Screening Loss Averse Consumers∗ Jong-Hee Hahn†

Jinwoo Kim‡

Sang-Hyun Kim§

Jihong Lee¶

February 1, 2012

Abstract Sellers often discriminate heterogeneous consumers with just a few products. This paper proposes an explanation for such coarse screening, based on consumer loss aversion. In our model, a seller offers a menu of bundles before a consumer learns his willingness to pay, and the consumer experiences gain-loss utility with reference to his prior (rational) expectation ´ a la K˝oszegi and Rabin (2006). For the case of binary consumer types, we show that the seller finds it optimal to offer a pooling menu under an intermediate range of loss aversion if the likelihood of low willingness-to-pay consumer is sufficiently large. We also identify sufficient conditions under which partial or full pooling dominates screening for the case of continuous types. JEL Classification: D03, D82, D86 Keywords: Reference-dependent preferences, loss aversion, price discrimination, screening menu, pooling menu

∗

The authors thank Yeon-Koo Che, Paul Heidhues, Navin Kartik, Fuhito Kojima, Tracy Lewis, Matthew

Rabin, Luis Rayo, Tim Van Zandt, and seminar participants at the Gerzensee, KEA/KAEA conferences, Seogang, and Yonsei for their comments. Jinwoo Kim acknowledges the financial support from the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology through its World Class University Grant (R32-2008-000-10056-0). Jihong Lee’s work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2011-327-B00094). † School of Economics, Yonsei University, Seoul 120-749, Korea; [email protected] ‡ School of Economics, Yonsei University, Seoul 120-749, Korea; [email protected] § Department of Economics, Michigan State University, East Lansing, MI 48824, U.S.A.; [email protected] ¶ Department of Economics, Seoul National University, Seoul 151-746, Korea; [email protected]

1

1

Introduction

When facing heterogeneous buyers, price discrimination allows a seller to capture a larger portion of the total market surplus than offering a single product quality. Price discrimination is indeed prevalent but sellers often employ just a small number of product types, despite our casual and statistical observations that suggest significant heterogeneity among buyers’ willingness to pay.1 Such simple or coarse screening has been commonly attributed to the existence of some fixed costs of launching products of different qualities. In many instances (e.g. cable TV or software packages), however, these costs tend to be small, which makes it difficult to justify the relative lack of quality diversity by resorting to such costs alone. This paper aims to offer another explanation for coarse screening, based on consumer loss aversion, by introducing K˝oszegi and Rabin’s (2006) expectation model of referencedependent preferences into a standard screening model ´a la Mussa and Rosen (1978).2 Our motivation for this approach to the question of coarse screening is as follows.3 First, in addition to the large existing literature documenting empirical support for loss aversion in a variety of economic situations, several recent studies point to the specific role played by expectations in formation of reference points (Mas (2006), Crawford and Meng (2011) and Abeler, Falk, G¨otte and Huffman (2011), Gill and Prowse (2010)).4 Second, it is common that consumers learn their own demands only after observing sellers’ products. We usually know the types of airline seat, rental car and hotel room that are available before discovering the specific conditions that determine our next holiday preferences. Given such a time lag between menu offer and realization of uncertain demand, it may then be the case that when the buyer learns his own type and makes a consumption choice, he compares his choice to what he could have consumed. 1

Crawford and Shum (2007) find that the US cable television industry largely consists of firms offering a

single product quality. Leslie (2004) estimates many different consumer valuations for a Broadway play but observes that the majority of performances come with just two seat qualities. 2 K˝ oszegi and Rabin (2007, 2009) extend their previous model to incorporate risky and intertemporal decisions. Other models of expectation-based reference-dependent preferences are analyzed by Bell (1985), Loomes and Sugden (1986), Gul (1991) and Shalev (2000). 3 See Ellison (2006) for a broad perspective on studying firm behavior under less than fully rational consumers. 4 Evidence suggest that loss aversion can account for diverse economic phenomena, from the equity premium puzzle (Benartzi and Thaler, 1995) to the seller behavior in Boston housing market (Genesove and Mayer, 2001). See Camerer (2006) for a survey.

2

Moreover, in these situations, it is plausible to think that the buyer formulates some contingent consumption plan ex ante. Consider another example, a subscriber of cable TV who has to choose among multiple packages of channels. A parent’s demand for the Disney package depends on the level of distraction that it creates for her child, which is uncertain before the child enters school; a sports fan’s willingness to pay for the NFL package is influenced by the performance of his favorite team. One may reasonably expect such a consumer to devise a future plan to renew the premium package if and only if the child turns out to be focused on school work or the favorite team ends up having a promising pre-season. We consider a setup in which a monopolist seller offers a menu of bundles before a buyer privately observes his willingness to pay and decides whether or not to make a purchase. As in K˝oszegi and Rabin (2006), henceforth referred to as KR, the buyer anticipates his future consumption choice for each possible contingency and experiences “gain-loss utility” with reference to his own past expectation of contingent consumption, in addition to standard “consumption/intrinsic utility.” Furthermore, the expectation must be correct; that is, it must be consistent with the buyer’s optimal consumption choice in each realization of uncertainty. This requirement of rational expectation, or personal equilibrium, implies that the menu must satisfy incentive compatibility and (ex-post) participation constraints that account for reference-dependent preferences and loss aversion. We show that loss aversion can indeed be responsible for the optimality of pooling menu in the above setup, where buyers with standard preferences would be separated via a menu with strictly increasing quality-price schedule. However, what turns out to be also important for endogenizing coarse screening is the distribution of consumer demand. Our results suggest that given a (sufficient) level of loss aversion, pooling is more likely to arise in markets with large population of consumers with low willingness to pay. This is consistent with our casual observation on the degree of price discrimination for some real-world products. For example, buses and motels usually offer a single type of seats and rooms, and this contrasts with pricing practices of airlines and hotels whose markets frequently serve business travelers. Our main message is most clearly conveyed in the case of binary consumer types, where the effect of loss aversion manifests itself in two ways. First, when each consumer compares the alternative of non-participation to the bundle of his choice ex post, he experiences a loss on quality and a gain in money. Thus, as the consumer becomes more loss averse, this participation effect makes him willing to pay more for a given quality, which implies that the seller can profitably increase the quality for the type whose participation constraint is binding (i.e.

3

the low willingness-to-pay type). Second, for the consumer who acquires an information rent (i.e. the high willingness-to-pay type), deviation to lower quality-price bundle also entails a loss on quality and a gain in money. However, in this case, the comparison is weighted by the likelihood of the alternative event. Given loss aversion, the deviation incentive is greater when the low willingness-to-pay type, and thus a lower price, was anticipated with a larger probability. The combination of the above two effects generates the following: when the likelihood of low willingness-to-pay consumer is sufficiently large and the degree of loss aversion lies in an intermediate range, the seller’s optimal menu is to offer the same bundle to both types.5 The optimality of pooling remains valid under a wide range of parameters if the monopolist’s contract must additionally satisfy an ex-ante participation constraint, in which case the consumer’s ex-ante insurance motive works further in favor of pooling menu. In fact, the monopolist’s profit can be maximized by pooling menu even when the consumer is capable of choosing his favorite personal equilibrium from a given menu, or preferred personal equilibrium (PPE) proposed by KR. We further extend our main analysis to the case of continuous types in which full separation is optimal under standard preferences. Here, we establish conditions under which partial or even full pooling is optimal among menus with monotone quality and price. These results, above all, contribute to the literature that has long been concerned with the question of why we often observe simple contracts (see, for instance, Holmstr¨om and Milgrom (1987) for an early account). Recently, Herweg, M¨ uller and Weinschenk (2010) and Herweg and Mierendorff (2011) investigate the role of reference dependence and loss aversion in KR’s framework in endogenizing simple contracts.6 Herweg, M¨ uller and Weinschenk (2010) consider an incentive design problem with moral hazard and show that the optimal contract may involve only binary payment schemes.7 Similarly to our paper, Herweg and 5

When the buyer is sufficiently loss averse, a reverse-screening menu, where the low consumer type

purchases a higher quality-price bundle than the high consumer type, can be made incentive feasible and even optimal. However, this result does not hold with continuous types. 6 Also, Cabarjal and Ely (2010) apply their generalized methodology of Mirrlees representation to study the design of optimal selling mechanism when the buyer has a reference-dependent preference with loss aversion. Their analysis and results differ from ours due to a couple of crucial aspects in their model: First, the buyer’s utility from the monetary dimension is not reference-dependent; second, the reference quality is assumed to be exogenously given. 7 In an alternative effort provision setup, Hart and Moore (2008) also derive the role of reference dependence in generating simple optimal contracts.

4

Mierendorff (2011) study a price discrimination setup with uncertain demands. Restricting attention to two-part tariffs, their results demonstrate the optimality of flat-tariffs under loss aversion. In contrast to our paper, however, their setup only considers situations with ex-ante participation requirement; moreover, their main analysis assumes that the consumer experiences gain-loss utility only in the money dimension and not in the quality dimension. Our paper also adds to the growing literature on firm behavior under biased consumer preferences (see Armstrong and Huck (2010) for a recent survey). Within this literature, we are most closely related to Heidhues and K˝oszegi (2005, 2008) who show that consumer loss aversion in the framework of KR’s expectation-based reference point can explain why a monopolist may adopt a sticky pricing schedule and firms with differentiated products and heterogeneous costs may end up charging a uniform price. Other lines of enquiry that can be broadly categorized under this literature include pricing for consumers with timeinconsistent preferences or self-control problems (e.g. DellaVigna and Malmendier (2004), Eliaz and Spiegler (2006), Esteban, Miyagawa and Shum (2007) and Heidhues and K˝oszegi (2010)) and for overconfident consumers (e.g. Eliaz and Spiegler (2008) and Grubb (2009)), among others. This paper is organized as follows. Section 2 lays out a price discrimination setup with KR’s reference-dependent preferences for the binary-type case, We then characterize in Section 3 the optimal menu under various constraints faced by the seller. In Section 4, the main analysis is extended to the continuous type case. Finally, Section 5 offers some concluding comments. All proofs are relegated to the Appendix unless mentioned otherwise. We also present the details of some omitted results and proofs in a Supplementary Material.

2

The Setup

2.1

Price Discrimination with Loss Averse Consumers

Consider a market that consists of a monopolistic seller of some product and its buyer. Let b = (q, t) denote a ‘bundle’ in which the product of quality q is sold for the payment of t, and let ∅ = (0, 0) denote the null bundle. The seller’s profit from a bundle b = (q, t) is t − cq, where c > 0 is the constant marginal cost of production. We assume that the buyer has reference-dependent preferences, which consist of “consumption” (or intrinsic) utility and “gain-loss” utility. First, the buyer’s consumption utility

5

from a bundle b = (q, t) is given by m(b; θ) := θv(q) − t, where θ is a measure of willingness-to-pay and v(·) is a (differentiable) function satisfying v(0) = 0, v 0 (·) > 0, v 00 (·) < 0, limv 0 (q) = ∞ and lim v 0 (q) = 0. We assume that θ is q→0

q→∞

randomly drawn and becomes privately known to the buyer when it is realized. For now, θ is assumed to take binary values, θL > 0 and θH > θL with probability p ∈ (0, 1) and 1 − p, respectively. Later in Section 4, we will consider a continuum-of-types case. Next, we explain the gain-loss utility. The situation we have in mind is as follows. The seller commits to a menu of bundles, M , which is observed by the buyer before his willingness-to-pay, θ, is known. The buyer then anticipates his choice of bundle for each possible contingency of the realization of his type, and this expectation serves as his reference point when actual consumption takes place after the uncertainty becomes resolved. We will later require that this expectation be consistent with the buyer’s actual choice; that is, the expectation should be rational. The following timeline will be useful to illustrate the model and compare it with the standard screening model. -Time

The seller offers a menu; The buyer forms a reference point

θ realized

The buyer chooses/consumes a (no) bundle from the menu

In the standard contract theory, the type parameter θ usually represents the agent’s inherent preference for consumption which is realized and privately observed prior to the contracting stage. In our setup, θ can be interpreted as a random utility component that becomes known to the buyer only when the uncertainty, which determines the buyer’s willingness to pay, gets resolved around the time of consumption (e.g. the level of distraction from school work or the pre-season performance of favorite team in the aforementioned cable TV example). Prior to learning θ and making the actual purchase, the buyer has time to observe the seller’s menu already existing in the market (and possibly being bought by other buyers) and forms a contingent consumption plan that will serve as reference point. As argued in the Introduction, we consider the above timeline as plausible in many real-world situations.8 8

It is known that sometimes random contracts can improve the seller’s performance in standard settings.

6

Formally, let ri = (qir , tri ) denote the bundle that the buyer expects to choose if his type is realized to be θi , i = H, L. Given R := {rL , rH }, type-θ buyer’s gain-loss utility from a bundle b = (q, t) takes the following form: r n(b; θ, R) := p [µ(θv(q) − θL v(qLr )) + µ(trL − t)] + (1 − p) [µ(θv(q) − θH v(qH )) + µ(trH − t)] ,

(1) where µ is an indicator function such that, for any k1 , k2 ∈ R+ , ( k1 − k2 if k1 ≥ k2 µ(k1 − k2 ) := λ(k1 − k2 ), λ > 1 if k1 < k2 . The term µ(θv(q) − θL v(qLr )) in the RHS of (1), for instance, captures type θ’s gain-loss from consuming quality q relative to qLr , the level that he would have consumed had the realization of uncertainty been θL , which is then weighted by probability p with which the buyer had expected θL to occur. Other terms can be explained similarly. Here the parameter λ measures the degree of loss aversion; λ > 1 means that the buyer is loss averse, i.e. more sensitive to loss than to gain. Following Tversky and Kahneman (1991) and K˝oszegi and Rabin (2006), we assume that the gain-loss utility is additively separable across the two consumption dimensions, quality and monetary transfer. Note that this is a key feature of the reference-dependent preference theory that successfully explains the endowment effect (e.g. the gap between sellers’ willingness to accept and buyers’ willingness to pay for the same object), which has regularly been observed in a number of empirical and experimental studies but cannot be explained by the standard theory.9 Given the reference point R (expected choices of bundles), a type-θ buyer’s overall utility Also, Heidhues and K˝ oszegi (2005) discuss how a monopolist facing loss averse consumers may be able to employ random pricing to increase profits. In contrast to these models, the seller in our setup offers a contract before the realization of uncertainty and, more importantly, the formation of expectation. Random contracts will therefore have no role in our setup. 9 An alternative formation would be to apply the gain-loss utility to the total utility, θv(q) − t. Such a model, however, does no better at explaining the endowment effect than the standard theory. moreover, it is shown that in our setup it produces results no different from the standard model (see Section S.3 of the Supplementary Material for details).

7

from b = (q, t) is the sum of consumption and gain-loss utilities:10 u(b|θ, R) := m(b; θ) + n(b; θ, R).

2.2

Truthful Personal Equilibrium

We now introduce the notion of personal equilibrium proposed by KR, which incorporates the idea that the reference point formed by an economic agent should be in accordance with his actual choices. Definition 1. Given any menu M , R = {ri }i=H,L ⊆ M is a personal equilibrium (PE) if u(ri |θi , R) ≥ u(b|θi , R), ∀b ∈ M ∪ {∅}, ∀i = H, L.

(2)

Furthermore, R = {ri }i=L,H is a truthful personal equilibrium (TPE) if it is a PE given M = R. Condition (2) requires that each bundle ri in the PE be optimal for type θi with R as the reference point so that ri is the bundle the buyer actually chooses if his type turns out to be θi . Note that the equilibrium utility of each type must be no lower than its utility from choosing the null option since the buyer can always opt out after the realization of his type. In the case of a TPE, the reference point itself is offered as a menu and therefore each type only needs to prefer his choice of bundle over the other type’s bundle or the null bundle. That is, R = {ri }i=L,H is a TPE if and only if the incentive compatibility and individual rationality requirements hold as follows: for each i = H, L, u(ri |θi , R) ≥ u(r−i |θi , R)

(ICi )

u(ri |θi , R) ≥ u(∅|θi , R).

(IRi )

Since these inequalities, henceforth referred to as (IC) and (IR) constraints, are implied by (2), the following result is immediate. Proposition 1. Suppose R = {ri }i=L,H is a personal equilibrium (PE) for a given M . Then, R is a truthful personal equilibrium (TPE). 10

To adjust the magnitude of gain-loss utility relative to consumption utility, we could introduce a pa-

rameter, say β, and multiply it to the gain-loss utility term. Here, we set β equal to 1 for simplicity; the qualitative features of our results remain the same for any β provided it is not too small.

8

Given this result, there is no loss in restricting ourselves to menus that are themselves TPEs, when searching for the seller’s optimal menu. We shall sometimes refer to such a menu simply as a TPE menu and let M denote the set of all TPE menus. There is, however, a potential caveat in using TPE as a solution concept. Given a TPE menu, another PE may exist in which the buyer plays untruthfully (i.e. some type θi chooses r−i or ∅ rather than ri ) and sometimes the buyer might be better off under the non-truthful equilibrium than under the truthful one. In particular, the TPE might generate a negative ex-ante expected utility while it is also a PE that the buyer does not buy at all. In such a case, it may not be reasonable to expect the TPE to prevail in the market. Our approach to this issue is two-fold. First, while the seller’s menu may admit multiple PEs, the consumer may not be able to choose his best reference point. This could arise, for instance, if the consumer is not capable of translating the full consequences of his future behavior into current payoffs, or if the reference point is formed not through correct calculation of future incentives but as a result of some naive response to the seller’s marketing effort and/or the consumer’s own search activity (in principle, the reference point may even involve consuming an alternative product). In our setup, the consistency of consumer expectation with actual purchases could then be attributed to the seller’s design of bundles that are ex post incentive-compatible and individually rational. Second, if the consumer is sophisticated enough to compute the ex-ante payoff of a PE and hence select his best PE from a give menu, it is reasonable to think that he should also be able to commit not to buy whenever the best PE offers a negative ex-ante payoff. For instance, the parent in the aforementioned cable TV example could simply get rid of the TV itself if he correctly anticipates that he will end up renewing the subscription against his benefit. In order to address the possibility of consumer commitment, in Section 3.3, we refine our solution concept by (i) adding an ex-ante participation constraint and (ii) considering menus that are not only TPEs but also maximize the buyer’s ex-ante payoff among all PEs; that is, we require the optimal menu to constitute a preferred personal equilibrium (PPE), as proposed by KR.

2.3

The Seller’s Problem

The seller’s problem, denoted as [P ], is given by max

{(qL ,tL ),(qH ,tH )}∈M

p(tL − cqL ) + (1 − p)(tH − cqH ), 9

[P ]

where M denotes the set of TPE menus, i.e. menus satisfying (IC) and (IR) constraints. Under the reference-dependent preference framework, a broader class of menus can be supported as TPEs, compared to the standard screening model. In particular, it is possible to have the low-type buyer choosing a higher quality-price pair instead of the high type. This is because, given such menu, the high type may suffer a loss from deviating to mimic the low type and paying more than anticipated, which renders unavailable the usual incentive compatibility argument for the necessity of quality monotonicity of a feasible menu. One class of menus that can be easily ruled out is one where one type of buyer receives a lower quality but pays more than the other type (including the case of a higher payment for the same quality or the same payment for a lower quality). The reason is simple: if the former type deviates to the latter’s bundle, then he will enjoy a higher gain-loss utility as well as a higher intrinsic utility. We are therefore left with the following three classes of menus to consider. 1. Pooling menu: qH = qL and tH = tL 2. Screening menu: qH > qL and tH > tL 3. Reverse-screening menu: qH < qL and tH < tL We let MP , MS and MR denote the set of pooling, screening and reverse-screening menus, respectively, that satisfy the (IC) and (IR) constraints. For the full expressions of relevant (IC) and (IR) constraints, we refer the reader to Section S.1 of the Supplementary Material.

3 3.1

Optimal Menu with Binary Types Symmetric Information Benchmark

Before the main analysis, we examine the optimal menu when the seller and buyer are symmetrically informed. This will give us an insight into how the informational asymmetry interacts with loss aversion to generate the optimality of pooling. Consider a profit-maximizing seller who is symmetrically informed of θ and thus can commit to a menu ex ante such that she imposes (qi , ti ) upon observing each type θi being realized. Specifically, we modify the seller’s problem [P ] by dropping the (IC) constraints. Let us denote by [P s ] the seller’s profit maximization problem among contracts that satisfy the (IR) constraints only.

10

The following result gives a necessary condition for the optimal menu with symmetric information. Lemma 1. The solution to [P s ] must be such that θH v(qH ) ≥ θL v(qL ) and tH ≥ tL . Using the above Lemma and the fact that both (IR) constraints are binding,11 we obtain

tL =

θH v(qH ) − θL v(qL ) (λ + 1) θL v(qL ) and tH = tL + . 2 B(p, λ)

(3)

Assuming θH v(qH ) > θL v(qL ) at the optimum,12 we can plug (3) into the objective function and take the first-order conditions to obtain [(λ + 1)B(p, λ) − 2(1 − p)] θL 2pB(p, λ) L) c θH = , 0 v (qH ) B(p, λ) c

v 0 (q

=

(4) (5)

where B(p, λ) :=

1 + (1 − p) + pλ . 1 + p + (1 − p)λ

(6)

Note from (4) and (5) that qL ≥ qH if and only if (λ + 1)B(p, λ) − 2(1 − p) θH ≥ , 2p θL

(7)

which holds for λ exceeding some threshold since (λ+1)B(p, λ) strictly increases in λ without bound. Thus, with λ high enough to satisfy (7), the symmetrically informed seller can maximize profit by endowing the low type with a higher quality but charging the high type with a larger transfer (see (3)). Note that the optimal qualities are the same across the two types only when (7) holds as equality, which is a knife-edge phenomenon. Furthermore, the same quality does not necessarily mean the same transfer. This implies that pooling menu, which is the main focus of our analysis, does not arise when the buyers are loss averse but do not hold private information. Neither does it emerge as a consequence of asymmetric information alone, as Mussa and Rosen (1978) show. The optimality of pooling is indeed a consequence of the interplay between loss aversion and asymmetric information, as we shall demonstrate in later sections. Intuitively, pooling will emerge as the optimal menu when the quality reversal is desirable due to loss aversion but is not feasible in the presence of asymmetric information. 11 12

See Section S.1 of the Supplementary Material for the full expressions of these constraints. It is possible to have θH v(qH ) = θL v(qL ) at the optimum. We ignore this case to ease the exposition.

11

3.2

Main Results

We now turn to the analysis of [P ], i.e. finding an optimal menu when the seller and buyer are asymmetrically informed. A unified analysis of all possible menus is not available since different classes of menus entail different forms of gain-loss utility. So our analysis below considers each class separately to identify an optimal menu within that class, which will then lead us to characterize the overall optimal menu. Note that any pooling menu lies on the boundary of the set of feasible screening menus (MS ) or reverse-screening menus (MR ). The optimality of pooling will thus arise if two inequality constraints, qH ≥ qL and qH ≤ qL , which we impose to find an optimal menu within MS and MR , turn out to be binding. In what follows, whenever we mention an “optimal screening (pooling or reverse-screening) menu”, it will mean optimality within the set of screening (pooling or reverse-screening) menus. 3.2.1

Pooling Menu

We begin by characterizing the seller’s profit-maximizing choice within the class of the pooling menu. Consider a pooling menu R = {r = (q, t)} ∈ MP . Clearly, the (IRH ) constraint is implied by the (IRL ) constraint since, if both types choose the same bundle, type θH is better off in terms of both intrinsic and gain-loss utilities while the outside payoff is typeindependent. Now, (IRL ) can be written as u(r|θL , R) = θL v(q) − t − (1 − p)λ(θH − θL )v(q) ≥ u(∅|θL , R) = p[t − λθL v(q)] + (1 − p)[t − λθH v(q)], or after rearrangement, t≤

(λ + 1) θL v(q). 2

(8)

Clearly, (8) must be binding at the optimum. The following result is then immediate from the first-order condition of the seller’s profit-maximization. Proposition 2. The optimal pooling menu, {(q p , tp )}, is such that θL v 0 (q p ) =

2c . λ+1

Thus, the seller finds it optimal to sell a higher quality as the buyer gets more loss averse, i.e. λ gets bigger. This is because the buyer wants to avoid the loss from non-participation and, therefore, is willing to pay more for a given amount of consumption if he is more loss averse, as can be seen in (8) above. 12

3.2.2

Screening Menu

Consider a screening menu R = {rL = (qL , tL ), rH = (qH , tH )} ∈ MS where qL < qH and tL < tH . As in the standard screening model, we can show that (ICH ) and (IRL ) constraints are binding at the optimum while the other constraints are not. Using a similar derivation to (8), the binding (IRL ) constraint can be written as tL =

λ+1 θL v(qL ). 2

(9)

Thus, for the same reason as in the optimal pooling menu above, the optimal quality for the low type increases with loss aversion. We will refer to this as the participation effect of loss aversion, meaning that a greater aversion to the loss resulting from comparison with non-participation enables the seller to charge more and thus increase the quality for the low type consumer. Next, write the (ICH ) constraint as u(rH |θH , R) = θH v(qH ) − tH + p[θH v(qH ) − θL v(qL ) − λ(tH − tL )] ≥ u(rL |θH , R) = θH v(qL ) − tL + p(θH − θL )v(qL ) + (1 − p)[(tH − tL ) − λθH (v(qH ) − v(qL ))], which can then be rewritten as [1 + (1 − p) + pλ](tH − tL ) ≤ [1 + p + (1 − p)λ]θH [v(qH ) − v(qL )].

(10)

The benefit of type θH deviating to rL , captured by LHS of (10), consists of reduced payment, tH − tL , and its positive impact on the gain-loss utility, (1 − p + pλ)(tH − tL ). To understand the latter, note first that the gain from paying tL instead of tH is weighted by probability 1 − p with which the buyer has expected the payment to be tH according to the reference point. At the same time, by the deviation, the high type avoids the loss equal to λ(tH − tL ) that he would have incurred from sticking with his equilibrium choice, which is weighted by probability p with which θL would have occurred. The cost of deviation, captured by the RHS of (10), results from a reduced quality from qH to qL and can be explained similarly. Notice from (10) that higher λ amplifies both the benefit and cost of deviation. When binding, (10) can be written as tH = tL +

θH [v(qH ) − v(qL )] , B(p, λ) 13

(11)

where B(p, λ) =

1+(1−p)+pλ 1+p+(1−p)λ

as define previously in (5). Thus, the relative magnitude of the

effect of higher loss aversion on deviation incentives is determined by B(p, λ). If a higher λ makes B(p, λ) larger (smaller), then the loss aversion makes screening less (more) effective in enabling the extraction of more payment from the high type. We will refer to this as the screening effect of loss aversion, which could be favorable or adverse to the seller depending on the value of p. Also, (11) implies that, for a fixed λ, the effectiveness of screening is decreasing in the likelihood of low type, i.e. B(p, λ) is increasing in p. Now, we describe the optimal screening menu and compare it with the optimal pooling menu. Proposition 3.

(a) The optimal screening menu, {(qLs , tsL ), (qLs , tsH )}, is such that   c (λ + 1)B(p, λ)θL − 2(1 − p)θH = max ,0 v 0 (qLs ) 2pB(p, λ) c θH = , s 0 v (qH ) B(p, λ)

(12) (13)

s where qLs , if not equal to 0, increases in λ and qH decreases (increases) in λ if p >

1 2

(p < 21 ). (b) Any screening menu is dominated by the optimal pooling menu if and only if   θH λ+1 ≤ B(p, λ), (14) θL 2     θH θH S S which in turn holds if and only if λ ≥ λ p, θL for some threshold λ p, θL > 1 that decreases in p and increases in

θH . θL

In part (a) of Proposition 3, the optimal quality qL increasing with λ should be expected from the participation effect. The behavior of qH is related to the fact that B(p, λ) increases with λ if and only if p > 21 : That is, a higher λ means the adverse (favorable) screening
effect if p > 21 p < 21 . Part (b) states the condition under which pooling dominates screening. The inequality (14) holds when the participation effect, measured by

λ+1 2

(see (9) above), is large and/or

when the screening effect works against the profitability of screening as B(p, λ) gets large. There are a couple of noteworthy observations here. First, with sufficiently large λ, the dominance of pooling over screening remains even when p <

1 2

such that the screening effect

works favorably for the screening seller. This is because the participation effect dominates the screening effect, namely,

λ+1 2

increases with λ faster than B(p, λ) does. Second, the 14

  threshold, λS p, θθHL , is decreasing in p, and this implies that screening is less attractive relative to pooling when the low-type consumers are more abundant. This follows from the fact that a higher (ex-ante) likelihood of θL generates a greater deviation incentive for the high type via the gain-loss utility ( ∂B(p, λ)/∂p > 0). 3.2.3

Reverse-Screening Menu

Let us consider next a reverse-screening menu R = {rL = (qL , tL ), rH = (qH , tH )} ∈ MR such that qL > qH and tL > tH , satisfying the (IC) and (IR) constraints. The reversescreening menu is a useful device to exploit the aforementioned participation effect by giving a higher quality to the low type. Giving a higher quality to the low type, however, may create a deviation incentive for the high type. This incentive can be curbed should the high type suffer a sufficient loss from a higher deviation price. How this loss is affected by the parameters in our model will determine when the reverse-screening menu is optimal. We first provide a couple of necessary conditions for reverse-screening menu to be feasible or optimal. Lemma 2.

(a) A reverse-screening menu can be a TPE only if θH λ+1 ≤ . θL 2

(15)

(b) Any optimal reverse-screening menu must satisfy θH v(qH ) ≥ θL v(qL ). Part (a) states that loss aversion must be high enough to sustain a reverse-screening menu as a TPE. According to part (b), the seller does not want to reverse the qualities to the extent that the utility from quality consumption is reversed. We now compare reverse-screening and pooling menus. Proposition 4. Any reverse-screening menu is dominated by the optimal pooling menu if and only if θH 1 + p + (1 − p)λ ≥ , (16) θL 2     which in turn holds if and only if λ ≤ λR p, θθHL for some threshold λR p, θθHL that increases in p and

θH . θL

Thus, if λ is large enough to violate (16), reverse-screening in fact dominates pooling. This arises due to the participation effect that makes the increase in qL , rather than qH , more 15

effective in extracting transfers. Since the high-type consumer derives a higher level of utility from any given contract and therefore cares less about an improvement in quality than the low-type consumer, the attractiveness of exploiting the high type’s higher marginal intrinsic utility can be outweighed by the participation effect when the consumer is significantly loss averse. Condition (16) shows that pooling tends to dominate reverse-screening as p gets larger. The logic is similar to that behind part (b) of Proposition 3: a higher p makes it more tempting for the high type to deviate. When the realization of the low type has been anticipated to be more likely, under screening, the high type experiences a greater loss from sticking to rH that involves a higher payment while, under reverse-screening, the same consumer finds it less costly to deviate to rL . 3.2.4

Optimal Menu

We are now ready to characterize the menu that maximizes the seller’s expected profit among all TPE menus.     Theorem 1. There exists some p¯ ∈ (0, 1) such that λS p, θθHL ≤ λR p, θθHL if and only if p ≥ p¯. Then, the solution to [P ] is given by (a) a pooling menu if p ≥ p¯ and λ ∈ [λS , λR ]; (b) a screening menu if λ < min{λR , λS }; (c) a reverse-screening menu if λ > max{λR , λS }; (d) either screening or reverse-screening menu if p < p¯ and λ ∈ [λR , λS ]. Proof. First, it is straightforward to see that 2θH lim λS = ∞ > − 1 = lim λR p→0 p→0 θL r θH lim λS = 2 − 1 < ∞ = lim λR . p→1 p→1 θL Thus, by the mean value theorem and the monotonicity of λS and λR , we can find p¯ ∈ (0, 1) such that λS ≥ λR if and only if p ≥ p¯. Then, parts (a) to (d) of the claim immediately follow from combining part (b) of Proposition 3 and Proposition 4. Pooling can therefore be optimal if there is enough mass of low types and the consumer is sufficiently, but not too, loss averse. Otherwise, a screening or reverse-screening menu is 16

optimal. In the latter case, there is a region of parameters, as shown in part (d), in which we have not been able to fully sort between screening and reverse-screening menus, but in most cases we expect the screening (reverse-screening) menu to be optimal if λ is low (high). The central message of Theorem 1 is the optimality of pooling. Although reverse screening can be optimal with sufficiently large degree of loss aversion, we shall later show that this is not a robust result: with continuous types, the reverse-screening menu can never be optimal (Proposition 7). Also, even in the binary-type case, the reverse-screening menu can only be optimal with a very large degree of loss aversion, as demonstrated in Example 1 below. The following example illustrates how the optimal menu varies with the parameter values. Here, pooling is optimal for a wide range of parameter values, while reverse screening requires λ to be larger than 2.13 Example 1. Suppose that θL /θH = 1.5. The following Figure 1 divides the space of (λ, p) into four regions according to Theorem 1 and illustrates the type of optimal menu in each region. p 1.0

HaL Pooling

0.8

ΛR

0.6

0.4

0.2

HcL Reverse-Screening

HbL Screening

0.0 1.0

ΛS

HdL Screening or Reverse-Screening

Λ 1.5

2.0

2.5

3.0

3.5

4.0

Figure 1 It can be shown, though only numerically, that in the region (d), there is a threshold value of λ for each p below (resp. above) which the screening (resp. reverse-screening) menu is optimal. 13

Estimates of loss aversion have been obtained in a variety of contexts, ranging from 1.3 to 2.7; see

Camerer (2006). However, these estimates do not translate directly to values of λ in our setup since they are measured only in terms of money.

17

3.3

Consumer Commitment

The optimal menu described in Section 3.2 has the feature that the consumer often incurs an ex-ante loss, that is, the buyer’s ex-ante expected utility (including anticipated gain-loss) falls below zero.14 This can be problematic for the seller if the consumer can calculate the ex-ante loss and hence finds some commitment device to stay away from the menu altogether. In fact, such commitment may come from the menu itself without an external action if the menu admits an alternative PE that generates a higher payoff than the TPE. In order to accommodate such a sophisticated consumer, we next modify the seller’s problem in two ways, by first introducing an ex-ante participation constraint and then adding the notion of preferred personal equilibrium (PPE) ´a la K˝oszegi and Rabin (2006) . We shall below establish that the optimality of pooling menu is robust to these additional requirements. Given a TPE menu R = {(qL , tL ), (qH , tH )}, let U (R) denote the buyer’s ex-ante expected utility, which is given by U (R) := pu(rL ; θL , R) + (1 − p)u(rH ; θH , R).

(17)

Then, an ex-ante participation constraint can be written as follows: U (R) ≥ 0.

(EA)

Let Me denote the set of all TPE menus satisfying (EA). We first consider the following program: max

{(qL ,tL ),(qH ,tH )}∈Me

p(tL − cqL ) + (1 − p)(tH − cqH ).

[P e ]

Our next result provides the optimality of pooling menu within the set Me . Proposition 5. Suppose that case (a) in Theorem 1 holds (so that the solution to [P ] is a pooling menu). Then, under the constraints of the program [P e ], (a) any screening menu is dominated by the optimal pooling menu; (b) if p(λ − 1) ≤ 1, any reverse-screening menu is dominated by the optimal pooling menu. 14

In fact, it is straightforward to show that the optimal pooling or reverse-screening menu characterized

in Theorem 1 always involves ex-ante loss.

18

In sum, if the case (a) in Theorem 1 holds and p(λ − 1) ≤ 1, then the solution to [P e ] is a pooling menu. Proof. See Section S.2 of the Supplementary Material. To understand this result, note that a negative ex-ante utility in the solution to [P ] arises in part because the optimal menu is designed to exploit the buyer’s loss aversion to extract more payment, especially from the low type via the participation effect mentioned earlier. Assuming that p is close 1, for instance, the buyer’s ex-ante utility consists mostly of the low type’s intrinsic utility, which, given the transfer tL in (9), is equal to (1 −

λ+1 )θL v(qL ) 2

< 0.

From this, one might reason that the ex-ante loss problem could be handled by reducing the low type’s transfer/quality, which would in turn restore the optimality of screening menu. What is also responsible for the ex-ante loss problem, however, is the gap between the two types’ utilities from quality/transfer consumption that worsens the gain-loss utility and thereby reduces the ex-ante utility. The pooling menu alleviates this effect better than the screening menu and, thus, continues to dominate the screening menu when it does so without the ex-ante participation constraint. Without the extra condition, p(λ − 1) ≤ 1, reverse-screening would be preferred to the pooling since the former is more effective than the latter at reducing the utility loss in quality dimension in case λ is large. Next, we incorporate the possibility that a TPE menu may entail other non-truthful PEs yielding higher ex-ante payoffs. Such a problem can be alleviated if the TPE menu guarantees the highest ex-ante payoff among all its PEs menus. To this end, let P(R) denote the set of all PEs that can arise when the seller offers a menu R; that is, R0 = {ri0 }i=H,L belongs to P(R) if R0 ⊂ R ∪ {∅} and R0 ∈ M, where M denotes the set of all TPE menus satisfying the (IC) and (IR) constraints.15 Definition 2. A TPE menu R ∈ M is a preferred personal equilibrium (PPE) if U (R) ≥ U (R0 ) for all R0 ∈ P(R). Our next result shows that if a pooling menu solves the seller’s problem with the ex-ante participation constraint, it must also maximize the consumer’s ex-ante payoff. Proposition 6. Suppose that case (a) in Theorem 1 holds and also that p(λ − 1) ≤ 1. Then, the optimal pooling menu that solves [P e ] is a PPE. 15

We note that R0 need not satisfy (EA).

19

Thus, by choosing the pooling menu, the seller can satisfy the PPE requirement without affecting her profit. Intuitively, an additional merit of pooling is that it allows for just a few potential deviations by the buyer (only to the null bundle in each state). In contrast, a screening or reverse-screening TPE menu is more likely to admit some non-truthful PE that makes the buyer better off. If this were the case, the requirement of PPE would clearly reduce the seller’s profit. Thus, when pooling is optimal, there cannot be another class of menu that dominates it, and Proposition 6 implies the following corollary. Corollary 1. If the case (a) in Theorem 1 holds and p(λ − 1) ≤ 1, then the optimal pooling menu that solves [P e ] is (weakly) optimal among all PPE menus in Me . The fact that the PPE menu constraint is non-binding once (EA) is imposed greatly simplifies our analysis. Without (EA), however, a TPE pooling menu that maximizes the seller’s profit sometimes fails to be a PPE. This makes the analysis much more complicated since then one has to characterize the set of all PEs for each menu in each of the three classes and then select the profit-maximizing alternative among them. The number of potentially feasible contracts that need to be considered to characterize an optimal PPE menu increases to an intractable level even with binary types. Our next numerical example illustrates how the results of this Section compare to those of Section 3.2. Here, we observe that optimal pooling is in fact supported under a wider set of parameter values with the additional constraints. Example 2. Suppose that θH /θL = 1.5 (as in Example 1), c = 0.5, and v(q) =

√ q. The

dashed curves in Figure 2 below are reproduced from Figure 1. With (EA) constraint imposed additionally, pooling dominates screening (resp. reverse-screening) if (λ, p) is located to the right (resp. left) of the solid curve labeled λSEA (resp. λR EA ). Thus, the pooling menu is an optimal PE, and also PPE, in the parameter range between the two solid lines.

20

p 1.0

R ΛEA

0.8

ΛR 0.6

0.4

S ΛEA

ΛS 0.2

1.5

2.0

2.5

3.0

3.5

4.0

Λ

Figure 2

4

Optimal Menu with Continuous Types

We now extend our analysis to the case in which there is a continuum of consumer types over ¯ The type distribution is given by a cdf F , which has a strictly positive the interval [θ, θ]. ¯ and continuously differentiable pdf f . Define the “virtual value” function as J(θ) := θ −

1 − F (θ) , f (θ)

and assume that it is strictly increasing. As is well known, without loss aversion, this assumption leads to the full separation of types; that is, the optimal quality schedule that is strictly increasing where the virtual value is not negative. Hence, any pooling of types at some positive quality in the optimal menu can be associated with the effect of loss aversion. What is crucial for our analysis of optimal menu is to understand how the virtual value function and its monotonicity are affected by the loss aversion. ¯ → R+ × R denote a menu offered by the seller. Given q(·), let V (θ) := Let (q, t) : [θ, θ] ¯ θv(q(θ)). For simplicity, we assume that q(·) and t(·) are continuous.16 We shall restrict our attention to two classes of menu: (i) both q(·) and t(·) are non-decreasing; and (ii) both q(·) and t(·) are non-increasing while V (·) is non-decreasing. With some abuse of terminology, we shall refer to the former class of menus as screening menus and the latter as as reverse-screening menus.17 Then, the pooling menu refers to a menu with constant q(·) 16

If the optimal schedule involves some jump(s), then it will manifest itself as a boundary solution of the

optimization program since any such schedule can be approximated by a sequence of continuous schedules. 17 Note that this restriction on reverse-screening menu is parallel to the requirement in part (b) of Lemma 2 with binary types.

21

and t(·) as before. While there are many other menus that do not fall into either class (with non-monotone quality and/or transfer schedule), our restriction makes the analysis tractable; otherwise, the gain-loss utility can depend on the detailed pattern of quality/transfer schedule in a complex way beyond the scope of our analysis. We focus on TPE menus that satisfy (IC) and (IR) constraints. Given any such menu, let U (θ0 ; θ) denote the payoff of type θ reporting θ0 and let U (θ) := U (θ; θ). Then, the (IC) constraint can be written as U (θ) = max U (θ0 ; θ), ∀θ,

(18)

¯ θ0 ∈[θ,θ] ¯

while the (IR) constraint as Z

θ¯

U (θ) ≥

(t(s) − λV (s))dF (s), ∀θ.

(19)

θ ¯

In both screening and reverse-screening menus we consider, V (θ) := θv(q(θ)) is nondecreasing and, hence, we can define ˆ θ0 ) := sup{r ∈ [θ, θ] ¯ | V (s) ≤ θv(q(θ0 )), ∀s ≤ r}. θ(θ, ¯ Note that if type θ (mis)reports to be of type θ0 and receives q(θ0 ), then he will experience a utility gain (resp. loss) in quality dimension, compared to the types below (resp. above) ˆ θ0 ). This threshold type will be useful for calculating and expressing gain/loss utilities. θ(θ;

4.1

Sub-optimality of Reverse Screening

We first establish that all reverse screening menus are dominated by a pooling menu. Consider any menu where both q(θ) and t(·) are non-increasing while V (θ) = θq(θ) is nondecreasing. Then, U (θ0 ; θ) = θv(q(θ0 )) − t(θ0 ) +

"Z

0) ˆ θ(θ,θ

(θv(q(θ0 )) − V (s))dF (s) +

θ

"¯Z

#

θ0

Z

(t(s) − t(θ0 ))dF (s)

θ ¯ θ¯

−λ

(V (s) − θv(q(θ0 )))dF (s) +

0) ˆ θ(θ,θ

Z

θ¯

#

(t(θ0 ) − t(s))dF (s) ,

θ0

(20) where the expression in the first (second) square bracket corresponds to the gain (loss). Note that since t(·) is decreasing, any agent type, who reports θ0 , experiences a utility gain (resp. loss) in transfer dimension, compared to the types below (resp. above) θ0 . 22

Observe first that, using (20), one can obtain Z θ¯ Z θ(θ,θ 0) ˆ ∂U (θ0 ; θ) 0 0 v(q(θ0 ))dF (s) v(q(θ ))dF (s) + λ = v(q(θ )) + ∂θ 0) ˆ θ(θ,θ θ h ¯ i ˆ θ0 )) + λ(1 − F (θ(θ, ˆ θ0 ))) . = v(q(θ0 )) 1 + F (θ(θ, ˆ θ) = θ, the envelope theorem yields Given this and the fact that θ(θ, ∂U (θ0 ; θ) 0 U (θ) = = v(q(θ))(1 + F (θ) + λ(1 − F (θ))). ∂θ θ0 =θ

(21)

(22)

An alternative way to obtain U 0 (·) is to set θ0 = θ in (20) and differentiate the resulting expression with θ, which yields U 0 (θ) = (V 0 (θ) − t0 (θ)) [1 + F (θ) + λ(1 − F (θ))] .

(23)

Equating (23) with (22) yields t0 (θ) = θ(v(q(θ)))0 .

(24)

Note that this is the same marginal transfer expressed in terms of quality schedule as in the standard setup without loss aversion. Using this, it can be shown that the seller’s expected revenue is the expectation of the usual virtual value (weighted by the degree of loss aversion). This leads us to establish the following result. Proposition 7. Any reverse-screening menu is dominated by a pooling menu with constant q(·). Note that this result contrasts with Theorem 1 in the binary case which states that the reverse-screening menu can be optimal for some parameter values. In the binary case, the revenue impact of quality increase for type θL is more than marginal and moreover depends on the degree of loss aversion. By contrast, in the continuous case, the loss aversion does not play any role in determining the (marginal) relationship between quality and transfer schedules, as shown in (24). This is due to the fact that, given a reverse-screening menu, the gain-loss utility works in the same direction for the two dimensions, quality and money; that is, the utility gain for each type θ in both dimensions derives from the comparison with the types below. With the screening menu, as we shall see below, the gain-loss utility works in opposite directions across the two dimensions, and this makes the loss aversion play a non-trivial role in determining the virtual value. 23

4.2

Optimality of Pooling

We now turn our attention to screening menus in which both q(·) and t(·) are non-decreasing. The payoff of type θ from reporting θ0 is then given by "Z ˆ 0 θ(θ,θ )

U (θ0 ; θ) = θv(q(θ0 )) − t(θ0 ) +

(θv(q(θ0 )) − V (s))dF (s) +

(t(s) − t(θ0 ))dF (s)

θ0

θ

"¯Z

#

θ¯

Z

θ¯

−λ

(V (s) − θv(q(θ0 )))dF (s) +

Z

0) ˆ θ(θ,θ

#

θ0

(t(θ0 ) − t(s))dF (s) .

θ ¯

(25) As before, the envelope theorem yields U 0 (θ) = v(q(θ))(1 + F (θ) + λ(1 − F (θ))).

(26)

Also, setting θ0 = θ in (25) and differentiatting the resulting expression with θ yield U 0 (θ) = V 0 (θ) − t0 (θ) + V 0 (θ)F (θ) − t0 (θ)(1 − F (θ)) − λ[−V 0 (θ)(1 − F (θ)) + t0 (θ)F (θ)] = V 0 (θ) (1 + F (θ) + λ(1 − F (θ))) − t0 (θ) (2 − F (θ) + λF (θ)) = (v(q(θ)) + θ(v(q(θ)))0 ) (1 + F (θ) + λ(1 − F (θ))) − t0 (θ) (2 − F (θ) + λF (θ)) . Equating this with (26), we obtain t0 (θ) =

θ(v(q(θ)))0 (1 + F (θ) + λ(1 − F (θ))) = (v(q(θ)))0 G(θ, λ), 1 + (1 − F (θ)) + λF (θ)

(27)

where G(θ, λ) := θ

(1 + F (θ) + λ(1 − F (θ))) = θH(θ, λ). 1 + (1 − F (θ)) + λF (θ)

Note that the function H(θ, λ) is the continuous type counterpart of the inverse of B(p, λ) in (11) in that it affects the speed with which the payment increases as the consumer’s type, and thus its corresponding quality, marginally increases. Without reference-dependent utility, the rate of increase is proportional to G(θ, 1) = θ; this should be adjusted using H(p, λ) in the presence of reference-dependent utility. The numerator of G(θ, λ) measures the benefit of this marginal change due to the gainloss utility: the gain, which θ enjoys relative to the types below, increases by θ(v(q(θ)))0 F (θ) while the loss, which θ suffers relatives to the types above, decreases by λθ(v(q(θ)))0 (1−F (θ)). Thus, the marginal benefit is proportional to [1 + F (θ) + λ(1 − F (θ))]. This benefit does 24

not however translate entirely into the payment increase since a higher payment negatively impacts the gain-loss utility. If the payment increases by t0 (θ) for type θ, then the gain, which θ enjoys relative to the types above, decreases by t0 (θ)(1 − F (θ)) while the loss, which θ suffers relatives to the types below, increases by λt0 (θ)F (θ). To account for this impact, the payment increase associated with the quality increase should be adjusted downward as much as the denominator of G(θ, λ). Given this interpretation, we refer to G(θ, λ) as the “gain-loss adjusted type,” whose behavior will turn out to be crucial for determining the optimal quality schedule. Note that

G(θ, λ) > θ if θ < F −1 12 (and G(θ, λ) < θ if θ > F −1 12 ), so the gain-loss adjusted type is leveled out. Moreover, H(θ, λ) decreases in θ and does so faster with higher λ, which may cause G(θ, λ) = θH(θ, λ) to decrease. The equation in (27) results from the local incentive compatibility condition. While the global incentive compatibility is usually guaranteed by the nonnegative cross derivative ˆ θ0 ) is increasing of U (θ0 ; θ), the expression in (21) may not be increasing with θ since θ(θ, ˆ θ0 )) + λ(1 − F (θ(θ, ˆ θ0 ))), is with θ and thus the term in the square bracket, 1 + F (θ(θ, decreasing. We introduce the following assumption that is sufficient to ensure the global incentive compatibility. Assumption 1. θ(1 + F (θ) + λ(1 − F (θ))) is non-decreasing in θ. Lemma 3. Suppose that Assumption 1 holds. Given any non-decreasing quality schedule q(·) and the payment schedule satisfying (27), the (global) (IC) constraint (18) is satisfied. Proof. See Section S.2 of the Supplementary Material. Note that if the menu is fully pooling i.e. q(·) and t(·) are constant, then the (IC) constraint is trivially satisfied so Assumption 1 is unnecessary. We now turn to the analysis of finding the optimal menu. Given (27), the seller’s revenue can be expressed as follows: letting Gθ (θ, λ) := ! Z ¯ Z ¯ Z θ

θ

θ

t(θ)dF (θ) = θ ¯

θ ¯

Z

θ ¯ θ¯

= θ ¯

Z = θ ¯

θ¯

∂G(θ,λ) , ∂θ

Z

θ

t0 (s)ds + t(θ) dF (θ) ¯ !

(v(q(s)))0 G(s, λ)ds dF (θ) + t(θ) ¯ θ ¯ ! Z θ v(q(θ))G(θ, λ) − v(q(s))Gθ (s, λ)ds dF (θ) + t(θ) − v(q(θ))G(θ, λ) ¯ ¯ ¯ θ ¯

25

Z

θ¯

= θ



 1 − F (θ) (1 + λ) v(q(θ)) G(θ, λ) − Gθ (θ, λ) dF (θ) + t(θ) − θv(q(θ)) ¯ ¯ ¯ f (θ) 2

Z¯ θ¯ = θ ¯

v(q(θ))J(θ, λ)dF (θ) + t(θ) − θv(q(θ)) ¯ ¯ ¯

(1 + λ) , 2

(28)

where 1 − F (θ) Gθ (θ, λ), f (θ) and the second equality follows from integration by parts as does the third equality along J(θ, λ) := G(θ, λ) −

with the fact that G(θ, λ) = ( 1+λ )θ. 2 ¯ ¯ Let us refer to J(θ, λ) as “gain-loss adjusted virtual value,” which boils down to the usual virtual value J(θ) if λ = 1. The next result says that the transfer must be designed in such a way that the participation constraint is binding at the lowest type θ. ¯ Lemma 4. At the optimal menu, it must be that 1+λ t(θ) = ( )θv(q(θ)), ¯ ¯ 2 ¯

(29)

which implies that the (IR) constraint (19) is satisfied and, moreover, binding at θ = θ. ¯ Proof. See Section S.2 of the Supplementary Material for proof. Thus, the seller’s problem, denoted by [P c ], can be written as Z

[v(q(θ))J(θ, λ) − cq(θ)] dF (θ),

max q(·)

θ¯

[P c ]

θ ¯

subject to q(·) being non-decreasing.18 The shape of the optimal quality schedule will depend on the behavior of J(·, λ), the gain-loss adjusted virtual value. While J(·, λ) may behave in a complicated way depending on the distribution F (·), it can be decreasing over some or entire range for high enough λ, which implies that the optimal menu must involve some pooling. To see this, we can obtain Jθ (θ, λ) := 18

1 − F (θ) ∂J(θ, λ) = J 0 (θ)Gθ (θ, λ) − Gθθ (θ, λ), ∂θ f (θ)

(30)

Note that global incentive compatibility is ignored here. It can be verified by showing that Assumption

1 holds, or checking directly the optimality of θ0 = θ in the (IC) constraint (18) by plugging in the solution of [P c ]. Also, if the optimal schedule turns out to be constant, the global incentive compatibility is trivially satisfied.

26

where Gθθ (θ, λ) :=

∂G(θ,λ)2 . ∂2θ

Two important terms here are Gθ (θ, λ) and Gθθ (θ, λ). As

mentioned above, it is possible to have Gθ (θ, λ) < 0 as λ increases. Given this, (30) tells that J(θ, λ) can decrease if Gθθ (θ, λ) > 0. Note that the last term of (30) concerns the impact of loss aversion on the information rent.19 While Gθθ (θ, 1) = 0, having Gθθ (θ, λ) > 0 with λ > 1 means that loss aversion aggravates the information rent problem, which may cause the gain-loss adjusted virtual value to decrease and, hence, some pooling to arise. Indeed, this can happen if λ is high enough, as the following result shows. Theorem 2. Consider menus with non-decreasing q(·). Then, the optimal menu has the following properties: (a) Suppose Assumption 1 holds and ¯ highest type θ.

λ2 +2λ−3 2(λ+1)

>

1 ¯ (θ) ¯ . θf

Then, pooling occurs around the

¯ >1 (b) Suppose θ > 0, θf (θ) > F (θ) ∀θ, and f 0 (θ) ≤ 0 ∀θ. Then, there exists some λ ¯ ¯ pooling occurs over the entire interval [θ, θ]. ¯ such that, for any λ > λ, ¯ ¯ λ) < 0, The inequality condition in part (a) above is equivalent to requiring that Gθ (θ, i.e. the gain-loss adjusted type decreases with the original type around the top. Without having to concern with information rent at the top, this means that the gain-loss adjusted virtual value also decreases, leading to pooling at the top. Note that the inequality in (a) never holds if λ = 1. Part (b) gives a set of conditions sufficient for full pooling to be optimal. The first condition, θ > 0, prevents the optimal menu from excluding the bottom type, as required ¯ by a full pooling menu. To understand the second condition, let us first note lim G(θ, λ) = θ

λ→∞

1 − F (θ) . F (θ)

Thus, for sufficiently high λ, the gain-loss adjusted type decreases going from θ to θ¯ while ¯ it may not in-between. Then, the condition that θf (θ) > F (θ) ∀θ ensures that this expression monotonically decreases over the entire interval so that Gθ (θ, λ) is always negative for sufficiently high λ. The last condition, f 0 (θ) ≤ 0, ensures (along with the second condition) that Gθθ (θ) ≤ 0 for sufficiently high λ, which means worsening of the information rent problem due to loss aversion. Note that this condition is consistent with the observation in the 19

In the standard screening model, the expression

1−F (θ) f (θ)

represents the information rents that have to

be given up to the types above θ if an extra unit of good is to be sold to θ.

27

previous binary type analysis that the screening effect adversely affects the profitability of a screening menu when the low type is abundant. The following examples demonstrate that λ need not be very high in order for some or full pooling to arise and also that a diverse pattern of pooling can emerge depending on the value distribution. Example 3. Suppose that θ is uniformly distributed on [1, 2] so F (θ) = θ−1.20 The following Figure 3 draws J(·, λ) in the left panel and its “ironing-out” obtained using the technique of Myerson (1981) or Toikka (2011) in the right panel: Λ=2.6 2.0

2.0

Λ=2.3 1.5

1.5

Λ=2

Λ=2 1.0

1.0

Λ=1.7

Λ=1.7 Λ=1.4

Λ=1.4 1.2

Λ=2.6 Λ=2.3

1.4

1.6

1.8

2.0

1.2

1.4

1.6

1.8

2.0

Figure 3 Pooling does not arise if λ = 1.4, and does arise in the interval [1.674, 2] if λ = 1.7, in the larger interval [1.236, 2] if λ = 2 and over the entire interval if λ = 2.3 or higher. Example 4. Suppose that θ is distributed on the interval [1, 2] with F (θ) = 1 − (2 − θ)n for n ≥ 1. Note that, as n grows, F (·) becomes more concave so that weights are shifting toward lower types. The following Figure 4 illustrates J(·, 1.75) in the left panel and its ironing in the right panel: 1.5

n=1

1.5

n=1

1.4

n=1.4

1.4

n=1.4

1.3

n=3

1.3 1.2

1.2

n=3

1.1

1.1

n=6

1.0

1.0

1.2

1.4

1.6

1.8

2.0

1.2

Figure 4 20

n=6

One can verify that Assumption 1 holds if λ ≤ 2.

28

1.4

1.6

1.8

2.0

Thus, pooling occurs in the upper interval for n = 1, in the middle interval for n = 1.4, and in the lower interval if n = 3 or 6. As n grows, a pooling area shifts down as probability weights do. This can be understood from the fact that screening types with higher probability weights has a greater impact on the gain-loss utility.

5

Conclusion

We often find sellers offering menus with just a small number of bundles. This paper demonstrates that such observations are consistent with profit-maximizing firms that face loss averse consumers. We show that, in the binary-type case, a pooling menu is the seller’s optimal menu under a range of loss aversion parameter if the low willingness-to-pay consumers are sufficiently abundant. This result arises as a consequence of the interplay between loss aversion and asymmetric information. We also identify conditions under which partial or even full pooling dominates screening for the seller facing a continuum of consumer types. The analysis of price discrimination with loss averse consumers opens up some new research questions. In particular, our results suggest several potentially interesting channels of empirical study. For instance, since the optimality of pooling requires a sufficiently large mass of consumers with low demand, one may explore if indeed coarse screening occurs more often when the demand estimates point to more consumers with low willingness to pay. Another suggestion from our model is that reverse-screening can be optimal if the consumer is significantly loss averse (at least with only few consumer types), but this is difficult to justify in practice. Measuring the degree of consumer loss aversion in price discrimination settings with utilities derived from quality consumption as well as money may shed insights into this dichotomy. Our model abstracts from many aspects that might be present in reality. First, while we take the uncertainty to affect willingness to pay directly, variations in willingness to pay may arise from other sources, such as income shocks. In such a case, however, the buyer should also realize gain-loss utility in that uncertain monetary dimension. Second, the degree of loss aversion may vary among different consumers. The seller may then be able to employ the varying magnitudes of loss aversion as an extra screening device. Finally, there are factors other than willingness to pay that influence the form of optimal menu (for instance, liquidity constraints as considered by Che and Gale (2000)). How those factors interact with loss aversion in determination of optimal menus poses another interesting avenue for future

29

research.

Appendix Proof of Lemma 1: The proof consists of two claims. Claim 1. If the optimal menu satisfies θH v(qH ) ≥ θL v(qL ), then it must be that tH ≥ tL . Proof. Suppose to the contrary that tL > tH . Clearly, we must have both (IR) constraints binding or u(∅|θH , R) = u(rL |θL , R) = θL v(qL ) − tL − (1 − p)λ[θH v(qH ) − θL v(qL ) + tL − tH ]

(A.1)

u(∅|θL , R) = u(rH |θH , R) = θH v(qH ) − tH + p[θH v(qH ) − θL v(qL ) + tL − tH ].

(A.2)

Since u(∅|θH , R) = u(∅|θL , R), we can equate LHS of (A.1) and (A.2) to obtain after rearrangement [1 + p + (1 − p)λ](tL − tH ) = [1 + p + (1 − p)λ](θL v(qL ) − θH v(qH )), which is a contradiction since tL − tH > 0 but θL v(qL ) − θH v(qH ) ≤ 0. Claim 2. It is never optimal to offer a menu with θL v(qL ) > θH v(qH ). Proof. Suppose to the contrary that θL v(qL ) > θH v(qH ) at the optimum. A similar argument to that in the proof of Claim 1 can be used to show that tL ≥ tH . Given this, we can write the (IR) constraints as (λ + 1) θH v(qH ) 2 θL v(qL ) − θH v(qH ) tL ≤ tH + . B(p, λ)

tH ≤

Since both constraint must clearly be binding at the optimum, we can substitute these into the objective function and take the first-order conditions as follows: c

θL B(p, λ) L) c [(λ + 1)B(p, λ) − 2p] θH = . 0 v (qH ) 2(1 − p)B(p, λ) v 0 (q

=

This, however, yields a contradiction since c c (λ + 1)B(p, λ)θH − 2 [pθH + (1 − p)θL ] [(λ + 1)B(p, λ) − 2] θH − = ≥ ≥ 0, v 0 (qH ) v 0 (qL ) 2(1 − p)B(p, λ) 2(1 − p)B(p, λ) where the last inequality holds since (λ + 1)B(p, λ) ≥ (λ + 1)B(0, λ) = 2, ∀λ, p. 30

Clearly, combining Claim 1 and 2 leads to the desired result. Proof of Proposition 3: We consider the problem of maximizing the seller’s profit under the (IC) and (IR) constraints and under the quality constraint, qH − qL ≥ 0. We show the following: When the quality constraint is not binding, the optimal qualities must be given by (12) and (13); the quality constraint is binding if (14) holds, which means any screening menu is dominated by the optimal pooling menu. First, one can easily check that (IRH ) is implied by (IRL ) and (ICH ) since u(rH |θH , R) ≥ u(rL |θH , R) ≥ u(rL |θL , R) ≥ u(∅|θL , R) = u(∅|θH , R),

(A.3)

where the second inequality holds since if two types choose the same bundle, rL , then θH is better off in terms of both intrinsic and gain-loss utilities. Next, after rearrangement, we can write (ICL ) and (ICH ) (whose full expressions are given in Section S.1 of the Supplementary Material) as θL [v(qH ) − v(qL )] θH [v(qH ) − v(qL )] ≤ tH − tL ≤ , B(p, λ) B(p, λ) where B(p, λ) =

1+(1−p)+pλ 1+p+(1−p)λ

as defined in (6). By the usual argument, (IRL ) and (ICH ) must

be binding.21 Given the binding (ICH ) and the above equations for the (IC) constraints, (ICL ) is satisfied given the constraint qH ≥ qL . Using the two binding constraints, we obtain (9) and (11) for tL and tH , respectively. Substitute these into the objective function, the seller’s problem is max p(tL − cqL ) + (1 − p)(tH − cqH )

{qL ,qH }

=

λ+1 θH [v(qH ) − v(qL )] θL v(qL ) + (1 − p) − pcqL − (1 − p)cqH 2 B

subject to qH ≥ qL . Ignoring the quality constraint for the moment, the first order conditions with respect to qL and qH yield (12) and (13), as can be checked easily. Given this, one can check that the RHS of (13) is no larger than that of (12) if and only if the inequality (14) holds, which means that the quality constraint is binding in such case. To obtain the comparative statics for qL , qH , and λS , let us first observe the following facts: (i) B(p, λ) increases with λ > 1 if and only if p > 21 ; (ii) B(p, λ) increases with p if λ > 1; and (iii) (λ + 1)B(p, λ) increases from 1 to infinity as λ increases starting from λ = 1. 21

If (IRL ) is not binding, then the seller can slightly increase both tL and tH by the same amount. If

(ICH ) is not binding, then the seller can increase tH slightly.

31

The comparative statics for qH directly follows from (i) and the fact that

c v 0 (·)

is increasing.

As for the comparative statics regarding qL , rewrite the maximand in (12) as θL − [2(1 − p)θH ] / [(λ + 1)B(p, λ)] , 2p/(λ + 1) whose numerator increases with λ by (iii), while its denominator decreases. So the optimal qL , if not equal to 0, must increase with λ. The existence and properties of λS (p, θH /θL ) follow from (ii) and (iii). Before proving Lemma 2 and Proposition 4, we write here the (IC) and (IR) constraints for the reverse-screening menu, whose forms differ depending whether θL v(qL ) ≥ θH v(qH ) or θL v(qL ) ≤ θH v(qH ). (For the full expressions, see Section S.1 of the Supplementary Material.) In case θL v(qL ) ≥ θH v(qH ), [1 + (1 − p) + pλ]θH (v(qL ) − v(qH )) − p(λ − 1)(θH − θL )v(qL ) 1 + p + (1 − p)λ [1 + (1 − p) + pλ]θL (v(qL ) − v(qH )) + (1 − p)(λ − 1)(θH − θL )v(qH ) ≥ tL − tH 1 + p + (1 − p)λ tL − tH ≥

θH (λ + 1)v(qH ) ≥ 2tH

0 (ICH )

(ICL0 ) 0 ) (IRH

[1 + (1 − p) + pλ]θL v(qL ) + (1 − p)(λ − 1)θH v(qH ) ≥ [1 + p + (1 − p)λ]tL − (1 − p)(λ − 1)tH (IRL0 ) while in case θL v(qL ) ≤ θH v(qH ), [1 + p + (1 − p)λ](tL − tH ) ≥ 2θH [v(qL ) − v(qH )]

00 (ICH )

(λ + 1)θL [v(qL ) − v(qH )] ≥ [1 + p + (1 − p)λ](tL − tH )

(ICL00 )

[1 + p + (1 − p)λ]θH v(qH ) + p(λ − 1)θL v(qL ) ≥ 2tH

00 (IRH )

(λ + 1)θL v(qL ) ≥ [1 + p + (1 − p)λ]tL − (1 − p)(λ − 1)tH .

(IRL00 )

Proof of Lemma 2: To prove (a), let us consider both cases of reverse-screening menu. 00 In case θL v(qL ) ≤ θH v(qH ), the LHS of (ICL00 ) being greater than the RHS of (ICH ) yields 0 (15) after rearrangement. In case θL v(qL ) ≥ θH v(qH ), combining (ICL0 ) and (ICH ) yields

(λ + 1)v(qH ) − 2v(qL ) ≥ 0, which implies

λ+1 2

≥

v(qL ) v(qH )

≥

(A.4)

θH . θL

To prove (b), consider the problem of maximizing the seller’s profit under the (IC 0 ) and (IR0 ) constraints and the constraint that θL v(qL ) − θH v(qH ) ≥ 0. We aim to show that the 32

last constraint must be binding at the optimum. First, the same inequalities as in (A.3) can 0 0 be used to show that (IRH ) is implied by (ICH ) and (IRL0 ). Next, to identify the binding 0 constraints, we depict the set of (tL , tH ) in Figure 5 that satisfy (ICH ), (ICL0 ), and (IRL0 ) 0 0 for any given qL and qH . (The line labeled ICH , for instance, is where (ICH ) is binding.) 0 Clearly, (ICH ) and (IRL0 ) must be binding at the optimum.22

tH

0 : slope= 1+p+(1−p)λ > 1 IRL (1−p)(λ−1) 0 : slope=1 ICH

ICL0 : slope =1

seller’s iso-profit line where ptL + (1 − p)tH =constant tL

Figure 5 Combining the two binding constraints, we obtain (λ + 1)θH v(qH ) − 2(θH − θL )v(qL ) (A.5) 2 [1 + (1 − p) + pλ]θH [v(qL ) − v(qH )] − p(λ − 1)(θH − θL )v(qL ) tL (qL , qH ) = + tH (qL , qH ). 1 + p + (1 − p)λ (A.6)

tH (qL , qH ) =

0 Note that given (ICH ) is binding, (ICL0 ) is satisfied if and only if (A.4) is satisfied so we can

replace (ICL0 ) by the inequality constraint (A.4). Now, using (A.4) and θL v(qL )−θH v(qH ) ≥ 0 as constraints, the Lagrangian for the seller’s maximization problem can be written as L(qL , qH , γ, µ) = p(tL (qL , qH ) − cqL ) + (1 − p)(tH (qL , qH ) − cqH ) + γ [(λ + 1)v(qH ) − 2v(qL )] + η [θL v(qL ) − θH v(qH )] .

(A.7)

where γ and µ are nonnegative multipliers. Suppose, to the contrary, that θL v(qL ) − θH v(qH ) ≥ 0 is not binding at the optimum so η = 0. Substituting (A.5) and (A.6) into 22

A formal proof is a straightforward translation of graphical illustration and is thus omitted.

33

(A.7), the first-order conditions w.r.t. qL and qH are given by c 2γ [1 + p − p2 + (1 − p + p2 )λ]θL − (1 − p)(λ + 1)θH + = =: ΨL v 0 (qL ) p p[1 + p + (1 − p)λ] γ(λ + 1) [1 − 2p + 2(1 + p)λ + λ2 ]θH c − = =: ΨH . v 0 (qH ) 1−p 2[1 + p + (1 − p)λ]

(A.8) (A.9)

One can verify that ΨH > ΨL .23 Given this and γ ≥ 0, (A.8) and (A.9) requires qH > qL , which is a contradiction. Proof of Proposition 4: Thanks to Lemma 2, we can focus on the case θH v(qH ) − θL v(qL ) ≥ 0. Consider the problem of maximizing the seller’s profit under the constraints, (IC 00 ), (IR00 ), and θH v(qH ) − θL v(qL ) ≥ 0 along with the quality constraint, qL ≥ qH . We aim to show that if (16) holds, then the quality constraint must be binding at the optimum, which means that any reverse-screening menu is dominated by the optimal pooling menu. 00 00 Fist, we can ignore (IRH ) since it is implied by (ICH ) and (IRL00 ) for the same reason

as in (A.3). To identify the binding constraints, with qL and qH fixed, we depict the set of 00 (tL , tH ) satisfying the constraints, (ICH ), (ICL00 ), and (IRL00 ), to obtain the same graph as 00 Figure 5. From this, it is immediate that (IRL00 ) and (ICH ) are binding, which gives us

(λ + 1)θL v(qL ) − θH [v(qL ) − v(qH )] 2 (λ + 1)θL v(qL ) (1 − p)(λ − 1) tL (qL , qH ) = − θH [v(qL ) − v(qH )]. 2 1 + p + (1 − p)λ

tH (qL , qH ) =

(A.10) (A.11)

Ignoring the constraint qL ≥ qH for the moment, we set up the Lagrangian L(qL , qH , γ, µ) = p(tL (qL , qH ) − cqL ) + (1 − p)(tH (qL , qH ) − cqH ) + γ [(λ + 1)v(qH ) − 2v(qL )] + η [θH v(qH ) − θL v(qL )] , whose first-order conditions w.r.t. qL and qH are 2γ + ηθL [1 + p + (1 − p)λ](λ + 1)θL − 2(1 − p)(λ + 1)θH e L (A.12) = =: Ψ p 2p[1 + p + (1 − p)λ] L) c γ(λ + 1) + ηθH (λ + 1)θH eH. − = =: Ψ (A.13) v 0 (qH ) 1−p 1 + p + (1 − p)λ c

v 0 (q

23

+

To see this, suppose ΨH ≤ ΨL , which can be rewritten as 2 − p − 2p2 + 2(1 + p2 )λ + pλ2 θL ≤ (< 1). 2 2 2[1 + p − p + (1 − p + p )λ] θH

This inequality cannot hold since one can check that its LHS is greater than 1 given λ > 1.

34

eH ≥ Ψ e L if (16) holds, which implies that given γ, η ≥ 0, (A.12) and One can check that Ψ (A.13) can only be satisfied when qH ≥ qL . This means the constraint qL ≥ qH must be binding if (16) holds. Now rewrite (16) as 1 λ≤ 1−p



 2θH − (1 + p) := λR (p, θH /θL ). θL

It is straightforward to check that λR (p, θH /θL ) increases with both arguments. This completes the proof of Proposition 4. Proof of Proposition 6: Let R be the reference where both types of consumers make a purchase, given a pooling menu R = {(q, t)} that solves [P e ]. Similarly, let Ri , i = H, L denote the reference where only θi type makes a purchase. Our proof consists of two steps: Step 1. Given any pooling menu, RL is not a PE. Proof. For RL to be a PE, the following two conditions must be satisfied. L

p [t − λθL v(q)] ≥ θH v(q) − t + p(θH − θL )v(q) + (1 − p) [θH v(q) − λt]

(IRH )

p [t − λθL v(q)] ≤ θL v(q) − t + (1 − p) [θL v(q) − λt] ,

(IRLL )

L

L where (IRH ) is the negation of (IRH ), the participation constraint for θH type given RL .

Rearranging the terms, the conditions can be restated as [1 + p + (1 − p)λ] t ≥ [2θH + p(λ − 1)θL ] v(q) [1 + p + (1 − p)λ] t ≤ [2θL + p(λ − 1)θL ] v(q), which can not be true at the same time because θH > θL .   Step 2. Given a pooling menu that solves [P e ], R is prefered to RH if λ > λS p, θθHL and p(λ − 1) ≤ 1. Proof. First, one can write the ex-ante expected utility with reference R as A U (RA ) = pU (rLA |θL , RA ) + (1 − p)U (rH |θH , RA )

= p [θL v(q) − t − (1 − p)λ(θH − θL )v(q)] + (1 − p) [θH v(q) − t + p(θH − θL )v(q)] = {(1 − p) [1 − p(λ − 1)] θH + p [1 + (1 − p)(λ − 1)] θL } v(q) − t   θH ≡ D p, λ, v(q) − t. θL 35

Recall that the binding (IRL ) constraint given a pooling menu is 0=

λ+1 θL v(q) − t. 2

Thus, at the optimum of [P e ], either (EA) or (IRL ) has to be binding depending on the size   of D p, λ, θθHL and λ+1 θL . 2 Suppose, first, (IRL ) is binding. Given the reference RH , the paritcipation constraint for H type θH , (IRH ) is

θH v(q) − t + p [θH v(q) − λt] ≥ (1 − p) [t − λθH v(q)] which can be rewritten as θH v(q) ≥ B (p, λ) t. Using binding (IRL ) constraint, it is restated as   θH λ+1 ≥ B (p, λ) , θL 2   θH S which contradicts with the assumption λ > λ p, θL . In other words, RH can not be a PE   if λ > λS p, θθHL . Next, we assume (EA) is binding at the optimum. It suffices to show U (RH ) ≤ 0 given the pooling menu which satisfies 

θH t = D p, λ, θL

 v(q).

The ex-ante expected utility with reference RH is H |θH , RH ) U (RH ) = pU (rLH |θL , RH ) + (1 − p)U (rH

= p(1 − p) [t − λθH v(q)] + (1 − p) [θH v(q) − t + p(θH v(q) − λt)] = (1 − p) {[1 − p(λ − 1)] θH v(q) − [1 + p(λ − 1)] t} . Incorporating binding (EA), this can be further simplified as 1 ×U (RH ) = [1 − p(λ − 1)] [1 − (1 − p)(λ − 1)] θH −[1 + p(λ − 1)] [1 + (1 − p)(λ − 1)] θL . p(1 − p)v(q) To show U (RH ) ≤ 0 under the given assumption, note first that given our assumption p(λ − 1) ≤ 1, we clearly have U (RH ) ≤ 0 if either p(λ − 1) = 1 or p(λ − 1) < 1 and

36

(1 − p)(λ − 1) ≥ 1. Suppose thus that p(λ − 1) < 1 and (1 − p)(λ − 1) < 1 so λ − 1 < 1 min{ p1 , 1−p } ≤ 2, i.e. λ < 3. Then, rewrite the condition U (RH ) ≤ 0 as

θH [1 + p(λ − 1)][1 + (1 − p)(λ − 1)] ≤ =: τ (p, λ). θL [1 − p(λ − 1)][1 − (1 − p)(λ − 1)] λ+1 2

This inequality is implied by (14) since τ (p, λ) ≥




B(p, λ) =

λ+1 2




2−p+pλ 1+p(1−p)λ



, which

can be seen from observing that for any given λ ∈ (1, 3), τ (p, λ) is minimized at p =   1 (1 + λ)2 λ+1 (λ + 1)2 τ ( , λ) = = max B(p, λ). ≥ p∈[0,1] 2 (3 − λ)2 4 2

1 2

and

Proof of Proposition 7: Given (24), the seller’s expected revenue can be expressed as ! Z ¯ Z ¯ Z θ

θ

θ

t0 (s)ds + t(θ) dF (θ) ¯ θ θ ¯ ¯   Z θ¯ 1 − F (θ) = v(q(θ)) θ − dF (θ) + t(θ) − V (θ) ¯ ¯ f (θ) θ Z¯ θ¯ = v(q(θ))J(θ)dF (θ) + t(θ) − V (θ), ¯ ¯ θ

t(s)dF (θ) = θ ¯

(A.14)

¯

where the second equality follows from (24) and integration by parts. Next, the participation constraint for the type θ, can be written as ¯ "Z ¯ # Z θ¯ θ U (θ) = V (θ) − t(θ) − λ (V (s) − V (θ))dF (s) + (t(θ) − t(s))dF (s) ¯ ¯ ¯ ¯ ¯ θ θ ¯ ¯ Z θ¯ Z θ¯ ≥ t(s)dF (s) − λ V (s)dF (s), θ ¯

θ ¯

or λ−1 t(θ) − V (θ) ≤ ¯ ¯ λ+1

Z

θ¯

! t(s)dF (s) .

θ ¯

Thus, from (A.14) and (A.15), we have Z

θ¯

Z

θ¯

t(s)dF (θ) ≤ θ ¯

v(q(θ))J(θ)dF (θ) + t(θ) − V (θ) ¯ ¯ ! Z θ¯ Z θ¯ λ−1 ≤ v(q(θ))J(θ)dF (θ) + t(s)dF (s) , λ+1 θ θ θ ¯

¯

¯

37

(A.15)

or Z θ ¯

θ¯

Z

λ+1 t(s)dF (θ) ≤ 2

!

θ¯

v(q(θ))J(θ)dF (θ) . θ ¯

Thus, the seller’s profit is bounded above by Z θ ¯

θ¯ 

 λ+1 v(q(θ))J(θ) − cq(θ) dF (θ). 2

(A.16)

Now we show that this expression is maximized by setting q(·) constant. To do so, consider R θ¯ any non-increasing q(·) and let q¯ denote its expected value i.e. q¯ = θ q(θ)dF (θ). Then, we ¯

must have ! Z ¯ !  Z θ¯   Z θ¯  θ λ+1 λ+1 v(q(θ))J(θ) − cq(θ) dF (θ) ≤ v(q(θ))dF (θ) J(θ)dF (θ) 2 2 θ θ θ ¯ ¯ ¯ Z θ¯ − cq(θ)dF (θ) θ ¯ !   Z θ¯ λ+1 v(¯ q) J(θ)dF (θ) − c¯ q ≤ 2 θ ¯  Z θ¯  λ+1 v(¯ q )J(θ) − c¯ q dF (θ), = 2 θ ¯

where the first inequality follows from the fact that v(q(·)) is non-increasing while J(·) is increasing, and the second from the Jensen’s inequality. Thus, (A.16) is maximized by a constant q(·). Given the constant quality q¯ that maximizes (A.16), the upper bound of the seller’s profit can be achieved by setting t(θ) = Z θ ¯

θ¯

λ+1 θv(¯ q ), ∀θ 2 ¯

since

λ+1 λ+1 v(¯ q )J(θ)dF (θ) = θv(¯ q) = 2 2 ¯

where the first equality follows from the fact that check that the (IR) constraint is satisfied.

R θ¯ θ ¯

Z

θ¯

t(θ)dF (θ), θ ¯

J(θ)dF (θ) = θ. It is straightforward to ¯

Proof of Lemma 4: Note first that the lowest type θ participates only if ¯ Z θ¯ Z θ¯ U (θ) = V (θ) − t(θ) + (t(s) − t(θ))dF (s) − λ (V (s) − V (θ))dF (s) ¯ ¯ ¯ ¯ ¯ θ θ ¯ ¯ Z θ¯ Z θ¯ ≥ t(s)dF (s) − λ V (s)dF (s) θ ¯

θ ¯

38

(A.17)

or V (θ)(1 + λ) − 2t(θ) ≥ 0, ¯ ¯ which implies that (28) is maximized by setting t(θ) = ( 1+λ )V (θ) for any q(θ) chosen. Given 2 ¯ ¯ ¯ this, it is easy to verify that the participation constraints for all other types are also satisfied since, from (26), U 0 (θ0 ) ≥ 0 or the equilibrium payoff U (·) is increasing while the outside payoff is constant irrespective of type realization, as shown in (A.17). Proof of Theorem 2: We first derive the following lemma whose proof can be found in Section S.2 of the Supplementary Material: Lemma 5. A quality schedule that solves [P c ] must be constant wherever J(·, λ) is decreasing. To make use of Lemma 5, it suffices to check that Jθ (θ, λ) is negative in the desired range. To do so, let us do some tedious calculation to obtain Jθ (θ, λ) = J 0 (θ)Gθ (θ, λ) −

1 − F (θ) Gθθ (θ, λ), f (θ)

where Gθ (θ, λ) =

(λ2 + 2λ)[F (θ)(1 − F (θ)) − θf (θ)] + 2λ + (1 + F (θ))(2 − F (θ)) + 3θf (θ) (2 − F (θ) + λF (θ))2 (A.18)

and Gθθ (θ, λ) =

(λ − 1)(λ + 3) n λ[2θf (θ)2 − F (θ)(2f (θ) + θf 0 (θ))] (2 − F (θ) + λF (λ))3 o − 2θf (θ)2 − (2f (θ) + θf 0 (θ))(2 − F (θ)) .

(A.19)

¯ λ) = J 0 (θ)G ¯ θ (θ, ¯ λ) so the result will obtain if Gθ (θ, ¯ λ) < 0 To prove (a), note that Jθ (θ, ¯ It is straightforward to check from (A.18) and thus, by continuity, Jθ is negative near θ. ¯ λ) < 0 is equivalent to requiring that Gθ (θ,

λ2 +2λ−3 2(λ+1)

>

1 ¯ (θ) ¯ . θf

Note that any solution of [P c ]

is guaranteed to satisfy the global incentive compatibility given Assumption 1. ¯1 To prove (b), we check that under the stated conditions, (i) Gθ (θ, λ) < 0, ∀θ if λ > λ ¯ 1 ; (ii) Gθθ (θ, λ) > 0, ∀θ if λ > λ ¯ 2 for some λ ¯2. for some λ For (i), note first that the coefficient for the quadratic term λ2 in the numerator of (A.18) is negative since F (θ)(1 − F (θ)) ≤ F (θ) < θf (θ) , where the inequality is due to the condition θf (θ) > F (θ). This implies that one can find sufficiently large λ, say

39

¯ 1 (θ), such that Gθ (θ, λ) < 0 if λ > λ ¯ 1 (θ). Clearly, λ ¯ 1 (θ) is continuous in θ so we can let λ ¯ 1 = maxθ∈[θ,θ]¯ λ ¯ 1 (θ). λ ¯

For (ii), let us first see that the expression in the square bracket in (A.19) is positive: 2θf (θ)2 − F (θ)(2f (θ) + θf 0 (θ)) ≥ 2θf (θ)2 − 2F (θ)f (θ) = 2f (θ)(θf (θ) − F (θ)) > 0, where the weak inequality follows from f 0 (θ) ≤ 0. Thus, one can find sufficiently large λ, ¯ 2 (θ), such that Gθθ (θ, λ) > 0 if λ > λ ¯ 2 (θ). Also, λ ¯ 2 (θ) is continuous in θ so we can let say λ ¯ 2 = maxθ∈[θ,θ]¯ λ ¯ 2 (θ). λ ¯ ¯ = max{λ ¯1, λ ¯ 2 }, Jθ (·, λ) is negative everywhere if λ > λ. ¯ Then, by Lemma 5, Letting λ the optimal quality schedule must be constant everywhere. This schedule clearly satisfies the global incentive compatibility, which completes the proof.

References Abeler, J., A. Falk, L. G¨otte and D. Huffman (2011), “Reference Points and Effort Provision,” American Economic Review, 101, 470-492. Armstrong, M. and S. Huck (2010), “Behavior Economics as Applied to Firms: A Primer,” mimeo. Bell, D. E. (1985), “Disappointment in Decision Making under Uncertainty,” Operations Research, 33, 1-27. Benartzi, S. and R. H. Thaler (1995), “Myopic Loss Aversion and the Equity Premium Puzzle,” Quarterly Journal of Economics, 110, 73-92. Camerer, C. F. (2006), “Behavioral Economics,” in R. Blundell, W. K. Newey and T. Persson, eds., Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Vol. II, Cambridge University Press, New York. Carbajal J. C. and Ely, J. (2010), “Mechanism Design Without Revenue Equivalence,” mimeo. Che, Y.-K. and I. Gale (2000), “The Optimal Mechanism for Selling to a Budget-Constrained Buyer,” Journal of Economic Theory, 92, 198-233.

40

Crawford, G. S. and M. Shum (2007), “Monopoly Quality Choice in Cable Television,” Journal of Law and Economics, 50, 181-209. Crawford, V. P. and J. Meng (2011), “New York City Cabdrivers’ Labor Supply Revisited: Reference-Dependent Preferences with Rational-Expectations Targets for Hours and Income,” American Economic Review, 101, 1912-1932. DellaVigna, S. and U. Malmendier (2004), “Contract Design and Self-Control: Theory and Evidence,” Quarterly Journal of Economics, 119, 353-402. Eliaz, K. and R. Spiegler (2006), “Contracting with Diversely Naive Agents,” Review of Economic Studies, 73, 689-714. Eliaz, K. and R. Spiegler (2008), “Consumer Optimism and Price Discrimination,” Theoretical Economics, 3, 459-497. Ellison, G. (2006), “Bounded Rationality in Industrial Organization,” in R. Blundell, W. K. Newey and T. Persson, eds., Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Vol. II, Cambridge University Press, New York. Esteban, S., E. Miyagawa and M. Shum (2007), “Nonlinear Pricing with Self-Control Preferences,” Journal of Economic Theory, 135, 306-338. Genesove, D. and C. Mayer (2001), “Loss Aversion and Seller Behavior: Evidence from the Housing Market,” Quarterly Journal of Economics, 116, 1233-1260. Gill, D. and V. Prowse (2010), “A Structural Analysis of Disappointment Aversion in a Real Effort Competition,” forthcoming in American Economic Review. Grubb, M. D. (2009), “Selling to Overconfident Consumers,” American Economic Review, 99, 1770-1807. Gul, F. (1991), “A Theory of Disappointment Aversion,” Econometrica, 59, 667-686. Hart, O. and J. Moore (2008), “Contracts as Reference Points,” Quarterly Journal of Economics, 123, 1-48. Heidhues, P. and B. K˝oszegi (2005), “The Impact of Consumer Loss Aversion on Pricing,” mimeo. 41

Heidhues, P. and B. K˝oszegi (2008), “Competition and Price Variation when Consumers are Loss Averse,” American Economic Review, 98. 1245-1268. Heidhues, P. and B. K˝oszegi (2010), “Exploiting Naivete about Self-Control in the Credit Market,” American Economic Review, 100, 2279-2303. Herweg, F. and K. Mierendorff (2011), “Uncertain Demand, Consumer Loss Aversion, and Flat-Rate Tariffs,” forthcoming in Journal of the European Economic Association. Herweg, F., D. M´ uller and P. Weinschenk (2010), “Binary Payment Schemes: Moral Hazard and Loss Aversion,” American Economic Review, 100, 2451-2477. Holmstr¨om, B. and P. Milgrom (1987), “Aggregation and Linearity in the Provision of Intertemporal Incentives,” Econometrica, 55, 303-328. K˝oszegi, B. and M. Rabin (2006), “A Model of Reference-Dependent Preferences,” Quarterly Journal of Economics, 121, 1133-1165. K˝oszegi, B. and M. Rabin (2007), “Reference-Dependent Risk Attitudes,” American Economic Review, 97, 1047-1073. K˝oszegi, B. and M. Rabin (2009), “Reference-Dependent Consumption Plans,” American Economic Review, 99, 909-936. Leslie, P. (2004), “Price Discrimination in Broadway Theatre,” RAND Journal of Economics, 35, 520-541. Loomes, G. and R. Sugden (1986), “Disappointment and Dynamic Consistency in Choice under Uncertainty,” Review of Economic Studies, 53, 271-282. Mas, A. (2006), “Pay, Reference Points, and Police Performance,” Quarterly Journal of Economics, 121, 783-821. Mussa, M. and S. Rosen (1978), “Monopoly and Product Quality,” Journal of Economic Theory, 18, 301-317. Shalev, J. (2000), “Loss Aversion Equilibrium,” International Journal of Game Theory, 29, 269-287. Toikka, J. (2011), “Ironing with Control,” forthcoming in Journal of Economic Theory. 42

Tversky, A. and D. Kahneman (1991), “Loss Aversion in Riskless Choice: A ReferenceDependent Model,” Quarterly Journal of Economics, 106, 1039-1061.

43