Multiproduct Pricing in Oligopoly

Transcription

1 Multiproduct Pricing in Oligopoly Sandro Shelegia Department of Economics and Business Universitat Pompeu Fabra November 29, JOB MARKET PAPER Abstract This paper proposes a two-good model of price competition where some consumers visit all the shops and others visit only one. We find that, in the Nash Equilibrium, the information frictions lead to price dispersion. When the goods are complements their prices will be negatively correlated so if one of them is priced high then the other one is at a discount. This finding is supported by the empirical observation that simultaneous price discounts of complements are infrequent. If the goods are substitutes then their prices will be uncorrelated, rationalizing the lack of evidence on their co-pricing. When selling complements, retailers earn higher profits than in the one-good model by discriminating consumers through taxing captives with high overall price tag and attracting shoppers by pricing one of the goods aggressively. This strategy does not yield additional profits for substitutes because captives will switch to buying only the aggressively-priced good. JEL classification: D83, L11, L13 Keywords: Multiproduct Competition, Substitutes, Complements I am grateful to Antonio Cabrales for careful supervision. I am thankful to Andreu Mas-Colell, Patrick Rey, Karl Schlag and Xavier Vives for helpful discussions, and to participants of the Microeconomics Workshops at Universitat Pompeu Fabra and the European Winter Meeting of the Econometric Society at the University of Cambridge for their comments. Any remaining errors are my own. sandro.shelegia@upf.edu; Website: 1

2 1 Introduction Traditionally, a vast majority of goods are delivered to final consumers through retail stores. While small retailers could carry hundreds of items, the largest ones like Fnac or Tesco offer tens of thousands of goods for sale. Managers of such stores have to make pricing decisions for innumerable products bearing in mind that the price of nearly any of them has an impact on the demand for its substitutes and complements. In spite of being so widespread, multiproduct price competition (and multiproduct price dispersion even more so) has not received considerable attention in theoretical economics. This may be due to the fact that in frictionless world of perfect competition pricing in multiproduct setting is reduced to the pricing of each good separately. However, many frictions are well documented and are known to affect pricing. One of the most notable of them, incomplete information on prices, has not been studied rigorously in multiproduct models where its impact remains largely unknown. It has been shown by Varian (1980) and many others that, in the one-good setting, information frictions lead to price randomization in equilibrium so it seems natural that they will play an important role in the co-pricing of complements and substitutes. 1 In this regard, it is interesting that empirical marketing literature has documented negative relationshietween the prices of complementary goods within shops which is hard to account for with deterministic pricing subject to idiosyncratic demand or supply shocks. Surprisingly, no clear pattern has been identified in the co-pricing of substitutes. This paper aims to bridge the gaetween theory and evidence by rationalizing the empirical findings and providing insights on multiproduct pricing that can be used in future empirical research. We model price competition between retailers that all sell two interrelated but homogeneous products. We consider the full range of interdependency between the two goods from perfect complementarity to independent valuations to perfect substitutability. The information frictions are introduced by assuming two types of consumers - captives, who visit only one retailer and shoppers who visit two and buy each good at the lowest price. We show that, in the static Nash Equilibrium, retailers use mixed strategies for both prices leading to price dispersion in the market. The dichotomy between complements and substitutes is particularly noteworthy: if the goods are complements their prices within every shop will be negatively correlated, while if they are substitutes (independent valuations are included here), the two prices will be uncorrelated. The reason for the negative correlation is that if a price of one of the two complements 1 For one-good models with information frictions see Diamond (1971), Salond Stiglitz (1977) and Stahl, II (1989) among others. 2

3 is at its highest level then this good is never sold to shoppers because some other retailer is surely setting a lower price. If so, the retailer will want to increase this price further until even captive consumers are about to stouying the good. While doing so, the retailer has the incentive to combine the highest price for the first good with a relatively low price for the other to be able to increase the former as much as possible and yet sell both goods. Such reasoning leads to negative relationshietween prices of complements. The complementarity also has an interesting implication for the profitability of retailers. It turns out that when the two goods are complements, the multiproduct offering allows retailers to jointly discriminate between captives and shoppers. In the one-good version of this model (Burdett and Judd (1983)) retailers are setting only one price so when lowering it to attract shoppers they also lose the guaranteed profits they could earn from captives. In contast, if the retailers are selling two complements, they can keep the sum of the two prices constant at the joint reservation price of the two goods (ensuring that the profits earned from captives are maximized) and lower one of the prices, engaging aggressively in price competition for shoppers. This practice yields higher profits than selling one composite good because it allows selling one of the complements to shoppers even if the sum of the two prices is kept at the monopoly level. The rational for no correlation between the prices of substitutes is that unless the substitutability is near perfect, in which case retailers abandon the good with lower profit margin, retailers still want to sell both. In order to do so, they have to keeoth prices low because the two goods, as substitutes, compete with each other. When prices are capped in this way even the highest price ever charged for one of the substitutes can be combined with any price for the other one and, nevertheless, lead to selling both goods. As a result, there is no need to take into account the price of one of the substitutes when pricing the other. The lack of correlation between prices of substitutes does not imply that their prices are chosen as if retailers were selling only one of them. Quite to the contrary, equilibrium price ranges and distributions are determined by the characteristics of the entire product assortment, it is just that in equilibrium there is no correlation between the prices chosen from these distributions. We also demonstrate that the kind of joint discrimination that was possible with complements is not feasible if the two goods are substitutes. Substitutes compete with each other so retailer cannot keep the sum of their prices high and decrease one of them to attract shoppers because by doing so she induces captives to buy only the cheaper good. Not only are retailers incapable of earning additional profits through discriminating between the two groups of consumers, but they also earn lower profits than in the one-good model. When the consumers have to buy a composite good retailers 3

4 do not have to take into account the competition between the two substitutes within their store and are able to earn higher profits. This phenomenon is unrelated to the nature of competition and is also applicable to a monopolist. Related Literature Theoretical literature on multiproduct price competition is scarce. The inherent difficulty lies in the ability of consumers to mix and match goods offered by different retailers leading to a complicated pattern of competition. Nevertheless, several authors have addressed the issue. Armstrong and Vickers (2001) show that multiproduct competition can be modeled in terms of utility offers made by retailers. The authors provide a method for solving multiproduct models when assortments and prices differ across horizontally differentiated retailers while consumers have to buy everything at one shop. The aim of our study is distinct: we analyze pricing strategies of multiproduct retailers when some consumers are able to buy products at different shops while all the retailers offer the same assortment of homogenous goods. This practice appears to be widespread (e.g. savvy shoppers who buy a TV set at one shond a DVD player at another when both shops carry the same products) and is not addressed in Armstrong and Vickers (2001). Notably, when we allow shoppers to combine their purchases from different shops in equilibrium they frequently do so. There are several models of multiproduct price competition in differentiated goods where, as in our model, consumers can engage in mixing purchases between shops. 2 Lal and Matutes (1989) have studied price competition in a Hotelling duopoly selling two independent goods. They find that, in equilibrium, retailers charge different but Contrary to our model, in Lal and Matutes (1989) goods are differentiated, consumers are heterogeneous in transportation costs and willingness to pay. 3 deterministic prices and may even capture the entire consumer surplus of the rich consumers. Meanwhile, we assume that the two goods are homogeneous across the retailers and all the consumers have identical preferences which allows us to concentrate on the interplay of demand dependency between goods and information asymmetry among consumers. Even thought in Lal and Matutes (1989) the goods are assumed to be independent, the authors introduce complementarity between the goods by assuming that it is cheaper to buy both at one shop. Unlike our model, the complementarity is not carried along for 2 For an interesting model of multiproduct competition with perishable goods see Hosken and Reiffen (2007). 3 Consumer heterogeneity in preferences is a standard assumption in the multiproduct literature (see Whinston (1990), Chen (1997), Choi and Stefanadis (2001), Hosken and Reiffen (2007) and Denicolò (2000) among others). 4

5 mixed purchases. In our model both complementarity and substitutability are present within and as well as across shops and play crucial role for the equilibrium strategies. Due to this we are able to characterize the relationshietween prices within each shop and generate empirical predictions. Even though there are numerous empirical studies on one-good price dispersion (see Baye et al. (2006) for a survey of empirical literature on the subject), recently Hosken and Reiffen (2004) have argued that the multiproduct approach to modeling retail pricing is key to understanding important aspects of retail behavior. There is a consensus in marketing literature that if one of the complementary goods is on sale the other ones are unlikely to be at a discount as well. 4 Even though this negative relationshietween prices of complements had not been documented by the literature directly, it was so well accepted that some researchers investigated what happens if, unorthodoxly, prices of complements are reduced simultaneously. 5 The theoretical justification for reducing the price of only one complement, until now, has been grounded in the monopoly paradigm: if a store lowers the price for one of the goods the profit-maximizing price for the other rises as the demand for it increases. As a result, having both goods on sale cannot be optimal. It is not obvious why, in the first place, a monopolist would lower one of the prices from its optimal monopoly level. One would imagine that, in equilibrium, all observed prices should give equal expected profits so, unless one explicitly models multiproduct competition, the relationshietween prices of complements is ambiguous. Relatively recently, after scanned data from supermarkets became available, some authors started studying choices of individual consumers in response to price discounts rather then focusing on the overall store sales, as was common previously (Mulhern and Leone (1991)). Van den Poel et al. (2004) have analyzed consumer decisions based on their baskets of purchases in a large do-it-yourself retailer and found that simultaneous large discounts on two complementary products occur rarely. This is precisely what our model predicts, that is when one of the complements is at a discount the other one is priced high. To the best of our knowledge there are no clear empirical predictions for co-pricing of substitutes in a competitive environment. To a certain degree our model rationalizes this absence by predicting no correlation between the prices of substitutes. Moreover, we provide testable predictions that can be used in future empirical studies. This paper is organized as follows: in Section 2 we specify the model and analyze optimal behavior of the consumers and a hypothetical monopolist, in Section 3 we solve 4 See for example Mulhern and Leone (1991) and Mulhern and Padgett (1995). 5 Van den Poel et al. (2004) explore the possibility of simultaneous discounts on complements. 5

6 the oligopoly model for all the cases, in Section 4 we provide discussion of some extensions of the main model and in Section 5 we conclude. 2 Model 2.1 Assumptions Consider a market with two retailers selling two homogeneous goods labeled a and b. The marginal cost of production of good i (where i = a, b) is denoted by c i, is assumed to be constant, independent of the production level of the other good and equal among the retailers. The cost of producing one unit of each good is equal to c a + c b and will be denoted by c ab. 6 There are many consumers who have identical tastes and their mass per retailer is normalized to one. The consumers demand exactly one unit of each good and the triplet (v a, v b, v ab ) describes their reservation prices for one unit of good a, one unit of good b and a bundle composed of one unit of each good. We will assume that consumers get utility of zero if they do not consume anything and can freely dispose the goods they own so v ab max(v a, v b ). The triplet of unit valuations describes the entire possible set of demand interrelations between the two goods. If v a = v b = 0 then good a and good b are perfect complements, if v a + v b = v ab they have independent valuations, and if v ab = v a = v b then they are perfect substitutes. We allow the two goods to be asymmetric and without loss of generality we assume that: Assumption 1. v b c b v a c a. If the inequality holds strictly then we refer to good b as the good with higher margin or the more profitable good. While identical in tastes, consumers differ in their shopping behavior and come in two types: proportion θ of the consumers visits only one retailer at random (we refer to these consumers as captives) while the rest of the consumers (proportion 1 θ) visit both retailers (shoppers). 7 Shoppers can buy each good at the lowest price they observe without paying any extra transportation cost if they choose to buy the goods at different shops. 8 If two retailers charge the same price for a good shoppers are equally likely to purchase from either of them. 6 That is, we assume there are no economies of scope in production. 7 θ is given exogenously but it can easily be made endogenous as in Burdett and Judd (1983). 8 This assumption amounts to requiring free recall by shoppers, something widely used in the consumer search literature. For a more detailed discussion see Section

7 Finally, in order to simplify the treatment of border cases we assume that in the event of indifference the consumers respect the following order: buy both goods, buy only good b, buy only good a and do not buy anything. Firms compete by setting prices for the two goods simultaneously and we use static Nash Equilibrium as the solution method. We assume that the retailers will not, in addition, set a separate price for a bundle of the two goods. The implications for the model when the retailers are allowed to bundle the two goods are discussed in Section Consumer Behavior and Monopolist Since each retailer has monopoly power through captive consumers, the strategies employed in the oligopolistic equilibrium depend on the pricing behavior of a hyphotetical monopolist facing captive consumers. Before proceeding to solving the oligopoly model we will illustrate the optimal behavior of consumers facing any price pair and, subsequently, the profit-maximizing strategy of the monopolist. This section will demonstrate that the pricing by the monopolist is fundamentally different depending on whether the two goods are substitutes or complements and so we shall solve the oligopoly model for these two cases separately. Assume a consumer can buy the goods at a price pair (, ). For a captive consumer this pair is the one charged by the only retailer she visits while for a shopper each price is the minimum between the prices of each good from the two retailers. The consumer has a choice of buying both goods, only good a, only good b and none at all, and gets a surplus of v ab, v a, v b and 0, respectively. The consumer will buy both goods if and only if: v ab + (1) v ab + v b (2) v ab + v a. (3) She will buy only good i if v i p i, v i v j p i p j and v ab v i p j hold at the same time (when used along i, subscript j always denotes the other good). Figure 1 illustrates the consumer choices depending on the prices and the relation between v ab and v a + v b. If the two goods are complements the most the monopolist can earn when selling both good a and good b is v ab c ab. In this case Inequality 1 is binding so she can charge any point on the line connecting x 2 and x 3 in Figure 1 b). It is easy to see 7

8 v b + = v ab a x 2 v ab v a + = v ab a x 3 x 1 v ab v a a + b x 1 b x 3 v b a + b x 2 b 0 v ab v b v a 0 v a v ab v b a) b) Figure 1: Consumer choice when the goods are a) substitutes and b) complements. Labels a, b and a + b indicate the price pairs such that consumers buy only good a, only good b and both goods, respectively. These areas are delimited with solid lines. that this pricing is only feasible when the goods are complements. Figure 1 a) shows that if v ab = + then consumers will not buy the two substitutes together, hence the monopolist is unable to earn v ab c ab. When the monopolist aims to sell both substitutes Inequalities 2 and 3 bind and she earns (2v ab v a v b ) c ab < v ab c ab by charging the price pair x 1. The inability to earn v ab c ab is the result of substitutability between the goods. When good a and good b are substitutes they effectively compete with each other not allowing the monopolist to extract their joint value from the consumer. 9 If the monopolist sells only one of the substitutes then she should sell good b (recall Assumption 1). She will charge = v b and any v a to do so and will earn v b c b. Hence, if (2v ab v a v b ) c ab < v b c b v ab < v b (v a + c a ) the monopolist will choose to sell only B, otherwise she will sell both goods. Having verified the pricing by the monopolist we turn to our oligopolistic model. We will consider complements and substitutes separately as suggested by the analysis of this section. 9 In Section 4.2 we discuss the implications for the behavior of the monopolist if she can bundle substitutes. It is shown that the monopolist can sell both goods and still earn v ab c ab if she refuses to sell the goods separately. 8

9 3 Equilibrium 3.1 Complements In this section we assume that v ab > v a + v b. Previously, we have demonstrated that in this case the monopolist will charge a pair of prices such that their sum is equal to v ab (e.g. = v a and = v ab v a ). The competition will force the retailers to undercut each other from the monopoly prices. This pressure on prices is downwards so the retailers will nevertheless sell both goods to captives in equilibrium. Lemma 1. When the goods are complements, in equilibrium retailers set such prices that captives always buy both goods, that is, Inequalities 1-3 hold. Proof. Assume the opposite. For simplicity assume that good a is the one that captives do not buy. It should be clear that shoppers will not buy any good that captives do not buy so the retailer will not sell good a at all. If that is so, the retailer can lower the price of good a to a level such that it is still above c a and captives buy both goods, a strategy that increases profits. In terms of Figure 1, good b is the only good sold if the price vector is in region good b. For any point in this region the retailer can fix the price of good b and lower the price of good a before the price pair is in the region a + b. By doing so the retailer will increase her profit because she will be selling good a at a price v ab v b > c a. More formally good b is the only good sold to captives iff all of the following are true: v ab v b < v b v a + v b. Let the retailer, instead of, charge ˆ = v ab v b. One can show that at (ˆ, ) captive consumers will buy both goods and ˆ + c ab > c b so the profit earned from captives will increase. Shoppers were not buying good a before and by lowering its price the profit earned from them cannot decrease. Once we have verified that firms never charge prices that do not attract consumers we can move to characterizing the equilibrium pricing strategies. First we demonstrate that, in analogy to the one-good models of price dispersion, equilibrium distributions of both prices will be atomless and gapless, defined over a closed and connected support. 9

10 Lemma 2. In equilibrium, p i (i = a, b) will be randomized according to the continuous distribution function F i (p i ) defined over the interval [p i, p i ]. The reason why there are no atoms in the equilibrium distributions is clear: if some price was charged with a strictly positive probability, there would be a positive probability of a tie at that price and all retailers would have incentive to charge a slightly lower price with the same probability as the old one and serve all shoppers in the case of a tie by others. Moreover, there will be no gaps in the equilibrium price distributions. This is because charging the price at the lower edge of the gattracts shoppers with the same probability as charging the price at the upper edge of the gap does but the latter price yields higher profits. Proof. See proofs of Lemmas 3 and 8 from Varian (1980). Next we will argue that charging any p i along with p j such that both goods are sold to captive will give the same expected profits earned from selling i. Lemma 3. Expected profits earned from selling good i when charging any p i [p i, p i ] are constant for all such p i and are independent from the price charged for good j. Proof. See Appendix. Now we proceed to identifying F i in the equilibrium. We know that due to the fact that F i is atomless, if any retailer is charging p i for good i then she will not sell i to shoppers because other retailers will charge a lower price for it with probability one. Hence, she should increase p i until captive consumers are indifferent between buying the two goods and either buying only good j or not buying anything at all. Formally, p i = max{p i v ab v j p i and v ab p j p i }. (4) Using the last expression and Lemma 3 we conclude that the highest price for i will be charged along with the lowest price for good j when p j v j. If p j < v j then p i will be charged along with some p j v j and will be equal to v ab v j. Lemma 4. If p j v j then p i will be charged along with p j and their sum will be equal to v ab. Proof. Assume the opposite so that a pair (p i, ˆp j ) is charged such that ˆp j > p j when p j v j. The last two inequalities combined imply ˆp j > v j. In the maximization problem in Equation 4 the second restriction will bind so p i = v ab ˆp j. But then, the retailer 10

11 can charge the pair (v ab p j, p j ) and earn higher profits on i without changing the profits earned on j (Lemma 3). From the last argument it follows that if p j v j then p i = v ab p j. If the highest price of good i is restricted by the price of good j (in the sense of Equation 4) a retailer should always choose the lowest price for j in order to increase the highest price for i as much as possible. Now consider the case when p j v j. It should be clear that because the retailer wants to increase p i as much as possible she should always charge p i with some p j v j so because of Equation 4 we have p i = v ab v j. Lemma 5. If p j v j then p i is always charged along with some p j v j and p i = v ab v j. Proof. Assume the opposite so that a retailer charges p i with some ˆp j > v j in equilibrium. We know that p i + ˆp j v ab so v ab v j > p i. In this case the retailer can increase her profits by charging a price pair (v ab v j, v j ) instead. By doing so she will earn the same profits from selling good j (Lemma 3) but will earn strictly higher profits from selling good i, a contradiction. Lemmas 4 and 5 imply that there are four possible cases when good a and good b are complements: 1. v a and v b = = v ab and = v ab. In this case we refer to good a and good b as Strong Complements. 2. v a and v b = = v ab v b and = v ab v a. Weak Complements. 3. > v a and < v b = = v ab v b and = v ab. Intermediate Complements. 4. < v a and > v b = = v ab and = v ab v a. Intermediate Complements II, is impossible due to Assumption 1. Next we will consider each case separately Strong Complements In this case v b and v a. We will demonstrate that these two inequalities hold only if the complementarity is strong enough (i.e. v ab is large enough with respect to v a + v b ), hence the name for the case. When v ab is large, retailers increase p i up to the point where consumers are indifferent between buying both products or not buying anything at all (v ab = 0). Retailers never have to be concerned that by increasing 11

12 p i they may induce consumers to switch to buying only j because prices of both goods are above the individual valuations for the goods. Using p i v i (i = a, b) along with Lemma 4 gives: p i = v p j. (5) Since p i never attracts shoppers and p i attracts them with probability one, the expected profits from charging either of these two have to be equal so: 2(1 θ)(p i c i ) = θ(p i c i ). (6) Using the last equation along with Equation 5 we get: = 1 2 [θc a + (2 θ) (v ab c b )] (7) = 1 2 [(2 θ)c a + θ (v ab c b )] (8) = 1 2 [θc b + (2 θ) (v ab c a )] (9) = 1 2 [(2 θ)c b + θ (v ab c a )]. (10) Recall that v a and v b should hold in this case. After some algebra we are left with: v ab 2 (v a c a ) + c ab θ v ab 2 (v b c b ) + c ab. θ Assumption 1 implies that 2(va ca) θ + c ab 2(v b c b ) θ + c ab so if good a and good b are strong Substitutes it should be the case that: 2(v b c b ) θ + c ab v ab. (11) At this point we need to verify that captive consumers buy both goods at all the price pairs charged in equilibrium or: = 1 2 [θc a + (2 θ) (v ab c b )] v ab v b (12) = 1 2 [θc b + (2 θ) (v ab c a )] v ab v a. (13) 12

13 These reduce to: v ab 2 (v a c a ) + c ab θ (14) v ab 2 (v b c b ) + c ab, θ (15) precisely the two conditions we have obtained for Strong Complements. As mentioned above, we refer to this case as Strong Complements because if v ab is large enough the prices for both goods will always exceed their individual reservation values (p i v i ) and the price ranges and equilibrium strategies are independent of v a and v b. The expected profits for a retailer charging a price pair (, ) such that + v ab are given by π ab = π a + π b where π i (i = a, b) is: π i = [θ + 2(1 θ)(1 F i (p i ))] (p i c i ). (16) Given that + v ab, the expected profits from selling i is constant for all p i [p i, p i ] and is equal to π i = θ(p i c i ). As a result, the unique cumulative marginal distribution functions for prices of good a and good b in the equilibrium will be: respectively. F a ( ) = (2 θ) (2 v ab θ (2 θ)c a + θc b ) 4(1 θ) ( c a ) (17) F b ( ) = (2 θ) (2 v ab θ (2 θ)c b + θc a ), (18) 4(1 θ) ( c b ) What remains to be shown is that there exists a joint distribution function F (, ) such that + v ab for all price pairs and the derived marginal distributions are F a ( ) and F b ( ). There is no such range of and where the two prices can be randomized independently because for any p i [p i, p i ] the restriction p i + p j v ab is binding for some p j. This in particular implies that F (, ) = F a ( ) F b ( ) cannot be the equilibrium joint distribution. Next we will solve for a simple randomization rule: randomize the price of good a according to the marginal distribution function in Equation 17 and set according to some monotonically decreasing function b( ) such that the resulting marginal distribution of the price of good b is exactly as in Equation 18. Such function exists and 13

14 is unique, defined by the equation F a ( ) = 1 F b (b( )). This is because if is a decreasing function of then the probability a price of good a is below should be equal to the probability that a price of good b is above b( ). After some algebra one can check that the function + b( ) (the sum of prices) is decreasing at, increasing at (at both points it is equal to v ab ) and the derivative ( + b( ))/ changes its sign only once on the interval [, ] so + b( ) v ab for all [, ]. It is possible introduce noise to the function b( ) and obtain some other joint distribution function which has the necessary marginals so this could lead to a multiplicity of such functions. Notably, the joint distribution we have derived is unique in the class of monotone single-valued functions (that is when for every there is a unique ): Proposition 1. When v ab 2(v b c b ) θ + c ab the equilibrium marginal distributions given in Equations 17 and 18 are unique. Moreover, the joint distribution function defined by b( ) along with F a ( ) is unique in the class of single-valued monotone joint distribution functions. Proof. Lemmas 1 to 5 suffice to show that the equilibrium distribution functions given in Equations 17 and 18 are unique. We have already argued that there exists strictly decreasing function = b( ) such that if is randomized according to the marginal distribution function given in Equation 17 then the resulting marginal distribution for is exactly that in Equation 18 while + v ab holds for all [, ]. This is sufficient to prove that the joint distribution function described above satisfies all the conditions for the equilibrium. It is also unique in the class of monotone single-valued function becuase b( ) cannot be strictly increasing given that + > v ab. Equilibrium marginal densities and the function b( ) are illustrated in Figure 2. In the equilibrium, only the prices along the curve b( ) will be charged, starting from the price pair (, ) and ending with the pair (, ). It seems natural that any retailer, if possible, would want to have no coordination between the two pricing strategies. If that is not feasible then, as in Strong Complements, the retailer would find it easiest to draw one of the prices, say, from the equilibrium price distribution and set according to some monotonic function such that the resulting distribution of is the equilibrium one. In order to pick from potentially multiple equilibria we will introduce intuitive criterion that will be used throughout the paper: Assumption 2. When indifferent the retailers use the following ordering of the pricing strategies: they charge uncorrelated prices. 14 If the the latter is not possible then they

15 + = v ab fb(pb) v b b( ) f a ( ) 0 v a Figure 2: Strong Complements. The shaded area indicates price pairs that can be charged in equilibrium ( + v ab ). The blue axis are the marginal densities for the price of each good. randomize one of the prices and set the other one as a monotonic function of the first, and finally they jointly randomize both prices. Given Assumption 2 the joint distribution function defined by F a and b( ) will be unique. When the goods are Strong Complements retailers earn expected profits of π ab = θ(2 θ) (v ab c ab ) which are larger than the profits they would have obtained if they sold one good with a valuation v ab at a cost c ab (π = θ (v ab c ab )). Intuition for this profit bump is the following: when setting the sum of prices equal to v ab the retailers can surely sell one of the goods to shoppers by setting its price low enough. By doing so, they will earn θ (v ab c ab ) from captives and θ(1 θ) (v ab c ab ) from shoppers. If instead, they sold only the composite good, v ab would be the highest price ever charged and at that price only captives would buy the good giving the retailer excepted profits of only θ (v ab c ab ). As this case demonstrates, when the two goods are complements retailers can jointly discriminate between captives and shoppers and earn higher profits than in the one-good model Weak Complements Here we assume that p i v i for i = a, b. The goods in this section are called Weak Complements because their individual valuations are large enough (relative to v ab ) to 15

16 be higher than at least some prices charged for them. Unlike the case of Strong Complements, here the process of increasing p i stops when consumers are ready to switch to buying only good j and this is a concern because p j v j. We will show that here v ab has to be close enough to v a + v b so the case of independent valuations (v ab = v a + v b ) will be approached in the limit. As in the previous section it should be the case that expected profits at p i and p i are equal so (2 θ)(p i c i ) = θ(p i c i ). Given this and Lemma 5 the boundaries for price distributions will be given by: We impose v b and v b to get: = v v a (19) = v v b (20) = 2(1 θ)c a + θ (v ab v b ) 2 θ (21) = 2(1 θ)c b + θ (v ab v a ). (22) 2 θ v a 2(1 θ)c a + θ (v ab v b ) 2 θ v b 2(1 θ)c b + θ (v ab v a ). 2 θ Rewriting in terms of v ab and remembering Assumption 1 reduces the last two restrictions to: v ab (2 θ) (v b c b ) θ + v a + c b. (23) The marginal price distributions for good a and good b in the equilibrium can be derived as in Section using Equation 16 and we get: F a ( ) = v ab θ c a + θc a + θv b (1 θ) ( c a ) (24) F b ( ) = v ab θ c b + θc b + θv a. (25) (1 θ) ( c b ) As in the previous case, the equilibrium joint distribution function should satisfy the following conditions: the derived marginal distributions should coincide with the two we have obtained and for all equilibrium price pairs (, ) the sum of prices should be no larger than v ab ( + v ab ). Note that when v ab = v a + v b, that is when the goods are independent, = v a and 16

17 = v b so + v ab for all pairs. In this case there will be no restriction linking the marginal pricing strategies for the two goods so in the equilibrium the strategies can be independent and the joint distribution function can be written as a product of the marginal distributions: F (, ) = F a ( )F b ( ). 10 Prices of weak complements can be randomized independently, albeit for a subset of the equilibrium range, even when v ab > v a + v b. To see this note that for any p i v i the restriction p i + p j v ab is not binding for any p j [p j, p j ]. The last fact implies that if a price of one of the goods is set below its reservation price then the price of the other good can be randomized independently of the first price. When v ab approaches v a + v b the probability of p i v i approaches one allowing the randomization of the two prices independently as noted above. Figure 4 c) illustrates the set of possible price pairs in equilibrium for the case of Weak Complements. We will use Assumption 2 to identify single joint distribution function that supports the equilibrium. It is easy to see that both firms will randomize price of good a according to F a ( ) and will set price of good b from a monotonically decreasing function b( ) which is implicitly define by F a ( ) = 1 F b (b( )). Proposition 2. When v ab (2 θ)(v b c b ) θ + v a + c b the equilibrium marginal distributions given in Equations 24 and 25 are unique. Moreover, the joint distribution function defined by b( ) along with F a ( ) is unique in the class of single-valued monotone joint distribution functions. Proof. See proof of Proposition 1. For any retailer the expected profits in equilibrium will be equal to π ab = θ(2v ab v a v b c ab ) θ(v ab c ab ) so they are at least as large as the profits obtained in a one-good model. As expected, the profits converge to the sum of individual goods equilibrium profits when v ab approaches v a + v b. That is, the additional gain from discriminating between the two groups of consumers disappear as goods become independently valued Intermediate Complements Here we assume that < v b and > v a. The two goods are evidently asymmetric here and we will prove that the case of Intermediate Complements exists if and only if Assumption 1 holds with the strict inequality. The latter implies that selling good b yields more profits than selling good a holding the surplus obtained by consumers fixed. 10 When v ab = v a + v b the marginal distribution we derive are identical to those derived by Burdett and Judd (1983). 17

18 This case is a mixture of the previous two in the sense that is constrained by v ab v b while is constrained by. For this range of v ab the price charged for good a will always exceed its individual reservation price. One could think of a laptond a laptop bag as an illustrative example. In equilibrium the prices of laptoags will be always higher than their individual value, pricing that can be frequently observed. Lemmas 4 and 5 imply that = v ab and = v ab v b. Remembering that attracts only captives and attracts shoppers with probability one we write: From the previous equation we get = θ(v ab v b ) + 2(1 θ)c a. (26) 2 θ = v ab = 2(1 θ) (v ab c a ) + θv b 2 θ (27) = 2(1 θ) (v abθ + 2c b θ (c a + c b )) + θ 2 v b (2 θ) 2. (28) We have to impose the restrictions for Intermediate Complements to get: which simplify to v a < θ(v ab v b ) + 2(1 θ)c a 2 θ v b > 2(1 θ) (θv ab + 2c b θ (c a + c b )) + θ 2 v b (2 θ) 2 2 θ (v (2 θ) b c b ) + c a + c b > v ab > (v b c b ) + v a + c b. (29) θ Note that the goods are Intermediate Complements if and only if v b c b > v a c a. If the two goods are equally profitable then it is impossible that only one of the goods is always sold at a price above its individual reservation price. Possible price pairs for Intermediate Complements are illustrated in Figure 4 b). The marginal distribution functions for the price of good a and good b in the equilibrium will be: F a ( ) = v ab θ c a + θc a + θv b (1 θ) ( c a ) (30) F b ( ) = v ab (1 θ)θ c b + θ ((1 θ)c a + c b θv b ). (31) (1 θ) ( c b ) 18

19 The joint distribution function should have derived marginal distributions as in the previous two equations and for all pairs (, ) such that + v ab. Invoking Assumption 2 we conclude that the only equilibrium in this case is when both firms randomize price of good a according to F a ( ) and set price of good b from a monotonically decreasing function b( ) which is implicitly define by F a ( ) = 1 F b (b( )). Proposition 3. When (2 θ) θ (v b c b )+v a +c b < v ab < 2 θ (v b c b )+c a +c b the equilibrium marginal distributions given in Equations 30 and 31 are unique. Moreover, the joint distribution function defined by b( ) along with F a ( ) is unique in the class of singlevalued monotone joint distribution functions. Proof. See proof of Proposition 1. The equilibrium expected profits are equal to: π = θ 2 θ [(4 3θ)v ab (1 θ)(v b + c a )] (32) which are strictly larger than θ(v ab c ab ). In this case, as well as in the previous two, the retailers are able to increase their profits through joint discrimination of captive consumers. We have already exhausted all the cases of v ab > v a +v b. It is trivial to show that the case of Intermediate Complements II cannot occur. To see this one can derive similar conditions for this case and verify that they cannot be satisfied given Assumption Substitutes In this section we assume that v ab v a + v b. In Section 2.2 we have demonstrated that the monopolist compares 2v ab v a v b c ab and v b c b and prices accordingly. If v ab 1 2 (v a c a ) + v b the monopolist will charge = v ab v b and = v ab v a, a price pair at which captive consumers buy both goods. Instead, if v ab < 1 2 (v a c a ) + v b the prices charged will be = v b and v a and captive consumers buy only good b. It turns out that these two ranges for v ab are important even when the competition is present. We will call the two goods Weak Substitutes when all the retailers sell both goods which is the case when v ab 1 2 (v a c a ) + v b. When the goods are close enough to being independently valued, all the retailers still choose to sell them both. As the goods become better substitutes the retailers will find it less and less profitable to sell both as this requires lowering both prices and at some point they switch to selling only good b. When v ab < 1 2 (v a c a ) + v b the monopolist would sell only good b. We will show that in our model for only some part of this parameter range good b is the only good 19

20 sold while for the rest of the range retailers attach a positive probability to selling both goods. The former case will be referred to as Strong Substitutes while the latter as Intermediate Substitutes Weak Substitutes Here we assume that in the equilibrium retailers sell both goods to captives with probability one. We will prove that this is the case if and only if v ab 1 2 (v a + c a ) + v b, that is the two goods are relatively close to being independently valued. One could think of photo and video cameras as an example of weak substitutes. Both devices can perform overlapping tasks but their functions are distinct enough to induce consumers to buy both. Lemma 6. Both goods are sold to captives with the probability one iff v ab 1 2 (v a + c a )+ v b. Proof. Assume retailers sell both goods. If this is true then we can use Lemma 2 to argue that the distribution function F i (p i ) will be atomless and defined over a closed and connected support so the price of i will be randomized over an interval [p i, p i ]. In order for the retailers to sell both goods it has to be true that v ab v b and v ab v a. Note that for each price the condition of selling both goods depends only on the price of that good so provided these are true, the expected profits earned on each good will be independent of the price of the other. Since the distribution functions are atomless shoppers will not buy the good priced at p i. So any retailer will increase this price up to the maximum possible provided that both goods are sold, that is: = v ab v b (33) = v ab v a. (34) The expected profits earned in equilibrium will be π = θ(2v ab v a v b c ab ). We have to make sure no retailer wants to deviate and sell only one of the goods. It is obvious that if i is the only good sold then p i > p i, otherwise the retailer can decrease the price of the other good and sell both which leads to higher profit. But if only i is sold to captives when p i > p i it will never be sold to shoppers so p i = v i. If this is true the retailer will earn θ(v b c b ). Because we want both goods to be sold in equilibrium it has to be the case that θ(2v ab v a v b c ab ) θ(v b c b ) v ab 1 2 (v a + c a ) + v b. Now assume that v ab 1 2 (v a+c a )+v b. We will argue that in this case all the retailers will choose to sell both goods. Assume the opposite so retailers in the equilibrium sell 20

21 only good b. As before, we can argue that the highest price of good b is equal to v b and the profit earned will be θ(v b c b ). Charging = v ab v b and = v ab v a will result in more profit than θ(2v ab v a v b c ab ) which given v ab 1 2 (v a + c a ) + v b is larger than θ(v b c b ) so selling only good b brings strictly less profit than selling both goods at the price pair (v ab v b, v ab v a ). The lowest prices any retailer will charge for good a and good b are the ones that attract shoppers with probability one and give the same profits as charging the highest price for the good and attracting no shoppers so: = 2(1 θ)c a + θ (v ab v b ) 2 θ = 2(1 θ)c b + θ (v ab v a ) 2 θ (35) (36) In equilibrium, price of good i will be randomized in the interval [p i, p i ] according to the unique marginal distribution: F a ( ) = θv ab c a + θc a + θv b (1 θ) ( c a ) (37) for good a and good b, respectively. F b ( ) = θv ab c b + θc b + θv a, (38) (1 θ) ( c b ) The price ranges for Weak Substitutes are illustrated in Figure 5 a). Note that the marginal price distributions are identical to those from the case of Weak Complements but the marginal distribution functions in the latter case can never be independent. Proposition 4. When v ab 1 2 (v a +c a )+v b the equilibrium marginal distributions given in Equations 37 and 38 are unique. Given Assumption 2 the equilibrium joint distribution function will be the product of marginal distribution functions, that is F (, ) = F a ( ) F b ( ). Proof. The first part of the proposition results from Lemma 6 and the proof of Proposition 1. The second part follows from Assumption 2 and from the fact that + v ab. Equilibrium profits in this case (π = θ(2v ab v a v b c ab )) are less than in the one-good model (θ(v ab c ab )). The reason is that the goods are substitutes so there is no opportunity to discriminate between captives and shoppers. Monopoly profits from 21

22 captives obtain only for one price pair (v ab v b, v ab v a ) and as a result the retailers do not have the opportunity to keep the monopoly sum constant while lowering one of the prices Intermediate Substitutes In the previous section we demonstrated that both goods are always sold iff v ab 1 2 (v a c a ) + v b. So if v ab < 1 2 (v a c a ) + v b with some probability only one good will be bought by captives. In this section we consider the case when probability of selling both goods is still more than zero, albeit less than one. We will argue that in the presence of the price competition good a will never be the only good sold to captives. Lemma 7. When v ab < 1 2 (v a c a )+v b good a will never be the only good sold to captives. Proof. Assume the opposite so that for some (, ) good a is the only good sold. There are two possibilities: either v a > v ab v b and then it has to be true that > v b v a +, or v ab v b and then > v ab v a. Let us consider the latter case first. As before, good b will not be bought by shoppers so the retailer can decrease her price to v ab v a and earn strictly higher profits by selling both goods instead of selling only good a. If v a > v ab v b then decreasing the price of good b can only induce captives and shoppers to switch to buying good b but will never lead to selling both goods. Assume that shoppers in this case were buying good a with a probability λ a. Since > v ab v b in the case shoppers buy good a they do not buy anything else from the other retailers and they get overall surplus of v a. Now consider setting the price of good b at v b v a +. Then with the probability λ a shoppers will buy good b instead of good a. The expected profits will be at least (v b v a + c b )(θ + 2(1 θ)λ a ) which is larger than the previous profit of ( c a )(θ + 2(1 θ)λ a ), a contradiction. We have established that either both goods are bought or only good b is bought by captives. It should be clear that when retailers sell only good b ( > v ab v a and v b v a ) they will randomize in some interval [, v b ] where > v ab v a. Charging the retailer will sell good b to captives with the same probability they would sell both goods to them if they were to charge = v ab v b and = v ab v a so c b = 2v ab v a v b c ab. Since > v ab v a we can derive the first condition for Intermediate Substitutes which is given by: v ab > v b + c a. (39) 22

23 If the retailer charges the highest price for good b she will only sell good b and only to captives so = v b and the quilibrium profit of all the retailers is θ(v b c b ). So when > v ab v a the distribution function of is: F b ( ) = 2 2c b (1 θ) ( + v b )θ 2( c b )(1 θ) (40) while can be chosen arbitrarily provided that v b v a +. Note that the distribution function for coincides with the one from a one-good model. Now we will require that: θv b + 2(1 θ)c b. 2 θ The latter is necessary because no retailer aiming to sell only good b would ever charge a price below θv b+2(1 θ)c b 2 θ. Hence, the second condition for Intermediate Substitutes is 2v ab v a v b c ab θv b + 2(1 θ)c b 2 θ v ab v a + c a + 3c b 2 + v b c b 2 θ. (41) If v ab is more than the maximum between v b + c a and va+ca+3c b 2 + v b c b 2 θ goods will be sold in equilibrium with a positive probability. then both Now lets turn to price pairs such that captives buy both goods. This is the case when v ab v b and v ab v a. In this case, if the price of good a is such that it never attracts shoppers then it will be set to the maximum so = v ab v b. Now assume the retailer is charging the highest price for good b of those below v ab v a. This price will attract shoppers only when other retailers charge above v ab v a so the retailer will get the highest profit only when this price is equal to v ab v a. The expected profits from charging any price pair such that v ab v b and v ab v a should be equal so for such prices: ( c a ) [θ + 2(1 θ)(1 F a ( ))]+( c b ) [θ + 2(1 θ)(1 F b ( ))] = θ(v b c b ). (42) The expected profits from charging (v ab v b, v ab v a ) should be such that: θ(v b c b ) = [θ + 2(1 θ)(1 F b (v ab v a ))] (2v ab v a v b c ab ), (43) which defines F b (v ab v a ). We know that F b (v ab v a ) = F b ( ) and we verify that = 2v ab v a v b c a as derived before. If a retailer charges = v ab v a along with some v ab v b the profit earned 23