A Model of Recommended Retail Prices - Kelley School of Business [PDF]

Oct 31, 2016 - price recommendations and market outcomes has both been shown ... a price ceiling directly.1 In addition,

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A Model of Recommended Retail Prices Dmitry Lubensky∗ October 31, 2016

Abstract Manufacturers frequently post non-binding public price recommendations, but neither the rationale for this practice nor its impact on prices is well understood. I develop a model in which recommendations signal a manufacturer’s production cost to searching consumers, who then form beliefs about retail prices. Increasing search makes consumers reject offers for the manufacturer’s and competitors’ products more often, and I show that both consumers and the manufacturer prefer more search when the production cost is low and less search when it is high. With incentives thus aligned, manufacturer recommendations inform consumers via cheap talk and their removal harms both parties.

Keywords: consumer search, sequential search, search with uncertainty, manufacturer suggested retail prices, vertical markets, signaling and cheap talk



Indiana University, Kelley School of Business; [email protected].

Thanks to Josh Cherry, Kai-Uwe K¨ uhn, Francine Lafontaine, Stephan Lauermann, Doug Smith, and Mike Stevens for insightful feedback on the early version of this article. Thanks also to Mike Baye, Rick Harbaugh, John Maxwell, Jeff Prince, Michael Rauh, Eric Rasmusen, Collin Raymond, Babur de los Santos, Matthijs Wildenbeest and the participants of the Consumer Search and Switching Cost Workshops in 2013 and 2014 for helpful comments on this version.

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Introduction

Manufacturers use non-binding recommended retail prices in markets ranging from grocery products to big ticket items such as electronics, appliances, and cars. These recommendations, which come in a variety of forms such as list prices, manufacturer suggested retail prices (MSRPs), sticker prices, etc., are visibly printed on product packaging and often promoted by the manufacturer through costly advertising. The existence of a link between price recommendations and market outcomes has both been shown empirically (e.g. Faber and Janssen (2008), De los Santos et. al. (2016)) and also implicitly assumed in the myriad studies that use recommendations as a proxy for transaction prices (e.g. Berry et al. (1995)). There is also anecdotal evidence that recommendations can directly affect the decisions of market participants, for example, consumers often expect a discount off the MSRP when buying a new car and strategic dealers take this into account as they set prices. However, despite the evidence that price recommendations affect behavior, our understanding of how they do so is quite limited. Due to the fact that price recommendations are non-binding, the mechanism by which they have an impact and the motives of the manufacturer in making these recommendations are still not well understood. A common explanation is that recommendations act as price ceilings. This story is compelling because most products sell at or below MSRP and because the manufacturer’s rationale for imposing a price ceiling in order to reduce double marginalization is well established. Yet this explanation of recommendations is incomplete. First, price recommendations are not binding, at least in name, and thus in jurisdictions where resale price maintenance is legal it is not clear why a manufacturer would make a recommendation instead of imposing a price ceiling directly.1 In addition, recommendations often do not bind in practice, for instance most cars sell strictly below MSRP but very few sell at MSRP. Lastly, manufacturers publicize their recommendations and an explanation of recommendations as explicit price ceilings ignores the potential role played by consumers. This article presents an alternative explanation in which price recommendations affect consumer search. I develop a model in which a manufacturer faces consumers with unit demand who search among sellers of his product and sellers of competing products, observing a price and a utility shock at each visit. The manufacturer’s cost is uncertain and thus consumers do not know the distribution of downstream prices. In particular, a consumer visiting a seller does not know whether it is optimal to accept the current offer or continue searching. A price recommendation informs consumers of the manufacturer’s cost, and thus of the continuation value of rejecting. In this way recommendations directly affect consumers’ search decisions, and by extension the prices set by retailers. 1

For instance, MSRPs are commonly used in the United States where the Supreme Court ruled in 1997 in State Oil Co. versus Kahn that price ceilings are not inherently unlawful.

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Whereas it benefits consumers to incorporate informative recommendations into their search decisions, the incentives for a manufacturer to make these recommendations are not immediately clear. I demonstrate that in revealing his cost and thereby affecting search the manufacturer faces a tradeoff. Specifically, consider consumers A and B each currently marginal at a manufacturer’s retailer and at a competitor. By making both consumers more optimistic about the distribution of downstream prices and thus continue searching, the manufacturer reduces his probability of selling to A from one to the continuation probability and increases his probability of selling to B from zero to the continuation probability. More search thus benefits the manufacturer if and only if the continuation probability is sufficiently high. With these incentives for affecting search, the question is whether the manufacturer can use price recommendations to credibly reveal his costs. A price recommendation is pure cheap talk because it is not a falsifiable statement of fact2 and it is no costlier to recommend a higher price than a lower one. If the manufacturer had incentive to mislead consumers, they in turn would rationally ignore recommendations and no information could be transmitted. In order for non-binding non-falsifiable recommendations to have an effect, the manufacturer must prefer to report truthfully. The article’s main result is that with respect to search the interests of consumers and the manufacturer are aligned and thus credible signaling is possible. In the model the manufacturer’s cost has either a low or a high realization. When the cost is low the manufacturer sets a low wholesale price which leads to low retail prices, in turn making his product more attractive on average than the products of competitors. The low cost manufacturer thus faces a high continuation probability that a searching consumer eventually buys from him, and consequently prefers more search. Therefore, he truthfully reveals his low cost and leads consumers to expect low prices. Similarly, when his cost is high the manufacturer sets a high wholesale price, making his product on average less attractive than competing products and thus inducing a low continuation probability. In this case the manufacturer benefits from less search and reveals his cost truthfully to induce consumers to expect higher prices. Because the manufacturer has incentive to reveal truthfully in both states, he can inform consumers using cheap talk. This ability to credibly communicate the continuation value of searching is unique to the manufacturer. By contrast, a downstream seller always benefits by making visiting consumers maximally pessimistic about their outside options regardless of the true state. The model thus predicts that whereas manufacturer price recommendations influence consumer 2

It is difficult to hold a manufacturer liable when he recommends a price and a retailer ignores the recommendation. By contrast, other statements on product packaging, such as “New York Times Best Seller” on a book jacket, are factual and punishable by law if false.

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search, cheap claims by direct sellers about how their product compares to competitors should have little effect. Manufacturer price recommendations have traditionally received little explicit attention and instead have been lumped in as an instrument of resale price maintenance in the literature on vertical price restraints. However, there has been renewed interest in the topic and several recent articles have explored the mechanism behind the effects of these non-binding recommendations. Buehler and G¨artner (2013) shows that MSRPs can be used by a manufacturer to convey demand and cost information to a retailer. The manufacturer and retailer are engaged in an infinitely repeated game and there exists an equilibrium in which both aim to maximize joint profits. Because in this equilibrium incentives are aligned, the manufacturer can credibly inform the retailer using cheap talk. Although the authors establish that credible communication between a manufacturer and retailer is possible, it is not immediately clear why publicizing MSRPs is the preferred method for doing so. In particular, public price recommendations are also likely to affect the behavior of consumers and the authors abstract from this effect. A different approach is taken by Puppe and Rosekranz (2011), building on the theory in Thaler (1985) in which a price recommendation provides a behavioral reference point. Consumers’ demand is kinked above the MSRP so that the recommendation acts as a de-facto price ceiling. The present article can be thought of as providing a rational foundation for the reference point theory. I explicitly model how consumers incorporate the MSRP and their knowledge of the market to form price expectations, and derive their demand as the solution to the problem of optimal search. A consumer that refuses to accept a price above a certain level does so not only because it feels like a bad deal, but because it actually is a bad deal. The role of MSRPs as information for consumers contributes to the policy debate of whether manufacturer recommendations ought to be regulated. I show that removing recommendations harms both consumers and the manufacturer. In particular, I demonstrate that when recommendations are banned there exists an equilibrium with the same wholesale and retail prices as the equilibrium with recommendations, but in which consumers are less informed during search. In this equilibrium consumers learn about the manufacturer’s cost through the price charged by his retailers, however there is a group of consumers that has not yet visited any retailers and remains uninformed. Upon visiting a competitor these consumers search too little when the cost is low and too much when the cost is high, and are thus worse off in both states. Meanwhile the manufacturer faces a tradeoff with this group: he benefits from consumers being too selective at competitors when his cost is high and is harmed by consumers not being selective enough at competitors when his cost is low. Because the value of inducing these consumers to continue searching equals the continuation probability, the effect in the low cost state dominates and on net the manufacturer is worse off. Broadly speaking, taking away the ability of the manufacturer and consumers to coordinate harms 3

both because their interests are aligned. Whether sellers can cheaply communicate with searching consumers was studied in a related setting by Horowitz (1992), who provides evidence of a relationship between non-binding list prices and transaction prices in the market for residential housing. Kim (2012) then develops a theoretical foundation, showing that when sellers have private information about product quality, there exists a partially informative mixed strategy equilibrium in which sellers that report a high quality attract fewer visitors and command a higher price than sellers that report a low quality. In this equilibrium, a buyer’s expected utility from visiting a seller reporting a high quality equals that of visiting a seller reporting a low quality, thus misreporting does not improve a seller’s ranking among his competitors. By contrast, in the current context no such indifference can be obtained because a downstream seller always benefits from understating a visiting consumer’s future utility offers. Thus it is solely the manufacturer and not the downstream retailers that can credibly communicate the returns to search. This work also makes a methodological contribution. First, it provides a tractable model of a vertical market with sequential search and endogenous prices. By using the Wolinsky (1986) random utility framework, the equilibrium is in pure strategies and easily amenable to comparative statics. The closest work in this area is Janssen and Shelegia (2015), which instead uses the Stahl (1989) framework to obtain an equilibrium in mixed strategies. Janssen and Shelegia’s central result is that double marginalization is exacerbated in a setting in which consumers search, owing to a holdup argument similar to that in Diamond (1971), and this result also obtains quite naturally in the present model. The articles diverge in that I focus on the manufacturer communicating the realization of uncertain market conditions whereas in Janssen and Shelegia (2015) there is no aggregate uncertainty and thus nothing to communicate. In addition, I derive a simple and intuitive expression for the demand faced by an upstream manufacturer in a search environment, drawing a connection between models of search and contests. The rest of this article proceeds as follows. Section 2 presents the model, Section 3 characterizes the equilibrium when consumers are informed, and Section 4 demonstrates the main results that cheap signaling can credibly convey manufacturer costs to consumers, and that without recommendations consumers and the manufacturer are worse off. Section 5 provides a discussion of whether recommendations can inform consumers about product quality instead of manufacturer costs and draws a comparison between recommendations and resale price maintenance. Section 6 then concludes.

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2

Model

A manufacturer of a product has a constant but uncertain marginal cost, with a low realization cL and a high realization cH , both equally likely. The manufacturer sells his product through a collection of exclusive downstream sellers which face no additional costs. The downstream market also includes exclusive sellers of other products. I refer to the sellers of the manufacturer’s product as retailers and to the sellers of other products as competitors. The total number of downstream sellers is infinite, with retailers and competitors each comprising one half of the market. There is a measure one of consumers with unit demand. A consumer faces the set of downstream sellers, which she may visit sequentially at a cost of s > 0 per observation. A consumer that has visited k sellers and buys from seller j obtains a payoff uij = αi + ηj − pj − sk, in which αi is the quality of product i, ηj is a seller-specific utility shock, and pj is seller j’s price. Assume that both for the manufacturer’s and competitors’ products, αi is publicly observed by all market participants. To fix ideas one may think of the market for an economy sedan with buyers choosing among different cars in this class (i.e. Ford Focus, Nissan Sentra, Honda Civic, etc.) by sequentially visiting dealerships, each exclusive to a particular manufacturer. A consumer visiting a Ford dealership j knows that every Ford Focus in the lot has the same dimensions, weight, factory warranty and other features that are common across all versions of this vehicle, corresponding αF ord in the model. However other features such as color, leather seats, and a sunroof vary across vehicles, and the consumer is uncertain whether dealership j has her preferred combination of these features in stock, which is captured by ηj .3 The analysis focuses on the behavior of the manufacturer and his retailers, whereas competitors are in the model to provide consumers with a stochastic outside option. To this end, if the seller is a retailer then the shock ηj is drawn from distribution F with support [η, η¯] and the price pj is endogenously determined in equilibrium. Meanwhile, if the seller is a competitor the price is unmodeled and the value νi ≡ αi + ηj − pj is exogenous and drawn from distribution G with support [ν, ν¯]. Assume distributions F and G admit continuous and uniformly bounded densities f and g, that f is log-concave and non-decreasing, and that g(ν) the hazard ratio 1−G(ν) is non-decreasing. To ensure that an initial search is optimal assume 1 that s < 2 EG [νi ], so that a consumer’s payoff exceeds s even if she commits to searching 3 When a vehicle with a consumer’s preferred features is not in stock a dealer may order it from the manufacturer. However this is often associated with additional time and monetary costs, thus consumers with different preferences for features differentially value a dealership with a particularly inventory.

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Time Cost realized c ∈ {cL , cH }

Manufacturer w(c), σ(c)

Retailers p(w, σ)

Consumers search A(h, σ)

Figure 1: Model Timing exactly once and accepting at competitors and not at the manufacturer’s retailers. The game proceeds as follows. First, the manufacturer draws and privately observes production cost c ∈ {cL , cH } and chooses a price recommendation σ ∈ {σL , σH } and a wholesale price w ∈ R. Then, every retailer j observes σ and w and sets retail price pj . With retail prices fixed, each consumer observes the manufacturer’s recommendation but not his wholesale price and decides whether to begin searching.4 If she initially decides not to search, she exits and receives a utility of zero. Otherwise, she pays cost s and visits a randomly selected downstream seller, where she observes pj and ηj . The consumer can accept the offer, exit, or pay cost s again and visit a different randomly selected seller. During search a consumer can accept any previous offer at no additional cost and the process continues until she accepts an offer or exits. The search strategy is denoted as A(h, σ) ∈ {accept offer in h, search again, exit}, in which h is a vector of previously observed utility shocks and prices. The timing is summarized in Figure 1. The expected payoff of a retailer j when setting price pj is π(pj |w, σ) = (pj − w)q(pj |w, σ), in which q(pj |w, σ) is the expected demand faced by retailer j in equilibrium. To form this expectation the retailer can use recommendation σ to anticipate consumers’ search behavior and both σ and wholesale price w to anticipate other retailers’ prices. The function q(pj |w, σ) is an equilibrium object and is derived in the ensuing analysis. The manufacturer’s expected payoff is Π(w, σ |c) = (w − c)Q(w, σ), 4

When consumers observe the wholesale price, recommendations serve no purpose because they do not help predict retail prices. However, if the manufacturer had relevant information beyond the wholesale price, for instance about quality, recommendations can play a role even if wholesale prices are observed.

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in which Q(w, σ) is the manufacturer’s expectation of the sum of the sales by all his retailers, another object derived explicitly in equilibrium. The solution concept is perfect Bayesian equilibrium, namely a collection of strategies for the manufacturer (w(c), σ(c)) and retailers p(w, σ) and a search policy for consumers A(h, σ), all satisfying sequential rationality and beliefs that obey Bayes rule whenever possible. The present environment borrows from common constructions in the literatures on vertical relationships and search markets. The downstream game is analogous to the random utility search model in Wolinsky (1986) and Anderson and Renault (1999), and the uniform wholesale price is a contract that, although suboptimal, is frequently observed and widely analyzed, starting with Spengler (1950). There are several additional modeling assumptions which are important for the main result and I briefly elaborate on them here. First, I abstract from specifically modeling the behavior of competitors. It will be shown that when signaling his cost, the manufacturer is only concerned with how the distribution of utilities offered by his retailers compares with that offered by competitors; the mechanism by which the latter is determined is not relevant for his signaling decision. Consequently, the main result applies in a variety of market structures, including but not limited to a symmetric setting in which each downstream seller belongs exclusively to one of several competing manufacturers, all facing the decision problem described above. The key assumption is that G(νj ) is exogenous to the manufacturer, so that he cannot affect the distribution of outside options through his actions. In this sense, the model captures markets in which upstream decisions are made simultaneously or if the manufacturer is the last mover in a sequential game.5 Second, the manufacturer has exactly two cost states and exactly two possible messages. In any equilibrium in pure strategies allowing for as many messages as states is without loss of generality, however that there are exactly two cost outcomes is an assumption made for tractability. It will be argued that for any cost the manufacturer either prefers as much search as possible or as little search as possible. With this in mind, in any cheap talk equilibrium the manufacturer may only reveal whether his cost falls into the former category or the latter. When there are only two costs, one in each category, consumers become perfectly informed and face no aggregate uncertainty. However, if either category includes more than one cost, a searching consumer continues to learn during the search process. This makes her decision problem non-stationary and thereby significantly complicates the analysis. 5

Alternatively, if upstream competitors first observe the manufacturer’s recommendation then they would increase prices when he signals high and reduce prices when he signals low. Reporting truthfully is then even more beneficial when the manufacturer’s cost is high, but now requires a larger cost advantage when his cost is low.

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Third, that the recommendation is observed prior to search rather than during the first store visit is assumed mostly for tractability, else by chance some consumers may visit several competitors prior to their first retailer and tracking these search histories in order to express demand significantly adds to the model’s complexity. The assumption corresponds to a setting in which a manufacturer that already advertises his product decides whether or not to include a price recommendation in the advertisement. Furthermore, the assumption seems not to be crucial for the article’s main result of credible signaling. Specifically, the manufacturer’s tradeoff from inducing more search, that is reducing acceptance probabilities both for his product and for competing products, still remains even if MSRPs are discovered only during search and not beforehand. Lastly, consumers decide whether to accept or reject an offer but not which store to visit next. In a symmetric model this simplifying assumption is benign, and thus it is commonly used in the search literature. However, in the current environment consumers may learn whether the manufacturer’s retailers or competitors are likely to offer a better deal, and consequently may wish to direct their search. A manufacturer that signals a low cost may gain prominence in the search process, which has been shown to both directly increase his share of consumers and also potentially decrease competition (e.g. Armstrong et al. (2009), Wilson (2010), Arbatskaya (2007), Haan and Moraga-Gonz´ales (2011), Fishman and Lubensky (2016)). The random search assumption abstracts from this motive and is thus restrictive, though is justified in some situations. For instance, in a spatial model the order of search may simply be determined by store locations. Alternatively, if a consumer is unaware of which sellers carry which products then the decision of which store to visit next is independent of which product she is likely to prefer. The analysis proceeds as follows. I first solve the game in which the manufacturer’s cost is known and derive equilibrium strategies for the manufacturer, retailers, and consumers conditional on this cost. I demonstrate that when the manufacturer’s cost increases, he charges a higher wholesale price and consumers are willing to accept a lower level of utility. I then use this result as well as the explicitly derived structure of the equilibrium with informed consumers to prove that the manufacturer can communicate credibly via cheap talk. That is, I derive a condition so that for each cost outcome cL and cH , the manufacturer prefers to induce the search behavior corresponding to the true cost rather than that associated with the other cost.

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Equilibrium with Informed Consumers

In this section I characterize the equilibrium pricing strategies of the manufacturer and retailers and the search strategy of consumers given that the manufacturer’s cost c (but not his wholesale price) is publicly observed. Going forward I will refer to this as the informed 8

equilibrium. To keep notation consistent with the following section, let the manufacturer’s signal be σ(c) = c and let retailers and consumers condition strategies on σ. In addition, I suppress the subscript on the quality of the manufacturer’s product, now denoted by α.

Consumer Search and Retail Prices In deciding whether to search, a consumer evaluates the expected utility from future draws and faces two types of uncertainty. The first type includes the identity of the seller on the next draw, the realization of the outside option νj if the seller is a competitor, and the realization of the preference shock ηj if the seller is a manufacturer’s retailer. These sources of uncertainty are stationary, i.e. a consumer’s belief about future realizations is independent of the consumer’s previous search history. The second type is uncertainty over the price pj if the seller is a retailer. In the full model in which the manufacturer’s cost is uncertain, previously observed retail prices can shed light on the manufacturer’s cost, and in turn help refine a consumer’s expectation of what prices she would face when continuing to search. However, in the present environment the consumer is informed about the manufacturer’s cost and her history allows no additional inference. Hence a consumer faces a stationary decision problem and, by a standard result in Kohn and Shavell (1974), uses a stationary threshold policy u(σ) defined implicitly by # "Z Z 1 1 (ν − u(σ))dG(ν) + Epj |σ s= (ηj − (u(σ) + pj − α))dF (ηj ) 2 ν≥u(σ) 2 ηj ≥u(σ)+pj −α # "Z Z 1 1 (1 − G(ν))dν + Epj |σ (1 − F (ηj ))dηj . (1) = 2 ν≥u(σ) 2 ηj ≥u(σ)+pj −α The right hand side of the top line describes the option value of taking another draw when the currently available best option gives a payoff of u(σ), with the first term denoting the value from visiting a competitor and the second term denoting the expected value of visiting a retailer. The expectation allows for retail price pj to be uncertain conditional on recommendation σ.6 The second line follows from integration by parts. Next consider the strategy of a retailer, who observes recommendation σ and wholesale price w and chooses a retail price p. The probability a retailer makes a sale to a consumer upon her arrival is 1 − F (u(σ) + p − α), and if the sale is not made the consumer continues to search, never returning because there are infinitely many sellers. Taking as given the number of arriving consumers each retailer maximizes his per-consumer profit with a price 6

In the ensuing equilibrium all retailers charge the same price conditional on σ, however the expectation operator is useful in that (1) also describes the search threshold if retailers were heterogeneous and charged different prices, an extension discussed later in the article.

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that solves p(w, σ) = arg max (p − w)(1 − F (u(σ) + p − α)). p

(2)

Let x(w, σ) ≡ 1 − F (u(σ) + p(w, σ) − α) denote a retailer’s optimally chosen probability of sale to an arriving consumer, to which I will also refer as the retailer’s acceptance probability.

Manufacturer’s Wholesale Price The quantity sold by the manufacturer is determined by the equilibrium actions of retailers and consumers in the downstream market. In the present environment the manufacturer cannot directly affect the consumers’ search threshold u(σ), but can affect retail prices through his wholesale price. The manufacturer’s demand Q(w, σ) may be interpreted as the probability that a consumer eventually purchases from a retailer and not a competitor. This interpretation allows for Q(w, σ) to be expressed as   1 1 1 (1 − x(w, σ)) + G(u(σ)) Q(w, σ). Q(w, σ) = x(w, σ) + 2 2 2 The first term denotes the joint probability that the consumer’s initial draw is at a manufacturer’s retailer and that she accepts. The second term accounts for the probability that the consumer rejects her initial draw, at either type of seller, and continues to search. For her next draw the consumer faces the identical problem as for her first draw, which yields the recursive specification. Solving the above obtains Q(w, σ) =

x(w, σ) . x(w, σ) + 1 − G(u(σ))

(3)

The manufacturer’s demand resembles the success function in a ratio-form (Tullock) contest, and as with the ratio-form contest the analogy of a raffle applies. A consumer using threshold u(σ) continues to sample until reaching the first seller that meets the threshold, thus it is as if the consumer randomly samples exactly once but only from the set of qualifying sellers. The probability of sale by the manufacturer then just equals the proportion of all qualifying sellers that belong to the manufacturer. By inspection the manufacturer’s demand is concave in acceptance probability x(w, σ), which follows solely from the search process and not from any assumption on the underlying distribution of valuations. This fact is useful for technical reasons, as the concavity of the demand function is important for the existence of a unique optimizer. It also highlights that the extent to which a manufacturer is concerned about double-marginalization depends on how his retailers compare with the competitors. In a setting with a monopolist manufacturer with a single retailer, a one unit reduction in the sales of the retailer translates into a one unit 10

reduction in the sales of the manufacturer. In this search environment, however, a consumer that rejects an offer from a retailer continues to search. The higher is the acceptance probability at retailers relative to competitors, the higher the chance that a consumer induced into searching still buys from the manufacturer.7 The manufacturer solves a monopoly problem with demand Q(w, σ) and constant marginal cost c. His profit is Π(w, σ, c) = (w − c)Q(w, σ) = (w − c)

x(w, σ) , x(w, σ) + (1 − G(u(σ)))

and he chooses a wholesale price w which solves w(c) = arg max Π(w, σ, c). w

(4)

Informed Equilibrium and Properties I now demonstrate the existence and uniqueness of an equilibrium with informed consumers and establish several useful comparative statics.   Proposition 1 There exists a unique pure strategy equilibrium w(c), p(w, σ(c)), u(σ(c)) which solves equations (1), (2), and (4). The structure of the proof is to first solve the retailer’s problem taking as given wholesale price w and search threshold u and then to plug the solution into the the optimization decisions of the manufacturer and consumers, reducing the equilibrium characterization into a system of two equations with two unknowns w and u. It is then demonstrated that the manufacturer’s and consumers’ best responses must intersect exactly once. An important element of the argument is demonstrating the uniqueness and continuity of best responses. Whereas for the search threshold this is immediate and for retailers follows directly from the log-concavity of f , the manufacturer’s problem requires more work because the curvature of the manufacturer’s demand is a function of the curvature with respect to w of the endogenous acceptance probability x(w, σ). I demonstrate that a non-decreasing density f is sufficient to ensure that the manufacturer’s payoff is single peaked, thus establishing the existence and continuity of the optimal wholesale price and allowing it to be characterized by the first order condition. Finally, although the search threshold and wholesale price can be strategic complements, I demonstrate that when f is non-decreasing the consumer’s best response function is everywhere steeper than the manufacturer’s and must result in a 7

The observation that a manufacturer is less concerned with double-marginalization when consumers search is explored in Janssen and Shelegia (2015), which demonstrates that due to search a manufacturer sets an even higher wholesale price than he would in a classic setting.

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single crossing. See Appendix A1 for the detailed proof. Next let the equation 1 s= 2

Z

ν¯ uG

(1 − G(ν)) dν

(5)

implicitly define uG as the utility threshold of a consumer that never accepts offers from retailers. Proposition 2 In an informed equilibrium, the search threshold u weakly decreases in s, weakly increases in α − c, and is bounded in [uG , ν¯). Although it is intuitive that the equilibrium utility threshold u falls with the search cost s and increases with gains from trade α − c, to prove this formally one must account for equilibrium effects. For instance, whereas an increase in the search cost reduces u for a fixed wholesale price w, if the manufacturer lowers w in response then in turn a higher u is optimal, and it must be established that the initial direct effect dominates. For the bounds of the search threshold, that uG is a lower bound is immediate because, regardless of the set of utilities available at retailers, the return to search when acting optimally is at least as high as the return to search when committing to rejecting offers from all retailers. That the search threshold never exceeds ν¯ follows from a hold-up argument, similar to that in Janssen Shelegia (2015) and Diamond (1971), in that a manufacturer cannot commit to provide consumers with sufficient return to searching. The full proof can be found in Appendix A2.

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Equilibrium with Uncertain Costs

In this section I use the informed equilibrium of the preceding section to help describe an equilibrium when manufacturer costs are unobserved by consumers. I begin by describing the connection between the informed equilibrium and the Bayesian equilibrium, specifically focusing on consumer beliefs. Then I outline the intuition for why the manufacturer prefers more search when his cost is low and less search when his cost is high. Lastly, I demonstrate the article’s main results.

Consumer Beliefs with Uncertain Costs A consumer that has seen k retailers and l competitors has a history h = {σ, {(p1 , η1 ), ..., (pk , ηk )} ∪ {ν1 , ..., νl }} = {σ, ν}, and her strategy is a mapping from histories to a set of actions which includes accepting an available offer, searching, or exiting. With each history is associated a posterior µ(h) about 12

the probability that the manufacturer has a high cost, and this posterior must be formed using Bayes rule whenever possible. In a conjectured truth-telling equilibrium in which the manufacturer recommends σ(c) and retailers all set the full-information equilibrium price p(σ(c)) ≡ p(w(c), σ(c)), the consumer’s belief starts at either µ = 1 if σ = σH or µ = 0 if σ = σL prior to her first visit and remains constant throughout search. On the equilibrium path, the consumer does not expect her beliefs to change, so (1) still describes her optimal behavior. The issue, however, arises when evaluating retailers’ pricing decisions. Specifically, suppose that the manufacturer truthfully recommends σL but a retailer deviates to p˜ = p(σ(cL )) + ε. The combination of σL and p˜ is off the equilibrium path, thus Bayes rule alone does not guarantee that the consumer continues to believe that costs are low. If the retailer can successfully convince the consumer through this deviation that costs are high, the deviation may be profitable.8 To address this concern, I restrict beliefs so that ( 1 if σH ∈ h µ(h) = . (6) 0 if σL ∈ h Namely, if a consumer ever observes a retail price that is not consistent with the recommended price, she places all the weight on the price recommendation. Although this restriction on beliefs is stronger than what is necessary to sustain the cheap talk equilibrium and is chosen primarily for simplicity, it may also be supported in several ways. One justification is the ensuing cheap talk result itself, that the manufacturer’s and consumers’ interests about search are aligned whereas retailers all prefer less search, and therefore consumers ought to trust the manufacturer more than retailers. Furthermore, the beliefs above would obtain in a slightly richer setting in which retailers are heterogeneous, for example with respect to private idiosyncratic costs (e.g. Reinganum (1979)). In such a setting, instead of a single price, a set of prices is consistent with a manufacturer’s recommendation. This in turn implies that for a given retailer, deviation prices in the neighborhood of his equilibrium price are on the equilibrium path, along which the belief is restricted by Bayes rule. The setting with retailer heterogeneity highlights another important aspect of the present model. Currently in equilibrium a single retail price is associated with each manufacturer cost, thus in principle a consumer perfectly learns the cost from the retail price alone. However, this is an artifact of the simplifying assumption of homogeneous retailers. If retailers were heterogeneous, then a given retail price can be consistent with both states and, although the consumer could make statistical inference from the retail price, she would not in general be able to learn the state perfectly. Thus, demonstrating that the incentives of the 8

In fact, by the envelope theorem any marginal increase in belief would induce a deviation.

13

manufacturer and consumers are aligned with respect to search suggests that the manufacturer can convey information to consumers that otherwise would not be available. If beliefs are formed according to (6) then the consumers’ search strategy, the optimal retail price, and the optimal wholesale price described in Proposition 1 still constitute mutual best responses conditional on truthful reporting. It remains to be shown that conditional on these strategies, truthful reporting by the manufacturer is a best response.

Credible Signaling The preceding analysis of the informed equilibrium defines search thresholds uL ≡ u(σL ) and uH ≡ u(σH ) which are consistent with consumers being informed of the manufacturer’s true cost. The next step is to check whether the manufacturer can credibly provide this information via cheap talk. In such an equilibrium, regardless of the actual cost realization the manufacturer is able to induce either search threshold, and it needs to be shown that he prefers to induce the higher threshold uL when his cost is cL and the lower threshold uH when his cost is cH . In order to understand the manufacturer’s incentive for inducing search, consider two consumers, one currently on the margin for purchasing at a retailer and the other on the margin at a competitor. By increasing the search threshold the manufacturer faces a tradeoff: he decreases the likelihood of sale to the first consumer from one to the continuation probability and increases the likelihood of sale to the second consumer from zero to the continuation probability. When the continuation probability is high, that is when the utility threshold is more likely to be exceeded by a retailer than by a competitor, more search benefits the manufacturer. Conversely, when the continuation probability is low the manufacturer is better off with less search. This tradeoff can be formally seen by differentiating the manufacturer’s demand in (3) with respect to the search threshold u, which simplifies to ∂Q = Q(w, u)(1 − Q(w, u))(εG(u) − εx (w, u)), ∂u

(7)

g(u) u (w,u)| and εG (u) ≡ 1−G(u) are the elasticities of acceptance probabilities where εx (w, u) ≡ |xx(w,u) at retailers and competitors. Whether the manufacturer’s demand increases or decreases in u thus depends entirely on which of the two elasticities is larger,9 which in turn depends on the wholesale price charged by the manufacturer. If the wholesale price is low then the retailers’ acceptance probability x(w, u) is high, the elasticity εx (w, u) is low, and consequently an increase in search threshold u shifts out the manufacturer’s demand. Meanwhile the opposite 9

This is consistent with the interpretation of the manufacturer’s demand as a ratio-form contest – whether demand increases or decreases in u depends only on the change to the ratio of the acceptance probabilities, that is on the difference in their percent changes.

14

w α + η¯ − uH α + η¯ − uL w ˜

Q(w, uL ) Q(w, uH )

Q Figure 2: Effect of increased search on manufacturer demand

is true at high wholesale prices, where x(w, u) is low, εx (w, u) is high, and demand shifts in as u increases. Figure 2 illustrates the recommendation decision faced by the manufacturer, i.e. the choice between demand functions Q(w, uH ) and Q(w, uL ). For signaling to be credible, the manufacturer must prefer to charge a sufficiently low wholesale price when his cost is cL and a sufficiently high wholesale price when his cost is cH . Proposition 3 There exists a cheap talk equilibrium in which the manufacturer truthfully reveals his costs whenever α − cL is sufficiently high and α − cH is sufficiently low.

It needs to be shown that Π(wL , uL , cL ) ≥ Π(w(uH , cL ), uH , cL ) and that Π(wH , uH , cH ) ≥ Π(w(uL , cH ), uL , cH ), where w(u, c) is the profit maximizing wholesale price. For this, consider Figure 2 and let the two demand curves represent the informed outcomes in each state, intersecting at wholesale price w. ˜ If w(uH , cL ) < w˜ then even at the deviation price the manufacturer prefers more search (uL rather than uH ), which then implies Π(w(uH , cL ), uH , cL ) < Π(w(uH , cL ), uL , cL ) < Π(wL , uL , cL ). Therefore, signaling is credible in the low cost state whenever w(uH , cL ) < w, ˜ and by the same logic signaling is credible in the high cost state whenever w(uL , cH ) > w. ˜ Then, to show that w(uH , cL ) < w˜ < w(uL , cH ) I derive an expression for the manufacturer’s marginal profit at w˜ and demonstrate that it is negative when cL is sufficiently low and positive when cH is sufficiently high. Because the manufacturer’s profit is concave in the wholesale price this completes the proof. See Appendix A4 for details. The proof of Proposition 3 defines what it means for α − cL to be sufficiently large and α − cH to be sufficiently small conditional on search cost s to allow truthful reporting. However, as s gets smaller a given gains from trade advantage or disadvantage becomes more 15

meaningful. The following proposition demonstrates that for any given advantage in state cL and disadvantage in state cH , credible signaling is possible for sufficiently small search costs. Proposition 4 There exists a cheap talk equilibrium in which the manufacturer truthfully reveals his costs whenever α − cH < ν¯ − η¯ < α − cL and search cost s > 0 is sufficiently small. According to (7), the manufacturer prefers more search if his wholesale price induces a higher elasticity at competitors than at retailers and less search otherwise. As in Proposition 3, this implicitly defines a “matching” wholesale price w˜ where the elasticities are equal and the aim is to show that when the search cost is sufficiently small the manufacturer’s marginal profit at w˜ is negative when his cost is cL and positive when his cost is cH , even when misreporting. The key is that as s falls so too does the competitors’ acceptance probability and, by definition, also the retailers’ acceptance probability at w. ˜ This has two main implications: one that the manufacturer’s demand at w˜ becomes progressively more elastic, as a small decrease in w corresponds to a larger percentage increase in x, and two that the value of w˜ approaches α + η¯ − ν¯ ∈ (cL , cH ) at which retailers are fully squeezed. If the cost is cL then as s shrinks the manufacturer’s markup at w˜ is bounded from below whereas the demand elasticity approaches infinity. Thus when s is small enough, the marginal profit at w˜ becomes negative, implying that the manufacturer prefers more search and reports truthfully. If the cost is cH then at some sˆ > 0 the markup at w˜ becomes zero whereas the demand elasticity is finite. Thus for search costs below sˆ (and in a neighborhood above sˆ) the manufacturer’s optimal price is above w, ˜ he prefers less search and reports truthfully. See Appendix A5 for details. Although Propositions 3 and 4 require sufficiently low and high gains from trade or sufficiently low search costs, they do not not imply that credible signaling is only supported in the extreme limits of the parameter space or at corner solutions. For example, it can be seen that at the smallest allowable α − cL and at the largest allowable α − cH in Proposition 3 and at a small but positive search cost s in Proposition 4, both retailers and competitors make strictly positive sales. Furthermore the condition that w(uH , cL ) ≤ wˆ ≤ w(uL , cH ) is sufficient but not necessary for credibility. The proof of Proposition 3 makes it clear that if w(uH , cL ) = wˆ then Π(w(uH , cL ), uL , cL ) < Π(wL , uL , cL ), i.e. if at the deviation wholesale price the manufacturer is indifferent between the two search thresholds then he strictly prefers reporting truthfully, and similarly if w(uL , cH ) = w. ˆ By continuity, this implies credible signaling can be supported even if w(uH , cL ) > wˆ or w(uL , cH ) < w. ˆ On the other hand, it is also true that if the manufacturer is not sufficiently ahead of or sufficiently behind competitors, he will have incentive to misreport. For example, consider a rather extreme case in which the manufacturer is barely ahead in the low cost state and priced out in the high cost state, that is there is a small δ > 0 so that εx (w(uL , cL ), uL ) = εG (uL )−δ 16

and εx (w(uH , cH ), uH ) = ∞ > εG (uH ). By inspection of the expression in Lemma 10 of Appendix A4, in the low cost state misreporting a high cost is profitable because along the way from uL to uH , εx (w(cL , u), u) > εG (u) at almost every u. Put differently, misreporting and inducing less search makes it less costly for the manufacturer to increase his markup, and when the markup is relatively important, doing so is worthwhile.

Example with Uniform Distributions To get a better sense of the informed equilibrium and of when credible communication is possible, consider the following example. Suppose that ν, η ∼ U[0, 1], that α − cL = δ, α − cH = −δ, and that δ ∈ [0, 1]. Note that δ = 0 corresponds to no uncertainty and δ = 1 corresponds to zero maximal gains from trade when the cost is high. As in the main section, I first characterize the equilibrium when c is observed by consumers, study the comparative statics, and then establish the credible signaling result. With consumers informed of c, a retailer faces demand x(p, u) = α + 1 − u − p and his profit maximizing price is p(w, u) = 12 (α + 1 − u + w), with corresponding acceptance probability x(w, u) = 12 (α + 1 − u − w). The manufacturer’s profit function is thus Π(w, u, ci) = (w − ci )

α+1−u−w x(w, u) = (w − ci ) , x(w, u) + 1 − G(u) α + 3 − 3u − w

with corresponding optimal wholesale price ( p α + 3 − 3u − 2(1 − u)(α − ci + 3 − 3u) if ci < α + 1 − u w(u, ci ) = . ci if ci ≥ α + 1 − u

(8)

The profit-maximizing wholesale price solves the first order condition whenever the maximal gains from trade between the manufacturer and consumers are greater than u. Otherwise, the manufacturer cannot make sales with a positive markup, and it is assumed that he sets a price equal to his cost.10 A consumer faces utility offers uniformly distributed on [0, 1] at competitors and uniformly distributed on [α−p(w, u), α+1−p(w, u)] at retailers. His best response threshold u satisfies s=

1 2

Z

1 u

(1 − ν) dν +

1 2

Z

1 u+p(w,u)−α

(1 − η) dη

1 1 = I(u ≤ 1) · (1 − u)2 + I(u ≤ α + 1 − w) · 4 4 10



2 1 (α + 1 − u − w) . 2

When ci ≥ α + 1 − u, the manufacturer is priced out and any wholesale price w ≥ ci leads to the same equilibrium outcomes.

17

Taking into account the indicator functions and solving for u obtains   ( p √ 1 2 if w ≤ α + 2 s 1 − 5 w − α + 2 20s − (w − α) u(w) = √ √ if w > α + 2 s. 1−2 s

(9)

To interpret the above, recall that if a consumer commits to always rejecting offers from Z 1 √ (1 − ν) dν ⇔ uG = 1 − 2 s. retailers then his utility threshold uG satisfies s = 12 uG

If the manufacturer’s wholesale price makes it impossible to exceed this threshold, that is if √ comes only from visiting competitors w > α + 1 − uG = α + 2 s, then the value to searching √ and u(w) = uG . If on the other hand w ≤ α + 2 s then there is value in visiting retailers as well and u(w) > uG .

Proposition 5 There exists a unique equilibrium (w ∗ , u∗ ) solving (8) and (9), with retail price p∗ = 21 (α + 1 − u∗ + w ∗) and comparative statics as follows: i.

du∗ dc

≤ 0,

dw ∗ dc

≥ 0,

dp∗ dc

≥ 0, and

ii. there exist 0 < s1 < s2 so that unless α > c and s < s1 .

du∗ ds

≤ 0,

dw ∗ ds

≥ 0 unless α > c and s < s2 , and

dp∗ ds

≥0

The intersection of the manufacturer’s best response in (8) and the consumer’s best response in (9) defines the informed equilibrium, and although there is no closed form solution, the slopes of these functions shed light on the equilibrium’s characteristics. The consumer’s threshold u(w) is always decreasing because a higher w leads to a lower distribution of utilities during search, whereas whether the manufacturer’s optimal wholesale price w(u) is increasing or decreasing depends on parameters. Specifically, as in (7) recall that an increase in u reduces both retailers’ and competitors’ acceptance probabilities. When the manufacturer’s cost c is low (and thus his equilibrium wholesale price is low), retailers are less affected than competitors in percent terms and thus an increase in u shifts out the manufacturer’s demand and increases the optimal wholesale price. When c is high the reverse is true, a higher u shifts the manufacturer’s demand inward, and the optimal wholesale price falls. These two scenarios are depicted in Figure 3. Observe that in both panels the consumers’ best response intersects the manufacturer’s best response from above, and in Lemmas 3 and 4 of Appendix A1 this is shown to be the case more generally. Having established the above, the comparative statics follow directly. First, an increase in production cost c causes an upward shift of w(u), which in either panel of Figure 3 results in a lower u∗ and a higher w ∗ , both of which contribute to a higher retail price p∗ . On the other hand, an increase in s causes an inward shift of u(w) and the effects of this depend on whether c is high or low. If c is high (right panel of Figure 3) then u∗ decreases whereas w ∗ 18

Figure 3: Best response functions for the manufacturer and consumers in the uniform example with α = 1, s = 0.01, and c = 0 (left panel) or c = 1 (right panel). The search threshold u(w) is always decreasing whereas the wholesale price w(u) is decreasing when c high and increasing when c is low. increases, again resulting in a higher retail price p∗ . However, if c is low (left panel of Figure 3) then both w ∗ and u∗ decrease, and the retail price p∗ increases only if u∗ falls faster, i.e. < 1. The details of the proof can be found in Appendix A3. only if ∂w ∂u With the equilibrium with informed consumers thus described, I now focus on the article’s main result to check when communication is credible. Doing so requires explicitly computing equilibrium and deviation profits and because no closed form solution to (8) and (9) exists, I do so numerically. I fix α = 1 (and in turn cL = 1 − δ and cH = 1 + δ) and solve for the informed equilibrium on a grid of δ ∈ [0, 1] and s ∈ [0, 1/4], in which δ = 1 is the maximal value at which gains from trade are possible in the high cost state and the upper bound s = 1/4 ensures uG ≥ 0. Having values (uL , wL , uH , wH ) for each combination of (δ, s), the maximized profit from misreporting in the low cost state Π(w(uH , 1 − δ), uH , 1 − δ) and in the high cost state Π(w(uL , 1 + δ), uL , 1 + δ) can be computed and compared to the profit from truthful reporting, as in Figure 4. In this specification, it is always profitable to report truthfully when the cost is cH , whereas it is credible to report truthfully when cost is cL as long as either δ is sufficiently large or s is sufficiently small, as per Propositions 3 and 4. Observe that communication is credible for the majority of the considered parameter space.

Comparison to No Communication The question of what ensues if the manufacturer does not make recommendations is important for two reasons. First, as with any cheap talk environment, there always exists a babbling equilibrium in which consumers ignore manufacturer recommendations and recommendations are uncorrelated with the realization of manufacturer’s costs (for instance, the 19

Figure 4: Parameters where signaling is credible (light gray) and not credible (dark gray)

signal “no recommendation” is always used). Given that in most markets it is optional11 and often costly for the manufacturer to print and publicize price recommendations, it helps to argue that the informative equilibrium is more profitable for the manufacturer than the babbling equilibrium. Second, being able to compare the informative equilibrium with an equilibrium without MSRPs is important from a policy perspective in determining whether manufacturer recommendations should be regulated. To build some intuition, recall the logic by which the manufacturer’s signal of his cost is credible. For a consumer A who is currently marginal at a retailer, if the true state is cL then uncertainty makes her overly pessimistic and deters her from search, thus increasing the likelihood that she buys from the manufacturer from the continuation probability to one. On the other hand, if the true cost is cH uncertainty makes her overly optimistic, causing her to search rather than accept and reducing the likelihood she buys from the manufacturer from one to the continuation probability. The effect is stronger when the cost is cH because the continuation probability is lower, and therefore on net the manufacturer is harmed by A’s uncertainty. For consumer B currently marginal at a competitor, if the manufacturer’s cost 11

An exception is the mandatory Monroney sticker required on all new cars in the US by the Automobile Information Disclosure Act (1958). However, while the MSRP is included it is not necessarily informative – a manufacturer can make the same recommendation for two vehicles he expects to sell for different prices.

20

is cL then uncertainty makes B overly pessimistic, causing him to accept the competitor’s offer rather than search and decreasing his likelihood of buying from the manufacturer from the continuation probability to zero. If instead the cost is cH the reverse ensues and the likelihood increases from zero to the continuation probability. Here the effect is stronger when the cost is cL because the continuation probability is larger, so again on net the manufacturer is harmed by B’s uncertainty. To operationalize this intuition one must formally describe how the absence of recommendations creates uncertainty, and in particular account for consumers’ ability to infer the manufacturer’s cost from the prices of retailers. In the extreme, if retail prices perfectly reveal the manufacturer’s cost, as can be the case in the current setting with completely homogeneous retailers, then having seen a retailer’s price consumer A behaves the same as when informed by a recommendation. On the other hand, if consumer B has not yet visited any retailers then she remains uncertain about the state. Then by the previous argument the manufacturer benefits from B’s uncertainty when the cost is cH , is harmed when the cost is cL , and is worse off on net. This is formalized in the following proposition. Proposition 6 Given cL < α + η¯ − ν¯, for a sufficiently high cH there exists an equilibrium without recommendations in which wholesale and retail prices in both states are the same as in the informed equilibrium. Furthermore, the payoffs of the manufacturer and consumers are lower in this equilibrium than in the equilibrium with recommendations. The key to demonstrating both that there exists an equilibrium with the same prices and that the manufacturer and consumers are worse off in this equilibrium is understanding the consumers’ search strategy is this new environment. A consumer that has visited a retailer at any point is fully informed and uses the same threshold as in (1), whereas a consumer whose history consists only of visits to competitors holds the prior of µ = 1/2 and uses threshold u(1/2) ∈ (uH , uL ). When this consumer eventually visits a retailer, she learns the true state and either becomes more selective with threshold uL or less selective with threshold uH , in   the latter case potentially returning to accept a previous competitor’s offer ν ∈ uH , u(1/2) .

With this non-stationary search strategy, the manufacturer’s demand is no longer expressed as simply as in (3). However using a recursive technique I show that the manufacturer’s demand in each state is a scalar multiple of the demand when consumers are informed. This establishes that the profit-maximizing wholesale price remains unchanged in each state relative to when consumers are informed, and by the logic in the discussion preceding the proposition, that the demand is lower when the cost is cL and higher when the cost is cH . Because demand is a scalar multiple so are maximized profits, and thus as cL is fixed and cH is increased the benefit of uncertainty shrinks toward zero whereas the cost remains fixed, eventually making the manufacturer worse off on net. Finally, because consumers face the same prices but make less informed decisions they too are worse off in this setting. For the detailed proof see Appendix A6. 21

Figure 5 demonstrates the difference in the manufacturer’s profit with and without recommendations for the uniform example previously considered. The benefit of having informed consumers grows as the states are farther apart, as captured by the fact that the difference increases in δ and decreases in s. In fact, in this uniform specification the profit at the full information prices is higher when consumers are informed than when they are uninformed for every (δ, s) combination.

Figure 5: The difference in the manufacturer’s profit between when consumers are informed and uninformed

5

Discussion

A natural alternative specification of the main model is one in which consumers are uncertain about product quality rather than the manufacturer’s cost, and here I briefly explore this setting. Despite the result of Lemma 9 (in the Appendix) that in the informed setting there is no meaningful distinction between quality and cost, it is not immediate that credible signaling ensues in the Bayesian environment and I examine this more carefully. Then, because MSRPs have been viewed as a means for manufacturers to exert influence over retail prices, I contrast their effect to the effect of maximum resale price maintenance.

Uncertain Quality Suppose there is no uncertainty about the manufacturer’s cost c, but instead the product quality α has either a high realization αH or a low realization αL , both equally likely. The manufacturer privately observes the product’s quality and consumers may only infer the 22

quality by visiting retailers. Recall that the consumer’s utility is u = α + ηj − pj . To capture the idea that quality may be inferred but not directly observed, let zj ≡ α + ηj be the full consumption utility of the product at retailer j and suppose that the consumer observes zj but not its individual components.12 Although Lemma 9 demonstrates that the full information game is equivalent to when costs are uncertain, the beliefs in the Bayesian game must be revisited. In the specification with uncertain costs, it was assumed that when the price recommendation and retail price are inconsistent with the equilibrium, beliefs place full weight on the recommendation. Now the consumer observes three objects: the recommendation, the retail price, and his consumption utility zj . If the manufacturer deviates then σ is sometimes inconsistent with zj . For instance, suppose the quality is αL but the manufacturer misreports σ(αH ). If the realized private shock is η < η + (αH − αL ) then the consumer must infer that the quality is αL with certainty. The payoff to the manufacturer from misreporting is now difficult to evaluate. Not only would consumers become aware of the true state with some probability and thereby change their search strategy, but also retailers having observed the manufacturer’s deviation would need to anticipate the change in consumers’ search and react accordingly. However, this is more of a technical issue than a qualitative one. It is still the case that the manufacturer’s incentives for search can be aligned with the consumers’ – he wants more search when he offers a higher range of utilities than do competitors and less search otherwise. The issue is then not in the manufacturer’s incentives to increase or decrease search but the particular mechanism by which he can achieve this. In addition, this concern can be resolved on a technical level by allowing the distribution of private shocks F (η) to have full support on [−∞, ∞], in which case all observed realizations of zj are consistent with both of the manufacturer’s recommendations. Thus, given the underlying incentive structure, it is likely that the manufacturer’s recommendations can also be used to convey information about product quality.

Comparison to Resale Price Maintenance In practice, retail prices rarely exceed recommendations and consequently MSRPs are often viewed as price ceilings. With explicit price ceilings, a retailer faces a penalty for setting a price above the ceiling that is typically enforced by the manufacturer. Similarly, MSRPs have been conjectured to provide a similar penalty but enforced instead by consumers, who refuse to pay above the recommended price, or more generally have a kink in their demand at this price (Thaler (1985), Puppe and Rosenkranz (2011)). Such theories predict that the presence of MSRPs reduces demand only at prices that exceed the recommendation. In 12

In contrast to standard quality signaling models, here the consumer knows his utility for a product prior to deciding about whether to purchase it, and his interest in quality is only in forming expectations about the returns to searching.

23

contrast, the present theory of MSRPs as information for consumers predicts that demand is potentially affected at all prices, including those below MSRP. Specifically, in the model the per-person demand is x(p, u) = 1 − F (p + u − α), thus an increase in search threshold u corresponds to an inward shift of demand for every price. Recent evidence in De los Santos et. al. (2016), which studies the effects of a ban of price recommendations on packaged foods in South Korea, suggests that although MSRPs lower prices they do not act as price ceilings. In particular, the authors find that more than 96% of prices charged are at least 10% below the recommended price, that there is no bunching around the recommended price when it is present, and that when an MSRP is removed there is no visible increase in the number of prices charged above it. One may also consider the US auto market as an anecdotal example. The vast majority of cars sell not just below MSRP but strictly below, so that in most transactions the MSRP does not bind. In principle, it is possible that consumers use some fraction of the MSRP rather than the MSRP itself as a reference price, so for instance the true price ceiling is at 90% of the recommended price. However, in the auto market it is commonly known that more popular models sell closer to MSRP, and very popular models may even sell above MSRP. For the theory to apply, it must be that a consumer’s reference price is informed by a product’s popularity or its other market characteristics. And this in fact is the mechanism in the present article, in which consumers use MSRPs to set a reference utility which in equilibrium is optimal under current market conditions. Put differently, the present search model can be used to rationalize the existing reference point theories of price recommendations. As a conduit of information, price recommendations provide the manufacturer with an instrument that is both similar to and different from resale price maintenance. On one hand, a consumer’s search threshold determines the highest price she is willing to accept for a given utility shock, and thus making a price recommendation is analogous to choosing a price ceiling. However, providing consumers with information accomplishes more than indirectly restricting retail prices. By affecting search, the manufacturer affects the elasticity of demand, and in doing so affects the nature of both downstream and upstream competition.

6

Conclusion

Despite the prevalence of manufacturer suggested retail prices, why manufacturers make recommendations and what effect they have on market outcomes such as prices and sales is still not well understood. I present a model in which a manufacturer uses recommendations to affect consumers’ search decisions. Consumers do not know the distribution of retail prices due to uncertain manufacturer costs and rely on the recommendation to determine whether to purchase at a given price or to continue shopping. By using a suggested price to induce 24

more search, the manufacturer pressures retailers to reduce double marginalization but also loses consumers that search and purchase from a competitor. The manufacturer profits from more search when he offers consumers more value than competitors; otherwise he prefers less search. Consumers benefit from more search when higher utilities are available, thus their interests are aligned with the manufacturer’s. I demonstrate that manufacturer recommendations can be used as cheap-talk signals to credibly inform consumers of the manufacturer’s costs. Recommended prices help manufacturers and consumers coordinate and thereby can improve the welfare of both. It is likely that the manufacturer’s incentive for truthful reporting applies beyond the current setting of uncertainty over costs and instead can convey information about product quality. In principle, it is difficult for a seller to convey the value of his product to a buyer using cheap talk. The intuition is simple – if a consumer’s willingness to pay for a product can be increased by a cheap message, this message is used regardless of the product’s value (e.g. Milgrom and Roberts (1986)). I demonstrate that in a vertical market this intuition no longer holds; instead the manufacturer can affect consumers’ willingness to pay by informing them of the set of other options available through search. In particular, the manufacturer can sometimes benefit by informing consumers that the product he supplies retailers is not of high quality on average, thereby reducing the perceived return to searching and making them more willing to accept the current offer. Because the manufacturer does not always benefit from overstating the value of his product, he is able to communicate credibly. The main results demonstrate that signaling is credible only when there is sufficient uncertainty about the distribution of prices. This is consistent with the fact that MSRPs are typically found on products that are purchased infrequently, such as cars, electronics or a particular book, and not found on products that are purchased frequently, such as milk or laundry detergent. By contrast, models of MSRPs as explicit resale price maintenance make no such distinction. In addition, in practice there is variation in the way actual prices relate to price recommendations. For instance, books often sell for exactly their jacket price whereas cars tend to sell for strictly less than MSRP. Even within the car market, how far below MSRP a car sells varies by the popularity of that vehicle, and in fact some cars sell above MSRP. Imposing that recommendations are price ceilings or some other exogenously determined restraints precludes the analysis from explaining such variation. Because in my model a recommendation is just a cheap message, I allow for the consumers’ interpretation of this message to vary with market conditions and thus can accommodate the varying relationship between prices and recommendations. The goal of this article is to highlight a mechanism by which price recommendations serve a purely informational role and still influence market outcomes. Towards this end, the model I 25

develop is quite stylized and thus has limitations for direct use in assessing policy. Nonetheless, the message that emerges is that ceteris paribus price recommendations help consumers, allowing them to make more informed decisions and saving search costs. Hence, an antitrust policy discussion of the merits of price recommendations should keep these informational benefits in mind.

Appendix A1

Proof of Proposition 1: Existence and uniqueness of informed equilibrium

I first demonstrate that for retailers taking as given search threshold u and wholesale price w there is a unique optimal price p(α − u + w) and corresponding acceptance probability x(u + w − α). Plugging this into the manufacturer’s and consumers’ decisions reduces the characterization into a system of two equations with w and u as the unknowns. I then use a monotonicity argument to show that exactly one solution to this system exists.   Lemma 1 There exists a unique maximizer p(α − u+ w) of (2) so that p′∈ 0, 12 and the resulting acceptance probability x(u + w − α) ≡ 1 − F u + p(α − u + w) − α satisfies x′ < 0 and x′′ < 0 whenever w < α + η¯ − u (i.e. whenever retail sales are feasible).

Proof of Lemma: The log-concavity of f guarantees the concavity of π(p) = (p − w)(1 − F (u + p − α)) and thus a unique maximizer p(α − u + w). Furthermore, taking the first order = 0 = 1 − F (u + p − α) − (p − w)f (u + p − α) and totally differentiating with condition dπ dp respect to p and w yields

f , 2f + (p − w)f ′ h i and thus because f ′ ≥ 0 it follows that pw ∈ 0, 21 . Differentiating with respect to w once more and performing several steps of algebra yields  ′   ′ f f ′′ f 2 pw − 1 + (p − w) − ′ . pww = (pw ) f f f pw =

Turning now to the acceptance probability x(·), I first show it can be expressed as a function of a single argument. Recall that a retailer’s probability of sale is x(p, u) = 1 − F (u + p − α) and define the inverse demand as A(x) ≡ F −1 (1 − x) − u + α. The retailer’s payoff can be re-written as π(x) = x(A(x) − w) with corresponding first order condition π ′ (x) = 0 = A + xAx − w = F −1 (1 − x) + x 26

 d F −1 (1 − x) − (u + w − α), dx

the solution to which can be expressed as x(u + w − α) = 1 − F (u + p(α − u + w) − α). Then, x′ = xw = −f pw < 0 and plugging in pww from above yields ′′



2

x = xww = −f (pw ) − f pww

  ′  f f ′′ = −f (pw ) pw + (p − w) < 0, − ′ f f ′

2

where the sign of x′′ follows because pw ≥ 0 and

f′ f



f ′′ f′

≥ 0 by log concavity.

Having described the retailer’s decision, to find an equilibrium it suffices to find search threshold u and wholesale price w that are mutual best responses conditional on x(u+w−α). The following lemma demonstrates that the manufacturer has a unique best response w(u) and the ensuing discussion describes its intersection with the consumer’s best response u(w). Lemma 2 If u ≤ α + η¯ − c then sales are feasible for the manufacturer and there exists a unique continuous profit maximizing w(u) which is the solution to the first order condition from (4). If u > α + η¯ − c then sales are not feasible for the manufacturer and any w(u) ≥ α + η¯ − u, which results in no sales, is a best response. Proof of Lemma: If u > α + η¯ −c then for any w ≥ c the manufacturer makes no sales, and thus any wholesale price at which he makes no sales is optimal. If u ≤ α + η¯ − c then there is a region of wholesale prices above his cost at which the manufacturer can make positive sales, and the aim is to show his profit is single-peaked in this region. First, to establish that x(u+w−α) the manufacturer’s demand Q = x(u+w−α) is concave in w, taking the derivative + (1−G(u)) twice and simplifying yields 1 − G(u) (x + (1 − G(u)))2   −2x′ x′′ = Qw + ′ . x + (1 − G(u)) x

Qw = x′ Qww

Because x′ < 0 (and thus Qw < 0) and x′′ < 0, it follows that Qww < 0. Finally, the profit function (w − c)Q(w) is concave whenever Q is concave, and thus is single peaked with a unique continuous maximizer. Lemma 3 If u ≤ α + η¯ − c (i.e. sales are feasible) then

dw du

> −1.

Proof of Lemma: Differentiating in (4) obtains the manufacturer’s first order condition     1 − G(u) εx (u + w − α) , (10) Πw = 0 = Q(w, u) 1 − (w − c) x(u + w − α) + 1 − G(u) 27



(u+w−α)| . Now suppose that dw = −1, so that when u increases where εx (u + w − α) ≡ |xx(u+w−α) du the sum u + w remains fixed. Then in (10) it follows that as u increases (w − c) decreases,   1−G(u) decreases because x remains unchanged and 1 − G falls, and εx remains x(u+w−α) + 1−G(u) unchanged. But then Πw > 0, which by the concavity of Π(w) implies the equilibrium w is larger and thus that dw > −1. du

Turning now to the consumer, let u(w) define the solution to (1) and let ϕ(u) ≡ u−1 (u) denote the wholesale price for which u is the consumer’s best response. Observe this inverse function is not well-defined for u < uG because no wholesale price can justify a search threshold below uG , and that ϕ(uG ) is multi-valued because any ϕ ≥ α + η¯ − uG results in the consumer expecting never to buy from the manufacturer. Lemma 4 The consumer’s inverse best response ϕ(u) satisfies

dϕ du

≤ −1 whenever u ≥ uG .

Proof of Lemma: Differentiating (1) with respect to u and ϕ obtains   0 = − (1 − G(u)) − (1 − p′ )(1 − F (u + p − α)) du − p′ (1 − F (u + p − α))dϕ 1 − G(u) 1 − p′ dϕ =− ′ − du p (1 − F (u + p − α)) p′

  Because we previously established in Lemma 1 that p′ ∈ 0, 21 , it follows that

dϕ du

< −1.

I now demonstrate the main statement of the proposition. Lemma 5 If 0 ≤ c < α + η¯ − uG then the manufacturer makes positive sales and there is a unique equilibrium (u, w) ∈ (uG , ν¯) × (c, α + η¯ − ν¯). Else if c ≥ α + η¯ − uG then the manufacturer is priced out, u = uG and w ≥ α + η¯ − ν¯. Proof of Lemma: Recall that uG is the consumers’ search threshold when never buying form the manufacturer, and suppose first that 0 ≤ c < α + η¯ − uG . Observe that ϕ(uG ) ≥ α + η¯ − uG ≥ w(uG ). The first inequality follows because it is only rational to never buy from retailers if the wholesale price ϕ ≥ α + η¯ − uG , so that retailers can never make sales. The second inequality follows from the fact that it is not rational for the manufacturer to fully squeeze retailers and make zero sales because by assumption a positive markup is available. Also, note that ϕ(¯ ν ) < α + η¯ − ν¯ < w(¯ ν ). For the first inequality, a consumer that uses threshold ν¯ never buys from competitors and thus must expect that when visiting retailers her utility strictly exceeds ν¯ with a strictly positive probability, i.e. ϕ < α + η¯ − ν¯. For the second inequality, Lemma 6 argues the manufacturer can deviate to a slightly higher w and 28

increase his markup without losing any sales, thus w(¯ ν ) > α + η¯ − ν¯. Now, because w(u, c) − ϕ(u) starts negative at uG , ends positive at ν¯, and is continuous and by Lemmas 3 and 4 monotonically increasing, there exists a unique intersection at some u∗ ∈ [uG , ν¯) at which w(u, c) − ϕ(u) = 0. Then, if c ≥ α + η¯ − uG then the manufacturer is priced out. That is, because Lemma 6 demonstrates that u ≥ uG , it follows that w ≥ c ≥ α + η¯ − uG ≥ α + η¯ − u, which in turn implies that the manufacturer fully squeezes retailers. Therefore in equilibrium u = uG and w ≥ α + η¯ − uG , with the exact value of w irrelevant for payoffs. This concludes the proof of the Proposition.

A2

Proof of Proposition 2: Comparative statics of informed equilibrium

The proposition is proven in four separate lemmas, establishing the bounds of the search threshold u in Lemma 6, the comparative statics of u with respect to production cost c in Lemma 7 and with respect to search cost s in Lemma 8, and then establishing in Lemma 9 that when an increase in c is combined with a corresponding increase in quality α the equilibrium remains unchanged except for nominal prices. Lemma 6 For any c, u(σ(c)) ∈ [uG , ν¯). Proof of Lemma: That uG is a lower bound is immediate because, regardless of the set of utilities available at retailers, the return to search when acting optimally is at least as high as the return to search when committing to rejecting offers from all retailers. That the search threshold never exceeds ν¯ follows from a hold-up argument, similar to that in Janssen Shelegia (2015) and Diamond (1971), in that a manufacturer cannot commit to provide consumers with sufficient return to searching. To see this, conjecture that the consumer uses a threshold u > ν¯. In such a situation, searching consumers accept only at retailers and never at competing sellers, which allows the manufacturer to raise the wholesale price without losing sales. The manufacturer’s best response is a wholesale price at which the acceptance probability at retailers is arbitrarily close to zero, which in turn makes the expected return to searching close to zero and leads to a contradiction. Lemma 7 If c < α + η¯ − uG then u(σ(c)) is decreasing in c. If c ≥ α + η¯ − uG then u(σ(c)) = uG . Proof of Lemma: Using the proof of Proposition 1 an equilibrium occurs where 0 =

29

w(u, c) − ϕ(u), and differentiating the equilibrium condition with respect to c yields  d w(u(c), c) − ϕ(u(c)) dc ∂w du ∂w ∂ϕ du + − = du dc ∂c ∂u dc

0=

∂w du = − ∂w ∂c ∂ϕ . dc − ∂u ∂u

First note that the numerator is positive because the   manufacturer’s payoff has increasing d2 differences in w and c. That is, dw dc (w − c)Q(w, u) = − dQ = − ∂Q x′ ≥ 0 because ∂Q ≥0 dw ∂x ∂x and x′ ≤ 0. Next as established in the proof of Proposition 1 the denominator is positive as well, and therefore the right hand side is negative. Lemma 8 Threshold u(σ) decreases in the consumers’ search cost s and lims→0 u(s) = ν¯. Proof of Lemma: As in the preceding proof I differentiate the equilibrium condition 0 = w(u) − ϕ(u, s) with respect to s and obtain du = ds

∂ϕ ∂s ∂w ∂u



∂ϕ ∂u

.

First to demonstrate that the numerator is negative, totally differentiating the consumer’s 2 search decision from (1) obtains ∂ϕ = − p′ F (u+p−α) ≤ 0 because p′ ≥ 0 by Lemma 1. Then, ∂s because the denominator was shown in the proof of Proposition 1 to be positive, it follows ≤ 0. that du ds Finally, to demonstrate that lims→0 Ru(s) = ν¯, note that u(s) ∈ [uG (s), ν¯] (by Lemma 6) ν¯ and that uG (s), which solves s = 21 uG (1 − G(ν))dν, approaches ν¯ as s approaches zero. Lemma 9 If (w, p, u) constitutes an informed equilibrium given α and c, then for any α ˆ= α + δ and cˆ = c + δ, (w + δ, p + δ, u) is an informed equilibrium and all participants’ payoffs are unchanged. Proof of Lemma: The idea is that if consumers’ willingness to pay α and production cost c increase by the same amount then there is no real change to the market except for a relabeling. To this end, I suppose that (w, p, u) describes the equilibrium for parameters α and c, conjecture that (w, ˆ pˆ, uˆ) = (w + δ, p + δ, u) describes an equilibrium with parameters α ˆ = α + δ and cˆ = c + δ, and then verify that indeed every participant best responds.

30

Beginning with consumers, plugging in the new values α ˆ = α + δ and pˆ = p + δ into (1) keeps the right hand side unchanged, implying the same search threshold u remains optimal. For retailers, A1 establishes that the optimally chosen retail price induces an acceptance probability x(u + w − α) which is expressed in terms of only a single argument. At the new values the optimal acceptance probability x(u + (w + δ) − (α  + δ)) remains  unchanged, which in turn implies that for pˆ to be optimal it must satisfy 1−F u+ pˆ−(α+δ) = 1−F (u+p−α), implying pˆ = p + δ. Finally recall that the manufacturer chooses w to solve w = arg max (w − c) w

x(u + w − α) = arg max (w − c)Q(u + w − α, u), w x(u + w − α) + (1 − G(u))

with the ensuing first order condition 0 = Q(u + w − α, u) − (w − c) · Q1 (u + w − α, u). It follows that if the first order condition holds initially then it also holds when u remains unchanged, wˆ = w+δ, cˆ = c+δ, and α ˆ = α+δ. This concludes the proof of the Lemma. Taken together, Lemmas 6-9 imply the statement of Proposition 2.

A3

Proof of Proposition 5: Equilibrium in uniform example

With uniqueness following from Proposition 1, I demonstrate the comparative statics of the equilibrium given by (8) and (9). Focusing on interior solutions (i.e. when the manufacturer makes positive sales) and differentiating the top line of (9) obtains ! ∂u 1 2(w − α) p − 1 ≤ 0, (11) = ∂w 5 20s − (w − α)2

√ with the inequality ensuing by plugging in the corresponding condition w ≤ α + 2 s. Thus the consumer’s best response is downward sloping. Meanwhile, performing the same exercise for the manufacturer’s best response in the top line of (8) obtains ∂w α − c + 6 − 6u , = −3 + p ∂u 2(1 − u)(α − c + 3 − 3u)

31

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the sign of which is ambiguous, by a logic similar to that of Figure 2, and as depicted in the two panels of Figure 3. Turning first to the manufacturer’s production cost, that ∂w du∗ = − ∂w ∂c ∂ϕ ≤ 0 dc − ∂u ∂u

was already established by Lemma 7 more generally. Then for the wholesale price, ∂w ∂ϕ dw ∗ ∂ϕ du∗ ∂w ∂w du∗ = + = − ∂w∂c ∂u∂ϕ = − ≥ 0, dc ∂c ∂u dc ∂u dc − ∂u ∂u

≥ 0 from (11). Lastly for the retail price p∗ = with the inequality now following from ∂ϕ ∂u p(α − u∗ + w ∗ ) and   ∗ dp∗ du∗ dw ′ ≥ 0, =p − dc dc dc with the inequality following from p′ ≥ 0 in Lemma 1.

Next for the search cost, as shown more generally in Lemma 8 du∗ = ds

∂ϕ ∂s ∂w ∂u



∂ϕ ∂u

≤ 0.

For the wholesale and retail prices, dw ∗ du∗ ∂w = ds ds ∂u  ∗ ∂w dp∗ ′ du =p −1 ds ds ∂u with the signs of each depending on the slope of the manufacturer’s best response function √ ∂w . Recalling that in any equilibrium u ∈ [uG , ν¯) = [1 −2 s, 1), observe in (12) that if α < c ∂u > 0 for any feasible u. However, if α > c then by inspection of (12) the second term then ∂w ∂u increases in u and approaches infinity as u approaches 1. Thus there exist u2 < u1 < 1 so that ∂w ≥ 0 for all u ≥ u2 and ∂w ≥ 1 for all u ≥ u1 . Because we have already established ∂u ∂u ∗ that u decreases in s and approaches 1 as s approaches zero, there exist 0 < s1 < s2 that ∗ ∗ ≥ 0 unless α > c and s ≤ s2 and dp ≥ 0 unless α > c correspond to u1 and u2 . Thus, dw ds ds and s ≤ s1 .

32

A4

Proof of Proposition 3: Credible signaling when α − cL large and α − cH small

To show that signaling is credible, it needs to be established that the manufacturer’s informed equilibrium payoff exceeds his maximized payoff from misreporting in both the low cost state (Π(w(uL , cL ), uL , cL ) − Π(w(uH , cL ), uH , cL ) ≥ 0) and the high cost state (Π(w(uH , cH ), uH , cH ) ≥ Π(w(uL , cH ), uL , cH )). To do this, I first establish a simplified expression for these profit differences (Lemma 10), demonstrating that whether the manufacturer prefers more or less search depends on search elasticities εx (w, u) and εG (u) at retailers and competitors. Then I establish that the search elasticity at retailers increases monotonically in the wholesale price (Lemma 11), which implies for each search threshold u the existence of a wholesale price w(u) ˜ below which the manufacturer prefers less search and above which he prefers more search. I then show that for all u ∈ [uH , uL ], when the manufacturer’s cost cH is sufficiently high his marginal profit at w(u) ˜ is positive, which implies that at his optimal wholesale price he induces a high retailer search elasticity and prefers less search. Because the manufacturer would set a high wholesale price for every u ∈ [uH , uL ], he prefers threshold uH and thus reports truthfully. Then the same argument is made when cL is sufficiently low, showing that for every u ∈ [uH , uL ] the marginal profit at w(u) ˜ is negative, the manufacturer prefers a low wholesale price, inducing a low retailer search elasticity and a preference for more search. To begin, I establish the expression used to evaluate the difference in the manufacturer’s maximized payoff from inducing either threshold uH or uL . Using the fact from Lemma 1 that the acceptance probability can be expressed as x(u + w − α), define the elasticities of ′ (u+w−α) g(u) acceptance probabilities as εx (u + w − α) ≡ −x and εG (u) ≡ 1−G(u) at retailers and x(u+w−α) competitors. Lemma 10 Let w(u, c) denote the profit-maximizing wholesale price. Then   Z uL εG (u) Π(w(uL , c), uL , c) − Π(w(uH , c), uH , c) = − 1 du. Q(w(u, c), u) εx (u + w(u, c) − α) uH Proof of Lemma: The manufacturer’s maximized profit changes with respect to u at a rate dΠ(w(u, c), u, c) ∂Π(w(u, c), u, c) ∂w(u, c) ∂Π(w(u, c), u, c) = + du ∂w ∂u ∂u ∂Q = 0 + (w(u, c) − c) ∂u   = w(u, c) − c Q(1 − Q) εG (u) − εx (u + w(u, c) − α) , 33

(13)

where the first term on the right hand side of the first line equals zero by the envelope theorem and the expression for ∂Q comes from (7). Also, because w(u, c) is optimal for the ∂u manufacturer it satisfies the first order condition ∂Q ∂w = 1 − (w(u, c) − c)(1 − Q)εx (u + w(u, c) − α) ⇔ 1 (w(u, c) − c)(1 − Q) = , εx (u + w(u, c) − α) 0 = Q + (w(u, c) − c)

where the second line comes from solving for the right hand side back into (13) obtains

∂Q ∂w

from (3) and simplifying. Then, plugging

  dΠ(w(u, c), u, c) εG (u) = Q(w(u, c), c) −1 . du εx (u + w(u, c) − α) The expression in the statement of the lemma then immediately follows by the fundamental theorem of calculus. Lemma 10 yields an easily stated condition for the manufacturer’s preference for search based solely on search elasticities. It follows that if for every u ∈ [uH , uL ] the manufacturer’s optimal wholesale price satisfies εx (u + w(u, cH ) − α) ≤ εG (u) ≤ εx (u + w(u, cL ) − α), then signaling is credible. Doing this by explicitly characterizing w(u, c) and directly evaluating εx (u + w(u, c) − α) is not feasible without further functional form assumptions. Instead, I demonstrate that w(u, c) falls in a range of wholesale prices in which more search is preferred when c = cL and less search is preferred when c = cH . Toward this end, the following lemma defines these ranges by identifying a threshold price w(u) ˜ below which more search is preferred and above which less search is preferred. Then, I show that the manufacturer’s marginal profit evaluated at w(u) ˜ is negative in the low cost state and positive in the high cost state, which by the fact that the manufacturer’s profit is single peaked shows the main result. Lemma 11 For each u ∈ [uG , ν¯) there exists a unique w(u) ˜ such that εx (u + w(u) ˜ − α) > ( ( 0. Because x > 0, x′ < 0, and as was proven in Appendix A1 x′′ < 0, it follows that dε dw ′ Furthermore, limw→α+¯η−u εx = ∞ because x approaches zero and x is bounded because f (η) is bounded. In addition, limw→−∞ εx = 0. This follows by considering the retailer’s marginal profit

πp = 1 − F (u + p − α) + (p − w)f (u + p − α)   1 − F (u + p − α) = f (u + p − α) · − (p − w) . f (u + p − α) Even at p = α + η − u, at which the acceptance probability is one, the marginal profit with 1 − (α + η − u − w) and is negative when w is sufficiently small (and respect to price is f (η) potentially negative). Thus for low wholesale prices, x(u + w − α) = 1 and x′ (u + w − α) = 0, implying that εx (u + w − α) = 0. Therefore, it has been established that for any u ∈ [uG , ν¯), retailers’ elasticity εx (u+w −α) is increasing in w, and for every ε ∈ [0, ∞) there exists a unique w such that εx (u + w − α) = ε. Finally, note that εG (u) is increasing and bounded on [εG (uG ), ∞) ⊂ [0, ∞) when u ∈ [uG , ν¯). Thus, for every u ∈ [uG , ν¯) there exists a unique w(u) ˜ so that εx (u + w − α) > ( ( 0, I demonstrate here that for a fixed search cost s signaling is credible if cL is sufficiently small and cH is sufficiently large. In the following proposition I then fix cL < α + η¯ − ν¯ < cH and show signaling is credible when search cost s is sufficiently small. Note that although the role of these parameters in evaluating the sign of expression (14) for a given threshold u is straightforward, the task is more complicated when taking into account all u ∈ [uH , uL ]. For example, when search cost s decreases then the marginal profit in (14) remains unchanged at any particular u. What changes, however, are the full information thresholds uL and uH , and thus the set of search thresholds over which the sign of the marginal profit must be evaluated. Explicitly keeping track of how uH and uL change with respect to the parameters of interest is difficult and instead I use a bounds approach. Namely, because Lemma 6 establishes that [uH , uL ] ⊂ [uG , ν¯) for all parameter values, it is sufficient to show that the marginal profit has the correct sign for all u ∈ [uG , ν¯). Lemma 12 For a given cL , there exists c¯H (cL ) < α + η¯ − uG so that whenever cH > c¯H (cL ) signaling is credible in the high cost state, i.e. Π(wH , uH , cH ) ≥ Π(w(uL , cH ), uL , cH ). Proof of Lemma: It needs to be shown that expression (14) is positive at all u ∈ [uH , uL ]. First, εG (u) is bounded above by εG (uL ) < ∞ for all u, which follows from the fact that g(u) εG (u) is increasing and uL < ν¯. Next, g(u)+|x′ (u+ is bounded above by some γ¯ , which is w(u)−α)| ˜ guaranteed to be finite because f (η) and g(ν) are bounded. Finally, w(u) ˜ is bounded above by α + η¯ − uG , because for any higher w retailers do not make sales, which corresponds to an elasticity of εx = ∞, which is a contradiction of the definition of w. ˜ Putting this together,       ∂Π(w(u), ˜ u, cH ) g(u) Sign = Sign 1 − (w(u) ˜ − cH ) εG (u) ∂w g(u) + |xu (u + w(u) ˜ − α)|   ≥ Sign 1 − (α + η¯ − uG − cH )¯ γ ε¯G (uL ) . ˜ H) > 0 for all u ∈ [uH , uL ] if cH > c¯H (cL ) ≡ α + η¯ − uG − Thus ∂Π(w(u),u,c ∂w enters implicitly in threshold uL .

1 , γ ¯ εG (uL )

where cL

One way to understand this result is to consider an extreme case in which cH > α + η¯ − uL . Here, if the manufacturer misreports, he makes it so consumers’ maximum willingness to pay is below his cost and thus he reports truthfully to avoid being forced out of the market. This same logic applies without the manufacturer being fully priced out – misreporting and 36

reducing the search threshold reduces the manufacturer’s sales disproportionately at higher wholesale prices, thus as his cost increases misreporting becomes less attractive. Lemma 13 There exists cL so that Π(wL , uL , cL ) ≥ Π(w(uH , cL ), uH , cL ) whenever cL < cL . Proof of Lemma: Here the aim is to show that expression (14) is negative at all u ∈ [uH , uL ]. First, εG (u) ≥ εG (uG ) for all u ∈ [uH , uL ] because εG (u) is increasing and because g(u) u ≥ uG . Also, g(u)+|x′ (u+ is bounded from below by γ > 0, which again follows from w(u)−α)| ˜ the fact that f (η) and g(ν) are finite and have full support. The key is to demonstrate that w(u) ˜ is bounded from below. The object of interest is w(εx , u), the wholesale price which induces an elasticity εx at retailers when the search threshold is u. The proof of Lemma 11 implies that εx (u + w − α) is monotonically increasing in its argument and has a range [0, ∞) for u ∈ [uG , ν¯) and w ∈ R. Therefore w(εx , u) is well-defined on [0, ∞) × [uG , ν¯), increasing in εx , and decreasing in u. Then, let w ≡ w(εG (uG ), ν¯) and observe that w(u) ˜ ≥ w for all u ∈ [uH , uL ]. That is, the wholesale price that equates retailers’ and competitors’ elasticities for threshold u is higher than the wholesale price that induces retailers that face consumers with the highest possible utility threshold ν¯ to have the lowest feasible competitor elasticity εG (uG ). Then, returning to the expression for the manufacturer’s marginal profit,       g(u) ∂Π(w(u), ˜ u, cL ) = Sign 1 − (w(u) ˜ − cL ) εG (u) Sign ∂w g(u) + |x′ (u + w(u) ˜ − α)|  ≤ Sign 1 − (w − cL )γεG (uG ) . Thus, if cL ≤ cL ≡ w −

1 γεG (uG )

then Π(wL , uL , cL ) ≥ Π(w(uH , cL ), uH , cL ).

The quantities α − cL in Lemma 13 and α − c¯H (cL ) in Lemma 12 define the minimal and maximal gains from trade in the low and high states required for credible signaling, as per the statement of Proposition 3. Observe that by Lemma 9, if signaling is credible with parameters (α, cL , cH ) then it also credible with parameters (α + δ, cL + δ, cH + δ), because the latter is equivalent up to nominal price changes. Therefore the minimal and maximal gains from trade required for credible signaling are independent of α. In turn, if costs are required to be non-negative then there is a minimal α at which the required minimal gains from trade in the low cost state are feasible.

A5

Proof of Proposition 4: Credible signaling when s is small

Most of the work has already been done in the proof of the previous statement. Again, the strategy is to evaluate the manufacturer’s marginal profit at w(u) ˜ in (14) and to show that 37

Πw (w(u, ˜ cL ), u, cL ) ≤ 0 ≤ Πw (w(u, ˜ cH ), u, cH ) for all u ∈ [uH , uL ]. To build intuition, recall from Lemma 6 that as s approaches zero uG approaches ν¯ and thus all u ∈ [uG , ν¯) also approach ν¯, with εG (u) aproaching infinity. By definition, at the wholesale price w(u) ˜ the search elasticity for retailers also approaches infinity, and thus the retailers’ acceptance probability must approach zero, which has two important implications. First, as acceptance probabilities are both shrinking toward zero the manufacturer’s demand at w˜ becomes highly elastic, because a small change in w results in a dramatic percent change in x and thus Q. Second, because the retailers’ acceptance elasticity approaches zero and thus w˜ approaches α + η¯ − ν¯, the manufacturer’s markup approaches α + η¯ − ν¯ − c. If c = cH then as s approaches zero w(u) ˜ − cH approaches α + η¯ − ν¯ − cH , which is strictly negative by assumption. There is then some sˆ so that if s ≤ sˆ then w(u ˜ G ) − cH ≤ 0, which ˜ implies w(u) ˜ − cH ≤ 0 for all u ∈ [u(σH ), uL ]. Thus the marginal profit dΠ(w(u),u)) > 0 for all dw u ∈ [uG , ν¯) and reporting cH truthfully is a best response. If c = cL then w(u) ˜ − cL approaches α + η¯ − ν¯ − cL which is bounded above zero by assumpg(u) tion. The term g(u)+|xu (u+ is bounded above zero and εG (u) monotonically approaches w(u)−α)| ˜   g(u) εG (u) apinfinity because u approaches ν¯. Thus the expression (w(u) ˜ − c) g(u)+x′ (u+w(u)−α) ˜ proaches infinity as s approaches zero, which implies that for small s the marginal profit dΠ(w(u),u)) ˜ < 0 for all u ∈ [uG , ν¯), and therefore reporting cL truthfully is a best response. dw

A6

Proof of Proposition 6: Manufacturer prefers recommendations to no recommendations

I characterize the consumer’s optimal search behavior conditional on the manufacturer and retailers setting informed equilibrium prices wL and pL in the low cost state and wH and pH in the high cost state, and then show that conditional on the consumers’ behavior these prices are in fact best responses. Then I show that the manufacturer and consumers are both better off in an equilibrium with informative recommendations than in an equilibrium with no recommendations. To save notation I assume throughout that quality α = 0. Beliefs Consumer beliefs off the equilibrium path play an important role here because they determine deviation profits for retailers and by extension for the manufacturer. The main model focuses on an equilibrium in which off the path beliefs depend only on the manufacturer’s recommendation and a retailer cannot affect consumers’ beliefs with his price. To make a comparison in a setting with no recommendations, I specify off the path beliefs to attempt 38

to replicate these retailer incentives.13 To this end, in a separating equilibrium with retail prices pL and pH , for any history h = {{(p1 , η1 ), ..., (pk , ηk )} ∪ {ν1 , ..., νl }} let  1   2 if ∄ p1 (i.e. no retailers visited) . (15) µ(h) = 0 if p1 < pH   1 if p1 ≥ pH For histories on the equilibrium path, it is easily verified that µ(h) is simply computed using Bayes rule: beliefs start with the prior µ = 21 and remain fixed only until the first retail price is observed, thereafter either moving up to µ = 1 if that price is pH and µ = 0 if that price is pL . For histories off the equilibrium path, beliefs are determined entirely by the first observed retail price and follow a step function. Because a retailer benefits from reducing the utility threshold, making the step in beliefs at as high a price as possible minimizes the retailer’s incentive to manipulate beliefs. With this structure, a retailer can alter a consumer’s belief only if he is the first retailer in the consumer’s history and if the deviation in price is large. Search strategy and manufacturer demand In the conjectured equilibrium with retail prices p(c) from the informed equilibrium, a consumer starts search with belief µ = 12 and maintains this belief until visiting a retailer, where the price p(c) perfectly reveals the state and induces him to search with correct belief µ(c) thereafter. In this environment a threshold strategy u(µ) is still optimal for the consumer during any particular visit, and it is easily verified that u(1) < u(1/2) < u(0). However, because learning takes place the consumer may sometimes return to accept previous offers. For example, suppose a consumer has thus far visited only competitors and received a highest utility offer uˆ. Upon visiting a retailer for the first time and observing price p(cH ), the consumer reduces his threshold from u(1/2) to u(1) and may return if u(1) ≤ uˆ < u(1/2). The manufacturer’s demand must reflect this possibility, and thus requires keeping track of search histories. Let Q(µ, c, u ˆ) denote the continuation probability that a consumer with belief µ and current best available option uˆ eventually buys from the manufacturer when the true cost is c. Then,   1 1 ˆ < u(µ(c)) Q (1/2, c, 0) = x(c) + (1 − x(c))Q µ(c), c, u 2 2 1 + G (u (1/2)) E [Q(1/2, c, ν) | ν < u(1/2)] . 2 13

Replicating the incentives perfectly is not possible without recommendations. For example, in any separating equilibrium it must be that µ(pL ) = 0 and µ(pH ) = 1, thus in the low cost state a retailer that deviates from pL to pH can reduce the consumer’s threshold from u(0) to u(1) and thus shift out demand.

39

The left hand side is the continuation probability of a new consumer, whose belief is the prior µ = 21 and whose outside option of zero corresponds to exit without purchasing. If the first visit is to a retailer, the consumer learns the state by observing the retail price and forms correct belief µ(c) with corresponding search threshold u(µ(c)), accepts with probability x(c) ≡ 1 − F (u((µ(c)) + p(c)), else rejects and continues to search with correct belief µ(c) thereafter. If the first visit is to a competitor, the consumer rejects with probability G(u(1/2)) and continues search with the same belief µ = 12 and an option to return and accept his current offer ν. Observe that when c = cH the consumer may exercise the option to return to the competitor because upon learning the true state his threshold falls, whereas when c = cL the consumer never returns because upon learning the true state his threshold increases.   x(c) To simplify the above expression, note that Q(c) ≡ Q µ(c), c, u ˆ ≤ u(µ(c)) = x(c)+1−G(u(µ(c))) , because a consumer that has learned the true state at the first retailer and rejected acts exactly like an informed consumer in the model with recommendations. In addition, letting | ν≤u(1/2)] , the manufacturer’s demand simplifies to γ ≡ E[Q(1/2,c,ν) Q(1/2,c,0) Q(1/2, c, 0) =

2 − G(u(µ(c))) Q(c). 2 − γG(u(1/2))

(16)

If the manufacturer’s cost is cL , the fact that the consumer never returns to accept previous offers implies that γ = 1 and consequently Q(1/2, cL , 0) =

2 − G(u(0)) Q(cL ). 2 − G(u(1/2))

Because u(1/2) < u(0), sales in the low cost state are lower than in the informed equilibrium. Meanwhile, if the manufacturer’s cost is cH , the fact that a consumer may return to accept previous offers implies γ ≤ 1, which yields the following upper bound for sales: Q(1/2, cH , 0) ≤

2 − G(u(1)) Q(cH ). 2 − G(u(1/2))

Because u(1) < u(1/2), this upper bound is higher than the quantity sold in the high cost state in the informed equilibrium. Retailer and Manufacturer strategies Having established the consumers’ search strategy and the ensuing demand function faced by the manufacturer, I now show that it is a best response for the manufacturer and retailers to charge the same prices as when consumers are informed. Lemma 14 In the low cost state with no recommendations, the full information wholesale 40

price wL and retail price pL are consistent with equilibrium whenever cH is sufficiently large. Proof of Lemma First, it must be shown that pL is a best response for retailers. In the low cost state a retailer faces two groups of consumers. The first group is those that have visited at least one retailer, learned the state from price pL , and formed threshold u(0). This group is homogeneous and acts exactly like the consumers informed by recommendations, buying from the retailer if and only if η ≥ u(0)−p. The second group consists of consumers that have visited no retailers but possibly some competitors, and have received a best offer uˆ up to this point, with uˆ < u(1/2) because they continued search. This group forms a threshold u(µ(p))   based on the price at the current retailer, and accepts if and only if η ≥ max uˆ, u(µ(p)) +p. If the retailer charges a price p < pH then he sells to both groups if and only if η ≥ u(0) + p, the same demand as under full information, and given that he faces the same wholesale price wL the optimal price in this range is pL . If the retailer charges a price p′ ≥ pH , he still sells to the first group if and only if η ≥ u(0) + p′ but now sells to the second group if and only if η ≥ u(1) + p′ . That is, by charging p′ the retailer makes the second group more pessimistic thereby shifting out demand. To ensure that the payoff from pL is larger, observe that π(p′ ) ≤ (p′ − cL )(1 − F (u(1) + p′ )) ≤ (p′ − cL )(1 − F (uG + cH )). The first inequality comes from assuming that every consumer has the pessimistic threshold u(1) rather than just the second group. The second inequality follows from the fact that p′ ≥ pH ≥ cH and that u(1) ≥ uG , recalling that uG was earlier defined as the threshold when a consumer commits to never accepting retailers’ offers. As cH is increased the right hand side of the second inequality approaches zero whereas the payoff to charging pL remains fixed at some strictly positive amount. Thus, for a high enough cH , it is a best response for the retailer to charge pL when facing wholesale price wL . Next it needs to be demonstrated that wL is best for the manufacturer. The set of wholesale prices can be categorized by the retail prices p(w) that they induce. Namely, there exists some w¯ so that p(w) < pH when w < w¯ and p(w) ≥ pH when w ≥ w. ¯ In the first group of lower wholesale prices the manufacturer induces retail prices that reveal the low cost state, consequently consumers never return during search and the manufacturer’s profit is given by Π(w < w) ¯ = (w − cL )Q(1/2, cL , 0)    2 − G(u(0)) 1 − F (p(w) + u(0)) = (w − cL ) 2 − G(u(1/2)) 1 − F (p(w) + u(0)) + 1 − G(u(0)) 2 − G(u(0)) = ΠM SRP (w, cL ), 2 − G(u(1/2))

41

with demand Q(1/2, 0, cL ) given in (16). Because the profit is simply a scalar multiple of the full information profit it has the same maximizer. Therefore, wL is the best option of all wholesale prices below w. ¯ To show that prices above w¯ make the manufacturer worse off, observe from (16) that    1 − F (p(w) + u(1)) 2 − G(u(1)) Π(w ≥ w) ¯ = (w − cL ) 2 − γG(u(1/2)) 1 − F (p(w) + u(1)) + 1 − G(u(1))    2 − G(u(1)) 1 − F (cH + uG ) ≤ (w − cL ) . 2 − γG(u(1/2)) 1 − F (cH + uG ) + 1 − G(u(1)) The inequality follows again from the fact that p(w) ≥ pH ≥ cH and that u(1) ≥ uG . As cH increases the term in the third parentheses of the last line approaches zero whereas the term in the first parentheses remains unchanged and the term in the second parentheses falls because u(1/2) falls. Thus, as cH is increased Π(w ≥ w) ¯ approaches zero. Meanwhile, 2−G(u(0)) Π(wL ) = 2−G(u(1/2)) ΠM SRP (wL , cL ) also falls with cH as u(1/2) falls, however is bounded above by optimal.

2−G(u(0)) ΠM SRP (wL ) 2

> 0. Thus for large enough cH the wholesale price wL is

Lemma 15 In the high cost state with no recommendations, the full information wholesale price wH and retail price pH are consistent with equilibrium whenever cH is sufficiently large. Proof of Lemma The retailer again faces two groups of consumers, those who have visited other retailers and hold a threshold u(1) and those who have not visited any retailers but possibly some competitors and hold a threshold u(1/2) with best previous offer uˆ < u(1/2). For prices p ≥ pH the retailer sells to  the first  group if and only if η ≥ u(1) + p and to the second group if and only if η ≥ max uˆ, u(1) + p. The retailer thus faces an average utility threshold that is higher than under full information, and because profits are single-peaked, strictly prefers pH to all higher prices. For prices p < pH , the first group continues to accept if and only if η ≥ u(1) + p whereas the second group forms the optimistic belief µ(p) = 0 and adopts a higher utility threshold, accepting if only if η ≥ u(0) − p. Demand is thus lower at all prices p < pH than under full information, and because this deviation was not profitable under full information, it continues not to be so. To demonstrate that wH is a best response for the manufacturer, observe again that there exists some w¯ so that p(w) < pH if w < w¯ and p(w) ≥ pH if w ≥ w. ¯ The latter case is analogous to truthful reporting in that consumers form the correct belief µ(p(w)) = 1 after visiting a retailer, and consequently may return to accept a previous offer. Following (16),

42

the manufacturer’s profit function is Π(w ≥ w) ¯ = (w − cH )Q(1/2, cH , 0)    1 − F (p(w) + u(1)) 2 − G(u(1)) = (w − cH ) 2 − γG(u(1/2)) 1 − F (p(w) + u(1)) + 1 − G(u(1)) 2 − G(u(1)) = ΠM SRP (w, cH ). 2 − γG(u(1/2)) The current profit function simply a scaling of the full information profit ΠM SRP (w, cH ), thus the full information maximizer wH is also optimal here. Therefore, wH is the most profitable of the wholesale price above w. ¯ Charging a wholesale price w < w¯ is analogous to misreporting as it induces the incorrect belief µ(p(w)) = 0 when consumers visit a retailer. To demonstrate that wH yields a higher profit than wholesale prices in this low range, note that    2 − G(u(1)) 1 − F (p(w) + u(0)) Π(w < w) ¯ = (w − cH ) 2 − γG(u(1/2)) 1 − F (p(w) + u(0)) + 1 − G(u(0))    2 − G(u(1)) 1 − F (p(w) + u(0)) ≤ (¯ η − u(0) − cH ) , 2 − γG(u(1/2)) 1 − F (p(w) + u(0)) + 1 − G(u(0)) in which the inequality follows from plugging in w ≤ p(w) ≤ η¯ − u(0) into the first term. Then, as cH grows u(0) remains unchanged and the first term falls toward zero and eventually becomes negative, thus falling below the weakly positive profit from charging wH . Payoff comparison to making recommendations Having thus shown that the informed equilibrium values wL , pL and wH , pH are consistent in an equilibrium with no recommendations, I now demonstrate the manufacturer and consumers are worse off. The fact that in both states demand is proportional to the informed equilibrium level facilitates the comparison of the manufacturer’s profit across the two settings as follows:  1 (wH − cH )(Q(1/2, cH , 0) − Q(cH )) + (wL − cL )(Q(1/2, cL , 0) − Q(cL )) − ΠM SRP ≤ 2   1 = ΠM SRP (cH )(G(u(1/2)) − G(u(1))) − ΠM SRP (cL )(G(u(0)) − G(u(1/2))) . 4 − 2G(u(1/2)) (17)

ΠN o

M SRP

Because u(1) < u(1/2) < u(0), removing recommendations benefits the manufacturer in proportion to his informed profit in the high cost state and harms him in proportion to the informed profit in the low cost state. For fixed values of cL < α + η¯ − ν¯ and 43

s < 12 E[ν] (as assumed in the main model), as cH is increased ΠM SRP (cH ) falls toward zero, G(u(1/2)) − G(u(1)) grows but is bounded above by one, ΠM SRP (cL ) remains unchanged, and G(u(0)) − G(u(1/2)) grows. Therefore, once cH is sufficiently large the difference in (17) becomes negative. Note that this is not a limit argument, that is there is a positive measure set of values of cH for which ΠM SRP (cH ) > 0 and the result holds. Finally, given that the consumers face the same prices as with recommendations but are less informed, they must necessarily be worse off.

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