Beliefs, Market Size and Consumer Search

Beliefs, Market Size and Consumer Search Maarten Janssen and Sandro Shelegia March 30, 2014 Abstract We analyze how the market power of firms in consumer search markets depends on how consumers form beliefs about the offers they encounter on their next search. Using the Wolinsky model, we show that under symmetric beliefs, firms have much more market power and set higher prices. Pricing behavior is a non-monotonic function of search cost. We show that symmetric beliefs naturally arise in an environment where retailers common costs are unknown to consumers and either follow a random process or are determined by an upstream firm. The first environment allows us to provide an alternative explanation for sales (compared to Varian s model of sales). In the second environment, we show that the manufacturer has the incentives and the ability to prevent the market from breaking down, providing an alternative channel through which the Diamond paradox can be solved. JEL Classification: D40; D83; L13 Keywords: Vertical Relations, Consumer Search, Double Marginalization, Product Differentiation 1 Introduction When consumers search, they have to form beliefs about what offers they can expect to get at a next search. If they are pessimistic that the next search yields a good offer, they are more likely to accept the current offer, giving firms an incentive to set higher prices. Thus, market power in an industry where search is important crucially depends on consumer beliefs. This insight is important in any search market, but particularly so in markets where there is additional uncertainty concerning factors that influence market prices. For example, if consumers know that prices depend on firms marginal cost, but are uninformed about them, then beliefs about offers consumers can expect on the next search may depend on what consumers currently observe at the firm they are visiting. The consumer search literature, by and large, has side-stepped issues regarding consumer beliefs and implicitly or explicitly assumes that consumers hold passive beliefs, i.e., no matter what they observe at a particular search round, consumers believe that at a next search round they encounter firms that stick to their equilibrium strategies. There is, however, no reason not to consider alternative beliefs. For example, consumers could hold symmetric beliefs (see, Department of Economics, University of Vienna and State University Higher School of Economics. Email: maarten.janssen@univie.ac.at Department of Economics, University of Vienna. Email: sandro.shelegia@univie.ac.at 1

for instance, Hart and Tirole (1990), McAfee and Schwarz (1994) and more recently Pagnozzi and Piccolo (2012)), i.e., after observing in a search round that a firm has deviated from its equilibrium strategy, consumers could believe that other firms (that they would encounter on a next search) have also deviated in the same way. In this paper we show that symmetric beliefs emerge naturally in two important search settings and they lead to equilibrium predictions that are qualitatively and quantitatively different from predictions obtained under traditional passive beliefs. The first setting is one where firms have a common cost that fluctuates randomly and is not observed by consumers. The second considers a vertical industry where retailers cost is determined by an upstream monopoly. We take the now standard Wolinsky (1986) model of sequential search where cost is known and consumers have passive beliefs as our starting point. 1 To fully understand the impact of beliefs on market prices, and the price increasing effect of symmetric beliefs, we analyze the Wolinsky model for small and large search cost. Most papers that build on the Wolinsky model (see, e.g., Anderson and Renault (2006), Armstrong, Vickers and Zhou (2009) and Bar-Isaac, Caruana and Cuñat (2011)) take search costs to be relatively small so that the consumers reservation utility is not binding for the firms optimization problem. The behavior of the model for larger search cost is relatively under-explored. While it is known that, when the search cost is such that prices exceed consumers reservation utility, firms set the monopoly price (that is independent of the search cost), the literature has not looked int the equilibrium market demand that drops discretely at the critical level of the search cost. The reason is that once search cost is so high that prices are above the reservation utility, consumers are not willing to pay the cost to obtain a new utility draw in case the first one was unsatisfactory and drop out of the market. This leads to a drop in demand from those consumers who would have purchased at the second firm, but never arrive there. This unexplored market size effect is the second important ingredient our our results in the setting with vertical market structure. With symmetric beliefs in the Wolinsky model, it is not difficult to see that firms will set higher equilibrium prices as consumers are less willing to continue to search after observing a deviation to a higher price than the equilibrium price under passive beliefs. This price increasing effect drives our first main result, namely that under symmetric beliefs, prices are non-monotonic in search cost. The reason can be understood as follows. For large search cost, as consumer behavior is not driven by their beliefs about prices at the other firms, price equals the monopoly price. For very small search cost, the equilibrium price remains smaller than consumers reservation utility and here prices are increasing in search cost and larger than under passive beliefs. As the reservation utility is independent of whether consumers have passive or symmetric beliefs, this utility becomes binding at lower levels of the search cost. At these intermediate levels of search cost where the reservation utility is binding, equilibrium prices are equal to the reservation utility and as this utility is decreasing in search cost. Thus, for intermediate levels of the search cost, equilibrium prices are decreasing in search cost. 1 In the context of the alternative, homogeneous goods search model by Stahl (1989), Dana (1994), and more recently, Tappata (2009), Chandra and Tappata (2011) and Janssen, Pichler and Weidenholzer (2011) have analyzed the consequences of asymmetric information on firms common marginal cost. Janssen and Shelegia (2014) study retailers common cost being determined by an upstream firm and the consequences of this wholesale price not being observed by consumers. These models use passive beliefs, because in a homogeneous goods model reservation price equilibrium does not exist under symmetric beliefs. 2

These observations are of importance for understanding the implications consumers not having information about a common cost component of firms. This situation is relevant in retail gasoline markets, for example, where retail gasoline stations know today s world oil market price (as this is reflected in the common cost they have), but consumers probably do not know current world oil market prices (due to the fact that they are highly volatile). Symmetric beliefs are natural in such a setting where retailers have a common cost as after observing a retail price by one firm, consumers form beliefs about the cost that is common to all retailers. Along the equilibrium path, upon observing a high price, consumers will infer that common cost is high and that the other retailer will also have a high price. Given the observation on the Wolinsky model with symmetric beliefs, it is not surprising that in this setting with small search cost there exists a symmetric equilibrium where for each cost realization, prices (and firms profits) are strictly larger than in the corresponding model where cost is known to consumers and they have passive beliefs. Symmetric beliefs make continuing searching less attractive, providing firms with more market power and an incentive to raise prices. Interestingly, for some level of search cost, the equilibrium price can be as high as the joint profit maximizing price! When this happens, the joint profit maximizing price equals consumers reservation utility. It also follows that for a given cost realization, retail price may be decreasing in search cost when search cost is at intermediate values. An interesting feature that plays an important role in the vertical relations model discussed below is that for most values of the search cost, the equilibrium pricing strategy has a flat part where price does not vary with cost, apart from parts where price is increasing in cost. Thus, for many parameter values the equilibrium is neither pooling, nor separating, but semi-separating. The reason is that there is a region of cost levels where the equilibrium price equals consumers reservation utility and this reservation utility is independent of cost. An interesting aspect of this equilibrium characterization is that it provides an alternative explanation for the common observation that retailers often follow a pricing strategy for their products that is characterized by a regular price and a sales price at irregular moments in time (as in Varian (1980) s model of sales). Unlike the existing literature, our model can explain sales in terms of a pure strategy equilibrium where a firm s price is a function of its cost (with horizontal parts where price is independent of cost). We then turn to the results of our paper dealing with vertical relations, where the retailers common cost is set by an upstream firm. We show that in this vertical relations model with search, a combination of interesting features that have been discussed above in isolation can arise. To do so, we add to the Wolinsky model an upstream firm setting the wholesale price. In the benchmark vertical model consumers observe this wholesale price. We then compare the equilibrium of that model with the case where the wholesale price remains unknown to consumers. When wholesale price is observed, but search cost is small, we have a model where the upstream firm chooses its profit maximizing price given the margins that downstream firms will charge. For small search costs downstream prices are smaller than the consumers reservation utility and increasing in search cost, even though upstream prices are falling. Once the search cost crosses a critical level, the retail price becomes larger than the consumers reservation price (with the corresponding drop in market demand as a consequence) in case the manufacturer does 3

not drastically adjust its behavior. As the manufacturer has a clear incentive that consumers who are not satisfied at the first retailer continue to search to see whether the product of the second retailer better fits their taste, it will prevent the market demand to drop by lowering its price significantly. This leads to retail prices also falling in search cost. It is interesting to observe that this creates a second channel through which the Diamond paradox can be solved. Wolinsky (1986) solves the Diamond paradox by introducing product differentiation. However, when search cost becomes large and the first search is costly, the market will still break down in his model as retail firms individually cannot prevent this from happening. By being able to influence both retail prices, the manufacturer has different incentives and has the ability to keep the market active for (much) higher levels of search cost. Of course, (depending on whether or not the first search is for free) when search cost rises further there is a point where also the manufacturer has to set such a low price to prevent the market demand to fall that it is not in its interest to do so. When the first search cost is not for free, this will happen at the point where the manufacturer charges a price equal to its marginal cost. When the search cost is for free, at this level of search cost, wholesale and retail prices drastically increase to the classic double marginalisation level. Consumers stop searching at this point and for all larger search cost the equilibrium prices are constant. When retail costs are not observed by consumers, there are multiple equilibria depending on to whom consumers attribute a deviation from the conjectured (equilibrium) retail price. As retail prices are now the outcome of the combination of a retail pricing strategy and the actual upstream price chosen, a consumer cannot know whether the upstream firm or the retailer has deviated after observing a deviation and both passive and symmetric beliefs can be rationalized. Under passive beliefs, the equilibrium structure is very similar to the case where wholesale price is observed. The only difference is that the manufacturer may have an incentive to charge a different price due to the fact that its deviations are not observed. As this effect on the unobservability of the wholesale price to consumers is explored in Janssen and Shelegia (2014), we focus here on the impact of consumer beliefs. With symmetric beliefs, for a given wholesale price downstream behavior is identical to that behavior in the random cost model. This fact creates interesting incentives for the manufacturer. In particular, at a certain level of search cost, as the retail price is independent of retailers cost, the manufacturer has an incentive to increase price and the equilibrium wholesale price increases discontinuously in search cost. With wholesale price being unobserved by consumers, to the market size effect also provides the manufacturer with incentives to reduce its price for larger search cost. The range of parameters for which the manufacturer can prevent the market from breaking down is reduced, however. Thus, in the case where the wholesale arrangement is unobserved by consumers, wholesale prices can be increasing in search cost, decreasing in search cost, and may perform discontinuous jumps. Retail prices fall with search costs in a surprisingly large interval and the quantitive effects we show in numerical examples can be quite substantial. Thus, the forces created by symmetric beliefs and the market size effect can be empirically important. This paper is related to several strands of literature. In the context of the homogeneous goods search model by Stahl (1989), Dana (1994), and more recently, Tappata (2009), Chandra and Tappata (2011) and Janssen, Pichler and Weidenholzer (2011) have analyzed an asymmetric 4

information model where firms know their common cost and consumers do not. 2 These papers show that for the same average cost realization equilibrium prices in the asymmetric information model are higher than in the setting where the cost is known to be equal to the average cost of the cost uncertainty model. Our results in this paper for the asymmetric information model are stronger in that we show that for all cost realizations prices are higher. Moreover, in our asymmetric information model prices are non-monotonic in search cost, whereas they are monotonically increasing in the homogeneous goods search models. The difference in results is related to the difference in underlying mechanism: in our paper the main mechanism builds on symmetric beliefs and the market size effect, while symmetric beliefs would destroy the equilibrium analysis in the homogeneous goods search literature. The importance of beliefs in a vertical structure has recently also been stressed by Pagnozzi and Piccolo (2012). In that paper, the authors study the reason for manufacturers to sell to consumers via an independent retailer rather than directly. In their framework, it is retailers that have to form beliefs about the wholesale prices that manufacturers charge to their competitors. They show that the nature of the vertical channel depends on whether retailers hold passive or symmetric beliefs. Janssen and Shelegia (2014) perform the vertical analysis of the Stahl (1989) model where retailers cost is determined by a manufacturer maximizing profits. They consider two different scenarios, one where consumers do not observe retailers cost and one where they do. show that the unobservability of retailers cost has severe implications for the prices charged in the market due to the fact that market demand is much more inelastic when the wholesale price is unobserved. The analysis of the vertical structure in the current paper mainly relies on mechanisms (symmetric beliefs and the market size effect) that are absent in Janssen and Shelegia (2014). Other explanations for prices being declining in search cost differ sharply from the ones that are given in earlier contributions. In Janssen et al. (2005) a participation effect is responsible for prices being declining in search cost. In that model the equilibrium fraction of consumers participating in the market is determined by a participating constraint and if search cost are large that constraint is binding. Zhou (2014) studies a multi-product search environment where with one search, consumers can buy multiple products. In this environment products become complements and if search cost become higher, firms may compete more intensely to keep the consumers from searching further. We provide two alternative reasons why retail prices may be declining in search cost. First, in the absence of vertical considerations, there is an intermediate region of search cost where retail prices are higher than the monopoly price and the consumer reservation utility is binding and this reservation utility is decreasing in search cost. Second, this effect is significantly strengthened under vertical relations when the manufacturer has an additional incentive to prevent retail prices to be larger than the reservation utility. The rest of the paper is organized as follows. they The next section sets out the model we use to analyze the pricing implications of the different environments we study. Section 3 then analyzes the implications of symmetric beliefs in the Wolinsky model and it shows the underexplored market size effect. Section 4 discusses the implications of asymmetric information 2 In the context of different models, Benabou and Gertner (1993) and Fishman (1996) have also studied the effects of asymmetric information about retailers cost in consumer search models. 5

concerning retailers common cost. Interestingly, the increased price effect does not disappear if cost uncertainty vanishes. Section 5 then analyzes the effects of asymmetric information concerning retailers common cost in an environment where this cost is not a random variable, but strategically chosen by an upstream firm. The last section concludes. 2 The models The retail side of the three models is exactly the same as in Wolinsky (1986). There are two retail firms, 1 and 2, who can acquire some input at a common cost c. The retailers transform the input costlessly into a final differentiated good, using a one for one technology. There is a unit mass of consumers per firm. Utility to a consumer from buying the good at firm i is v i. This utility is drawn according to some distribution function G(v i ), which is defined over the interval [v, v]. The valuation of a consumer for the product of firm 1 is independent of his valuation for the product of firm 2. Each consumer costlessly visits one of the downstream firms at random and finds out v i and p i. After visiting the first firm, consumers have to decide whether to buy there, search the second retail firm, or stop searching altogether. The additional search comes at a cost s. The consumer can always go back to the first firm at no additional cost (free recall). In the search literature based on the Wolinsky model, it is common to denote by w the expected utility of searching another firm including the search cost. Consumers visit the first firm for free, 3 and after observing the match value v i and the price p i they decide whether or not to visit the second firm. If they do so, they make their purchase at the best available surplus v i p i. If they decide not to continue searching, they buy at the first firm if v i p i 0. In general, an individual retailer i s demand depends on the price p i it chooses, the price p j of firm j and on consumers beliefs p e j about the price of firm j. The latter may depend on the marginal cost c and price charged by firm i. For now we will keep the formulation general, and specify model-specific beliefs later. Consider a consumer who visits firm i. To determine the reservation utility w we have to determine the benefit from searching firm j. Under free recall, the reservation utility w if prices are equal at both firms, is the solution to: v w (v w )f(v) dv = s. If such a solution does not exist, then w = v. This means that a consumer who draws v i and p i will search the other firm if v i < w p e j + p i. For consumers to ever continue to search we need that w > p e j. If this inequality is not satisfied, a consumer will not continue to search. Given the optimal search behavior from above, firm 1 s demand is given by Q 1 (p 1 ) = (1 G(w p e 2 + p 1 )) + G(w p e 1 + p 2 )(1 G(w p e 1 + p 1 )) (1) + w p e 2 +p 1 p 1 G(p 2 p 1 + v)g(v)dv + w p e 1 +p 1 p 1 G(p 2 p 1 + v)g(v)dv. 3 Here, we follow most of the consumer search literature. When desired we show, however, how our results depend on this assumption. 6

The first term is the demand from consumers who visit firm 1 and buy outright because their v 1 falls below the threshold w p e 2 + p 1. The second term is the demand from consumers who visit firm 2, draw v 2 lower than the threshold w p e 1 + p 2 move on to firm 1, encounter there v 1 more than w p e 1 + p 1 and buy at firm 1. The first integral is the demand from those who move on from firm 1 to firm 2, but come back to firm 1. Finally, the second integral is the demand from those consumers who first visit firm 2, move on to firm 1 and buy there. To understand the role of beliefs in the Wolinsky model, we formally define the equilibrium notion below. As, depending on the out-of-equilibrium beliefs, consumers beliefs about the price to be observed at the next search may depend on the price p 1 observed at the first search, consumers expected utility (or reservation utility) of searching another firm may depend on the price observed at the first firm, and we write w (p 1 ). Definition 1. A symmetric perfect Bayesian equilibrium of the Wolinsky model is a price p (the same for both firms) and a reservation utility w (p) such that 1. Each firm i chooses p i = p to maximize its expected profit given the price of the other firm and consumers reservation utility; i.e., firm i will set a price that solve the first order condition where Q 1 is given in (1); p 1 = c Q 1 Q 1 (p 1), 2. Consumers follow an optimal search rule given their beliefs and the match value v i and the price p i they observe at firm i; 3. Consumers beliefs are updated using Bayes Rule when possible, i.e., whenever they observe p 1 = p at their first search, they believe that the other firm has also set a price p. Outof-equilibrium beliefs p e 2 of the price set by the firm that is not yet visited are either i. (passive beliefs) such that p e 2 = p if p 1 p, or ii. (symmetric beliefs) such that p e 2 = p 1 if p 1 p. In principle, one could study the implications of many different formulations of out-ofequilibrium beliefs. For example, one could specify a convex combination of passive and symmetric beliefs or one could have a different belief for different out-of-equilibrium prices. Passive beliefs is the common assumption in the search literature and we want to study the impact of that assumption by studying the implications of another belief formation process that is considered in another literature (see, e.g., Hart et al. (1990) and Mcafee and Schwartz (1994) in the context of vertical relations between a manufacturer and two retailers, something we also consider later). As explained in the Introduction, we also (and mainly) want to supplement the Wolinsky model with settings where the common cost is determined. First, we consider a market where the common cost c follows a random distribution with support [c, c]. Firms know the realization of c, but consumers do not, so that we have asymmetric information between firms and consumers. One can think of this as the common cost being determined in a competitive (upstream) market 7

environment where price is determined by supply and demand. Consumers think it is highly unlikely (a probability 0 event) that cost is below c or above c. A second environment we consider is where the common cost is determined by an upstream monopolist. The upstream firm U produces the essential input at a marginal cost of zero and charges c for one unit to the downstream competitors. In this second environment we study a benchmark model where retailers cost is observed by consumers and a model where it is not observed. The timing of these two games is that the common cost is determined either by Nature or by an upstream firm. The choice of c is observed by the downstream firms (and, in the benchmark vertical relations model, but only there, also by consumers). After c is determined, we have the same interaction between retailers and consumers as described above. In the asymmetric information model where retailers cost is randomly determined, retailers (symmetric) strategy is given by a function p(c), whereas the search strategy is characterized by w (p 1 ). A symmetric equilibrium for this model is then given as follows. Definition 2. A symmetric perfect Bayesian equilibrium in continuous strategies of the search model with asymmetric information about cost is a continuous pricing strategy p (c) with reach P and a reservation utility w (p) such that 1. for every cost realization c, each firm i chooses p i (c) = p (c) to maximize its expected profit given the pricing strategy p (c) of the other firm and consumers reservation utility w (p); 2. Consumers follow an optimal search rule given their beliefs and the match value v i and the price p i they observe at firm i; 3. Consumers beliefs are updated using Bayes Rule when possible, i.e., whenever they observe p 1 P at their first search, they believe that the other firm has also set a price p e 2 = p 1. Out-of-equilibrium beliefs p e 2 of the price set by the firm that is not yet visited are symmetric, i.e., p e 2 = p 1 if p 1 / P. Note that in the asymmetric information model, for any equilibrium price p i P, consumers have to have beliefs that satisfy p e j = p i. The reason is simple: if equilibrium is symmetric, and consumers observe one (or possible many) equilibrium prices, they have no other choice but to believe that the firm they visited first has chosen a price according to the equilibrium strategy, update their beliefs about cost and thus believe the other firm has the same cost and charges the same price. Again, there are in principle no restrictions on beliefs after observing out-of-equilibrium prices. The problem with passive beliefs is, however, that it is typically not clear what passive beliefs mean in this context. In equilibrium, all prices in P may be observed and if another price is observed upon the first search, then passive beliefs are only meaningful in a pooling equilibrium where P is a singleton. We therefore assume that out-of-equilibrium beliefs are similar to equilibrium beliefs and are symmetric: p e j = p i. In the model where retailers cost are set by an upstream firm and the decision of this upstream firm is not observed by the consumer, a symmetric equilibrium is defined in a similar way. Of course, the equilibrium for this setting has to introduce the manufacturer as a separate player. In addition, we can have both passive and symmetric out-of-equilibrium beliefs as 8

consumers may either blame the retailer that was visited for having deviated (assuming the wholesale price is at its equilibrium level) resulting in passive beliefs, or they may blame the manufacturer (assuming the retailers choose their equilibrium strategies and simply react to a change in the wholesale price) resulting in symmetric beliefs. Definition 3. A symmetric perfect Bayesian equilibrium of the search model with vertical relations is a wholesale price c, a retail pricing strategy p (c) and a reservation utility w (p 1 ) such that 1. the manufacturer chooses c so as to maximize its profit given the pricing strategy p (c) of the retailers and consumers reservation utility w (p 1 ); 2. each retailer i chooses p i (c) = p (c) to maximize its expected profit given the pricing strategy p (c) of the other retailer and consumers reservation utility w (p 1 ); 3. Consumers follow an optimal search rule given their beliefs and the match value v i and the price p i they observe at firm i; 4. Consumers beliefs are updated using Bayes Rule when possible, i.e., whenever they observe p 1 = p (c ) at their first search, they believe that the other firm has also set a price p e 2 = p (c ). Out-of-equilibrium beliefs p e 2 of the price set by the firm that is not yet visited are either i. (passive beliefs) such that p e 2 = p (c ) if p 1 p (c ), or ii. (symmetric beliefs) such that p e 2 = p 1 if p 1 p (c ). When, as in the benchmark vertical relations model, the consumer observes the common retailer s cost, the consumer search strategy can depend on this cost and the expected utility of searching another firm and is denoted by w (p, c). Other aspects of the equilibrium definition remain the same. 3 The Wolinsky Model The reference point for the environments we consider in this paper is the Wolinsky model where two retailers have the same cost c. In this section, we reconsider the Wolinsky model, show how under passive beliefs the market size effect comes about and then analyze the implications of symmetric beliefs. In order to find the symmetric equilibrium price p for the Wolinsky model under passive beliefs, we use (1) and adopt it to the Wolinsky model. In particular, we then have that firm 2 charges the equilibrium price, and consumers who visit firm 1 or firm 2 first also expect the other firm to charge the equilibrium price no matter what the price is that the consumer observes, so p 2 = p e 2 = pe 1 = p. When the equilibrium price satisfies w p, expected demand for firm 1 then simplifies to: Q 1 = (1 G(w p + p 1 ))(1 + G(w )) + 2 9 w p +p 1 p 1 G(p p 1 + v)g(v) dv.

The first order condition for firm 1 s profit maximization, along with the equilibrium condition p 1 = p gives then the following price setting rule for both retailers: p 1 G(p ) 2 (c) = c + 2 w p g(v) 2 dv + 2G(p )g(p ) + (1 G(w ))g(w ). (2) where we note that p is an increasing function of c. Anderson and Renault (1999) show that the equilibrium price is increasing in search cost s. For large enough s, however, p > w and no consumer will choose to search beyond the first firm. Each firm is then a single-good monopolist that faces demand 1 G(p). It is easy to see that the resulting profit (p c)(1 G(p)) is maximized at the price p m that solves p m (c) = c + 1 G(pm ) g(p m. (3) ) The threshold search cost level s is such that w = p m at which point (3) and (2) coincide. Formally, s solves v p m (v p m )g(v) dv = s. (4) So for s s, the equilibrium price solves (2), for s > s the equilibrium price equals the monopoly price p m (c). Figure 1 shows that the equilibrium price is continuous in search cost s and strictly increasing for small s and constant for s > s. Note that after s crosses the threshold value s equilibrium demand decreases discretely. For s > s, consumers only buy when their match value at the first firm is large enough so that the market size equals 1 G(p m ). For smaller s, consumers will search on and total demand equals (1 G(p)) (1 + G(p)). As equilibrium price is continuous in search cost, the market size (and thus firms profits) in equilibrium is discontinuous at s = s. Figure 1 also displays this discontinuous jump in market size. s **** *** INSERT FIGURE 1 HERE: Wolinsky passive beliefs prices and demand as function of On may wonder why retailers cannot prevent the market size from decreasing discretely by abstaining from increasing their prices above w ( s) (the reservation utility corresponding to s)? The answer is the following. Once s > s, even though both firms would prefer to collectively charge p 1 = p 2 = w (s), individually they have an incentive to increase their price to a level above w (s). This happens because once consumers believe prices do not exceed w they search, and thus firms have the incentive to increase prices to the levels given by 2, but at such prices consumers should not search, and thus in equilibrium both firms charge p m (c) and no consumer searches for s > s. Note that this explanation for the discrete fall in equilibrium market demand is related to the Diamond paradox (Diamond, 1971), but as the first search is costless the market does not break down here. As is known in the search literature (see, e.g., Wolinsky (1986)), note also that when the first search is costly, instead of having a discrete fall in market size, the market would break down completely for high search cost. The reason is that when consumers expect prices to be larger than their reservation utilities, they will not search in the first place if the first search is costly. 10

Symmetric Beliefs In the previous sections we have argued that consumers could also have symmetric out-ofequilibrium beliefs, i.e., if consumers that have visited firm 1 observe it has deviated to p 1 p they expect firm 2 has deviated to the same price. We now proceed to constructing a symmetric equilibrium p(c) under symmetric beliefs, where to make the transition to the next section easier we already take into account that the equilibrium price may depend on cost (which is here known to consumers). For low enough s, so that p(c) < w, expected demand for firm 1 which charges p 1, while firm 2 charges the equilibrium price p(c) and all consumers expect the firm they have not visited first to charge the price they observed at the firm they just visited is given by: + w Q 1 (p 1 )G(w p(c) + p 1 ) + w p(c)+p 1 p 1 G( p(c) p 1 + v)g(v)dv p 1 G( p(c) p 1 + v)g(v)dv. (5) The first-order condition for profit maximization given the demand in (5), is then given by 1 G( p(c)) 2 p(c) = c + 2 w p g(v) 2 dv + 2G( p(c))g( p(c)). (6) Recall that (6) is derived under the assumption that p(c) < w. As w is decreasing in s, for large enough s we will have p(c) = w and p(c) = w = c + 1 G( p(c))2 2g( p(c))g( p(c)). (7) It is not difficult to see that this price actually is the profit-maximizing price for a monopolist selling both products. Such a monopolist would set p 1 = p 2 = p (NEEDS TO BE PROVEN) and maximize (1 G(p) 2 )(p c), which is maximized at p jm that solves p jm (c) = c + 1 G(pjm ) 2 2g(p jm )G(p jm ). (8) This price is always higher than the single-product monopoly price p m because the joint profit maximization takes into account that some consumers who do not buy product 1 will buy product 2. Thus for some value of the search cost, the two firms set prices that are higher than the single-good monopoly price and as high as the price of a multi-good monopolist! So the largest search cost at which our solution in (6) still holds is where w (s) = p jm. Denote by s the search cost that solves v p jm (v p jm )g(v) dv = s. (9) Then for s s, the equilibrium price is given by (6). As p jm > p m, it is clear that s < s. What about equilibrium prices for s > s? First, note that in this case w < p(c). This means that whatever the symmetric equilibrium price, it cannot be smaller than w because in 11

that case each firm wants to deviate to a larger price (as the symmetric equilibrium price is still given by p(c)), so they will at least charge w if consumers still search. Can it be the case that firms charge equilibrium prices above w? In this case consumers do not search and firm 1 s demand is the monopoly demand 1 G(p). For this demand profit is maximized at p m. Now depending on whether w is above or below p m we will have two cases. First, consider w p m, which happens when s < s < s. In this case firms will never want to charge any price above w, because they have an incentive to charge at most w. So the only possible candidate for equilibrium is p 1 = p 2 = w. We know that deviations to higher prices, where the profit function equals the monopoly profit function, are not profitable. Deviations to lower prices require to look at the quantity sold by the deviating firm. This quantity is given by (5) since both prices are at most w. In this case we know that profit is increasing in p 1 because the symmetric solution to the FOC is p that is larger than w. Thus, in this case p(c) = w and as w is decreasing in s, we have that the equilibrium price is decreasing in search cost. Now consider the case where w < p m, which happens when s > s. Here, for the same reasons as explained in the Wolinsky model, even though firms suffer from a discrete drop in demand, they cannot stop themselves from deviating upwards to prices larger than w when consumers do search, and thus consumers stop searching and the equilibrium price becomes p m. Thus we have the following proposition. Proposition 1. Under symmetric beliefs, the Wolinsky model has a unique equilibrium where the retailers price p(c) is implicitly given by (6) for s < s, is equal to w for s s s, and is equal to p m (c) for s < s. Figure 2 illustrates for a given c how the equilibrium price varies with search cost in the Wolinsky model under symmetric beliefs. For small search cost prices first rise, until the joint monopoly price, and then fall in s until price equals the monopoly price, and price is then constant for even larger values of s. Note that the non-monotonicity of the equilibrium price in search cost, does not depend on whether or not the first search is costly. With costly first visits, the monopoly pricing part of the Figure would disappear as the market breaks down (as before), but the part of the equilibrium construction where s < s < s would not be affected as expecting these prices, consumers would still consider making a first search, and given that consumers search, firms do not have an incentive to charge higher prices. 4 Asymmetric Information about common cost In this Section we study an environment where firms have a common cost that is known to the firms, but unknown (and uncertain) to consumers. The effect of common cost uncertainty has been studied in the context of the Stahl (1989) model (see, e.g., Dana (1994), Tappata (2009) and Janssen et al. (2011)), but has not been incorporated in the Wolinsky model. The timing of the model and the equilibrium concept we use has been described in Section 2. As explained in that section, a symmetric equilibrium p (c) is such that if consumers observe a price p on the equilibrium path, then they have to update their beliefs about cost in such a way that they 12

p w p jm (c) p(c) p m (c) p (c) p d (c) s s s Figure 1: Equilibrium prices in Wolinsky (red) and random cost (blue) models for equal c. Dashed line depicts w while p d is the oligopoly price when consumers know both prices and their valuations for both products. believe that the other firm charges the same price p. Thus, along the equilibrium path, the equilibrium notion requires that consumers have symmetric beliefs. As explained in Section 2, we will restrict ourselves to out-of-equilibrium beliefs that are also symmetric. To characterize the possible equilibria in this game, we first show how the equilibrium price p(c) in the Wolinsky model with symmetric beliefs (but without asymmetric information about cost) varies with c. Subsequently, we characterize the equilibrium of the asymmetric information model. Let p(c) and p m (c) be given as in (6), respectively (3). It is clear that p(c) and p m (c) are increasing without bound. Thus, for any given s, there exist c and c such that p( c) = w (s) and p m ( c) = w (s). Moreover, as for any given c it is the case that p(c) = w (s) when p(c) = p jm (c) > p m (c), it follows that c < c. Figure 3 then summarizes the dependence of the equilibrium price of the Wolinsky model with symmetric beliefs on c. For c < c, the equilibrium price is given by p(c) which is increasing in c. For c < c < c, the equilibrium price is equal to w (s), which does not depend on c. Finally, for c > c the equilibrium price is given by p m (c) which is again increasing in c. *** INSERT FIGURE EQUILIBRIUM PRICE DEPENDENCE ON C ******* As these prices are derived under the condition of symmetric beliefs and the equilibrium of the asymmetric information model is defined under symmetric beliefs, the next proposition characterizing the equilibrium of the asymmetric information model immediately follows: Proposition 2. For any given s, let c and c be defined as above. Then, depending on the range of cost uncertainty [c, c] the equilibrium of the asymmetric information model takes one of the following forms: a. If, c < c the equilibrium is fully separating and the equilibrium prices are given by (6). 13

b. If, c < c < c, then the equilibrium is partially separating with equilibrium prices being given by (6) for c < c, while for c > c c the equilibrium price is independent of c and equal to w (s); If, in addition c > c, then for cost levels c > c, the equilibrium price is given by (3). c. If, c < c < c < c, then the equilibrium is pooling with equilibrium prices being given by w (s); d. If, c < c < c, then the equilibrium is partially separating with equilibrium prices being equal to w (s) for c < c < c, while for c < c the equilibrium price is given by (3); If, in addition c < c, then for cost levels c < c, the equilibrium price is given by (6). e. If, c > c the equilibrium is fully separating and the equilibrium prices are given by (3) The equilibrium nature can be fully understood by looking at Figure 3 As that figure has increasing and flat parts, the equilibrium can be partially or fully separating and also pooling depending on how cost uncertainty interacts with the search cost s. For instance, if the exogenous cost uncertainty is small, the equilibrium is likely to be fully separating or fully pooling, whereas if the range of cost uncertainty becomes larger, it will be partially separating. 4 It is interesting to relate some aspects of this equilibrium characterization to the two independent literatures: (i) the literature on sticky prices and (ii) the literature explaining sales. There is a large literature trying to explain why firms do not adjust their prices to variations in their cost. SHOULD WE SAY MORE ON THIS: IF SO; ALSO IN ABSTRACT; INTRO (AND I ALREADY DID CONCLUSION). Following the seminal article by Varian (1980), the literature on sales provides an explanation for the common observation that retailers often follow a pricing strategy for their products that is characterized by a regular price most of the time and a sales price at random moments in time, where the discount given on the regular price is also subject to large variations (see, also, e.g., Narasimhan (1988)). This literature explains this phenomenon by an asymmetric mixed strategy equilibrium with a mass point for the regular price and a continuous price distribution for the sales prices. From the viewpoint of the consumers (or the empirical economist who does not have data on cost, but only on retail prices) the equilibrium under b) in the Proposition can also be characterized as an equilibrium with sales. For many cost levels, firms (regular) price is independent of cost, but when cost becomes low enough, the retailers give a discount. When cost is unobserved by the empirical economist, the firm acts as if it is giving a random price promotion, but in fact (in our model) this sales arise as the realization of a pure strategy that is symmetric across firms. WE SHOULD BE CAREFUL HERE. IN OUR MODEL ALL FIRMS REDUCE PRICES, SO THAT AN ECONOMETRICIAN CAN OBSERVE (AND THEY RARELY TO I THINK). BUT, IT LOOKS LIKE A PERIODIC PRICE WAR, AND THERE S SOME DYNAMIC COL- LUSION LITERATURE ON THAT. Empirically, Pesendorfer (2002) finds evidence for the fact that a firm has a regular price and then provides discounts at irregular points in time. Note, however, that in line with d) in 4 If the first search would not be for free, then the equilibrium characterization would be similar. the only difference would be that at cost levels c > c, the market would break down. 14

our Proposition, a firm my also have a regular price with irregular price movements going up and down. This is a pricing pattern that is found in Hosken and Reiffen (2004). 5 Vertical Industry Structure We now turn to the environment where retailers cost is determined by an upstream firm and consider how consumer beliefs and the market size effect influence the pricing decisions of the upstream firm. We also consider the impact on upstream and downstream profits. we start with a relatively straightforward adaptation of the Wolinsky model as our benchmark analysis for the vertical industry structure. In the benchmark, we have an upstream firm choosing the wholesale price, retailers and consumers observing this wholesale price and consumers having passive beliefs when searching. the benchmark model thus simply adds an upstream layer to the standard Wolinsky model. The market size effect already plays an important role in the benchmark model. We then consider a model where consumers do not observe the wholesale arrangement between manufacturer and retailers and have symmetric beliefs, i.e., if they observe a retail price that is different from the one they had anticipated in equilibrium, they think that it is the manufacturer that has deviated so that the second retail firm they have not yet visited will charge the same price as the price they have observed at their first search. 5.1 The benchmark vertical model For any c that the upstream firm may charge, we know from Section 3 what happens downstream. This is because when consumers observed the choice of the upstream firm and have passive beliefs, any deviation from the retail price consumers expect to see in equilibrium is interpreted as a unilateral deviation by that retailer. Thus, for search cost and upstream price such that s < s(c) we have that prices are given by (2), while for larger search cost the downstream price is p m (c). Given this behavior downstream, we consider the manufacturer s optimal choice. One important consideration is that (downstream and upstream) demand is discontinuous at a critical level of search cost, which we have denoted in Section 3 by s. Note, however, that as s is determined by p m (c) this critical level of search cost depends on c. In Section 3, we have not made that dependance explicit as c was exogeneously given. As the manufacturer chooses the retail production cost c we now make that dependance explicit and write s(c). It is clear from definition (4) that s(c) is declining in c. Thus, for a given s if the manufacturer chooses c low enough, s < s(c), (upstream) demand is given by 1 G(p) 2, while for higher c demand falls. If the first search is for free demand falls to to 1 G(p) as consumers stop searching beyond the first retailer. If the first search is costly demand drops to 0 as consumers do not visit any firm. As the manufacturer chooses c, it has an incentive to accommodate larger search cost by reducing its price to prevent the fall in demand due to downstream prices becoming larger than w. To solve the model, let us start with the case where s is relatively small, so the above consideration does not apply. Then both retailers react to a choice of c by setting p (c) as in the Wolinsky model with passive beliefs. The manufacturer s demand consists of the demand at both 15

retailers, and consumers buy at one of the two retail firms if, and only if, max(v 1, v 2 ) > p (c), and thus buy with probability 1 G(p (c)) 2. The upstream firm maximizes (1 G(p (c)) 2 )c, and the optimal wholesale price (retail cost) c o when consumers observe this price is given by: c o = 1 G(p ) 2 1 2G(p )g(p ) p c. (10) The derivative p c (co ) can be derived from (2). This price remains the optimal price for any s where w is not binding and the resulting p (c o ) is smaller than w. Next, we consider such search cost s that for the upstream price that solves (10) we have p (c o ) > w. In this case consumer would not search, and thus the optimal price does not equal (10). In fact, downstream firms would revert to pricing at the monopoly price (corresponding to c) and upstream profits would suffer a discrete decline because consumers search less and buy less. This means that the optimal upstream price has to be such that p m (c) = w. In this case consumers (just barely) prefer to search, and downstream firms charge p = w. Since as s increases, w falls, this accommodation of downstream search by the upstream firm cannot continue forever. At some search cost, if the first search is for free it will become profitable to let consumers stop searching, and charge the optimal price given that downstream firms charge a monopoly price for that c. From that moment on the equilibrium characterization is independent of s and we are in the world on classical double marginalisation prices. Formally, let us define s o as the search cost such that at the wholesale price c o that solves (10) we have p (c o, s o ) = w (s o ). Note that as the Wolinsky price is increasing in search cost and the reservation utility is decreasing in search cost, s o is uniquely defined. Also, let c m denote the upstream price that maximizes (1 G(p m (c)))c. It follows that c m is given by c m = 1 G(pm (c m )). g(p m (c m )) pm c This is the wholesale price that would be set in the classic double marginalisation problem. Finally, let s o denote the search cost such that (1 G(w ( s o )) 2 )c = (1 G(p m (c m )))c m where c is such that downstream firms charge w. Charging such a c or charging c m is equally profitable at s o. Now we can formally describe the equilibrium characterization. Proposition 3. In case consumers observe the price set by the upstream firm, the equilibrium is given by: (i) upstream price is given by (10) and downstream prices are equal to p (c) if s < s o ; (ii) upstream price solves p m (c) = w and downstream prices are equal to w for s o s s o ; (iii) upstream price is c m and downstream prices are p m (c m ) for s > s o. SHOULD WE FORMALLY PROVE IN AN APPENDIX THAT s o s o AND THAT c m > 16