ESSAYS ON INFORMATION AND STRATEGY By Se Hoon Bang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Economics 2011 ABSTRACT ESSAYS ON INFORMATION AND STRATEGY By Se Hoon Bang Chapter 1: Reverse Price Discrimination and Information Discrimination with Bayesian Buyers: The Case of Monopoly The paper studies price discrimination under the situation where buyers' prior valuations are initially observable to a seller but the buyers receive further information about a product or service which remains private thereafter. The buyers interpret new information via Bayes' rule. We show that, in this environment, prices are not monotone in buyers' prior valuations. Interestingly, this results in the possibility that a seller intentionally offers a higher price to a low valuation buyer rather than a high valuation buyer (Reverse Price Discrimination), which sharply contrasts with the standard result of price discrimination. We further explore the seller’s incentives to discriminate in provision of information. Chapter 2: Reverse Price Discrimination with Competition: a Hotelling Duopoly We extend the previous analysis of Reverse Price Discrimination to a duopoly market using a standard Hotelling model. We show that the equilibrium prices can be nonmonotone in the buyer’s prior valuation even with competition. The buyers with relatively “weak” prior beliefs are charged higher prices, since they are more likely to view the products of two sellers vertically differentiated when receiving different signals from them, which mitigates sellers’ price competition. Chapter 3: Price Discrimination via Information Provision: Online vs. Offline Shoppers We study a new type of second-degree price discrimination where a different price is offered as a bundle with a different level of information about a product. We consider the situations where buyers are uncertain about the value of the product, and may update their expected valuation in a Bayesian way after observing a signal coming from the product while shopping at a store. When an online store provides less information on the product than an offline (brick-and-mortar) store, the seller's price discrimination induces high valuation buyers to shop at the online store and low valuation buyers at the offline store. Furthermore, the buyers' incentive compatibility constraints ensure that the price should be lower at online stores than at offline stores when both stores incur the same transportation (shopping) costs. The conditions under which it is more profitable for the seller to sell at both online and offline stores than to sell only at either of the stores are examined. We then explore the case in which buyers are sophisticated enough that they may "milk" the information from one store and use it at the other store (Information Arbitrage). It is also discussed that, in an ex post sense, the low valuation buyer's purchase is socially optimal, whereas the high valuation buyer's purchase is suboptimal due to excessive consumption. To my beloved wife, Suzie, and children, Ikhan and Amy iv ACKNOWLEDGMENTS First and foremost I am truly indebted to my advisor, Jay Pil Choi, for his invaluable guidance and support throughout the writing of this dissertation. This dissertation would not have been completed without his help and encouragements in every step. I am also grateful to all my committee members, Carl Davidson, Anthony Creane, and Steven Wildman, for their thoughtful comments and suggestions. They have shared their deep insights and knowledge without reservation. I am heartily thankful to Jaesoo Kim, who as a good friend, was always willing to help and give his best suggestions. It has been a great pleasure working with him. Last but not least, I owe my deepest gratitude to my parents for their lifelong guidance and endless love. Especially, I would like to thank my wife, Suzie, and my two children, Ikhan and Amy for their love, support, and patience during the course of my studies. v TABLE OF CONTENTS LIST OF FIGURES ...................................................................................................... vii CHAPTER 1: REVERSE PRICE DISCRIMINATION AND INFORMATION DISCRIMINATION WITH BAYESIAN BUYERS: THE CASE OF MONOPOLY ............1 1.1 Introduction ...................................................................................................1 1.2 The Model.....................................................................................................5 1.3 Monopoly ......................................................................................................9 1.4 Concluding Remarks ..................................................................................22 Appendix ..........................................................................................................25 Bibliography ..................................................................................................... 33 CHAPTER 2: REVERSE PRICE DISCRIMINATION WITH COMPETITION: A HOTELLING DUOPOLY .............................................................................................35 2.1 Introduction .................................................................................................35 2.2 The Model...................................................................................................37 2.3 Duopoly: Hotelling Model ............................................................................39 2.4 Concluding Remarks ..................................................................................45 Appendix ..........................................................................................................47 Bibliography ......................................................................................................51 CHAPTER 3: PRICE DISCRIMINATION VIA INFORMATION PROVISION: ONLINE VS. OFFLINE SHOPPERS .........................................................................................52 3.1 Introduction .................................................................................................52 3.2 The Model...................................................................................................55 3.3 Online or Offline ..........................................................................................58 3.4 Online and Offline: Second-Degree Price Discrimination ...........................60 3.5 Optimal Choice of Marketplace...................................................................65 3.6 Extensions: Sophisticated Buyers and Milking Strategy .............................66 3.7 Concluding Remarks ..................................................................................69 Bibliography ......................................................................................................72 vi LIST OF FIGURES 1.1 Demand ...................................................................................................................7 1.2 Timing .....................................................................................................................8 1.3 (a) Linear demand function (b) The optimal price ..................................................20 2.1 Demand in the Hotelling model .............................................................................40 2.2 Equilibrium prices in duopoly .................................................................................43 3.1 Timeline .................................................................................................................58 3.2 ICs and IRs ...........................................................................................................62 3.3 PD with Sophisticated Buyers ...............................................................................68 vii Chapter 1 Reverse Price Discrimination and Information Discrimination with Bayesian Buyers: The Case of Monopoly 1.1 Introduction When you open the door to a car dealership, a salesperson will welcome you and ask how you are. At this moment, the salesperson may try to …gure out your willingness to pay by looking at your appearance such as your gender, race, and age, and further by asking some related questions from which he could infer your income and so forth. Then the salesperson may o¤er you a personalized price based on this observation. This looks like a typical third-degree price discrimination: a seller is able to observe a buyer’ willingness to pay and charges the s optimal price using that information. However, the seller’ pricing can be di¤erent from the one in a standard (third-degree) s price discrimination if buyers’valuations can change depending on private information they 1 additionally gather. For example, car buyers can do test drives and home buyers can visit houses before their purchase. These advance activities allow consumers to learn more about the products in which they are interested. In this situation, although buyers’prior willingness to pay might be observable to a seller, their posterior valuations formed by the new information they acquire may remain private. A key feature is that buyers update their valuations in a Bayesian way in the sense that buyers’posterior valuations are systematically dependent on their prior beliefs through Bayes’rule. The analysis of the optimal pricing in this situation is of particular interest because it begins by departing from the standard classi…cation of price discrimination. As is well known, third-degree price discrimination is charging di¤erent prices based on some observable characteristics, whereas second-degree price discrimination considers the case where the buyers’ types are not observable. Our model considers the intermediate case in the following sense. Although a buyer’ prior valuation is perfectly observable to the seller, the buyer receives s new private information and updates his valuation before purchasing. As a consequence, the seller has to choose the optimal price based only on the buyer’ prior valuation, but not s on the posterior valuation because it is not observable. Then, what would be the optimal pricing rule for the seller who faces this environment? We address this question in a monopoly setup, and we show that prices can be nonmonotone in buyers’prior valuation. In other words, interestingly, there is a situation where the seller o¤ers a higher price to low valuation buyers than to high valuation buyers. Our result seems contradictory to the general conjecture that it would be optimal to charge a higher (lower) price if buyers are with high (low) valuation. Here is the intuition underlying our result. When the seller makes a decision on the prices, she faces a traditional trade-o¤ between getting a higher margin and getting a greater market share. This trade-o¤ depends crucially on the price elasticity of demand, which is essentially determined by the interaction between a buyer’ prior valuations and the precision of information available to the buyer. When s 2 information precision is not extreme (i.e., neither too accurate nor too vague), we …nd that the demand is elastic (inelastic) when the buyer’ prior valuation is high (low). Thus, if the s buyer’ prior valuation is high, the seller chooses to o¤er a low price because the buyer’ s s updated valuation does not decline signi…cantly even when he receives unfavorable private information. On the other hand, if his prior valuation is low, it is more pro…table for the seller to charge a high price and target only the buyer who receives favorable private information because the price under which the buyer with unfavorable information is covered is too low to be pro…table. This leads to the result that a higher price can be o¤ered to buyers with lower willingness to pay and vice versa. Our main results suggest that the seller can charge a higher price to buyers with lower willingness to pay as a pro…table strategy. We refer to this outcome as reverse price discrimination. In particular, the process we suggest as a cause of reverse price discrimination consists of the following three simple steps: i) consumers have prior valuation, ii) consumers update their belief using Bayes rule after observing a private signal, and iii) consumers have private posterior valuation. We believe that this is a quite general process that we experience frequently in our everyday life. Our model shows how this common process provides a suf…cient condition for which reverse price discrimination can arise. In other words, our model shows how the posterior valuations which yield reverse price discrimination are endogenously formed in terms of information quality. Our approach enables us to answer the following interesting questions: i) what product markets are more likely to give rise to reverse price discrimination? or ii) what would be the characteristics of products for which we would expect reverse price discrimination more often? We will highlight that information should be neither too precise nor too imprecise for reverse price discrimination to arise. Thus, reverse price discrimination can be prevalent in a market for professional services such as the ones provided by lawyers, car mechanics, and home improvement contractors. Consumers in these markets are de…nitely uncertain of the quality of services, so they try to gather additional private information. Of course, the 3 sellers in these markets often o¤er personalized prices to di¤erent individuals. The result derived in our model may provide an alternative explanation to the discrimination in car sales observed by Ayres (1995) and Ayres and Siegelman (1995).1 They tested how new-car dealerships quoted di¤erently across customers’races. They found that dealers o¤ered signi…cantly lower prices to white buyers than to non-white buyers, even though white people are believed to have a higher willingness to pay than non-white people, which is why price discrimination appears to be the racial discrimination.2 However, our model suggests the possibility that a salesperson may o¤er a higher price to the group which is more likely to have a lower willingness to pay because it is pro…table to target only those who have favorable posterior impressions. Our result can also explain reward programs for frequent customers. When a seller can identify repeated customers, she regards them as consumers with a high willingness to pay. The recent literature on purchase-based price discrimination has studied this issue extensively.3 A common …nding in this literature is the so called "ratchet e¤ect," which describes price discrimination against loyal customers by charging a higher price. However, it is also not hard to …nd numerous opposite examples where sellers often use price discrimination favorable to loyal customers. For example, airline companies provide discounts and free upgrades to frequent customers. Automobile companies o¤er loyalty rebates to customers who currently own the same brand car. In our model, a buyer’ prior valuation can be thought of s as a brand loyalty which can be assessed by a seller through a customer’ purchase history. s Our model then provides one reasoning about why a seller may o¤er discount prices to the loyal customers. The topic of third-degree price discrimination, which considers the case where the char1 See Yinger (1998) for a good survey and summary of the discrimination literature in consumer markets. 2 White people are believed to have a higher willingness to pay because of their greater ability to pay from the greater average income and the higher opportunity costs of search time. 3 See Armstrong (2006) and Fudenberg and Villas-Boas (2005) for the review of the literature on behavior-based price discrimination. 4 acteristics of buyers (who have no private information) are observable, by a monopolist has been covered extensively in literature after the seminal work of Robinson (1933).4 In particular, Borenstein (1985), Holmes (1989), Corts (1998), and Chen (1999) are the recent papers which extend the analysis of third-degree price discrimination to the setting of oligopoly. Price discrimination with incomplete information has been studied in the environment of screening or self-selection mechanism followed by Mussa and Rosen (1978) and Maskin and Riley (1984). In particular, Lewis and Sappington (1994) and Courty and Li (2000) are closely related to ours because they also study the environment where consumers are initially uncertain of their valuations that are in part revealed afterwards. Both papers study second-degree price discrimination in which …rms are assumed not to be able to observe consumers’expected valuations. On the other hand, in our paper, the seller is able to observe consumers’expected valuations and price discriminate based on them. Most signi…cantly, to the best of our knowledge, our paper is the …rst study of reverse price discrimination. The rest of this article proceeds as follows. Section 1.2 describes the structure and assumptions of the basic model. Section 1.3 studies the monopolist’ price discrimination. s We …rst consider the case where a buyer has a unit demand and receives a binary signal. We then relax these two key assumptions and show the robustness of our main result even after incorporating a non-unit demand function and general information structure. Section 1.4 concludes. 1.2 The Model In this section, we explain the basic setup of the model which will be used throughout the paper. The additional assumptions or setups needed in each scenario will be introduced as needed in each section. Players. The buyers have a unit demand for the good, which is solely supplied by seller(s). 4 Armstrong (2006), Stole (2007) and, Varian (1989) are the excellent survey papers about price discrimination. 5 The true value of the good depends on the match between a buyer’ preference and the s features of the good, which is denoted by v 2 fH; Lg. If v = H, the good is a good match with a buyer and if v = L, it is a bad match. A buyer’ prior belief for v = H is denoted s as 2 [0; 1]. We normalize the buyer’ valuation for the good to be 1 for v = H and 0 for s v = L. Accordingly, can be thought of as the buyer’ prior valuation for the good and s therefore as the willingness to pay. The buyer’ type s is public information and observable to the seller(s). For simplicity, the reservation value of seller(s) for the good is assumed to be 0. Information. The buyers receive private information about the good while they are inspecting the good before purchase. Although the buyers are ex ante homogeneous, they may draw di¤erent binary signals, s 2 fsH ; sL g; on their match value. The realization of a signal is privately observed by the buyers, so it is private information. As is standard, the signals partially reveal the true match value of the good in the sense of Blackwell, Pr(sH jv = H) = Pr(sL jv = L) = Pr(sL jv = H) = Pr(sH jv = L) = 1 where 2 1; 1 2 without loss of generality. is often interpreted as the precision or the quality of a signal and is common knowledge. In addition, one interesting interpretation of is related to the products classi…cation. Nelson (1970) classi…es products into two categories: search goods and experience goods. He de…nes a search good as one whose qualities can be easily evaluated by the consumer before purchase. Similarly, he de…nes an experience good as one whose qualities are di¢ cult to observe before purchase. According to this de…nition, = 1 is comparable to a search good because the buyers are able to observe the quality of a good completely before their purchase. By contrast, = 1=2 represents an experience good because the buyers are not 6 1 pH (θ , α ) θ pL (θ , α ) 1 Pr( sH ) pH (θ , α ) pH (θ L , α ) pL (θ , α ) 1 pL (θ H , α ) Pr( sL ) 0 0 θH θL ~ θ H (θ L ) 1 Figure 1.1: Demand (For interpretation of the references to color in this and all other …gures, the reader is referred to the electronic version of this dissertation.) able to observe the quality in advance.5 Thus, if 2 1 ; 1 , the good can be thought of 2 as one which is neither perfectly a search good nor perfectly an experience good. In reality, there would be many goods and services which fall somewhere between those two extreme types of goods, and those are our main interest in this paper. As the buyer receives a signal s 2 fsH ; sL g, he updates his beliefs on the match value for the good. Let us refer to pH ( ; ) and pL ( ; ) as the buyer’ posterior valuation when s the buyer with a prior 2 (0; 1) draws a signal sH and sL , respectively. Then, Bayes’rule leads to pH ( ; ) = pL ( ; ) = + (1 )(1 (1 ) (1 ) + (1 ) ) and (1.1) (1.2) 1 = 2 , it means that the private signal s does not provide any information about the true value. 5 If 7 θ 2 1 the seller observes buyer's type the seller offers a good and provides information the buyer draws a private signal the seller makes take-it-or-leaveit offer to the buyer Figure 1.2: Timing because we normalize the buyer’ valuation for the good to be 1 for v = H and 0 for v = L. s The buyer’ posterior valuation is a mean-preserving spread of his prior belief. A signal s disperses the prior belief to two-point distribution: pH ( ; ) with the probability Pr(sH ) and pL ( ; ) with the probability Pr(sL ). The probabilities that the buyer receives a good signal and bad signal are given by Pr(sH ) = X v2fL;Hg Pr(sH jv) Pr(v) and Pr(sL ) = X v2fL;Hg Pr(sL jv) Pr(v); respectively. In this information structure, the following properties are well-known. First, the posterior valuations pH ( ; ) and pL ( ; ) are increasing in at a decreasing rate and at an increasing rate respectively: @pH ( ; ) @p ( ; ) @ 2 pH ( ; ) @ 2 pL ( ; ) > 0, L > 0, < 0, and >0 @ @ @ @ (1.3) as represented in Figure 1:1. Second, pH ( ; ) is an increasing function and pL ( ; ) is a decreasing function in : @pH ( ; ) @p ( ; ) > 0 and L <0 @ @ (1.4) Timing. The timing of the game is summarized in Figure 1:2. In the …rst stage, the seller 8 can observe the buyer’ prior valuation s perfectly. At the same time, the buyer draws a private signal about the match value of the good. In the second stage, the seller quotes price P to the buyer and makes a take-it-or-leave-it o¤er. We assume that the buyer prefers to purchase the good when he is indi¤erent between buying and not buying the good. 1.3 Monopoly 1.3.1 Basic model Now we consider the case where there is only one seller. In this section, we assume that there are two types of buyers and type-i buyer’ prior valuation is i 2 f L ; H g where s 0 1 L< 2 < H 1.6 We de…ne the buyer as a type-H if i = H and a type-L if i = L . Given the information structure, we …nd the optimal prices that the seller o¤ers to each type of buyer. Note that when buyer i receives a good signal, he will purchase the good if and only if pH ( i ; ) Pi 0. Likewise, when he receives a bad signal, he will purchase if and only if pL ( i ; ) Pi 0. Hence, the expected demand is the equilibrium probability that buyer i accepts an o¤er for a given price Pi , which is given by 8 > > > 1 > < D(Pi ) = Pr(sH ) = > > > > : 0 if Pi i + (1 )(1 i) pL ( i ; ); if pL ( i ; ) < Pi pH ( i ; ); if Pi > pH ( i ; ): Figure 1:1 illustrates the demand function depending on the realization of a private signal. For each type of buyer, the seller has to choose between two alternative prices: (i) a high price at which only the buyer who receives a good signal s = sH can a¤ord to buy, i.e., Pi = pH ( i ; ) and (ii) a low price at which everyone including the buyer who receives a bad h 6 This can also be interpreted as follows: a seller regards the buyer with 2 0; 1 as 2 i 1 ; 1 as a type-H. This assumption simpli…es our analysis a type-L and the one with 2 2 without a signi…cant change in our main results. 9 signal s = sL can a¤ord to buy, i.e., Pi = pL ( i ; ). When a seller charges pH ( i ; ) to a type-i buyer, she can sell the good with probability Pr(sH ) which is the probability that the buyer receives a signal s = sH . On the other hand, she can sell the good with probability 1 if she charges pL ( i ; ). Keep in mind that the seller can observe the buyer’ type ( i , prior s valuation) and therefore can o¤er di¤erent prices to two di¤erent types of buyers. As in any model of price discrimination, we rule out a resale possibility. Let H ( i ; ) and L ( i ; ) be the seller’ expected pro…t when she charges pH ( i ; ) s and pL ( i ; ) to a type-i buyer. Then, we obtain from (1.1) and (1.2) H ( i ; ) = pH ( i ; ) Pr(sH ) = L ( i ; ) = pL ( i ; ) 1 = i and (1 (1 ) i i ) + (1 (1.5) ) i : (1.6) Comparing H ( i ; ) and L ( i ; ) determines the optimal price the seller charges to type-i buyer.7 The tension in determining the price is the trade-o¤ between getting a higher margin by charging pH ( i ; ) and getting a greater market share by charging pL ( i ; ). The seller’ s pricing decision is determined by the price elasticity of demand. Now, let us show that the elasticity systematically depends on the interaction between a buyer’ prior valuation s and the precision of information. There are three possible cases. The following proposition summarizes the result. 7 In fact, the comparison tells us whether the expected demand is inelastic, unit-elastic, or elastic. Since the expected demand is a two-point distribution, we shall use the midpoint pH ( i ; )+pL ( i ; ) 2 ; for calculating the price elasticity of 1+Pr(sH ) 2 demand. Then, we obtain H ( i ; ) R L ( i ; ) as "p Q 1. The seller charges a higher price, pH ( i ; ); when the demand is inelastic and a lower price, pL ( i ; ); when the demand is method, "p = Pr(sH ) 1 pH ( i ; ) p L ( i ; ) elastic. 10 Proposition 1 Let 1 = ( i = L ) and 2 = ( i + 1) ( i) = q 2 (2 i ( i = H ) where 2 i 6 i+5 1) : Then, given H > 1=2 > L , there exist 1 ; 2 2 1 ; 1 such that the following results hold. 2 (i) If 1 2 2 ; 1 , a seller charges pL ( i ; ) for each type i. (ii) If 2 [ 1 ; 2 ], a seller charges pH ( L ; ) for a type-L buyer and pL ( H ; ) for a type-H buyer. (iii) If 2 ( 2 ; 1), a seller charges pH ( i ; ) for each type i. Proof of Proposition 1 In the appendix. If information is su¢ ciently precise (case iii), the demand becomes inelastic. In this case, the seller prefers getting a higher margin to getting a greater market share because she has to lower the price too much in order to sell to the buyer with a bad signal. In other words, the price pH ( i ; ) is high enough to compensate the loss in pro…t from losing the buyers with a bad signal. Hence the seller charges pH ( i ; ) to each type of buyer. By contrast, if information is too vague (case i), the demand is elastic, so the seller prefers to have the large market share and serve all buyers by charging pL ( i ; ). In these two cases, the elasticity is solely determined by the quality of information. The most interesting case is the one where information quality is intermediate and the buyer’ prior valuation determines the elasticity (case ii). When a buyer is type-H, it is s more pro…table to increase a market share because a type-H buyer’ posterior valuation even s with s = sL is relatively high enough to make the seller o¤er pL ( L ; ) and serve all type-H buyers. On the other hand, when a buyer is type-L, the price for serving the buyers with s = sL should be signi…cantly low. Hence, increasing a market share is not attractive, so 11 the seller charges the price pH ( L ; ) to type-L buyers. Now we compare the prices o¤ered to each type of buyer. Proposition 2 If 2 1; 1 2 or 2 ( 2 ; 1), the price o¤ered to a type-H buyer is higher than that o¤ered to a type-L buyer. If information quality is su¢ ciently precise, i.e., 2 ( 2 ; 1), the seller charges pH ( i ; ) to each type i. Since pH ( i ; ) is an increasing function in i from (1.1), we obtain pH ( H ; ) > pH ( L ; ). On the other hand, if information quality is su¢ ciently imprecise, i.e., 2 1; 2 1 , the seller charges pL ( i ; ) for each type i. Again, we obtain pL ( H ; ) > pL ( L ; ) because pL ( i ; ) is also an increasing function in i from (1.2). Hence, if information quality is either su¢ ciently precise or imprecise, the price o¤ered to a type-H buyer should be higher than that for a type-L buyer. This implies that the standard result of price discrimination holds if the goods are close to search goods or experience goods. Now, the striking result is that we …nd the possibility of reverse price discrimination for the case of in-between goods where 2 [ 1 ; 2 ]. The price o¤ered to a type-L buyer can be higher than that o¤ered to a type-H buyer. Since pH ( i ; ) is concave in i and pL ( i ; ) is convex in i , we can easily denote the case in which pH ( L ; ) > pL ( H ; ) by looking at a certain L and H , as shown in Figure 1:1. We summarize this result more formally in the following proposition. Proposition 3 Suppose that 2 [ 1 ; 2 ]. Let us de…ne eH ( L ) such that pL (eH ( L ); ) = pH ( L ; ) for given L and eL ( H ) such that pH (eL ( H ); ) = pL ( H ; ) for given H . We obtain the reverse price discrimination, pH ( L ; ) > pL ( H ; ); if H < eH ( L ) = 2 2 L +(1 L )2 (1 L ; or equivalently eL ( H ) = ) 12 1)2 H 2 H 2 H+ ( < L. Proof of Proposition 3 In the appendix. According to Proposition 3, for the reverse price discrimination to arise, the di¤erence between two prior valuations L and H should be bounded above. Also note that @ eH ( L ) @ and @ eL ( H ) @ That is, for given L , as 2( L = 2 = 1) ( L L 2 (1 2 1) >0 2 2 + 2+1 H) ( H L 1) H H < 0: 2 2 increases, the value of eH ( L ) below which pH ( L ; ) > pL ( H ; ) increases. Also for given H , as increases, the value of eL ( H ) above which pH ( L ; ) > pL ( H ; ) decreases. These imply that as information quality increases (decreases), the parameter set of L and H for which the reverse price discrimination is derived becomes larger (smaller). Here is the reason. As information quality increases, the type-L buyer’ s posterior valuation, after observing a good signal, becomes relatively high enough to compensate the initial low prior belief. Also, although the type-H buyer’ initial prior belief is s high, his posterior belief after observing a bad signal becomes relatively low. These mean that the curvatures of both functions pH ( L ; ) and pL ( H ; ) become larger as increases, as seen in Figure 1:1. Hence, even though the prior valuations di¤er much, as information quality increases, the reverse price discrimination is more likely to arise. Our results say that the information quality should be intermediate, i.e., 2 [ 1 ; 2 ] for the reverse price discrimination to arise. In other words, the signals buyers observe should be neither too precise nor too vague. We believe this is more applicable to many cases because a large number of goods are neither perfectly search goods nor perfectly experience goods. Consider a car sales market. Consumers are usually given some opportunities to inspect and test drive a car before the purchase is made, but it is still hard for them to 13 completely …gure out how well the car …ts their taste. In this sense, the car sales market is a good example where reverse price discrimination may occur. In sum, our theory implies that while standard result of price discrimination applies to search goods and experience goods, the reverse price discrimination can arise for intermediate types of goods. We believe that our result may provide an alternative explanation for Ayres and Siegelman (1995). According to their …ndings, the car dealers o¤ered the $1; 061 higher price to non-white buyers rather than to white buyers, although the non-white buyers are believed to have a lower willingness to pay than white buyers.8 Our model suggests that their …ndings can be due to the dealer’ strategy, which is to target only the buyers having a higher s posterior valuation in the case of non-white buyers. In general, the statistical survey shows that the average income of white American is higher than that of non-white American. If this statistical information induces the dealer to have a perception that the non-white buyers have relatively low initial willingness to pay, the dealer may think that he has to lower the price signi…cantly to serve everyone including the ones with low posterior valuation.9 Thus, it may yield a larger pro…t for the seller to target only the ones with high posterior valuation. On the other hand, if the dealer deals with the white buyers who are believed to have relatively high willingness to pay, the seller may want to o¤er a relatively low price in order to entice all the buyers regardless of the signals they receive, since even the ones getting a bad signal still have a relatively high posterior 8 Ayres and Siegelman (1995) discuss the following three hypotheses: i) More people in minority groups are not aware of the fact that the sticker price is negotiable. ii) They are more likely to be averse to conducting negotiations. iii) Black Americans might have higher willingness to pay in terms of search costs. Especially, iii) is directly related to price discrimination based on the di¤erence in consumers’ willingness to pay. As to this, our hypothesis is that salespeople may charge a higher price to black Americans even though they are more likely to have lower willingness to pay. 9 In our basic model, the buyer’ initial willingness to pay is de…ned as the buyer’ initial s s valuation of whether the given good would be a good match or not. However, it would be natural to conjecture that the income level is the other important factor which determines the buyer’ willingness to pay. s 14 valuation.10 Until now, the information structure has been given exogenously. What happens if we relax this assumption so that the seller can choose the level of information to some extent? In particular, it would be interesting to ask whether the seller has incentives to engage in information discrimination. If we restrict our attention to the relevant case for reverse price discrimination, it can be easily understood that the seller has opposite incentives in providing information to di¤erent people. While H ( L ; ) is increasing in ; L ( H ; ) is decreasing in : That is, the seller would provide as much (precise) information as possible to the L buyer, whereas she would provide as little (precise) information as possible to the H buyer. Since the seller’ optimal strategy for L buyers is to charge a high price and serve only those s who receive a good signal, she wants to allow them to update their beliefs more precisely so that they have a higher posterior valuation upon a good signal. On the contrary, for H buyers, since the seller charges a low price and targets all buyers including the ones with a bad signal, the more precise information would only make the buyers with a bad signal more disappointed.11 This result may be able to explain one interesting observation in Ayres and Siegelman (1995). They tested whether salespeople might discriminate between white and non-white buyers simply because of their animus or hostility. However, they found that salespeople actually spent more time with non-white consumers while o¤ering higher prices, which is consistent with our argument.12 10 We do not claim that our reasoning is the most proper explanation for the …nding in Ayres and Siegelman (1995). We believe that price discrimination in car sales can stem from a complex interaction of all factors discussed in Ayres and Siegelman (1995) and our complementary reasoning. 11 Technically, this argument is reminiscent of Johnson and Myatt (2006). They show that pro…ts are "U-shaped" in the dispersion of demand. When the marginal consumer is above average, as in the case of the L buyer, pro…ts increase as the demand curve is more dispersed. In contrast, when the marginal consumer is below average, as in the case of the H buyer, the less dispersion of the demand curve raises pro…ts. In our model, the demand curve is dispersed by more precise information. A major di¤erence in our model is that the degree of dispersion is systemically dependent on a buyer’ prior valuation through Bayes’ s rule. Thus, we show that the seller provides the L buyer with more precise information to induce the dispersion of the demand curve, whereas she provides the H buyer with less precise information to reduce the dispersion of the demand curve. 12 If buyers can decide whether to receive more information or not, there is no reason for 15 In the following two sections, we relax two key assumptions underlying our basic model, the unit demand and the binary signals, to check the robustness of the main results. In each case, we retain the other assumption to isolate the e¤ect of each assumption. 1.3.2 General Information Structure Now, we incorporate a general information structure. The buyer observes a signal s 2 [0; 1]. The signal is drawn from one of two distributions: FH (s) (with density fH (s)) if v = H, or FL (s) (with density fL (s)) if v = L: Both densities are bounded and twice di¤erentiable and the likelihood ratio M (s) fL (s)=fH (s) is decreasing in s. This monotone likelihood ratio property (MLRP) implies that higher signals are more associated with the high quality. The support of fL (s) is de…ned to be the set, fs : 0 de…ned to be the set, fs : s s s sg; whereas the support of fH (s) is 1g with s > s: Accordingly, we must have lim M (s) = 1 and lim M (s) = 0: s!0 s!1 This implies that the extreme signals are perfectly informative. When s > s, buyers can be sure that the quality of the good is certainly H, while when s < s, it is certainly L. In the s case when s 2 [s; s], the quality of the good is unclear and the buyer’ posterior valuation is p( ; s) = fH (s) =1 fH (s) + (1 )fL (s) 1+ (1 ) M (s) : (1.7) It is important to note that the two density functions should overlap partially to have reverse price discrimination. What if two density functions do not overlap as in a case of s s? In this case, the information structure is perfect because buyers can surely tell whether the good is H or L. This case corresponds to = 1 in our basic model. On the other hand, what if two density functions overlap substantially by having the same support L buyers to refuse further information since they will end up with zero net utilities after all regardless of the level of information they obtain. 16 as in a case of s = 1 and s = 0? This case is now equivalent to = 1=2: That is, our modelling strategy again captures the notion that the good is neither perfectly search good nor perfectly experience good. Given the buyer’ prior belief and a signal about the quality, he will make a decision on s whether to buy or not. The buyer decides to purchase the good if he receives a signal better than his standard. Since the buyer’ expected net payo¤ from buying the good is p( ; s) s he purchases if and only if p( ; s) P , i.e., M (s) (1 P, 1 P . The cuto¤ signal is de…ned ) P as s( ; P ) b min s 2 [0; 1] M (s) 1 (1 ) P P : (1.8) If the buyer receives a signal better (worse) than the standard s( ; P ), he decides to buy (not b buy) the good or service. s( ; P ) is decreasing in , i.e., the buyer with a more optimistic b belief will set a lower standard because p( ; s) is increasing in both and s. Obviously, the buyer sets a higher standard for a higher price. We turn to the seller’ problem, where she chooses the price P to maximize her pro…ts. s There are two possible cases. First, the seller can choose P = 1 only to target the buyer with s > s. That is, when P = 1, the fraction of buyers (1 The corresponding pro…t is H ( ) = (1 FH (s)) purchase the good. FH (s)) 1. On the other hand, the seller can choose P < 1: In this case, the buyer purchases the good when he receives a signal s the expected demand can be written by D(b) = [1 s FH (b)] + (1 s )[1 FL (b)]: s The seller maximizes M ax P L = P f (1 s:t: M (b) = s FH (b)) + (1 s 1 (1 ) P P 17 ; )(1 FL (b))g s s, b and from the analysis, the following proposition summarizes the possibility of reverse price discrimination. Proposition 4 If P = 1 if fL (s) < (1 M 0 (s) b; and P < 1 if FH (s)), there exists at least one cuto¤ value b so that > b. Proof of Proposition 4 In the appendix. The necessary condition for reverse price discrimination is intuitively appealing. First, s should not be too large because when s is close to 1, the right-hand side 1 FH (s) approaches to 0. Second, s should not be too small. As s decreases, M 0 (s) in the left-hand side will be decreasing as well. Combining the two arguments, we can conclude that the two density functions should not overlap too much. That is to say, information should not be too imprecise. In addition, when we consider the two types of buyers …xed as in the baseline model, the cuto¤ b must exist between L and H : Then, the type-L buyer is o¤ered a higher price than the type-H buyer. Intuitively, for b to be smaller than H ; s should not be too s close to s. It is because the seller’ incentive to choose P = 1 is greater as the probability of drawing a signal s 2 (s; 1] becomes larger. On the other hand, the seller’ incentive to s P < 1 is smaller because the probability of drawing a signal s 2 [s; s] becomes smaller. Thus, information should not be too precise either. As an example, we can consider the case in which a bad signal follows the uniform distribution on [0; s], and a good signal follows the uniform distribution on [s; 1], where s < s. Note that when s 2 [s; s], the likelihood ratio 1 s s is constant and buyers purchase the good as long as s s + (1 s)(1 ) P: When P = 1, the buyer purchases the good with probability 1 s . This is the probability 1 s that buyers draw a signal from (s; 1]. On the other hand, when P = s +(1 ss)(1 18 ) , the fraction of buyers + (1 ) s s s purchase the good. This is the probability that buyers draw a signal from [s; 1]. We can easily compute the seller’ pro…t at each price as follows. s 8 > < > : 1 s H = 1 s s +(s s)(1 L = s +(1 s)(1 at P = 1, and ) ) at P = s +(1 ss)(1 ) . Both H and L are monotonically increasing in : We can …nd b = that H ( ) R L ( ) as (1 s)(1+s 2s) such s(s s) Q b, where H (b) = L (b). For b to exist between 0 and 1, we should have 1 + s > 2s: Again, this condition implies that the two density functions should not overlap too much. Now, consider H and L …xed. Then, for b to be smaller than H ; s s should be large enough. That is, the two density functions should not overlap too little. The result is very similar to what has been shown in our basic model. When the buyer has a relatively low valuation, the seller charges the maximum price 1 and serves only the buyer who can be sure of the high quality. By contrast, when the buyer’ valuation is greater s than the threshold b, the seller o¤ers a lower price to serve the buyers who are uncertain of the quality. The reverse price discrimination, P ( L) = 1 > s H + (1 s H s)(1 H) = P ( H ); arises if L 2 [0; b] and H 2 (b; 1]. Of course, this is possible only when information is neither too precise nor too imprecise. 1.3.3 Non-unit Demand: Linear Demand Case We now relax the buyer’ unit demand to a linear demand function. A buyer’ demand s s function is given by D(P ) = P where is the buyer’ prior valuation about the good. s Depending on what signal s 2 fsH ; sL g the buyer draws, his posterior demand function is either D(P ) = pH ( ; ) P or D(P ) = pL ( ; ) P . As s 2 fsH ; sL g is the buyer’ private s information, a seller does not know the actual demand and therefore should calculate the 19 Figure 1.3: (a) Linear demand function expected demand. If P (b) The optimal price pL ( ; ), the expected demand is D(P ) = Pr(sH ) (pH ( ; ) which turns out to be D(P ) = P ) + Pr(sL ) (pL ( ; ) P); P . On the other hand, if P > pL ( ; ), the buyer with a bad signal does not want to purchase and thus the seller faces D(P ) = pH ( ; ) P with probability Pr(sH ). Then, the seller’ expected demand with the Bayesian buyer is inwardly s kinked as shown in Figure 1:3 (a). D(P ) = 8 > > > > < if P P Pr(sH ) (pH ( ; ) > > > > : 0 P) pL ( ; ); if P 2 (pL ( ; ); pH ( ; )]; if P > pH ( ; ): Since we assume zero production cost, the optimal price should be either P = pH ( ; )=2 or =2. With a linear demand curve, the monopoly price is simply half of the price intercept. The analysis yields that the optimal price P depends on the value of 20 and as follows. Lemma 1 (1) Suppose that p 5 1 2 ; 1 . If 2 2 1; 2 p 5 1 . Then, P 2 = =2. (2) Suppose that 2 0; 2+ 1 1 , P = pH ( ; )=2 and if 2+ 1 2 1 ;1 , P 2 2 = 2. Proof of Lemma 1 In the appendix. If the information quality is relatively low, i.e., 2 1 2; p 5 1 , the optimal price is a 2 monotone increasing function of . On the other hand, if the information quality is relatively high, i.e., 2 p 5 1 2 ; 1 , there exist two optimal prices contingent on the prior valuation . The seller’ decision is whether to charge a low price, =2, for the sake of obtaining a high s demand or charge a high price, pH ( ; )=2, for the sake of earning a high margin. Keep in mind (1.4): pH ( ; ) is increasing in and pL ( ; ) is decreasing in . When is relatively low, the demand with a bad (good) signal is not much lower (higher). In this case, charging the low price for serving a buyer with a bad signal is more pro…table. On the other hand, when is relatively high, the demand with a bad (good) signal becomes low (high) enough so that sometimes charging a high price can be more pro…table. More precisely, charging a high price is optimal if is relatively low. Otherwise, with a high , the demand with a bad signal is not likely to be low enough for the high price to be optimal.13 Figure 1:3 (b) demonstrates the optimal price P , when 2 p 5 1 2 ; 1 , as a function of . There is a downward jump in the optimal price at a certain threshold. As a result, the optimal price is not monotone with respect to . Then we can show the parameter set of for which the reverse price discrimination is obtained.14 Proposition 5 Suppose that + 2 1; _ 2 1 where _ = (2 L 2 L p 5 1 2 ;1 . L +1) If L 2 + 2 1; + 2 1 2 1 +2 2 1 and H 2 , then P ( L ) > P ( H ) for L < H . 13 If is su¢ ciently high, we need very high prior valuation in order to sustain this low optimal price. 14 In other words, here we …nd the necessary condition for reverse price discrimination, which says that information should not be too imprecise. When we consider H and L …xed, we can …nd the su¢ cient condition for which should not be too large. This can be + 2 1 ; _ in this case. easily seen when is cloase to 1; because H 2 = 2 21 1 1.4 Concluding Remarks The model presented in this paper studies price discrimination under partially incomplete information in the sense that a buyer’ prior valuation can be observed by the seller(s) but s the buyer further draws a private signal which may give him or her additional information about a product sold by the seller(s). In this environment, we demonstrate the possibility that the buyer with a higher willingness to pay is o¤ered a lower price and vice versa, since the seller only targets the buyers who draw a good signal by charging a high price when they are of a low type, whereas she wants to serve all of the buyers by o¤ering a low price when they are of a high type. A possible criticism to our main …nding is that reverse price discrimination is not surprising because it is an existing result in the literature of statistical discrimination. According to the reasoning of statistical discrimination, the price reversion is mainly induced by the di¤erence in the variance of posterior distributions.15 But this is simply not true in our model because the optimal pricing is determined by the price elasticity of demand, not by the variance. Another criticism is that one can easily construct an example of two distributions where the optimal price is non-monotone in the mean valuation. That is, it can be argued that reverse price discrimination can arise without information inference. However, only providing a necessary condition is not the main point of our paper. We are not trying to …nd the arbitrary distributions, without economic motivation, for which reverse price discrimination can be derived. Our model shows how the posterior valuations generating reverse price discrimination are formed endogenously in terms of information quality. In particular, our paper answers what product markets and what characteristics of products 15 This idea originated from Phepls (1972) that proposes the seminal model in which group average is used as a proxy for unobserved exogenous di¤erence that are relevant to economic outcomes. See Fang and Moro (2011) for an excellent survey of the theories of statistical discrimination. In fact, Ayres and Siegelman (1995) borrow this idea to explain discrimination in car sales markets and argue "it may be pro…table for dealers to o¤er higher prices to a group of consumers who have a lower average reservation price, if the variance of reservation prices within the group is su¤ciently large." Certainly, this is possible. But this is not the mechanism of our model. For example, there is a case where the optimal price is higher with a lower variance in our model. 22 give rise to reverse price discrimination. 23 APPENDIX 24 Appendix Proof of Proposition 1 H ( i; ) R L ( i ; ) =) 0. Let f ( ) = i 2 (1 (1 iR (1 ) i i )+(1 2 i+ i 2 i) + ) i =) i 2 (1 2 i) + 2 i+ i iR i. 1 = H>2 Case (1) Then, (a) f ( ) is a concave function, (b) f ( ) attains max value at = H +1 2 (2 H >1 1) and f (c) f 2 = H +1 2 (2 H = 1) ( H 5) ( H 1) H > 0; 4 (2 H 1) 1 =1 = 1) > 0. So 9 2 such that if H < 0, (d) f ( = 1) = H( H 2 4 i i 1 1; , f( ) 0 and if 2 ( 2 ; 1), f ( ) > 0. This implies that if 2 2 ; 2 , 2 2 L ( H ; ) and if H( H; ) Case (2) 2 ( 2 ; 1), H ( H ; ) > L ( H ; ). = L<1 2 Then, (a) f ( ) is a convex function, (b) f ( ) attains min value at = L+1 2 (2 L 1) < 1 2 and f (c) f 2 =1 2 = 1 4 = L+1 2 (2 L L < 0, (d) f ( 1; 2 1 , f ( ) < 0 and if H ( L; ) < 1) L ( L ; ) and if = ( L 5) ( L 1) L < 0; 4 (2 L 1) = 1) = 2 [ 1 ; 1), f ( ) 2 [ 1 ; 1), H ( L ; ) 25 L( L 1) > 0. So 9 1 such that if 0. This implies that if L ( L ; ). 2 1; 2 1 , Now let us de…ne 1 ( i) = 2 i 1 The computation yields that 2 = @( Here, 6 +5 q 1 6 i+ 2+5+ i 2 : ( i = L ). Also q 7+3 q 2 1) > 0: 5 i 2 (2 i p 3 2 1 2 ( i = H ) and 1 = ( i )) = @ i 1 2 i 2 2 i 2 i 6 i+5 6 i+5 5 )2 = ( 4) (2 (7 1)2 < 0; p 5 because 3 2 6 + 5 > 0 and 7 5 > 0 for @ ( ( i )) 2 [0; 1]. As the numerator is negative, > 0. Then, this implies that 2 > 1 @ p which implies 3 2 6 +5 < 7 because H > 1 > L . Then, (i) If 2 H ( L; ) > L ( i ; ). L ( L ; ) and Note that i 1; 2 1 , 2 H( H; ) < H ( i; ) < L ( H ; ). L ( i ; ). (ii) If (iii) If 2 ( 2 ; 1), 2 [ 1 ; 2 ], H ( i; ) > H ( i ; ) ( L ( i ; )) is the pro…t when a seller charges pH ( i ; ) (pL ( i ; )) for type i. Then this proves Proposition 1. Proof of Proposition 3 Let us de…ne eH ( L ) such that pL (eH ( L ); ) = pH ( L ; ) for given L . Then, pL ( H ; ) < pH ( L ; ) for L < H < eH ( L ). By solving L L + (1 )(1 L) = (1 in terms of eH ( L ), we obtain eH ( L ) = )eH ( L ) eH ( L )) + (1 )eH ( L ) (1 2 2 L + (1 L )2 (1 L) : Or if we de…ne eL ( H ) such that pH (eL ( H ); ) = pL ( H ; ) for given H , pL ( H ; ) < 26 pH ( L ; ) for eL ( H ) < L < H . By solving (1 ) H H ) + (1 (1 L = ) H L + (1 )(1 L) ; we obtain ( eL ( H ) = H 1)2 H : 2 H+ 2 Proof of Proposition 4 Note that we are considering the case where s 2 [s; s]. When P = 1, H( ) = (1 FH (s)). When P < 1, the seller solves M ax P L = P f (1 FH (b)) + (1 s 1 s:t: M (b) = s (1 ) P P )(1 FL (b))g s : We can …nd the optimal price from the …rst-order condition, D(b) s P [ fH (b) + (1 s )fL (b)] s @b s = 0: @P (1.9) Now, what we want to show is that there must exist a cuto¤ value b such that H ( ) R L ( ) as Q b. First, H ( = 0) = lim L ( ) = 0 and both H ( ) and L ( ) are non-decreasing !0 in . It is easy to see that H ( ) is linearly increasing in : To investigate the slope of L ( ); we use the envelop theorem and obtain @ L =P @ fFL (b) s FH (b)g s f fH (b) + (1 s )fL (b)g s This is always non-negative. The …rst term in the bracket, FL (b) s @b s @ 0: (1.10) FH (b), represents the s …rst-order stochastic dominance which always holds under the MLRP. The second term is 27 @b negative because of @ s Second, when 0: is close to 1; we have H ( ) < L ( ). Let us show that lim L > H ( = 1) = 1 !1 It can be shown that lim L = 1: Intuitively, as FH (s): is close to 1; the posterior belief (1.7) !1 approaches to 1. In other words, when buyers believe almost surely that the good is of high quality, the seller’ optimal price is 1 and buyers decide to purchase the good. Formally, let s us show that lim L = !1 lim P lim D(b(P )) s !1 !1 It is immediate to obtain lim D(b(P )) = 1 s !1 = 1: b FH (s) = 1 because lim s = s: To prove !1 lim P = 1; let us show that the …rst-order condition is always positive as !1 approaches to 1: Applying the implicit function theorem, the …rst-order condition (1.9) can be rewritten as D(b) + s 1 1 1 [ fH (b) + (1 s M 0 (b) P s )fL (b)] = 0: s Note that the second term is indeterminate in the limit because 1 goes to in…nity and M 0 (b) goes to 1. Once we apply L’ s hopital’ rule, it can be shown that the second term s turns out to be 0. Then, as the …rst-order condition is always positive, the optimal pricing is to increase the price as much as possible. Hence, P = 1. Next, let us …nd the condition for @ L < (1 !0 @ lim This implies that when of (1.10) at ! 0 as FH (s)): is close to 0, we must have H ( ) > L ( ). We …nd the limit value @ L = !0 @ lim 28 fL (s) : M 0 (s) This can be derived in the following steps. As approaches to 0, the posterior belief (1.7) becomes closer to 0 as well and so P must be 0. The formal proof is analogous to the one given for the case of ! 0, the …rst term becomes 0 because it goes to ! 1. If P fFL (b) FH (b)g. Now, let us investigate the second term, s s P f fH (b)+(1 s )fL (b)g @ s : s @b Using the implicit function theorem, @b s = @ (1 1 1 P : )2 M 0 (b) P s The second term is rewritten as f fH (b) + (1 s As )fL (b)g s (1 1 (1 )2 M 0 (b) s P ): approaches to 0; s becomes s: Consequently, this term turns out to be b fL (s) : Note M 0 (s) that the sign of this term is positive because the likelihood ratio, M (s), is decreasing. As a result, if fL (s) is smaller than (1 M 0 (s) such that H R L as Proof of lemma 1 Q b. FH (s)); at least there should exist one cuto¤ value b D Pr(sH ) Case (1) P = pH ( ) In this case, the optimal price is p ( ; ) P = H = 2 2 (2 and it is binding only if pL ( ; ) < P + 1) < pH ( ; ) is satis…ed. It is always true that P < pH ( ; ) because P pH ( ; ) = 2 ( (2 29 1) + (1 )) < 0: Also P where 2 < 0 for all 2+2 2 +4 5 2 ( (2 1) + (1 pL ( ; ) = 1 2 2 ; 1 and 2 )) (2 2 ) 2 (0; 1). The analysis of the numerator yields the following result. Result 1. pL ( ; ) < P holds if Case (2) P = 2 2 p 2; 1 and 2 0; 2 4 +2 . 2 2 5 +2 D In this case, the optimal price is P = 2 and it is binding only if (1 ) (1 ) + (1 0 < P < pL ( ; ) = ) is satis…ed. Here, P where 2 pL ( ; ) = (2 3 2 (2 + 2) ; ) < 0 always. Then the analysis of the numerator yields the following result. Result 2. P < pL ( ; ) holds in following two cases: (i) If or (ii) if 1 2 2; 2 3 and 2 2 ; 1 and 3 3 2 2 2; 1 1 2 1; 1 . 2 Then, from Results 1 and 2, the optimal price P can be described as follows. Result 3. (1) If 2 1 2; 2 p 2 , P = 2 . (2) Suppose that 2 2 p 2; 2 . (i) 3 2 4 +2 , there exist two candidates P = 2(2 and 2 . (ii) If +1) 2 2 5 +2 2 4 +2 2 3 2 2 ; 1 , P = 2 . (3) Suppose that 2 3 ; 1 . (i) If 2 0; 2 1 , P = 2 2 5 +2 2 , (ii) if 2 3 2 ; 2 4 +2 , there exist two candidates: P = 2(2 2 1 2 2(2 +1) +1) 5 +2 2 4 +2 and 2 , (iii) if 2 ; 1 , P = 2. 2 2 5 +2 If 2 0; Now, for given two candidates P = 2(2 +1) 30 and 2 , we check which price yields a greater pro…t to the seller. When P = 2(2 +1) , the pro…t is denoted by 2 2 p ( ) (P = H )= 2 4 (2 + 1) and when P = 2 , the pro…t is denoted by 2 (P = ) = : 2 4 The comparison of both pro…ts yields the following result. Result 4. (1) If < p p 5 1 2 , (P p ( ; ) = H2 ) < (P = 2 ) =) P 5 1 + 2 1 , (P = pH ( ; ) ) > (P = ) =) P 2 , if 2 0; 2 1 2 2 2 1 pH ( ; ) + ) < (P = 2 ) =) P = 2 . 2 2 1 ; 1 , (P = 2 > if = 2 . (2) When = 2(2 +1) and We apply Result 4 to 2-(i) and 3-(ii), Result 3 can be simpli…ed as Lemma 3. (1) Suppose that (2) Suppose that 2 + 2 1; 1 , P 2 1 p 1 2 2 ; 5 1 . Then P = 2 . 2 2 p 5 1 2 ;1 . If 2 = 2. 31 2 0; + 1 1 , P 2 = 2(2 +1) and if BIBLIOGRAPHY 32 Bibliography [1] Armstrong, Mark., "Recent Development in the Economics of Price Discrimination," Advances in Economics and Econometrics: Theory and Applications: Ninth World Congress, Blundell eds., Newey and Persson, Cambridge University Press, 2006 [2] Ayres, Ian., "Further Evidence of Discrimination in New Car Negotiations and Estimates of its Causes," Michigan Law Review, 1995, 94(1), pp. 109-147 [3] Ayres, Ian and Siegelman, Peter., "Race and Gender Discrimination in Bargaining for a New Car," American Economic Review, June 1995, 85(3), pp. 304-332 [4] Borenstein, Severin., "Price Discrimination in Free-Entry Markets," RAND Journal of Economics, 1985, 16(3), pp. 380-397 [5] Chen, Yongmin., "Oligopoly Price Discrimination and Resale Price Maintenance," RAND Journal of Economics, 1999, 30(3), pp. 441-455 [6] Corts, Kenneth S., "Third-Degree Price Discrimination in Oligopoly: All-Out Competition and Strategic Commitment," RAND Journal of Economics, 1998, 29(2), pp. 306-323 [7] Courty, Pascal and Li, Hao., "Sequential Screening," Review of Economic Studies, 2000, 67, pp. 697-717 [8] Fang, Hanming and Moro, Andrea., "Theories of Statistical Discrimination and A¢ rmative Action: A Survey," Handbook of Social Economics, North-Holland, 2011, Vol I. [9] Fudenberg, Drew and Villas-Boas, Miguel J., "Behavior-Based Price Discrimination and Customer Recognition," Handbook on Economics and Information Systems, 2005, Hendershott eds., North-Holland, Amsterdam [10] Holmes, Thomas J., "The E¤ects of Third-Degree Price Discrimination in Oligopoly," American Economic Review, March 1989, 79(1), pp. 244-250 [11] Johnson, Justin P. and Myatt, David P., "On the Simple Economics of Advertising, Marketing, and Product Design," American Economic Review, June 2006, 96(3), pp. 756-784 33 [12] Lewis, Tracy R. and Sappington, David E., "Supplying Information to Facilitate Price Discrimination," International Economic Review, May 1994, 35(2), pp. 309-327 [13] Maskin, Eric and Riley, John, "Monopoly with Incomplete Information," RAND Journal of Economics, 1984, 15(2), pp. 171-196 [14] Mussa, Michael and Rosen, Sherwin, "Monopoly and Product Quality," Journal of Economic Theory, 1978, 18, pp. 301-317 [15] Nelson, Phillip, "Information and Consumer Behavior," Journal of Political Economy, 1970, 78, pp. 311- 329 [16] Phelps, Edmund S., "The Statistical Theory of Racism and Sexism," American Economic Review, 1972, 62, pp. 659-661 [17] Robinson, Joan, "The Economics of Imperfect Competition," Macmillan, 1933 [18] Stole, Lars A., "Price Discrimination and Competition," Handbook of industrial organization, Volume 3, Armstrong and Porter eds., North-Holland, 2007 [19] Varian, Hal, “Price discrimination,” Handbook of industrial organization, Volume 1, Schmalensee, Richard and Willig eds., North-Holland, 1989 [20] Yinger, John, "Evidence on Discrimination in Consumer Markets," Journal of Economic Perspectives, 1998, 12, pp. 23-40 34 Chapter 2 Reverse Price Discrimination with Competition: A Hotelling Duopoly 2.1 Introduction In the previous chapter, we show that prices can be non-monotone in buyers’prior valuation - the seller may o¤er a higher price to low valuation buyers than to high valuation buyers. Our result seems contradictory to the general conjecture that it would be optimal to charge a higher (lower) price if buyers are with high (low) valuation. The intuition underlying this result is as follows. When the seller makes a decision on the prices, she faces a traditional trade-o¤ between getting a higher margin and getting a greater market share. This trade-o¤ depends crucially on the price elasticity of demand, which is essentially determined by the interaction between a buyer’ prior valuations and the precision of information available to s the buyer. When information precision is not extreme (i.e., neither too accurate nor too vague), we …nd that the demand is elastic (inelastic) when the buyer’ prior valuation is s high (low). Thus, if the buyer’ prior valuation is high, the seller chooses to o¤er a low price s because the buyer’ updated valuation does not decline signi…cantly even when he receives s unfavorable private information. On the other hand, if his prior valuation is low, it is more pro…table for the seller to charge a high price and target only the buyer who receives favorable 35 private information because the price under which the buyer with unfavorable information is covered is too low to be pro…table. This leads to the result that a higher price can be o¤ered to buyers with lower willingness to pay and vice versa. We now want to explore a similar phenomenon in a duopoly market to see if reverse price discrimination still occurs even with competition between sellers. Consider that the competing sellers share buyers’prior valuations and the buyers gather further private information about the product or service provided by each seller. Again, we show that reverse price discrimination can arise in this case as well. In the duopoly case, new information plays a role of di¤erentiating two products. Especially when the buyer gathers opposite information about each product (for example, the buyer receives a good signal from one seller while obtaining a bad signal from the other seller), he perceives that two products are more di¤erentiated than before and thus the price competition between the sellers can be mitigated. We …nd that the degree of di¤erentiation becomes higher, and so does the equilibrium price, when the buyer has an intermediate prior valuation. The intuition is straightforward. The buyer with intermediate prior valuation is not certain whether the product would be a good match or a bad match for him, thus he is quite sensitive to the new information he gathers. Hence, when he receives opposite information from the two products he tends to be strongly biased toward the product from which he obtains relatively favorable information. This, in turn, gives the sellers a weaker incentive to o¤er a lower price to attract them, since the buyer with biased preferences is less likely to change his minds by price di¤erential. For the opposite reason, the buyer with extreme (either high or low) prior valuation has the advantage of being o¤ered a lower price, since he is likely to perceive that the two products are still similar even after updating his valuations. As a result, the equilibrium prices do not change monotonically with the buyer’ prior valuation. This yields the possibility that s the buyer with lower prior valuation is charged the higher price than the buyer with higher prior valuation. The topic of third-degree price discrimination, which considers the case where the characteristics of buyers (who have no private information) are observable, by a monopolist has 36 been covered extensively in literature after the seminal work of Robinson (1933).1 In particular, Borenstein (1985), Holmes (1989), Corts (1998), and Chen (1999) are the recent papers which extend the analysis of third-degree price discrimination to the setting of oligopoly. Price discrimination with incomplete information has been studied in the environment of screening or self-selection mechanism followed by Mussa and Rosen (1978) and Maskin and Riley (1984). In particular, Lewis and Sappington (1994) and Courty and Li (2000) are closely related to ours because they also study the environment where consumers are initially uncertain of their valuations that are in part revealed afterwards. Both papers study second-degree price discrimination in which …rms are assumed not to be able to observe consumers’ expected valuations. On the other hand, in our paper, the sellesr are able to observe consumers’expected valuations and choose their prices based on them. The rest of this article proceeds as follows. Section 2.2 describes the structure and assumptions of the basic model. Section 2.3 shows the possibilities of reverse price discrimination in duopoly case. Section 2.4 concludes. 2.2 The Model In this section, we explain the basic setup of the model, which is consistent with the monopoly case in the previous chapter. Players. The buyers have a unit demand for the good, which is supplied by two sellers. The true value of the good depends on the match between a buyer’ preference and the features s of the good, which is denoted by v 2 fH; Lg. If v = H, the good is a good match with a buyer and if v = L, it is a bad match. A buyer’ prior belief for v = H is denoted as s 2 [0; 1]. We normalize the buyer’ valuation for the good to be 1 for v = H and 0 for s v = L. Accordingly, can be thought of as the buyer’ prior valuation for the good and s therefore as the willingness to pay. The buyer’ type s is public information and observable 1 Armstrong (2006), Stole (2007) and, Varian (1989) are the excellent survey papers about price discrimination. 37 to the sellers. For simplicity, the reservation value of sellers for the good is assumed to be 0. Information. The buyers receive private information about the good while they are inspecting the good before purchase. Although the buyers are ex ante homogeneous, they may draw di¤erent binary signals, s 2 fsH ; sL g; on their match value. The realization of a signal is privately observed by the buyers, so it is private information. As is standard, the signals partially reveal the true match value of the good in the sense of Blackwell, Pr(sH jv = H) = Pr(sL jv = L) = Pr(sL jv = H) = Pr(sH jv = L) = 1 where 2 1; 1 2 without loss of generality. is often interpreted as the precision or the quality of a signal and is common knowledge. As the buyer receives a signal s 2 fsH ; sL g, he updates his beliefs on the match value for the good. Let us refer to pH ( ; ) and pL ( ; ) as the buyer’ posterior valuation when s the buyer with a prior 2 (0; 1) draws a signal sH and sL , respectively. Then, Bayes’rule leads to pH ( ; ) = pL ( ; ) = + (1 )(1 (1 ) (1 ) + (1 ) and ) because we normalize the buyer’ valuation for the good to be 1 for v = H and 0 for v = L. s The buyer’ posterior valuation is a mean-preserving spread of his prior belief. A signal s disperses the prior belief to two-point distribution: pH ( ; ) with the probability Pr(sH ) and pL ( ; ) with the probability Pr(sL ). The probabilities that the buyer receives a good signal and bad signal are given by Pr(sH ) = X v2fL;Hg Pr(sH jv) Pr(v) and Pr(sL ) = 38 X v2fL;Hg Pr(sL jv) Pr(v); respectively. Timing. The timing of the game is as follows. In the …rst stage, the sellers can observe the buyer’ prior valuation s perfectly. At the same time, the buyer draws a private signal about the match value of the good from each of two sellers. In the second stage, the sellers simutaneously choose their prices, and then the buyers make a purchasing decision given the prices. 2.3 Duopoly: Hotelling Model We have shown in the previous chapter that reverse price discrimination can be derived in the monopoly case. Then, can it also be the case even when sellers compete with each other? In this chapter, we extend our analysis to a duopoly market and consider the case where two sellers are located at two end points on the Hotelling line of unit length. We also show that the equilibrium prices can be non-monotone in the buyer’ prior valuation in the duopoly s case. Two sellers, A and B, supply di¤erentiated products A and B respectively. As in a standard Hotelling model, buyers are uniformly distributed and indexed as x 2 [0; 1] which denotes each buyer’ location or brand preference. Buyers purchase either one unit of a s good from only one seller or nothing. Type ( ; x) buyer’ value is w + s and w + t(1 x) for good B. Note that w + tx for good A is the intrinsic value of consuming the product. While w captures the product value with no uncertainty, represents the value with uncertainty. We assume that w is su¢ ciently large so that the market is fully covered. The parameter t > 0 re‡ ects the degree of product di¤erentiation. To keep consistency with the monopoly case, we assume that the buyers’prior valuations are public information and known to sellers. For analytical simplicity, we assume that the symmetric information structure is exogenously given to the sellers. Now the buyers independently receive a private signal from each seller. They face 39 Pr( sH ) 2 Pr( sL ) 2 Pr( sH ) Pr( s L ) Pr( sH ) Pr( s L ) 0 xHH 1 0 xLL 1 0 xHL 0 1 1 xLH Group HH Group LL Group HL Group LH Figure 2.1: Demand in the Hotelling model one of four possible cases according to the combinations of signal realizations sA ; sB , where sA ; sB 2 fsH ; sL g denote the signals from sellers A and B. As both signals sA and sB are buyers’ private information, each seller expects the possible outcome as follows: (i) sA ; sB = (sH ; sH ) with probability Pr(sH )2 , (ii) probability Pr(sL )2 , (iii) sA ; sB sA ; sB = (sL ; sL ) with = (sH ; sL ) with probability Pr(sH ) Pr(sL ), and (iv) sA ; sB = (sL ; sH ) with probability Pr(sH ) Pr(sL ). Figure 2.1 illustrates the demand structure as a result of the realization of private signals. If the buyers receive the same signals from both sellers, i.e., sA ; sB = (sH ; sH ) or (sL ; sL ), the private signals do not provide additional information about which good would be more preferable than the other. As their relative preference between two goods does not change, there is no di¤erence from the standard Hotelling model in this case. Hence the location of marginal buyers who are indi¤erent between the two goods is denoted by 1 PB PA xHH = xLL = + 2 2t (2.1) where the subscription for x denotes the group as illustrated in Figure 4, and P A and P B are the prices o¤ered by sellers A and B, respectively.2 2 We only consider the interior solution such as 0 < x HH ; xLL < 1. This implies that 40 On the other hand, if the di¤erent signals are drawn from two sellers, the marginal buyers for the case where sA ; sB = (sH ; sL ) and (sL ; sH ) are, respectively, xHL = where ( ; ) 1 PB + 2 PA + 2t ( ; ) pL ( ; ). Here, pH ( ; ) 1 PB and xLH = + 2 PA 2t ( ; ) (2.2) ( ; ) represents a buyer’ bias after observing s di¤erent signals. Note that depending on the size of the bias, the marginal buyers may not exist in the Hotelling line. That is, xHL can be greater than 1 and/or xLH can be smaller than 0. Lemma 2 (1) If then xHL ( ; ) < t, then 0 < xHL < 1 and 0 < xLH < 1. (2) If 1 and xLH ( ; ) t, 0. Proof of Lemma 2 In the appendix. When ( ; ) < t, seller A0 s demand function is written by DA = (Pr(sH )2 + Pr(sL )2 )xHH + Pr(sH ) Pr(sL )(xHL + xLH ): In this case, since the bias is small, the marginal buyers exist and stand on the interior point in the Hotelling line. By contrast, when t, seller A0 s demand function is given by ( ; ) DA = (Pr(sH )2 + Pr(sL )2 )xHH + Pr(sH ) Pr(sL ): the prices di¤erence is less than the product di¤erentiation. For example, if P A 1 + P B P A < 0 and if P B 2 2t B A P B > t, P A > t, 1 + P 2tP > 1. That is, we exclude the possibility 2 that seller j 2 fA; Bg dominates the market of group HH and LL only due to the price e¤ects such that P j is far lower than P j . This also implies that we focus on the symmetric equilibrium although there may exist the asymmetric cases where 0 < xHL < 1 and xLH < 0 and xHL > 1 and 0 < xLH < 1. 41 Now, the bias is large enough relative to the price di¤erence, so the two sellers can avoid competition for the two groups of buyers, groups HL and LH. That is, all buyers in group HL purchase good A, while all buyers in group LH purchase good B. We can write down seller B 0 s demand function in a similar way. Then, for j 2 fA; Bg, each seller j’ demand s function can be derived as follows. Dj = 8 > t+P j P j < if > Pr(s )2 + Pr(s )2 : H L 1 + P j Pj 2 2t + Pr(sH ) Pr(sL ) ( ; ) < t; if 2t ( ; ) t: In turn, solving the two sellers’maximization problems, the symmetric equilibrium prices can be readily shown as follows. Lemma 3 In the symmetric equilibrium, each seller’ optimal price is as follows. s P ( ) = PA = PB = where Pr(sH ) = + (1 )(1 8 > t > < > t > : (Pr(sH )+Pr(sL ))2 Pr(sH )2 +Pr(sL )2 ) and Pr(sL ) = (1 ! if ( ; ) < t; if ( ; ) ) + (1 t; ) . Proof of Lemma 3 In the appendix. When the bias is less than the product di¤erentiation parameter t, the equilibrium prices are t. However, when the bias is greater than t, the equilibrium prices become greater than t. Now, let us describe the condition, ( ; ) R t, as a function of in detail in order to show the relationship between the buyer’ prior valuation and the equilibrium prices. s Proposition 6 (1) For t > 1, P A = P B = t for all 2 (0; 1). (2) For 0 < t < 1, if 1 t+1 A = P B = t for all 2 (0; 1). On the other hand, if 2 t+1 ; 1 , there 2; 2 , P 2 1 and 2 1 ; 1 such that if 2 (0; ) or 2 ( ; 1), P A = P B = t and if exist 2 0; 2 2 (Pr(sH )+Pr(sL ))2 2 [ ; ], P A = P B = t. Pr(sH )2 +Pr(sL )2 2 42 2t P (θ ) * t 0 θ 1/2 1 θ θ Figure 2.2: Equilibrium prices in duopoly Proof of Proposition 6 In the appendix. When t > 1, the bias is always less than the product di¤erentiation due to ( ; ) = pH ( ; ) pL ( ; ) < 1: Thus, the equilibrium price is t as in the standard Hotelling model. The interesting case is the one where 0 < t < 1, in which the bias can be either greater or less than t. In this case, the information quality quality is relatively low, i.e., matters in determining the equilibrium. If the information 1 2 2 ; t+1 , its e¤ect is insigni…cant and the equilibrium price 2 is still P A = P B = t. On the other hand, if the information quality is relatively high, i.e., 2 t+1 ; 1 , the private signal may have a dramatic e¤ect on the buyer’ posterior s 2 valuation, especially when the prior valuation is intermediate, thereby leading to changes in the equilibrium prices. In the case that buyers have extreme (i.e., 43 2 (0; ) or 2 ( ; 1)), they are stubborn about their prior valuation no matter what signals they draw. Thus, they hardly become inclined toward one seller over the other even if they receive di¤erent signals from the sellers. Because of this insensitivity to the signals, the equilibrium prices are still determined to be t, as in the standard Hotelling model. On the other hand, in the case that buyers have intermediate (i.e., 2 [ ; ]), new information starts to play a role in buyer’ relative s preferences. Note that these buyers can be regarded as not so obstinate about their prior beliefs on the match value of the product: the probability that their match value turns out to be either v = H or v = L is similar. Therefore, they are relatively sensitive to the new information, and are easily swayed by the signals they receive. In particular, we …nd that the equilibrium prices are increasing in in 2 1 2; , and maximized at that the bias 2 1; 2 ( ; ) = pH ( ; ) 2 ; 1 , decreasing 2 = 1 . The intuition to understand this is simple. Recall 2 pL ( ; ) is increasing in 2 ;1 2 and decreasing in . This implies that buyers become signal-sensitive as the prior valuation is closer to 1=2. As the buyers are more (less) signal-sensitive, they become less (more) sensitive to the price di¤erential, which, after all, mitigates (intensi…es) price competition between the sellers. Hence the equilibrium prices are higher as is closer to 1=2. To put it simply, the easier consumers are swayed by the signals, the more they are exploited by the …rms. As a result, the equilibrium prices are not monotone with respect to as in Figure 2.2. This non-monotonicity suggests that, even in the duopoly market, there is a possibility that sellers o¤er a higher price to the buyer with lower willingness to pay than to the one with higher willingness to pay. With slight abuse of notation, we reuse the notations H and L to show the reverse price discrimination explicitly. Proposition 7 Suppose that 0 < t < 1 and 2 t+1 ; 1 . Then we obtain the reverse price 2 discrimination, P ( L ) > P ( H ) for L < H ; if (1) L 2 ;1 2 and H 2 (1 L ; 1) or (2) L 2 44 1; 2 and H 2 ( L ; 1). Our paper may provide complementary but sharply contrary results to Armstrong (2006). He shows that it has no e¤ect on the …rms’prices and pro…ts in the Hotelling model even though the …rms can observe a consumer’ valuation and target a personalized price to the s consumers. In our setting with private information, however, the sellers may o¤er di¤erent personalized prices based on buyers’prior valuations.3 2.4 Concluding Remarks The paper studies price discrimination of two competing sellers under partially incomplete information where a buyer’ prior valuation can be observed by the sellers but the buyer s further draws private signals which may give him additional information about a product sold by the sellers. We show that the buyer with a higher willingness to pay may be o¤ered a lower price than the one with a lower willingness to pay, even when there is price competition between sellers. In the monopoly market, the reverse price discrimination occurs due to the elasticity of posterior demand, which was determined by the prior beliefs of buyers and the precision of signals. To illustrate, the seller only targets the buyers who draw a good signal by charging a high price when they are of a low type, whereas she wants to serve all of the buyers by o¤ering a low price when they are of a high type. In the duopoly market, however, the sellers charge a higher price to the buyers with intermediate prior valuations than to the buyers with extreme prior valuations, since the formers are likely to perceive that two competing products are more di¤erentiated when they receive di¤erent signals from the sellers. 3 Damiano and Li (2007) study very similar issues such as price competition for privately informed buyers. A crucial di¤erence is that they focus on the case in which the prior valuation is …xed as = 1=2. In other words, price discrimination is not an issue in their paper. On the other hand, our focus is to …nd what prices two sellers o¤er to buyers based on . In addition, it is worthwhile to explain the di¤erence of the modeling strategy between the two papers. In their model, the two goods are ex ante identical. Thus, there is no pure-strategy Nash equilibrium, and it is hard to characterize and compare the equilibrium prices with our general prior . This is the reason why we model the duopoly market by using the Hotelling model of product di¤erentiation. 45 APPENDIX 46 Appendix Proof of lemma 2 P B P A+ ( ; ) . 2t (i) xHL > 0 =) P B P A > (1) xHL = 1 + 2 ( ; ), which is always true under our assumption. t ( ; ). xLH = 1 + 2 (ii) xHL < 1 =) P B PA < t (iii) xLH > 0 =) P B PA > (iv) xLH < 1 =) P B PA < t + Then, from (ii) and (iii), ( ; ) =) ( ; ) ( ; )t Seller j’ problem is s M ax j = Pj 1 P j Pj + 2 2t Pr(sH )2 + Pr(sL )2 ! ! + Pr(sH ) Pr(sL ) P j : Then, the …rst-order condition is @ j 1 t = Pr(sH )2 + Pr(sL )2 j 2t @P 2P j + P j + Pr(sH ) Pr(sL ) = 0: Again, solving the two …rst-order conditions, we obtain PA = PB = (Pr(sH ) + Pr(sL ))2 Pr(sH )2 + Pr(sL )2 ! t: Proof of Proposition 6 Let us check the condition ( ; ) ( ; ) R t. Here, pH ( ; ) pL ( ; ) = ( (2 1) (2 1) + 1) (2 ) ; then we obtain (2 @ ( ; ) = @ (2 Case (1) 1) ( 1) (2 1) 2 )2 + 1) (2 : >1 2 @ ( ; ) < 0, so ( ; ) is a decreasing function @ (ii) = 1; = (2 1) > 0 2 (i) (iii) 2 [ ; 1), ( = 1; ) = 0. So if (2 2 ( 1 ; ], 2 ( ; ) < t. On the other hand, if (2 Case (2) 1) > t, 9 such that if 1) < t, ( ; ) < t for all <1 2 48 1 2 2; 1 . ( ; ) > t and if (i) @ ( ; ) > 0, so @ (ii) ( = 0) = 0 (iii) = 1; 2 ( ; ) is an increasing function 2 [ ; 1 ), 2 = (2 1). So if (2 1) > t, 9 s.t. if 2 (0; ], ( ; ) < t and if ( ; ) > t. On the other hand, if (2 condition that (2 1) < t, 1) R t =) ( ; ) < t for all R t+1 matters. Note that 2 t+1 2 2 0; 1 . In both cases, the 2 1 1 = t>0 2 2 and t+1 2 1= 1 (t 2 1) : Then we have the following result. (1) if t > 1, always for all if t < 1, 9 hand, if and if = t+1 such that if 2 2 t+1 ; 1 , there exist 2 2 ( ; ), 2 1 ; t+1 , for all 2 2 and 2 (0; 1), such that if ( ; ) > t. We already know that if P A = P B = t and if ( ; ) 2 (0; 1), ( ; ) < t. On the other 2 (0; ) or (Pr(sH ) + Pr(sL ))2 Pr(sH )2 + Pr(sL )2 49 2 ( ; 1), ( ; ) < EU = pN tN ; if buys at online store, i i > EU = (1 p ) t ; if buys at o- ine store. : i i F F 8 See, among many others, Wilde and Schwartz (1979) and Stahl (1989) 9 We assume that it is impossible for the buyers to return the product once they purchase it. 10 That is, we consider that browsing the pictures on website was initially available to every potential buyer with no extra e¤orts, and that he can be perfectly informed of the match value of the product when inspecting the product face-to-face. 57 Figure 3.1: Timeline Timeline. The timing of the game is as follows. First, buyers form their prior beliefs about their match value of the product. Second, the monopolist seller determines marketplace(s) to sell her product: selling only at online, only at o- ine, or both at online and o- ine stores. Third, the seller quotes price(s) which immediately become observable to all potential buyers. Last, the buyers make a purchase decision knowing the price(s): where to go shopping and whether to buy or not. 3.3 Online or O- ine To begin with, we study the benchmark case where the product is available only at either online or o- ine stores. In either regime, the seller’ price choice is to decide whether to serve s only type-H buyers or both types of buyers. Only online store. The type-i buyer goes to the online store and buys if i If the seller charges L Whereas, if he charges pN tN < pN L pN tN 0. tN , then only type-H buyers will purchase. H tN , then both types of buyers will buy. The optimal price 58 and corresponding pro…ts are pN = H tN and N = H pN = L tN and N = 2( L tN ; tN ); if H > 2 L tN ; (Case 1) if H tN : (Case 2) 2 L It is noteworthy that, since buyers’valuation does not change when they visit the online store, the buyers who decide to shop at the online store always end up buying the product –otherwise they would not have come at the …rst place. Thus, the seller can simply charge the expected gross utility of the targeted type of buyers, i , but the compensation for the transportation cost tN , as is re‡ ected on the optimal prices above. Only o- ine store. Now, the type-i buyer goes to the o- ine for shopping if i (1 pF ) tF 0. Only type-H buyers will go shopping at the o- ine store if the price is 1 tF = L < pF 1 tF = H ; but both types of buyers will do so if pF 1 tF = L : Thus, the optimal price and the corresponding pro…ts are now pF = 1 tF = H and F = H pF = 1 tF = L and F = ( H + L )(1 tF ; if H > 2 =tF ; (Case 3) L tF = L ); if H 2 =t : (Case 4) L F Similarly to the previous cases, the optimal price chosen by the seller is essentially comprised of the buyer’ gross utility and the compensation for the transportation cost. When s the seller operates an o- ine store, however, the only ones receiving a good signal will decide to buy the product at last, but the others receiving a bad signal will walk away empty- 59 handed. Since the compensation for the transportation cost can only be granted via the discount of price, those who leave empty-handed have no way to receive the compensation. Considering this risk from a buyer’ perspective, the seller should compensate more than the s transportation cost itself in order to entice all the potential buyers to visit the o- ine store. Therefore, the optimal price would incorporate the gross utility of the actual buyers and the compensation for the transportation costs of all visitors including even "window shoppers." 3.4 Online and O- ine: Second-Degree Price Discrimination Let us consider the case that the seller runs both online and o- ine stores and quotes prices at each store. There are possibly two self-selective ways in terms of buyers’choices of stores.11 First, the self-selection may occur in the way that type-L buyers shop at the online store and type-H buyers at the o- ine store. However, this case never arises in the equilibrium. Proposition 8 There exists no pair of prices, (pN ; pF ), which induce type-L buyers to shop at online and type-H buyers at o- ine stores. Proof. Let us assume that there exists a pair of prices, (pN ; pF ), where type-L buyers shop _ _ at online and type-H buyers at o- ine stores. Then, it must be true for pN and pF that _ _ L pN _ tN L (1 pF ) _ tF H pN _ tN : and H (1 pF ) _ tF 11 We shall focus on the cases in which each type of buyers is to choose di¤erent store. Otherwise, the analysis becomes exactly the same as only online (or o- ine) store is available. 60 Note these conditions ensure that type-L prefers online to o- ine stores but type-H prefers o- ine to online stores. Combining two conditions, we get pN _ _ L pF tF tN pN _ _ H pF ; which is not feasible for L < H : The intuition to understand this result is quite simple. Type-H buyers are more optimistic about the match value of the product, and thus they are more likely to prefer shopping at the online store with no further information to shopping at the o- ine store than type-L buyers are. Hence, whenever type-L buyers …nd the price of online store a better deal than that of o- ine store, so do type-H buyers. The other possible scenario is that type-L buyers shop at the o- ine store but type-H buyers at the online store. The seller maximizes the pro…t function with the following four constraints. M ax (3.1) pF L + pN pF ;pN subject to [IRL ] L (1 [IRH ] H [ICL ] L (1 [ICH ] H pF ) pN pF ) pN tF tN tF tN () pF 0 () pN 0 L H (1 pN tF = L H tN () tF tN pN L pF tF () tF tN pN H pF tN pF ) 1 Recall that type-L buyers, who shop o- ine, purchase the good with probability L , i.e., only when they receive a good signal after examining the product at the store. On the other hand, type-H buyers make the purchase with probability 1, as long as they decide to visit the online store given the price. Thus, the pro…t function to maximize is pF L + pN . The …rst two individual-rationality [IR] constraints ensure that the buyers’ expected net 61 Figure 3.2: ICs and IRs utility should be non-negative. The other two constraints are the incentive-compatible [IC] constraints which require type-L buyers to choose the o- ine over the online store, and vice versa. As usual, [IRH ] is always satis…ed: [ICH ] and [IRL ] together imply pN then pN H H (1 tF = L ) + tF tN ; tN holds given that L < H . In addition, one can easily show that [ICL ] is not binding whenever both [ICH ] and [IRL ] hold with equality. Then the maximization problem reduces to maximize (3.1) subject to [IRL ] and [ICH ]. Solving it leads to the following proposition. Proposition 9 Suppose the seller operates both online and o- ine stores and employs e¤ective price discrimination. Then there exists an optimal pair of prices, pF = 1 b 62 tF = L and pN = H (1 b tF = L ) + (tF tN ); for which type-H buyers shop at the online and type-L buyers at the o- ine stores. The corresponding pro…ts are bN F = (tF tN ) + ( H + L )(1 tF = L ): The intuitive explanation on the optimal prices above is as follows. The type-H buyers will purchase at the o- ine store paying pF , whenever they …nd the product matching well b with their tastes. Since this probability is H for type-H buyers, the expected price they are paying at the o- ine store is H pF . Knowing this, the seller can entice type-H buyers b to choose the online store instead of o- ine store, if she sets the price of the online store as low as their expected price of the o- ine store. In addition, she can charge their savings in transportation costs, (tF tN ), on top of that. From the proposition above, we …nd some interesting observations on the optimal prices and welfare implications, which are summarized in the following two corollaries. Corollary 1 Suppose the seller operates both online and o- ine stores and employs e¤ective price discrimination. Then the optimal price o¤ered at the online store is lower than that at the o- ine store, i.e., pN < pF ; if (tF b b tN ) < (1 H )(1 tF = L ) Corollary 2 Suppose the seller operates both online and o- ine stores, and sets the pair of prices, pF = 1 b tF = L and pN = H (1 b tF = L ) + (tF tN ). The type-H buyers with low match value will su¤er from a severer negative ex post net surplus than the type-L buyers with low match value. Price discrimination through information provision shows several interesting di¤erences from that through quality or quantity. Note …rst that, without informational discrepancies between online and o- ine stores, the price at the online store is supposed to be higher than 63 at the o- ine store due to a lower transportation costs associated with the online store. When shopping at the online store, however, buyers have to take risks of buying an undesirable product with zero match value, and the risks are larger to type-L buyers than type-H buyers. Thus, the seller can screen out type-H buyers from type-L buyers by lowering the price at the online store where no further information on products is available. Such strategic e¤ects to separate out two types of buyers may push down the price at the online store so dramatically that type-H buyers may pay a lower price than type-L buyers after all.12;13 Second, type-H buyer may end up having a larger negative net surplus than type-L buyers, in an ex post sense. In a traditional second-degree price discrimination model, low valuation consumers have no net surplus, whereas high valuation consumers always enjoy a positive net surplus. This is because a seller must leave some rents to high valuation consumers in order to prevent them from buying the bundle designed for low valuation consumers. In our settings, however, type-H buyers will su¤er from a negative net surplus with probability (1 14 H ). Last but not the least, if there is a slight production cost, type-L buyer’ purchase s is socially optimal while type-H buyer’ purchase is suboptimal, which is contrary to the s common results in the literature on second-degree price discrimination. Since type-L buyers are perfectly informed of the characteristics of the product at the o- ine store prior to purchase decisions, only people who …nd a good match will buy the product. However, type-H buyers purchase the product without knowing how well it matches their preferences, their purchase may not be socially desirable with probability (1 H ) if a seller has to incur positive production costs.15 12 If there is no di¤erence in the transportation costs of online and o- ine stores, i.e., tF = tN , then only the strategic price discriminating e¤ects will prevail. Therefore, in this case, the price is always lower at online stores than at o- ine stores. 13 Brynjolfsson and Smith (2000) …nd that prices on the Internet are 9-16% lower than prices in conventional outlets. See also Carlton and Chevalier (2001), Goolsbee (2001), Brown and Goolsbee (2002), and Chevalier and Goolsbee (2003) for empirical studies on online prices. 14 Of course, type-H buyers should enjoy strictly positive ex ante net surplus on average. 15 More precisely, the social desirability would be determined by the relative size of production and transportation costs. Note that the statement is true given the assumption 64 3.5 Optimal Choice of Marketplace Comparing the optimal pro…ts in three cases – selling at the online, at the o- ine, or both stores, we obtain the following results. t Proposition 10 (i) If 1 L =N tF L 2 L H tN or 2 L tN < H 2 =t , then L F it is optimal for the seller to operate both online and o- ine stores and to employ price discrimination. (ii) If H > maxf2 L tN ; 2 =tF g, then it is optimal to operate only an L t tN ; 1 L =N g, then tF L it is optimal to operate only an online store and to sell to both types of buyers. online store and to sell only to type-H buyers. (iii) If H < minf2 L Proof. (i) If H 2 L now we should compare L tN 1 tF = L H . If 2 L (ii) If H > maxf2 L H > F < 2 L It can be easily shown that 2 =t . L F bN F when H N bN F when tN ; 2 =tF g, then maxf N ; N g = N and maxf F ; F g = F . L N , we compare 2 =t . L F (iii) If H N and bN F . tN < H , then maxf N ; N g = N . Similarly, we compare N and bN F , then we get N Since tN , then maxf N ; N g = N . Since F < N and F < bN F , N and bN F . It is straightforward to see bN F < N for tN , then maxf N ; N g = N . Since F < N and F < bN F , we t compare N and bN F , then we get bN F < N for H < 1 L =N . tF L When the di¤erence between H and L is su¢ ciently large (case ii), the seller is better o¤ by targeting only type-H buyers. Due to the advantage of transportation costs of online stores, he will want to operate only online stores and sell only to type-H buyers. On the other hand, when the di¤erence is small (case i and iii), the seller is drawn to sell to both types of buyer. Note that it becomes more di¢ cult (or costly) for the seller to separate out two types as H and L come closer, i.e., their incentives become similar. Therefore, the seller will be better o¤ running both stores and adopting price discrimination if the di¤erence tN tF . 65 is intermediate (case i), but he will rather opt to run only an online store if it is too small (case iii). 3.6 Extensions: Sophisticated Buyers and Milking Strategy Until now, we implicitly assumed that buyers are somewhat naive that they only can purchase at the store where they receive the signal. If buyers are sophisticated enough, however, the buyers might consider visiting one store to get information about the product and then purchasing from the other store (milking strategy). The buyers will be tempted to do so especially when they …nd out that the other store is o¤ering a lower price than the store from which they gather the product information. In order to prevent buyers’such milking strategies, the seller’ choice of prices must satisfy the following "arbitrage(milking)-proof" s conditions. [APL ] L (1 [APH ] H pF ) pN tF tN L (1 H (1 pN pN tN ) tN ) tF () pN tF () pN pF tN tF =(1 H) tN [APL ] implies that, for type-L buyers, buying at the o- ine store is better than switching to the online store after examining the product at the o- ine store. This constraint boils down to tN pF pN , (3.2) which is intuitively appealing because the type-L buyers would marginally compare (i) the extra transportation costs incurred from an additional visit to the online store with (ii) the bene…ts from paying a lower price at the online store.16 Similarly, [APH ] assures that type-H buyers do not visit the o- ine store for the purpose of collecting information before making a 16 If the price is higher at the online than the o- ine store, then it is senseless for buyers to switch to the online store after collecting the information from the o- ine store. 66 purchase at the online store. When solving the pro…t maximization problem, we can ignore the latter condition [APH ] since it is already implied by [ICH ] and [APL ] together. The intuition is simple. [APL ] tells us that even if the buyer arrives at the o- ine store and …nds the product perfectly matching his tastes, the price advantage of the online over the o- ine store is not prominent enough to provoke him an another trip to the online store. That is, buyers have no incentives to switch to the online store no matter what signal they receive from the o- ine store. Even for type-H buyer, the additional information from signals would not change his purchase decision on "where to buy," and therefore he is only concerned about "where to go shopping," online or o- ine stores, which is determined by [ICH ] constraint. Considering the two additional incentive compatibility constraints, [APL ] and [APH ], we obtain the following condition for which the optimal pair of prices derived in the previous section still can be an e¤ective price discrimination. Proposition 11 Suppose that the seller operates both online and o- ine stores. When consumers are sophisticated enough to use milking strategy, the optimal pair of prices, pF = b 1 tF = L and pN = H (1 b tF = L ) + (tF tN );are sustainable if tF L (1 H ) . L +(1 H ) Seemingly, the milking strategy can be prevented when tN is large –say, if tN is negligible, whenever type-L buyer receives a good signal from the o- ine store, he may want to switch to the online store to buy it despite the additional transportation costs tN . However, the "milking-proof" condition above is only a function of (tF ; L ; H ), but not tN .17 The reason is straightforward from the price o¤ered at online stores (bN ). When tN becomes smaller, p the seller can raise pN as much without increasing the deviation incentives of type-H buyers, b while keeping pF as it is. Therefore, the changes in tN do not a¤ect type-L buyer’ incentives b s for milking strategy as the both sides of (3.2) would equally change. By contrast, when tF gets smaller, the seller is tempted to raise the price for the o- ine store (bF ), but not so p 17 For this reason, we may freely normalize as t = 0 without a¤ecting any qualitative N results. 67 Figure 3.3: PD with Sophisticated Buyers much for the online store since the advantage in transportation costs of online over o- ine store, tF tN , becomes less. This, in turn, makes the online store more attractive for type-L buyers, i.e., the only right-hand side of (3.2) increases. As a result, type-L buyers are more likely to deviate from the o- ine store, when tF is smaller. Proposition 12 Suppose that the seller operates both online and o- ine stores, and that L (1 H ) , then L +(1 H ) t the optimal pair of prices which can separate types of buyers are pF = 1 F and pN = b b H 1+ L H tN +(tF tN ) , and the corresponding pro…ts are b tF tN . NF = 1 1 H H consumers are sophisticated enough to use milking strategy. If tF < 68 3.7 Concluding Remarks In this paper, we study a new type of second-degree price discrimination where a di¤erent price is o¤ered as a bundle with a di¤erent level of information about a product. When buyers are uncertain about their match value of the product, they may update their expected valuation in a Bayesian way after observing a signal coming from the product. The precision of the signal, or equivalently the amount of information buyers can collect from o- ine stores may be superior to that from online stores since the buyers can inspect and examine the product’ characteristics more thoroughly in the former than the latter. In this environment, s the seller may raise pro…ts by using price discrimination in the way that high valuation buyers choose to shop at the online stores but low valuation buyers at the o- ine stores. This may result in the distortion of welfare: the low valuation buyer’ purchase is socially s optimal, but the high valuation buyer’ is suboptimal, since the high valuation buyers make s a purchase decision without perfectly knowing their match value, i.e., their consumption may be excessive in retrospect. In turn, the high valuation buyers would su¤er from a severer negative ex post net surplus when they turn out to have a low match value than the low valuation buyers would. We further study buyers’ reaction against seller’ such price discrimination. If buyers s are sophisticated enough, they may "milk" the information from o- ine stores and make a purchase at online stores, when the online stores o¤er a better price deal. We …nd that, facing buyers’milking strategy, the seller’ price discrimination can be successful only when s the transportation costs for o- ine stores are su¢ ciently high. If the costs are low, the seller will have to lower the prices to e¤ectively separate out the buyers, which makes the price discrimination no longer pro…table at some point. There can be some ways for the seller to prevent buyers from using milking strategy. One immediate way is to manipulate o- ine transportation costs either by relocating o- ine stores to remote sites or by reducing the number of o- ine stores.18 An alternative way can be that 18 Forman et al. (2009) empirically examine the trade-o¤ between the bene…ts of online and o- ine shopping and show that o- ine transportation costs matter. 69 she di¤erentiates products sold in online and o- ine stores. If the seller provides a slightly di¤erent version in online from o- ine, the product information buyers gather from o- ine stores will not perfectly show the match value of the product being sold in online stores. This would give an interesting explanation for marketplace-speci…c product versioning by which some products are only made available for either online or o- ine purchase. 70 BIBLIOGRAPHY 71 Bibliography [1] Bar-Isaac, Heski, Caruana, Guillermo and Cunat, Vicente "Information Gathering and Marketing," Journal of Economics and Management Strategy, 2010, 19(2), pp. 375– 401 [2] Brown, Je¤rey and Goolsbee Austan "Does the Internet Make Markets More Competitive? Evidence from the Life Insurance Industry," Journal of Political Economy, 2002, 110(3), pp. 481-507 [3] Brynjolfsson, Erik and Smith, Michael D. "Frictionless Commerce? A Comparison of Internet and Conventional Retailers," Management Science, 2000, 46(4), pp. 563-585 [4] Carlton, Dennis W. and Chevalier, Judith A. "Free Riding and Sales Strategies for the Internet," Journal of Industrial Economics, 2001, 49(4), pp. 441-461 [5] Chevalier, Judith A. and Goolsbee Austan "Measuring Prices and Price Competition Online: Amazon.com and BarnesandNoble.com," Quantitative Marketing and Economics, 2003, 1, pp. 203-222 [6] Courty, Pascal and Li, Hao "Sequential Screening," Review of Economic Studies, 2000, 67, pp. 697-717 [7] Creane, Anthony "A Note on Welfare-Improving Ignorance About Quality,", Economic Theory, 2008, 34(3), pp. 585-590 [8] Forman, Chris, Ghose, Anindya and Goldfarb, Avi "Competition Between Local and Electronic Markets: How the Bene…t of Buying Online Depends on Where You Live," Management Science, 2009, 55(1), pp. 47-57 [9] Ghose, Anindya "Internet Exchanges for Used Goods: An Empirical Analysis of Trade Patterns and Adverse Selection," MIS Quarterly, 2009, 33(2), pp. 263-291 [10] Grubb, Michael D. "Selling to Overcon…dent Consumers," American Economic Review, 2010, 99(5), pp. 1770-1807 [11] Lewis, Tracy R. and Sappington, David E. "Supplying Information to Facilitate Price Discrimination," International Economic Review, May 1994, 35(2), pp. 309-327 [12] Maskin, Eric and Riley, John "Monopoly with Incomplete Information," RAND Journal of Economics, 1984, 15(2), pp. 171-196 72 [13] Miravete, Eugenio J. "Screening Consumers Through Alternative Pricing Mechanisms," Journal of Regulatory Economics, 1996, 9, pp. 111-132 [14] Mussa, M. and Rosen, S. "Monopoly and Product Quality," Journal of Economic Theory, 1978, 18, pp. 301-317 [15] Nocke, Volker, Peitz, Martin and Rosar, Frank "Advance-Purchase Discounts as a Price Discrimination Device," Journal of Economic Theory, 2011, 146, pp. 141-162 [16] Shapiro, Dmitry and Shi, Xianwen "Market Segmentation: the Role of Opaque Travel Agencies," Journal of Economics and Management Strategy, Winter 2008, 17(4), pp. 803837 [17] Stahl, Dale O. "Oligopolistic Pricing with Sequential Consumer Search," American Economic Review, 1989, 79(4), pp. 700-712 [18] Wilde, Louis L. and Schwartz, Alan "Equilibrium Comparison Shopping," Review of Economic Studies, 1979, 46(3), pp. 543-553 73