Competitive Pricing with Product Differentiation and Ordered Consumer Search

Transcription

1 Competitive Pricing with Product Differentiation and Ordered Consumer Search Jian Shen October 21, 213 Abstract The Internet has made price search a lot easier; consequently, consumers are now mainly searching for the best match. I incorporate this change into a sequential search model by focusing on consumers product search behaviors with the price information being costlessly observed. Since consumers now use the price information to determine the product search order, firms set prices strategically to influence their ranking decisions. I also assume consumers are endowed with ex-ante "brand" preferences before search. Depending on the shape of the preference distribution, a decrease in the search cost can lead to an increase or decrease of the equilibrium price, which is in sharp contrast to previous theoretical predictions. Hence, the impact of the Internet on product prices is ambiguous. I further investigate the welfare impacts of search costs. When there are enough, but not too many, brand-neutral consumers, both the consumer and total surplus can decline as search costs decrease. This result is driven by the possibility that a lower search cost can reduce the market coverage, since firms concentrate on serving consumers with better matches. Keywords: Product search, ordered search, search costs, brand preferences, the Internet. JEL: D43, D83, L13, L86 I am grateful to Dan Levin, Lixin Ye, and Huanxing Yang for their advice, guidance and comments. I would also like to thank Jim Peck, Jingfeng Lu, Matt Lewis, Andrzej Skrzypacz, and participants of the NASM(USC, 213), Midwest Economic Theory Conference(MSU, 213), MEA Annual Conference(Columbus, 213), Oligo workshop(corvinus University of Budapest, 213), and the micro lunch at OSU (212) for their helpful comments and suggestions. All remaining errors are mine. Department of Economics, The Ohio State University, 41 Arps Hall, 1945 North High Street, Columbus, OH address: shen.179@osu.edu 1

2 1 INTRODUCTION The Internet is changing consumers search behavior. Before the birth of the Web, consumers spent most of the time searching for price information or the presence of a particular product. With the developments of powerful web search engines, such information becomes a lot easier to find, and today what consumers are mainly searching for is the details of a product. As Brynjolfsson et al. (21) finds in an online book market, consumers face significant search costs and they put relatively more weight on non-price factors when searching on a shopbot. Considering books are a nearly homogeneous good, their findings are quite surprising. In a differentiated products market, like automobiles, we should expect to see similar patterns as well. However, such a significant change in consumers search behavior is not properly reflected in the existing search literature, because consumers are always assumed to learn both the price and how much they like the product at the same time. It is therefore necessary to incorporate this trend into a search model and study what impacts this change will cause. To this end, I set up a simple sequential search model in which consumers are mainly searching for how well the product fits their tastes or needs, the match value 1, while the price information is costless to observe. Therefore, in my model, price becomes the determining factor in the order of consumers search for a particular product. Firms now care about the search order and set prices strategically to influence consumers ranking decisions. I call this effect the order effect, which is not present in the current literature and has a big impact on the market equilibrium as well as the social welfare. The separation of the product search from the price search highlights this new effect which firms must take into account when making their pricing decisions. Meanwhile, the magnitude of this effect depends on some other factors, one of which being the pre-search preferences. For example, when an Apple product user needs to buy a new smartphone, he or she is more likely to try the new iphone before the Android phones. This order will not be changed by a moderate price cut from an Android phone manufacturer. To capture such preferences, I study my search model in a spatial competition context. Each consumer is endowed with ex-ante preference for firms, which can be interpreted as brand preference. With this modeling, I can avoid marginalcost pricing and focus on pure strategy equilibria, which provides clearer interpretations. More 1 It is worth noting that match values are idiosyncratic, meaning that they are independent across consumers and firms. Product quality is therefore not part of the match value. If quality was included in the model and was observable to the consumers, the analysis and main results in this paper will still hold. If quality was not observable, the price could be used as a signaling device, which would interfere with the effect of search costs on the equilibrium price. For this reason, quality has been intentionally excluded from my model so that it better serves the purposes of this study. 2

3 importantly, the brand preference 2 distribution greatly influences the market performance. This relationship has never been studied before; hence this article will shed some new light on the relationship between prices and search costs. The first major result in my research therefore concerns the impact of search costs on the equilibrium price. Standard economic intuition suggests that prices would be higher when it becomes more difficult for consumers to search, which boils down to the well-known lock-in effect: consumers search less when search costs are increased. However, the order effect in my model provides an opposite prediction: when search costs are higher, consumers tend to stay with their initial picks and it becomes more important for firms to attract consumers to visit their stores first, which leads to lower prices. In which direction the equilibrium price responds to a decrease in search cost depends on whether the lock-in effect or the order effect dominates. In my model the brand preference distribution is the determining factor. If most consumers are brand neutral, the order effect will dominate because these consumers search decisions are mostly affected by price changes. As a result, the equilibrium price will increase if the search cost is lower. In contrast, if most consumers have strong brand preferences, the lock-in effect becomes more pronounced. The equilibrium price then will exhibit an inverted-u pattern as the search cost falls gradually: the price will first increase and then decrease. Suppose that initially the search cost is so large that a change in it will affect only those brand-neutral consumers search decisions. Firms therefore concentrate on these consumers, and the situation is very similar to the case when there are a lot of brand-neutral consumers. The order effect thus dominates and the price increases following a search cost reduction. As the search cost continues to decrease, more and more consumers will consider another search, and at some point even consumers with strong brand preferences may try their luck and visit other firms. However, since these consumers are loyal to their favorite brands, it is easy for firms to re-capture them by slightly reducing the price. Given that most consumers are of this type, the lock-in effect becomes dominant and the equilibrium price decreases when the search cost is reduced. Moraga- González et al. (212) studies the Dutch automobile market and is the only empirical paper that specifically looks at the product search cost. Their simulation result suggests that car prices would go up if there were no search costs, which can be properly explained by my result. This price result leads us to reconsider how the Internet affects product prices. First, the Internet brings down the price search cost to a negligible level, which results in a lower price level. However, the size of the impact is not uniform across markets. In particular, I show that firms in markets requiring smaller search costs suffer less. Second, the Internet also reduces the product search cost. Indeed, many product characteristics can be quantified. When consumers 2 In what follows, we use brand preference and horizontal (product) differentiation interchangeably. 3

4 can find such information online, the time needed for them to physically test or inspect a product will be shortened because they already know the product quite well. Given my price result, the price drop resulting from the widespread use of the Internet will be less drastic. Moreover, a consumer s ex-ante preference may also be altered by the Internet (see, for example, Danaher et al. (23) and Shankar et al. (23), for empirical evidence). It has been recognized in the literature that advertising and social network affect consumers preferences (see Bagwell (27) and Jackson (21) for references), and the Internet is changing both elements in a revolutionary way. Besides, it is easy for people to read product reviews online, which will update their evaluations of a product before search. If the preference distribution shifts in a way that there are more brand-loyal consumers, the competition will be softened and the Internet will effectively increase product prices in some markets. Given the novel finding regarding the equilibrium price, it is important to investigate the welfare impact from a reduction in search cost. While firms usually gain from a search cost reduction, whether consumers and the whole society benefit becomes ambiguous since the prices are higher in equilibrium. My analysis shows that the welfare impact of the search cost also heavily depends on the shape of the brand preference distribution. In particular, when consumers search decisions are not constrained by the search cost 3, the welfare level monotonically increases with the search cost if there are enough, but not too many, brand-neutral consumers. In my model, the search cost therefore is more than just the friction in the market and it offers at least the following two benefits to the whole society. First, it intensifies price competition through the order effect, and the resulting lower price increases the market coverage. Second, it reduces the negative externality occurred among consumers. Indeed, after the initial search, two types of consumers emerge: those who have received positive utilities and those who have received negative utilities. Since the unfortunate consumers will continue to search irrespective of the size of search cost, a change in the search cost affects only the fortunate consumers. When the search cost is reduced, the fortunate consumers search more as they expect to get higher gains, which supports higher market prices and imposes negative externalities on the unfortunate consumers. The society as a whole is then worse off. A higher search cost thus reduces such negative externalities and improves social welfare. This paper is the first one that highlights the important differences between product search and price search; hence, it will add to the vast literature on consumers searching for variety of goods. Starting from Wolinsky (1986), there has been a number of variants of his model that study various phenomena. Bakos (1997) looks at several different models to study the implica- 3 It means that the search cost is low enough so that all consumers can afford multiple searches if they want to get better matches. 4

5 tions for electronic marketplaces. Anderson and Renault (1999) investigates the effect of product diversity on prices, and later they consider situations in which a fraction of consumers are aware of the match values but not the prices in Anderson and Renault (2). In an alternative scenario of his sequential procurement model, Wolinsky (25) considers the possibility that a price quote has no cost for the consumers. He finds that the equilibrium outcome is not efficient; a parametric example shows that the equilibrium price either increases with search costs or is zero. More recently, Bar-Isaac et al. (212) and Yang (213) both focus on explaining the long-tail phenomenon. None of these papers consider the impacts of search order or the effects of ex-ante product differentiation. My paper is closely related to the growing literature on ordered search. Arbatskaya (27) is the first in this strand, who studies a sequential search model in which the search order is predetermined. Zhou (211) adds product fitness and finds that prices rise in the order of search. Armstrong et al. (29) explores the effect of prominence in a market where all consumers will visit the prominent firm first. In Wilson (21), the order of search is endogenously determined by the size of each firm s search cost, while Haan and Moraga-González (211) considers the effects of advertising in which consumers first visit the firm that advertises most. In this research, I model the search order as a response to both the prices and ex-ante preferences. There are two papers on homogeneous goods predicting that prices can decrease with search costs. Janssen et al. (25) modifies Stahl (1989) s model to allow for partial consumer participation and concludes that expected price decreases in search cost. Moraga-González et al. (213) considers the dispersion of search costs and finds that higher search costs can result in lower prices when the search cost distribution has a decreasing elasticity with respect to the parameter that shifts the distribution. In essence, both results are driven by the possibility that higher search costs may deter consumer participation. I show that the price may still decrease with search cost even when all consumers participate in the market due to the presence of the order effect. In the next section, I lay out the structure of the model, and the consumer behavior is derived in Section 3. I briefly study the random search model in a spatial competition context in Section 4 for comparisons. The ordered search model is studied in detail in Section 5. Discussions and extensions are contained in Section 6, which concludes the paper. All proofs are moved to the Appendix. 5

6 2 THE MODEL Consider a duopoly model in a differentiated products market, in which firm 1 and firm 2 are respectively located at the two endpoints of the classic linear city á la Hotelling. A continuum of consumers with measure one is distributed along the city. As is common in the literature, I view this Hotelling line as a product space instead of a physical space. Therefore, a consumer s location represents her brand preference: she prefers firm i s product before search if she is closer to firm i, i = 1,2. The consumers know their brand preferences before the game starts. Besides the brand preferences, a consumer also attaches a match value to the product sold by each firm. The match is learned only through costly search. I interpret the per-search cost c as an aggregate term, which includes the travel cost as well as the time and energy associated with testing the product. The match values are idiosyncratic, meaning that they are independent across consumers and firms. A consumer then can be characterized by a triple (x;ɛ 1,ɛ 2 ). Here x [,1] denotes the distance to firm 1, and ɛ i represents the match value attached to firm i s product. x and ɛ s are assumed to be independently distributed. For tractability, I place additional structures on the distribution functions: ɛ 1 and ɛ 2 are identically and independently distributed according to the uniform distribution on [, 1], and x follows the cumulative distribution function H s (x) with density h s (x). H s (x) is twice differentiable and is symmetric with respect to x = 1/2. Formally, H s (x) = 1 H s (1 x) with the density following h s (x) = h s (1 x). The parameter s characterizes the shape of the distribution and a change in it induces a rotation of the density function 4. In what follows, I focus on the region where x 1/2 as the other half is a mirror image. For s [s, s], there is a family of rotation points x s [,1/2] such that dh s (x) when x x s ds. As a result, an increase in s is equivalent to a counter-clockwise rotation of the density function. Without loss of generality, I let s = 1 represent the uniform distribution. Hence, when s > 1 the density h s (x) is a bell-shaped function meaning that more consumers are around the center, while s < 1 describes a U-shaped one meaning that more mass are around the two firms. By symmetry and differentiability, H s (1/2) = 1/2 and h s (1/2) = for s [s, s]. Given her type (x;ɛ 1,ɛ 2 ), a consumer s utility (before search cost) is u i (x) = ɛ i p i k x i + 1 if she purchases from firm i = 1,2 at price p i. Here the last term captures the total disutility a 4 I borrow this concept from Johnson and Myatt (26) and modified it to serve my purposes here. 6

7 consumer suffers when not consuming her ideal brand. 5 The parameter k > is referred to as the degree of brand preference, which measures how ex ante differentiated the market is. When k is large, the consumers suffer a lot for buying a product that they don t like, which suggests that the two firms products are quite distinct. A consumer s type (x;ɛ 1,ɛ 2 ) is her private information and is not observable to the firms, while the uniform per-search cost c is common knowledge. Each consumer demands up to one unit of the product. The outside option is normalized to zero. The marginal cost of production is constant and the same across the two firms, which is normalized to zero. After production, each firm posts its price p i, i = 1,2, simultaneously, which will be fixed for the rest of the game. In the following stage, the consumers need to decide whether to participate in the market and, if they do participate, which firm to visit first. Consumers have the option to also search the second firm. Hence, after the first search consumers can buy at the firm, visit the second firm, or simply exit the market without making a purchase. Search is with perfect recall, meaning that consumers can always go back to the first firm without paying the search cost c again. Following the literature (e.g. Wolinsky (1986) and Robert and Stahl (1993)), I assume the consumers must search to buy. I focus on symmetric equilibria in which the two firms follow the same pricing strategy. In this paper, I consider the model under two different information structures. The random search model resembles the classic model, in which the price information and the match value are bundled and therefore will be discovered at the same time. The search order is then exogenously given and consumers search randomly. In the ordered search model, the two pieces of information are separated, and the cost of finding out the prices is negligible, which I assume to be zero for simplicity. In essence, the price information becomes transparent, and consumers search based on the actual prices charged by the firms. The ordered search case is rarely addressed in the literature and is the focus of this research. A comparison between the two models is helpful to find out the real impact of the Internet on product prices. 3 MARKET DEMAND In this section, I study consumers search and purchasing behaviors and then derive the demand function for firms. The derivations are very similar regardless of the information consumers can observe because in equilibrium they either observe or correctly anticipate the prices. I first look at consumers participation decisions. A rational consumer considers not only the 5 The disutility is sometimes referred to as the transport cost, which should not be confused with the search cost. The disutility is incurred if and only if the consumer makes the purchase, while the search cost arises whenever the consumer conducts a search at a new firm. 7

8 first search outcome but also the possibility for finding a better match from the other firm when making her participation decision. In the ordered search case, consumers can see the prices p 1 and p 2. In the random search case, consumers do not observe the prices, but they form a belief about the equilibrium price p E. Therefore, p 1 = p 2 = p E in the case where prices are not observable. Next I calculate the expected utility from participation for the consumers. If a consumer decides to participate, she will choose to start with the firm that provides her the higher total expected utility. Since match values are independent and identically distributed in my model, consumers will first visit the firm closer to their ideal spots. Formally define ˆx = p 2 p 1, and consumers will visit firm 1 first if x ˆx and firm 2 otherwise. 6 In what follows I focus on the consumer with x ˆx. By symmetry, consumers on the other half behave similarly. After paying a visit to firm 1, the consumer learns the match value ɛ 1 (and the actual price p 1 when it is not observable). She then needs to select one from the following three options: to buy directly at firm 1, to continue the search at firm 2, or to quit, depending on her match value from firm 1. If ɛ 1 p 1 kx <, the consumer gets negative utility from firm 1. She then either stops and exits the market or continues her search. The utility she derives in each case is given by: u 1 (x) = c 1 p 2 +k(1 x) (ɛ 2 p 2 k(1 x)) dɛ 2 2c if the consumer stops; if the consumer continues to search. If ɛ 1 p 1 kx, the consumer has the same options except that she will buy at firm 1 if she stops: ɛ 1 p 1 kx c if the consumer purchases; u 1 (x) = 1 ɛ 1 +p 2 p 1 +k(1 2x) (ɛ 2 p 2 k(1 x)) dɛ 2 + ɛ 1 +p 2 p 1 +k(1 2x) (ɛ 1 p 1 kx)dɛ 2 2c if the consumer continues to search. I introduce a new parameter, a, termed the reservation value, that satisfies the following equation: 1 a ɛ adɛ = c. We see that a is uniquely determined by c and is decreasing with c on the unit interval [,1]. The LHS represents the expected gain from another search and a is a threshold of the current 6 After deriving the expected utilities, it is not difficult to see that ˆx is the right threshold dividing the consumers. 8

9 best offer in the following sense: if one s current offer is lower than a, it is more likely for her to benefit from a second search. Therefore, a smaller a implies a higher bar and that there are less searches on the market. In what follows, for convenience I will use a as the (inverse) proxy for the search cost c. In the case of ɛ 1 p 1 kx <, the best offer the consumer has is the outside option:. She thus will continue the search if and only if which is equivalent to 1 p 2 +k(1 x) (ɛ 2 p 2 k(1 x)) dɛ 2 c, x x 1 1 a p 2. k Therefore, consumers who are close enough to firm 1 will give up their opportunities to search twice because they are too far away from firm 2. When ɛ 1 p 1 kx, the consumer receives a positive utility from firm 1, and the condition for her to visit firm 2 is given by: 1 ɛ 1 +p 2 p 1 +k(1 2x) {ɛ 2 [ɛ 1 + p 2 p 1 + k(1 2x)]} dɛ 2 c, which can be simplified as ɛ 1 a p 2 + p 1 k(1 2x). We can see that as the consumer is located further from firm 1, she needs a higher match value to stay with the firm since her expected gain from another search improves. Combining the nonnegative-utility requirement, I derive the following cutoff for x: x 1 a p 2 k so that both inequalities hold. Note that this cutoff coincides with x 1 derived in the previous case. Therefore, only consumers with an x x 1 will ever search twice. A similar cutoff x 2 = (a p 1 )/k can be found on the other half of the city. Even though the prices charged by the two stores may be different, x 1 and x 2 are symmetrically located around ˆx, as long as they are within the unit interval. I use the following Figure 1 to better illustrate my demand analysis. This figure illustrates the case when both firms post the same price. The horizontal axis represents the Hotelling city, while the match values each consumer may get from the two firms are denoted on the two vertical axes, respectively. The vertical dashed line x = ˆx separates the consumers visiting firm 1 first from those visiting firm 2 first. The solid line, ɛ 1 = p 1 +kx, represents 9

10 Figure 1: An Illustration of Consumers Choices the zero-utility condition, and the dashed line, ɛ 1 = a p 2 + p 1 k(1 2x), determines whether the consumers want to continue the searching conditional on them receiving nonnegative utilities. Note that the dashed line always hits ɛ 1 = a when evaluated at ˆx regardless of the prices charged by the two firms. These two lines combined with x = x 1 divide the left half space into four regions. In region I, consumers get a high enough match value to buy right away at firm 1. In region II, though receiving positive utilities, these consumers still want to try their luck at firm 2. In region III, consumers are getting negative utilities from firm 1, but they will continue the search because firm 2 is not too far away. In region IV consumers get low match values from firm 1, but they choose to exist the market instead of visiting firm 2. It is worth noting from Figure 1 that changes in the reservation value a can cause x 1 to be less than or greater than ˆx. If x 1 > ˆx, that is a < k/2 + (p 1 + p 2 )/2, both regions II and III disappear and no one in the market will ever search twice. A gap will emerge between firm 1 s market coverage and firm 2 s and consumers from the city center will not participate at all. Each seller then effectively becomes a local monopolist, which is not of interest here. 7 As a result, I restrict the attention to situations in which a k/2 + p in equilibrium, where p denotes the symmetric equilibrium price. On the other hand, if a k + p 2 so that x 1, then everyone getting a negative utility from firm 1 will continue to search since the search cost is relatively 7 Anderson and Renault (26) studies the monopoly case in detail with a focus on search and advertising. 1

11 low (that is, region IV disappears). The demand functions are different in these two cases, which greatly affect the firms pricing strategies. I will therefore divide the discussion according to the value of a. Without loss of generality, I assume a k for the rest of the paper so that both cases can occur in my model. 8 Now we are ready to calculate the expected utility for each consumer. When a k + p 2, for a consumer who is located at x < x 1, her expected utility from participation can be calculated as follows: If x x 1, the expected utility is given by 1 EU 1 = (ɛ 1 p 1 kx)dɛ 1 c. (1) p 1 +kx EU 2 = 1 + (ɛ 1 p 1 kx c)dɛ 1 a p 2 +p 1 k(1 2x) a p2 +p 1 k(1 2x) p 1 +kx EU (1) s2 dɛ 1 + p1 +kx EU (2) s2 dɛ 1, (2) where the expected utility from searching firm 2 given ɛ 1 is EU s2 = EU (1) s2 = 1 ɛ 1 +p 2 p 1 +k(1 2x) (ɛ 2 p 2 k(1 x))dɛ 2 + ɛ 1 +p 2 p 1 +k(1 2x) (ɛ 1 p 1 kx)dɛ 2 2c when ɛ 1 p 1 + kx; EU (2) s2 = 1 p 2 +k(1 x) (ɛ 2 p 2 k(1 x))dɛ 2 2c when ɛ 1 < p 1 + kx. Since x x 1, continuing the search is (weakly) better than staying with firm 1, and it is easy to check that EU (1) s2 > ɛ 1 p 1 kx c and EU (2) s2 > c. Therefore the expected utility when x x 1 must be at least 1 (ɛ 1 p 1 kx)dɛ 1 c, p 1 +kx which is also the minimum for all consumers with x < ˆx. Given the fact that x < ˆx, we see that which implies that p 1 + kx < p 1 + k ˆx = k 2 + p 1 + p p 1 +kx (ɛ 1 p 1 kx)dɛ 1 c >. Therefore, participating in the market is always beneficial to every consumer, as long as the search cost is not too large so that there exists direct competition between the two firms. Note that the above analysis holds for both the random search case and the ordered search case. 8 If a < k, only the case a k + p will arise and very little will be changed. All the results will still hold. a, 11

12 Following this argument, when a k + p 2, every consumer will search at least once and full participation is ensured as well. I next derive the demand function for each firm. Figure 1 suggests that a firm s demand is composed of three different groups. First, there are consumers from region I who directly purchase at firm 1 without visiting firm 2. I call them the loyal customers. Second, for some of the consumers from region II, they continue to search firm 2 but will come back to purchase. They are called the returning customers. Finally, there are consumers whose first choice is not firm 1. However, some of them will search twice and may eventually purchase at firm 1. These consumers are called the escaping customers. The total demand is thus the sum of the demands from these three groups. Because the random search case has a different information structure, p 1 and p 2 enter the calculation of demand in a very different way from the ordered search case. When consumers cannot observe the actual prices and search randomly, the cutoff ˆx always equals 1/2 in a symmetric equilibrium. After visiting the first firm, consumers are still unaware of the other firm s actual price. Following the literature, I assume they hold the same belief as before even if the price they already discovered is not the equilibrium one. Therefore, the horizontal cutoff is x 1 = 1 (a p E )/k in this case. In contrast, in the ordered search case all the price information is transparent and consumers make decisions according to the actual prices posted by the firms. Next I derive firm 1 s demand function D 1 for each model. Firm 2 s demand is simply a mirror image. Let s start with the random search model in which consumers expect the two firms to charge the same price p E. Suppose firm 1 deviates and actually charges p 1 p E. Since the second search decision only depends on the expected prices, x 1 = 1 ( a p E) /k and x 2 = ( a p E ) /k. When a p E + k, the demand from the loyal consumers is given by x1 S 1 = (1 p 1 kx) h s (x)dx + x 1 (1 a + k(1 2x) + p E p 1 ) h s (x)dx. The first integral comes from the consumers who are very close to firm 1. They will purchase as long as they get positive utilities from firm 1. In the second term, the consumers are further away from firm 1 and have higher opportunity costs to buy directly from firm 1, meaning that they need higher match values to be loyal. firm 2. If the consumers are not satisfied with the match values they get from firm 1, they will visit demand Some of them will become returning customers, from whom we derive the following S 2 = a k(1 2x) p E +p 1 x 1 p 1 +kx (ɛ 1 + k(1 2x) p 1 + p E )dɛ 1 h s (x)dx. 12

13 S 3 = For the escaping consumers, their demand can be calculated as follows x2 1/2 t=1 x ==== x 1 [ p E +k(1 x) (1 p 1 kx) dɛ 2 + a+k(1 2x) [ p E +kt [1 p 1 k(1 t)] dɛ 2 + p E +k(1 x) a k(1 2t) p E +kt [1 ɛ 2 + k(1 2x) + p E p 1 ] dɛ 2 ] h s (x)dx [1 ɛ 2 k(1 2t) + p E p 1 ] dɛ 2 ] h s (t)dt. We see that S 3 contains two parts. The first term inside the big squared brackets comes from the consumers who receive negative utilities from firm 2. They will purchase at firm 1 as long as the firm provides positive utilities for them. In the second term, these consumers get positive utilities from firm 2 and need higher match values from firm 1 to make their switches. I change the variable here in S 3 so that the domain of integration is consistent with S 1 and S 2, which helps later derivations of the profit function and the pricing strategy. The total demand D 1 = S 1 + S 2 + S 3. When a p E + k, even consumers at x = or 1 may visit the other firm. Replacing x 1 with in D 1, we get the new demand function. Now we move on to the ordered search model in which the price information is observable. In this scenario, the cutoff ˆx = 1/2+(p 2 p 1 )/ with x 1 = 1 (a p 2 )/k and x 2 = (a p 1 )/k. When a p + k in equilibrium, the demand functions for firm 1 are given as follows: S 1 = S 2 = S 3 = t=2 ˆx x ==== x1 ˆx (1 p 1 kx)h s (x)dx + ˆx a p2 +p 1 k(1 2x) x 1 x2 p 1 +kx [ p2 +k(1 x) ˆx a p1 +p 2 +k(1 2x) + p 2 +k(1 x) ˆx p1 +kt + x 1 ˆx a p2 +p 1 k(1 2t) x 1 p 1 +kt x 1 (1 x + k a + p 2 p 1 )h s (x)dx; (ɛ 1 p 1 + p 2 + k(1 2x))dɛ 1 h s (x)dx; (1 p 1 kx) dɛ 2 [1 ɛ 2 + p 2 p 1 + k(1 2x)] dɛ 2 ] h s (x)dx [1 p 2 k(1 t)] dɛ 2 h s (2 ˆx t)dt [1 ɛ 2 + p 1 p 2 k(1 2t)] dɛ 2 h s (2 ˆx t)dt. In the case of a p + k, we need to substitute x 1 with and x 2 with 1 in the above expressions. From the demand functions, we can see the different role p 1 and p 2 play in each model. In the random search model, the actual prices can affect consumers purchase decisions only. Indeed, consumers form a belief about the market prices before searching, which will not be changed even when firms charge prices different from the belief. As a result, firms lack incentives to 13

14 lower their prices to attract consumers. On the contrary, in the ordered search model, firms can effectively influence consumers both purchase and search decisions because p 1 and p 2 enter into the calculations of ˆx and x i. This difference in the information structure changes the firms pricing strategies significantly. In the random search model, as the search cost increases, firms gain greater market powers as it is more difficult for consumers to visit a second firm. However in the ordered search model, in addition to the market power generated from the search cost, firms also need to consider how to attract consumers to their stores, which complicates their pricing decisions. In the next section, I study the equilibrium in the random search model as a baseline. The ordered search model will be discussed in detail in Section 5. 4 THE RANDOM SEARCH MODEL I use this section to illustrate that the inclusion of ex-ante brand preference will not change the prediction of standard economic intuition. This implies that the difference in the information structure is the reason for the different predictions between the classic search model and mine. The equilibrium price derived in this model increases with the search cost, which is consistent with the existing literature and comes from the absence of the order effect. To keep the analysis simple and to the point, I use the uniform distribution for H s (x) (that is s = 1). The equilibrium concept we use in this section is the symmetric oligopolist equilibrium defined in Wolinsky (1986). As mentioned before, though consumers cannot see the actual prices, in equilibrium they form correct beliefs about the prices charged by the two firms. The real price and the match value are discovered at the same time once the search cost c is paid. Given the demand functions derived from the last section, firm 1 s profit function is π 1 = p 1 D 1 (p 1 ). The equilibrium price p is then derived from the following first order condition by imposing symmetry: D 1 (p 1 ) + p 1 D 1 p 1 =. (3) I drop the superscript for the rest of the paper as long as it does not cause confusion. For convenience, let G 1 (p) D 1 / p 1 p1 =p 2 =p. For the case when a p + k, the derivative is given by D 1 p 1 = 1 2 x 1 (a k(1 2x))dx, which is negative because a k(1 2x) for x x 1. Since x 1 is independent of p 1, D 1 is linear in p 1 and thus the profit function π 1 is strictly concave in p 1. A unique equilibrium then must 14

15 exist. When a p + k, the derivative is the same as the above expression except that x 1 is replaced by : D 1 p 1 = 1 2 (a k(1 2x))dx <, and there exists a unique symmetric equilibrium in this case as well. How the equilibrium price responds to a change in the search cost will be summarized later in Proposition 1. Now the question is how the above two cases are connected when a moves continuously from k to 1. Suppose in the case of a p + k there exists some a L such that when a a L the equilibrium price p a L k, and similarly for the case of a p + k we have some a H such that when a a H the equilibrium price p a H k. In the random search case, a L = a H. To see this, note that a L and a H can be found out in condition (3) by inserting p = a k. Since D 1 / p 1 is the same in both cases when x 1, the two cutoffs derived from condition (3) will be the same as well. There is no jump in the equilibrium price when a moves from k to 1. Proposition 1. There exists a unique Nash equilibrium in the random search model. Moreover, the equilibrium price increases with the search cost c. The proof is moved to the Appendix. Basically, the effect of the search cost on the equilibrium price is reflected in d p/da, which I derive in the following equation: ( D1 (p) p +G 1 (p) + p G ) 1(p) d p p da = D 1(p) p G 1(p) a a. (4) In the proof I show that the RHS is positive while the expression in the brackets on the LHS is negative, which implies d p/da < so that the equilibrium price increases with c. Intuitively, when the cost of search is increased, it is less likely for the consumers to search on and they are locked in to one firm. Therefore, high search costs help to build up firms market powers and high prices thus can be sustained. In a differentiated products market, there exists a force opposing the lock-in effect that mitigates its influence. Indeed, an increased search cost means that consumers are less likely to find out the best match values, which lowers their willingness to pay. However, this fitting effect is not strong enough to offset the lock-in effect. In the end, the equilibrium price increases with search costs in the random search model. This result is overturned as online search technology develops. Importantly, the change in consumers search behaviors brings on a new force, the order effect, which greatly affects firms pricing strategies and market performance. In the next section I will focus on the impacts of the search cost associated with the product information and discuss what role the ex-ante brand preference distribution plays in the analysis. 15

16 5 THE ORDERED SEARCH MODEL I go back to the general distribution H s (x) in this section as it greatly influences the magnitude of each effect in the model and thus the market equilibrium as well as social welfare. The equilibrium concept used here is subgame perfect. The consumer behavior has been discussed in Section 3 and here I focus on the firms pricing strategies. 5.1 The Symmetric Equilibrium I derive the equilibrium conditions in this subsection and characterize what an equilibrium looks like. At the end of the subsection, I provide the existence result in the case when H s (x) is uniform. Given firm 2 s price p 2, firm 1 s profit function is given by π 1 (p 1 ) = p 1 D 1 (p 1 ), where D 1 (p 1 ) = S 1 + S 2 + S 3 as derived in Section 3. The equilibrium price p is then derived from the first order condition (3) when evaluated at p 1 = p 2. Again I should break the analysis into two cases according to the size of the reservation value a. First when a is relatively small, or equivalently the search cost c is relatively large, we have x 1. In equilibrium, p 1 = p 2 = p and ˆx = 1/2. Note that h s (x) is symmetric, S 3 then can be written as S 3 = = p1 +kx + [1 p 2 k(1 x)] dɛ 2 h s (1 x)dx x 1 a p2 +p 1 k(1 2x) x 1 p 1 +kt p+kx + x 1 a k(1 2x) x 1 p+kx [1 ɛ 2 + p 1 p 2 k(1 2x)] dɛ 2 h s (1 x)dt [1 p k(1 x)] dɛ 2 h s (x)dx [1 ɛ 2 k(1 2x)] dɛ 2 h s (x)dx. Firm 1 s equilibrium demand function is then given by the following simple form: D 1 = (1 p kx)h s (x)dx + x 1 [1 p k(1 x)](p + kx)h s (x)dx. (5) As for dd 1 /d p 1 in equilibrium, it is calculated as follows: dd 1 d p 1 p1 =p 2 =p = ( ds1 + ds 2 + ds 3 d p 1 d p 1 d p 1 = 1 2 h s(1/2) [ 1 ) p 1 =p 2 =p ( p + k 2 ) 2 ] + x 1 [1 p k(1 x)] h s (x)dx (6) 16

17 + [ p+kx x 1 = 1 2 h s(1/2) a k(1 2x) [1 p k(1 x)] dɛ 2 + p+kx (1 a) 2 h s( x 1 ) (1 a)(2p + k a) k ] h [1 ɛ 2 k(1 s (x) 2x)] dɛ 2 k dx x 1 (1 k(1 2x)) h s (x)dx. The last line is derived by integration by parts. Again let G 1 (p) dd 1 /d p 1 p1 =p 2 =p. The equilibrium price p is then derived from D 1 (p) + pg 1 (p) =. (7) When a p + k, the equilibrium demand D 1 and dd 1 /d p 1 are given by: G 1 (p) dd 1 d p 1 p1 =p 2 =p D 1 = (1 p kx)h s (x)dx + = 1 2 h s(1/2) + + a k(1 2x) [ ( k p (1 p k(1 x)) h s (x)dx + p+kx = 1 2 h s(1/2) (1 a) 2 (1 p k(1 x))(p + kx)h s (x)dx (8) ) 2 ] + p k (1 k p) + 1 k p+kx h s [1 ɛ 2 k(1 2x)] dɛ (x) 2 k dx a k p (1 ɛ 2 k)dɛ 2 h s (1 p k(1 x)) dɛ (x) 2 k dx (1 k(1 2x)) h s (x)dx. (9) Plugging the above two expressions in condition (7), we will obtain the equilibrium price. Note that both G 1 (p) s are less than zero as seen from (6) and (9). As in the random search model, next I look at how these two cases are connected in a unified model when a moves continuously from k to 1. Unlike the previous scenario, a gap exists between these two cases in which a corner solution arises. Suppose in the case of a p + k there exists an a L (k) such that the interior equilibrium price p(a L ) derived from condition (7) satisfies p(a L ) = a L k. Similarly, we see an a H (k) in the a p+k case such that the equilibrium price p(a H ) = a H k when a = a H (k). Since a L (k) never equals a H (k), a gap must appear between the two cases. Indeed, if we denote the LHS of condition (7) by LHS L(H) (p), in which the subscript L(H) represents the case a ( )p + k, then a L(H) is the solution to LHS L(H) (p = a k) =. Since LHS L (p = a k) = LHS H (p = a k) p h s() (1 a)(2p + k a), k we must have LHS L (p = a H k) < and LHS H (p = a L k) >. Therefore, a H (k) falls outside the range of a p + K, and vice versa. The corner solution p = a k thus arises when a falls between a L (k) and a H (k). 17

18 Whether a L < a H or a L > a H depends on the distribution H s (x) and k. Because LHS L(H) (p = a k) is a cubic function of a, it is possible that there is more than one solution to LHS L(H) (p = a k) =. However, if there is indeed only one solution on a [k,1], then a L must be strictly smaller than a H since LHS L(H) (p = a k) > at a = k and that LHS L (p = a k) lies below LHS H (p = a k). If there are multiple solutions, then as a moves from k to 1 we pass the corner solution area more than once, and a L might be greater than a H. Besides these two cutoffs a L and a H, I also need a to be large enough so that the local monopoly case would not occur. Similar to the above analysis, in the local monopoly case, D 1 = (a p)/k (1 p kx)h s (x)dx and dd 1 /d p 1 = 1/2 (1 a)h s (1/2)/k when p 1 = a k/2. Simple comparison with (6) shows that there exists an a which separates the model from the local monopoly case. I therefore require a a. However, I keep k as the lower bound of a since imposing a does not change my analysis or results. 9 Next I show the existence and uniqueness of the equilibrium for the uniform distribution. Proposition 2. When s = 1, there exists a unique symmetric equilibrium for a [k,1] 1. Specifically, there exist a L (k) (k,1] and a H (k) ( a L (k),1 ] such that: when a a L, the equilibrium price p a k; when a a H, p a k; when a (a L, a H ), p = a k. If k >.653, then only the case a a L arises. The detailed proof is involved and can be found in the Appendix. The sketch of proof has two steps. First I show that there exists only one solution to LHS L(H) (p = a k) =, meaning that a L < a H. Then I move on to show the existence and uniqueness in the interior solution cases: a a L and a a H. In the second step, the existence result is derived from Kakutani s fixed-point theorem by showing the log-concavity of the demand function and the uniqueness comes from Proposition 6 in Caplin and Nalebuff (1991). I also take care of the crazy pricing issue in the proof. For example, in the case when a a L (k) so that x 1(2) [,1] with p a k, if firm 1 prices too differently from p it is possible that x 2 exceeds the interval [ ˆx,1]. The demand function then needs to be modified, and it might not be log-concave globally. Nevertheless, I managed to show that it is still in firm 1 s interest to charge p even if such crazy pricing behaviors are allowed. Proposition 2 describes the unique symmetric equilibrium for the uniform distribution as a increases, or equivalently as the search cost c decreases. When the search cost is relatively large, x 1 and x 2 fall inside the unit interval, and some consumers who are close to the firms will exit the market without searching twice. As it gets easier to search, x 1 and x 2 move towards the boundary of the city, meaning that more consumers are willing to search twice. Finally, when 9 When a > k, what hold for a k continue to hold for a a. When k a, the local monopoly case will not arise as long as we impose a k. 1 It can be easily verified that the local monopoly case never occurs in the s = 1 case as long as a k. 18

19 the search cost becomes small enough, all the consumers in the market can afford to visit both firms. Between the partial-search case and the full-search case, there is a transition region of consumers search cost in which the equilibrium price p = a k. Interestingly, the price increases with a, meaning that it decreases as search costs get larger, and this result holds for any horizontal distribution H s (x). Most previous studies suggest that a larger search cost helps firms to build up their market powers because consumers tend to search less. However, in my ordered search model the two firms face more intense competition due to the Internet and here the conventional wisdom no longer holds. The impact actually extends to areas other than the transition region, and in the next subsection I will focus on the two interior-solution cases. For expositional purposes, I denote by a a L the case a p + k and by a a H the case a p + k. 5.2 The Equilibrium Price and Profit In this subsection, I show that a decrease in the search cost can lead to an increase or decrease of the equilibrium price. The brand preference distribution turns out to be the deciding factor for determining the direction of the price adjustment. Firms in my setting are likely to earn higher profits if the search cost is reduced, which has some implications on the business strategies observed in actual markets. In what follows, I study the a a L case first and then the a a H case. When a a L, there are some consumers who can afford only one search since x 1(2) 1. A decrease in the search cost then affects the demand in the following ways. 11 First, more consumers who could have made purchases without searching twice decide to enter the re-search market. The proportion of loyal consumers thus drops and charging a lower price seems reasonable. Second, firms now receive more patronage from consumers who visit the opponents first, which might provide cushions for an increased price. Therefore, how a firm should adjust its price becomes uncertain given that both the size and the composition of demand have changed. Meanwhile, the search order certainly becomes less important since most consumers will search twice. In my ordered search model, firms lack the incentives to use prices to attract consumers. Next I formalize the analysis and show that which force dominates in the model depends on the shape of h s (x). I further impose the following restrictions on the distribution to simplify the analysis: for x [,1/2], h s (x) when s 1 (bell-shaped) and h s (x) when s 1 (U-shaped). For this a a L case, I also drop the subscript s. 11 Recall that all the consumers will search at least once in my model, I therefore focus on their second search decisions. 19

20 The equilibrium price is given by the equilibrium condition (7): D 1 (p) + pg 1 (p) =. D 1 and G 1 are given by expressions (5) and (6), respectively. d p/da is then derived from: in which we have ( D1 p +G 1 + p G ) 1 d p p da + D 1 a + p G 1 a =, D 1 p +G 1 + p G 1 p D 1 a + p G 1 a = 1 2 h(1/2) (1 a)2 h ( x 1 ) k 2 p(2p + k a)(1 a) h( x 1) k [2(1 a)(2p + k a) + p(1 2p k)] H( x 1 ) (1 k(1 2x))h(x)dx (2p + k) x 1 [ h( x1 ) = (1 a) k (2p + k a) + h ( x 1 ) k 2 ( ) 1 2 H( x 1). (1) ] p(2p + k a) + h(1/2) p k ; (11) The absense of a closed-form solution for the equilibrium price makes the analysis difficult. Even though I have imposed additional assumptions on h(x), a complete result is still not available. I thus concentrate on deriving sufficient conditions to illustrate the point. The analysis is divided into two subcases: s 1 and s < 1. When s 1, there are (weakly) more consumers who are brand neutral in the market, and we expect to see more intense competition from the two firms. By contrast, when s < 1 many consumers have strong brand preferences, which will affect the price competition in a different way. Before moving on with the discussion of these two subcases, I first determine the sign of expression (1). Specifically, conditions are derived in the following lemma to ensure expression (1) is negative. A negative (1) suggests that p 1 / p 2 < 1 at the equilibrium point, which is a reasonable requirement. Indeed, expression (1) is equivalent to 12 It is derived as follows: ( SOC 1 p ) (11) p 2 p 1 =p 2 =p = = D 1 p +G 1 + p G 1 p [ D 1 + D 1 + D 1 2 D 1 2 ] D 1 + p 1 p 1 p 2 p 1 p 2 + p 1 p 1 1 p 2 p1 =p 2 =p 2 D1 2 D 1 + p 1 p 1 p 2 + D 1 2 D 1 + p 1 p 1 2 p 1 p 2 }{{}}{{} SOC SOC p 1 / p 2 p1 =p 2 =p 2

21 Here SOC stands for the second order condition, which must be negative in equilibrium. The sign of (1) is therefore determined by p 1 / p 2. I want to focus on the case expression (1) being negative because it seems natural that p 1 / p 2 < 1 so that a firm would not increase his price faster than the opponent. Lemma 1. When a a L, D 1 p + G 1 + p G 1 p < if the following set of conditions holds for x [,1/2]: a 3+k+ (3+k) 2 8k 8 ; h (x)a(1 a)(2a k) ( k 2 + h( 1 2 )k(1 a2 ) ) (12). Condition (12) suggests a should be relatively small and that h(x) should not be too U-shaped. Otherwise, the model may enter either the a a H case or the local monopoly case. In what follows, I will continue the analysis taking condition (12) as given. Given Lemma 1, to determine the sign of d p/da, I need to focus on expression (11) only. When s 1 we must have D 1 / a + p G 1 / a >. To see this, note that 2p + k a > since x 1 in equilibrium. Plus, by assumption, h (x) for x [,1/2]. We therefore have d p/da > in this case, meaning that the equilibrium price decreases with the search cost. 13 Intuitively, more brand-neutral consumers means a higher degree of competition between firms. As the search cost increases, there are more consumers searching only once. By lowering the price, a firm can not only attract more escaping consumers from the opponent, but more importantly also draw the attention of brand-neutral consumers. Indeed, a price drop will change these consumers minds to first visit the firm with a lower price, who will very likely to keep these consumers given the higher search cost level. The big increase in demand thus provides enough incentives for firms to lower their prices, and the result follows. The analysis is less clear when we turn our attention to the subcase s < 1, in which more consumers possess strong brand preferences, as x 1 now comes into play. When x 1 is close to 1/2, a change in the search cost affects mostly the brand-neutral consumers who are the most price-sensitive. On the other hand, when x 1 is close to, consumers who are close to the firms start to reconsider their search decisions. They form a large proportion of the total population when s < 1. Since the two types of consumers react very differently towards a price change, the equilibrium price and the search cost will exhibit an inverted-u relationship as x 1 moves from 1/2 to. Next, I will look at the two extreme cases: x 1 1/2 and x 1 and then study what will happen when in between. = SOC ( 1 p ) 1. p 2 p 1 =p 2 =p 13 For the case when s = 1, the sufficient condition (12) is redundant and the equilibrium price always decreases with the search cost. The proof is available upon request. 21