A Model of Recommended Retail Prices

Dmitry Lubensky∗

October 31, 2016

Abstract

Manufacturers frequently post non-binding public price recommendations, but neitherthe rationale for this practice nor its impact on prices is well understood. I develop amodel in which recommendations signal a manufacturer’s production cost to search-ing consumers, who then form beliefs about retail prices. Increasing search makesconsumers reject offers for the manufacturer’s and competitors’ products more often,and I show that both consumers and the manufacturer prefer more search when theproduction cost is low and less search when it is high. With incentives thus aligned,manufacturer recommendations inform consumers via cheap talk and their removalharms both parties.

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

∗Indiana University, Kelley School of Business; dmitry.lubensky@gmail.com.

Thanks to Josh Cherry, Kai-Uwe Kuhn, Francine Lafontaine, Stephan Lauermann, Doug Smith, and MikeStevens 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, MatthijsWildenbeest and the participants of the Consumer Search and Switching Cost Workshops in 2013 and 2014for helpful comments on this version.

1 Introduction

Manufacturers use non-binding recommended retail prices in markets ranging from groceryproducts to big ticket items such as electronics, appliances, and cars. These recommenda-tions, which come in a variety of forms such as list prices, manufacturer suggested retailprices (MSRPs), sticker prices, etc., are visibly printed on product packaging and often pro-moted by the manufacturer through costly advertising. The existence of a link betweenprice recommendations and market outcomes has both been shown empirically (e.g. Faberand Janssen (2008), De los Santos et. al. (2016)) and also implicitly assumed in the myriadstudies 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 ofmarket participants, for example, consumers often expect a discount off the MSRP whenbuying 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 howthey do so is quite limited. Due to the fact that price recommendations are non-binding, themechanism by which they have an impact and the motives of the manufacturer in makingthese recommendations are still not well understood.

A common explanation is that recommendations act as price ceilings. This story is com-pelling because most products sell at or below MSRP and because the manufacturer’s ratio-nale for imposing a price ceiling in order to reduce double marginalization is well established.Yet this explanation of recommendations is incomplete. First, price recommendations arenot binding, at least in name, and thus in jurisdictions where resale price maintenance islegal it is not clear why a manufacturer would make a recommendation instead of imposinga price ceiling directly.1 In addition, recommendations often do not bind in practice, for in-stance most cars sell strictly below MSRP but very few sell at MSRP. Lastly, manufacturerspublicize their recommendations and an explanation of recommendations as explicit priceceilings ignores the potential role played by consumers.

This article presents an alternative explanation in which price recommendations affect con-sumer search. I develop a model in which a manufacturer faces consumers with unit demandwho search among sellers of his product and sellers of competing products, observing a priceand a utility shock at each visit. The manufacturer’s cost is uncertain and thus consumersdo not know the distribution of downstream prices. In particular, a consumer visiting a sellerdoes not know whether it is optimal to accept the current offer or continue searching. A pricerecommendation informs consumers of the manufacturer’s cost, and thus of the continuationvalue of rejecting. In this way recommendations directly affect consumers’ search decisions,and by extension the prices set by retailers.

1For instance, MSRPs are commonly used in the United States where the Supreme Court ruled in 1997in 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 searchdecisions, the incentives for a manufacturer to make these recommendations are not imme-diately clear. I demonstrate that in revealing his cost and thereby affecting search the man-ufacturer faces a tradeoff. Specifically, consider consumers A and B each currently marginalat a manufacturer’s retailer and at a competitor. By making both consumers more optimisticabout the distribution of downstream prices and thus continue searching, the manufacturerreduces his probability of selling to A from one to the continuation probability and increaseshis probability of selling to B from zero to the continuation probability. More search thusbenefits the manufacturer if and only if the continuation probability is sufficiently high.

With these incentives for affecting search, the question is whether the manufacturer canuse price recommendations to credibly reveal his costs. A price recommendation is purecheap talk because it is not a falsifiable statement of fact2 and it is no costlier to recommenda higher price than a lower one. If the manufacturer had incentive to mislead consumers, theyin turn would rationally ignore recommendations and no information could be transmitted.In order for non-binding non-falsifiable recommendations to have an effect, the manufacturermust prefer to report truthfully.

The article’s main result is that with respect to search the interests of consumers and themanufacturer are aligned and thus credible signaling is possible. In the model the manufac-turer’s cost has either a low or a high realization. When the cost is low the manufacturersets a low wholesale price which leads to low retail prices, in turn making his product moreattractive on average than the products of competitors. The low cost manufacturer thusfaces 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 leadsconsumers to expect low prices. Similarly, when his cost is high the manufacturer sets a highwholesale price, making his product on average less attractive than competing products andthus inducing a low continuation probability. In this case the manufacturer benefits from lesssearch and reveals his cost truthfully to induce consumers to expect higher prices. Becausethe manufacturer has incentive to reveal truthfully in both states, he can inform consumersusing cheap talk.

This ability to credibly communicate the continuation value of searching is unique to themanufacturer. By contrast, a downstream seller always benefits by making visiting con-sumers maximally pessimistic about their outside options regardless of the true state. Themodel thus predicts that whereas manufacturer price recommendations influence consumer

2It is difficult to hold a manufacturer liable when he recommends a price and a retailer ignores therecommendation. By contrast, other statements on product packaging, such as “New York Times BestSeller” 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 competitorsshould have little effect.

Manufacturer price recommendations have traditionally received little explicit attention andinstead have been lumped in as an instrument of resale price maintenance in the literature onvertical price restraints. However, there has been renewed interest in the topic and severalrecent articles have explored the mechanism behind the effects of these non-binding recom-mendations. Buehler and Gartner (2013) shows that MSRPs can be used by a manufacturerto convey demand and cost information to a retailer. The manufacturer and retailer areengaged in an infinitely repeated game and there exists an equilibrium in which both aim tomaximize joint profits. Because in this equilibrium incentives are aligned, the manufacturercan credibly inform the retailer using cheap talk. Although the authors establish that cred-ible communication between a manufacturer and retailer is possible, it is not immediatelyclear why publicizing MSRPs is the preferred method for doing so. In particular, publicprice recommendations are also likely to affect the behavior of consumers and the authorsabstract from this effect.

A different approach is taken by Puppe and Rosekranz (2011), building on the theory inThaler (1985) in which a price recommendation provides a behavioral reference point. Con-sumers’ demand is kinked above the MSRP so that the recommendation acts as a de-factoprice ceiling. The present article can be thought of as providing a rational foundation for thereference point theory. I explicitly model how consumers incorporate the MSRP and theirknowledge of the market to form price expectations, and derive their demand as the solutionto the problem of optimal search. A consumer that refuses to accept a price above a cer-tain 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 whethermanufacturer recommendations ought to be regulated. I show that removing recommenda-tions harms both consumers and the manufacturer. In particular, I demonstrate that whenrecommendations are banned there exists an equilibrium with the same wholesale and retailprices as the equilibrium with recommendations, but in which consumers are less informedduring search. In this equilibrium consumers learn about the manufacturer’s cost throughthe price charged by his retailers, however there is a group of consumers that has not yetvisited any retailers and remains uninformed. Upon visiting a competitor these consumerssearch too little when the cost is low and too much when the cost is high, and are thus worseoff in both states. Meanwhile the manufacturer faces a tradeoff with this group: he benefitsfrom consumers being too selective at competitors when his cost is high and is harmed byconsumers not being selective enough at competitors when his cost is low. Because the valueof inducing these consumers to continue searching equals the continuation probability, theeffect in the low cost state dominates and on net the manufacturer is worse off. Broadlyspeaking, taking away the ability of the manufacturer and consumers to coordinate harms

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both because their interests are aligned.

Whether sellers can cheaply communicate with searching consumers was studied in a relatedsetting by Horowitz (1992), who provides evidence of a relationship between non-bindinglist prices and transaction prices in the market for residential housing. Kim (2012) thendevelops a theoretical foundation, showing that when sellers have private information aboutproduct quality, there exists a partially informative mixed strategy equilibrium in whichsellers that report a high quality attract fewer visitors and command a higher price thansellers that report a low quality. In this equilibrium, a buyer’s expected utility from visitinga seller reporting a high quality equals that of visiting a seller reporting a low quality, thusmisreporting does not improve a seller’s ranking among his competitors. By contrast, in thecurrent context no such indifference can be obtained because a downstream seller alwaysbenefits from understating a visiting consumer’s future utility offers. Thus it is solely themanufacturer and not the downstream retailers that can credibly communicate the returnsto search.

This work also makes a methodological contribution. First, it provides a tractable model of avertical 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 com-parative statics. The closest work in this area is Janssen and Shelegia (2015), which insteaduses the Stahl (1989) framework to obtain an equilibrium in mixed strategies. Janssen andShelegia’s central result is that double marginalization is exacerbated in a setting in whichconsumers search, owing to a holdup argument similar to that in Diamond (1971), and thisresult also obtains quite naturally in the present model. The articles diverge in that I focuson the manufacturer communicating the realization of uncertain market conditions whereasin Janssen and Shelegia (2015) there is no aggregate uncertainty and thus nothing to com-municate. In addition, I derive a simple and intuitive expression for the demand faced byan upstream manufacturer in a search environment, drawing a connection between modelsof search and contests.

The rest of this article proceeds as follows. Section 2 presents the model, Section 3 charac-terizes the equilibrium when consumers are informed, and Section 4 demonstrates the mainresults that cheap signaling can credibly convey manufacturer costs to consumers, and thatwithout recommendations consumers and the manufacturer are worse off. Section 5 providesa discussion of whether recommendations can inform consumers about product quality in-stead of manufacturer costs and draws a comparison between recommendations and resaleprice 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 real-ization cL and a high realization cH , both equally likely. The manufacturer sells his productthrough a collection of exclusive downstream sellers which face no additional costs. Thedownstream market also includes exclusive sellers of other products. I refer to the sellers ofthe 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 com-prising one half of the market.

There is a measure one of consumers with unit demand. A consumer faces the set ofdownstream 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’sprice. Assume that both for the manufacturer’s and competitors’ products, αi is publiclyobserved by all market participants.

To fix ideas one may think of the market for an economy sedan with buyers choosing amongdifferent cars in this class (i.e. Ford Focus, Nissan Sentra, Honda Civic, etc.) by sequentiallyvisiting dealerships, each exclusive to a particular manufacturer. A consumer visiting a Forddealership j knows that every Ford Focus in the lot has the same dimensions, weight, factorywarranty and other features that are common across all versions of this vehicle, correspond-ing αFord in the model. However other features such as color, leather seats, and a sunroofvary across vehicles, and the consumer is uncertain whether dealership j has her preferredcombination of these features in stock, which is captured by ηj .

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The analysis focuses on the behavior of the manufacturer and his retailers, whereas com-petitors are in the model to provide consumers with a stochastic outside option. To thisend, 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 isa competitor the price is unmodeled and the value νi ≡ αi + ηj − pj is exogenous and drawnfrom distribution G with support [ν, ν]. Assume distributions F and G admit continuousand uniformly bounded densities f and g, that f is log-concave and non-decreasing, and thatthe hazard ratio g(ν)

1−G(ν)is non-decreasing. To ensure that an initial search is optimal assume

that s < 12EG[νi], so that a consumer’s payoff exceeds s even if she commits to searching

3When a vehicle with a consumer’s preferred features is not in stock a dealer may order it from themanufacturer. However this is often associated with additional time and monetary costs, thus consumerswith different preferences for features differentially value a dealership with a particularly inventory.

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Time

Cost realizedc ∈ {cL, cH}

Manufacturerw(c), σ(c)

Retailersp(w, σ)

Consumers searchA(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 produc-tion cost c ∈ {cL, cH} and chooses a price recommendation σ ∈ {σL, σH} and a wholesaleprice w ∈ R. Then, every retailer j observes σ and w and sets retail price pj. With retailprices fixed, each consumer observes the manufacturer’s recommendation but not his whole-sale price and decides whether to begin searching.4 If she initially decides not to search, sheexits and receives a utility of zero. Otherwise, she pays cost s and visits a randomly selecteddownstream seller, where she observes pj and ηj . The consumer can accept the offer, exit, orpay cost s again and visit a different randomly selected seller. During search a consumer canaccept any previous offer at no additional cost and the process continues until she accepts anoffer 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 issummarized 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 thisexpectation the retailer can use recommendation σ to anticipate consumers’ search behaviorand 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 expectedpayoff is

Π(w, σ |c) = (w − c)Q(w, σ),

4When consumers observe the wholesale price, recommendations serve no purpose because they do nothelp 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 themanufacturer (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 verti-cal relationships and search markets. The downstream game is analogous to the randomutility search model in Wolinsky (1986) and Anderson and Renault (1999), and the uniformwholesale price is a contract that, although suboptimal, is frequently observed and widelyanalyzed, starting with Spengler (1950). There are several additional modeling assumptionswhich 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 shownthat when signaling his cost, the manufacturer is only concerned with how the distributionof utilities offered by his retailers compares with that offered by competitors; the mechanismby which the latter is determined is not relevant for his signaling decision. Consequently, themain result applies in a variety of market structures, including but not limited to a symmet-ric setting in which each downstream seller belongs exclusively to one of several competingmanufacturers, all facing the decision problem described above. The key assumption is thatG(νj) is exogenous to the manufacturer, so that he cannot affect the distribution of outsideoptions through his actions. In this sense, the model captures markets in which upstreamdecisions are made simultaneously or if the manufacturer is the last mover in a sequentialgame.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 lossof generality, however that there are exactly two cost outcomes is an assumption made fortractability. It will be argued that for any cost the manufacturer either prefers as muchsearch as possible or as little search as possible. With this in mind, in any cheap talk equi-librium the manufacturer may only reveal whether his cost falls into the former category orthe latter. When there are only two costs, one in each category, consumers become perfectlyinformed and face no aggregate uncertainty. However, if either category includes more thanone cost, a searching consumer continues to learn during the search process. This makes herdecision problem non-stationary and thereby significantly complicates the analysis.

5Alternatively, if upstream competitors first observe the manufacturer’s recommendation then they wouldincrease prices when he signals high and reduce prices when he signals low. Reporting truthfully is theneven more beneficial when the manufacturer’s cost is high, but now requires a larger cost advantage whenhis cost is low.

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Third, that the recommendation is observed prior to search rather than during the firststore visit is assumed mostly for tractability, else by chance some consumers may visit sev-eral competitors prior to their first retailer and tracking these search histories in order toexpress demand significantly adds to the model’s complexity. The assumption correspondsto a setting in which a manufacturer that already advertises his product decides whether ornot to include a price recommendation in the advertisement. Furthermore, the assumptionseems not to be crucial for the article’s main result of credible signaling. Specifically, themanufacturer’s tradeoff from inducing more search, that is reducing acceptance probabilitiesboth for his product and for competing products, still remains even if MSRPs are discoveredonly during search and not beforehand.

Lastly, consumers decide whether to accept or reject an offer but not which store to visitnext. In a symmetric model this simplifying assumption is benign, and thus it is commonlyused in the search literature. However, in the current environment consumers may learnwhether the manufacturer’s retailers or competitors are likely to offer a better deal, and con-sequently may wish to direct their search. A manufacturer that signals a low cost may gainprominence in the search process, which has been shown to both directly increase his shareof consumers and also potentially decrease competition (e.g. Armstrong et al. (2009), Wil-son (2010), Arbatskaya (2007), Haan and Moraga-Gonzales (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 searchmay simply be determined by store locations. Alternatively, if a consumer is unaware ofwhich sellers carry which products then the decision of which store to visit next is indepen-dent of which product she is likely to prefer.

The analysis proceeds as follows. I first solve the game in which the manufacturer’s cost isknown and derive equilibrium strategies for the manufacturer, retailers, and consumers con-ditional on this cost. I demonstrate that when the manufacturer’s cost increases, he chargesa higher wholesale price and consumers are willing to accept a lower level of utility. I thenuse this result as well as the explicitly derived structure of the equilibrium with informedconsumers to prove that the manufacturer can communicate credibly via cheap talk. Thatis, I derive a condition so that for each cost outcome cL and cH , the manufacturer prefers toinduce the search behavior corresponding to the true cost rather than that associated withthe other cost.

3 Equilibrium with Informed Consumers

In this section I characterize the equilibrium pricing strategies of the manufacturer andretailers and the search strategy of consumers given that the manufacturer’s cost c (but nothis wholesale price) is publicly observed. Going forward I will refer to this as the informed

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equilibrium. To keep notation consistent with the following section, let the manufacturer’ssignal be σ(c) = c and let retailers and consumers condition strategies on σ. In addition, Isuppress 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 drawsand faces two types of uncertainty. The first type includes the identity of the seller on thenext draw, the realization of the outside option νj if the seller is a competitor, and therealization of the preference shock ηj if the seller is a manufacturer’s retailer. These sourcesof uncertainty are stationary, i.e. a consumer’s belief about future realizations is independentof the consumer’s previous search history. The second type is uncertainty over the price pjif 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 helprefine 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’scost and her history allows no additional inference. Hence a consumer faces a stationarydecision problem and, by a standard result in Kohn and Shavell (1974), uses a stationarythreshold policy u(σ) defined implicitly by

s =1

2

∫

ν≥u(σ)

(ν − u(σ))dG(ν) +1

2Epj |σ

[

∫

ηj≥u(σ)+pj−α

(ηj − (u(σ) + pj − α))dF (ηj)

]

=1

2

∫

ν≥u(σ)

(1−G(ν))dν +1

2Epj |σ

[

∫

ηj≥u(σ)+pj−α

(1− F (ηj))dηj

]

. (1)

The right hand side of the top line describes the option value of taking another draw whenthe currently available best option gives a payoff of u(σ), with the first term denoting thevalue from visiting a competitor and the second term denoting the expected value of visitinga retailer. The expectation allows for retail price pj to be uncertain conditional on recom-mendation σ.6 The second line follows from integration by parts.

Next consider the strategy of a retailer, who observes recommendation σ and wholesaleprice w and chooses a retail price p. The probability a retailer makes a sale to a consumerupon her arrival is 1− F (u(σ) + p− α), and if the sale is not made the consumer continuesto search, never returning because there are infinitely many sellers. Taking as given thenumber of arriving consumers each retailer maximizes his per-consumer profit with a price

6In the ensuing equilibrium all retailers charge the same price conditional on σ, however the expectationoperator is useful in that (1) also describes the search threshold if retailers were heterogeneous and chargeddifferent prices, an extension discussed later in the article.

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that solves

p(w, σ) = argmaxp

(p− w)(1− F (u(σ) + p− α)). (2)

Let x(w, σ) ≡ 1− F (u(σ) + p(w, σ)− α) denote a retailer’s optimally chosen probability ofsale 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 retailersand consumers in the downstream market. In the present environment the manufacturer can-not directly affect the consumers’ search threshold u(σ), but can affect retail prices throughhis wholesale price. The manufacturer’s demand Q(w, σ) may be interpreted as the prob-ability that a consumer eventually purchases from a retailer and not a competitor. Thisinterpretation allows for Q(w, σ) to be expressed as

Q(w, σ) =1

2x(w, σ) +

(

1

2(1− x(w, σ)) +

1

2G(u(σ))

)

Q(w, σ).

The first term denotes the joint probability that the consumer’s initial draw is at a manu-facturer’s retailer and that she accepts. The second term accounts for the probability thatthe consumer rejects her initial draw, at either type of seller, and continues to search. Forher next draw the consumer faces the identical problem as for her first draw, which yieldsthe 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 thresholdu(σ) continues to sample until reaching the first seller that meets the threshold, thus it is asif 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 qualifyingsellers that belong to the manufacturer.

By inspection the manufacturer’s demand is concave in acceptance probability x(w, σ), whichfollows solely from the search process and not from any assumption on the underlying distri-bution of valuations. This fact is useful for technical reasons, as the concavity of the demandfunction is important for the existence of a unique optimizer. It also highlights that the ex-tent to which a manufacturer is concerned about double-marginalization depends on how hisretailers compare with the competitors. In a setting with a monopolist manufacturer witha single retailer, a one unit reduction in the sales of the retailer translates into a one unit

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reduction in the sales of the manufacturer. In this search environment, however, a consumerthat rejects an offer from a retailer continues to search. The higher is the acceptance prob-ability at retailers relative to competitors, the higher the chance that a consumer inducedinto searching still buys from the manufacturer.7

The manufacturer solves a monopoly problem with demand Q(w, σ) and constant marginalcost 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) = argmaxw

Π(w, σ, c). (4)

Informed Equilibrium and Properties

I now demonstrate the existence and uniqueness of an equilibrium with informed consumersand 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 wholesaleprice w and search threshold u and then to plug the solution into the the optimization de-cisions of the manufacturer and consumers, reducing the equilibrium characterization intoa system of two equations with two unknowns w and u. It is then demonstrated that themanufacturer’s and consumers’ best responses must intersect exactly once.

An important element of the argument is demonstrating the uniqueness and continuity ofbest responses. Whereas for the search threshold this is immediate and for retailers followsdirectly from the log-concavity of f , the manufacturer’s problem requires more work becausethe curvature of the manufacturer’s demand is a function of the curvature with respect to wof the endogenous acceptance probability x(w, σ). I demonstrate that a non-decreasing den-sity f is sufficient to ensure that the manufacturer’s payoff is single peaked, thus establishingthe existence and continuity of the optimal wholesale price and allowing it to be character-ized by the first order condition. Finally, although the search threshold and wholesale pricecan be strategic complements, I demonstrate that when f is non-decreasing the consumer’sbest response function is everywhere steeper than the manufacturer’s and must result in a

7The observation that a manufacturer is less concerned with double-marginalization when consumerssearch is explored in Janssen and Shelegia (2015), which demonstrates that due to search a manufacturersets 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

s =1

2

∫ ν

uG

(1−G(ν)) dν (5)

implicitly define uG as the utility threshold of a consumer that never accepts offers fromretailers.

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 costs and increases with gains from trade α − c, to prove this formally one must account forequilibrium effects. For instance, whereas an increase in the search cost reduces u for afixed wholesale price w, if the manufacturer lowers w in response then in turn a higher u isoptimal, and it must be established that the initial direct effect dominates. For the boundsof the search threshold, that uG is a lower bound is immediate because, regardless of theset of utilities available at retailers, the return to search when acting optimally is at least ashigh as the return to search when committing to rejecting offers from all retailers. That thesearch threshold never exceeds ν follows from a hold-up argument, similar to that in JanssenShelegia (2015) and Diamond (1971), in that a manufacturer cannot commit to provideconsumers with sufficient return to searching. The full proof can be found in Appendix A2.

4 Equilibrium with Uncertain Costs

In this section I use the informed equilibrium of the preceding section to help describe anequilibrium when manufacturer costs are unobserved by consumers. I begin by describingthe connection between the informed equilibrium and the Bayesian equilibrium, specificallyfocusing on consumer beliefs. Then I outline the intuition for why the manufacturer prefersmore search when his cost is low and less search when his cost is high. Lastly, I demonstratethe 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 anavailable offer, searching, or exiting. With each history is associated a posterior µ(h) about

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the probability that the manufacturer has a high cost, and this posterior must be formedusing Bayes rule whenever possible.

In a conjectured truth-telling equilibrium in which the manufacturer recommends σ(c) andretailers all set the full-information equilibrium price p(σ(c)) ≡ p(w(c), σ(c)), the consumer’sbelief starts at either µ = 1 if σ = σH or µ = 0 if σ = σL prior to her first visit and remainsconstant throughout search. On the equilibrium path, the consumer does not expect herbeliefs to change, so (1) still describes her optimal behavior.

The issue, however, arises when evaluating retailers’ pricing decisions. Specifically, supposethat 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 notguarantee that the consumer continues to believe that costs are low. If the retailer cansuccessfully convince the consumer through this deviation that costs are high, the deviationmay be profitable.8 To address this concern, I restrict beliefs so that

µ(h) =

{

1 if σH ∈ h

0 if σL ∈ h. (6)

Namely, if a consumer ever observes a retail price that is not consistent with the recommendedprice, she places all the weight on the price recommendation. Although this restriction onbeliefs is stronger than what is necessary to sustain the cheap talk equilibrium and is cho-sen primarily for simplicity, it may also be supported in several ways. One justification isthe ensuing cheap talk result itself, that the manufacturer’s and consumers’ interests aboutsearch are aligned whereas retailers all prefer less search, and therefore consumers ought totrust the manufacturer more than retailers. Furthermore, the beliefs above would obtainin a slightly richer setting in which retailers are heterogeneous, for example with respect toprivate idiosyncratic costs (e.g. Reinganum (1979)). In such a setting, instead of a singleprice, a set of prices is consistent with a manufacturer’s recommendation. This in turn im-plies that for a given retailer, deviation prices in the neighborhood of his equilibrium priceare on the equilibrium path, along which the belief is restricted by Bayes rule.

The setting with retailer heterogeneity highlights another important aspect of the presentmodel. Currently in equilibrium a single retail price is associated with each manufacturercost, thus in principle a consumer perfectly learns the cost from the retail price alone. How-ever, this is an artifact of the simplifying assumption of homogeneous retailers. If retailerswere heterogeneous, then a given retail price can be consistent with both states and, al-though the consumer could make statistical inference from the retail price, she would not ingeneral be able to learn the state perfectly. Thus, demonstrating that the incentives of the

8In 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 manufac-turer 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 retailprice, and the optimal wholesale price described in Proposition 1 still constitute mutual bestresponses conditional on truthful reporting. It remains to be shown that conditional on thesestrategies, 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’strue cost. The next step is to check whether the manufacturer can credibly provide thisinformation via cheap talk. In such an equilibrium, regardless of the actual cost realizationthe manufacturer is able to induce either search threshold, and it needs to be shown thathe 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 con-sumers, one currently on the margin for purchasing at a retailer and the other on the marginat a competitor. By increasing the search threshold the manufacturer faces a tradeoff: hedecreases the likelihood of sale to the first consumer from one to the continuation probabilityand increases the likelihood of sale to the second consumer from zero to the continuationprobability. When the continuation probability is high, that is when the utility threshold ismore likely to be exceeded by a retailer than by a competitor, more search benefits the man-ufacturer. Conversely, when the continuation probability is low the manufacturer is betteroff with less search. This tradeoff can be formally seen by differentiating the manufacturer’sdemand in (3) with respect to the search threshold u, which simplifies to

∂Q

∂u= Q(w, u)(1−Q(w, u))(εG(u)− εx(w, u)), (7)

where εx(w, u) ≡ |xu(w,u)|x(w,u)

and εG(u) ≡ g(u)1−G(u)

are the elasticities of acceptance probabilitiesat retailers and competitors. Whether the manufacturer’s demand increases or decreases in uthus depends entirely on which of the two elasticities is larger,9 which in turn depends on thewholesale 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 in-crease in search threshold u shifts out the manufacturer’s demand. Meanwhile the opposite

9This is consistent with the interpretation of the manufacturer’s demand as a ratio-form contest – whetherdemand 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

QQ(w, uH)

α+ η − uH

Q(w, uL)

α+ η − uL

w

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 inas u increases.

Figure 2 illustrates the recommendation decision faced by the manufacturer, i.e. the choicebetween demand functions Q(w, uH) and Q(w, uL). For signaling to be credible, the man-ufacturer must prefer to charge a sufficiently low wholesale price when his cost is cL and asufficiently high wholesale price when his cost is cH .

Proposition 3 There exists a cheap talk equilibrium in which the manufacturer truthfullyreveals 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, con-sider 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 man-ufacturer 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 statewhenever w(uH, cL) < w, and by the same logic signaling is credible in the high cost statewhenever w(uL, cH) > w.

Then, to show that w(uH, cL) < w < w(uL, cH) I derive an expression for the manufac-turer’s marginal profit at w and demonstrate that it is negative when cL is sufficiently lowand positive when cH is sufficiently high. Because the manufacturer’s profit is concave inthe 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 andα− cH to be sufficiently small conditional on search cost s to allow truthful reporting. How-ever, 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 statecL and disadvantage in state cH , credible signaling is possible for sufficiently small searchcosts.

Proposition 4 There exists a cheap talk equilibrium in which the manufacturer truthfullyreveals his costs whenever α − cH < ν − η < α − cL and search cost s > 0 is sufficientlysmall.

According to (7), the manufacturer prefers more search if his wholesale price induces a higherelasticity at competitors than at retailers and less search otherwise. As in Proposition 3, thisimplicitly defines a “matching” wholesale price w where the elasticities are equal and the aimis to show that when the search cost is sufficiently small the manufacturer’s marginal profitat 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 defi-nition, also the retailers’ acceptance probability at w. This has two main implications: onethat the manufacturer’s demand at w becomes progressively more elastic, as a small decreasein w corresponds to a larger percentage increase in x, and two that the value of w approachesα + η − ν ∈ (cL, cH) at which retailers are fully squeezed. If the cost is cL then as s shrinksthe manufacturer’s markup at w is bounded from below whereas the demand elasticity ap-proaches 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 cHthen at some s > 0 the markup at w becomes zero whereas the demand elasticity is finite.Thus for search costs below s (and in a neighborhood above s) the manufacturer’s optimalprice 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 suf-ficiently low search costs, they do not not imply that credible signaling is only supportedin the extreme limits of the parameter space or at corner solutions. For example, it can beseen that at the smallest allowable α− cL and at the largest allowable α− cH in Proposition3 and at a small but positive search cost s in Proposition 4, both retailers and competitorsmake strictly positive sales. Furthermore the condition that w(uH, cL) ≤ w ≤ w(uL, cH) issufficient but not necessary for credibility. The proof of Proposition 3 makes it clear thatif w(uH, cL) = w then Π(w(uH, cL), uL, cL) < Π(wL, uL, cL), i.e. if at the deviation whole-sale price the manufacturer is indifferent between the two search thresholds then he strictlyprefers reporting truthfully, and similarly if w(uL, cH) = w. By continuity, this implies cred-ible signaling can be supported even if w(uH, cL) > w or w(uL, cH) < w.

On the other hand, it is also true that if the manufacturer is not sufficiently ahead of orsufficiently behind competitors, he will have incentive to misreport. For example, consider arather extreme case in which the manufacturer is barely ahead in the low cost state and pricedout 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 Ap-pendix A4, in the low cost state misreporting a high cost is profitable because along the wayfrom uL to uH , εx(w(cL, u), u) > εG(u) at almost every u. Put differently, misreporting andinducing less search makes it less costly for the manufacturer to increase his markup, andwhen 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 ispossible, 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 = 1corresponds 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 con-sumers, study the comparative statics, and then establish the credible signaling result. Withconsumers informed of c, a retailer faces demand x(p, u) = α + 1 − u − p and his profitmaximizing price is p(w, u) = 1

2(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)x(w, u)

x(w, u) + 1−G(u)= (w − ci)

α + 1− u− w

α + 3− 3u− w,

with corresponding optimal wholesale price

w(u, ci) =

{

α + 3− 3u−√

2(1− u)(α− ci + 3− 3u) if ci < α + 1− u

ci if ci ≥ α + 1− u. (8)

The profit-maximizing wholesale price solves the first order condition whenever the maximalgains 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 setsa price equal to his cost.10

A consumer faces utility offers uniformly distributed on [0, 1] at competitors and uniformlydistributed on [α−p(w, u), α+1−p(w, u)] at retailers. His best response threshold u satisfies

s =1

2

∫ 1

u(1− ν) dν +

1

2

∫ 1

u+p(w,u)−α(1− η) dη

= I(u ≤ 1) · 14(1− u)2 + I(u ≤ α+ 1− w) · 1

4

(

1

2(α+ 1− u− w)

)2

.

10When ci ≥ α + 1 − u, the manufacturer is priced out and any wholesale price w ≥ ci leads to the sameequilibrium outcomes.

17

Taking into account the indicator functions and solving for u obtains

u(w) =

{

1− 15

(

w − α + 2√

20s− (w − α)2)

if w ≤ α + 2√s

1− 2√s if w > α + 2

√s.

(9)

To interpret the above, recall that if a consumer commits to always rejecting offers from

retailers then his utility threshold uG satisfies s = 12

∫ 1

uG

(1 − ν) dν ⇔ uG = 1 − 2√s.

If the manufacturer’s wholesale price makes it impossible to exceed this threshold, that is ifw > α+1−uG = α+2

√s, then the value to searching comes only from visiting competitors

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 retailprice p∗ = 1

2(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 du∗

ds≤ 0, dw∗

ds≥ 0 unless α > c and s < s2, and

dp∗

ds≥ 0

unless α > c and s < s1.

The intersection of the manufacturer’s best response in (8) and the consumer’s best responsein (9) defines the informed equilibrium, and although there is no closed form solution, theslopes of these functions shed light on the equilibrium’s characteristics. The consumer’sthreshold u(w) is always decreasing because a higher w leads to a lower distribution ofutilities during search, whereas whether the manufacturer’s optimal wholesale price w(u) isincreasing or decreasing depends on parameters. Specifically, as in (7) recall that an increasein u reduces both retailers’ and competitors’ acceptance probabilities. When the manufac-turer’s cost c is low (and thus his equilibrium wholesale price is low), retailers are less affectedthan competitors in percent terms and thus an increase in u shifts out the manufacturer’sdemand and increases the optimal wholesale price. When c is high the reverse is true, ahigher 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 and4 of Appendix A1 this is shown to be the case more generally.

Having established the above, the comparative statics follow directly. First, an increasein production cost c causes an upward shift of w(u), which in either panel of Figure 3 resultsin a lower u∗ and a higher w∗, both of which contribute to a higher retail price p∗. On theother hand, an increase in s causes an inward shift of u(w) and the effects of this depend onwhether 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 exam-ple with α = 1, s = 0.01, and c = 0 (left panel) or c = 1 (right panel). The search thresholdu(w) is always decreasing whereas the wholesale price w(u) is decreasing when c high andincreasing when c is low.

increases, again resulting in a higher retail price p∗. However, if c is low (left panel of Figure3) then both w∗ and u∗ decrease, and the retail price p∗ increases only if u∗ falls faster, i.e.only if ∂w

∂u< 1. The details of the proof can be found in Appendix A3.

With the equilibrium with informed consumers thus described, I now focus on the article’smain result to check when communication is credible. Doing so requires explicitly computingequilibrium 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 theinformed equilibrium on a grid of δ ∈ [0, 1] and s ∈ [0, 1/4], in which δ = 1 is the maximalvalue at which gains from trade are possible in the high cost state and the upper bounds = 1/4 ensures uG ≥ 0. Having values (uL, wL, uH , wH) for each combination of (δ, s), themaximized profit from misreporting in the low cost state Π(w(uH, 1 − δ), uH , 1 − δ) and inthe high cost state Π(w(uL, 1 + δ), uL, 1 + δ) can be computed and compared to the profitfrom truthful reporting, as in Figure 4. In this specification, it is always profitable to reporttruthfully when the cost is cH , whereas it is credible to report truthfully when cost is cL aslong 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 im-portant for two reasons. First, as with any cheap talk environment, there always exists ababbling equilibrium in which consumers ignore manufacturer recommendations and recom-mendations 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 helpsto argue that the informative equilibrium is more profitable for the manufacturer than thebabbling equilibrium. Second, being able to compare the informative equilibrium with anequilibrium without MSRPs is important from a policy perspective in determining whethermanufacturer recommendations should be regulated.

To build some intuition, recall the logic by which the manufacturer’s signal of his cost iscredible. For a consumer A who is currently marginal at a retailer, if the true state is cLthen uncertainty makes her overly pessimistic and deters her from search, thus increasing thelikelihood that she buys from the manufacturer from the continuation probability to one. Onthe other hand, if the true cost is cH uncertainty makes her overly optimistic, causing her tosearch rather than accept and reducing the likelihood she buys from the manufacturer fromone to the continuation probability. The effect is stronger when the cost is cH because thecontinuation probability is lower, and therefore on net the manufacturer is harmed by A’suncertainty. For consumer B currently marginal at a competitor, if the manufacturer’s cost

11An exception is the mandatory Monroney sticker required on all new cars in the US by the AutomobileInformation 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’soffer rather than search and decreasing his likelihood of buying from the manufacturer fromthe continuation probability to zero. If instead the cost is cH the reverse ensues and thelikelihood increases from zero to the continuation probability. Here the effect is strongerwhen the cost is cL because the continuation probability is larger, so again on net the man-ufacturer is harmed by B’s uncertainty.

To operationalize this intuition one must formally describe how the absence of recommen-dations creates uncertainty, and in particular account for consumers’ ability to infer themanufacturer’s cost from the prices of retailers. In the extreme, if retail prices perfectlyreveal the manufacturer’s cost, as can be the case in the current setting with completelyhomogeneous retailers, then having seen a retailer’s price consumer A behaves the same aswhen informed by a recommendation. On the other hand, if consumer B has not yet visitedany retailers then she remains uncertain about the state. Then by the previous argumentthe manufacturer benefits from B’s uncertainty when the cost is cH , is harmed when the costis 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 equilibriumwithout recommendations in which wholesale and retail prices in both states are the same asin the informed equilibrium. Furthermore, the payoffs of the manufacturer and consumersare lower in this equilibrium than in the equilibrium with recommendations.

The key to demonstrating both that there exists an equilibrium with the same prices andthat the manufacturer and consumers are worse off in this equilibrium is understanding theconsumers’ search strategy is this new environment. A consumer that has visited a retailer atany point is fully informed and uses the same threshold as in (1), whereas a consumer whosehistory consists only of visits to competitors holds the prior of µ = 1/2 and uses thresholdu(1/2) ∈ (uH , uL). When this consumer eventually visits a retailer, she learns the true stateand either becomes more selective with threshold uL or less selective with threshold uH , inthe 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 expressedas simply as in (3). However using a recursive technique I show that the manufacturer’sdemand 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 staterelative to when consumers are informed, and by the logic in the discussion preceding theproposition, 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 cHis 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 thesame prices but make less informed decisions they too are worse off in this setting. For thedetailed proof see Appendix A6.

21

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

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

5 Discussion

A natural alternative specification of the main model is one in which consumers are uncertainabout product quality rather than the manufacturer’s cost, and here I briefly explore thissetting. Despite the result of Lemma 9 (in the Appendix) that in the informed setting thereis no meaningful distinction between quality and cost, it is not immediate that crediblesignaling 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 overretail 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 productquality α has either a high realization αH or a low realization αL, both equally likely. Themanufacturer 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 capturethe idea that quality may be inferred but not directly observed, let zj ≡ α + ηj be the fullconsumption utility of the product at retailer j and suppose that the consumer observes zjbut not its individual components.12

Although Lemma 9 demonstrates that the full information game is equivalent to when costsare uncertain, the beliefs in the Bayesian game must be revisited. In the specification withuncertain costs, it was assumed that when the price recommendation and retail price areinconsistent with the equilibrium, beliefs place full weight on the recommendation. Now theconsumer observes three objects: the recommendation, the retail price, and his consump-tion utility zj . If the manufacturer deviates then σ is sometimes inconsistent with zj . Forinstance, suppose the quality is αL but the manufacturer misreports σ(αH). If the realizedprivate shock is η < η+ (αH −αL) then the consumer must infer that the quality is αL withcertainty. The payoff to the manufacturer from misreporting is now difficult to evaluate. Notonly would consumers become aware of the true state with some probability and therebychange their search strategy, but also retailers having observed the manufacturer’s deviationwould 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 themanufacturer’s incentives for search can be aligned with the consumers’ – he wants moresearch when he offers a higher range of utilities than do competitors and less search other-wise. The issue is then not in the manufacturer’s incentives to increase or decrease searchbut the particular mechanism by which he can achieve this. In addition, this concern can beresolved on a technical level by allowing the distribution of private shocks F (η) to have fullsupport on [−∞,∞], in which case all observed realizations of zj are consistent with bothof the manufacturer’s recommendations. Thus, given the underlying incentive structure, itis likely that the manufacturer’s recommendations can also be used to convey informationabout product quality.

Comparison to Resale Price Maintenance

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

12In contrast to standard quality signaling models, here the consumer knows his utility for a product priorto deciding about whether to purchase it, and his interest in quality is only in forming expectations aboutthe returns to searching.

23

contrast, the present theory of MSRPs as information for consumers predicts that demandis potentially affected at all prices, including those below MSRP. Specifically, in the modelthe per-person demand is x(p, u) = 1− F (p+ u−α), thus an increase in search threshold ucorresponds 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 ofprice recommendations on packaged foods in South Korea, suggests that although MSRPslower prices they do not act as price ceilings. In particular, the authors find that morethan 96% of prices charged are at least 10% below the recommended price, that there isno bunching around the recommended price when it is present, and that when an MSRP isremoved there is no visible increase in the number of prices charged above it. One may alsoconsider the US auto market as an anecdotal example. The vast majority of cars sell notjust 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 theMSRP itself as a reference price, so for instance the true price ceiling is at 90% of the recom-mended price. However, in the auto market it is commonly known that more popular modelssell closer to MSRP, and very popular models may even sell above MSRP. For the theoryto apply, it must be that a consumer’s reference price is informed by a product’s popularityor 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 optimalunder current market conditions. Put differently, the present search model can be used torationalize the existing reference point theories of price recommendations.

As a conduit of information, price recommendations provide the manufacturer with an in-strument 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 agiven utility shock, and thus making a price recommendation is analogous to choosing aprice ceiling. However, providing consumers with information accomplishes more than indi-rectly restricting retail prices. By affecting search, the manufacturer affects the elasticity ofdemand, 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 rec-ommendations and what effect they have on market outcomes such as prices and sales is stillnot well understood. I present a model in which a manufacturer uses recommendations toaffect consumers’ search decisions. Consumers do not know the distribution of retail pricesdue to uncertain manufacturer costs and rely on the recommendation to determine whetherto 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 alsoloses consumers that search and purchase from a competitor. The manufacturer profits frommore search when he offers consumers more value than competitors; otherwise he prefers lesssearch. Consumers benefit from more search when higher utilities are available, thus theirinterests are aligned with the manufacturer’s. I demonstrate that manufacturer recommen-dations can be used as cheap-talk signals to credibly inform consumers of the manufacturer’scosts. Recommended prices help manufacturers and consumers coordinate and thereby canimprove the welfare of both.

It is likely that the manufacturer’s incentive for truthful reporting applies beyond the currentsetting 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 usingcheap talk. The intuition is simple – if a consumer’s willingness to pay for a product canbe 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 nolonger holds; instead the manufacturer can affect consumers’ willingness to pay by informingthem of the set of other options available through search. In particular, the manufacturercan sometimes benefit by informing consumers that the product he supplies retailers is notof high quality on average, thereby reducing the perceived return to searching and makingthem more willing to accept the current offer. Because the manufacturer does not alwaysbenefit 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 un-certainty about the distribution of prices. This is consistent with the fact that MSRPs aretypically found on products that are purchased infrequently, such as cars, electronics or aparticular book, and not found on products that are purchased frequently, such as milk orlaundry detergent. By contrast, models of MSRPs as explicit resale price maintenance makeno such distinction.

In addition, in practice there is variation in the way actual prices relate to price recom-mendations. For instance, books often sell for exactly their jacket price whereas cars tend tosell for strictly less than MSRP. Even within the car market, how far below MSRP a car sellsvaries by the popularity of that vehicle, and in fact some cars sell above MSRP. Imposingthat recommendations are price ceilings or some other exogenously determined restraintsprecludes the analysis from explaining such variation. Because in my model a recommen-dation is just a cheap message, I allow for the consumers’ interpretation of this message tovary with market conditions and thus can accommodate the varying relationship betweenprices and recommendations.

The goal of this article is to highlight a mechanism by which price recommendations serve apurely 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. Nonethe-less, 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 antitrustpolicy discussion of the merits of price recommendations should keep these informationalbenefits 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 pricew there is a unique optimal price p(α − u + w) and corresponding acceptance probabilityx(u + w − α). Plugging this into the manufacturer’s and consumers’ decisions reduces thecharacterization into a system of two equations with w and u as the unknowns. I then usea 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 ordercondition dπ

dp= 0 = 1− F (u+ p− α)− (p− w)f(u+ p− α) and totally differentiating with

respect to p and w yields

pw =f

2f + (p− w)f ′,

and thus because f ′ ≥ 0 it follows that pw ∈[

0, 12

]

. Differentiating with respect to w once

more and performing several steps of algebra yields

pww = (pw)2

(

f ′

f

)(

pw − 1 + (p− w)

(

f ′

f− f ′′

f ′

))

.

Turning now to the acceptance probability x(·), I first show it can be expressed as a functionof 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 bere-written as π(x) = x(A(x)− w) with corresponding first order condition

π′(x) = 0 = A+ xAx − w = F−1(1− x) + xd

dx

(

F−1(1− x))

− (u+ w − α),

26

the solution to which can be expressed as x(u+w−α) = 1−F (u+p(α−u+w)−α). Then,

x′ = xw = −fpw < 0

and plugging in pww from above yields

x′′ = xww = −f ′(pw)2 − fpww = −f ′(pw)

2

(

pw + (p− w)

(

f ′

f− f ′′

f ′

))

< 0,

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 searchthreshold 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 aunique continuous profit maximizing w(u) which is the solution to the first order conditionfrom (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, andthus any wholesale price at which he makes no sales is optimal. If u ≤ α+ η − c then thereis a region of wholesale prices above his cost at which the manufacturer can make positivesales, and the aim is to show his profit is single-peaked in this region. First, to establish thatthe manufacturer’s demand Q = x(u+w−α)

x(u+w−α) + (1−G(u))is concave in w, taking the derivative

twice and simplifying yields

Qw = x′ 1−G(u)

(x + (1−G(u)))2

Qww = Qw

( −2x′

x + (1−G(u))+

x′′

x′

)

.

Because x′ < 0 (and thus Qw < 0) and x′′ < 0, it follows that Qww < 0. Finally, the profitfunction (w − c)Q(w) is concave whenever Q is concave, and thus is single peaked with aunique continuous maximizer.

Lemma 3 If u ≤ α+ η − c (i.e. sales are feasible) then dwdu

> −1.

Proof of Lemma: Differentiating in (4) obtains the manufacturer’s first order condition

Πw = 0 = Q(w, u)

(

1− (w − c)

(

1−G(u)

x(u+ w − α) + 1−G(u)

)

εx(u+ w − α)

)

, (10)

27

where εx(u + w − α) ≡ |x′(u+w−α)|x(u+w−α)

. Now suppose that dwdu

= −1, so that when u increases

the sum u + w remains fixed. Then in (10) it follows that as u increases (w − c) decreases,(

1−G(u)x(u+w−α) + 1−G(u)

)

decreases because x remains unchanged and 1−G falls, and εx remains

unchanged. But then Πw > 0, which by the concavity of Π(w) implies the equilibrium w islarger and thus that dw

du> −1.

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 in-verse function is not well-defined for u < uG because no wholesale price can justify a searchthreshold below uG, and that ϕ(uG) is multi-valued because any ϕ ≥ α + η − uG results inthe consumer expecting never to buy from the manufacturer.

Lemma 4 The consumer’s inverse best response ϕ(u) satisfies dϕdu

≤ −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

dϕ

du= − 1−G(u)

p′(1− F (u+ p− α))− 1− p′

p′

Because we previously established in Lemma 1 that p′ ∈[

0, 12

]

, it follows that dϕdu

< −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 thereis a unique equilibrium (u, w) ∈ (uG, ν) × (c, α + η − ν). Else if c ≥ α + η − uG then themanufacturer is priced out, u = uG and w ≥ α+ η − ν.

Proof of Lemma: Recall that uG is the consumers’ search threshold when never buy-ing form the manufacturer, and suppose first that 0 ≤ c < α + η − uG. Observe thatϕ(uG) ≥ α+ η−uG ≥ w(uG). The first inequality follows because it is only rational to neverbuy from retailers if the wholesale price ϕ ≥ α + η − uG, so that retailers can never makesales. The second inequality follows from the fact that it is not rational for the manufacturerto fully squeeze retailers and make zero sales because by assumption a positive markup isavailable.

Also, note that ϕ(ν) < α + η − ν < w(ν). For the first inequality, a consumer that usesthreshold ν never buys from competitors and thus must expect that when visiting retailersher utility strictly exceeds ν with a strictly positive probability, i.e. ϕ < α + η − ν. For thesecond 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 continuousand by Lemmas 3 and 4 monotonically increasing, there exists a unique intersection at someu∗ ∈ [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 Lemma6 demonstrates that u ≥ uG, it follows that w ≥ c ≥ α+ η − uG ≥ α+ η − u, which in turnimplies that the manufacturer fully squeezes retailers. Therefore in equilibrium u = uG andw ≥ α + η − uG, with the exact value of w irrelevant for payoffs. This concludes the proofof the Proposition.

A2 Proof of Proposition 2: Comparative statics of informed equi-

librium

The proposition is proven in four separate lemmas, establishing the bounds of the searchthreshold u in Lemma 6, the comparative statics of u with respect to production cost c inLemma 7 and with respect to search cost s in Lemma 8, and then establishing in Lemma9 that when an increase in c is combined with a corresponding increase in quality α theequilibrium 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 setof utilities available at retailers, the return to search when acting optimally is at least ashigh as the return to search when committing to rejecting offers from all retailers. That thesearch threshold never exceeds ν follows from a hold-up argument, similar to that in JanssenShelegia (2015) and Diamond (1971), in that a manufacturer cannot commit to provideconsumers with sufficient return to searching. To see this, conjecture that the consumeruses a threshold u > ν. In such a situation, searching consumers accept only at retailers andnever at competing sellers, which allows the manufacturer to raise the wholesale price withoutlosing sales. The manufacturer’s best response is a wholesale price at which the acceptanceprobability at retailers is arbitrarily close to zero, which in turn makes the expected returnto 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 thenu(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

0 =d

dc

(

w(u(c), c)− ϕ(u(c)))

=∂w

du

du

dc+

∂w

∂c− ∂ϕ

∂u

du

dcdu

dc= −

∂w∂c

∂w∂u

− ∂ϕ∂u

.

First note that the numerator is positive because the manufacturer’s payoff has increasing

differences in w and c. That is, d2

dw dc

(

(w− c)Q(w, u))

= −dQdw

= −∂Q∂xx′ ≥ 0 because ∂Q

∂x≥ 0

and x′ ≤ 0. Next as established in the proof of Proposition 1 the denominator is positive aswell, 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’ssearch decision from (1) obtains ∂ϕ

∂s= − 2

p′F (u+p−α)≤ 0 because p′ ≥ 0 by Lemma 1. Then,

because the denominator was shown in the proof of Proposition 1 to be positive, it followsthat du

ds≤ 0.

Finally, to demonstrate that lims→0 u(s) = ν, note that u(s) ∈ [uG(s), ν] (by Lemma 6)and that uG(s), which solves s = 1

2

∫ ν

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 =α+ δ and c = c+ δ, (w+ δ, p+ δ, u) is an informed equilibrium and all participants’ payoffsare unchanged.

Proof of Lemma: The idea is that if consumers’ willingness to pay α and productioncost c increase by the same amount then there is no real change to the market except for arelabeling. To this end, I suppose that (w, p, u) describes the equilibrium for parameters αand c, conjecture that (w, p, u) = (w + δ, p + δ, u) describes an equilibrium with parametersα = α + δ and c = c+ δ, and then verify that indeed every participant best responds.

30

Beginning with consumers, plugging in the new values α = α + δ and p = p + δ into(1) keeps the right hand side unchanged, implying the same search threshold u remains op-timal.

For retailers, A1 establishes that the optimally chosen retail price induces an acceptanceprobability x(u+w− α) which is expressed in terms of only a single argument. At the newvalues the optimal acceptance probability x(u+(w+ δ)− (α+ δ)) remains unchanged, which

in turn implies that for p to be optimal it must satisfy 1−F(

u+p−(α+δ))

= 1−F (u+p−α),

implying p = p+ δ.

Finally recall that the manufacturer chooses w to solve

w = argmaxw

(w − c)x(u+ w − α)

x(u+ w − α) + (1−G(u))= argmax

w(w − c)Q(u+ w − α, 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 remainsunchanged, w = w+δ, c = c+δ, and α = 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 theequilibrium given by (8) and (9). Focusing on interior solutions (i.e. when the manufacturermakes positive sales) and differentiating the top line of (9) obtains

∂u

∂w=

1

5

(

2(w − α)√

20s− (w − α)2− 1

)

≤ 0, (11)

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 exercisefor the manufacturer’s best response in the top line of (8) obtains

∂w

∂u= −3 +

α− c+ 6− 6u√

2(1− u)(α− c+ 3− 3u), (12)

31

the sign of which is ambiguous, by a logic similar to that of Figure 2, and as depicted in thetwo panels of Figure 3.

Turning first to the manufacturer’s production cost, that

du∗

dc= −

∂w∂c

∂w∂u

− ∂ϕ∂u

≤ 0

was already established by Lemma 7 more generally. Then for the wholesale price,

dw∗

dc=

∂w

∂c+

∂w

∂u

du∗

dc= −

∂w∂c

∂ϕ∂u

∂w∂u

− ∂ϕ∂u

= −∂ϕ

∂u

du∗

dc≥ 0,

with the inequality now following from ∂ϕ∂u

≥ 0 from (11). Lastly for the retail price p∗ =p(α− u∗ + w∗) and

dp∗

dc= p′

(

dw∗

dc− du∗

dc

)

≥ 0,

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∗

ds=

du∗

ds

∂w

∂udp∗

ds= p′

du∗

ds

(

∂w

∂u− 1

)

with the signs of each depending on the slope of the manufacturer’s best response function∂w∂u. Recalling that in any equilibrium u ∈ [uG, ν) = [1−2

√s, 1), observe in (12) that if α < c

then ∂w∂u

> 0 for any feasible u. However, if α > c then by inspection of (12) the second termincreases in u and approaches infinity as u approaches 1. Thus there exist u2 < u1 < 1 sothat ∂w

∂u≥ 0 for all u ≥ u2 and ∂w

∂u≥ 1 for all u ≥ u1. Because we have already established

that u∗ decreases in s and approaches 1 as s approaches zero, there exist 0 < s1 < s2 thatcorrespond to u1 and u2. Thus, dw∗

ds≥ 0 unless α > c and s ≤ s2 and dp∗

ds≥ 0 unless α > c

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 in-formed equilibrium payoff exceeds his maximized payoff from misreporting in both thelow 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 simplifiedexpression for these profit differences (Lemma 10), demonstrating that whether the man-ufacturer prefers more or less search depends on search elasticities εx(w, u) and εG(u) atretailers and competitors. Then I establish that the search elasticity at retailers increasesmonotonically in the wholesale price (Lemma 11), which implies for each search threshold uthe existence of a wholesale price w(u) below which the manufacturer prefers less search andabove which he prefers more search. I then show that for all u ∈ [uH , uL], when the man-ufacturer’s cost cH is sufficiently high his marginal profit at w(u) is positive, which impliesthat at his optimal wholesale price he induces a high retailer search elasticity and prefers lesssearch. Because the manufacturer would set a high wholesale price for every u ∈ [uH , uL], heprefers threshold uH and thus reports truthfully. Then the same argument is made when cLis 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 anda preference for more search.

To begin, I establish the expression used to evaluate the difference in the manufacturer’smaximized payoff from inducing either threshold uH or uL. Using the fact from Lemma 1that the acceptance probability can be expressed as x(u + w − α), define the elasticities of

acceptance probabilities as εx(u+ w − α) ≡ −x′(u+w−α)x(u+w−α)

and εG(u) ≡ g(u)1−G(u)

at retailers andcompetitors.

Lemma 10 Let w(u, c) denote the profit-maximizing wholesale price. Then

Π(w(uL, c), uL, c)−Π(w(uH, c), uH , c) =

∫ uL

uH

Q(w(u, c), u)

(

εG(u)

εx(u+ w(u, c)− α)− 1

)

du.

Proof of Lemma: The manufacturer’s maximized profit changes with respect to u at arate

dΠ(w(u, c), u, c)

du=

∂Π(w(u, c), u, c)

∂w

∂w(u, c)

∂u+

∂Π(w(u, c), u, c)

∂u

= 0 + (w(u, c)− c)∂Q

∂u=(

w(u, c)− c)

Q(1−Q)(

εG(u)− εx(u+ w(u, c)− α))

, (13)

33

where the first term on the right hand side of the first line equals zero by the envelopetheorem and the expression for ∂Q

∂ucomes from (7). Also, because w(u, c) is optimal for the

manufacturer it satisfies the first order condition

0 = Q+ (w(u, c)− c)∂Q

∂w= 1− (w(u, c)− c)(1−Q)εx(u+ w(u, c)− α)

⇔

(w(u, c)− c)(1−Q) =1

εx(u+ w(u, c)− α),

where the second line comes from solving for ∂Q∂w

from (3) and simplifying. Then, pluggingthe right hand side back into (13) obtains

dΠ(w(u, c), u, c)

du= Q(w(u, c), c)

(

εG(u)

εx(u+ w(u, c)− α)− 1

)

.

The expression in the statement of the lemma then immediately follows by the fundamentaltheorem of calculus.

Lemma 10 yields an easily stated condition for the manufacturer’s preference for searchbased solely on search elasticities. It follows that if for every u ∈ [uH , uL] the manufacturer’soptimal wholesale price satisfies εx(u+ w(u, cH)− α) ≤ εG(u) ≤ εx(u+ w(u, cL)− α), thensignaling is credible. Doing this by explicitly characterizing w(u, c) and directly evaluatingεx(u + w(u, c) − α) is not feasible without further functional form assumptions. Instead, Idemonstrate that w(u, c) falls in a range of wholesale prices in which more search is pre-ferred when c = cL and less search is preferred when c = cH . Toward this end, the followinglemma defines these ranges by identifying a threshold price w(u) below which more searchis preferred and above which less search is preferred. Then, I show that the manufacturer’smarginal profit evaluated at w(u) is negative in the low cost state and positive in the highcost state, which by the fact that the manufacturer’s profit is single peaked shows the mainresult.

Lemma 11 For each u ∈ [uG, ν) there exists a unique w(u) such that

εx(u+ w(u)− α) > (<) εG(u) if and only if w > (<) w(u).

34

Proof of Lemma: First I show that εx increases monotonically in w for w ≤ α + η − u.

dεxdw

= −x′′x− (x′)2

x2

= −x′

x

(

x′′

x′− x′

x

)

.

Because x > 0, x′ < 0, and as was proven in Appendix A1 x′′ < 0, it follows that dεxdw

> 0.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 marginalprofit

πp = 1− F (u+ p− α) + (p− w)f(u+ p− α)

= f(u+ p− α) ·(

1− F (u+ p− α)

f(u+ p− α)− (p− w)

)

.

Even at p = α+ η − u, at which the acceptance probability is one, the marginal profit with

respect to price is 1f(η)

− (α + η − u − w) and is negative when w is sufficiently small (and

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) isincreasing 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) > (<) εG(u) ifand only if w > (<) w(u).

Thus for signaling to be credible it needs to be established that w(u, cL) ≤ w(u) ≤ w(u, cH)for all u ∈ [uH , uL]. Given that the manufacturer’s profit is single-peaked (Lemma 2), itis sufficient to check the sign of the manufacturer’s marginal profit at w(u), which can besimplified as follows:

∂Π(w(u), u, c)

∂w= Q(w(u), u) + (w(u)− c) ·Qw(w(u), u)

= Q(w(u), u)

(

1− (w(u)− c)(1−Q(w(u), u))εx(u+ w(u)− α)

)

= Q(w(u), u)

(

1− (w(u)− c)

(

g(u)

g(u) + |x′(u+ w(u)− α)|

)

εG(u)

)

. (14)

The second line follows from the definition of Q(w, u) = x(u+w−α)x(u+w−α) + 1−G(u)

and the third

line uses the fact that εx(u + w(u) − α) = εG(u) to substitute for 1 − Q(w(u), u) and for

35

εx(u+ w(u)− α).

Expression (14) is the key for demonstrating the existence of an informative cheap talkequilibrium. The manufacturer can credibly inform consumers if for all u ∈ [uH , uL] the ex-pression is negative when c = cL and positive when c = cH . Taking as given the distributionG of outside options and the quality α > 0, I demonstrate here that for a fixed search costs signaling is credible if cL is sufficiently small and cH is sufficiently large. In the followingproposition I then fix cL < α + η − ν < cH and show signaling is credible when search costs 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 intoaccount all u ∈ [uH , uL]. For example, when search cost s decreases then the marginal profitin (14) remains unchanged at any particular u. What changes, however, are the full infor-mation thresholds uL and uH , and thus the set of search thresholds over which the sign ofthe marginal profit must be evaluated. Explicitly keeping track of how uH and uL changewith 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 issufficient to show that the marginal profit has the correct sign for all u ∈ [uG, ν).

Lemma 12 For a given cL, there exists cH(cL) < α+ η− uG so that whenever cH > cH(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

εG(u) is increasing and uL < ν. Next, g(u)g(u)+|x′(u+w(u)−α)|

is bounded above by some γ, which is

guaranteed to be finite because f(η) and g(ν) are bounded. Finally, w(u) is bounded aboveby α+ η−uG, because for any higher w retailers do not make sales, which corresponds to anelasticity of εx = ∞, which is a contradiction of the definition of w. Putting this together,

Sign

(

∂Π(w(u), u, cH)

∂w

)

= Sign

(

1− (w(u)− cH)

(

g(u)

g(u) + |xu(u+ w(u)− α)|

)

εG(u)

)

≥ Sign(

1− (α + η − uG − cH)γεG(uL))

.

Thus ∂Π(w(u),u,cH)∂w

> 0 for all u ∈ [uH , uL] if cH > cH(cL) ≡ α + η − uG − 1γεG(uL)

, where cLenters implicitly in threshold uL.

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 topay 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 higherwholesale 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

u ≥ uG. Also, g(u)g(u)+|x′(u+w(u)−α)|

is bounded from below by γ > 0, which again follows from

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 isw(εx, u), the wholesale price which induces an elasticity εx at retailers when the searchthreshold is u. The proof of Lemma 11 implies that εx(u + w − α) is monotonically in-creasing in its argument and has a range [0,∞) for u ∈ [uG, ν) and w ∈ R. Thereforew(εx, u) is well-defined on [0,∞)× [uG, ν), increasing in εx, and decreasing in u. Then, letw ≡ w(εG(uG), ν) and observe that w(u) ≥ w for all u ∈ [uH, uL]. That is, the wholesaleprice that equates retailers’ and competitors’ elasticities for threshold u is higher than thewholesale price that induces retailers that face consumers with the highest possible utilitythreshold ν to have the lowest feasible competitor elasticity εG(uG).

Then, returning to the expression for the manufacturer’s marginal profit,

Sign

(

∂Π(w(u), u, cL)

∂w

)

= Sign

(

1− (w(u)− cL)

(

g(u)

g(u) + |x′(u+ w(u)− α)|

)

εG(u)

)

≤ Sign(

1− (w − cL)γεG(uG))

.

Thus, if cL ≤ cL ≡ w − 1γεG(uG)

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

The quantities α − cL in Lemma 13 and α − cH(cL) in Lemma 12 define the minimal andmaximal gains from trade in the low and high states required for credible signaling, as perthe statement of Proposition 3. Observe that by Lemma 9, if signaling is credible withparameters (α, cL, cH) then it also credible with parameters (α + δ, cL + δ, cH + δ), becausethe latter is equivalent up to nominal price changes. Therefore the minimal and maximalgains from trade required for credible signaling are independent of α. In turn, if costs arerequired to be non-negative then there is a minimal α at which the required minimal gainsfrom 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, thestrategy 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 andthus all u ∈ [uG, ν) also approach ν, with εG(u) aproaching infinity. By definition, at thewholesale price w(u) the search elasticity for retailers also approaches infinity, and thus theretailers’ acceptance probability must approach zero, which has two important implications.First, as acceptance probabilities are both shrinking toward zero the manufacturer’s demandat w becomes highly elastic, because a small change in w results in a dramatic percent changein x and thus Q. Second, because the retailers’ acceptance elasticity approaches zero andthus 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 strictlynegative by assumption. There is then some s so that if s ≤ s then w(uG)− cH ≤ 0, which

implies w(u)− cH ≤ 0 for all u ∈ [u(σH), uL]. Thus the marginal profit dΠ(w(u),u))dw

> 0 for allu ∈ [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 assump-

tion. The term g(u)g(u)+|xu(u+w(u)−α)|

is bounded above zero and εG(u) monotonically approaches

infinity because u approaches ν. Thus the expression (w(u)−c)(

g(u)g(u)+x′(u+w(u)−α)

)

εG(u) ap-

proaches infinity as s approaches zero, which implies that for small s the marginal profitdΠ(w(u),u))

dw< 0 for all u ∈ [uG, ν), and therefore reporting cL truthfully is a best response.

A6 Proof of Proposition 6: Manufacturer prefers recommenda-

tions to no recommendations

I characterize the consumer’s optimal search behavior conditional on the manufacturer andretailers setting informed equilibrium prices wL and pL in the low cost state and wH andpH in the high cost state, and then show that conditional on the consumers’ behavior theseprices are in fact best responses. Then I show that the manufacturer and consumers areboth better off in an equilibrium with informative recommendations than in an equilibriumwith 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 deter-mine deviation profits for retailers and by extension for the manufacturer. The main modelfocuses on an equilibrium in which off the path beliefs depend only on the manufacturer’srecommendation and a retailer cannot affect consumers’ beliefs with his price. To make acomparison 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 retailprices pL and pH , for any history h = {{(p1, η1), ..., (pk, ηk)} ∪ {ν1, ..., νl}} let

µ(h) =

12

if ∄ p1 (i.e. no retailers visited)

0 if p1 < pH

1 if p1 ≥ pH

. (15)

For histories on the equilibrium path, it is easily verified that µ(h) is simply computed usingBayes rule: beliefs start with the prior µ = 1

2and 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 ispL. For histories off the equilibrium path, beliefs are determined entirely by the first observedretail price and follow a step function. Because a retailer benefits from reducing the utilitythreshold, making the step in beliefs at as high a price as possible minimizes the retailer’sincentive to manipulate beliefs. With this structure, a retailer can alter a consumer’s beliefonly 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 con-sumer starts search with belief µ = 1

2and 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 consumerduring 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 high-est utility offer u. Upon visiting a retailer for the first time and observing price p(cH), theconsumer reduces his threshold from u(1/2) to u(1) and may return if u(1) ≤ u < u(1/2).The manufacturer’s demand must reflect this possibility, and thus requires keeping track ofsearch histories. Let Q(µ, c, u) denote the continuation probability that a consumer withbelief µ and current best available option u eventually buys from the manufacturer when thetrue cost is c. Then,

Q (1/2, c, 0) =1

2x(c) +

1

2(1− x(c))Q

(

µ(c), c, u < u(µ(c)))

+1

2G (u (1/2))E [Q(1/2, c, ν) | ν < u(1/2)] .

13Replicating the incentives perfectly is not possible without recommendations. For example, in anyseparating equilibrium it must be that µ(pL) = 0 and µ(pH) = 1, thus in the low cost state a retailer thatdeviates 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 theprior µ = 1

2and 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 andforms correct belief µ(c) with corresponding search threshold u(µ(c)), accepts with proba-bility x(c) ≡ 1 − F (u((µ(c)) + p(c)), else rejects and continues to search with correct beliefµ(c) thereafter. If the first visit is to a competitor, the consumer rejects with probabilityG(u(1/2)) and continues search with the same belief µ = 1

2and an option to return and

accept his current offer ν. Observe that when c = cH the consumer may exercise the optionto return to the competitor because upon learning the true state his threshold falls, whereaswhen c = cL the consumer never returns because upon learning the true state his thresholdincreases.

To simplify the above expression, note thatQ(c) ≡ Q(

µ(c), c, u ≤ u(µ(c)))

= x(c)x(c)+1−G(u(µ(c)))

,

because a consumer that has learned the true state at the first retailer and rejected acts ex-actly like an informed consumer in the model with recommendations. In addition, lettingγ ≡ E[Q(1/2,c,ν) | ν≤u(1/2)]

Q(1/2,c,0), the manufacturer’s demand simplifies to

Q(1/2, c, 0) =2−G(u(µ(c)))

2− γG(u(1/2))Q(c). (16)

If the manufacturer’s cost is cL, the fact that the consumer never returns to accept previousoffers implies that γ = 1 and consequently

Q(1/2, cL, 0) =2−G(u(0))

2−G(u(1/2))Q(cL).

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 acceptprevious offers implies γ ≤ 1, which yields the following upper bound for sales:

Q(1/2, cH, 0) ≤2−G(u(1))

2−G(u(1/2))Q(cH).

Because u(1) < u(1/2), this upper bound is higher than the quantity sold in the high coststate in the informed equilibrium.

Retailer and Manufacturer strategies

Having established the consumers’ search strategy and the ensuing demand function facedby the manufacturer, I now show that it is a best response for the manufacturer and retailersto 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 lowcost state a retailer faces two groups of consumers. The first group is those that have visitedat least one retailer, learned the state from price pL, and formed threshold u(0). This groupis homogeneous and acts exactly like the consumers informed by recommendations, buyingfrom the retailer if and only if η ≥ u(0)−p. The second group consists of consumers that havevisited no retailers but possibly some competitors, and have received a best offer u up to thispoint, with u < 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(

u, 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 pricewL the optimal price in this range is pL. If the retailer charges a price p′ ≥ pH , he still sellsto the first group if and only if η ≥ u(0)+ p′ but now sells to the second group if and only ifη ≥ u(1) + p′. That is, by charging p′ the retailer makes the second group more pessimisticthereby 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 thresholdu(1) rather than just the second group. The second inequality follows from the fact thatp′ ≥ pH ≥ cH and that u(1) ≥ uG, recalling that uG was earlier defined as the thresholdwhen a consumer commits to never accepting retailers’ offers. As cH is increased the righthand side of the second inequality approaches zero whereas the payoff to charging pL remainsfixed at some strictly positive amount. Thus, for a high enough cH , it is a best response forthe 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 wholesaleprices can be categorized by the retail prices p(w) that they induce. Namely, there existssome w so that p(w) < pH when w < w and p(w) ≥ pH when w ≥ w. In the first group oflower 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)

= (w − cL)

(

2−G(u(0))

2−G(u(1/2))

)(

1− F (p(w) + u(0))

1− F (p(w) + u(0)) + 1−G(u(0))

)

=2−G(u(0))

2−G(u(1/2))ΠMSRP (w, cL),

41

with demand Q(1/2, 0, cL) given in (16). Because the profit is simply a scalar multiple ofthe full information profit it has the same maximizer. Therefore, wL is the best option ofall wholesale prices below w. To show that prices above w make the manufacturer worse off,observe from (16) that

Π(w ≥ w) = (w − cL)

(

2−G(u(1))

2− γG(u(1/2))

)(

1− F (p(w) + u(1))

1− F (p(w) + u(1)) + 1−G(u(1))

)

≤ (w − cL)

(

2−G(u(1))

2− γG(u(1/2))

)(

1− F (cH + uG)

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. AscH increases the term in the third parentheses of the last line approaches zero whereas theterm in the first parentheses remains unchanged and the term in the second parenthesesfalls because u(1/2) falls. Thus, as cH is increased Π(w ≥ w) approaches zero. Meanwhile,

Π(wL) = 2−G(u(0))2−G(u(1/2))

ΠMSRP (wL, cL) also falls with cH as u(1/2) falls, however is bounded

above by 2−G(u(0))2

ΠMSRP (wL) > 0. Thus for large enough cH the wholesale price wL isoptimal.

Lemma 15 In the high cost state with no recommendations, the full information wholesaleprice 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 visitedother retailers and hold a threshold u(1) and those who have not visited any retailers butpossibly some competitors and hold a threshold u(1/2) with best previous offer u < 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(

u, 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 acceptif and only if η ≥ u(1)+p whereas the second group forms the optimistic belief µ(p) = 0 andadopts a higher utility threshold, accepting if only if η ≥ u(0)− p. Demand is thus lower atall prices p < pH than under full information, and because this deviation was not profitableunder full information, it continues not to be so.

To demonstrate that wH is a best response for the manufacturer, observe again that thereexists some w so that p(w) < pH if w < w and p(w) ≥ pH if w ≥ w. The latter case isanalogous to truthful reporting in that consumers form the correct belief µ(p(w)) = 1 aftervisiting 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)

= (w − cH)

(

2−G(u(1))

2− γG(u(1/2))

)(

1− F (p(w) + u(1))

1− F (p(w) + u(1)) + 1−G(u(1))

)

=2−G(u(1))

2− γG(u(1/2))ΠMSRP (w, cH).

The current profit function simply a scaling of the full information profit ΠMSRP (w, cH), thusthe full information maximizer wH is also optimal here. Therefore, wH is the most profitableof the wholesale price above w.

Charging a wholesale price w < w is analogous to misreporting as it induces the incor-rect belief µ(p(w)) = 0 when consumers visit a retailer. To demonstrate that wH yields ahigher profit than wholesale prices in this low range, note that

Π(w < w) = (w − cH)

(

2−G(u(1))

2− γG(u(1/2))

)(

1− F (p(w) + u(0))

1− F (p(w) + u(0)) + 1−G(u(0))

)

≤ (η − u(0)− cH)

(

2−G(u(1))

2− γG(u(1/2))

)(

1− F (p(w) + u(0))

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 eventuallybecomes 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 consis-tent in an equilibrium with no recommendations, I now demonstrate the manufacturer andconsumers are worse off. The fact that in both states demand is proportional to the in-formed equilibrium level facilitates the comparison of the manufacturer’s profit across thetwo settings as follows:

ΠNo MSRP −ΠMSRP ≤ 1

2

(

(wH − cH)(Q(1/2, cH , 0)−Q(cH)) + (wL − cL)(Q(1/2, cL, 0) −Q(cL)))

=1

4− 2G(u(1/2))

(

ΠMSRP (cH)(G(u(1/2)) −G(u(1))) − ΠMSRP (cL)(G(u(0)) −G(u(1/2))))

.

(17)

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

43

s < 12E[ν] (as assumed in the main model), as cH is increased ΠMSRP (cH) falls toward zero,

G(u(1/2)) − G(u(1)) grows but is bounded above by one, ΠMSRP (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 measureset of values of cH for which ΠMSRP (cH) > 0 and the result holds.

Finally, given that the consumers face the same prices as with recommendations but areless informed, they must necessarily be worse off.

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