Consumer Obfuscation by a Multiproduct Firm

Transcription

1 Consumer Obfuscation by a Multiproduct Firm Vaiva Petrikaitė 13 March Abstract A multiproduct profit-maximizing firm internalizes the externalities that its products confer upon one another. As a result, the prices of a multiproduct firm, which sells substitutes, are above the prices that would be chosen if the decision maker did neglect the relationships between the products. In a market where consumers search to find satisfactory products, I show that a multiproduct firm has an incentive to obfuscate some of its products in order to weaken the externalities that they impose on each other. Additionally, by obfuscating some products, i.e. by making it costly to learn their characteristics, the seller screens out consumers better and extracts more consumer surplus. The extent to which products are obfuscated depends on whether the seller is a monopolist or faces competition. Keywords: Sequential search, obfuscation, horizontal differentiation, search costs, multiproduct firm JEL classification: D1, D43, D83 1 Introduction A profit-maximizing firm that sells an array of products must internalize the externalities which its products confer upon one another. As a result, a multiproduct firm, which sells substitutes, raises the prices of its products over and above the prices that would be chosen if the decision maker did neglect the relationships between the products. In a context where consumers must search to find satisfactory products, I demonstrate that a multiproduct firm has incentives to deliberately obfuscate some of its products, which weakens the externalities that they impose on each other. Further, by obfuscating some products, i.e. by making it costly to inspect the characteristics of Comments are welcome. I am grateful to José Luis Moraga-González, Shu Yu, Wim Siekman and Dominic Hauck for their helpful comments and suggestions, and Žanna Naidionova for her complaints about messy shops and drawing my attention to the layout of discounted items. University of Groningen, Department of Economics and Business, v.petrikaite@rug.nl 1

2 some products, the seller, in effect, screens out consumers better. Furthermore, the extent to which products are obfuscated depends on whether the seller is a monopolist or faces competition. To study product obfuscation by multiproduct firms, I adopt the random utility framework of Perloff and Salop 1985). Firstly, I consider obfuscation under monopoly. In the benchmark nonobfuscation equilibrium, consumers observe the match values and the prices of all the products and choose the most satisfactory one. When the monopolist obfuscates some of its products, consumers can observe the characteristics of those products only by incurring positive costs. Artificially creating search frictions among products turns out to be profitable. In fact, consumers who find the nonobfuscated products satisfactory enough do not enquire further about other products, which is costly, and buy one of the non-obfuscated products right away. Relative to the benchmark case of nonobfuscation, this gives the monopolist an incentive to raise the price of the non-obfuscated products. By contrast, consumers who are unsatisfied with the non-obfuscated products choose to incur the cost of inspecting the obfuscated products, which gives the monopolist an incentive to lower the price of its obfuscated products. These price adjustments result in higher profits for the monopolist. I also study the choice of optimal obfuscation strategy by a monopolist. I show that the profits of the monopolist are monotonic in obfuscation intensity, which implies that the monopolist has an incentive to continue to raise the search cost for the obfuscated products. These costs are raised until consumers which are unsatisfied with the non-obfuscated varieties are indifferent between searching the obfuscated product and leaving the market without buying anything. As a result, the monopolist can treat the unsatisfied searching consumers as a separate market and charges them the monopoly price. In addition, I show that, for a given obfuscation intensity, and keeping the total number of products fixed, the profits of the monopolist are first increasing and then decreasing in the number of obfuscated products. As a result, the profit of the monopolist is maximized at an intermediate number of obfuscated varieties. I finally consider obfuscation by a multiproduct firm that faces competition by a single-product firm. I show that the impact of obfuscation on the profits of both sellers is similar to the effects of a horizontal merger between the firms. Obfuscation softens competition among products, which results in an increase in the prices of both firms products. The single product firm takes a free ride in the sense that its price rises less than the multiproduct firm s prices. As a result, it gains more from the obfuscation than the multiproduct firm itself. As obfuscation intensity increases, the singleproduct firm lures away more customers from the multiproduct firm, which has a negative effect on the multiproduct firm s profit. As a result, vis-à-vis the monopoly case, the optimal obfuscation intensity is lower under competition. My results resemble some observed retailing business practices. For example, clothing shops often arrange their products in such a way that discounted products are placed at the end of the shops and consumers have to walk away from more expensive items prior to finding the cheapest ones. 1 In 1 Advices for starters suggest that a salesman should start by offering the most expensive product and then offer more affordable options accessed 1 Sept. 1)

3 this connection, Dréze et al. 1994) observe in their shelf reorganization experiment that firm profits increased after raising the distance between different categories of bath tissues in a way that made the comparison of products more complicated for consumers. Apple, in its American online store obfuscates some special deals by placing a link at the left-bottom of the page of the online shop, as opposed to the regular items, which appear at the top centre of the screen. These business practices are in line with my results. My model of obfuscation is in the spirit of Ellison and Wolitzky 1) and Wilson 1) in that firms can deliberately increase the costs of searching their products. In the set-up of Ellison and Wolitzky 1) firms choose obfuscation intensity which is unobservable to consumers a priory and has an effect on the expected search costs for other firms. They find that homogeneous products selling firms earn more profits if they obfuscate. Wilson 1) shows that it is profitable for homogeneous products selling firms to obfuscate even if consumers observe search cost per firm before they start searching. In my model, obfuscation serves to screen low valuation consumers out. In this sense, it is related to the model of damaged goods of Deneckere and McAfee 1996). In their paper, a monopolist decreases the quality of one version of the product in order to be able to raise the price of the standard version of the product. In a way, this is similar to what happens in my model: if the monopolist obfuscates some of its products, it decreases the ex-ante expected utility of an obfuscated variety and increases the price of non-obfuscates products. Nevertheless, the models are clearly different because my paper deals with horizontal product differentiation; in fact, an obfuscated variety can ultimately provide the consumer with the highest utility ex post, whereas a damaged good is always qualitatively inferior. My model is also similar to the works of Caminal 1996), Anderson and Renault 6) and Wang 11). In the paper of Caminal 1996), a single product monopolist chooses whether to advertise its price, and production costs are not known to consumers. In the model of Anderson and Renault 6), consumers have different valuations for the variety of a monopolist. Hence, the firm can choose whether to advertise its price and a match value. In the paper of Wang 11) a single product monopolist chooses whether to advertise a match value while the price is observed by consumers. If a monopolist sells only one variety then price and or) match value advertising has an effect on whether a consumer arrives at the shop or not. Then the firm chooses the advertising option which gives the highest possible consumer traffic. In the model with an obfuscating multiproduct monopolist, less consumers observe obfuscated varieties. This draws a parallel between the obfuscating multiproduct monopolist and the advertising single product monopolist. However, there is a major difference between the two models: the single product monopolist sells less if less consumer search, whereas the multiproduct monopolist gains from the consumers who decide not to search because they buy more expensive non-obfuscated varieties. or merchandise items that... have the greatest margins on in [most visual spots], move the lower priced budget minded items to the base deck accessed 1 Sept. 1). 3

4 By obfuscating some of its products, the monopolist controls the order in which products are sampled. In this sense, my model is also related to the literature on ordered-search. Arbatskaya 7) shows that when consumers shop for homogeneous products equilibrium prices must fall as a consumer proceeds searching for lower prices. Zhou 11) demonstrates that when consumers shop for differentiated products equilibrium prices increase instead. Armstrong, Vickers, and Zhou 9) explore the role of prominence in search markets and show that a prominent product is sold at a lower price relative to non-prominent products. Haan and Moraga-González 11) and Armstrong and Zhou 11) present alternative ways in which firms can become prominent. The main difference between these papers and mine is that I focus on the role of frictions among the products of a single firm. Zhou 9) also notices that a multiproduct monopolist obtains higher profits if one of its varieties is prominent. My monopoly analysis extends his result by allowing the monopolist to choose the obfuscation level and the number of obfuscated varieties. Additionally, I analyse the incentives of a multiproduct firm to obfuscate part of its varieties when there is competition in the market. The role of search costs in multiproduct settings has also been studied by Rhodes 1), Shelegia 11), Zhou 1) and Moraga-González and Petrikaitė 13). Rhodes 1) and Zhou 1) analyse the cases when a consumer buys more than one variety and the items of a multiproduct seller are neither complements nor substitutes. Shelegia 11) looks at a duopoly case where the varieties of one multiproduct firm are either complements or substitutes and both firms sell identical items. Meanwhile, in my model, a multiproduct firm sells horizontally differentiated substitutes, which makes my set-up similar to the analysis of Moraga-González and Petrikaitė 13). However, differently from Moraga-González and Petrikaitė 13), in my model, a multiproduct firm sets different obfuscation intensity and, consequently, prices for its varieties, whereas all the varieties of a merged entity are symmetric in the model of Moraga-González and Petrikaitė 13). If a multiproduct obfuscating firm faces competition then my model has some similarities with the work of Ellison 5). Two competing firms sell horizontally and vertically differentiated products in the model of Ellison 5). Ellison 5) allows the firms to hide the prices of higher quality items with add-ons) and argues that it is profitable for the sellers to do so if high valuation consumers are unsophisticated, which implies that they are not good at making price comparisons across firms and... are also easier to talk into buying add-ons at the point of sale. The obfuscating multiproduct firm sets the price of its non-obfuscated variety less than the price of its obfuscated varieties if there is a competing seller in the market. This makes my model to resemble to the addon pricing model with competing sellers. However, the equilibrium price order in the obfuscating multiproduct monopolist model arises only due to search order effects, whereas add-ons increase the quality of a product in the model of Ellison 5). The rest of the paper is organized as follows. Basic assumptions are listed in section. The analysis of the obfuscating monopolist with two varieties is in section 3. In section 4, the effect of competing sellers on the obfuscation decision is explored. I address the optimal number of obfuscated varieties in the monopoly market in section 5. Finally, some concluding remarks are in section 6. 4

5 The proofs of propositions and other more complicated derivations can be found in the appendix of the paper. Model There is a mass of consumers and I normalize this mass to one. A typical consumer has unit demand and wants to buy one of n < horizontally differentiated varieties of a product. The utility of consumer l who buys variety i equals the difference between a match value, ε il, and the price of the product: u il = ε il p i The match value measures the valuation of variety i by consumer l. Tastes of consumers differ. Therefore, the same variety is valued differently by distinct buyers, which implies ε il ε ik, l k. Match values are distributed independently among consumers and varieties according to a continuous log-concave distribution function F ε) with a positive density f ε) in the interval [, ε]. In order to assure the concavity of the pay-off functions I require f ε) f ε) y f ε), y [, ε]. 3 All n varieties are sold by a single seller. The firm has constant unit production costs, which are the same for all the varieties, and I normalize them to zero. The seller knows the distribution of the match values and cannot observe any ε. Hence, all consumers pay the same price for a given variety. The monopolist chooses its profit maximizing prices) and the layout of its varieties. The layout of the varieties determines the ease with which a consumer can observe a variety and learn its utility. I assume that the monopolist can choose the layout of its goods such that either consumers observe all utilities immediately upon entering the shop or consumers observe the utilities of 1 m < n varieties immediately upon entering the shop but have to incur a positive search cost s for every other of n m 1 varieties. The varieties which utilities are observed after entering the shop are referred to as prominent varieties; and the other varieties for which a consumer has to pay search cost are called obfuscated varieties. Suppose the monopolist obfuscates some of its varieties. When a consumer enters the shop and observes both the match values and the prices of the prominent varieties then she has to decide whether to search for the utilities of the obfuscated varieties, and how many of them to search. Buyers search the obfuscated varieties sequentially with costless recall and I assume that consumers follow the optimal stopping and search rules that have been given in Weitzman 1979). Consider a The terms a buyer, a consumer, a customer are used as synonyms. 3 The condition f ε) f ε) y f ε) holds if ε follows a power distribution F ε) = ε α and α 1, an exponential distribution F ε) = e λε which is truncated to ε [, ε] and ελ, a Gumbel distribution F ε) = exp { exp { ε µ β }}, which is truncated to [, ε] and ε β. 5

6 consumer who has observed the utility of variety j, denoted by u j, and the utility of variety j is the highest observed utility so far. The consumer is contemplating whether to search variety k. If the consumer searches and it happens that u k > u j then the utility of variety k is a new reference point, which is used to decide whether the buyer searches further or stops searching after variety k has been searched. If u j > u k then the consumer continues to use the utility of variety j as a reference point for further decisions. The consumer is indifferent between searching variety k and terminating her search at variety j if the following equality is satisfied: ˆ ū u j u k df u k ) + ˆ uj u j df u k ) s = u j 1) where ū is the upper bound of u. After substituting u with ε p and rearranging the integrals in 1), I obtain ˆ ε ε k p k ε j + p j ) df ε k ) = s ) ε j p j +p k I define x ε j p j + p k such that ) is satisfied. Then the difference x p k is the reservation utility of variety k. If a consumer sees that the expected gains from searching are less than the search cost then the consumer stops and buys variety j. However, if ε j p j < x p k then it is worthwhile for the consumer to search variety k. An optimally searching consumer ranks all the varieties according to their reservation utilities in a decreasing order and starts searching from the top of the list. In order to assure that consumers always search at least once, I set the maximum search cost to be equal to s, where s = ˆ 1 z ε z) df ε) and z is defined by 1 F z) n m n m) F z) n m 1 f z) z =. The value of z equals the symmetric equilibrium price of a monopolist which sells n m varieties in a non-obfuscating equilibrium. 4 3 Monopolist with two varieties The simplest setting in which the main point of my analysis can be made is where the monopolist sells two varieties. The firm may choose to let consumers see both utilities at no costs, or obfuscate one of its varieties, say the second. 5 4 The value of z is unique because the derivative ) of 1 F z) n m n m) F z) n m 1 f z) z with respect to z is negative: n m) F z) n m 1 f z) + zf z) <. The LHS of the equation equals 1 if z = and it is negative if z = ε. 5 If the monopolist obfuscates both varieties by introducing the same search cost s per variety then the equilibrium price of both varieties is the same and the equilibrium pay-off of the firm does not depend on the search cost, i.e. it is the same as if there were no obfuscation. The monopolist can also choose to obfuscate both varieties by obfuscating 6

7 3.1 No obfuscation If the monopolist chooses not to obfuscate then consumers observe the utilities of both varieties and buy one of them, provided that the highest observed utility is positive. Suppose that the monopolist considers to set price p 1 for its first variety and p for its second variety. Without loss of generality I assume that p 1 p. Then the demand for the first variety is the probability that ε 1 p 1 > max {ε p, }, which equals d 1 = 1 F ε p + p 1 ) + ˆ ε p +p 1 p 1 F ε p 1 + p ) f ε) dε The demand for the second variety is the probability that ε p > max {ε 1 p 1, }, which gives d = F ε p + p 1 ) f ε) dε p Then the profit of the monopolist is ˆ ε π = p 1 d 1 + p d The first order conditions which determine optimal p 1 and p are 1 F ε p + p 1 ) + ˆ ε p +p 1 p 1 F ε p 1 + p ) f ε) dε ˆ ε p 1 p ) f ε p + p 1 ) f ε) dε p 1 F p ) f p 1 ) = p 3) ˆ ε p F ε p + p 1 ) f ε) dε p p 1 ) ˆ ε p f ε p + p 1 ) f ε) dε p F p 1 ) f p ) = 4) In the supplementary appendix of the paper, I show that when ε is distributed uniformly in the interval [, 1], the pay-off function of the monopolist is globally quasi-concave. Numerical simulations show that the same result also holds for other distribution functions that satisfy the properties defined in section. Hence, from here on I assume that the pay-off function of the non-obfuscating monopolist is globally quasi-concave. Then the solution to the system of first order conditions 3) and 4) is unique. as 5). I impose symmetry by setting p 1 = p = p, and both first order conditions turn to be the same one variety less than another. The monopolist can always set the search cost for less obfuscated variety such that the reservation utility of the less obfuscated variety is higher than the reservation utility of a more obfuscated variety. Then the less obfuscated variety becomes prominent and the pay-off function of the firm does not depend on the obfuscation level of the prominent variety, i.e. it is as if the prominent variety is not obfuscated at all. 7

8 1 F p ) p F p ) f p ) = 5) The LHS of 5) decreases with p, it is negative if p = ε and equals one if p =. Therefore, the solution p to 5) exists. After both prices are set equal to p then the non-obfuscating equilibrium profit of the monopolist is π = p 1 F p ) ) 3. Obfuscation effects Now suppose that the monopolist decides to obfuscate its second variety. For the moment I assume that s is exogeneous and focus on the effects of obfuscation on prices. Later, in section 3.3, I study how obfuscation intensity affects profits and the optimal level of obfuscation. Consumers observe the utility of the prominent variety at no cost but they have to pay positive search cost s if they want to learn the utility of the obfuscated variety. The expectations of consumers about the obfuscated variety s utility affect whether the buyers search the obfuscated variety or buy the prominent variety right away. I assume that consumers hold equilibrium beliefs about the price of the obfuscated variety. Hence, they expect that the monopolist charges p for its second variety. The firm takes the expectations of consumers into account and considers two profitable prices 1 x + p p 1 and 1 x + p p p. 6 Consider a consumer who arrives at the firm and observes ε 1 p 1. The buyer has to decide whether to search the second variety and learn u or buy the first variety right away. The consumer expects that the price of the second variety is p. Therefore, she buys the first variety without searching the second variety if ε 1 p 1 > x p. The probability of this event is 1 F x p + p 1 ). If the consumer searches the second variety then she observes p and buys the prominent variety when ε 1 p 1 > max {ε p, }. The probability of this event is p +p 1 p 1 F ε p 1 + p ) f ε) dε If the consumer searches the obfuscated variety then it must be the case that ε 1 p 1 < x p. Hence, if the buyer observes ε p > x p then she buys the obfuscated variety. The joint probability that the consumer searches the second variety and ε p > x p is F x p + p 1 ) 1 F x p + p )). If ε p x p then the consumer buys the second variety provided that ε p > max {ε 1 p 1, }. The probability of this event is 6 The restriction on the upper bound of the deviation price assures that some consumers buy the prominent variety without searching the obfuscated variety. If p 1 > 1 x + p then the pay-off function of the monopolist is the same as if there was no obfuscation. The same holds if p > 1 x + p. Additionally, in equilibrium 1 p 1 > x p. 8

9 p +p p F ε p + p 1 ) f ε) dε Putting together these probabilities, the pay-off function of the obfuscating monopolist becomes π o = p 1 [1 F x p + p 1 ) + p +p 1 + p [F x p + p 1 ) 1 F x p + p )) + p 1 ] F ε p 1 + p ) f ε) dε p +p p ] F ε p + p 1 ) f ε) dε In equilibrium the monopolist sets p 1 = p 1 and p = p. Therefore, the system of the first order conditions is 7 1 F x p + p 1) + p +p 1 p p 1) 1 F x)) f x p + p 1) p 1 p ) p 1 F ε p 1 + p ) f ε) dε p 1F p ) f p 1)+ p f ε p + p 1) f ε) dε = 6) p 1 p ) p f ε) f ε + p 1 p ) dε + F x p + p 1) 1 F x)) + p F ε p + p 1) f ε) dε p F p 1) f p ) = 7) Proposition 1. There is at least one pair of prices p 1, p ), which satisfies the system of equations 6) and 7). If p 1, p ) is a solution to the system of equations 6) and 7) then p 1 > p and p > p. If F ε) = ε α where α 1 then the solution is unique. Learning the utility of the prominent variety is free, whereas learning the utility of the obfuscated variety is costly. Then if the prices of both the prominent variety and the obfuscated variety were the same, the ex ante expected utility of the obfuscated variety would be smaller because of the search cost. Therefore, obfuscation introduces a product differentiation aspect, which is similar to vertical product differentiation. This product differentiation exists only ex ante, i.e. before a consumer searches the obfuscated variety, and thereby it serves to introduce a search order which redistributes consumers among the two varieties and prompts the monopolist to set two different prices. I use Figure 1 to illustrate how the customers of the monopolist are redistributed with obfuscation. In Figure 1a, the monopolist sells both its varieties at price p. In Figure 1b, the second variety is obfuscated. In a non-obfuscating equilibrium, the monopolist charges the same price for both its 7 I show in the supplementary appendix that the pay-off function of the obfuscating monopolist is globally concave if s s, it is locally concave around p 1, p ) if F ε) = ε α, and numerical simulations show that the solution to the system of first order conditions gives the highest pay-off in p 1, p ) [, 1] [, 1]. This final remark is proven analytically if match values are distributed uniformly. 9

10 varieties. Then the first variety is bought by the consumers whose pairs of valuations for the first variety and the second variety are in the dotted area to the left of the 45-degree line in Figure 1a. If the second variety was obfuscated and the monopolist kept the same price p for both its varieties then all the consumers with valuations for both products above line x would buy the first variety without searching the second variety. Hence, the consumers with valuations for both products in the triangle ABC would switch from the second variety to the first variety. This is because the prominent variety is good enough for these consumers and they do not have any reason to search for the utility of the second variety. As a result, obfuscation increases the average valuation of the consumers who buy the first variety while it decreases the average valuation of the consumers who buy the second variety. These effects prompt the monopolist to increase the price of the first variety and to decrease the price of the second variety. If the monopolist sets p 1 > p then the threshold value of ε 1 increases from x to x p + p 1 Figure 1b). As a result some consumers who would not have searched the second variety if p 1 = p search the second variety. Part of these additional searchers switch to the second variety and part of them still buy the first variety. Since the line ε p + p 1 shifts to the left if p 1 > p, some consumers who search both varieties and who would have bought the first variety if p 1 = p switch to the second variety. The monopolist loses from switching consumers, because they pay a lower price. However, the firm also has gains from setting two different prices. More particularly, the monopolist receives additional income from the first variety buyers, who pay the higher price, and from its new customers because p < p. The monopolist sets p 1 exactly at the level where the loss of increasing the price equals the gains from doing so. Similarly, the price of the obfuscated variety is set at the level where the gains from decreasing this price are equal to the loss from decreasing it. The equilibrium price ranking of proposition 1 is different from the equilibrium price ranking when the two varieties are sold by distinct firms. Armstrong et al. 9) and Zhou 11) show that the price of a prominent firm is lower than the price of a non-prominent firm, which is the opposite to my finding. This difference appears because a multiproduct monopolist has more incentives to increase the price of its prominent variety than a single variety prominent firm. Moreover, the incentives of a multiproduct monopolist to decrease the price of its obfuscated variety are stronger than those of a single product non-prominent firm. To see this, suppose that a prominent firm increases its price p 1. Then some consumers switch to a non-prominent firm. Hence, the net gain of the prominent firm is the additional revenue that it gets from staying customers minus the loss of the revenue from switching customers. For the monopolist the net gain also equals its additional revenue that comes from the first variety buyers minus the loss of revenue from switching customers. However, the loss of the monopolist is smaller because the switching customers switch to a product which is also sold by the monopolist. By contrast, the prominent firm does not get anything from the switching customers. Therefore, the incentives for the single product prominent firm to decrease its price are stronger than for the multiproduct monopolist. Similarly, if p increases then a non-prominent firm gains from the consumers who buy its variety 1

11 a) No obfuscation b) With obfuscation Figure 1: Demand decomposition and loses from the consumers who cease buying its variety because of either switching to other varieties or leaving the market. If the monopolist increases the price of its obfuscated variety then the firm loses less from the switching consumers because they switch to its prominent variety. However, the firm experiences an additional loss that comes from all prominent variety buyers. If the price of the obfuscated variety increases then the mean valuation of the prominent variety buyers decreases, which prompts the monopolist to decrease the price of the prominent variety. The seller wants to extract as much surplus as possible from its high valuation customers. Therefore, it is profitable for the firm to shift its low valuation customers to the obfuscated variety. As a result, the monopolists decreases the price of the obfuscated variety. The extent to which the prominent variety price increases and the obfuscated variety price decreases in an obfuscating equilibrium depends on the obfuscation intensity. The obfuscation intensity is determined by the size of the search cost for the obfuscated variety. If the search cost increases then the obfuscation intensity increases. An increase in the search cost lowers the ex ante expected utility of the obfuscated variety. Then it becomes easier to prevent high valuation consumers from taking the cheaper but obfuscated variety, which gives the monopolist incentives to increase p 1. It is shown in proposition that if match values are distributed uniformly and the search cost increases then the prominent variety s price goes up. Numerical simulation results show that the same holds if F ε) = ε α where α > 1 Figure a). On the contrary, the price of the obfuscated variety decreases if it is more costly for consumers to search it. This happens because the monopolist wants to shift its low valuation customers to the obfuscated variety. Consequently, the firm must increase search intensity, which implies that p must go down as the search cost increases. Then the difference between p 1 and p increases. Proposition. Assume that F ε) = ε α, α 1 then the price of the obfuscated variety decreases 11

12 with the search cost. If α = 1 then the price of the prominent variety increases with the search cost. a) Prominent variety b) Obfuscated variety Figure : Obfuscation equilibrium solid) and non-obfuscating dashed) equilibrium prices for different s and α values. 3.3 Choice of obfuscation intensity No mater the extent of obfuscating, the firm earns more with obfuscation that without obfuscation. In order to show this, one should set the deviation prices p 1 and p equal to p in the pay-off function π o. After it is done, π o equals π, which together with the fact that p 1 p p implies that the firm earns more in an obfuscating equilibrium. If the monopolist prefers obfuscation to non-obfuscating then it remains to find the optimal obfuscation intensity for the monopolist. If the monopolist obfuscates one of its varieties then less consumers observe this variety. Because of that it is as if the seller sacrifices the obfuscated variety for the sake of increasing the profits from the prominent variety. As the obfuscation intensity is increased, the threshold utility value for the prominent variety or the reservation utility of the obfuscated variety) decreases. 8 Then the valuations of consumers who search the obfuscated variety are closer to the left-bottom corner in Figure 1a. On the contrary, the consumers who buy the prominent variety without searching the obfuscated product are the consumers whose valuations are closer to the top-right corner in Figure 1a. As a result, the monopolist screens out its lowest valuation customers better if the firm increases obfuscation intensity. If the lowest valuation customers are better screened out then the firm increases the price for the prominent variety buyers. The gain in profits that comes from increasing the price of the prominent variety is more than the loss in profits due to decrease in the price of the obfuscated variety. Thus, the monopolist chooses the maximum search cost for the obfuscated item. If the search cost equals s then x = p. As a result, the threshold match value for the prominent variety equals its price p 1. If the match value for a variety is less than its price then a consumer 8 x p decreases with s because, by Claim in the appendix, p / < 1. 1

13 Figure 3: Difference between π o and π for different s and α values never buys this variety. Hence, only the consumers who definitely do not buy the prominent variety search the obfuscated variety. If the monopolist increased the obfuscation intensity further then the firm would remove its second variety from the market, because consumers would not search it. It is not optimal for the monopolist because then the lowest valuation consumers would get into the same pool with the highest valuation consumers as without obfuscation. Moreover, the expected maximum match value with one variety is smaller that with two varieties, which results in a decrease in profits. Proposition 3. If F ε) = ε α, α 1 then the profit of the monopolist increases with obfuscation intensity. Hence, the firm chooses the highest obfuscation level. Consumer surplus and total welfare decrease if the monopolist obfuscates. The profit of the monopolist increases with obfuscation for all α values. However, the obfuscation gain is higher the higher the value of α is Figure 3). Higher α values mean that the density of the match values is more left-skewed, which implies lower product differentiation. Additionally, the share of high valuation consumers is higher for higher α values. Then a particular separation level of the lowest valuation consumers is achieved at lower search cost values if α increases. Hence, the difference between πo and π increases with obfuscation more for more left-skewed density functions. When the monopolist obfuscates then total consumer surplus is as follows CS o = ˆ ε x p +p 1 + F x p + p 1) F x p + p 1) s ε p 1) f ε) dε + ˆ ε p +p 1 p 1 ε p ) f ε) dε + x p ε p 1) F ε p 1 + p ) f ε) dε ε p ) F ε p + p 1) f ε) dε 8) The sum in the first line of 8) is the surplus of the prominent variety buyers. The surplus of the obfuscated variety buyers is in the second line of the expression. Consumers search the obfuscated variety if ε < x p + p 1. As a result, F x p + p 1) s are total search costs. If the monopolist sets 13

14 the search cost equal to s then the sum of the second and the third lines in 8) equals zero. Hence, the total consumer surplus equals the surplus of prominent variety buyers. Consumer surplus in an obfuscating equilibrium is smaller than without obfuscation because p 1 > p and some prominent variety buyers would get better match values with the second variety if they searched. Although the obfuscating monopolist earns more profit, an increase in profits is not sufficient to compensate the decrease in consumer surplus and total welfare decreases if the monopolist obfuscates. 4 Competition In this section, I introduce a second firm in the market. This firm sells only one variety. Then consumers can choose from three horizontally differentiated varieties where two out of three varieties are offered by one firm. The two variety seller is called a multiproduct firm and the seller with one variety is referred to as a single product firm. Consumers observe the utility of the single product firm s variety at no costs. Meanwhile, the multiproduct firm allows consumers to observe the utility of its first variety for free and obfuscates the second variety by introducing a positive search cost. In order to simplify algebraic expressions I assume that the match values are distributed uniformly in the interval [, 1]. Consumers incur no costs for the entering a shop. 9 A consumer observes the utilities of the prominent variety and the variety of the single product firm right away. Then the buyer makes the decision whether to search the obfuscated variety. I again assume that consumers hold equilibrium beliefs about the price of the obfuscated variety, which is p. The firms know the beliefs of consumers and look for their profits maximizing prices. The single product firm sets 1 x + p p s and the multiproduct firm contemplates about 1 x + p p 1 and 1 x + p p p. 1 Without loss of generality I assume that p s < p 1. A consumer observes ε s p s and ε 1 p 1 immediately and makes the decision whether to search for the utility of the obfuscated variety. If the maximum of ε 1 p 1 and ε s p s is more than the reservation utility of the obfuscated variety, x p, then a consumer does not search the obfuscated variety. She buys the variety of the single product seller if ε s p s > max {ε 1 p 1, x p }. This event happens with probability 1 1 p 1 + p s ) + ˆ 1 p1 +p s x p +ps ε p s + p 1 ) dε = p 1 p s + ˆ 1 p1 x p ε + p 1 ) dε 9 In this case the search cost has nothing to do with the transportation costs, which a consumer incurs due to moving from one shop and another. Alternatively, one could assume that both firm advertise their products by supplying consumers with the full information. Then the decision to obfuscate would be a decision not to advertise. Additionally, firms could sell their products in one selling platform and a multiproduct firm may decide to withdraw one of its varieties and sell it in a separate shop. 1 I show in the supplementary appendix that the single product firm will never set p s > 1 x+p and π s is concave in p s if p s < 1 x + p. Assuming that p 1, p ) is the pair of the multiproduct firm s prices which is given by the first order conditions, the pay-off function of the multiproduct seller is concave at this point if s s and s. Numerical simulations show that the pay-off function of the multiproduct firm is quasi-concave with other values of s for p 1, p ) [, 1] [, 1]. 14

15 If the consumer searches the obfuscated variety then it must be the case that max {ε 1 p 1, ε s p s } < x p. The customer observes ε p but may still decide to buy from the single product firm. This happens if x p > ε s p s > max {ε 1 p 1, ε p, }, which gives the returning demand of the single product firm p +p s p s ε p s + p 1 ) ε p s + p ) dε = p As a result, the pay-off function of the single product shop is π s = p s p 1 p s + ˆ 1 p1 x p ε + p 1 ) dε + p ε + p 1 ) ε + p ) dε ε + p 1 ) ε + p ) dε Suppose that ε 1 p 1 > max {ε s p s, x p }. Then a consumer does not search the obfuscated variety and buys the prominent variety. The probability of this event equals ˆ 1 x p +p 1 ε p 1 + p s ) dε The consumer may decide to buy the prominent variety even if she searches the obfuscated variety. The probability of this event is Pr [max {, ε s p s, ε p } < ε 1 p 1 < x p ] = = p +p 1 p 1 p ) ε p 1 + p ) ε p 1 + p s ) dε ε + p ) ε + p s ) dε If the consumer searches the obfuscated variety and observes that ε p > x p then she buys the second variety because max {ε 1 p 1, ε s p s } < x p. Then the joint probability that a consumer searches the obfuscated variety and finds its utility above the reservation utility x p is x p + p 1 ) x p + p s ) 1 x p + p )). Suppose that ε p < x p. Then the buyer still may find the obfuscated variety worthwhile buying. This happens if x p > ε p > max {ε 1 p 1, ε s p s, } and the probability of this event is p +p p ε p + p 1 ) ε p + p s ) dε = p ε + p 1 ) ε + p s ) dε 15

16 The pay-off of the multiproduct firm is the total revenue that it gets from selling two varieties: π o = p 1 ˆ 1 x p +p 1 ε p 1 + p s ) dε + p ε + p ) ε + p s ) dε + p x p + p 1 ) x p + p s ) 1 x p + p ) + p ) ) ε + p 1 ) ε + p s ) dε In equilibrium, the single product seller sets its price equal to p s, the price of the prominent variety is p 1 and the price of the obfuscated variety is p. These prices are defined by the following system of first order conditions p 1 p s + ˆ 1 p 1 x p ε + p 1) dε + p ε + p 1) ε + p ) dε = 9) ˆ 1 x p +p 1 ε p 1 + p s) dε p p s p 1) + p ε + p ) ε + p s) dε+ p x p + p s) 1 x) = 1) x p + p 1) x p + p s) 1 x p ) + p ε + p 1) ε + p s) dε = 11) Proposition 4. There is a triplet of prices p 1, p, p s), which satisfies the system of equations 9), 1) and 11). Furthermore p 1 < p. The presence of a competing firm alters the equilibrium price order of the prominent and the obfuscated varieties. This happens because the multiproduct firm gets less incentives to increase the price of the prominent variety and decrease the price of the obfuscated variety. If there is a competing firm around and the multiproduct seller increases the price of its prominent variety then some prominent variety buyers will switch to the obfuscated variety and some consumers will switch to the single product seller. The multiproduct firm gets no revenue from the consumers who switch to the single product firm. Therefore, the loss from increasing the prominent variety s price is higher comparing to the case where the multiproduct firm is a monopolist. After the second variety is obfuscated, some high valuation consumers, who used to buy the second variety before obfuscation and do not search after obfuscation, switch to the single product firm instead of to the prominent variety. Therefore, the multiproduct firm finds it less profitable to increase the price of its prominent variety. As a result, with a competitor, the multiproduct firm increases the price of its prominent variety less than in a monopoly case. In a non-obfuscating equilibrium, the price of the single variety seller is less than the price of the multiproduct firm because the former does not internalize any pricing externalities. As a result, after obfuscation and an increase in the price of the prominent variety some lower valuation 16

17 Figure 4: The price of the prominent, the price of the obfuscated variety and the price of the single variety seller for different s values consumers switch to the single product firm instead of searching for the utility of the obfuscated variety. Therefore, the incentives for the multiproduct firm to decrease the price of its obfuscated variety are lower if there is a competitor in the market. Moreover, the multiproduct seller always has incentives to increase the price of the obfuscated variety as a non-prominent firm does. If the multiproduct firm is not a monopolist then these incentives are stronger than the incentives to decrease the price and the price of the obfuscated variety becomes higher than the price of the prominent variety. Proposition 5. Starting at maximal obfuscation intensity, p 1 and p s decrease and p increases if obfuscation intensity decreases. Starting at the non-obfuscating equilibrium, all three equilibrium prices increase as obfuscation intensity goes up. The price of the prominent variety increases with obfuscation intensity as in the monopoly case Figure 4). However, that increase is slower. If the multiproduct seller obfuscates one of its varieties then the single product shop faces less competition. Hence, the price of the single product seller also increases with obfuscation intensity. Meanwhile, the price of the obfuscated variety increases with obfuscation intensity if the search cost is sufficiently low and decreases if the search cost is sufficiently high. If the search cost is small and increases then the reservation utility of the obfuscated variety decreases and some of its former buyers switch either to the prominent variety or to the single product shop. As it has been discussed above the screening out of low valuation customers has a smaller effect on the incentives to decrease p when there is a competing firm. As a result, if the search cost goes up then the incentives to increase the obfuscated variety s price more than offset the incentives to decrease the price and p increases with obfuscation intensity. When obfuscation intensity is relatively large then the reservation utility of the obfuscated variety is very low, which means very low demand for this variety. A slow increase in the price of the prominent variety does not compensate the loss in income from the obfuscated variety. Then the multiproduct firm decreases p if the search cost is large and increases further. 17

18 a) Profits per firm b) Profits per product Figure 5: Profits of the multiproduct firm and a single product seller with different s It is shown in the appendix that if p decreases with s then it does it slower than x does, which implies that the reservation utility of the obfuscated variety always decreases with s. However, this decrease becomes slower if obfuscation intensity is large. Additionally, if s is large then the increase in the price of the obfuscated variety does not have a significant effect on the incentives to increase p 1. This happens because the reservation utility of the obfuscated variety is already sufficiently low and further sharp decrease would switch consumers more to the single product firm than to the prominent variety. Therefore, the multiproduct seller decreases the reservation utility of its obfuscated variety slower if the search cost is high and increases further. Proposition 6. Starting at maximal obfuscation intensity the profit of the multiproduct firm increases if obfuscation intensity decreases. Starting at non-obfuscating equilibrium, and increase in obfuscation intensity raises the profit of the multiproduct firm. If there is no obfuscation to start with, and the multiproduct firm introduces a small search cost then relatively little consumers switch to the single product firm. Hence, the gains of the multiproduct seller from obfuscation are similar to the gains of the monopolist: the profit increases with obfuscation intensity. On the contrary, if the search cost is already large and further increases then the loss of the buyers who stop buying the second variety has a significant effect on the pay-off of the multiproduct firm. Moreover, the price of the prominent variety increases with obfuscation intensity too slowly so as to compensate for that profit loss. Therefore, the profit of the multiproduct firm decreases with obfuscation intensity if s is already large. This suggests that there is an optimal obfuscation level for the multiproduct firm which has one single variety competitor. The simulation results suggest that the profit of the multiproduct firm increases with obfuscation intensity if s <.45 and starts decreasing thereafter. The single product seller benefits if the multiproduct competitor obfuscates. Obfuscation decreases competition among products. Therefore, the single variety seller earns more if the search cost increases. 18

19 5 Monopolist with more than two varieties The multiproduct firm which sells more than two varieties can choose both the obfuscation intensity and the share of obfuscated varieties. In this section, I assume that the multiproduct seller sells more than two varieties. The firm makes m varieties prominent and obfuscates n m varieties by setting the same search cost s for each. In order to simplify the analysis I assume that match values for the varieties are distributed uniformly in the interval [, 1] and the multiproduct firm is a monopolist. I start the derivation of the pay-off function of the multiproduct monopolist by deriving the demand for its prominent varieties and later continue with the derivation of the demand for the obfuscated varieties. Because of the same search cost consumers expect that for all the obfuscated varieties the monopolist sets the same price p. Prominent varieties The monopolist considers two vectors of profitable deviation prices {p 11,.., p 1m } for its prominent varieties and { } p 1,..., p n m) p for its obfuscated varieties. Because of the symmetry of varieties within the groups, the monopolist charges the same price for all the varieties within a group in equilibrium. Namely, the equilibrium price of the prominent varieties is p 1 and the monopolist charges p for every obfuscated variety. Then, I derive the pay-off function of the monopolist by assuming that all the prominent varieties except variety l are priced at p 1 and all the hidden varieties except variety k have price p. 11 The price of variety l is p l < p 1 and the price of variety k is p k p < 1 x + p. A consumer observes the utilities of m prominent varieties immediately and makes the decision whether to search the first obfuscated variety or to buy one of the prominent varieties. The buyer terminates her search and buys variety l if ε l p l > max { x p, ε i p 1} where i [1, m] \l. This happens with probability ˆ 1 p f p l = p 1 +p l 1 p l + ε p l + p 1) m 1 dε x p +p l If the consumer searches the first obfuscated variety then max {ε i p 1, ε l p l } i [1,m]\l < x p. Therefore, if she buys variety l then she does it only after she observes all the varieties in the shop and decides that ε l p l > max i [1,m]\l,s [1,n m]\k {ε i p 1, ε s p, ε k p k, }. Thus, the returning demand for variety l is r p l = p +p l p l ε p l + p 1) m 1 ε p l + p ) n m 1 ε p l + p k ) dε 11 The application of a partial symmetry at this stage does not change the outcome, which would be obtained if the vector of deviation prices was used instead. After the symmetry is applied on the first order conditions then the first order conditions of all the prominent varieties are the same. The same holds for the obfuscated varieties. Therefore, I need to derive one first order conditions with respect to the price of one prominent variety and one first order condition for an obfuscated variety. 19