Product Warranties and Double Adverse Selection

Transcription

1 rodut Warranties and Double Adverse eletion David A. oberman Assistant rofessor of Marketing INEAD Boulevard de Constane Fontainebleau Cede, Frane The author thanks rofessors Andy Mithell, Jak Mintz, Frank Mathewson, and Ralph Winter of the University of Toronto, rofessor ubert Gatignon of INEAD and rofessor Ganesh Iyer of Washington University at t. ouis for their omments and suggestions. In addition, I thank Angela Yau for her eellent researh assistane. I remain responsible for any errors.

2 rodut Warranties and Double Adverse eletion Abstrat There is etensive literature analysing the use of warranties and their appliation as a marketing tool. A frequently ited role for warranty poliy is to failitate prie disrimination when onsumer-types are unobservable. This issue is analysed by researhers inluding Kubo (1986, Matthews and Moore (1987, and admanabhan and Rao (1993. The use of warranties for prie disrimination is analogous to the model for monopoly priing of produts of differing quality developed by Mussa and Rosen (1978. A seond role for warranties is to signal produt quality to onsumers when quality is not observable. Nelson (1974 refers to produts where buyers annot evaluate the quality prior to buying as `eperiene' goods. When the quality of an `eperiene' good is variable, a long warranty an be used to signal better quality beause sellers of premium quality have a ost advantage over sellers of lower quality in terms of offering warranty protetion. This aspet of warranties is addressed by pene (1977, Grossman (1981, Emons (1988, utz (1989 and Gal-Or (1989. Frequently, a warranty does play one role or the other but there are markets in whih there is a need for it to play both of these roles simultaneously. This need eists when buyers annot observe produt quality and sellers annot identify buyers who have heterogeneous valuations for the produt. Markets that meet these riteria inlude the used ar market and the IBM-loned personal omputer market. The warranty literature has yet to onsider the possibility of warranties ating as both a sreen and signal simultaneously. Aordingly the objetive of this study is to develop a model whih eamines optimal warranty poliy under these onditions. The model involves a seller who desires to sell a produt and an optional etended warranty to heterogeneous onsumers. peifially, the seller hooses a base prie and warranty for his produt and the duration and priing for optional etended overage. The seller knows the quality of the produt and the buyer does not. Using the terminology of Akerlof (1970, this situation of asymmetri information is desribed as a `lemons market' and it ours when there is variability in the quality of a good and buyers annot evaluate it before buying. ellers are assumed to know the quality of the produt and this provides them with an informational advantage over the buyer. The objetive is to understand how sellers set warranty menus to maimise profit when the only potential signal of quality to buyers are the pries and warranty lengths hosen by sellers. The main results are first, a seller of premium quality an use a warranty menu to signal premium quality and prie disriminate simultaneously under most onditions. owever, when the inremental benefit of premium quality to buyers is high, the warranty menu annot play both roles simultaneously. The menu an signal premium quality to buyers, but it loses its ability to 1

3 sreen. eond, when the inremental benefit assoiated with premium quality is above a ertain level, a seller of premium quality will alter the menu in order to simultaneously maimise his profit and deter sellers of standard quality from mimiking. The optimal ation for the seller of premium quality invariably involves offering longer warranties to both types of buyers and generally harging higher pries. owever, when the inremental benefit of premium quality is relatively low, the menu hosen by a seller of premium quality is unaffeted by the eistene of sellers of standard quality. Finally, under onditions of two sided adverse seletion, there are several equilibrium onditions that an our in whih sellers do not offer omplete menus (a prie/warranty bundle designed for eah type of onsumer. When the inremental benefit for premium quality is high, a seller of premium quality may be fored to offer a `ollapsed' menu in whih both types of buyers are offered the same bundle. In addition, there are regions in the feasible parameter spae where a seller of standard quality may provide warranty overage for high valuation buyers only (the base warranty is zero or none at all (the base warranty is zero and etended warranties are not offered. Key Words: etended warranty, signalling, sreening, unobservable quality, adverse seletion, prie disrimination, menu of ontrats

4 rodut Warranties and Double Adverse eletion 1.0 Introdution The following quote appeared in a brohure published by Compaq Computer Corp. in 1994 to assist rookies in buying a first personal omputer: "When omparing omputer features, reliability is diffiult to assess. But the length of the warranty is a lue to the dependability of its omputers. Remember, it osts the ompany money to repair omputers under warranty. A longer warranty period reflets the ompany's onfidene that its produts will last." Even when onsumers do not have the time or epertise to assess the quality of produts, they an make useful inferenes about a produt's quality from the length of its warranty. uite simply, when warranties at as signals, a longer warranty signals a better produt. On the other hand, a feature of buying produts suh as major applianes, power tools and stereo equipment is the persistent effort of a salesperson to sell some form of etended warranty protetion 1. Clearly, retailers are making money by onvining some onsumers to buy etra warranty overage. With heterogeneous onsumers, a retailer an inrease profit by offering different prie/warranty ombinations and having buyers hoose the ombination that best suits them (in the eonomis literature, this proess is alled "sreening". But why are retailers so aggressive in the marketing of etended warranties? The answer lies in their tremendous profitability. Business Week (January 14, 1991 reports that ½ of the operating profits for big onsumer-eletroni hains ome from etended warranties and more reently, an artile in the New York Times (July 3, 1995 indiates that retailers may earn as muh as 75% of their gross inome from the sales of etended warranty and servie ontrats. These observations about the marketing of warranties underline two important fats. First, warranties (or ommitments by the seller to repair defetive produts for a speifi period of time are an inreasingly important element in the marketing mi for durable produts. Not only are manufaturers suh as amsung (a major eletronis manufaturer investing substantial funds ($15 million to enhane their warranty programmes (Business Korea, Vol. 1, No. 1, July 1994, but onsumers too are buying etended warranties on a wider array of goods than ever before. eond, warranties an do a lot more than provide etra value to purhasers of durable produts by insuring them against failure. The two roles ited above, signalling and sreening, underline the apaity warranties have to play different roles in different situations. Given that warranties an either signal or sreen, a question that omes to mind is do 1 ears reports that its sales of etended warranties on durable goods eeeds $1 billion (an Franiso Chronile, January 0,

5 warranties ever sreen and signal simultaneously?. Researhers have eamined how warranties signal produt quality when buyers are unertain about it and there is also etensive literature on the use of warranties to sreen onsumers when they have different unobservable `valuations' for warranty overage. Interestingly, researhers have yet to onsider a situation in whih warranty poliy both signals and sreens simultaneously. A first step in addressing this question is to ask whether the need for a warranty to play these roles simultaneously eists in observed markets? eond, assuming that there are markets where this need eists, an and how does warranty poliy sreen and signal simultaneously? In the following setion, we will disuss two well-known markets where there appears to be a need for warranties to signal and sreen simultaneously. That is, there is both variability in the quality of produts (and sellers know more about the quality of produts than buyers and buyers want some degree of warranty protetion (and sellers annot tell whih buyers are most interested in buying warranty protetion. The remainder of the paper is devoted to answering an and how warranty poliy plays these roles simultaneously..0 Bakground Adverse seletion in the eonomis literature refers to a lass of problems where preontratual opportunism by parties possessing private information leads to ineffiieny in the operation of a market. The term is from the insurane industry where insurane ompanies fae "adverse seletion" in the sales of insurane poliies. The problem is that the people most likely to purhase insurane poliies are unfortunately those who are most likely to make laims (Rothshild and tiglitz, When sellers know more about the quality of produts than buyers, we learly have an adverse seletion problem for buyers. owever, in the onditions desribed in setion 1.0, sellers also lak information that is important for ontrating: namely, information on the preferenes of buyers. Aordingly, we desribe this as a ondition of double adverse seletion beause both parties (the buyer and the seller posses private information that has the potential to reate ineffiienies. Two markets haraterised by double adverse seletion are the non-branded personal omputer market and the used ar market 3. In the market for `IBM lone' omputers, a number of manufaturers import omponents from the Far East and assemble personal omputers in North Ameria. When a onsumer onsiders a purhase of a loned omputer, he/she annot neessarily depend on a reliable brand name, Consumer Reports or advertising. Consumers would like a redible signal that might ee Mas-Colell, Whinston and Green (1995 for a review of adverse seletion and its importane in the eonomi literature. 3 In the United tates, the used ar market is estimated to be 100% larger than the new ar market (Bennet, James, "eond-hand ars go first lass", Globe and Mail, Tuesday, August, 1994, B7. 4

6 provide them with useful information. Beause it is more epensive for a firm with low quality omputers to offer the same length of warranty as a firm with better quality omputers, a firm with better quality omputers may signal its higher quality by offering a longer warranty. At the same time, all firms (regardless of the quality of omputer they are selling have an inentive to offer an etended warranty option to etrat additional profit from those purhasers who are more risk averse. This is a situation where onsumers annot be identified a priori and onsumers are not able to observe the quality of the produt in question. Another eample of double adverse seletion ours in the used ar market. Etended warranties are popular beause used ars frequently need repairs. In addition, the Consumer Reports 1997 Buying Guide provides evidene of variability in automobile quality, aross manufaturers, within manufaturers and even within make. Dealers usually have different qualities available beause they purhase their stok through the wholesale market where quality is diffiult to determine. Genesove (1993 points out the mehanis of the wholesale automobile aution provide little time for buyers to eamine the ars and the reputation effets of sellers are not strong. While dealers obtain a range of qualities from the aution, by the time the ar is available for sale, dealers are invariably better informed than buyers about a ar s ondition (dealers have both eperiene and omputer systems that are important in evaluating the ondition of ars. The problem for a potential buyer is that the value of a ar depends on its quality, an attribute of the produt that she annot observe. Coneivably, this problem ould be quikly resolved if onsumers ould ostlessly test ars to measure their quality (many dealers do allow potential buyers to take the ars to independent diagnosti entres for testing. The problem is that signifiant time and osts are assoiated with suh testing and as a result, only a small minority of used ar buyers atually pursue this option for getting better information about a ar's quality. To see how warranty poliy might be affeted by onditions of double adverse seletion, we onduted a survey of used ar dealers in a major metropolitan area. Following the motivation for the problem, information was olleted by a prospetive used-ar buyer for ars in five ategories that were ostensibly in the same ondition. The survey involved disussions with 109 dealers and foussed on ars that were five years old (see Table In general, etended warranties for 5 year old used ars over the power train and have a dedutible of $50. The results of the survey are similar if the data is grouped by manufaturer or by speifi model (however, the data points per ell are signifiantly redued. 5

7 Table 1 urvey of Base and Etended Warranties in the Used Car Market Car Class base warranty 1 yr. etended warranty yr. etended warranty 3 yr. etended warranty ength (months tandard Deviation Average rie tandard Deviation Average rie tandard Deviation Average rie tandard Deviation Domesti Compat Imported Compat Domesti Midsize Imported Midsize Minivans By definition, used ar dealers are "sreening" beause not all ustomers buy etended warranties (they are a useful but epensive option. ubstantial variane in both the length of base warranties and the priing of etended warranties was observed aross all five ategories. This variane ould not be eplained by differenes in the base prie, the features, the options or the appearane of the ars. In addition, differenes of up to 50% in the priing of etended warranties were observed for idential ars of the same year. This suggests that warranties and priing are being used for more than just sreening. It also suggests that the entire menu (the base prie, the base warranty and the priing of etended warranties may have a role in helping buyers to learn about the hidden quality of a potential purhase. Of note, double adverse seletion an be a problem in any seond-hand market for used durable goods where different onsumers desire different levels of warranty protetion. These eamples show that onditions of double adverse seletion do eist in markets where warranties are important. In addition, the information from the used ar market suggests that under these onditions, warranties an and do have a role in both sreening onsumers and signalling quality. The objetive of this paper is to provide insight about how this happens using a game theoreti approah. A model is developed to analyse the optimal ontrats offered by sellers to buyers where the ontrat is omprised of a prie for a produt with a warranty of finite duration. The seller provides the produt and will repair any breakdowns that our during the warranty period. We now review the relevant literature on warranties. 3.0 Bakground iterature There is a rih literature in both eonomis and marketing that relates to the marketing of warranties on durable goods. The objetive of this setion is to review two subsets of this literature: first, the literature that analyses the use of warranties for prie disrimination and seond, the 6

8 literature that onsiders the use of warranties to signal quality to potential buyers. Warranties to rie Disriminate or reen In situations where a firm has market power, it an etrat additional surplus from the onsumer by inluding a repair warranty with the produt. This is the basi bundling result as disussed by Tirole (1990. An etension to this idea is that a warranty an be used to sreen onsumers that have heterogeneous valuations for warranty protetion; the length of warranty is a proy for quantity in a typial seond degree prie disrimination model (onsumers who have higher valuations for a produt buy more of it but at a lower ost per unit. ine 1986, several papers have appeared whih disuss this use of warranties. Kubo (1986, for eample, shows how a monopolist an inrease its profits through the use of an optional quality guarantee when onsumers are heterogeneous. Matthews and Moore (1987 etend this problem to a situation in whih a monopolist has three deision variables: prie, quality (whih is fully observable, and warranty level, instead of two (prie and warranty level. admanabhan and Rao (1993 show how ustomer heterogeneity an arise from risk toleranes whih vary aross onsumers. Given this heterogeneity, sellers an inrease their profits by offering a menu of prie/warranty bundles. Warranties as ignals ignalling is important when one agent to a ontrat is unfamiliar with the quality of the other agent (or his produt and that agent's quality annot be observed prior to ontrating. In the ontet of durable goods, the prinipal (the buyer selets an agent (the seller to perform a task (provide him/her with a durable good but annot observe the harateristis of the good before purhase. As noted by Bergen, Dutta and Walker (199, this problem (whih is also known as a problem of `idden Information' applies to many situations in marketing. With many durable produts, quality annot be evaluated prior to purhase and it beomes evident only after prolonged use. Nelson (1974 refers to these produts as "eperiene goods". Akerlof (1970 underlines how unobservable attributes an interfere with the operation of markets. In the used ar market, he predits that only poor quality ars (lemons will be traded beause buyers have poor information about individual ars that are offered for sale (and learn little by visually inspeting them. ignalling an be used to mitigate the problem noted by Akerlof. When sellers of high quality have a ost advantage in providing warranty protetion over sellers of low quality then warranty poliy an failitate the ehange of higher quality produts. The use of warranties as signals is analysed by pene (1977 who finds the amount of overage offered by manufaturers is a perfet signal of quality in a ompetitive market. This result ours in pene's model due to profit maimising behaviour by firms and the requirement that onsumers' beliefs about quality be 7

9 onsistent with what firms atually offer 5. As previously mentioned, no author has onsidered a situation in whih warranty poliy plays the dual role of sreening and signalling at the same time. The rest of the paper is organised as follows. In setion 4.0, we present the modelling framework that we use to analyse the problem of simultaneous signalling and sreening. In setion 5.0, present the findings of the study and disuss them in the ontet of the underlying assumptions. Finally, in setion 6.0, we disuss the managerial impliations of the study and provide a brief onlusion. 4.0 Overview of the model As noted previously, the objetive of this analysis is to identify the optimal ontrats that a seller of durable goods (with warranty overage will offer to a buyer under onditions of double adverse seletion. The market we onsider is one in whih a buyer purhases no more than 1 unit of produt. We further assume that the seller has a degree of prie setting ability and this follows from a situation in whih buyers have positive searh osts 6. For eample, when a buyer looks for a produt (like a used ar, she usually looks for a ertain brand, with ertain features (4 door for eample, standard transmission, CD player, olour, in a ertain prie range. One she finds a ar that meets these riteria (at either the first or the seond dealer or the nth dealer she visits, the dealer selling the ar has a degree of prie setting ability beause the buyer will need to spend time going to more dealers to find an alternative. It should be noted that there is no guarantee (ahead of time that the produt found by the buyer will be of a ertain quality. everal key assumptions underlie our analysis and we disuss these before proeeding with an eposition of the model. Assumption 1. ellers are risk neutral. This assumption allows us to fous our analysis on the problem of sreening and signalling without inorporating risk-sharing onsiderations. Moreover, sellers make numerous sales and this minimises their aversion to the risk assoiated with an individual transation. Assumption. roduer and purhaser moral hazard are insignifiant. roduer moral hazard an be a fator in the ontet of warranties; however, in the markets where 5 In related work, Grossman (1981 disusses the informational ontent of warranties when statements about quality annot be verified and Gal-Or (1989 eamines the signalling role of warranties in the ontet of differentiated produts. 6 Diamond (1971 notes that a market whih has subsequent searh osts greater than zero results in monopoly-like priing. 8

10 simultaneous sreening and signalling take plae, the seller has limited ability to affet the performane of the produt after it has been sold. Downstream moral hazard is frequently limited with dedutibles and maintenane programmes that indue areful behaviour on the part of purhasers 7. This assumption is also made to fous our analysis on the problem of sreening and signalling. Assumption 3. roduts an be either premium ( or standard ( quality and buyers annot determine quality prior to buying. As disussed in the ontet of the used ar market, this assumption is the basis for the onsumer s lemon problem. If a seller has a premium quality produt, his objetive is to ommuniate the premium quality to onsumers as heaply as possible 8. Assumption 4. Consumers differ in their valuation for warranty overage and sellers annot observe these valuations. Etended warranties are effetively insurane against unertain repair osts. Consumers frequently plae different values on these warranties and hene, we observe signifiant differenes in the amount of warranty overage purhased by onsumers 9. In our simplified market, we assume two different kinds of buyers: the first (type plaes higher value on warranty overage than the seond (type. Assumption 5. Over time, a premium quality produt is assoiated with lower repair osts and higher onsumer satisfation. A premium quality produt is assumed to fail less than a standard quality produt and aordingly, the epeted ost for a seller of providing a repair ontrat (in this ase, an etended warranty is lower. Without loss of generality, we assume linear repair osts with an epeted repair ost per unit time that is lower for premium quality produts i.e. p<s. Given the differene in repair osts, the marginal value of warranty overage to a buyer for a standard quality produt is also higher (the warranty insures against greater potential ependitures. We also assume there are negative aspets to owning a standard quality produt (suh as the time spent getting repairs and poorer performane beyond the ependitures on repairs. Thus, assuming 7 everal papers onsider warranties in the ontet of produer and purhaser moral hazard inluding Cooper and Ross (1985 and Dybvig and utz ( From a modeling perspetive, the need is to have two distint levels of quality. We use the terms premium and standard here instead of the usual high and low beause we wish to avoid onfusion with the two types of buyers (high and low that we have in our market. 9 eterogeneous preferene for warranty protetion an arise from differenes in inome [haked and utton (198] or differenes in risk aversion [admanabhan and Rao, 1993]. 9

11 equivalent warranty protetion, a premium quality produt is assumed to provide greater benefit to a buyer than a standard quality produt. Assumption 6. The value of a produt and hene warranty overage delines over time. Virtually all durable goods depreiate over time due to wear and tear or obsolesene. Aordingly the value of a warranty whih guarantees that broken produts are repaired also delines over time. Using these assumptions as a basis, we now outline the details of the model as it relates to the buyer s deision and the seller s deisions. We then disuss the informational assumptions and the etensive form of the model. The Buyer s Deision We assume that utility derived by a buyer from purhasing a produt of known quality an be represented by a quasi-linear funtion in whih there are three main omponents. The first omponent (B is the benefit that a buyer obtains from a produt of quality (= or without warranty protetion. The seond omponent is the benefit that the buyer obtains from the warranty offered with the ar. The final omponent ( is the total prie paid for the produt and the assoiated warranty overage. The utility funtion for buyers (where is the length of the warranty is: U(,,, = B V( (1 The seond term is the produt of three items. Following from Assumption 5, the first item is a parameter used to apture the assumption that the marginal value of warranty overage for a premium quality produt is less then the marginal value of warranty overage for a standard quality produt i.e. item <. The seond item T is a valuation parameter, whih is different for the two types of buyers in our model. Consistent with Assumption 4, the valuation parameter for buyers who plae a higher value on warranty overage (type is larger >. We assume that a fration `' of buyers are type and `1-' are type. V( is a funtion that allows us to reflet Assumption 6 (the value of warranty overage delines over time. Mathematially, this implies that V( has the following properties: V (>0, and V (<0. We further assume that a warranty of zero length has no value i.e. V(0=0. To simplify the analysis, we utilise the following form for V(: 1- (1- V(= ( This funtion ehibits the required properties for ε [0,1] and satiation at =1. ine most durable goods have a finite life, we model the warranty has having a limit beyond whih it is of little value. aving hosen this funtional form, we an now speify the onstraint implied by Assumption 5. T 10

12 When B B > BMIN =, all buyers prefer premium quality to standard quality given equivalent pries and warranty overage. Two further omments are neessary to fully eplain the buyer s deision proess. First, a buyer will not purhase unless the offering (i.e. the prie, warranty overage and quality is epeted to yield a pre-determined level of reservation utility (this is referred to as the Individual Rationality Constraint. The level of reservation utility an be any number that reflets the outside options of buyer but to simplify the analysis, we assume that reservation utility is zero 10. A seond issue onerns the deision proess of the buyer if he faes two options that both provide positive utility. We assume that the buyer hooses the option that yields the maimum utility and this is aptured mathematially through an Inentive Compatibility Constraint. ellers The epeted profits of the seller are a funtion of the distribution of buyers in the market and the marginal ost of the produt is assumed to be zero 11. Thus, a seller will maimise the following funtion: π (,,, = [ - ] (1 - [ - ] (3 The warranty/prie bundles (, and (, are purhased by the high and low valuation buyers respetively and is the ost per unit time of providing warranty overage for a seller of quality `' (= or. The warranty length an be thought of as the base warranty and (- is the length of the etended warranty that an be purhased for (-. The two options available to onsumers (the produt without an etended warranty, the produt with an etended warranty are referred to as two points (, and (, in warranty length/prie spae. Consistent with Assumption 5, the repair osts inurred by sellers are a linear funtion of the lengths of warranty overage. Two other onditions are worth mentioning in the ontet of the menus offered by sellers. In order for a seller to have an inentive to offer a positive warranty, the most willing buyer must be willing to pay more for the warranty than it osts the seller to provide it 1. This redues to simple onditions: >, > (4 For the low valuation buyer to purhase warranty protetion, these onditions must also be satisfied for. 10 This assumption implies that high and low valuation buyers have an inentive to purhase regardless of whether they believe the produt to be standard or premium quality. Choie of a positive reservation utility allows for a ase in whih a low valuation onsumer walks away from a seller that offers standard quality. 11 Assuming zero marginal ost does not affet the results; prodution ost drops out of the first order onditions. 11

13 A seond issue onerns the distribution of buyers in the marketplae. As we will see later, a key onstraint faed by a seller in designing a menu for two types of buyers is that he annot harge the high valuation buyer the maimum prie he is willing to pay. Thus, when the perentage of low valuation buyers beomes suffiiently low, it an be optimal for a buyer to offer warranty overage to high valuation buyers only (the low valuation buyer is offered the produt without warranty overage. We allow for this possibility in our analysis but fous on the interesting ase when both low and high valuation buyers are offered positive levels of warranty protetion 13. Informational Assumptions ellers' ost strutures and buyers' utility funtions are assumed to be ommon knowledge. As previously mentioned, the buyer annot tell a priori whether the seller is offering premium or standard quality and the seller annot tell a priori whether the buyer has a low or high valuation for warranty overage. In order for the buyer to figure out whether the seller is offering high or low quality, the buyer forms beliefs about the epeted ations of a seller of premium quality versus a seller of standard quality. The Cho-Kreps riterion (1987 is used to find a set of beliefs (for buyers that are reasonable and it is these beliefs that allow the derivation of a unique separating equilibrium. Etensive Form of the Game Although the game is modelled as a simultaneous single shot game in whih the players hoose optimal strategies, there is an implied order of play: tage 1. The seller hooses a menu of prie/warranty bundles and will announe them to any buyer who has interest in the partiular produt that the seller is offering. tage. A buyer arrives at the seller's plae of business and shows interest in the produt that the seller is offering. tage 3. The buyer deides whether to purhase any of the bundles that the seller announes. Following from our previous disussion, the seller s prie setting ability is aptured by having him hoose pries and warranty lengths first. For the buyer, the eistene of outside options is reognised through her Individual Rationality Constraint i.e. the more attrative (ompetitive outside options are, the more likely it is that a buyer will leave the seller without making a purhase. 1 This ondition is evaluated at =0, the point at whih the marginal value of warranty protetion is highest. 13 As shown in the tehnial appendi, a neessary ondition for low valuation buyers to be offered positive 1

14 We now derive the equilibrium warranty menus in a market where onditions of double adverse seletion are present. The approah we use is to first desribe the outomes that obtain in our model when produt quality is fully observable. We then desribe the outomes that result when the seond level of information asymmetry is added. This allows us to isolate and understand the impat that produt quality unertainty has on the ations of a seller that is using warranty poliy to sreen buyers. 5.0 Equilibrium Contrats in a Market under Conditions of Double Adverse eletion First, we onsider the ation that a seller would take were produt quality observable. Clearly the buyer's valuation of a given warranty/prie bundle depends on the quality of the produt. When quality is observable, the seller knows the value that the buyer will plae on a produt with a given warranty length. With observable quality, we solve a onstrained optimisation problem with the objetive funtion shown in equation 3. This objetive funtion is based on buyers seleting the appropriate bundle: (, or (, depending on their type ( or. ine all buyers have the option of buying both bundles, it is ritial that buyers voluntarily selet the bundle that is designed for them. This leads to two Inentive Compatibility Constraints, one for eah type of buyer. For highs For lows B B V( - B V( - (5 V( - B V( - (6 As previously disussed, we also assume that both buyer types realise positive surplus by purhasing and this is refleted through Individual Rationality Constraints. For highs B V( - 0 (7 For lows B V( - 0 (8 Finally, we onstrain pries to be positive and warranty lengths to values between zero and one. The key elements of the solution are that equations 5 and 8 bind with strit equality. This means that a high valuation buyer is indifferent between the bundle designed for her (, and the bundle designed for a low valuation buyer (, and a low valuation ustomer is indifferent between purhasing her bundle and not buying at all 14. We summarise the nature of the solution to this problem in roposition 1. warranty protetion is < (- /(-. 14 In spite of the high valuation buyer s indifferene between the two bundles, she is assumed to hoose the bundle designed for her. This is a typial assumption in self-seletion problems justified beause the high valuation buyer prefers her bundle stritly with an infinitesimal redution in. 13

15 roposition 1 When produt quality is observable, the profit maimising ation for the seller is to offer a menu of prie/warranty ombination where eah buyer-type self selets to a bundle designed for her. roposition 1 onfirms the earlier results of Mussa and Rosen (1978 and Maskin and Riley (1984 that relate to seond degree prie disrimination (how an a firm prie disriminate when onsumers have unobservable heterogeneity. The basi hallenge for the seller is that he wishes to serve two types of buyers, one of whih is willing to pay more for warranty overage than the other. On the one hand, he ould hoose to serve only the type who is willing to pay the most for warranty overage (type but then he would serve only % of the market and leave a signifiant number of profitable buyers unserved. On the other hand, he ould set a suffiiently low prie on a single bundle suh that all buyers happily purhase. The problem with this approah is first, that high valuation buyers would obtain a benefit that the seller ould potentially harge for and seond, the warranty overage would be signifiantly less than the optimal overage that ould be sold to high valuation buyers. Essentially, the seller is looking for a way to sell high valuation buyers more warranty overage at a higher prie and still keep low valuation buyers in the market. The solution to this dilemma is to offer a number of warranty/prie bundles to eah buyer and onstrut the menu in suh a way that high valuation buyers hoose a more epensive bundle with more warranty overage. If we restrit our attention to values of suh that <, the seller has an inentive to offer positive warranty lengths to both types and the optimal menu for the seller is: (, = 1 - (,, B = 1- ( 1 ( 1 1 ( ( (1,B (1 1 ( (9 (10 Mathematial omparisons lead us to roposition whih underlines the prinipal elements of the seller s sreening menu when produt quality is observable. roposition When quality is observable, the profit maimising menu for the seller has: (a A bundle designed for the high valuation buyer with warranty protetion of effiient length and a prie whih leaves her with stritly positive utility. (b A bundle for the low valuation buyer with a warranty that is shorter than the effiient length and a prie whih leaves her indifferent between buying or not. 14

16 roposition underlines three key aspets of the optimal sreening menu for a seller of known quality. First, offering two bundles to buyers ertainly inreases profit for the seller but it is not ideal (theoretial profit is left downstream with buyers. In a sense, the seller needs to provide high valuation buyers with a subsidy (or lower prie to buy the more epensive bundle. This situation obtains beause regardless of how short the warranty is in the low valuation buyer s bundle, it will be attrative to a high valuation buyer sine she is willing to pay more for warranty overage than a low valuation buyer. Any bundle designed for the high valuation buyer must provide at least the benefit that the high valuation buyer would obtain from the low valuation buyer s bundle. The seond point is that this subsidy (or prie redution to high valuation buyers is in some sense pre-determined (or fied by the harateristis of the low valuation buyer s bundle. Aordingly, the seller maimises his profit on sales to high valuation buyers by maimising the differene between the prie he an harge for the high valuation buyer s bundle and his epeted osts. This ours at the warranty length where the marginal benefit of additional warranty overage to the buyer is eatly equal to the marginal ost of providing warranty overage: i.e. when (1-=. Thus, even though, the seller is unable to etrat the maimum prie from the high valuation ustomer, he nonetheless hooses a warranty length for her that is effiient. Finally, a key drawbak in meeting the Inentive Compatibility Constraint for the high valuation buyer is the ost of the subsidy. The longer is the warranty overage in the low valuation buyer s bundle, the greater is the subsidy (or prie redution needed in the high valuation buyer s bundle. This eplains why the optimal menu for the low valuation buyer involves a warranty that is stritly shorter than the effiient length. The seller gives up some profit on low valuation buyers in order to redue the subsidy that he needs to pay high valuation buyers to ensure self-seletion. Naturally, the degree of this redution is a funtion of the distribution of types in the market. As the fration of low valuation buyers in the market inreases (i.e. dereases, the redution in warranty overage ( from an effiient level gets smaller. The Impat of uality on the reening Menu when uality is Observable When buyer types are not observable but the quality of the produt is, the seller of premium quality offers stritly more epensive bundles than the seller of standard quality to eah type of purhaser. This is illustrated graphially in Figure 1 where the menu offered by the seller of premium quality lies above the menu offered by the seller of standard quality. The high valuation buyer reeives the surplus maimising bundle (i.e. this is refleted by the tangeny of the isoprofit lines and indifferene urves at regardless of whether she finds herself dealing with a seller of premium or standard quality. On the other hand, the low valuation buyer reeives a bundle whih lies on her reservation utility indifferene urve but is stritly shorter than her surplus maimising bundle. (Figure 1 15

17 There are primarily two fators, whih affet the relative loation of the two menus in warranty length/prie spae. The first is the ratio of to. When the ratio is large, the isoprofit lines of the premium quality seller are muh flatter than those of the standard quality seller. This will move the premium quality seller s menu to the right, as he will find it advantageous to offer his ustomers longer warranty protetion. On the other hand when the differene between B and B is large (the fied benefit differene between premium and standard quality, the pries a premium quality seller an harge are muh higher and this tends to move the premium quality sellers upwards. A seller of standard quality would love to obtain higher pries for his produt/warranty ombinations but on the other hand, he does not want to provide etra warranty overage sine his osts of doing so ( are high. We an interpret this in the ontet of the Figure 1. When the menu of the premium quality seller lies signifiantly to the right (with long warranties, the standard quality seller is unlikely to want to mimi the premium quality menu. While he would get higher pries, he would also have to finane longer warranty overage. In ontrast, when the menu of the premium quality seller lies above the menu of the standard quality seller, the standard quality seller would love to offer the warranty/prie ombinations that are optimal for the premium quality seller. This is the essene of the double adverse seletion problem. When quality is not observable, a wary buyer will not pay premium quality pries for a produt/warranty ombination unless she is sure that standard quality seller would be unwilling to offer her the same deal. Optimal reening Menus when uality is Unobservable A simple way of desribing the problem reated when produt quality is unobservable is that we are looking for a market outome in whih sellers maimise profit and buyers atually obtain the quality that they think they are buying at the time of purhase. In other words, a situation in whih a buyer thinks that she is purhasing premium quality but atually obtains standard quality is not an equilibrium 15. Based on the disussion in the preeding paragraph, we need to address two situations. In the first, were quality observable, the premium quality seller s menu would lie in a region where the standard quality seller does not have an inentive to pretend to be a seller of premium quality i.e. the premium quality produt is offered with long warranties. In the seond, were quality observable, the premium quality seller s menu would lie in a region where a standard quality seller does have an inentive to pretend to be a seller of premium quality i.e. the premium quality seller s menu has high pries. The objetive is first, to desribe the boundary that delineates the two situations and seond, to identify equilibrium ation for the premium quality seller in eah situation. 15 Tehnially, it would not be an equilibrium for a buyer to epet standard quality before purhasing and atually reeive premium quality. owever, in our model, this situation is irrelevant sine a premium quality 16

18 The Non-Mimi Constraint The key problem when quality is unobservable rests with buyers who lak information about the quality of the produts that they are buying. To help buyers make deisions, we assume that buyers form beliefs about produt quality and these beliefs are based on prior information and ations (suh as the warranty/prie menus that are announed. This is the essene of signaling models in whih uninformed players (i.e. the buyers make inferenes about informed players (i.e. sellers who know the quality they are selling based on ations taken by informed players. The tehnial approah to this problem involves Bayesian equilibria, in whih the ations and beliefs (of players are speified as reasonable outomes in games of inomplete information. Following Mas-Collell, Whinston and Green (1995, the equilibrium onept to apply is that of the erfet Bayesian Equilibrium (BE. As noted by Fudenberg and Tirole (199 and Mas-Collell, Whinston and Green, this onept imposes restritions on the beliefs of players for ations that are part of the equilibrium. owever, there are no restritions on the beliefs of players for ations that are not part of the equilibrium. As a result, multiple separating equilibria as well as pooling equilibria (where both premium and standard quality sellers take idential ations are possible in signalling problems under the BE onept. owever, in many ases, these equilibria are supported by off equilibrium beliefs that are inherently unreasonable. To address this problem Cho and Kreps (1987 propose the Intuitive Criterion whih plaes a restrition on the beliefs of buyers off the equilibrium path. Essentially, a buyer annot attribute positive probability to a seller-type offering a given warranty/prie menu if the warranty/prie menu would make the seller-type worse off 16. In addition, the Intuitive Criterion points to the equilibrium where the premium quality seller has no alternate ation (i.e. a warranty/prie menu, that yields a higher payoff given the restrition on buyer beliefs. The net effet of the Intuitive Criterion is to eliminate both pooling equilibrium and separating equilibrium with ineffiient amounts of signalling. In fat, the only equilibrium not rejeted by the Intuitive Criterion is a separating equilibrium with the least amount of ineffiient signalling. Going bak to our problem, when the premium quality seller s optimal menu under observability is unattrative to the seller of standard quality, the least amount of ineffiient signalling for the premium quality seller is zero (i.e. his ations are unaffeted. This obtains sine a buyer s beliefs about the quality will be inferred from the menu that is offered and a standard quality seller would lose money by doing the same thing. In ontrast, when the premium quality seller s optimal menu under observability is attrative to the seller of standard quality, the premium quality seller needs to signal premium quality through his ations. ere we use the Intuitive Criterion to identify a unique separating seller never has an inentive to pretend to be standard quality. 16 The tehnial term for this restrition is that a player annot attribute positive beliefs to a type taking an ation for whom the said ation is equilibrium dominated. 17

19 equilibrium. Following the Criterion, we need to find a warranty/prie menu for the premium quality seller where no alternative will provide him with higher profit given reasonable beliefs by the buyers. This menu an be identified by introduing a onstraint, the `no mimi' onstraint, to the premium quality seller s optimisation problem. This onstraint simply restrits the premium quality seller s hoies to menus that are unattrative to a seller of standard quality. When the `no mimi' onstraint binds, a seller of premium quality hooses a menu for whih the seller of standard quality is indifferent between mimiking and not mimiking. The no mimi onstraint implies that if a standard quality seller hooses to offer the menu,, and (the menu hosen by the premium quality seller, he will make less profit than he makes by offering his optimal menu as a known seller of standard quality ( π ma 17. ma π > ( ( 1 ( (11 The problem for a seller of premium quality is analogous to the problem onsidered when quality is observable subjet to the additional onstraint shown in equation 11. Thus, we maimise the objetive funtion (equation 4 subjet to inentive ompatibility and individual rationality onstraints (equation 5-8 and the no-mimi onstraint (equation 11. The solution in the tehnial appendi leads to the following proposition: roposition 3 When produt quality is unobservable: (a and the fied benefit B of a premium quality produt is less than Bˆ, the warranty/prie menu offered by a seller of premium quality is idential to the one offered when produt quality is observable. (b and the fied benefit B of a premium quality produt is greater than Bˆ, the prie/warranty menu offered by a seller of high quality is different from the one offered when produt quality is observable. MAX (1 Where: = ~ Bˆ π ( ~, = and MAX π is the profit realized by a seller of standard quality when quality is observable. The epression for MAX π (1 ~ is given by: (1 ~ (1 ~ MAX π = B 1 Before providing the intuition for this result, we provide insight about the epression for Bˆ, whih appears very omple. Bˆ is a funtion of the relative osts of premium and standard quality sellers ( and, the relative values of warranty overage for buyers of premium and 17 ine there are only types of sellers in this model, a buyer believes that a seller is standard quality unless 18

20 standard quality respetively ( and and B the fied benefit assoiated with standard quality. It desribes the premium level B above whih the menu of a premium quality seller under observability will be attrative to a seller of standard quality. The intuition behind roposition 3 is that when the fied benefit for premium quality is low i.e. B { BMIN,Bˆ }, the ost advantage of the premium quality seller (i.e. the degree to whih is less than overshadows higher pries that buyers are willing to pay for premium quality (pries are a diret funtion of the premium B 18. As a result, the optimal menu hosen by the seller of premium quality is epensive for the standard quality seller to mimi (B is not suffiiently high in relation to B. In fat, when buyers have a high fied benefit for standard quality (B or when the differene between the marginal osts of providing overage is high ( versus, it is unlikely that a standard quality seller will mimi the optimal menu of the premium quality seller (under observability. We should also disuss the role of the and (the marginal value of warranty protetion parameters in the equilibrium. First, the higher is (the marginal value of warranty overage on standard quality produts, the higher is Bˆ beause a higher stritly inreases MAX π. The effet of an inrease in (the marginal value of warranty overage on premium quality produts on Bˆ is ambiguous beause it has two effets. The first is to ause the premium quality seller to offer more overage making the menu (under observability less attrative to a seller of standard quality. The other is to inrease the pries in the menu of the premium quality seller and this of ourse, makes the menu attrative to the seller of standard quality. imulations suggest that for most values of, an inrease in raises Bˆ and redues the likelihood of the premium quality seller s menu being affeted. owever, when either the differene between and is small or is high, the effet an be negative. When B>Bˆ, the menu that would be hosen by a seller of premium quality (when quality is observable is attrative to a seller of standard quality. In this situation, the solution to the onstrained optimisation problem for the premium quality seller yields a positive agrangean multiplier on the no-mimi onstraint (equation 11. This has three important eonomi interpretations. First, the solution to the problem involves the premium quality seller hoosing a menu for whih the seller of standard quality is preisely indifferent between mimiking and not mimiking. This menu is different from the menu that the premium quality seller would offer under onditions of observable quality. eond, relaing the no-mimi onstraint by one unit (for eample by inreasing B by one, the profit of the premium quality seller would inrease by an amount equal to the value of the multiplier. Finally, the profit of the premium quality seller is stritly redued due to the unobservability of quality (profit will inrease when the onstraint is relaed and the profit under observability is obtained by solving the problem with the onstraint fully relaed. We now onsider roposition 4 whih onsiders the speifi ations of the premium quality he has reason to believe otherwise. 18 As noted earlier in the paper, B MIN is the minimum value of B onsistent with our basi assumptions. 19