Working Papers R & D SIMULTANEOUS SIGNALLING AND SCREENING D. A. SOBERMAN* 2001/07/MKT

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1 Working Papers R & D SIMUTANEOUS SINAIN AND SCREENIN WIT WARRANTIES by D. A. SOBERMAN* 001/07/MKT * Assistant Professor of Marketing at INSEAD, Boulevard de Constance, Fontainebleau Cedex, France. A working paper in the INSEAD Working Paper Series is intended as a means whereby a faculty researcher s thoughts and findings may be communicated to interested readers. The paper should be considered preliminary in nature and may require revision. Printed at INSEAD, Fontainebleau, France.

2 Simultaneous Signalling and Screening with Warranties David A. Soberman* February 001 *The author is an Assistant Professor at INSEAD, Boulevard de Constance, Fontainebleau Cedex, France. I am grateful to Frank Mathewson, Jack Mintz, Andy Mitchell, and Ralph Winter for their advice and many helpful suggestions. In addition, the support provided through the S.S..R.C., the Faculty of Management and the Institute of Policy Analysis at the University of Toronto and the INSEAD R&D Committee is gratefully acknowledged. I would also like to thank Erin Anderson, Markus Christen, Anne Coughlan, ubert atignon, anesh Iyer, and Craig Smith for their comments and Angela Yau for her excellent research assistance. Any errors remain my responsibility.

3 Simultaneous Signalling and Screening with Warranties Abstract It is well known that sellers can use warranties to screen consumers and increase profits. The ability of warranties to signal is also well accepted. This paper considers a situation where a high quality seller needs warranty policy to do both. Using an analytical model, our objective is to identify the optimal strategy for a high quality seller who offers a base warranty and optional extended warranty for a product where product quality is not observable to buyers. We find that signalling can limit the ability of a seller to screen especially when buyers are willing to pay a significant premium for higher quality. To signal, the seller generally lengthens base warranties and shortens optional coverage making the bundles for each type of buyer more and more similar. We also provide an empirical application of the model in the Toronto used car market. Key Words: extended warranties, signalling, screening, unobservable quality, price discrimination, menu of contracts, used cars.

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5 1.0 Introduction 1.1 Background Consider the following quote from a brochure published by Compaq Computer Corp. to assist novices in buying a first personal computer: "When comparing computer features, reliability is difficult to assess. But the length of the warranty is a clue to the dependability of its computers. Remember, it costs the company money to repair computers under warranty. A longer warranty period reflects the company s confidence that its products will last." Even when consumers do not have the time or expertise to assess the quality of products, they can make useful inferences about a product s quality from the length of its warranty. uite simply, when warranties act as signals, a longer warranty signals a better product. On the other hand, a defining feature of buying products from major appliances to power tools is the persistent (if not aggressive) effort of salespeople to sell some form of extended warranty protection. For example, Sears reported sales of more $1 billion for extended warranties on durable goods (San Francisco Chronicle, January 0, 199). Economic theory predicts that sellers can increase profits by offering different price/warranty combinations to heterogeneous buyers and having each buyer choose a preferred combination (this is also known as "screening"). Empirical evidence confirms this prediction as Business Week (January 14, 1991) reports that ½ of the operating profits for big consumer-electronic chains come from extended warranties. In addition, an article in the New York Times (July 3, 1995) indicates that retailers may earn as much as 75% of their gross income from the sales of extended warranties and according to a large Canadian retailer, margins on extended warranties range from 60 to 70 per cent. These observations underline two important facts. First, warranties are an important part of the marketing mix for durable products. Manufacturers such as Samsung (a major electronics manufacturer) have invested substantial funds ($15 million) to enhance their warranty programmes (Business Korea, Vol. 1, No. 1, July 1994) and consumers are buying extended warranties on a wider array of goods than ever before. Second, warranties can do more than provide extra value to purchasers of durable products by insuring against product failure. The two roles cited above, signalling and 1

6 screening, underline the capacity warranties have to play different roles in different situations. In the used car market, warranty protection is popular because used cars frequently need repairs. owever, there are two further characteristics of the used car market that make the selling of warranty protection quite interesting. The first is the existence of information asymmetry with respect to the quality of cars that are for sale. In general, a dealer is better informed than the buyer about the condition of a car that he has for sale and a buyer will obviously be disappointed if she mistakenly overestimates the quality of a car. This underlies Akerlof s (1970) observation that it is difficult for a seller with a high quality car to charge a higher price since a seller with a low quality car (a lemon) has an incentive to do exactly the same thing. Thus, signalling higher quality with warranty policy may be a very an interesting option for a seller to convince buyers about a car s quality. The second characteristic of the used car market is significant heterogeneity in buyers evaluations for warranty coverage. As a result, we observe a preponderance of menu-based selling of extended warranties for used cars. Buyers are faced with choosing both the length of coverage (1, 4, or 36 months) and the nature of coverage (power train, bumper to bumper) whenever, they buy a used car. Interestingly, the cost of such coverage is not found on the printed brochures in the showroom (as is sometimes the case with new appliances) but is quoted verbally by the salesperson. These facts suggest it is highly plausible that sellers use warranty policy to signal and screen simultaneously in the used car market. In fact, in any market where there is information asymmetry about quality and buyers have heterogeneous preferences for warranty coverage, it is plausible that warranty policy might play this dual role. The objective of this paper is to ask what happens when warranty policy plays this dual role? The literature provides clear guidance on how a warranty can be used to signal (Spence 1977) i.e. the warranty for a high quality product will be longer. The literature also indicates how warranty policy can be used to screen (Mussa and Rosen 1978 and Padmanabhan and Rao 1991) i.e. a set of options are constructed such that buyer types that desire more warranty coverage select longer coverage and pay more. But the literature does not address a situation in which both signalling and screening occur simultaneously, nor can we easily combine the predictions of signalling and screening to predict what will happen. Accordingly, this paper seeks to provide answers to the following two questions:

7 1. When a seller finds a potential buyer, is his ability to screen the buyer with warranty policy affected when he also uses the policy to signal quality to the buyer?. When a seller uses warranty policy to signal quality to consumers, how is the information transmitted to the buyer (through the base warranty, through the extended warranties or both) and does the answer to this question have implications for the selling process in a market where signalling and screening take place? 1. Framework and Results We develop a model with a monopolistic seller who wishes to sell a product and an optional extended warranty to a heterogeneous market with two types of buyers. The model is designed to capture two features of a market where signalling and screening occur simultaneously: the first is the non-observability of the product s quality for buyers and the second is the nonobservability of the buyer s type for sellers. This situation creates a need for the seller to signal with his warranty policy and design a menu where a buyer reveals her type by selecting an option that is designed for her. With only two types of buyers in this stylised market, the seller chooses a base price and base warranty for a candidate product and unique choice of extended coverage. The objective is to isolate the impact that signalling has on the warranty menu that is chosen by a high quality seller. Our analysis shows that when quality is unobservable, a high quality seller sells longer warranty coverage than a low quality seller to every buyer. Moreover, when the premium that buyers are willing to pay for high quality is small, a high quality seller may naturally choose a price/warranty menu that a low quality seller would not want to mimic. owever, when the premium for high quality is sufficiently high, the seller s actions are affected by the need to signal and this leads to our first important finding: a high quality seller s ability to screen is reduced by signalling. In particular, the high quality seller will lengthen the base warranty and shorten the optional coverage making the bundles for each type of buyer more similar. In fact, when the premium that buyers place on high quality is large, the high quality seller loses the ability to screen completely and he sells the same price/warranty combination to both buyers. Our second important finding is that a high quality seller maximises profit by signalling with all elements of his warranty menu. Therefore, even a buyer who has no interest in buying an extended warranty, will use the pricing of an extended coverage to learn about a product s quality. This finding has implications for the used car selling process and suggests that optional warranty coverage will receive strong emphasis early in the sales pitch. This stands in contrast to 3

8 the approach that is employed for selling extended warranty coverage with new cars and appliances. To provide empirical support for the model, we discuss a survey of the used car market. The information collected provides evidence of significant unexplained variability in the length and prices of warranty coverage across very similar vehicles. Consistent with the model s prediction that the entire menu is used to provide information to buyers, used car dealers volunteer complete details about warranty policy in a significant majority of selling encounters. Finally, as would be suggested by the model, we find preliminary evidence of a number of sellers attempting to shift consumers to longer warranty lengths. 1.3 Related Research There is a rich literature in both economics and marketing that relates to the marketing of warranties on durable goods. The objective of this section is to review two subsets of this literature: first, the literature that analyses the use of warranties to signal quality to potential buyers and second, the literature that considers the use of warranties for price discrimination. There is also a significant literature on the use of warranties under conditions of moral hazard (for example, Cooper and Ross 1985 and Dybvig and utz 1993). owever, in these models, at least one of the parties takes non-contractible action that affects the other (the seller chooses quality and/or the buyer exerts effort in caring for the product). In contrast, our model considers a context where no action is taken by either the seller or the buyer. The problem we address is entirely one of hidden information and not hidden action. Warranties as Signals Signalling is important when one agent to a contract is unfamiliar with the quality of the other agent (or his product) and that agent's quality cannot be observed prior to contracting. In the context of durable goods, the principal (the buyer) selects an agent (the seller) to perform a task (provide him/her with a durable good) but cannot observe the characteristics of the good before purchase. With many durable products, quality cannot be evaluated prior to purchase and becomes evident only after prolonged use. Nelson (1974) refers to these products as "experience goods" and Akerlof (1970) underlines how unobservable attributes can interfere with the operation of markets. In these conditions, signalling is a mechanism that can facilitate the exchange of products of varying quality at different prices. Potential signals include advertising (Klein and effler 1981, Schmalensee 1978 and Milgrom and Roberts 1986), price (Bagwell and 4

9 Riordan 1986, Srinivasan 1991, Balachander and Srinivasan 1994), sale signs (Anderson and Simester, 1998) and distribution strategy (Chu and Chu 1994). Because a fundamental dimension of quality is performance (higher quality products either fail less, cost less to fix, or perform better), the use of performance warranties as signals has also received attention (Spence 1977, rossman 1981, utz 1989, and al-or 1989). With performance as a primary dimension of quality, it costs a seller less to guarantee performance for a high quality product. Thus, a seller can afford to offer a longer warranty to signal higher quality. The general conclusion that this literature makes is that signalling causes warranties for high quality products to be longer. owever, the signalling literature does not provide insight about how a warranty menu (that is being used to screen) will be modified. Warranties to Price Discriminate or Screen In situations where a firm has market power, it can extract additional surplus from the consumer by including a repair warranty with the product. This is the basic bundling result as discussed by Tirole (1990). An extension to this idea is that a warranty can be used to screen when consumers have heterogeneous valuations for a product. ere, the degree of warranty coverage is essentially a proxy for quantity in a typical second degree price discrimination models such as Mussa and Rosen (1978) and Maskin and Riley (1984). This use of warranties is discussed by Kubo (1986) who shows how a monopolist can increase its profits when consumers are heterogeneous through the use of an optional quality guarantee. Matthews and Moore (1987) extend this idea to a more complicated scenario in which the monopolist constructs a screening menu with to three decision variables: price, quality (which is fully observable), and warranty level instead of two (price and warranty level). Similarly, Padmanabhan and Rao (1993) show how a menu of price/warranty bundles can increase seller profits when customer heterogeneity arises from risk tolerances. The principal insight provided by this literature is that a seller with price setting ability can increase profit by constructing a set of price/warranty coverage options for buyers. owever, in contrast to our problem, this literature is restricted to situations where quality is either fully observable or fixed. Papers that Consider both Signalling and Screening In spite of the absence of signalling and screening together in the warranty literature, there are two papers that consider both practices in a different context. Chu (199) presents a model with which compares screening and signalling as mechanisms for resolving one-sided information 5

10 asymmetry (the manufacturer is assumed to have knowledge about the product demand that the retailer does not have). This contrasts significantly with the situation we consider where there is two-sided information asymmetry and both screening and signalling occur simultaneously. Balachander and Srinivasan (1994) consider a situation in which screening and signalling happen in the same model. They consider an incumbent who has the potential to screen heterogeneous consumers with different price/quality combinations and may wish to send signals to an entrant about his cost structure. This situation is different from the problem we consider. First, the target of the signal in Balachander and Srinivasan (B & S) is a third party (a potential entrant) not the party being screened (buyer). Second, B & S consider a two-period model in which the objective of the incumbent s first period signal is to deter entry in the second period. ater in the paper, these differences help us to better understand the results generated by our analysis. In the next section, we present an analytical model that is used to identify equilibrium warranty policy of a high quality seller in a market where warranties play a dual role. In section 3.0, we present the analysis and outcomes generated by the model and in section 4.0, we present empirical information to test some of the model s predictions. We conclude in section Overview of the Model We focus our analysis on the problem faced by a seller of high quality because he is the type of seller who can gain by signalling higher quality. The objective is thus, to identify the optimal warranty menu that a seller of high quality offers to a buyer when i) a buyer is uncertain about the seller s quality and cannot evaluate quality easily and ii) buyers have heterogeneous valuations for warranty coverage (that cannot be recognised by sellers). The market we consider is one in which a buyer purchases no more than 1 unit of product. We further assume that the seller has a degree of price setting ability and this follows from a situation in which buyers have positive search costs (Diamond 1971). Of course, price setting ability is affected by the proximity of competing sellers and the closeness of their options to those offered by the focal seller. Several key assumptions underlie our analysis and we discuss these before proceeding with an exposition of the model. 6

11 Assumption 1. Sellers are risk neutral. This assumption allows us to focus on the problem of signalling and screening without incorporating risk-sharing considerations. Assumption. Producer and purchaser moral hazard are insignificant. This assumption also allows us to focus the analysis on simultaneous signalling and screening. Moreover, the assumption is frequently justified in markets where simultaneous screening and signalling take place because: a) Sellers have limited ability to affect the performance of a product after its sale and, b) Downstream moral hazard is frequently limited with deductibles, exclusions and maintenance programmes that induce careful behaviour on the part of purchasers. In addition, unless downstream moral hazard were systematically worse for higher or lower quality, it would not have a material effect on the analysis. Assumption 3. Consumers differ in their valuation for warranty coverage and sellers cannot observe these valuations. Extended warranties are effectively insurance against uncertain repair costs. Consumers frequently place different values on these warranties and hence, we observe significant differences in the amount of warranty coverage purchased by consumers. In our simplified market, we assume there are two types of buyers: hedgers (type ) place higher value on warranty coverage than gamblers (type ). Assumption 4. The value of a product and hence warranty coverage declines over time. Many durable goods depreciate over time due to wear and tear or obsolescence. Accordingly the value of a warranty to repair broken products also declines over time. In addition, the extended warranty provides benefits related to future consumption. In general, economists have assumed that people place greater value on consumption, the nearer it is to the present (Silberberg 1990 and Fisher 1970). 7

12 Assumption 5 and 6 deal specifically with how quality is represented. Following from the earlier discussion, the objective is to represent a dimension of quality that is known to sellers but uncertain to buyers. Assumption 5. Products can be either high () or low () quality. Buyers prefer high to low quality but they cannot determine quality by inspection. There are only two levels of product quality in the model. Buyers prefer high to low quality ceteris paribus but will not pay higher prices for a product unless they are certain that the quality is high. Assumption 6. A high quality product () fails more than a low quality product (). We do not model failure rates directly. owever, because we assume that a low quality product fails more frequently: i) The expected cost to provide an extended warranty for low quality products is higher. We assume linear repair costs with an expected repair cost per unit time that is higher for low quality products i.e. c >c. ii) Without warranty protection, a buyer needs to finance the repair of a breakdown herself. Since low quality products fail more, the value of warranty coverage for a low quality product is higher. Assumption 6 is summarised in Figure 1. (Figure 1) We now present a mathematical framework based on these assumptions to represent a high quality seller s problem. The Buyer s Decision We assume that utility derived by a buyer from purchasing a product of known quality can be represented by a quasi-linear function in which there are three components. The first component (B ) is the benefit that a buyer obtains from a product of quality (= or ) without warranty protection. The second component is the benefit that the buyer obtains from warranty coverage on the product. The final component (P) is the total price paid. The utility function for buyers (where x is the length of the warranty) is: 8

13 U( θ,,x,p ) = B + θ V( x ) P (1) The second term is the product of three items. The parameter is used to ensure that the marginal value of warranty coverage for a high quality product is less then the marginal value of warranty coverage for a low quality product i.e. <. (Assumption 6, point ii). This assumption, while important to reflect the empirical reality of the problem, is not required in order for a high quality seller to be able to either signal or screen. The second item θ T is a valuation parameter, which is different for the two types of buyers in our model. Consistent with Assumption 4, the valuation parameter for buyers who place a higher value on warranty coverage (type ) is larger θ >θ. We assume that a fraction λ of buyers are type and 1-λ are type. V(x) is a function used to reflect the declining value of warranty coverage over time (Assumption 4). Mathematically, we need V (x)>0, and V (x)<0 and we also assume that a warranty of zero length has no value i.e. V(0)=0. We utilise the following form for V(x): 1 - (1- x ) V(x)= () This function exhibits the required properties for x ε [0,1] and satiation at x=1. Since most durable goods have a finite life, we model the warranty has having a limit beyond which it is of little value. aving chosen this functional form, we can now specify the constraint implied by Assumption 5. When B > BMIN = B + θ, both types of buyers (hedgers and gamblers) prefer high to low quality given equivalent prices and warranty coverage. Two further comments are necessary to fully explain the buyer s decision process. First, a buyer will not purchase unless the offer (i.e. the price, warranty coverage and quality) provides a minimum level of reservation utility. Reservation utility represents the outside options of the buyer and to simplify, we assume that reservation utility is zero. When the buyer faces two options that both provide more than reservation utility, we assume that the buyer chooses the option that yields the maximum utility. T The Seller s Problem Our objective is to identify the equilibrium strategy for a high quality seller. This is related to the action and profits realised by a low quality seller. To simplify the exposition, we consider the general problem faced by a seller of quality. The expected profits of the seller are a function of the distribution of buyers, the cost of warranty provision and the marginal cost of the product 9

14 (which we can normalize to zero because it does not affect the optimal warranty menu). If we assume the marginal cost of the product is zero, a seller maximises the following: π ( x,p, x,p )= λ[p - c x ] +(1- λ )[ P - c x ] (3) The warranty/price bundles (x, P ) and (x, P ) are purchased by hedgers and gamblers respectively and c is the cost per unit time of providing warranty coverage for a seller of quality (= or ). The warranty length x can be thought of as the base warranty and (x -x ) is the length of the extended warranty that can be purchased for (P -P ). The two options available to consumers (the product without an extended warranty, the product with an extended warranty) are referred to as two bundles (x,p ) and (x,p ) in warranty length/price space. Two conditions are worth mentioning in the context of the menus offered by sellers. In order for a seller to have an incentive to offer a positive warranty, the most willing buyer must be willing to pay more for the warranty than it costs the seller to provide it at x=0 (where the marginal value of warranty protection is highest). This reduces to simple conditions: θ > c, θ > c (4) For gamblers to purchase warranty protection, these conditions must also be satisfied for θ. A second issue concerns the distribution of buyers in the marketplace. When the percentage of gamblers becomes sufficiently low, it can be optimal for a seller to offer warranty coverage to hedgers only (gamblers will be offered the product without warranty coverage). We allow for this possibility in our analysis but focus on the interesting case where both buyer-types are offered positive levels of warranty protection i.e. when λ< ( θ -c )/( θ -c ). Informational Assumptions Sellers' cost structures and buyers' utility functions are assumed to be common knowledge. As previously mentioned, the buyer cannot tell a priori whether the seller is high or low quality and the seller cannot tell a priori whether the buyer has a low or high valuation for warranty coverage. In order for the buyer to figure out whether the seller is offering high or low quality, the buyer forms beliefs about the expected actions of a high quality seller. To identify the equilibrium, we apply a refinement of the Perfect Bayesian Equilibrium (PBE) known as the Intuitive Criterion (Cho and Kreps, 1987 and Fudenberg and Tirole, 1991). The refinement allows the identification of a unique outcome in certain signalling games by arguing that certain types (of sellers) will not use certain strategies. 10

15 Extensive Form of the ame The game is a simultaneous single-shot game with an implied order of play: Stage 1. Nature chooses the seller type ( or ) and the seller chooses a menu of price/warranty bundles and announces them to any buyer who has interest in the seller s product. Stage. Nature chooses the buyer type ( or ) who has interest in the product that the seller is offering. Stage 3. The buyer decides whether to purchase any of the bundles that the seller announces. We first describe the outcomes that obtain when quality is observable. We then describe the outcomes that obtain when quality is not observable to buyers. This allows us to isolate the impact that signalling has on the actions of a high quality seller. 3.0 Analysis and Discussion 3.1 Product uality is Observable We first consider the action that a seller would take were product quality observable. The buyer's valuation of a given warranty/price bundle depends on the quality of the product. When buyers can observe quality, we solve a constrained optimisation problem with the objective function shown in equation 3. This objective function is based on buyers voluntarily selecting the appropriate bundle: (x,p ) or (x,p ) depending on their type ( or ). This leads to two Incentive Compatibility Constraints, one for each type of buyer. For hedgers For gamblers B B + θ V( x ) - P B + θ V( x ) - P (5) + θ V( x ) - P B + θ V( x ) - P (6) As previously discussed, we also assume that both buyer types realise positive surplus by purchasing and this is reflected through Individual Rationality Constraints. For hedgers B + θ V( x ) - P 0 (7) For gamblers B + θ V( x ) - P 0 (8) 11

16 Finally, we constrain prices 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 strict equality. This means that a hedger is indifferent between the bundle designed for her (x,p ) and the bundle designed for a gambler (x,p ) and a gambler is indifferent between purchasing her bundle and not buying at all 1. We summarise the nature of the solution to this problem in Proposition 1. Proposition 1a When product quality is observable, the profit maximising action for the seller is to offer a menu of price/warranty combinations where each buyer-type self selects a bundle designed for her. Proposition 1a follows directly from the results of Mussa and Rosen (1978) and Maskin and Riley (1984). The basic challenge for the seller is that he wishes to serve two types of buyers, one of which is willing to pay more for warranty coverage than the other. Similar to Mussa and Rosen, the solution is to offer different warranty/price bundles to each buyer-type and construct the menu in such a way that hedgers choose a more expensive bundle with more warranty coverage. If we restrict our attention to values of λ such that θ λ < θ seller has an incentive to offer positive warranty lengths to both types and the optimal menu for the seller is: c c, the ( x,p c )= 1- θ, B θ ( 1 λ ) c c θ ( 1 λ ) c + 1 ( θ λθ ) θ ( θ λθ ) + (9) ( x,p )= 1- ( 1 λ )c θ λ θ,b θ + ( 1 λ ) c 1 ( θ λθ ) (10) Proposition 1b underlines the principal elements of the seller s screening menu when product quality is observable. 1 In spite of the hedger s indifference between the two bundles, she is assumed to choose the bundle designed for her. This is a typical assumption in self-selection problems justified because the hedger prefers her bundle strictly with an infinitesimal reduction in P. 1

17 Proposition 1b When quality is observable, the profit maximising menu for the seller has: (a) A bundle designed for hedgers with warranty protection of efficient length and a price which leaves a hedger with strictly positive utility. (b) A bundle for gamblers with a warranty that is shorter than the efficient length and a price which leaves a gambler indifferent between buying or not. Proposition 1b implies that the seller needs to provide high types (i.e. hedgers) with a subsidy (or lower price) to buy the extended coverage. This situation obtains because regardless of how short the warranty is in the low type s bundle (i.e. the gambler), it will be attractive to a high type since she is willing to pay more for warranty coverage than a low type. Any bundle designed for a high type must provide at least the benefit a she could obtain by not purchasing extended coverage. The second point is that this subsidy (or price reduction) to high types is determined by the characteristics of the low type s bundle. Accordingly, the seller maximises his profit on sales to high types by maximising the difference between the price he can charge for the high type s bundle and his expected costs i.e. at the efficient length where the marginal benefit of additional warranty coverage to hedgers, θ (1-x ), equals the marginal cost of providing warranty coverage, c. Nonetheless, the price that hedgers pay is constrained by the gambler s bundle: the longer is the warranty coverage in the gambler s bundle, the greater is the price reduction needed in the hedger s bundle. Thus, the optimal menu for the gambler entails a warranty that is shorter than the efficient length for gamblers. Of course, as the fraction of gamblers in the market increases (i.e. λ decreases), the reduction in warranty coverage (x ) from an efficient length gets smaller. When quality is observable, an example of optimal menus for a high and low quality seller is shown in Figure. (Figure ) The Impact of uality on the Screening Menu when uality is Observable In Figure, the high quality seller is observed to offer a menu where both buyer types pay higher prices and are offered longer warranty lengths. Assumption 5 implies that buyers always pay more for higher quality; however, when quality is observable, the warranty coverage offered by a high quality seller can be less than the coverage offered by a low quality seller (consider the 13

18 lengths x and x when c c < ). Because c >c, this is perhaps uncommon, yet a low value of may create this situation. For example, the value of warranty coverage to buyers could be quite low in a market where high quality products rarely fail (this would imply a low value of ). Clearly, the ratio c c : is important in determining the relative location of the menus of high and low quality sellers in warranty length/price space. When the ratio is large, the high quality seller s menu is well to the right of the low quality seller s menu. In contrast, when the ratio is smaller, the appearance of the menus is similar to that shown in Figure. The difference between B and B (the fixed benefit difference between high and low quality) also has a significant effect on the relative location of the two menus. When the difference is large, the prices a high quality seller can charge are high and this tends to moves the high quality seller s menu upward in warranty length/price space. What happens when quality is unobservable? A seller of low quality would like to obtain higher prices for his product/warranty combinations but on the other hand, he does not want to provide extra warranty coverage since his costs of doing so (c ) are high. We can interpret this in the context of the Figure. When the menu of the high quality seller lies significantly to the right (with long warranties), the low quality seller would be unlikely to mimic the high quality seller s menu. e would get higher prices but he would also have to finance longer warranty coverage. As a result, a high quality seller can send an informative signal to potential buyers without altering his behaviour from what he would do were quality observable. In contrast, when the menu of the high quality seller lies above the menu of the low quality seller, the low quality seller would gladly offer the warranty/price combinations that are optimal for the high quality seller under observability. In this situation, when quality is not observable, a high quality seller needs to alter his menu since a wary buyer will not pay high quality prices for a price/warranty combination unless she is sure that a low quality seller would not offer her the same deal. The following section is devoted to reflecting this situation and identifying the optimal action for a high quality seller when he needs to alter his menu. 3. The Optimal Menu for the igh uality Seller when uality is Unobservable In simple terms, the objective is to identify a market outcome in which sellers maximise profit and buyers actually obtain the quality that they think they are buying (a separating 14

19 equilibrium). A situation in which a buyer thinks that she is purchasing high quality but actually obtains low quality is not an equilibrium. Following the discussion in the preceding paragraph, there are two situations. In the first situation, were quality observable, the high quality seller s menu lies in a region where the low quality seller does not have an incentive to pretend to be a seller of high quality, i.e. the high quality product is offered with long warranties. In the second situation, were quality observable, the high quality seller s menu lies in a region where a low quality seller does have an incentive to pretend to be a high quality seller i.e. the high quality seller s menu has high prices. The No Mimic Constraint The key problem when quality is unobservable rests with buyers who lack information about the quality of the products. To make decisions, buyers are assumed to form beliefs about product quality and these beliefs are based on prior information and actions (i.e. the warranty/price menus that are announced). Similar to a typical signalling model, the uninformed player (i.e. the buyer) makes an inference about the informed player (i.e. the seller) based on the informed player s actions. To find the solution to this game, we need to specify both the actions and beliefs (of uninformed players). ere, the Perfect Bayesian Equilibrium (the PBE) is useful because it imposes a rule of logical consistency on the beliefs of uninformed players (Fudenberg and Tirole 1991). That is, beliefs must be consistent with Bayes' aw given what has happened prior to the point where the uninformed player takes a decision. owever, the PBE imposes logical consistency on the beliefs of players for actions that are on the equilibrium path only. There are no restrictions on the beliefs of players for actions that are off the equilibrium path. In signalling games, the PBE can generate multiple separating equilibria or pooling equilibria supported by off equilibrium beliefs that attribute positive probability to a seller choosing a profit-reducing strategy (recall, the PBE imposes no restrictions on the off equilibrium beliefs). The Intuitive Criterion reduces the solution space for this type of problem and eliminates all equilibria supported by buyer beliefs that attribute positive probability to a low quality seller playing a strategy that reduces profits i.e. that is equilibrium dominated (Cho and Kreps 1987). et a 1 be the menu offered by the seller i.e. a 1= { x,p ),( x,p )} ( ; let µ(a 1 ) be the beliefs of buyers that a seller offering a 1 is high quality; and let s T be the strategy of buyer type 15

20 T. For each a 1, define J(a 1 ) as the set of seller types for which: π ( ) > maxπ ( a,s,s, ). An equilibrium fails the Intuitive Criterion if there exists an a 1 for: * 1 s i T A. The Seller of uality (): Such that: π < min π { a1,s,s, ) µ ( )}. et a 1 be the set of menus where J(a 1 )=. * s t i The equilibrium fails the criterion if high quality seller has an alternate strategy (i.e. a warranty/price menu) that yields greater profit than the expected profit with buyer beliefs and buyer strategies as follows: B. The Beliefs of Buyers: µ ( ) { µ 1 µ ( a1 ) = 0}. The Intuitive Criterion requires checking the condition in point A, given that buyer s beliefs assign zero probability to a low quality seller playing a strategy that is equilibrium dominated. C. The Buyer s Optimal Strategy For buyer type T, st * = arg max U T ( a1, θ T, µ ( )). This means that buyers self select the bundle (x, P) that is optimal for them. s T The solution to the game is a collection of optimal strategies for each seller-type {( x,p ),( x, P )}, each buyer type * s T and buyer beliefs µ ( ) that do not fail the Intuitive Criterion. With only two types of informed players (high and low quality) and a onedimensional signal (warranty length), the only equilibrium not rejected by the Intuitive Criterion is a separating equilibrium with minimal inefficient signaling (Fudenberg and Tirole 1991). With the assumptions we have made, a high quality seller (the informed party) will always signal high quality; the downside of not doing so is that buyers will believe quality is low and the high quality seller will make strictly less profit. In contrast, with two period models such Balachander and Srinivasan (1994), an informed party with an advantage may not signal. With two period models, there is often less relation between the cost and benefits of signaling than there is in a one period model. For example, in B & S, the downside of not signaling (in period 1) for the incumbent is that he faces competition in the second period. It is quite possible for the expected gains in period (by deterring entry) to be With two seller types, the beliefs µ( ) over all actions that a seller is high quality are complete because 1-µ( ) are the beliefs that the buyer is low quality. 16

21 less than the cost of signaling in period 1. In these conditions, an incumbent with an advantage will not signal. The mathematical construction of the problem outlined by points A, B, and C for the high quality seller is as follows. Point C implies that any solution to the problem involves each buyer-type choosing the price warranty bundle that optimises her surplus. Of course, without further restrictions this is the problem described in section 3.1. Point B restricts the high quality seller s strategies to a set of strategies a 1 where J(a 1 )=, i.e. a 1 where π * ( ) > maxπ ( a1, s, s, ). This constraint implies that if a low s i t quality seller offers a menu for which J(a 1 )=, he will make less than the guaranteed level of profit, π * ( ) made by offering the optimal screening menu for a low quality seller. This is reflected in the high quality seller s problem as a `no mimic' constraint: π ( ) > λ( P x c ) + ( 1 λ )( P x c ) (11) * When the high quality seller s optimal menu under observability is unattractive to the low quality seller, this constraint is non-binding. In contrast, when the high quality seller s menu under observability is attractive to the low quality seller, this constraint binds and the actions of the high quality seller are affected by the constraint. Point A obliges us to look for an equilibrium (with action a 1 * for the high quality seller), where there exists no action a 1 (satisfying J(a 1 )=), that yields higher profit than the profit with action a 1 *. We identify the action a 1 * by solving a constrained optimisation problem for the seller with self selection and individual rationality imposed on buyers and the no-mimic constraint imposed by point B. In other words, the problem is effectively the high quality seller s screening problem subject to one additional constraint (equation 11). The solution leads to Proposition. Proposition When product quality is unobservable: (a) and the fixed benefit B of a high quality product is less than Bˆ, the warranty/price menu offered by a seller of high quality is identical to the one offered when product quality is observable. 17

22 (b) and the fixed benefit B of a high quality product is greater than Bˆ, the price/warranty menu offered by a seller of high quality is different from the one offered when product quality is observable. Where: Bˆ = π * ( ) + c θ ( c c ( ) 1 λ ) ~ c θ λc + θ ~ = and, θ θ λθ π * ( ) is the profit realized by a low quality seller when quality is observable. The expression for π * ( ) is given by: π * ( ) = B λ θ + ( ~ ) 1 λ c θ c θ ( c ~ ) c c ( ~ ) c θ 1 λ 1 λ λ θ θ θ Before providing the intuition for this result, we discuss the expression for Bˆ. Bˆ is a complex function of the relative costs of high and low quality sellers (c and c ), the relative values of warranty coverage for buyers of high and low quality respectively ( and ), and B, the fixed benefit for low quality. It describes the B above which the menu of a high quality seller under observability will be attractive to a low quality seller. The intuition behind Proposition is that when the fixed benefit for high quality is low i.e. B { B,Bˆ } MIN, the cost advantage of the high quality seller (i.e. the degree to which c is less than c ) overshadows higher prices that buyers are willing to pay for high quality (prices are a direct function of the premium B ) 1. As a result, the optimal menu chosen by the high quality seller is too expensive for the low quality seller to mimic. In fact, when buyers have a high fixed benefit for low quality (B ) or when the difference between the marginal costs of providing coverage is high (c versus c ), it is unlikely that a high quality seller s actions will be affected by the need to signal. The marginal value of warranty protection parameters ( and ) also have a significant effect on the optimal actions for the high quality seller. First, the higher is (the marginal value of warranty coverage on low quality products), the higher is Bˆ because a higher strictly increases π * ( ). The effect of an increase in (the marginal value of warranty coverage on high quality products) on Bˆ is ambiguous because it has two effects. The first is to cause the high quality seller to offer more coverage making the menu (under observability) less attractive to a low quality seller. The other is to increase prices in the menu of the high quality seller and this of course, makes the menu attractive to the low quality seller. For most values of, an increase in raises Bˆ and reduces the likelihood 18

23 that a high quality seller will have to offer a menu that is different from what he would offer were quality observable. owever, an increase in can have the opposite effect on Bˆ (when either the difference between c and c is small or is high). When B > Bˆ, the no mimic constraint (1) binds and the solution to the constrained optimisation yields a positive agrangean multiplier on the no-mimic constraint. This has two economic interpretations. First, the equilibrium involves the high quality seller choosing a menu for which the low quality seller is indifferent between mimicking and not mimicking. This menu is different from the menu that the high quality seller would offer when quality is observable. Second, relaxing the no-mimic constraint by one unit (for example, increasing B by one), the profit of the high quality seller would increase by an amount equal to the value of the multiplier. Thus, the profit of the high quality seller is strictly reduced due to the unobservability of quality (profit increases when the constraint is relaxed and the profit under observability is obtained by solving the problem with the constraint fully relaxed). Proposition 3 considers the specific actions of the high quality seller. Proposition 3 When product quality is unobservable and B > Bˆ, the high quality seller s menu involves more warranty protection and higher prices for all buyers (the high quality seller earns strictly greater profit by signalling with all elements of his warranty menu). Additionally, when B > Bˆ, signalling causes a high quality seller to offer strictly shorter extended warranties ( x -x ). The intuition for Proposition 3 is that a high quality seller lengthens the warranties to make the menu unaffordable to a low quality seller. The high quality seller charges higher prices but the higher prices do not compensate for the added cost of providing the coverage (this difference is the cost of signalling for the high quality seller). The reason that both x and x are longer is that the high quality seller minimises his cost of signalling by making as small movements as possible from the optimal screening menu under observability. In essence, the signalling cost is shared by the two bundles in the high quality seller s menu. This has important implications for a high quality seller who wishes to both signal and screen using warranty policy. First, the result shows that a seller makes strictly greater profit by signalling through all elements of his warranty menu and not just the base warranty. When signalling is done by the base warranty alone and B is sufficiently high, the no-mimic constraint is: 19

24 π * ( ) > P x c (1) owever, when B is low enough, the solution to the problem with equation 1 (as the nomimic constraint) yields a base warranty x that is less than c 1 (the efficient warranty θ length for hedgers with a high quality product). ere, a high quality seller could increase profit by selling an optional extension to hedgers (increasing coverage to the efficient length). If we assume that a mimicking low quality seller is constrained to act like a high quality seller for the entire selling process, equation 1 needs to be modified because we must account for the money that is lost or gained when the extension is sold. The revised constraint is: θ c 1 (1 x ) cc π * ( ) > λ (1 )( ) P c θ P x c λ (13) θ θ When signaling is done with only the base warranty, it is easy to show that the profit for the high quality seller is strictly lower than when his actions are constrained by the no-mimic constraint of equation 11. In particular, all solutions to the problem where the base warranty alone signals (with either equation 1 or equation 13 as a constraint) satisfy equation 11. owever, the optimal base warranty, x in these conditions is strictly different than the base warranty predicted when the problem is solved with the no-mimic constraint of equation 11. Thus, profit is strictly reduced when signaling is done entirely with the base warranty. A second implication of Proposition 3 is that extended warranties will be shortened when a high quality seller needs to signal. This occurs because the marginal value of warranty coverage is higher for gamblers than it is for hedgers under the screening menu of section 3.1 (gamblers obtain less than efficient coverage). Thus, when a high quality seller is obliged to lengthen warranty coverage increases in the base warranty generate greater price increases than increases in the extended warranty. Consequently, the optimal distortion involves greater increases in the base warranty ( x hedgers ( x ) and extended warranties ( x - x ) are shorter. ) than in the coverage purchased by Proposition 4 shows that the reduction of extended warranties due to signalling is extreme when the fixed benefit for high quality is above a certain limit. 0

25 Proposition 4 When product quality is unobservable and the fixed benefit for a high quality exceeds θ B MAX = π * ( ) + c, signalling considerations eliminate the ability of a high quality seller to sort buyers and a maximum warranty of 1 is offered to all buyers. When signalling requirements are severe, the inability of a high quality seller to sort buyers (by offering different levels of warranty coverage) occurs because of satiation in the buyers utility functions. As discussed earlier, we believe this to be a reasonable property for the utility function given that most durable products have a finite life. The intuition for this finding is that θ when B >B MAX, both types of buyers are more than willing to pay B MAX + for a high quality product with a full warranty of x=1. Thus, the individual rationality constraints (equations 7 and 8) hold for neither buyer type (both obtain positive surplus by buying). A high quality seller is prevented from charging more for his menu because of the no mimic constraint. This contrasts with the menu that is chosen when product quality is unobservable and B {B MIN,B MAX }. When B falls in this range, a hedger always receives positive surplus and a gambler s individual rationality constraint always binds (and hence she receives zero surplus). When B >B MAX, the need to signal not only distorts the high quality seller's menu, it also leaves a gambler strictly better off. The analysis shows that there are three potential zones for B given B, λ, θ, θ,, c and c. and the menu offered by the high quality seller depends on which zone B lies. owever, the restrictions on buyer beliefs imposed by the Intuitive Criterion apply to the equilibrium observed in any of the potential zones for B. The first is the zone { B MIN,Bˆ } where the high quality seller s action is unaffected by the unobservability of quality. The second is the zone { Bˆ, B MAX } where the high quality seller needs to lengthen warranties to signal higher quality. In the third zone, where B P >B MAX, the high quality seller offers one option with maximum warranty protection to everyone. It is important to note that for certain sets of parameters, B, λ, θ, θ,, c and c, there may be less than three potential zones for B. If we examine the expression for Bˆ, a low value of has no effect on the first two terms, drives the third term to zero, and causes the final term to become large and negative. As a consequence, it is possible that Bˆ could be a value less than B MIN for a given set of parameters B, λ, θ, θ,, c and c. When this happens, a high quality seller will be obliged to alter 1