Consumer Search and Asymmetric Retail Channels

Transcription

1 Consumer Search and Asymmetric Retail Channels Daniel Garcia Maarten Janssen March 15, 2016 Abstract We study vertical relations in markets with a monopoly manufacturer and consumer search. The monopoly manufacturer chooses the contracts with two retailers optimally and the main question we address concerns the wholesale contracts between manufacturer and retailers that emerge. We consider this question in different settings, with and without commitment of the manufacturer and with and without consumers who can buy at zero search cost. We show that under different settings, the manufacturer is substantially better off if he chooses his wholesale prices with some degree of positive or negative correlation. JEL Classification: D40; D83; L13 Keywords: Vertical Relations, Retailer Channel Optimization, Consumer Search, Diamond Paradox Department of Economics, University of Vienna. daniel.garcia@univie.ac.at Department of Economics, University of Vienna and Higher School of Economics, Moscow. maarten.janssen@univie.ac.at 1

2 1 Introduction Consider a manufacturer selling via retailers to final consumers. One of his main considerations is to make sure that retailers do not make unnecessary margins as these margins reduce his demand without any compensating effect in terms of his own margin. Retail margins, of course, depend on the degree of competition between retailers. Typically, competition is strengthened if competitors face more symmetric costs. Thus, the manufacturer would have an incentive to set identical wholesale prices across the different retailers that sell his product to consumers. This paper shows, however, that in consumer search markets there is an additional force that may result in a manufacturer setting different wholesale prices to different retailers. In general, it may be that the manufacturer wants to correlate his wholesale prices in order to give consumers incentives to search. This consumer search effect is an additional stimulus for retailers to compete. Thus, our paper investigates how the manufacturer is able to set his wholesale prices, thereby organizing his retail channel, in an optimal way. When some consumers incur a search cost, it is inevitable that retail margins are positive. Consider first a market where all consumers incur a positive search cost. In this case, if the manufacturer sets identical wholesale prices across retailers, the maximal profit he can hope to get is given by the double marginalization profit. The reason is that there is a Diamond paradox at the retail level (cf., Diamond (1971)) and for any wholesale price, retailers will always choose the retail monopoly price. If consumer search cost is large, this is actually the optimal profit he can make. Our first result is that if the manufacturer can commit to a wholesale pricing structure with negative correlation across prices, he can do strictly better if consumer search cost is small. The reason is as follows. If search cost is small, the manufacturer is tempted to squeeze retailers by setting a wholesale price that is equal to the consumer reservation price. If consumers expect both retailers to get the same wholesale price, they are, however, willing to buy at an even higher price, and retailers can still make a considerable positive margin, making the deviation not profitable. If the manufacturer chooses a negative correlation between prices, then consumers observing the higher price infer there is a reasonably high chance that the other retailer sets a lower price as the latter probably received the lower of the prices set by the manufacturer. This will make consumer more eager to search, putting downward pressure on retail margins. Interestingly, one of the two prices the manufacturer may choose with positive probability can be quite low and even below cost (and not profit maximizing on its own) somewhat similar to 2

3 the literature on loss leaders. Here, however, a manufacturer only sells one product and he sells to one retailer at a price below cost in order to restrict the competiting retailer to exploit his market power due to consumer search cost. Thus, in our model, selling at price below cost is a way to stimulate consumers to search. Numerically, we show that the effect on profits can be very substantial: especially when search costs are low, the manufacturer can increase profits by more than 30% compared to the double marginalization profit. When the manufacturer cannot commit to prices, we show that the manufacturer cannot make more profits than double marginalization when all consumers incur a search cost. We next consider the Stahl (1989) solution to the Diamond paradox, where consumers are heterogeneous in their search cost and some have a positive search cost, whereas others (the shoppers) have zero search cost. In that model, the manufacturer can benefit choosing a correlated wholesale pricing structure even if it cannot commit to it as the manufacturer optimally trade-offs the benefits from having retailers competing more fiercely for the shoppers if retailers buy at identical prices and the benefits from having retailers not being able to employ their market power over the non-shoppers with positive search cost. We cover a variety of different market circumstances, where the manufacturer may or may not be able to commit to his price and consumers may be more or less homogeneous with respect to their search cost. We also discuss an extension to allowing for non-linear prices. The literature on consumer search in vertical markets is small (Janssen and Shelegia (2015), Lubensky (2013) and Garcia et al. (2015)). In the monopoly manufacturer model of Janssen and Shelegia (2015) the incentive to squeeze retailers leads to a nonexistence of reservation price equilibrium for certain parameter values. In Garcia et al. (2015) this same incentives leads to the existence of an equilibrium where oligopolistic manufacturers randomize over two prices leading to an equilibrium with a bimodal price distribution. The current paper sheds light on the nonexistence result in Janssen and Shelegia (2015) bz allowing a manufacturer to set individualised prices to retailers. At the same time, the paper investigates to what extent the randomization in the oligopolistic model of Garcia et al. (2015) can be imitated by a monopoly manufacturer. The paper also relates to literature on sales (see, e.g. Varian (1980)). This paper is the first to be able to generate sales that are both negatively and positively correlated across different retailers. MORE Most manufacturers use a variety of retail channels and, presumably, offer them 3

4 different contractual arrangements. The literature so far has focused on the initial conditions that determine these asymmetries, such as a-priori differences in market access, technology or bargaining power power. 1 Our study takes a different approach and gives a rationale for asymmetric treatment of ex-ante homogeneous retailers as a way to extract rents in vertically related industries. There is a substantial literature that studies retail pricing strategies and, in particular, the issue of sales. A smaller literature, however, deals with the issue of cross-retailer pricing of a given product. First, there is ample evidence that a large chunk of the within-product variation in prices occurs at the store level (Kaplan et al. (2016)). Whether this is due to only to demand-side heterogeneity or directly imputable to vertical relations is still an open question. Second, an stylized fact in the empirical sales literature is that sales are almost independent across stores (Pesendorfer (2002)). Since the returns of a discount for the seller depend on the discounts offered by competitors, this policy seems paradoxical. Our model predicts that, depending on the specific setup, prices across stores may be arbitrarily correlated. The rest of the paper is organized as follows. Section 2 discusses the model and equilibrium concept. In section 3 we discuss the case of homogeneous consumers, while in section 4 we treat the heterogeneous consumer case. Both sections deal with the situation where the manufacturer may and may not be able to commit to his pricing decisions. Section 5 deals with nonlinear prices. 2 Retail Search Markets with a Monopoly Manufacturer The model we study is very similar in nature to the one in Janssen and Shelegia (2015) and introduces a wholesale level in the search model developed by Stahl (1989), whereby a monopolistic manufacturer sells an homogenous product to two retailers. The difference with Janssen and Shelegia (2015) is that we explicitly allow the manufacturer to choose different prices to different retailers and that we explicitly distinguish between different information and commitment scenarios. A single manufacturer has the technology to produce a certain good. The manufacturer chooses (linear) prices (w 1, w 2 ) for each unit it sells to retailer 1 and 2. 2 We allow 1 See Dukes et al. (2006) and Geylani et al. (2007) 2 Nonlinear prices are considered in Section 5. 4

5 the manufacturer to choose random prices, and the randomization may involve positive or negative correlation. The distribution of wholesale prices chosen by the manufacturer is denoted by G(w 1, w 2 ). For simplicity, we abstract away from the issue of how the cost of the manufacturer is determined and set this cost equal to 0. 3 Retailers take the wholesale price (their marginal cost) w i as given and compete in prices. The distribution of retail prices charged by the retailers is denoted by F (p; w i ) (with density f(p; w i )). Each firm s objective is to maximize profits given their beliefs about prices charged by other firms and the consumers behavior as given. On the demand side of the market, we have a continuum of risk-neutral consumers with identical preferences. A fraction λ [0, 1) of consumers, the shoppers, have zero search cost. Note that we explicitly allow for λ = 0, and the next Section is completely devoted to this case. Shoppers sample all prices and buy at the lowest price. The remaining fraction 1 λ of non-shoppers, have positive search cost s > 0 for every second search they make. 4 If a consumer buys at price p she demands D(p). In our general analysis we assume that the demand function is thrice continuously differentiable, and in addition satisfies the following regularity properties: (i) < D (p) < 0, (ii) for some finite P, D(p) = 0 for all p P and D(p) > 0 otherwise, (iii) for every w < p, π r (p; w) (p w)d(p) is strictly concave and maximized at p m (w), the retail monopoly price for a given wholesale price, and (iv) for all given w, (p w)π r(p)/π r (p) 2 is decreasing in p for all p (w, p m (w)]. 5 We define the wholesale monopoly price w m by w m = arg max w 0 wd(p (w)). In the special case of linear demand D(p) = 1 p we have p m (w) = 1+w and w m = 1/2. 2 The timing is as follows. First, the manufacturer chooses (w 1, w 2 ), where each retailer observes their own wholesale prices, but may, or may not know the price set for 3 To focus on the new insights derived from studying the vertical relation between retailers and manufacturers in a search environment, we assume all market participants know that the manufacturer s cost equals 0. Alternatively, the manufacturer s cost could be set by a third party or could be uncertain. However, this would create additional complexity that may obscure the results. 4 Our analysis for low search cost is unaffected when consumers also have to incur a cost for the first search. For higher search cost, the analysis would become more complicated (see, Janssen et al. (2005) for an analysis of the participation decision of non-shoppers in the Stahl model where the first search is not free). Janssen and Parakhonyak (2014) show that the Stahl equilibrium is unaffected by costly recall. 5 These are standard assumptions in this literature. For example, the last condition (accounting for the fact that retailers marginal cost is w) is used by Stahl (1989) to prove that the reservation price is uniquely defined. Stahl (1989) shows that the condition is satisfied for all concave demand functions and also for the demand function of the form D(p) = (1 p) β, with β (0, 1). 5

6 the other. Typically, consumers do not observe the individual wholesale prices, but we shall also study the case in which they do. Then, given w, each of the retailers i sets price p i. Finally, consumers engage in optimal sequential search given the equilibrium distribution of retail prices, not knowing the actual prices set by individual retailers. As consumers do not observe the wholesale price, the retail market cannot be analyzed as a separate subgame for a given w. To accommodate this asymmetric information feature we use Perfect Bayesian Equilibrium (PBE) as the solution concept, focusing on equilibria where buyers use reservation price strategies where non-shoppers buy at a retail price p ρ. The reservation price ρ is based on beliefs about (w 1, w 2 ), and in equilibrium, these beliefs are correct. As retailers know their wholesale price, their pricing is dependent on their wholesale price. PBE imposes the requirement that retailers respond optimally to any w, not only the equilibrium wholesale price. When there are shoppers, there may be price dispersion at the retail level and we denote by p(w i ) and p(w i ) the lower- and upper- bound of the equilibrium price support after observing wholesale price w i. 3 Retail Management in the absence of shoppers In this Section we consider markets where all consumers incur a positive search cost, i.e., there are no shoppers. Thus, the model in this Section corresponds to the monopolistic version of the model in Garcia et al. (2015). To understand the role of commitment in this market, we first consider the easiest case where the manufacturer cannot commit to any price level. 3.1 No Price commitment Without commitment to a pricing decision and once consumers have formed beliefs about the prices set by the manufacturer, the manufacturer may still change its prices. We start the analysis by showing that the optimal distribution of wholesale prices can only put probability mass on at most two prices, w m and a properly defined consumer reservation price ρ. It does not have any mass elsewhere. To see this, let w be the lower bound of the wholesale price distribution and let p(w) be the corresponding downstream price. As in any equilibrium the retail price reaction p(w ) must be at least weakly increasing, it is clear that all consumers buy at p(w). Thus, by a standard Diamond paradox argument, it must be that p(w) = p m (w). Given this, clearly w = w m. Now 6

7 consider any other price w < ρ. Either p(w) = ρ or p(w) < ρ. Clearly if p(w) = ρ, the manufacturer maximizes wd(ρ) so that she chooses w = ρ. If p(w) < ρ, then p(w) = p m (w) and the optimal wholesale price is w m. Thus, by using the notation π m = w m D(p m ) and π(ρ) = ρd(ρ) we can write expected manufacturer profits as Π = G(w m, w m )2π m + 2G(w m, ρ)[π m + π(ρ)] + G(ρ, ρ)2π(ρ), where (if G(ρ, ρ) + G(w m, ρ) > 0) G(w m, ρ) ρ D(p)dp = s. 6 G(ρ, ρ) + G(w m, ρ) p m In that case, it is easy to see that there exists a critical level of the search cost, denoted by s, such that if s > s, we should have G(w m, w m ) = 1 as there is no temptation for the manufacturer to deviate to ρ (as w m D(p m ) > ρd(ρ)). On the other hand, if s < s from the manufacturer s profit equation it follows that it has to be the case that π m = π(ρ) as otherwise, without commitment he can deviate and set both its prices equal to ρ. This condition uniquely determines ρ. Given that he is indifferent, the manufacturer my still randomize over w m and ρ and it may even choose to a form of affiliation between the two prices. In fact, there is a continuum of choices he can make, where the ratio of G(ρ, ρ) to G(w m, ρ) is determined by G(w m, ρ) ρ D(p)dp = s, G(ρ, ρ) + G(w m, ρ) p m and G(w m, w m ) + G(ρ, ρ) + 2G(w m, ρ) = 1 In particular, as will be important in the next Section, there are equilibria in which the manufacturer sets G(w m, w m ) = 0, but in principle, the manufacturer could optimally choose all G(w m, w m ), G(ρ, ρ) and 2G(w m, ρ) to be strictly positive. In all these equilibria, the manufacturer obtains the double marginalization profit level. We next show that this is substantially lower than what she may achieve with full commitment, especially if consumers search costs are sufficiently small. 6 If G(w m, w m ) = 1, then ρ is defined by ρ p m D(p)dp = s. 7

8 3.2 Full Commitment We start the analysis of the full commitment case by showing that the optimal distribution of wholesale prices can, as in the no commitment case, only put probability mass on at most two prices, w 0 and a properly defined consumer reservation price ρ. The difference with the no commitment case analyzed above is that it is now not clear that the lowest of the two prices, w 0, has to be equal to the wholesale monopoly price. Lemma 1. The optimal price distribution has mass in at most two price levels: w 0 w m and ρ. Proof. Let w be the lower bound of the wholesale price distribution and let p(w) be the corresponding downstream price. As in any equilibrium the retail price reaction p m (w) must be at least weakly increasing, it is clear that all consumers buy at p(w). Thus, by a standard Diamond paradox argument, it must be that p(w) = p m (w). Now suppose that w > w m and consider a deviation to w m. That is, take any initial distribution G and construct G so that for any w such that p m (w) < ρ, G (w w m ) = G(w w). Clearly, the reservation price induced by G (ρ(g )) is less than or equal to ρ and since wd(p m (w)) is decreasing for w > w m, G induces higher profits than G. On the other hand, for any ρ, if p(w) = ρ, the manufacturer maximizes wd(ρ) so that she chooses w = ρ. If p(w) < ρ, then p(w) = p m (w) and the optimal wholesale price is w m Denote by π(w) = wd( p(w)) where p(w) = min{p m (w), ρ}. Using this notation, we have that the manufacturer s profit can be written as Π = G(w 0, w 0 )2π(w 0 ) + 2G(w 0, ρ)[π(w 0 ) + π(ρ)] + G(ρ, ρ)2π(ρ), (1) where (if G(ρ, ρ)+g(w 0, ρ) > 0) and because in equilibrium it has to be that G(ρ, ρ) < 1 G(w 0, ρ) ρ D(p)dp = s. (2) G(ρ, ρ) + G(w 0, ρ) p m (w 0 ) The first trivial observation is that under commitment either G(w 0, w 0 ) = 0 or G(w 0, w 0 ) = 1 and w 0 = w m. To see this note that G(w 0, w 0 ) does not affect ρ and that the manufacturer can always guarantee himself a profit of w 0 D(p m (w 0 )). Note also that if the reservation price of consumers is sufficiently large (that is, s is sufficiently large) the manufacturer s profits cannot exceed the double marginalization profit as π(w 0 ) > π(ρ). 8

9 Therefore, in the rest of this subsection, we assume that s is below this threshold so that π(w 0 ) < π(ρ) for any w 0. When s is small enough, the next Proposition argues that it is optimal for the manufacturer to commit to a pricing structure with negative affiliation, where the consumer reservation price is set as a wholesale price to at least one retailer. The low price and the probability with which it is chosen mostly serve to lower the consumer reservation price to increase profits. In particular, the lowest of the two wholesale prices has to be smaller than the monopoly wholesale price, unlike the no commitment case. Proposition 2. If s < s, then G exhibits negative affiliation with G(w 0, w 0 ) = 0 and G(ρ, w 0 ) > 0. Moreover, w 0 < w m and ρ < p m (w 0 ). The next Proposition goes one step further and argues that when the consumer search cost becomes arbitrarily small, the manufacturer can obtain profits that are arbitrarily close to the monopoly profits. Given the Diamond paradox, this is surprising. In our context, the Diamond paradox would imply that for a given (identical) wholesale price, retailers could guarantee themselves retail monopoly profits for any positive search cost. The next Proposition shows that the manufacturer can prevent the Diamond paradox from arising at the retail level by committing to a pricing structure with negative affiliation where for small search cost a low price is charged with arbitrarily small probability. Proposition 3. As s 0, the manufacturer s profit converges to monopoly profits. Furthermore, w 0 < 0 with G(w 0, ρ) 0 so that the firm engages in loss-leader practices. This Proposition is also of interest for its new perspective on loss leaders. A retailer such as a supermarket may sell some products below cost in order to attract consumers that also buy other products when being in the shop. In our framework, it is the manufacturer that sells with a certain probability the product below cost (which is normalized to 0) to one of its retailer. The reason for doing so is to bring the consumer reservation price close to the monopoly price so that the manufacturer can extract maximal surplus most of the time. This result is visualized in Figure (??). It represents the optimal contract in case demand is linear (D(p) = 1 p) for different values of s. The red line represents the consumers reservation price. The green line represents the retailers monopoly price when observing w 0 (i.e. p m (w 0 )). The blue line represents the optimal mass at the lower price (q 0 ). Notice that the manufacturer engages in large losses when selling at 9

10 the lower price (w 0 < 1.7) but that this happens with small probability and he recoups these losses via a lower ρ. contract.pdf Figure 1: Optimal Contract 3.3 Partial Commitment In some circumstances, it is obviously not realistic to assume that a manufacturer can fully commit himself to a particular random pricing strategy. Without commitment, the manufacturer would, however, deviate and charge the same (reservation) price to both retailers to increase profits. obviously, this cannot be an equilibrium. Instead, what the manufacturer could commit to in practice is to have some retailers that are premium re-sellers and they are guaranteed to have a lower price than other retailers. If the manufacturer decides to have premium re-sellers, we will say that he is partially committed. In this Section we assume that retailers know whether they are chosen to be a premium re-seller or not, while consumers only know whether or not there are premium re-sellers, but do not know who they are. In the next section, where we also include shoppers into the analysis, we may interpret the shoppers as knowing who the premium re-sellers are. With two retailers, partial commitment implies that the manufacturer has one premium re-seller and and one other re-seller (retailer). in this case, the lower of the two 10

11 wholesale prices has to be equal to w 0 = w m. Thus, the manufacturer optimal policy under partial commitment is to offer two contracts, one to each retailer (at prices w m and ρ ) with ρ p m D(p)dp = s. (3) Profits of the manufacturer are then given by w m D(p m ) + ρ D(ρ ). Given the above analysis, if s < s, this profit is actually larger than the profit he gets by offering both retailers to buy at w m. Compared to a pricing structure under full commitment with 0 < G(ρ, w 0 ) < 1 the outcome is more efficient, and for two reasons. 2 First, there is an increase in the number of consumers buying at the lower price from G(ρ, w 0 ) to 1. Second, the higher price is now lower since G(ρ,w 0 ) G(ρ,w 0 ) Thus, there are two different sources of gains that the manufacturer cannot exploit with partial commitment. First, it cannot engage in loss-leading practices. Second, it can only offer one of its retailers the more profitable contract. As a result, the profit gains are much more modest than under full commitment. This can be seen in figure (??) which depicts the gains under full commitment (green line) and under partial commitment (red line) for linear demand (D(p) = 1 p) for different values of s. The no-commitment benchmark corresponds to 1/8. gains.pdf Figure 2: Gains from different levels of commitment 11

12 s q 0 ρ Π 0,01 0,1877 0,7679 0,168 0,05 0,407 0,781 0,152 0,1 0,5 0,7938 0,145 0,15 0,5 0,8197 0,139 0,25 0 0,8819 0,125 Table 1: Optimal Contract Nevertheless, when the search cost parameter is sufficiently small, a system with one premium re-seller and one ordinary retailer may still significantly increase manufacturer profit compared to the situation where he sell to both retailers at the same price. When search cost approach 0, the gains reach up to 24.1%. 4 Retail Management with competition We next allow for the presence of shoppers, which introduces a competition incentive between retailers. Competition between retailers generally will benefit the manufacturer as it leads to lower retail prices given fixed values of the wholesale prices. On the other hand, in the previous Section we have seen that the manufacturer may consciously create asymmetries between retailers in order to stimulate consumers searching, restraining retailers raising their prices. Under the interpretation that asymmetries between retailers are persistent (e.g. because some sellers are premium retailers), shoppers could be interpreted as consumers who are aware of the existence of different types of retailers. In this Section, we will consider what type of contracts the manufacturer may offer. By doing so, we will also shed new light on the equilibrium nonexistence result of?. While shoppers introduce competition between retailers, and retailers will price differently depending on their competitors cost, we have to be more careful about the information retailers have about the contract the manufacturer offers to their competitors. In what follows, we will distinguish again between the commitment and the no commitment case. In the former, it is more common to think that both consumers and retailers know about the commitment of the manufacturer and can respond accordingly. In the latter case, it is probably more realistic that wholesale contracts are private information between manufacturers and individual retailers and we will consider this case here. 12

13 4.1 Price commitment We show that the equilibrium we characterized in the previous Section for the price commitment case with s < s remains an equilibrium for small values of λ.to this end, we note that the Diamond paradox argument continues to hold in the presence of a sufficiently small share of shoppers if both retailers have unequal cost w 1 and w 2 (and they both know they have unequal cost): at relatively low retail prices and given the price of their competitor, both retailers would rather increase their price slightly so that non-shoppers continue to buy than to undercut and get a small extra share of shoppers. When they both set their respective retail monopoly prices, they make profits of (p m (w i ) w i )D(p m (w i )) per consumer. The higher cost retailer w 2 does not want to undercut, however, as long as λ is small enough so that 1 λ 2 (p m (w 2 ) w 2 )D(p m (w 2 )) 1 + λ (p m (w 1 ) w 2 )D(p m (w 1 )). 2 Knowing this, the manufacturer may choose asymmetric contracts and fully squeeze one of the retailers. the downside of this strategy is that he has to give monopoly rent to the other retailer. Thus, consider the situation where the manufacturer randomizes only over the states where at most one of the retailers gets a contract with a wholesale price w m, while maximally squeezing the other retailer by setting ρ. It is easy to see that all shoppers will then buy at p m (w m ). Using the notation of the previous Section, we can write manufacturer profits as Π = 2q 0 ( 1 + λ 2 π m + 1 λ ) π(ρ) + (1 2q 0 )2π(ρ). (4) 2 where the reservation price ρ continues to be determined by q 0 1 q 0 ρ p m D(p)dp = s. (5) Using, the same techniques as in Proposition 1, one can derive the optimal value of q 0 if the manufacturer wants to randomize this way. The main difference with the previous Section is that the per consumer manufacturer profit in case it chooses to contract both retailers at the same wholesale price w m (or any other wholesale price w < ρ) is different from, and actually larger than, π m. The reason is that in that case, retailers will start 13

14 competing for the shoppers. If we write π(w, w; λ) for the manufacturer profit in case he contracts both retailers at the same wholesale price w in a market with a fraction of λ shoppers, then we have that π(w m, w m ; 0) = π m, π(w, w; λ) is increasing in λ and π(w, w ; λ) π(w m, w m ; λ). As for all s < s, π(w m, w m ; 0) = π m < π(ρ), it follows that for λ small enough, the manufacturer finds it optimal to keep the asymmetric structure of retail contracts. Thus, we have the following proposition. Proposition 4. For any s < s, there exists a λ s > 0 such that for all λ < λ s there exists a unique optimal value of q 0 with 0 < q For all other parameter values, the manufacturer finds it optimal to have symmetric retailer contracts, where because of the shoppers the optimal level is different from w m. 4.2 No Price commitment Next, we consider the case where the manufacturer cannot commit to a certain pricing strategy. This is exactly the model that was analyzed in Janssen and Shelegia (2015). They concentrated on symmetric equilibria where the manufacturer chooses to offer both retailers the same linear contract. Importantly for our paper, they found that symmetric equilibria where the manufacturer chooses pure strategies do not always exist. We now show that if the manufacturer can discriminate among retailers, there is a continuum of equilibria, indexed by w. The worst equilibrium for the manufacturer is the double marginalization equilibrium, while the best equilibrium yields a payoff of g(λ)π, where π is the monopoly profit level and g(λ) was defined in Janssen and Shelegia (2015) for the case of linear demand and a non-binding reservation price. In general, it corresponds to the solution of max ( (1 λ) p(w) p(w) D(p)f(p; w) dp + 2λ p(w) p(w) s.t. F (p) = 1 + λ 2λ D(p)f(p; w)(1 F (p; w) dp ) w (1 λ)( p(w) w)d( p(w)) 2λ(p w)d(p) where p(w) = max{p m (w), z} and z is such that zd(z) = g(λ)π. Notice that g(λ) is increasing in λ and satisfies g(0) = πm π and g(1) = 1. 14

15 We are now in the position to characterize the set of equilibrium outcomes in this market. Proposition 5. For any π [π m, g(λ)π ), there exists an equilibrium in which the manufacturer obtains a profit of π. In these equilibria 1. The manufacturer randomizes with support in w and ρ, with joint distribution G(w 1, w 2 ). 2. Retailer i observes w i and chooses p i according to If w i < w, p i = p m (w i ). If w i = w, p i follows F (p; w ) If ρ > w i > w, p i = min{p m (w i ), ρ} If w i ρ, p i = w i 3. Consumers follow a reservation price strategy so that they buy a quantity D(p i ) right away iff p i ρ and they search otherwise. Notice that in a best equilibrium, the distribution of wholesale prices and, as a result, that of retail prices, is positively correlated. This is because the monopolist lacks commitment and, thus, has to be indifferent among all prices he charges in equilibrium. The advantage of negative correlation stems from inducing a more attractive reservation price by inducing consumers to search more actively. Such an advantage, however, requires substantial commitment power so that the manufacturer is willing to charge ex-post suboptimal prices with positive probability. On the other hand, notice that the price distribution that implements the best equilibrium is degenerate. This does not mean, however, that the ability to discriminate retailers is irrelevant for the manufacturer. Indeed, in order to implement such equilibria, retailers beliefs following a deviation must necessarily contemplate the possibility that they are being discriminated. As shown in Janssen and Shelegia (2015), if retailers believe that their competitor received the same treatment, many deviations are now profitable and this price may not be sustained as an equilibrium. 15

16 5 Two-Part Tariffs A reasonable concern from the previous analysis is the assumption that monopolists are restricted to linear contracts. On top of the classical justifications, we would like to notice that under a wide range of off-the-equilibrium path beliefs, there is no purestrategy equilibrium at the contracting stage in which at least two firms compete for shoppers, ex-post. To see this fix a certain equilibrium set of offers. Given a deviation to a lower unit price, as long as the retailers beliefs about the distribution of other offers does not change one-to-one with hers (i.e. she does not have symmetric beliefs) she will be willing to accept a higher lump-sum transfer. Indeed, her expected profits will increase by ɛd(p)(1 + λn) where ɛ is the cut in the unit price and p the lower bound in the price distribution. The monopolist can charge this to each seller so that he gets nɛd(p)(1+λ(n 1)). Since each seller has the same contract, however, the cost in terms of a lower retail price is simply nɛd(p) so that retailers lose n(n 1)λɛD(p) which goes directly to the monopolist. Notice that this logic would remain as long as the expected market share of a retailer who receives a lower unit price than the one specified in the putative equilibrium contract increases. Hence, an equilibrium exists if and only if retailers have symmetric (or wary) beliefs about the distribution of offers to other retailers. In such a case, however, the first best is still not implementable because retailers competition is either too strong (so that zero wholesale unit prices lead to excessively low prices) or too weak (so that zero lump-sum transfers yield the double marginalization problem). 5.1 Symmetric Equilibrium From now on consider the symmetric case and assume that the equilibrium is symmetric. Let {w, T } be the equilibrium contract. Given any w, firms expect their rivals to receive the same contract so that T must satisfy T (1 λ)π m (w) = (1 λ)d(p m (w))(p m (w) w) (6) since in equilibrium the upper-bound of the distribution of prices equals the retail monopoly price. Let F (p w) be such distribution and let F 2 (p w) denote the distribution of the minimum price among both retailers. The per-market profits of the 16

17 manufacturer equal Π = w{(1 λ) D(p)dF (p w) + λ D(p)dF 2 (p w)} + (1 λ)π m (w). (7) Let D(w) be the expected individual demand following a price w (i.e. the term in curly brackets). This is the case whether w is the equilibrium contract or some other contract, since beliefs are symmetric also off-the-equilibrium path. Then we have that in any optimal contract we have D(w) + w D(w) w (1 λ)d(pm (w)) = 0 (8) where the third term uses the envelope theorem given that p m (w) is chosen optimally. First thing to notice is that D(p) is decreasing and p m (w) is the upper bound of F (p w). Hence, w = 0 can only be a solution if λ = 0 so that F (p w) becomes degenerate at the upper bound. The second observation is that if λ = 1, then F (p w) collapses at w and the first order condition is satisfied for w = p m. Finally, the wholesale price in the case of linear contracts (assuming symmetry) would satisfy D(w) + w D(w) w = 0 (9) so that the optimal two-part tariff has a lower unit price than the optimal linear contract for any λ (0, 1). The comparative statics with respect to λ are, in general, hard to characterize since λ affects the distribution of retail prices for a given w. 6 Conclusion In this paper we have looked at the incentive of manufacturers to treat retailers that sell his product to consumers asymmetrically. In general, as manufacturers have an incentive to lower retailer margins, they face a trade-off. On one hand, manufacturers may sell to different retailers at different prices to stimulate consumer search. Consumers that observe a relatively high price may infer that the chance they a observe low price on the next search is considerably high and this will limit the market power of retailers. On the other hand, by creating an asymmetry retailers are competing less fierce for consumers who anyway consider all price offers before buying. This paper analyzes how manufacturers optimally deal with this trade-off in a variety of settings. 17

18 For simplicity, we first consider a situation where the competition aspect is absent as all consumers have to pay a search cost and typically will only visit one retailer. In that case we have analyzed three levels of commitment by the manufacturer: full commitment to the prices it hs chosen, no commitment and partial commitment. Without commitment, the manufacturer s maximal profit equals the double marginalization profit. he can reach this level of profit with a variety of pricing schemes, but the equilibrium profit is constant over the different schemes. With full commitment, the situation is very different for small search cost. In that case the manufacturer chooses a negative correlation structure between a very low price and the consumer reservation price. The purpose of the low price is to lower the consumer reservation price, which is the price that attracts most of the sales. When the search cost becomes arbitrarily small the manufacturer may find it optimal to choose the low price below cost, but with low probability. Doing so, the manufacturer prevents the Diamond paradox from arising and makes profits that are close to monopoly profits. Under partial commitment, the manufacturer may announce that one of the retailers get the status of premium reseller and that this retailer gets a lower wholesale price. Having a system of premium re-sellers may thus increase the manufacturer s profits and consumer welfare. With competition for shoppers the analysis becomes more complicated. In this case, the manufacturer may stimulate competition between retailers by choosing identical wholesale prices with some positive probability. Generally speaking, competition for shoppers gives manufacturers an incentive to choose prices with a positive affiliation. We show that by having some correlation between the wholesale prices resolves the non-existence problem that is present in? A Ommited Proofs Proof of Proposition 3. We can rewrite the profit level as Π = 2{π(ρ) G(w 0, ρ)(π(ρ) π(w 0 ))} (10) where G(w 0, ρ) ρ D(p)dp = s (11) 1 G(w 0, ρ) p m In order to increase the gains from search at lower prices, the manufacturer has two instruments. First, it may increase the probability that a single store offers the best 18

19 deal (G(w 0, ρ)). Second, it may reduce the lowest price (w 0 ) so that p m (w 0 ) yields a better deal for the consumer. Let q 0 = G(w 0, ρ) and notice that while In an interior solution we must have that Π = 2(1 q 0 )π (ρ) ρ 2(π(ρ) π(w 0 )) (12) q 0 q 0 Π = 2(1 q 0 )π (ρ) ρ + 2q 0 π (w 0 ). (13) w 0 w 0 π(ρ) π(w 0 ) 2q 0 π (w 0 ) = ρ w 0 ρ q 0. (14) It is obvious then that w 0 < w m as long as π(ρ) > π(w 0 ). This is the case if the manufacturer can secure a profit level above the double marginalization equilibrium. Notice also that and Combining all of these pieces we have ρ = D(pm (w 0 )) w 0 D(ρ) ρ q 0 = s q 2 0D(ρ) (15) (16) π(ρ) π(w 0 ) D(p m (w π 0 ) = s (17) (w 0 ) q 0 This condition depends on the demand function. For linear demand the LHS has a minimum in w 0 (0, 1/2) while the RHS is decreasing in w 0. For future reference, notice that w 0 = 0, LHS equals π(ρ).. In addition, we have that (1 q 0 ) π (ρ) D(ρ) = q π (w 0 ) 0 D(p m (w 0 )) (18) This condition can be understood as a marginal return of an increase in each of the two prices must be proportional to their impact on ρ. Proof of Proposition 4. For the first part, choose w0 < 0 so that p m D(p)dp = s p m (w0 ) 1 for some s 1 > 0 and consider the sequence of search costs {s n } = s 1 n. For each s n in the sequence, w 0 (n) = w 0 and ρ(n) = p m. Thus, q 0 (n) 0 so that profits converge to 19

20 monopoly. Now, to see that w m > w 0 0 is suboptimal for s small enough, consider first the case in which w 0 > w for any s. Clearly, the profit is a linear combination of w 0 D(p m (w 0 ) and ρd(ρ) and both would be bounded away from monopoly level. Hence, it must be that w 0 converges to zero as s vanishes. So assume that this is the case and notice that, along any such sequence of optimal contracts, it must be that s n /q 0 (n) = δ(n)π(ρ(n)) for some sequence δ(n) < 1 converging to 1. In such a case, the consumers reservation price is ρ D(p)dp = (1 q 0(n)s n (19) p q m 0 (n) (1 q 0 (n ))λ(n )π(ρ(n )) > w m D(p m (w m )) (20) for some n large enough. But then ρ is bounded away from p m (0) and so the manufacturer cannot obtain monopoly profits. Thus, w 0 = 0 is not a solution for the system. Proof of Proposition 6. First, it is clear that w and ρ have to give the same profits for the manufacturer to randomize. We can compute its profit using ( π(w ) = (1 λ) p(w ) D(p)f(p; w ) dp + 2λ p(w ) p(w ) p(w ) D(p)f(p; w )(1 F (p; w ) dp ) w. It must hold then that π(w ) = ρd(ρ). (21) For a given G(w 1, w 2 ), we have a unique F (p; w ) so that given w and G(w 1, w 2 ), there is a unique ρ that satisfies this relation. Importantly, F (p; w ) only depends on G(w, w ), while ρ has to satisfy G(w, ρ) G(w, ρ) + G(ρ, ρ) ρ D(t)dtf(p; w )dp = s (22) p So we can pick ρ by appropriately choosing the conditional probability that the other retailer observed w given that the retailer the consumer visited observed ρ. Moreover, given the strategy profile of retailers, the manufacturer has no incentive to deviate to any other price. 7 Similarly, consumers indifference condition is satisfied at 7 Notice, in particular, that the upper bound of the distribution of retail prices following w is no higher than p m (w and that pd(p) is decreasing in the relevant range 20

21 ρ. For simplicity, we assume that if the consumer observes any price outside the support of the equilibrium retail price distribution, he assumes that the retailer observed a price of ρ. Therefore, in order to ensure that he follows a reservation price strategy we need to verify that the gains from search following any price in the support of the price distribution renders search unprofitable. Notice that this is not guaranteed by the fact that the upper bound of the support is no higher than ρ since the beliefs are different. In particular, it must hold that G(w, w ) G(w, w ) + G(w, ρ) p(w) D(t)dtf(p; w )dp s. (23) p This implies that if p(w) = ρ, a necessary condition is that G(w, w ) G(ρ, ρ) G 2 (ρ, w ) (i.e., that the wholesale price distribution exhibits positive affiliation). p(w) < ρ, this condition is only sufficient but not necessary. We turn now to the problem of the retailers. By definition, F (p; w ) is sequentially optimal given G(w 1, w 2 ). Similarly, if w i ρ, p i = w i is sequentially optimal as long as retailers believe that the rival obtained a price of ρ. If w i (w, ρ) we impose the off-the-equilibrium path beliefs that the rival firm observed ρ if p m (w i ) < ρ and that she observed w if p m (w i ) > ρ. This ensures that the retailer is willing to follow the prescribed strategy. Finally, if w i < w we also assume that retailers believe that the rival observed ρ. We finally argue that the equilibrium payoff set equals [π m, g(λ)π ). It is clear that no equilibrium can exist whereby the monopolist obtains less than the double marginalization profit because by charging w m he can ensure that level. Similarly, we know from Janssen and Shelegia (2015) that no outcome exists in which the payoff exceeds g(λ)π. This equilibrium requires that G(w, w ) = 1, which we can implement exactly only if ρ > p m (w ). Otherwise, we can obtain an equilibrium that is arbitrarily close to it. 8 Consider the following class of equilibria that span the whole set. First, fix w = w(λ), the wholesale price that induces the highest possible profit level, and consider the equilibria indexed by G(w(λ), w(λ)) [q(λ), 1], where q(λ) is the lowest value such that (i) the expected profit at w(λ) is at least as high as the double-marginalization profit and (ii) G(w(λ), w(λ)) G(ρ, ρ) G 2 (ρ, w(λ)). It is clear that if G(w(λ), w(λ) = 0, then the profit level associated with this distribution is no higher than that of the doublemarginalization. By continuity, there exists a unique such value and because the profit 8 The problem is that if G(w, w ) = 1, observing ρ does not lead the consumer to believe that w i = ρ but rather than it equaled w. 21 If

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