Guided Search. Preliminary Version. Tobias Gamp. September 20, Abstract

Transcription

1 Guided Search Preliminary Version Tobias Gamp September 20, 2015 Abstract I study a monopolist who may aggravate the information acquisition of a consumer who inspects her products. I show that the monopolist can raise her profits at the expense of the consumer by obfuscating products. Obfuscation is a powerful tool as it allows the monopolist to guide the consumer s search in a way such that the consumer purchases in equilibrium the most expensive product which supplies positive utility. The monopolist achieves thus the highest profits attainable under the asymmetric information structure. The consumer s equilibrium utility, on the other hand, does not necessarily strictly exceed zero; there exists an equilibrium in which his expected information rent equals his expected search costs. I provide conditions under which the monopolist guides the consumer to polarizing products first, which are sold at the highest prices. Keywords: Information disclosure, monopoly, multiproduct, obfuscation, directed search, screening. JEL-Classifications: Financial support from the Bonn Graduate School of Economics and the German Science Foundation through SFB-TR 15 is gratefully acknowledged. For valuable discussions I am grateful to Daniel Krähmer, Tymon Tatur, Balasz Szentes as well as seminar participants at the EDP Jamboree (London). University of Bonn, Bonn Graduate School of Economics, Lennéstraße 43, Bonn, Germany; gamp@uni-bonn.de 1

2 1 Introduction A consumer often does not know which of the offered products is his preferred one and therefore has to search for a product that matches his taste. A monopolist can help the consumer to find a suitable product or impede his search by obfuscating her products. I show how a monopolist can raise her profits at the expense of the consumer by obfuscation. For instance, consider a firm which offers a menu of contracts (insurance policies, bank accounts, mobile phone contracts, etc.). When drafting the contracts, the firm makes a variety of decisions which determine whether a consumer can easily evaluate these. The firm chooses the payment schedules (simple or complex), the contract terms (long or short), and the forms of the contracts (helpful charts or not). Depending on these choices, the consumer decides which contracts to inspect first. Similarly, a firm which sells several variants of a technical product (microwaves, washing machines, computer parts, etc.) decides whether to provide understandable and comprehensive product descriptions, helpful illustrations, and whether to provide the opportunity to test the product at the store. The common feature of these motivating examples is the firm s ability to aggravate, or simplify, the information acquisition of consumers. This is the key ingredient of the search model which I study. A monopolist offers several heterogeneous products, which supply a utility that is ex ante unknown to the consumer and to the monopolist. The consumer may privately inspect products, sequentially, in which case he learns how much utility a specific product supplies. The novel aspect of the model is that the search costs, which the consumer incurs for each product inspection, are product-specific and a choice variable of the monopolist. This captures the monopolist s ability to impede the consumer s information acquisition. The consumer observes the monopolist s choices of prices and search costs, and then decides (i.) which products to inspect, (ii.) whether to buy a product or (iii.) whether to exit. This gives rise to new strategic opportunities for the monopolist: She can guide the consumer s search by her choices of prices and of search costs. I show that it is optimal for the monopolist to impede the information acquisition of consumers by setting strictly positive search costs. The consumer can thus 2

3 not find his preferred product without incurring search costs. In particular, the monopolist obfuscates each product except the most expensive one: Given its price, the monopolist sets search costs such that the consumer yet prefers inspecting the product to exiting right away. There are two main motives that lead the monopolist to obfuscate products. First, through obfuscation, the monopolists guides the consumer s search in her favor, so that the consumer prefers to inspect more expensive products first, as in the motivating examples. Second, through obfuscation, the monopolist reduces the benefits of continued search. The latter is profitable, as it results in the consumer ending his search in expectation sooner and purchasing one of the more expensive variants that he inspects first. The astonishing, resulting equilibrium outcome is that the consumer purchases, due to obfuscation, the most expensive product that supplies positive utility. It is therefore no surprise that the equilibrium outcome is bad news for the consumer. There even exists an equilibrium in which the consumer s expected utility is zero. In such an equilibrium the consumer does not derive any utility in excess of the utility that his outside option supplies and his expected information rent only compensates the search costs that he incurs in expectation. The monopolist, on the other hand, enjoys the powerful tool of guiding the consumer s search by obfuscation and achieves the highest, expected profits attainable under the informational asymmetry that emerges as learned valuations for products are the consumer s private information. The question arises to what kind of products the consumer is guided first by the monopolist. Recall that these products are as well the more expensive products in equilibrium. Intuitively, one might suspect that the quality of a product determines when it is to be inspected. This paper, however, clarifies that the property which shapes the consumer s search order is how polarizing a product is 1 whether the variance in the consumer s valuation for the product is high such that very high and very low valuations are likely to occur. I provide conditions under which the consumer inspects more polarizing products first in equilibrium. For instance, if 1 Formally, suppose two distribution functions, from which the consumer s valuations for two heterogeneous products are drawn, satisfy a single-crossing property. Then, the product whose distribution function is flatter at the point of crossing is referred to as the more polarizing product. 3

4 the monopolist offers an arbitrary number of products with uniformly distributed valuations, those products product with the higher variance are more polarizing and will be inspected first in any equilibrium. The intuition why the monopolist guides the consumer to polarizing products first is that she attempts to sell possibly highly valued products at high prices. In contrast, the monopolist guides the consumer to less polarizing products last, since she is less uncertain about the consumer s valuations for these products, which guarantees the monopolist that she can sell these products at moderate prices with high probability as part of a last resort safety strategy. The strategic use of obfuscation to guide the consumer s search is observed in practice. For instance, Ellison and Ellison [2009] describe an online market for computer parts, in which bait products, with which the consumer was lured to the retailer s webpage before, are available only when ordered over the phone. This way retailers achieve that some consumers, which consider such a phone call to be too annoying, take instead a closer look at other, more expensive product. The paper is related to a number of others. First, it illustrates how a monopolist can screen consumers and raise his profits by obfuscation. Thereby, it contributes to a large literature that explores selling techniques to claim larger shares of the surplus from trade if the consumer s valuations for products are his private information. 2 The importance of manipulating the consumer s search has been acknowledged before in competitive setting. In fact, the topic is at the very heart of the literature of advertising, where firms compete for the consumer s attention. 3 More closely related is a growing literature which studies the competition for positions in the consumer s search order. For instance, Chen and He [2011] and Athey and Ellison [2011] study how these positions can be sold via auctions and explicitly incorporate a model of sequential consumer search in their analysis.search engines The paper is related to the literature on consumer search with heterogeneous products. It shares the basic framework introduced by Wolinsky [1986]. Within this literature most related are Armstrong et al. [2009] and Zhou [2011], who examine 2 Classic papers include... 3 Formally, already Butters [1977] studies how firms advertise in order to affect the consumer s non-sequential search. 4

5 oligopolies, in which firms are visited in an exogenously given search order. The tenet of these papers is that, in contrast to my paper, later inspected products are more expensive, as firms correctly anticipate that the consumer s valuations for previously inspected products are low, conditionally on being visited. Finally, the paper contributes to the literature on obfuscation. Complementary to explanatory approaches based on the bounded rationality of consumers, 4 it provides a search-theoretic explanation for obfuscation. In that, it follows a line of research that puts forward arguments why it is individually rational for firms raise search costs in an environment of costly information acquisition. 5 Carlin [2009] argues that through obfuscation firms can affect the overall perceived complexity of a market, so that less consumer shop for better offers. Wilson [2010] shows that in a directed search model that firms split the market of heterogeneous consumers by differentiating in search costs. Ellison and Wolitzky [2012] point out that in a sequential search model firms have incentives to raise search costs in order to fatigue consumers, when search costs are convex, instead of linear. Gamp [2015] develops a search model which allows consumers to purchase products poorly informed and shows under which conditions firms impede the information acquisition of consumers. The paper is organized as follows: In section 2 the model is introduced. In section 3 the consumer s optimal search rule is determined. In section 4 the monopolist s best response to the consumer s search rule and the resulting equilibrium outcome are characterized. I conclude in section 5. 2 Model The market consists of a representative consumer (he) with unit demand and a profit-maximizing monopolist (she). The monopolist offers several heterogeneous 4 See Gabaix and Laibson [2006], Spiegler [2006], Carlin and Manso [2011], Piccione and Spiegler [2012] and Chioveanu and Zhou [2013] for explanatory approaches based on the bounded rationality of consumers. 5 This particular line of research is partially motivated by the observation that it is not individually rational for firms to raise search costs in classic search models as studied by Diamond [1971], Burdett and Judd [1983] and Stahl [1989] or Wolinsky [1986] and Anderson and Renault [1999]. 5

6 products. Her marginal costs of production are normalized to zero for each product. If the consumer buys product k {1,..., K} at price p k, his quasi-linear utility absent any search costs is u k = θ k p k, (1) where the match-value θ k is the consumer s valuation for the k-th product. Each θ k is the realization of an independent random variable ˆθ k, which is described by a distribution function F k. The interval support of F k is Supp (F k ) = [θ k, θ k ]. In order to simplify the exposition, assume that any product, absent prices, may be preferred to any other such that the supports of any two distributions overlap. Finally, let F k have no mass points and be twice continuously differentiable on the open interval (θ k, θ k ). Denote the reliability function with F k := 1 F k. Ex ante, the consumer only knows the products prices, but does not know realized match-values. He can, however, inspect products privately and sequentially in order to learn these. Each inspection is costly. Let c k 0 be the product-dependent search costs that the consumer incurs if he inspects product k. The key departure of my model is that search costs are a choice variable of the monopolist i.e. the monopolist may set search costs to zero such that the consumer can learn all match-values for free. Alternatively, the monopolist may aggravate the inspection of products by setting strictly positive search costs. The timing of events is as follows: First, the monopolist chooses publicly a price profile and a search cost profile: ( p, c) R K + R K +. The consumer then searches until he exits or purchases a product. The consumer s search proceeds as follows. At any time, the consumer can: i) inspect one product of his choice: learn θ k at cost c k, ii) purchase any previously inspected product k: end search and obtain the utility u k = θ k p k, iii) exit: end search and obtain a utility of zero. The equilibrium concept is Perfect Bayesian Equilibrium. The analysis is restricted to equilibria in pure strategies. 6

7 3 The consumer s search rule The problem the consumer faces is a so called Pandora s problem (Weitzman [1979]). He has to decide in which order to inspect products, possibly dependent on learned match-values, and when to end search in order to either purchase a product or to exit. As match-values are the realizations of independent random variables, the consumer s optimal search rule is the Weitzman rule. To begin with, I introduce the Weitzman reservation utility. Definition 1 (Weitzman reservation utility) For every (p k, c k ), the Weitzman reservation utility U res k (p k, c k ) is implicitly defined by: c k! = E ( max { (ˆθ k p k ) U res k (p k, c k ), 0 } ) The Weitzman reservation utility U res k (p k, c k ) equates the benefits of inspecting product k with its costs. The Weitzman reservation utility can thus be interpreted as the highest utility that a hypothetical outside option supplies such that the consumer yet prefers to inspect the considered product. Weitzman s rule states that the consumer inspects products with higher reservation utilities first, and ends his search, once the utility that his so far most preferred product supplies exceeds the highest reservation utility of the remaining products. Lemma 1 (Weitzman rule (1979)) An optimal consumer search rule is: i) Selection rule: If a product is to be inspected, it should be a product with the highest reservation utility that has not been inspected before. ii) Stopping rule: Terminate search, whenever the utility that an inspected product supplies or zero exceeds the highest reservation utility of all products that have not been inspected. In that case, purchase an inspected product that supplies the highest utility if the utility exceeds zero, and exit otherwise. The consumer s search rule is unique up to how indifferences are resolved. In a supplementary section at the end of the paper I show that in any equilibrium all payoffrelevant indifferences, those that have strictly positive probability mass, must be (2) 7

8 resolved in favor of the monopolist. In particular, the consumer inspects more expensive products first if several products have equal reservation utilities. Therefore, it is without loss of generality that in the following it is assumed that all indifferences are resolved in favor of the monopolist. 6 For each profile ( p, c), the Weitzman selection rule induces a search order that determines in which order the consumer inspects products. In order to satisfy the selection rule, products must be ordered by their reservation utilities. Definition 2 (Search order) A search order is a bijection φ : K K, where φ(k) denotes the k-th product that the consumer inspects if he does not end his search is de- earlier. A profile ( p, c) induces the search order φ if { U res φ(k) (p φ(k), c φ(k) ) } k K creasing. Alternatively, a search order can be understood via its inverse: Product k is the φ 1 (k)-th product to be inspected. 4 The monopolist s behavior 4.1 The monopolist s search costs strategy Intuitively, the consumer never purchases a product that supplies strictly less utility than the utility which his outside option supplies. As a consequence, for a price profile p the best that the monopolist can achieve is that the consumer purchases her most expensive product that supplies positive utility given the realized matchvalues. The corresponding expected profits π ( p) constitute hence an upper bound on her profits generated by the price profile p. Definition 3 (Upper-bound-profits) For a price profile p, the upper-bound-profits π( p) are defined as the expected profits that the monopolist obtains if the consumer purchases a most expensive product that supplies positive utility: π ( p) := E ( max k K { pk p k ˆθ k }) 6 Note that if indifferences are resolved in favor of the monopolists, the existence of an equilibrium in pure strategies is guaranteed, since the monopolist s profits π( p, c), given the consumer s search rule, are upper semi-continuous in ( p, c). 8

9 My first result is that the monopolist can guide the consumer s search and deteriorate his expected utility of continued search in a manner, so that for any price profile, she can achieve the corresponding upper-bound-profits π ( p). Beforehand, I introduce the obfuscation strategy of the monopolist. Definition 4 (Obfuscation) The monopolist obfuscates product k if she chooses the search costs c k (p k ), which are defined as: c k (p k ) := E (max { (ˆθ k p k ), 0 } ) (3) ( Equivalently, c k (p k ) solves U res k pk, c k (p k ) )! = 0. A product is thus obfuscated if its reservation utility is zero. Recall that the consumer never inspects a product that has a reservation utility that is strictly below zero, the utility which his outside option supplies. An alternative interpretation is hence that a product is obfuscated if the monopolist chooses, conditional on the product s price, the highest search costs such that the consumer yet prefers inspecting the product to exiting. Theorem 1 For any price profile p, the monopolist obtains the corresponding upperbound-profits π ( p) if all products are obfuscated. There are two reasons why obfuscation enables the monopolist to achieve the upperbound-profits. First, obfuscation results in the reservation utilities of all products being equal. This ensures that the consumer is indifferent about the order of inspection, so that it is as well optimal for the consumer to inspect more expensive products first. Second, due to obfuscation reservation utilities are equal to zero. As a consequence, as soon as the consumer finds a product for which his valuation is at least equal to its price, he purchases the product, since the utility it supplies exceeds the reservation utility of all remaining products. Thus, through obfuscation the monopolist deteriorates the consumer s benefits of continued search, so that the consumer purchases the first product that he finds that supplies positive utility. With theorem 1 in mind, the approach taken here to find the monopolist s equilibrium strategy profile becomes evident. I identified an upper bound on the monopolist s expected profits for each price profile and established that there exists 9

10 a search cost strategy that enables the monopolist to achieve these upper-boundprofits. It follows that in any equilibrium the monopolist must obtain the highest upper-bound-profits among all price profiles. This observation allows me to characterize the monopolist s strategy further. 4.2 Equilibrium characterization Consider as a first best case an informed monopolist and consumer who both know all match-values the consumer s valuations for the monopolist s products. The monopolist could then condition her prices on match-values. For instance, she could charge for each product a price equal to the highest match-value of all products. This would ensure that the first best allocation would be implemented, as the consumer would purchase a product that, absent prices, supplies the highest utility. The monopolist s first best profits are equal to the highest match-value of all products and the monopolist extracts the whole surplus generated from trade. 7 In my model, the monopolist is and remains uninformed, and thus cannot condition her prices on realized match-values. As the consumer only purchases a product that supplies positive utility, for any price profile the monopolist s expected profits therefore cannot exceed the corresponding upper-bound-profits. The highest profits attainable for the monopolist, referred to as second best profits, coincide, hence, with highest upper-bound-profits that are generated by some price profile. Definition 5 (Second best profits) The second best profits π are defined as: π := max p R K + π( p). The first best profits exceed the second best profits. As will be discussed later on, this can be partially explained by that the consumer obtains an information 7 If in my model an uninformed monopolist could choose any selling mechanism, then she could obtain in expectation the first best profits. The monopolist could charge the expected value of the highest match-value among all products as a lump-sum upfront fee that allows the consumer upon payment to learn all match-values of products for free, among which he then could pick his preferred product. The underlying reason is that the consumer is initially uninformed, so that the consumer consequently obtains no information rent. 10

11 rent, since learned match-values are his private information, so that the monopolist cannot extract the full surplus due to trade. 8 As indicated before, a corollary to theorem 1 is that a monopolist s strategy is an equilibrium strategy if and only if it yields the second best profits. Corollary 1 There exists an equilibrium. A monopolist s strategy profile is an equilibrium strategy if and only if it generates the second best profits π. One equilibrium strategy of the monopolist is to choose a price profile that maximizes π( p) and to obfuscate each product. This equilibrium has further noteworthy properties. Not only does the consumer purchase a most expensive product that supplies positive utility, but he also inspects more expensive products first, never returns to a previously inspected product, and obtains an expected utility of zero. Recall that the latter holds, since the Weitzman reservation utility of an obfuscated product is equal to zero, so that the consumer is ex ante indifferent about whether to exit or not. The next proposition clarifies to what extent these properties hold in general. Theorem 2 The following properties hold in any equilibrium: i) The consumer buys a most expensive product which supplies positive utility. ii) Each product is bought with strictly positive probability. iii) The consumer inspects more expensive products first. iv) Each product, except a most expensive one, must be obfuscated. v) The consumer never returns in order to purchase a strictly more expensive product that has been inspected before. Furthermore, there exists an equilibrium in which the consumer s expected utility is zero. The first result is an implication of the fact that the monopolist obtains the second best profits in equilibrium. The monopolist must consequently obtain the upperbound profits that correspond to the equilibrium price profile such that the consumer must purchase a most expensive product which supplies positive utility. 8 The other reason is that in equilibrium the first best allocation is not implemented. 11

12 Result ii) hinges on the assumption that each product, absent prices, may be preferred to any other. This assumption implies that the monopolist s optimal price profile must be such that each product is the most expensive product that supplies positive utility with strictly positive probability. Each product is consequently bought with strictly positive probability in any equilibrium. The results iii) and iv) capture that it is necessary equilibrium property that the consumer s search is guided by the monopolist to more expensive products first and that to intense consumer search is discouraged through obfuscation, as was argued before. Result v) is a corollary to the observation that almost all products are obfuscated. If the next product in the consumer s search order is obfuscated, then the consumer immediately purchases the product under consideration if it supplies positive utility. Since the consumer never purchases a product that supplies strictly negative utility, the consumer never returns. Result v) correctly suggests that all results would not change if the consumer had imperfect recall. 9 Finally, let me emphasize the last result, that there exists an equilibrium in which the consumer s expected utility is zero equal to the utility that his outside option supplies. 10 In my model this cannot be interpreted as usually as that the monopolist can extract the whole surplus from trade, which is here the match-value of the product that is purchased by the consumer. As the consumer purchases a product whose match-value strictly exceeds its price with probability one, the consumer obtains some part of the surplus. As this depends on that the monopolist is uninformed about the consumer s valuations for her products, this part of the surplus can be interpreted as the consumer s information rent. Consequently, an equilibrium in which the consumer expected utility is zero can be interpreted as an equilibrium in 9 If the consumer had imperfect recall, so that he could not return to any previously inspected product, all results remain valid. This is the case, since first the consumer evidently has no incentive to return in equilibrium, and second, by the definition of the second best profits, there cannot exists a profitable deviation for the monopolist from the equilibrium strategy. 10 The result remains valid if the consumer s outside option supplies a non-zero utility. In that case it is optimal for the monopolist to choose search costs such the reservation utility of all products is equal to the utility that the consumer s outside option supplies. Thus, more generally, there always exists an equilibrium in which the consumer does not obtain any utility in excess of the utility that his outside option supplies. 12

13 which the consumer s information rent is equal to his expected search costs. The equilibrium is not unique. Theorem 1, however, implies that the equilibrium is unique up to the search costs of the most expensive product if two conditions hold: argmax p R+ Kπ ( p) is a singleton and the most expensive product is unique; both is generically the case. Then, each product, but the most expensive one, is obfuscated, while the search costs for the most expensive product are arbitrary to the extent that they only have to satisfy that the corresponding reservation utility is positive. This suffices to ensure that it is inspected first. Note, however, that these search costs pin down the consumer s expected utility. Only if the monopolist obfuscates all products, then the consumer s expected utility is zero. 4.3 The monopolist s pricing strategy Two questions have not been answered so far. First, how does the exact equilibrium price profile look like? Second, to what kind of products is the consumer guided first by the monopolist? In other words, what is the equilibrium search order? The two questions are, however, linked to each other. Since in equilibrium the consumer inspects more expensive products first, the consumer s search order follows immediately from the monopolist s price profile. Vice versa, the equilibrium search order can be understood as a partial characterization of the equilibrium price profile, as it pins down which products are more expensive. In this subsection, I show that prices can actually be readily determined given the consumer s equilibrium search order. To begin with, I introduce some further notation. Define the monopoly profits π M (p k, F k, π cont ) as the profits that a monopolist obtains if she sells product k at p k, whenever the product supplies positive utility, and otherwise obtains the continuation profits π cont. π M (p k, F k, π cont ) := p k F k (p k ) + F k (p k ) π cont. Proposition 1 Let φ be the consumer s equilibrium search order. Then, the equilibrium price profile p is a solution to the recursive formula that is defined by the 13

14 equations (4), (5) and (6): π K+1 := 0, (4) π k := πm (p φ(k), F φ(k), π k+1 ), (5) p φ(k) argmax p R + π M (p, F φ(k), π k+1 ). (6) The monopolist s profits are π = π 1. The profits π k denote the monopolist s expected equilibrium continuation profits if the consumer does not purchase any of the first k 1 inspected products. If the consumer purchases no product at all, then the monopolist s profits are zero: π K+1 = 0, as stated in equation (4). This is the initial condition of the recursive equation (5), which determines the continuation profits. Its meaning becomes clear if one assumes that all products are obfuscated. Then, whenever the consumer does not purchase any of the first k 1 inspected products, the monopolist either sells the k-th inspected product at p φ(k), whenever it supplies positive utility, and otherwise obtains the continuation profits π k+1. Thus, π k := πm (p φ(k), F φ(k), π k+1 ). Finally, the price of the k-th inspected product, then, does not affect the consumer s demand for the first k 1 inspected products. The monopolist chooses hence the price of the k-th inspected product such that it maximizes the continuation profits π k 1, as stated in equation (6). Proposition 1 not only provides a recursive formula to find the equilibrium price profile given the equilibrium search order, but also suggests an algorithm that can be used to determine the equilibrium search order in the first place. Note that for an arbitrary, non-equilibrium search order φ a solution to the corresponding recursive formula, where in the equations (4), (5) and (6) φ is replaced by φ, is well-defined. The equilibrium search order can then be identified as the search order that yields the highest profits, as the following two-product example illustrates. 11 Example: Two products with uniformly distributed match-values 11 A proof of this claim can be found in a supplementary section at the end of the paper. More general, this procedure will not be analytically tractable, since the number of candidate search orders grows at factorial speed. 14

15 Consider a monopolist that offers two products. Suppose that the consumer s valuation for each product is distributed uniformly: F k (θ k ) = a kθ k. Suppose a 1 8 < a 2 < 8a Suppose that product 1 is inspected first in equilibrium. Then, solving the recursive formula of proposition 1 yields the equilibrium prices and profits: p 2 = 1 4a 2 (7) p 1 = 1 32a a 1 (8) π 2 = 1 16a 2 (9) π 1 = a a 1a a a 1 a 2 2 (10) where π = π 1 are the equilibrium profits of the monopolist. The equilibrium search order can be determined by solving for any search order the corresponding, recursive formula of propositon 1. For a 1 < a 2, simple algebra yields that in equilibrium the consumer is guided to product 1 first. 13 In the considered two product example with uniformly distributed match-values the product with the higher variance in the consumer s valuation is inspected first. This is not a coincidence, as will be shown in the following subsection. The notion of a more polarizing product will be introduced and conditions are provide which ensure that more polarizing products are inspected first. For instance, for an arbitrary number of products with uniformly distributed match-values, those products with a higher variance in the consumer s valuation are more polarizing and are always inspected first. 4.4 To which products is the consumer guided first? While so far no restrictive assumptions with regard to the heterogeneous products, that the monopolist offers, have been made, I now impose some structure on these, which allows me to characterize the equilibrium search order. 12 A technical remark: The assumption is a necessary and sufficient assumption which ensures that the maximization problem of the monopolist has an interior solution. 13 The profits if product 1 is inspected first are given by equation (10). Relabeling of a 1 and a 2 in equation (10) yields the profits if product 2 is inspected first. Some algebra then yields the desired inequality. 15

16 4.4.1 Two products I begin with an analysis of a monopolist that offers two products, of which one product is ex ante better than the other one in the sense that the corresponding distribution function first order stochastic dominates its counterpart. 14 Later on, I examine more general cases and consider as well an arbitrary number of products. These later results, however, will build on some of the ideas presented in the following. Lemma 2 Suppose the monopolist offers two products. Let ( p, c ) be the monopolist s equilibrium strategy profile. Suppose p 1 p 2 and let P Supp(F 1) Supp(F 2 ) be a price interval such that p i P for i {1, 2}. Suppose that product 1 is better than product 2 in the sense that F 1 (θ) F 2 (θ) for any θ P. i) The better product is inspected first if f k (θ)/f k (θ) is strictly increasing in k. ii) The better product is inspected second if f k (θ)/ F k (θ) is decreasing in k and increasing in θ. 15 Let me explain lemma 2 on the basis of two distinct two-product examples that are illustrated in figure 1. The example on the LHS satisfies the conditions of i) in lemma 2, so that the better product is inspected first. The example on the RHS satisfies the conditions of ii) in lemma 2, so that the better product is inspected second. 14 In the following, I say that a product k is better than a product l if F k (θ) F l (θ) for all valuations which are considered in the respective case. 15 The latter condition, f k (θ)/ F k (θ) increasing in θ, is the usual increasing hazard rate condition. There exists an elegant interpretation of the two conditions of ii) in Lemma 2. Suppose that the monopolist s demand for the two products is independent and described by the demand functions F i. Suppose furthermore that each F i has increasing hazard rates and that the monopolist faces equal marginal costs for both products. Then, f k (θ)/ F k (θ) decreasing in k is a necessary and sufficient condition to guarantee that the monopolist sells the better product at a lower price under the assumption of an interior solution. In other words, the conditions that ensure that the better product is inspected second and sold at a lower price, are as well sufficient and necessary conditions which guarantee that the monopolist would prefer to sell the product at a lower price if the demand for the two products was independent. 16

17 F k (θ) F k (θ) 1 F 2 1 F 2 F 1 F 1 0 p 2 p 1 θ 0 0 p 1 p 2 θ Figure 1: Uniform distributions & first order stochastic dominance Both plots show in different shades of blue the distribution function of two products, for which the consumer s valuations are uniformly distributed. In either example, product 1 is ex ante better than product 2 in the sense that F 1 first order stochastic dominates F 2. Equilibrium prices are shown on the horizontal axis and are indicated by full dots. The example shown on the LHS satisfies the conditions of i) in lemma 2, so that the better product is inspected first. The example shown on the RHS satisfies the conditions of ii) in lemma 2, so that the better product is inspected second. In either example, empty circles illustrate a particular non-equilibrium price profile for which red arrows illustrate a profitable deviation of the monopolist, a switch of prices, that is at the heart of the proof of lemma 2. Steeper - polarizing mention. Consider the example that is shown on the LHS of figure 1. Suppose that the better product was inspected second and hence sold at a lower price. For instance, as is indicated by empty circles, the true equilibrium prices were interchanged. This was an equilibrium in which the better product is sold at a lower price as part of a safety strategy, which ensures that the consumer purchases the product that he inspects last with a high conditional probability. In contrast, after a switch of prices, which is indicated by arrows, the better product is sold at a higher price as part of a of long-shot strategy, which increases the chances of selling the first inspected product at a high price. Whether this switch of prices a switch from a safety strategy to a long-shot strategy is profitable, intuitively, depends on whether F 2 (θ) F 1 (θ) is increasing or decreasing in θ. The example that is shown on the LHS of figure 1 illustrates the case in which F 2 (θ) F 1 (θ) is sufficiently increasing such that a switch from a safety strategy to a long-shot strategy is profitable. This implies that the better product must be 17

18 inspected first in equilibrium. More precisely, F 2 (θ) F 1 (θ) is sufficiently increasing if F 2 is relatively more increasing than F 1 : F 2 (θ)/f 1 (θ) strictly increasing in θ. The example that is shown on the RHS of figure 1 illustrates the opposite case. A switch from a long-shot strategy to a safety strategy is profitable if F 2 (θ) F 1 (θ) is sufficiently decreasing, which is the case if the reliability function F 2 is relatively more decreasing than its counterpart F 1 : F 2 (θ)/ F 1 (θ) strictly increasing in θ. The better product must, then, be inspected second. The intuition for both results is the following. The provided conditions imply that the distributions function of one product is relatively steeper than the one of the other product. Loosely speaking, this implies that a decrease in prices results in a higher relative increase in probability, that the consumer s valuations exceeds some price, for the product whose distribution function is steeper. As a consequence, it is more profitable to lower the price of the product whose distribution function is steeper such that this product is sold in equilibrium at a lower price. Lemma 2 already points at results which are yet to come. Note that a product whose distribution functions is steeper can be interpreted as being less polarizing. An alternative interpretation of lemma 2 is thus that the more polarizing product is inspected first Multiple products The first definition describes a single crossing property of several distribution functions. The property has already appeared before in Courty and Li [2000] and in Johnson and Myatt [2006] to describe demand rotations. 16 Definition 6 (Single Crossing Property) A family of distribution functions {F k } k K satisfies the single crossing property if there exists a single crossing (θ cross, F cross ) such that for k < l: θ θ cross F k (θ) F l (θ) 16 For a discussion of the connection to related properties as mean preserving spreads and second order stochastic dominance, I refer to the discussion in Johnson and Myatt [2006]. 18

19 This means that for each k < l the distribution F l can be generated by a counter clockwise rotation of F k, which is illustrated in figure 2. Figure 2: Uniform distributions The figure shows in different shades of blue three uniform distribution functions: F k (θ k ) = F cross + a k (θ k θ cross ), from which the consumer s valuations for three heterogeneous products are drawn. a k is increasing in k, so that {F k } k K satisfies the single crossing property, which is indicated by arrows. The example satisfies the conditions of theorem 3 such that more polarizing products are inspected first and are thus more expensive. F k (θ) 1 0 p 3 θ cross p 2 Crossing Point 0 p 1 F 3 F 2 F 1 θ I refer to a product whose distribution function is flatter at the point of crossing as more polarizing. Definition 7 (Polarizing product) Suppose {F k } k K satisfies the single crossing property. Then, product k is more polarizing than product l if k < l. This definition captures that for a more polarizing product very high and very low match-values are more likely to occur. In the following theorem, I provide conditions which guarantee that polarizing products are inspecteded first. Theorem 3 Let {F k } k K be generic and ordered by a sequence of rotations. If the following two conditions hold: i) f k (θ)/f k (θ) is strictly increasing in k for θ θ cross, 17 ii) f k (θ)/ F k (θ) is strictly increasing in k and in θ for θ θ cross, 18 then the consumer is guided to more polarizing products first. 17 The condition f k (θ)/f k (θ) strictly increasing in k is equivalent to F k (θ) strictly logsupermodular. 18 The condition f k (θ)/ F k (θ) strictly increasing in k is equivalent to F k (θ) strictly log-submodular. The condition f k (θ)/ F k (θ) increasing in θ is the usual increasing hazard rate condition. 19

20 The proof of theorem 3 consists of two steps, which I will outline in the following and which will provide intuition for the derivation of the theorem. A discussion follows at the end of this subsection after an example. The first step of the proof is to show that each product whose price exceeds θ rot is more polarizing than each other product whose price is below θ rot. The intuition why this must hold is simple. Suppose that product k is more polarizing than product l. Note that less polarizing products are better for prices below θ rot, while more polarizing products are better for prices above θ rot. As a consequence, if the price of product l was above θ rot and the price of product k below θ rot, it would be profitable to switch the prices of the two products, and consequently their positions in the consumer s search order. The second step builds on the ideas used to prove lemma 2. Consider those products whose prices exceed θ rot. Note that for prices above θ rot, more polarizing products are better. If more polarizing products were not inspected first, then there would exist two products which have neighboring positions in the consumer s search order and which satisfy that the less polarizing product is inspected first. This will allow us to apply a more general version of lemma 2, which considers pairwise switches of prices and positions in the consumer s search order for an arbitrary number of products. 19 This will yield that among those product whose price exceeds θ rot the better product, which is the more polarizing product, is inspected first if f k (θ)/ F k (θ) is strictlyincreasing in k for θ θ rot. The proof, that among those product whose price is below θ rot the more polarizing product is inspected first, uses a similar structure and builds as well on the ideas presented in lemma 2. Note that for prices below θ rot, less polarizing products are better. The following example describes a class of families of distribution functions that satisfy the conditions of theorem 3, so that the more polarizing products are inspected first. Example cont : An arbitrary number of heterogeneous products with uniformly distributed match-values. 19 In fact, lemma 2 is proven as a corollary to this auxiliary lemma, which is delegated to the appendix. 20

21 Consider a monopolist that sells an arbitrary, finite number of products, for which the consumer s valuations are uniformly distributed and satisfy the single crossing property. Formally, let the family of distribution functions {F k } k K satisfy F k (θ k ) = F cross +a k (θ k θ cross ), where a k is strictly increasing in k. Special cases are illustrated in figure 2 and in figure 1. The latter example illustrates that the family encompasses sets of products, for which the corresponding distribution functions can be ordered by first order stochastic dominance. The family {F k } k K satisfies the conditions of theorem 3: First, F k (θ k ) is strictly log-supermodular, since simple algebra yields d 2 da k dθ k logf k (θ k ) = F cross (F k (θ k > 0, which is equivalent to strict log-supermodularity, )) 2 d since a k is strictly increasing in k. Second, 2 da k dθ k log F k (θ k ) = (1 F cross) F k (θ k < 0 implies ) 2 analogously that F k (θ k ) is strictly log-submodular. Finally, the uniform distribution has increasing hazard rates, since its probability density function is log-concave. 20 Consequently, theorem 3 applies and more polarizing products are thus unambiguously inspected first. The intuition for theorem 3 is that the monopolist guides the consumer to polarizing products first in an attempt to sell possibly highly valued products at high prices. In contrast, the monopolist guides the consumer to less polarizing products last, since she is less uncertain about the consumer s valuation for these products. These product are consequently well suited to be used as part of a last resort safety strategy, which guarantees the monopolist that she can sell these products at moderate prices with high probability. An important takeaway from this last subsection is that, maybe in contrast to a first guess, the consumer should be guided to the best products nor the worst products first, but to those products which are polarizing and which have a high variance in the consumer s valuation. 5 Conclusion This paper has shown that a monopolist should and can guide the consumer s search by obfuscation. I identified two motives that render obfuscation profitable to the monopolist. First, through obfuscation the monopolist guides the consumer s search 20 For instance, see Bagnoli and Bergstrom [2005]. 21

22 to more expensive products first and second, she discourages to intense consumer search. This allows her to claim larger shares of the surplus from trade and enables her to achieve the highest attainable profits given that the consumer s learned valuations for her products are his private information. For the consumer obfuscation is bad. In one equilibrium his expected utility even does not exceed the guaranteed utility that his outside option supplies. The paper shows that the monopolist guides the consumer to more polarizing products first. She does so as part of a long-shot strategy an attempt to sell polarizing products, which are more likely to be highly valued, at high prices. In contrast, less polarizing products are lower priced as part of a last resort safety strategy that ensures that the consumer purchases some product before exiting empty handed. 22

23 References Simon P. Anderson and Régis Renault. Pricing, Product Diversity, and Search Costs: a Bertrand-Chamberlin-Diamond Model. The RAND Journal of Economics, 30(4): , Mark Armstrong, John Vickers, and Jidong Zhou. Prominence and Consumer Search. Rand Journal of Economics, 40(2): , Susan Athey and Glenn Ellison. Position Auctions with Consumer Search. The Quarterly Journal of Economics, 126(3): , Mark Bagnoli and Ted Bergstrom. Log-Concave Probability and its Applications. Economic Theory, 26(2): , Kenneth Burdett and Kenneth L. Judd. Equilibrium Price Dispersion. Econometrica, 51(4): , Gerard R. Butters. Equilibrium Distributions of Sales and Advertising Prices. Review of Economic Studies, 44(3): , Bruce I. Carlin. Strategic Price Complexity in Retail Financial Markets. Journal of Financial Economics, 91(3): , Bruce I. Carlin and Gustavo Manso. Obfuscation, Learning, and the Evolution of Investor Sophistication. The Review of Financial Studies, 24(3): , Yongmin Chen and Chuan He. Paid Placement: Advertising and Search on the Internet. The Economic Journal, 121: , Ioana Chioveanu and Jidong Zhou. Price Competition with Consumer Confusion. Management Sciene, 59(11): , Pascal Courty and Hao Li. Sequential Screening. The Review of Economic Studies, 67(4): , Peter A. Diamond. A model of price adjustment. Journal of Economic Theory, 3 (2): , June

24 Glenn Ellison and Sara Fisher Ellison. Search, Obfuscation, and Price Elasticities on the Internet. Econometrica, 77(2): , Glenn Ellison and Alexander Wolitzky. A Search Cost Model of Obfuscation. Rand Journal of Economics, 43(3): , Xavier Gabaix and David Laibson. Shrouded Attributes, Consumer Myopia, and Information Suppression in Competitive Markets. The Quarterly Journal of Economics, 121(2): , Tobias Gamp. Search, Differentiated Products, and Obfuscation. Bonn Mimeo, pages 1 51, Sanford J. Grossman. Warranties and Private Disclosure About Product Quality. Journal of Law and Economics, 24(3): , Justin P. Johnson and David P. Myatt. On the Simple Economics of Advertising, Marketing, and Product Design. The American Economic Review, 96(3): , Paul R. Milgrom. Good news and Bad News: Representation Theorems and Applications. The Bell Journal of Economics, 12(2): , Michele Piccione and Ran Spiegler. Price Competition under Limited Comparability. Quarterly Journal of Economics, 127:97 135, Ran Spiegler. Competition over Agents with Boundedly Rational Expectations. Theoretical Economics, 1: , Dale O II Stahl. Oligopolistic Pricing with Sequential Consumer Search. The American Economic Review, 79(4): , Martin L. Weitzman. Optimal Search for the Best Alternative. Econometrica, 47 (3): , Chris M. Wilson. Ordered Search and Equilibrium Obfuscation. International Journal of Industrial Organization, 28(5): ,

25 Asher Wolinsky. True Monopolistic Competition as a Result of Imperfect Information. The Quarterly Journal of Economics, 101(3): , Jidong Zhou. Ordered Search in Differentiated Markets. International Journal of Industrial Organization, 29(2): ,

26 6 Appendix Proof of lemma 1: A proof can be found in Weitzman (1979). Proof of theorem 1: Consider an arbitrary price profile p, and suppose that the monopolist obfuscates each product: c k = c k (p k ) for every k. Thus, the reservation utility of each product is zero: U res k (p k, c k ) = 0 for all k, and the consumer is indifferent about in which order to inspect products or whether to exit. Since indifferences are resolved in favor of the monopolist, the consumer inspects more expensive products first and the resulting price sequence, induced by the consumer s search order, is decreasing. If the consumer finds then a product that supplies positive utility, he ends his search and purchases the product. This holds, as the product supplies a utility that exceeds the highest reservation utility of all products that have not been inspected, which is equal to zero. Therefore, the consumer purchases the most expensive product that supplies positive utility, so that the monopolist s expected profits are π ( p). Proof of corollary 1: The claim is obvious, except that it remains to be shown that argmax p R K + π( p) is non-empty, which implies the non-emptiness of Σ and that π is well-defined. Let θ := max k K {θ k }. Define P 2θ := { p R+ p K k 2θ for every k} and note that it is a compact set. Then, since π is continuous, a maximum of π on P 2θ exists by the Bolzano-Weierstrass theorem. Moreover, the maximum is a global maximum, since π( p) = 0 for every p R K +\P 2θ. Proof of theorem 2: Let ( p, c ) be a monopolist s equilibrium strategy profile. i): By corollary 1, the monopolist obtains the second-best profits in equilibrium. The monopolist thus obtains the upper-bound profits π( p ) and therefore the consumer purchases one of the most expensive products which supply positive utility. ii): Assume the contrary i.e. that for some ( p, c ) product l is not bought in equilibrium. Since under p product l is not bought in equilibrium, it is never the only most expensive product that supplies positive utility by i) in theorem 2. If there exists a price profile p that only alters the price of product l and under 26