. Thus, the result immediately follows.  The intuition for the above result is that when inequity aversion is stronger the fair low type is provided with a higher V to compensate for the loss due to inequity aversion (individual rationality). This in turn provides the neutral low type with a higher net surplus, which makes it easier for the neutral high type to mimic the neutral low type. To prevent this from happening the monopoly increases the quantity difference between the two. The effects of private information and inequity aversion disentangled: At this point, we’d like to highlight the role of private information regarding the fairness type and separate it from the effect of existence of fairness types. To do this we compare the suboptimality involved in the third best with the one in second best II and second best I. Let b denote an element in {K, L, Kγ }. Note that K > L > Kγ .33 Then, we have

λθ1 (1 − K ) λθ1 (1 − L ) λθ1 (1 − K γ ) > > λθ2 (1 − K ) − (θ2 − θ1 ) λθ2 (1 − L ) − (θ2 − θ1 ) λθ2 (1 − K γ ) − (θ2 − θ1 ) λθ (1−b) λθ1 (1−K ) which follows from the fact that λθ (1−b1 )−(θ −θ ) is increasing in b. Thus, we have c > cˆ > c˜, where c = λθ (1−K )−(θ2 −θ1 ) c, 2 2 1 2 λθ1 (1−L ) λθ1 (1−K γ ) λθ1 and cˆ is either λθ (1−L )−(θ −θ ) c or λθ (1−K γ )−(θ −θ ) c, and c˜ = λθ −(θ −θ ) c. 2

2

1

2

2

1

2

2

1

Note that [1 + α (1 − λ )]k > 1. We will show a parallel and more general result in the continuum case with no exclusion possibility, in Section 4.1. 32 The indifference curve of the neutral high type does not shift as α changes. Although the indifference curve of the fair high type may shift as α changes, the fair high type’s bundle and the neutral high type’s bundle are the same at the optimal menu of bundles. This implies that the fair high type does not experience inequity at the optimal menu. Thus, looking at the indifference curve for the neutral high type suffices for the purpose of explaining the distortion created by the positive net surplus of neutral low type. 33 Recall, K = 1+αα(1(−1−λλ) ) and L = 1+αα(1(−1−λλ)γ)γ . Since 1+αα(1(−1−λλ)b)b is increasing in b and γ < 1, we get K > L. Also, L > Kγ immediately follows. 30

31

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Fig. 1. Distortion created by positive net surplus of neutral θ 1 type.

Fig. 2. Effects of private information and inequity aversion disentangled.

Recall that the suboptimality reflected in cˆ > c results in the second best where the demand type is not observed but the fairness type is observed.34 Thus, when we move to the third best, and have c , this change on the fair type’s quantity is due to the fairness type becoming incomplete information, the effect of which is reflected in cˆ < c . However, the fair low type † † † † receives a quantity q1 f such that θ1 u (q1 f ) = cˆ and the neutral low type receives a quantity q1 such that θ1 u (q1 ) = c˜, where

c˜ < cˆ. This implies that the effect of inequity aversion (between a neutral and a fair type when inequity aversion is complete information) is reflected in the inequality c˜ < cˆ. Finally, the effect of inequity aversion becoming private information on the neutral type is reflected in c˜ < c . This is summarized in Fig. 2 below. When the fairness type is observed and the demand type is not, the optimal nonlinear pricing for the neutral types is independent of the offers targeted at the fair types (as shown in the proof of Lemma 1), and then the neutral low type ends up with zero net surplus: V1 = 0. Thus, our result that the neutral low type is better off (V1 > 0) when fair types are introduced, is due to the private information on fairness types and to the fact that the fair low type must receive a positive

34

See Lemma 1.

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13

value, V1f , for her individual rationality to hold. This result is not surprising, because once the fairness type is observed, the monopoly can offer different quantities to fair low and neutral low types and can leave no surplus for the neutral low type. Thus, the neutral low type receives some information rent, when fairness type is not observed. However, we show below that as the degree of inequity aversion increases, the neutral low type’s net surplus also increases for a range of α values, under constant absolute risk aversion, extending our result for the neutral low type’s welfare to account for changes in the degree of inequity aversion. Corollaries 2 and 4 below consider the effect of the existence of inequity aversion on the neutral low type and neutral high type buyers, respectively. Corollary 3 provides the extended result we just mentioned: neutral low type is better off when the inequity aversion is stronger. Corollary 2. Under inequity aversion, the neutral low type buyer receives a higher net surplus and a bundle with lower quantity and lower total payment, relative to the case with no inequity aversion. Proof. The neutral low type’s higher net surplus relative to the case with no inequity aversion is directly implied by Proposition 1. For the latter part, we have c > c˜, that is, θ1 u (q∗1 ) > θ1 u (q1 ), where q1 is the optimal quantity for the low type in the case with no inequity aversion (see Lemma 1 for c˜). Concavity of u immediately implies that q∗1 < q1 . Since with q∗1 the neutral low type’s payoff is higher than with q1 , it follows that the payment under inequity aversion must be smaller, T1∗ < T1 .  The reason behind Corollary 2 is the following. We know that the distortion on the quantity for the low demand buyer is larger, through a smaller quantity. Also the neutral low type ends up with positive surplus when there are inequity-averse buyers, whereas she receives zero net surplus when there are no inequity-averse buyers. Thus, the total payment for this buyer should also be smaller, relative to the bundle she would get when there is no inequity-averse buyers. We now show that the surplus of the neutral low type buyer also increases as the degree of aversion increases. Corollary 3. The neutral low type’s net surplus increases as the degree of inequity aversion, α , increases, for a range of α values, under constant absolute risk aversion. Proof. The net surplus of the neutral low type is given by V1 = θ1 u(q1 ) − T1 where T1 =

V1 = θ1 u(q1 ) −

θ1 − K θ2 1−K

u ( q 1 ) = ( θ2 − θ1 )

1−K

u(q1 ). Thus,

K u(q1 ) = (θ2 − θ1 )(1 − λ )α u(q1 ) 1−K

The change in V1 with respect to α is given by



θ1 −K θ2

dV1 ∂ q1 = (θ2 − θ1 )(1 − λ ) u(q1 ) + α u (q1 ) dα ∂α



Since both (θ2 − θ1 ) and (1 − λ ) are positive, the sign of

dV1

is determined by the sign of the term in the brackets: dα ∂q ∂q )c [u(q1 ) + α u (q1 ) ∂α1 ]. Now we show that this term has a positive sign. First to get ∂α1 , we have u (q1 ) = λ(1−Kλ)(θ1−K = 2 − (θ2 −θ1 ) λc λc = . Taking the derivative of both sides with respect to α , we get: ( θ −θ ) λθ2 −(θ2 −θ1 )(1+α (1−λ )) λθ2 − 21−K1

u (q1 )

λc ∂ q1 ∂ λc(θ2 − θ1 )(1 − λ ) = = ∂α ∂α λθ2 − (θ2 − θ1 )(1 + α (1 − λ )) [λθ2 − (θ2 − θ1 )(1 + α (1 − λ ))]2

which can be written as

∂ q1 R =  ∂α u ( q1 ) λc(θ −θ )(1−λ )

where R = [λθ −(θ −2θ )(11+α (1−λ ))]2 . Then, we have 2 2 1

R du(q1 ) ∂ q1 u ( q1 ) = u ( q1 ) =  R=− dα ∂α u ( q1 ) r where r is the coefficient of constant absolute risk aversion. Now, the second derivative with respect to α is

d 2 u ( q1 ) = dα 2

∂  −R  ∂α r

∂ R > 0 by Assumption 1, we have ∂ ( −R ) < 0, that is, u(q (α )) is strictly concave in α (also decreasing in α ). Now Since ∂α 1 ∂α r consider the quantity when α is doubled: q1 (2α ). We have u(q1 (α )) + (2α − α )u (q1 (α )) > u(q1 (2α )) > 0 by concavity of ∂q u(·) in α .35 Now we have u(q1 (α )) + α u (q1 (α )) > 0, which is the term in the brackets above, u(q1 ) + α u (q1 ) ∂α1 and it is

positive. Thus, we get 35

dV1 dα

> 0, for all α < αmax 2 , which finishes the proof.



Here we assume that 2α < α max where α max is the maximum α that satisfies Assumption 1, which gives q1 (2α ) > 0, implying u(q1 (2α )) > 0.

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The intuition for Corollary 3 above is that as aversion gets stronger, that is, as α increases, the fair low type gets a bigger disutility from inequity. To sustain her individual rationality constraint, the monopoly needs to leave her a bigger consumption benefit net of payment. However, then the neutral low type can always pick the bundle for the fair low type, which causes her surplus to increase. The comparison for the neutral high type buyer is less straightforward, and we provide a comparison using CARA and CRRA utility functions. Corollary 4. Assume θ2 − θ1 > λ(θ2 − c ). For a CARA utility function in the form of u(q ) = 1 − e−q or for a CRRA utility function in the form of u(q ) = log(q + 1 ), under inequity aversion, the neutral high type buyer receives a lower net surplus, relative to the case with no inequity aversion. Proof. When there is inequity aversion, the neutral high type buyer receives θ2 u(q∗2 ) − T2∗ , and she receives θ2 u(q2 ) − T2 when there is no inequity aversion, where (q2 , T2 ) is the optimal bundle for the high type in the case with no inequity aversion. Since θ2 u(q∗2 ) = c and θ2 u(q2 ) = c in both cases, it suffices to show that T2∗ > T2 . We know that T2∗ = θ2 (u(q∗2 ) − u(q∗1 )) + T1∗ and T2 = θ2 (u(q2 ) − u(q1 )) + T1 . Thus, we need θ2 (u(q∗2 ) − u(q∗1 )) + T1∗ > θ2 (u(q2 ) − u(q1 )) + T1 , where T1∗ = θ1 −K θ2 1−K

u(q∗1 ) and T1 = θ1 u(q1 ). Inserting these and arranging, we get that T2∗ > T2 is equivalent to (θ2 − θ1 )u(q1 ) > (θ2 −

φ )u(q∗1 ), α (1−λ ) , 1+α (1−λ )

where φ =

θ1 −K θ2 1−K

and 1 − K =

. Note that θ2 − φ =

1 . 1+α (1−λ )

θ2 −θ1 1−K

. Thus, all we need to show is (1 − K )u(q1 ) > u(q∗1 ), where K =

Thus, we need to show that u(q1 ) > (1 + α (1 − λ ))u(q∗1 ). Let α (1 − λ ) = z, to ease nota-

) tion, that is we will show u(q1 ) > (1 + z )u(q∗1 ). Now, we know that u (q∗1 ) = λ(1−Kλ)(θ1−K c from Proposition 1. Arranging 2 − (θ2 −θ1 ) λ λ  ∗  this, we have u (q1 ) = λθ −(θ −θ )(1+z ) c. We also know u (q1 ) = λθ −(θ −θ ) c. 2

2

1

2

2

1

With CARA utility in the form of u(q ) = 1 − e−q , we have u(q ) = 1 − u (q ). Thus, we need to show



λc λc 1− > (1 + z ) 1 − λθ2 − (θ2 − θ1 ) λθ2 − (θ2 − θ1 )(1 + z )

Arranging this inequality, we get λ2 θ2 (θ2 − c ) < (θ2 − θ1 )[λθ2 (1 + z ) + λθ2 − (θ2 − θ1 )(1 + z )]. Note that λθ2 − (θ2 − θ1 )(1 + z ) > 0 by Assumption 1, and also z = α (1 − λ ) > 0. Thus, if λ2 θ2 (θ2 − c ) < (θ2 − θ1 )λθ2 then we are done. This inequality is satisfied by our assumption, θ2 − θ1 > λ(θ2 − c ). With CRRA utility in the form of u(q ) = log(q + 1 ), we have u (q ) = 1/(q + 1 ). Thus, we need to show

log(λθ2 − (θ2 − θ1 )) − log(λc ) > (1 + z )[log(λθ2 − (θ2 − θ1 )(1 + z )) − log(λc )] which is equivalent to zlog(λc ) > (1 + z )log(λθ2 − (θ2 − θ1 )(1 + z )) − log(λθ2 − (θ2 − θ1 )). We have

zlog(λc ) > zlog(λθ2 − (θ2 − θ1 )) = (1 + z )log(λθ2 − (θ2 − θ1 )) − log(λθ2 − (θ2 − θ1 )) > (1 + z )log(λθ2 − (θ2 − θ1 )(1 + z )) − log(λθ2 − (θ2 − θ1 )) The first inequality is by our assumption (θ2 − θ1 > λ(θ2 − c )) and the second inequality is because λθ2 − (θ2 − θ1 )(1 + z ) < λθ2 − (θ2 − θ1 ).  Corollary 4 shows that the neutral high type buyer ends up with a higher total payment, relative to the case with no inequity aversion. This is because the quantities are distorted further for the low type buyers and the monopoly leaves extra surplus to the neutral low type buyer. To compensate, the monopoly extracts more from the neutral high type (in terms of higher total payment, as the quantity for the neutral high type does not change), at the same time reducing the surplus difference between these two types, hence weakening the distortion. Proposition 3. Monopoly profit decreases as the degree of inequity aversion, α , increases. Proof. By Proposition 1, we know that the optimal bundles for the fair low type and the neutral low type are the same. Thus, the profit of the monopoly is given by

ineq = λγ [T f∗ − cq∗f ] + λ(1 − γ )[T1∗ − cq∗1 ] + (1 − λ )[T2∗ − cq∗2 ] = λ[T1∗ − cq∗1 ] + (1 − λ )[T2∗ − cq∗2 ] Using the optimal payments from Proposition 1 and arranging ineq , we get

ineq =

θ1 − K ( α ) θ2 ∗ u(q1 ) + (1 − λ )θ2 [u(q∗2 ) − u(q∗1 )] − [λcq∗1 + (1 − λ )cq∗2 ] 1 − K (α )

By Envelope theorem, we have d ineq d = dα dK

Thus, we get 36



θ1 − K θ2 1−K

d ineq dα

We already know

dK (α ) dα



dK (α ) d u(q∗1 ), where dα dK



θ1 − K θ2

< 0, which establishes our result36 .

1−K

=

−θ2 (1 − K ) + θ1 − K θ2 θ1 − θ2 dK (α ) = < 0 and >0 dα ( 1 − K )2 ( 1 − K )2



> 0 from the proof of Proposition 2.

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The fact that monopoly profit decreases as inequity aversion is stronger is quite intuitive. With stronger inequity aversion, there is an extra distortion in terms of individual rationality of the inequity averse low type, which generates a higher consumption benefit net of payment for the low types. But then this creates an additional difficulty in terms of preventing high types mimicking the low types. This makes providing the right incentives more costly, thus, monopoly achieves a lower profit level. An interesting feature of the optimal nonlinear pricing is that the quantities and the payments depend on α (through α (1−λ ) K = 1+ α (1−λ ) ), but they do not depend on γ , the fraction of the fair types. The reason is that once the monopoly serves all types of buyers, the quantity levels and the payment levels turn out to be the same for the fair low type buyers and neutral low type buyers, as it is too costly to offer three different bundles, and it is easier to match the bundles for the low types. But then, the proportion of fair buyers does not affect the monopoly’s overall profit level. Another implication of the fact that the optimal nonlinear pricing does not depend on the fraction of the fair types, γ , is as follows. Regardless of the fraction of the fair types, in the optimal nonlinear pricing, the bundles targeted at both the neutral low type and the fair low type are the same and independent of γ . This implies that if γ = 1, that is if all low types are fair, at the optimal nonlinear pricing, the fair types would get the same bundle as they do when γ is much smaller. Thus, when a very small fraction for fair types is introduced, the optimal nonlinear pricing is as if all low types are inequity-averse. 4. Continuum of types In this section, we assume that the demand type θ is distributed over an interval  = [θ , θ¯ ] according to the cumulative distribution function F(·) with a full support where the associated density function is f(·). Each buyer is either a fair type with probability γ or a neutral type with probability 1 − γ , regardless of her taste parameter. Thus, the type space is {[θ , θ¯ ]} × { f, n}. As in Section 2, the intrinsic utility of a neutral buyer with demand parameter θ , from a bundle b = (q, T ) is given by the function V (b, θ ) = θ u(q ) − T . A fair type with demand parameter θ receives the following net utility from a bundle b = (q, T ).

W (b, θ ) = V (b, θ ) − α + (1 − γ )

  θ¯ γ max{0, V (b f (θˆ ), θˆ ) − V (b, θ )} f (θˆ )dθˆ

 θ¯ θ

θ



max{0, V (bn (θˆ ), θˆ ) − V (b, θ )} f (θˆ )dθˆ

where b f (θˆ ) = (q f (θˆ ), T f (θˆ )) is the bundle (quantity and payment) for a fair type with demand parameter θˆ , and bn (θˆ ) = (qn (θˆ ), Tn (θˆ )) is the bundle (quantity and payment) for a neutral type with demand parameter θˆ . Therefore, utility functions of agents are as in the discrete case. 4.1. Optimal Nonlinear Pricing We consider the case in which the monopoly serves every type.37 Without the exclusion of any type, the monopoly picks a menu of contracts {qf (θ ), Tf (θ ), qn (θ ), Tn (θ )}θ ∈  to solve the following problem.

max{q f (θ ),Tf (θ ),qn (θ ),Tn (θ )}θ ∈

 θ¯ θ

[γ [T f (θ ) − cq f (θ )] + (1 − γ )[Tn (θ ) − cqn (θ )]] f (θ )dθ

subject to individual rationality and incentive compatibility constraints of each type. We begin by describing the individual rationality and the incentive compatibility constraints, which are all similar to the constraints in the discrete case. Individual Rationality: Each type of buyer should be at least as well off by choosing the bundle targeted at her as her outside option. Assuming outside options are zero, given the bundles for neutral and fair types bn (θ ) = (qn (θ ), Tn (θ )) and b f (θ ) = (q f (θ ), T f (θ )), individual rationality constraints for neutral and fair types are respectively as follows:

IRn (θ ) : V (bn (θ ), θ ) = θ u(qn (θ )) − Tn (θ ) ≥ 0   θ¯ IR f (θ ) : V (b f (θ ), θ ) − α γ max{0, V (b f (θˆ ), θˆ ) − V (b f (θ ), θ )} f (θˆ )dθˆ + (1 − γ )

 θ¯ θ

θ



max{0, V (bn (θˆ ), θˆ ) − V (b f (θ ), θ )} f (θˆ )dθˆ ≥ 0

37 We also provide an analysis, in Section 5.2, of the case in which the monopoly is allowed to choose whom to and whom not to serve, that is, where exclusion is possible. We show that the set of excluded fair types is larger than the set of excluded neutral types.

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Incentive Compatibility: Each type of buyer should be at least as well off by choosing the bundle targeted at her as choosing another bundle. A neutral buyer should not mimic other neutral types, ICn,n , nor other fair types, ICn,f .

θ u(qn (θ )) − Tn (θ ) ≥ θ u(qn (θ  )) − Tn (θ  ) IC (θ ) : θ u(qn (θ )) − Tn (θ ) ≥ θ u(q f (θ  )) − T f (θ  ) IC n,n (θ ) : n, f

for all θ , θ  ∈ [θ , θ¯ ]. A fair buyer should not mimic other fair types, ICf,f , nor other neutral types, ICf,n .

  θ¯ IC (θ ) : V (b f (θ ), θ ) − α γ max{0, V (b f (θˆ ), θˆ ) − V (b f (θ ), θ )} f (θˆ )dθˆ f,n

+ (1 − γ )

 θ¯ θ

θ

≥ V ( bn ( θ  ), θ ) − α + (1 − γ )

 θ¯ θ



max{0, V (bn (θˆ ), θˆ ) − V (b f (θ ), θ )} f (θˆ )dθˆ

  θ¯ γ max{0, V (b f (θˆ ), θˆ ) − V (bn (θ  ), θ )} f (θˆ )dθˆ θ



max{0, V (bn (θˆ ), θˆ ) − V (bn (θ  ), θ )} f (θˆ )dθˆ

  θ¯ IC (θ ) : V (b f (θ ), θ ) − α γ max{0, V (b f (θˆ ), θˆ ) − V (b f (θ ), θ )} f (θˆ )dθˆ f, f

+ (1 − γ )

 θ¯ θ

θ

≥ V ( b f ( θ  ), θ ) − α + (1 − γ )

 θ¯ θ



max{0, V (bn (θˆ ), θˆ ) − V (b f (θ ), θ )} f (θˆ )dθˆ

  θ¯ γ max{0, V (b f (θˆ ), θˆ ) − V (b f (θ  ), θ )} f (θˆ )dθˆ θ



max{0, V (bn (θˆ ), θˆ ) − V (b f (θ  ), θ )} f (θˆ )dθˆ

for all θ , θ  ∈ [θ , θ¯ ]. Thus, we have four different sets of incentive compatibility constraints. First, we reduce the monopoly’s problem as follows. Lemma 4. When intrinsic utility V((q, T), θ ) is strictly concave in q,

max{q f (θ ),Tf (θ )}θ ∈[θ ,θ¯ ]

 θ¯ θ

38

the monopoly’s problem reduces to

[T f (θ ) − cq f (θ )] f (θ )dθ

 θ¯ subject to IR f (θ ) : V (b f (θ ), θ ) − α [ θ (V (b f (θˆ ), θˆ ) − V (b f (θ ), θ )) f (θˆ )dθˆ ] = 0, and

IC n,n (θ ) : θ u (q f (θ )) − T f (θ ) = 0 f or all

θ ∈ [θ , θ¯ ].

Proof. This is proven through a series of lemmas in online Appendix C.



As common in optimal nonlinear pricing problems, the individual rationality constraint for the lowest type implies the individual rationality constraints for all higher types. We also have that the incentive compatibility constraint for one of the fair and neutral types implies the one for the other. Before we characterize the quantities in the optimal nonlinear pricing without exclusion, we make two assumptions. Assumption 2. The hazard rate of F increases with θ , that is,

f (θ ) 1−F (θ )

is increasing in θ .

Assumption 2 is fairly reasonable and common. It is satisfied by a number of distributions, including normal, uniform, logistic, exponential and any distribution with a nondecreasing density function. Assumption 3.

θ f (θ ) 1−F (θ )

> 1 + α.

Assumption 3 is a necessary condition when monopoly is assumed to serve all types, so its role is similar to that of f (θ ) Assumption 1 in the discrete case. Note that Assumptions 2 and 3 together imply 1θ−F (θ ) > 1 + α for all θ , which in turn implies θ − (1 + α )H (θ ) > 0 for any θ , where H (θ ) = 38

1−F (θ ) . f (θ )

We verify that this assumption actually holds under the optimal payment scheme T(q), right after the proof of Corollary 5 below.

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Proposition 4. In the optimal nonlinear pricing when exclusion is not allowed, the quantities are q∗n (θ ) = q∗f (θ ) = q∗ (θ ), where q∗ (θ ) solves u (q(θ ))[θ − H (θ ) − α H (θ )] = c, for every θ .

Proof. First, we have q∗n (θ ) = q∗f (θ ) = q∗ (θ ) from Lemma 14 in online Appendix C in online Appendix C. From the envelope

theorem and ICn,n (θ ), we have ∂∂θV = V  (θ ) = u(q(θ )). By integrating it, we get net surplus of any type using the net surplus of lowest type and the quantity scheme:

V ( q ( θ ), θ ) =

 θ θ

u(q(θˆ ))dθˆ + V (q(θ ), θ )

We also have W (θ ) = 0 since IRf (θ ) is binding. Then, we have the surplus of lowest type as

V ( q ( θ ), θ ) = =

α α



θ

θ¯



(V (q(θˆ ), θˆ ) − V (q(θ ), θ ) f (θˆ )dθˆ

 θ¯  θ

θˆ θ



u(q(t ))dt

f (θˆ )dθˆ = α

 θ

θ¯

 u(q(θ ))(1 − F (θ ))dθ

The above condition means the surplus of the lowest type should be equal to the total disutility she gets from fairness concerns, so that her net utility becomes zero. Since qn (θ ) = q f (θ ), we have equal payments T (q f (θ )) = T (qn (θ )). Using T (q(θ )) = θ u(q(θ )) − V (θ ), the profit of the monopoly becomes

=

 θ¯ θ

θ u(q(θ )) −

 θ θ

u(q(θˆ ))dθˆ − α [

 θ¯ θ



u(q(θ ))(1 − F (θ ))dθ ] − cq(θ ) f (θ )dθ

Using integration by parts, we get the second and third term of the integral.

 θ¯  θ θ

θ

 θ¯  θ¯ θ

θ



u(q(θˆ ))d (θˆ ) f (θ )dθ =

 θ¯ θ

u(q(θ ))(1 − F (θ ))dθ

u(q(θ ))(1 − F (θ ))dθ

f ( θ )d θ =

 θ¯ θ

u(q(θ ))(1 − F (θ ))dθ

Hence, combining them, the profit equation becomes

=

 θ¯ θ

θ u(q(θ )) − u(q(θ ))

(1 − F (θ )) (1 − F (θ )) − α u(q(θ )) − cq(θ ) f (θ )dθ f (θ ) f (θ )

1−F (θ ) , we maximize the integral pointwise at each θ . The first order condition with respect to q at a fixed f (θ )  gives θ u (q(θ )) − u (q(θ ))H (θ ) − α u (q(θ ))H (θ ) − c = 0. Rearranging, we get

Inserting H (θ ) =

θ

u (q(θ ))[θ − (1 + α )H (θ )] = c

(1)

which characterizes the quantity function in the optimal nonlinear pricing for each

θ .39



The above result says that there is pooling, that is, neutral and fair types of the same demand parameter receives the same quantity. The intuition is similar to that of the discrete case. Because of the inequity aversion of the fair type, it is costlier to give the right incentives for each buyer to self select, by offering different bundles. A direct corollary to Proposition 4 is that the larger the fairness concern, the larger the suboptimality involved in the optimal nonlinear pricing, which implies that the quantity offered decreases. Corollary 5. As α increases, the degree of suboptimality increases and the quantity offered q∗ (θ ) decreases for any θ . Proof. Consider the suboptimality involved in the quantities for the neutral types. By Eq. (1) we have θ u (q∗ (θ )) = c θ −(1+θα )H (θ ) > c. For any θ , the fraction θ −(1+θα )H (θ ) is positive by Assumptions 2 and 3 and increasing as α increases. Thus, the right hand side of the equation increases, which means a higher suboptimality. In other words, marginal utility deviates from the marginal cost even more with a higher α . Now consider the suboptimality involved in the quantities for the fair types. Since, qn (θ ) = q f (θ ) = q(θ ) by Proposition 4, and since V(q(θ ), θ ) is increasing in θ by Lemma 10 in online Appendix C, we have

W = V (q, θ ) − α

  θ¯   θ¯ γ [V (q(θˆ ), θˆ ) − V (q, θ )] f (θˆ )dθˆ + (1 − γ ) [V (q(θˆ ), θˆ ) − V (q, θ )] f (θˆ )dθˆ θ

θ

39 Note that when α = 0, Eq. (1) is equivalent to the one in the standard model. See equation (3.14) on page 156, in Tirole (1988). Also, note that Assumptions 2 and 3 ensure that θ − (1 + α )H (θ ) > 0, which is needed since u (q) > 0.

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= V (q, θ ) − α + (1 − γ )

  θ¯ γ V (q(θˆ ), θˆ ) f (θˆ )dθˆ − γ [θ¯ − θ ]V (q, θ )

 θ¯ θ

θ



V (q(θˆ ), θˆ ) f (θˆ )dθˆ − (1 − γ )[θ¯ − θ ]V (q, θ )

= (1 + α [θ¯ − θ ] )V (q, θ ) − α

 θ¯ θ

V (q(θˆ ), θˆ ) f (θˆ )dθˆ

The marginal benefit for a fair type buyer with θ is then given by

∂W ∂V θ = (1 + α [θ¯ − θ ] ) = (1 + α [θ¯ − θ ] )θ u (q∗ (θ )) = (1 + α [θ¯ − θ ] ) c = kc ∂qf ∂qf θ − ( 1 + α )H ( θ ) Since both (1 + α [θ¯ − θ ] ) and θ −(1+θα )H (θ ) are increasing in α , we conclude that the suboptimality, |k − 1|, increases in α . Finally, since θ u (q∗ (θ )) is larger when α is larger, we immediately get a smaller q∗ (θ ) for any θ since u is strictly concave.  Note that the above result also holds for α = 0. Thus, the suboptimality is larger when there is inequity aversion relative to the case where there is no inequity aversion. Verification of strict concavity of V((q, T), θ ) in q: To verify that V (q, θ ) = θ u(q ) − T (q ) is strictly concave in q under the optimal payment scheme T(q), given θ , first note that we have θ u (q(θ )) − T  (q(θ )) = 0 for all θ by incentive compatibility. Also, we can obtain the optimal quantity schedule q(θ ) by solving the equality, u (q )[θ − (1 + α )H (θ )] = c, for q. This quantity schedule is strictly increasing in θ .40 Therefore, using the incentive compatibility above, we can write T  (q ) = β (q )u (q ), where β (q) is the inverse of the quantity function, that is, β (q) is the type that consumes quantity q.41 Thus, for a fixed θˆ , we have V  (q ) = θˆ u (q ) − T  (q ) = θˆ u (q ) − β (q )u (q ) = [θˆ − β (q )]u (q ). Now, to show that V(q) is concave in q, we need to show that V  (q ) = [θˆ − β (q )]u (q ) is decreasing in q. We know that u (q) is decreasing in q since u is concave. We also know that as q increases, u (q) decreases, and for [θ −(1+cα )H (θ )] to decrease we need θ to increase. Thus, β (q) is increasing in q. Putting these together we get that [θˆ − β (q )]u (q ) is decreasing in q, proving that V is strictly concave in q.

Proposition 5. There exists a θ˜ ∈ (θ , θ¯ ], such that for all θ < θ˜ , neutral types are strictly better off when there are fair types relative to when there are none.  θ¯ Proof. From the proof of Proposition 4, we have V (q∗ (θ ), θ ) = α [ θ u(q∗ (θ ))(1 − F (θ ))dθ ] > 0.42 Note that when α = n 0, V (q(θ ), θ ) = 0 by IR (θ ), where q(·) is the optimal quantity function in this standard case. Thus, we have V(q∗ (θ ), θ ) > V(q(θ ), θ ). By Lemma 10 in online Appendix C, V(q∗ , θ ) is strictly increasing in θ . In the standard case, V(q, θ ) also strictly increases in θ . Regardless of which one increases faster, since both are strictly increasing and since V (q∗ (θ ), θ ) > 0 = V (q(θ ), θ ), there always exists a θ˜ ∈ (θ , θ¯ ], such that for all θ < θ˜ , V(q∗ , θ ) > V(q, θ ).43  The above proposition is the counterpart of the result in the discrete case that shows that the neutral low type buyer is better off. The intuition is that the existence of fair type buyers forces the monopoly to leave more positive information rent to those types that have low enough demand parameters, through pooling of bundles in order to decrease the disutility the fair types get. Proposition 6. Monopoly profit decreases as the degree of inequity aversion, α , increases. Proof. The final form of profit is

 θ¯ = u(q(θ ))[θ − (1 + α )H (θ )] − cq(θ ) f (θ )dθ θ

We show that u(q(θ ))[θ − (1 + α )H (θ )] − cq(θ ) is decreasing in α , for any given θ . First, note that [θ − (1 + α )H (θ )] is decreasing in α . Since the monopoly chooses quantity to maximize u(q(θ ))[θ − (1 + α )H (θ )] − cq(θ ) for a given θ , the maximum value has to decrease as the term in square brackets decrease. Formally,

u(q(θ ))[θ − (1 + α )H (θ )] − cq(θ ) = max{u(q )[θ − (1 + α )H (θ )] − cq} < max{u(q )[θ − (1 + α  )H (θ )] − cq} q

q

(θ ) 40 , which is strictly decreasing by Assumption 2. Thus, as θ increases, θ − (1 + α )H (θ ) also increases, Note that u (q ) = [θ −(1+cα )H (θ )] where H (θ ) = 1−F f (θ ) implying that u (q) decreases. Also, u’s concavity implies that q must increase. Therefore, q(θ ) is strictly increasing in θ . 41 Note that we have q f = qn .  θ¯ 42 This can also be seen by IRf (θ ) and Lemma 13 in online Appendix C, where we have W (q∗ (θ ), θ ) = V (q∗ (θ ), θ ) − α θ [V (q∗ (θˆ ), θˆ ) −  ¯ θ V (q∗ (θ ), θ )] f (θˆ )dθˆ = 0. Note that α θ [V (q∗ (θˆ ), θˆ ) − V (q∗ (θ ), θ )] f (θˆ )dθˆ > 0. This is implied by Lemma 10 in online Appendix C. Thus, we have V(q∗ (θ ), θ ) > 0. 43 For instance, if V(q∗ , θ ) increases faster than V(q, θ ), then θ˜ = θ¯ , and every neutral type is strictly better off when there are fair types in the environment, relative to the one with no fair types. Otherwise, θ˜ may be an intermediate threshold, that is, θ˜ ∈ (θ , θ¯ ), and for any neutral type with θ < θ˜ , it is better to have fair types in the environment.

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when α > α  . Hence, profit at each θ declines and consequently total profit falls as α rises.44

19



The above result generalizes Proposition 3 to the continuum type case through a similar intuition. Also, as we discussed at the end of Section 3, γ does not affect the optimal nonlinear pricing, when exclusion is not allowed. Note that γ is not present in Eq. (1). This is because, when the monopoly has to serve all types, the quantities turn out to be the same for fair and neutral types with the same demand parameter θ . Then, the proportion of fair type γ does not affect the optimal pricing and hence it disappears from the monopoly’s objective function. 5. Discussion and extensions In this section, we discuss a number of extensions and some issues, including the optimal nonlinear pricing when the monopoly is allowed to exclude consumers, an alternative participation constraint, the way disutility from inequity is evaluated and an interpretation regarding information structure. 5.1. Optimal nonlinear pricing with exclusion: Discrete case We focused on the case in which the monopoly serves all types without any exclusion.45 Now, we look into the exclusion possibilities and pin down sufficient conditions for exclusion to be optimal. If the monopoly excludes some types of buyers, it must exclude the low types, since high types have larger surplus to extract. Also, as we have shown earlier, monopoly profit declines with the existence of fair types. Therefore, we have two possible exclusion scenarios: either only fair low types are excluded or both fair and neutral low types are excluded. First consider the case in which only fair low types are excluded. In this case, the only remaining fair type is the high type and she is not affected by inequality since she gets the highest surplus together with neutral high type agent. Therefore, the problem reduces to the standard case without fairness. We know that the payment for the low type is Tˆ1 = θ1 u(qˆ1 ) and for the high type it is Tˆ2 = θ2 u(qˆ2 ) − θ2 u(qˆ1 ) + Tˆ1 = θ2 u(qˆ2 ) − (θ2 − θ1 )u(qˆ1 ) where qˆ1 and qˆ2 are quantities of low and high types, respectively. Therefore, by excluding fair low types, the monopoly receives profit

exc. f air.low = λ(1 − γ )(Tˆ1 − cqˆ1 ) + (1 − λ )(Tˆ2 − cqˆ2 ) = [(1 − λγ )θ1 − (1 − λ )θ2 ]u(qˆ1 ) − λ(1 − γ )cqˆ1 + (1 − λ )(θ2 u(qˆ2 ) − cqˆ2 ) We already know the monopoly profit without exclusion from Proposition 3. By rearranging it, we get



no.exc =

θ1 − K θ2 1−K



− (1 − λ )θ2 u(q∗1 ) − λcq∗1 + (1 − λ )(θ2 u(q∗2 ) − cq∗2 )

Note that both qˆ2 and q∗2 are the socially optimal levels. Thus, the relevant parts are those terms with qˆ1 and q∗1 . We claim that if (1 − λγ )θ1 >

λ ) θ2 ] u ( q 1 ) > [

θ1 −K θ2

θ1 −K θ2 1−K

then exc.fair.low > no.exc (Fig. 3). Whenever (1 − λγ )θ1 >

θ1 −K θ2 1−K

, we have [(1 − λγ )θ1 − (1 −

1−K − (1 − λ )θ2 ]u (q1 ) for any q1 . Also the cost line with exclusion (of fair low types only) is below the cost line with no exclusion for any q1 , since λ(1 − γ )c < λc. Using the fact that qˆ1 > q∗1 together with the tangency conditions at these optimal levels, we conclude that the profit from low types when fair low types are excluded exceeds the profit from low types when no type is excluded. θ −K θ λγ θ λ) Note that (1 − λγ )θ1 > 11−K 2 is equivalent to θ2 − θ1 > α (1−1λ ) , or θ1 < λγα+(α1− (1−λ ) θ2 . This is a sufficient condition for

exc.fair.low > no.exc . That is, for sufficiently large α or for sufficiently small λ, excluding fair low types is better for the

monopoly than serving all types. Now consider the case where both low types are excluded. In this case, the monopoly can extract all the surplus from the high types. Hence the monopoly’s profit from excluding all low types becomes

exc.al l .l ow = (1 − λ )(T 2 − cq2 ) = (1 − λ )(θ2 u(q2 ) − cq2 ) where quantities are again at the socially optimal level. When comparing this profit level, exc.all.low , to exc.fair.low , the only difference is the term with qˆ1 . Therefore, whenever the term [(1 − λγ )θ1 − (1 − λ )θ2 ] in exc.fair.low is negative, that is, whenever θ1 <

1−λ 1−λγ

θ2 , it is better to exclude both low types than to exclude only the fair low type: exc.all.low > exc.fair.low . λ , α (1−λ ) }θ . Excluding only the fair low type is Therefore, excluding both low types is optimal if θ1 < min{ 11−−λγ λγ +α (1−λ ) 2 λ) optimal when θ1 < λγα+(α1− θ holds and the profit from the neutral low types is positive at the same time, that is, (1−λ ) 2 44 Alternatively, one can apply Envelope theorem using the maximized profit of the monopoly together with the proof of Proposition 4, and show that the overall effect of α on monopoly profit is negative. 45 This requires Assumption 1, θ1 > (1 − 1+α (λ1−λ) )θ2 , to be satisfied, which translates into either a large enough λ or a small enough α . Given any λ, θ 1

λ )θ2 . This is because 1 − 1+α (λ1−λ) is strictly increasing in α . 1+α¯ (1−λ ) λ ) θ . So for all 0 < α < α¯ , we have θ1 > (1 − 1+α (λ1−λ) )θ2 . Alternatively, 2 1+α¯ (1−λ ) λ¯ λ ¯ ∈ (0, 1 ) such that θ1 = (1 − given any α , θ 1 and θ 2 , with θ 2 > θ 1 > 0, we can find a large enough λ ¯ ) )θ2 , since 1 − 1+α (1−λ ) is strictly decreasing in 1+α (1−λ λ λ. For all 1 > λ > λ¯ , we have θ1 > (1 − 1+α (1−λ) )θ2 . Therefore, when the monopoly serves all types, it must be that λ is large enough or α is small enough.

and θ 2 , with θ 2 > θ 1 > 0, we can actually find a threshold α¯ > 0 such that θ1 = (1 −

So, since θ 2 > θ 1 > 0, there exists a small enough α¯ > 0 such that θ1 = (1 −

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Fig. 3. Profit comparison between exclusion of fair low types and no exclusion.

[(1 − λγ )θ1 − (1 − λ )θ2 ]u(qˆ1 ) − λ(1 − γ )cqˆ1 > 0.46 In each case, the quantities and the payments are equivalent to the ones in the standard case, with no inequity aversion. 5.2. Optimal nonlinear pricing with exclusion: Continuum case Suppose the monopoly is allowed to exclude some of the types. Assume that the monopoly sells to the neutral types with demand type θ ∈ [θ˜n , θ ], excluding those neutral types with θ < θ˜n , and sells to the fair types with demand type θ ∈ [θ˜ f , θ ], excluding those fair types with θ < θ˜ . Here, θ˜n and θ˜ are chosen by the monopoly. We show that the set of excluded fair f

f

types is larger than the set of excluded neutral types: θ˜n < θ˜ f , that is, all neutral and fair types with θ < θ˜n are excluded, and those fair types with θ ∈ [θ˜n , θ˜ f ) are excluded, while such neutral types are not, and all types with θ ≥ θ˜ f are served. This is intuitive because fair types have harder IR constraints and if the monopoly is excluding some of the types, it will exclude more of the fair types than the neutral types. The reason for this is that if there is a demand type, for which the neutral type is excluded but the fair type is not, the neutral type mimics the fair type. Thus, it must be θ˜n ≤ θ˜ f . However, equality θ˜n = θ˜ f is also not possible, because otherwise there would be a neutral type with a θ , which is smaller than the threshold and is in a small neighborhood of the threshold. This type would mimic the threshold fair type, who receives a positive benefit V (q, θ˜ f ), which compensates the disutility from inequity to yield W (q, θ˜ ) = 0. We also characterize the optimal quantity schedules and optimal exclusion thresholds. f

The analysis and the proofs are in online Appendix D. 5.3. An alternative individual rationality constraint In our model, for a fair type buyer, we have considered an individual rationality constraint in which the outside option is normalized to zero. Thus, when a fair type buyer does not accept any of the offers, then she ends up with zero net payoff, and there is no additional disutility from other buyers having some positive consumption level. This assumption is motivated from the example we discussed in the introduction and in Section 3.1. Here we discuss an alternative individual rationality constraint where a buyer’s outside option (when she does not get any of the available offers) takes into account the disutility from at least some other buyer’s positive net surplus from some positive consumption level: when a buyer does not consume at all, she still feels disutility from others’ positive net surpluses. In the discrete case, such an alternative IR constraint for fair type buyers may be formalized as follows. A fair low type buyer’s IR is

V1 f − α [(1 − λ )γ max{V2 f − V1 f , 0} + λ(1 − γ ) max{V1 − V1 f , 0} + (1 − λ )(1 − γ ) max{V2 − V1 f , 0}] ≥ − α [(1 − λ )γ max{V2 f , 0} + λ(1 − γ ) max{V1 , 0} + (1 − λ )(1 − γ ) max{V2 , 0}] Note that by Lemma 2, which doesn’t use IR constraints, we have that V1 = V1 f and V2 = V2 f . We also have V2 ≥ V1 , otherwise neutral high type would deviate and get the bundle for the neutral low type. Thus, the alternative IR1f above becomes

V1 f − α [(1 − λ )γ (V2 f − V1 f ) + (1 − λ )(1 − γ )(V2 − V1 f )] ≥ −α [(1 − λ )γ V2 f + λ(1 − γ )V1 + (1 − λ )(1 − γ )V2 ] 46 This requires u(qˆ1 ) to be sufficiently larger than cqˆ1 , for which we need the functional form of u(·). To see this, note that the condition can be written as [(1 − λγ ) − (1 − λ )(θ2 /θ1 )]θ1 u(qˆ1 ) > λ(1 − γ )cqˆ1 . We know θ1 u(qˆ1 ) > cqˆ1 , otherwise neutral low type would be excluded. However, one can show that [(1 − λγ ) − (1 − λ )(θ2 /θ1 )] < λ(1 − γ ) using θ 1 < θ 2 . Thus, we need a large enough u(qˆ1 ) − cqˆ1 for this to hold.

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21

which is V1 f [1 + α (1 − λγ )] ≥ 0, or simply V1f ≥ 0. A fair high type buyer’s IR is

V2 f − α [λγ max{V1 f − V2 f , 0} + λ(1 − γ ) max{V1 − V2 f , 0} + (1 − λ )(1 − γ ) max{V2 − V2 f , 0}] ≥ −α [λγ max{V1 f , 0} + λ(1 − γ ) max{V1 , 0} + (1 − λ )(1 − γ ) max{V2 , 0} Again, using V1 = V1 f , V2 = V2 f and V2 ≥ V1 this IR2f becomes V2 f [1 + α (1 − λ )(1 − γ )] ≥ −αλV1 f , and this always holds whenever V2f ≥ V1f ≥ 0. Thus, with this alternative set of individual rationality constraints, the fair low type is essentially the same as the neutral low type: they have exactly the same IR and IC constraints. Also note that, IR2f and IR2 do not bind in our version by Lemma 3, and here with this alternative version, IR2f also does not bind. We will still have V1 = V1 f and V2 = V2 f , but now with V1 = V1 f = 0. And IC21 f will be the same as IC21 , since V2 = V2 f > 0 and fair high type does not experience any inequity. Thus, V2 will be such that IC21 binds. Therefore, the optimal nonlinear pricing will be the same as the one when there are no fair types, that is, this alternative outside option removes any possible effect of inequity aversion. The reason is simple: when a buyer is exposed to inequity regardless of whether she gets one of the offers or not, the disutility of inequity disappears from the IR constraints, and the monopoly can effectively see it as a pure neutral buyer environment.47 5.4. Evaluating the disutility from inequity When calculating a θˆ type’s disutility from possible inequities, we assumed that this type can evaluate the utility of every other type, V(q(θ ), θ ). This evaluation is feasible since she can technically learn which bundle is for which type by looking at the nonlinear pricing and she also knows the type space and the distribution function. Then, she compares V(q(θ ), θ ) with her utility V (q(θˆ ), θˆ ). An alternative way of thinking about how to calculate the disutility from possible inequities, is to let type θˆ buyer calculate the utility she would get from the bundle of other types, V (q(θ ), θˆ ), and then look at the difference V (q(θˆ ), θˆ ) − V (q(θ ), θˆ ). With this type of evaluation, the problem is that whenever V (q(θˆ ), θˆ ) − V (q(θ ), θˆ ) < 0, that is, when a fair type with θˆ suffers from inequity, she would deviate to get the θ type’s bundle, which not only gives her a higher utility, but also decreases her disutility from possible inequities. Thus, there would be no inequity at all, and the optimal nonlinear pricing would be the same as in the standard model with no fairness issues. In the flights with first-class cabin example we mentioned in the introduction, an economy seat owner’s net utility from her seat is larger than her net utility from a first-class seat, maybe because the price is too high and/or her valuation for a first-class seat is low. This is the reason she does not buy a first-class seat. Thus, calculating the disutility from inequity through comparing net utility at her own bundle and net utility levels at others’ bundles becomes irrelevant. Also, an economy seat owner understands that the price she is paying is lower than the one paid by a first-class seat owner, since the quality of a first-class seat is higher than an economy seat. Thus, it is also irrelevant to compare only the different utility levels a buyer would get from different quantity levels (here in this example it’s quality), θ [u(q ) − u(q )], as there is also a price dimension which is different. However, if she is receiving some disutility (hence the “air rage” mentioned in the introduction), it is possibly because she is left with less net surplus than those in the first-class cabin, which is due to the different offers made by the monopoly, which chooses to leave less surplus for some demand types than other, to maximize its profit. Then, it makes sense to take an approach where the disutility from inequity emerges from different net utilities each type ends up with, rather than from receiving a different net utility she would receive at others’ bundles, or from differences in quantities picked by each type. 5.5. Inequity aversion or lack of information Another interpretation of the distinction between fair and neutral type consumers can be their information status. Instead of assuming that people have different fairness concerns, we can think that they are all inequity-averse but they only learn other people’s consumption bundle with some probability, γ . For example, one would not be aware of the details of price discrimination in her flight if she does not check ticket prices at different times and locations or talk to other passengers. Therefore, even though one has fairness concerns, she may not be adversely affected by the inequity involved in price discrimination. This alternative interpretation of the parameter γ allows us to explain a different setup with the same model. 6. Conclusion We considered both discrete and continuum type environments where a monopoly is using a nonlinear pricing scheme facing buyers who may differ in their valuations and in their inequity preferences, in the sense that some are inequityaverse (fair types) while some are not (neutral types). We adopted the utility function that takes inequity aversion into account provided by Fehr and Schmidt (1999). We characterized the optimal nonlinear pricing assuming all types are served 47 This is similar to the analysis in the Supplementary Section S.5.3 of Hahn et al. (2018). They show that when not buying at all also has gain/loss utility, and gain/loss is with respect to the total net utility, the model simplifies to the standard one without loss-aversion.

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Please cite this article as: C.G. Destan and M. Yılmaz, Nonlinear pricing under inequity aversion, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.11.013