The Dark Side of Competitive Pressure

The Dark Side of Competitive Pressure ∗ Jason G. Cummins Ingmar Nyman Division of Research and Statistics Department of Economics Federal Reserve Board Hunter College, CUNY jason.g.cummins@frb.gov ingmar.nyman@hunter.cuny.edu August 25, 2002 Abstract One of the most basic principles in economics is that competitive pressure promotes efficiency. However, this pressure can also have a dark side because it makes (cid:222)rms reluctant to act on private information that is unpopular with consumers. As a result, (cid:222)rms that possess superior information about the consequences of their actions for consumers(cid:146) welfare may choose not to use it. Wedevelopthisideainasimplemodelofdelegatedinvestmentinwhichagents are fully rational and risk neutral, and agency problems are absent. We show that competitive pressure obliges (cid:222)rms to make inefficient decisions when their information advantage over consumers is relatively small. This result could be applied to a broad range of economically important situations. JEL Classi(cid:222)cation: D20; D40; D82. Keywords: Competition; Information Aggregation; Incentives. Wethank Darrel Cohen,AlessandroLizzeri, andseminarparticipants at theCUNY Graduate ∗ CenterandHunterCollegefortheircomments. Theviewspresentedaresolelythoseoftheauthors and do not necessarily represent those of the Federal Reserve Board or its staff.

1 Introduction In this paper, we point out a cost that can be caused by competitive pressure. Normally, the threat of competition forces (cid:222)rms to make decisions that are in the consumers(cid:146) best interest. But consider the situation in which (cid:222)rms are better informed than the consumers. In this case, competitive pressure makes it costly for (cid:222)rms to utilize their information if it contradicts consumers(cid:146) opinions. In effect, there(cid:146)s always another (cid:222)rm that stands ready to cater to my customer if I fail to do so. Hence, competitive pressure encourages (cid:222)rms to seek market share by pleasing the relatively uninformed consumers. This desire may prevent (cid:222)rms from utilizing their private information, which reduces social welfare. Agency problems (cid:150) the usual cause of inefficiency when there is private information (cid:150) play no role in the inefficiency we highlight because the inefficiency occurs even when the interests of (cid:222)rms and consumers are perfectly aligned. An example helps convey the intuition. Consider a benevolent investment manager who invests her client(cid:146)s resources with the aim of maximizing the expected return. There are two alternatives, buying equity in a dot-com or in a widgetmaker. The client believes that the dot-com is the superior investment, perhaps because of the recent history of share prices. However, the investment manager, who as an expert has better information than the client, believes that the widgetmaker is a better investment. In the absence of competition, the manager invests in the widget-maker, even though the client believes (cid:150) probably incorrectly (cid:150) that this is the wrong choice. However, if there is a second investment manager who believesthatthedot-comisabetterbet, thentheclient willinvestwiththatmanager. Knowing that competition poses this threat, the (cid:222)rst manager may be obliged to go against her better judgment and offer to invest in the dot-com in order to keep the client. From a social planner(cid:146)s perspective, it is undesirable to cater to the client in this way because the (cid:222)rm(cid:146)s private information is not fully utilized. 1

Thistypeofdelegatedinvestmentframeworkisthesettinginwhichweformalize our argument. In our model, we assume that all agents are fully rational and risk neutral, and that the interests of consumers and (cid:222)rms are perfectly aligned. In this setting, competitive pressure, by itself, can force investment managers to make inefficient investment decisions in order to appeal to consumers(cid:146) priors. Interestingly, making the consumer better informed can actually worsen the inefficiency. Since consumers are often less well-informed than (cid:222)rms, our insight is potentially important in many situations. We discuss possible applications after having introduced and discussed the model. The idea that pleasing the uninformed distorts the incentive to reveal and use privateinformationisattheheartoftheliteratureonmanagerialmyopia(seeGrant, King, and Polak 1996 for asurvey). Inthis literature, a corporate managerwhohas private information (cid:150) either in the form of a hidden action or hidden information (cid:150) and who cares about the current share price has an incentive to distort investment in an attempt to manipulate the stock market(cid:146)s information. In particular, the manager avoids investment that the stock market either does not want to see (hiddeninformation)orisunabletosee (hiddenaction). Forourpurposes, themost powerful analysis of this problem is due to Brandenburger and Polak (1996), who demonstrate that the manager avoids investment not because it reveals information that is objectively bad, but because it reveals information that contradicts the prior opinions of the stock market. As a consequence, the manager makes (cid:147)the decisions that the market wants to see.(cid:148) What drives Brandenburger and Polak(cid:146)s result is the assumption that managers care about others(cid:146) opinions. In our model, this type of concern is created endogenously by competitive pressure. Our work also complements Prendergast (1993), who shows that employees who arerewardedbasedonthesubjectiveperformanceevaluationoftheirsuperiorhavea tendency to become (cid:147)yes men.(cid:148) Prendergast frames his analysis as an inherent con- (cid:223)ictbetweenincentivesandtheuseofinformation,whichispreciselytheperspective we have. But while Prendergast makes his argument in a situation where incentives 2

are provided by a compensation contract, we argue that competition creates the same sort of tension between incentives and the use of information. Inadditionto competitionand explicit contracts, reputational concerns canalso provide incentives. Holmstro¤m (1982) (cid:222)rst pointed out that reputational considerationscandistortproductiondecisionsandleadtoinefficienciesbecauseagentstryto manipulate theinformationthat theyreveal. ScharfsteinandStein(1990) elaborate on this conclusion by showing that private information may be wasted when there are multiple producers who move sequentially. By ignoring their own information and instead mimicking the decisions of others, producers are trying to be perceived as capable. Morris (2001) demonstrates that the same type of wasteful disregard for private information can occur when the informedparty isanadvisorratherthan a decision maker and her worries about reputation are instrumental (i.e., driven by concern for future decision-making) rather than intrinsic (i.e., driven by concern for future perceptions of herself). The paper that bears the most immediate resemblance to ours is Heidhues and Lagerlo¤f (2002).1 They study an election game and show that a candidate with private information about different policy alternatives will pander to the electorate in order to achieve her sole goal of being elected. This is truly insightful application of the idea that information may be used inefficiently when the informed need to cater to the uninformed. The key difference between their analysis and ours lies in the relationship between the informed and the uninformed. Heidhues and Lagerl¤of assume that there(cid:146)s a con(cid:223)ict of interest that makes candidates use information inefficiently: their politicians care only about the uninformed short-term beliefs of voters and not at all about the real long-term consequences of their actions for the electorate. Formally, their model is a cheap-talk game in which signals are costless, and it is a standard result in this literature that the informed sender may change the signal in order to manipulate the receiver(cid:146)s information. By contrast, 1Thanks go to Alessandro Lizzeri, who brought this paper to our attention after we completed ours. 3

our model is not a cheap-talk game because we assume that (cid:222)rms have the best long-run interest of the consumers at heart. We show that even when interests are perfectly aligned in this way, the incentive to behave inefficiently in an attempt to manipulate the revelation of information may arise endogenously because of the presence of competitive pressure. The rest of the paper is organized as follows. In the next section, we develop a simple model to illustrate our argument. In section 3, we discuss the model and possible applications. The (cid:222)nal section concludes, and the two appendices contain technical details. 2 The Model Aftersettingupthemodelinthefollowingsubsection, weanalyzethesolutionwhen there is a single (cid:222)rm. This monopoly case serves as a benchmark for the main case withtwocompeting(cid:222)rms. Weclosethesectionwithadiscussionofthesolutionwith an arbitrary number of (cid:222)rms. The importance and restrictiveness of the numerous simplifying assumptions are discussed in Section 3. 2.1 Setup The model formalizesthe informational inefficiencyresultingfromcompetitivepressure in a highly stylized delegated investment problem, based loosely on BrandenburgerandPolak(1996). Inthemodel, asingleconsumerinvestsadollarwith(cid:222)rms that havesuperiorinformationabout theexpectedpro(cid:222)tabilityofdifferentprojects. The consumer is denoted by the subscript c and the (cid:222)rms are denoted by the subscripts i and j. There are two time periods. In the (cid:222)rst period, each (cid:222)rm offers an investment strategy. In the second period, the consumer allocates her investment between the projects and the investments generate a payoff. The consumer receives half of this payoff, while the (cid:222)rm (or (cid:222)rms) that made the investment receives the other half. 4

2.1.1 Investment Management There are two investment projects (cid:150) (cid:147)green(cid:148) and (cid:147)red(cid:148) (cid:150) denoted by z g,r . ∈ { } The investment payoff π depends on two possible states of the world (cid:150) (cid:147)Green(cid:148) and (cid:147)Red(cid:148) (cid:150) denoted by Z G,R . In the Green state the highest payoff is from ∈{ } the green project and in the Red state the highest payoff is from the red project. To simplify the analysis, the (cid:147)high(cid:148) and (cid:147)low(cid:148) payoffs (net of investment) are the same for both the green and red project. The high and low payoffs are normalized to one and zero, respectively. Formally, the net payoff function from the investment of one dollar is 1 if (g,G) or (r,R) π(z,Z) = 0 if (g,R) or (r,G). ‰ Without loss of generality, the prior distribution on the state space, which is known to everyone, is Pr(Z =G) =(cid:181) 1. ≥ 2 In the (cid:222)rst period, each (cid:222)rm offers an investment project with the objective of maximizing its own expected pro(cid:222)t.2 When making this choice, the (cid:222)rms have private information in the form of a signal about the state of the world. The signal process is the same for both (cid:222)rms and their signals are conditionally independent. The signals take on one oftwo realizations, γ orρ, and are denotedby s γ,ρ . A ∈{ } γ (ρ) signal is an indication that the state is Green (Red). Both signal realizations have the same accuracy, denoted by σ: Pr(γ G)=Pr(ρR) =σ > 1. The (cid:222)rms can | | 2 use mixed strategies with the probability that a (cid:222)rm chooses the green project conditional on its signal denoted by βs Pr(z = g s), where the superscript indicates ≡ | dependence on the realization of a (cid:222)rm(cid:146)s private signal. The normalization of the payoffs makes the expected value of the projects equal to the probability that the state is the matching one. To make the problem interesting, we assume that the (cid:222)rms(cid:146) private information is superior to the publicly available information, so that σ > (cid:181). This motivates the (cid:222)rms(cid:146) presence in the 2Wethus assumethat the(cid:222)rmscan credibly commit to acertain investment policy through an offer, which seems reasonable since both theofferand the investment are likely to be veri(cid:222)able. 5

model by allowing them to contribute something to the investment decision. It also makes it efficient for the (cid:222)rms to follow their signal. We denote the (cid:222)rst-best efficient investment strategy that maximizes the ex ante expected investment payoff by β = (βγ =1,βρ =0). In words, the (cid:222)rst ∗∗ best efficient strategy is the one in which the green (red) project is picked with certainty by a (cid:222)rm that receives a γ (ρ) signal. The resulting expected payoff is E[π(β )] = Pr(G)Pr(γ G)+Pr(R)Pr(ρR) = (cid:181)σ +(1 (cid:181))σ = σ. Hence, if a ∗∗ | | − (cid:222)rm invests efficiently, it creates an expected value-added equal to the information that its signal adds over and above the consumer(cid:146)s information, i.e., σ (cid:181). Finally, − the efficiency loss from any investment strategy, β, is denoted by W (β) and is de(cid:222)nedasthe shortfall inexpectedpayoff comparedtothat of the (cid:222)rst-bestefficient strategy: W (β) E[π(β )] E[π(β)]. ∗∗ ≡ − 2.1.2 Trading and Consumer Behavior After observing the investment projects offered by the (cid:222)rms, the consumer chooses how much to invest in each project with the objective of maximizing her expected payoff. When the consumer chooses a project, we assume that she receives half of thepayofffromthatinvestment, andthattheotherhalfgoestothe(cid:222)rmthatoffered theproject. Theassumptionthat theinvestorandthe(cid:222)rmsplit thereturnfromthe investment isimportant becauseit alignsthe long-runinterestsofthe consumerand the(cid:222)rm, whowouldbothpreferthattheinvestmentyieldahighpayoff. Thismeans that agency problems play no role in generating our conclusions and furthermore thattheyarerobusttotheintroductionofmechanismsthataddressagencyproblems such as explicit contracts or reputational concerns through repeated interaction. From the consumer(cid:146)s point of view, both investment projects always have a strictlypositiveexpectedpayoff. Notinvestingatallisthereforeastrictlydominated strategyfortheconsumer,sosheneedonlychoosethefractionofherwealthtoinvest in the green project, denoted by α [0,1]. Of course, the consumer can invest only ∈ in projects that are offered. As a result, if both (cid:222)rms offer green (red) projects, the 6

consumer(cid:146)s choice is restricted to α = 1 (α = 0). When this happens, we assume that the consumer splits her dollar evenly between the projects offered by the two (cid:222)rms. The purpose of this assumption is to eliminate any extraneous differences in (cid:222)rms(cid:146) competitiveness in order to focus on how competitive pressure, by itself, affects their choice of project type. 2.2 Equilibrium with Monopoly Turning to the analysis of the model, consider (cid:222)rst the case when there is a single (cid:222)rm. Theconsumergivesherdollartothe (cid:222)rmtoinvestnomatterwhatinvestment strategy the (cid:222)rm pursues. Therefore, the (cid:222)rm is unconcerned about making sure the consumer provides it with resources to invest. Instead, the (cid:222)rm can choose the investment project based solely on its long-run payoff. Since the (cid:222)rm gets a share of those long-run payoffs, it chooses the (cid:222)rst-best efficient investment strategy: π i (z =g | γ) = 1 2 Pr(G | γ)> 1 2 Pr(R | γ)=π i (z =r | γ) ⇒ βγ ∗ =1 π i (z =r | ρ) = 1 2 Pr(R | ρ)> 1 2 Pr(G | ρ) =π i (z =g | ρ) ⇒ βρ ∗ =0. The ex ante expected total payoff (cid:150) to be split evenly between the consumer and the (cid:222)rm (cid:150) is equal to σ. 2.3 Equilibrium with Two Firms Competing Suppose instead that two (cid:222)rms compete for the consumer(cid:146)s investment funds. This givesthe consumeranon-trivialchoicebetweentwoinvestment projects(ifbothare offered). The consumer always prefers the project with the greater expected payoff. Since the consumer cannot observe the (cid:222)rms(cid:146) signals, she must instead base her inference about the true state of the world on only her own prior beliefs, captured by (cid:181), and the (cid:222)rms(cid:146) choices of project. The key observation in this situation is that in a perfectly-separating efficient equilibrium, theconsumer(cid:146)sposteriorbeliefswilltypicallyfavoroneproject overthe other. Thissituationoccursgenerically, withthegreenprojecthavingacompetitive 7

advantage whenever (cid:181)> 1.3 When the consumer(cid:146)s preference is skewed in this way, 2 it enhances the incentive for a (cid:222)rm with a γ-signal to behave efficiently and choose the green project; not only does the green project yield a higher return than the red project, but if the other (cid:222)rm offers a red project, the (cid:222)rm offering the green project captures the entire market. If the other (cid:222)rm offers a green project as well, the consumer splits her dollar between the two green projects. But for a (cid:222)rm with a ρ signal the (cid:222)rst-best efficient strategy of investing in the red project now presents a dilemma: with this signal, the expected value of the red project is higher, but its expected market share is lower since the consumer prefers the green project when it is offered. This trade-off is determined by the relative strength of the consumer(cid:146)s prior that green is the better project, captured by (cid:181), and the ρ -type (cid:222)rm(cid:146)s conviction that it is not, captured by σ. If the ρ-type (cid:222)rm is sufficiently con(cid:222)dent that the state is Red (cid:150)i.e., σ is sufficiently large compared to (cid:181) (cid:150) then the red project will be chosen, which is efficient. But if the signal is weak (cid:150) i.e., σ is sufficiently small compared to (cid:181) (cid:150) then the ρ-type (cid:222)rm (cid:222)nds it optimal to ignore its private information and instead cater to the consumer(cid:146)s belief that the green project is the correct one. This intuition is summarized in Proposition 1 and illustrated in Figure 1. Proposition 1: With two (cid:222)rms, the (cid:222)rst-best efficient outcome is not an equilibrium when the consumer(cid:146)s prior belief favors one state over the other and the (cid:222)rms(cid:146) information advantage over the consumer is sufficiently small. Proof: See Appendix. Figure 1 illustrates the efficient and inefficient regions derived in Proposition 1. The basic idea is simple: the ρ-type (cid:222)rm may want to offer the green project to 3Eliminating the competitive advantage for one of the projects is even more difficult than this condition suggests. As Brandenburger and Polak (1996) show, in a perfectly-separating (cid:222)rst-best efficient equilibrium, the consumer has access to the (cid:222)rms(cid:146) signals, so her posterior beliefs depend on her own prior, the payoffs from the investment projects, and the accuracy of the (cid:222)rms(cid:146) signals. Our model has already imposed a non-genericsymmetry in investment payoffs and signalquality. 8

Figure 1: Efficient and inefficient regions 9

capture market share. For some combinations of σ and (cid:181), the incentive is strong enough to break the efficient equilibrium. For a given σ, pick a number like 3 in the 4 (cid:222)gure, considerincreasing(cid:181); eventually, theconsumer(cid:146)spriorgetsskewedenoughto break the efficient equilibrium. For a given (cid:181), again pick a number like 3, consider 4 increasing σ; eventually, the (cid:222)rm(cid:146)s private information becomes precise enough to restore the efficient equilibrium. In the inefficient region of the parameter space, the (cid:222)rms play a partially separatingequilibriuminwhichinvestmentisinefficient. Aγ-type(cid:222)rminvestsefficiently byalwayschoosingthegreenproject. Bycontrast, aρ-type(cid:222)rminvestsinefficiently, playingamixedstrategythatinvestsinthegreenprojectwithstrictlypositiveprobability. In equilibrium, the consumer is indifferent between the red and the green project, and splits her dollar between the two when they are both offered. The equilibrium is derived in Proposition 2. Proposition 2: In the inefficient region derived in Proposition 1: 1. A γ-type (cid:222)rm invests efficiently: βγ ∗ =1. 2. A ρ-type (cid:222)rm invests inefficiently: βρ ∗ =β ρ ≡ σ2 σ (1 (1 − (cid:181) σ ) )( ( 2 1 (cid:181) − σ 1 ) ) 2(cid:181) ∈ (0,1). − − − 3. The consumer splits her investment betwbeen the green and the red project when both are offered: α = σ 1 (cid:181) 1,1 . ∗ 1 σ −(cid:181) ∈ 2 − ‡ ·‡ · ¡ ¢ ρ 4. The efficiency loss due to competition is equal to W =(σ (cid:181))β . − b Proof: See Appendix. Figure 2 illustrates the economics of the inefficient equilibrium. In panel A, we graph the ρ-type (cid:222)rm(cid:146)s net bene(cid:222)t from choosing the green project when the γ-type (cid:222)rm invests in the green project with probability one, denoted Πρ . For i(g r) − low levels of βρ, the bene(cid:222)t is strictly positive because the green project has the ρ competitive advantage over the red one. However, at β , the consumer switches to b 10

Figure 2: Net bene(cid:222)t from the green project to the ρ-type (cid:222)rm and the consumer choosing the red project over the green one, which makes the net bene(cid:222)t from the green project strictly negative for high levels of βρ. The reason for the consumer(cid:146)s preference reversal is that by putting more weight on the green project, the ρ-type (cid:222)rm garbles the consumer(cid:146)s inference about the green project(cid:146)s pro(cid:222)tability. This ρ is illustrated in panel B: At β the consumer is indifferent between the green and ρ the red project, so any choice of α is best response to β . The choice of α by b ρ the consumer can therefore bridge the discontinuity at β in the ρ-type (cid:222)rm(cid:146)s net b bene(cid:222)t from choosing the green project. We call α the unique value of α that ∗ b makes the ρ-type (cid:222)rm indifferent between the green and the red project, so that ρ ρ β is a best response. Since β depends only on the information structure and ρ the (cid:222)rms are identical, they must both choose β in equilibrium if they receive a ρ b b ρ signal. Moreover, this equilibrium behavior by the (cid:222)rms, βγ ∗ = 1,βρ ∗ = β , makes b α = σ 1 (cid:181) . ∗ 1 σ −(cid:181) b − It‡is wo·rt‡h po·inting out that the ρ-type (cid:222)rm(cid:146)s equilibrium strategy has a discontinuity at the boundary in parameter space between the inefficient and efficient 11

regions (see Figure 1). When moving across the boundary from the inefficient to ρ the efficient region, βρ ∗ falls discontinuously from β > 0 to 0. This can be seen in Figure 2A, where a move toward the efficient region pushes the net bene(cid:222)t from b inefficient investment down. At the boundary, the segment in the upper-left-hand side, which is positive in the inefficient region, reaches zero, but the discontinuity in ρ the net bene(cid:222)t function is still interior, i.e., β is strictly positive at the boundary. The efficiency loss from competition is economically intuitive. The inefficiency b comes from (cid:222)rms failing to use completely their valuable private information when therealizationoftheρsignalputsthematoddswiththeconsumer(cid:146)spreconceptions. The efficiency loss is therefore the expected value of the (cid:222)rm(cid:146)s information, σ (cid:181), − ρ multiplied by the probability that it gets wasted by the ρ-type (cid:222)rm, β . The comparative staticanalysisoftheinefficient equilibriumrevealsthattheexb ρ tent to whichthe ρ-type (cid:222)rminvestsinefficiently, β , increases with(cid:181) anddecreases ρ with σ. The reason is that β re(cid:223)ects the dose of the ρ-type (cid:222)rm(cid:146)s contradictory b information that it takes to dissuade the consumer of her initial preference for the b green project. It is economically sensible that this required dose increases with the strength of the initial belief, (cid:181), and decreases with the accuracy of the new information, σ. By contrast, α decreases with (cid:181) and increases with σ. This is because ∗ the consumer(cid:146)s equilibrium allocation to the green project is determined by what makes the ρ-type (cid:222)rm indifferent between choosing the green and the red project. Therefore, the consumer(cid:146)s allocation to the green project must decrease in order to compensate for it becoming more attractive to the ρ-type (cid:222)rm when (cid:181) increases or σ decreases. As illustrated in Figure 3, the dependence of the efficiency loss, W, on the information structure is considerably more complicated. The most interesting observation about this picture is that the efficiency loss is non-monotonic in (cid:181). Hence, making the consumer better informed (cid:150) by increasing (cid:181) (cid:150) may make matters worse, increasing the inefficiency from competitive pressure. What happens in this 12

case is that (cid:222)rms waste more information by paying closer attention to the consumer(cid:146)s opinion, despite the fact that an increase in (cid:181) makes the (cid:222)rms(cid:146) information less valuable. 2.4 The Number of Firms It is natural to ask what happens to our results when the number of (cid:222)rms increases. Unfortunately, the general analysis is intractable. Nevertheless, to get a feel for the economic forces at work, we derived analytical solutions when there are three and four (cid:222)rms. For the most part, the economics of the analysis is intuitive, but the formalityofitisrathercumbersome(seeAppendixB). Thebasic conclusion(cid:150)that the inefficiency from competitive pressure diminishes with the number of (cid:222)rms (cid:150) is probably not surprising. There are two effects when the number of (cid:222)rms increases. First, the probability of being pivotal decreases. In a fully-separating efficient equilibrium, the consumer hasaccesstoallprivateinformation, soa(cid:222)rmcanbepivotalonlyifthesignalsofall the (cid:222)rms cancel out. Obviously, the probability of such an informational stalemate decreases with the number of (cid:222)rms.4 Second, when the number of (cid:222)rms increases, so does the amount of total private information. This makes it more costly for the ρ-type (cid:222)rm to disregard its own private information. It is intuitive that the inefficiency from competitive pressure shrinks with the number of (cid:222)rms. After all, we have already shown that the inefficiency disappears when the individual (cid:222)rm(cid:146)s information advantage over the consumer becomes sufficiently large. As the number of (cid:222)rms increases, their aggregate information advantage over the consumer increases, and because the information is correlated 4This also implies that when there is an odd number of (cid:222)rms, the inefficiency disappears: with anoddnumberof(cid:222)rms,alltheinformationcannotcancelout. Withthree(cid:222)rms,forexample,there areatleasttwoγ-type(cid:222)rmsoratleasttwoρ-type(cid:222)rms,sotheconsumerwouldalwayschoosethe projectthatisofferedbyalargernumberof(cid:222)rms. Asaconsequence,thepivotal(cid:222)rmdoesnothave to misrepresent its information in order to make the consumer choose its project. However, the distinctionbetweenanoddandanevennumberof(cid:222)rmsisnotarobustpieceofeconomicintuition, but rather a peculiarity of thespeci(cid:222)cmodeling approach that weuse. 14

across (cid:222)rms, the expected cost for each (cid:222)rm of disregarding its own information increases as well. Nevertheless, increasing the number of (cid:222)rms without contemplating any other changes may not be the best thought experiment. Consider, for example, the mutualfundindustry. Therearethousandsofcompetitorsbutthatdoesnotmeanthat thereiscompleteinformationaboutwhichfundisbest. Themainreasonwhycomes from ignoring product differentiation, which tends to shield (cid:222)rms from competitive pressure. More speci(cid:222)cally, if the differentiated characteristic has an objective component, then competition is localized; in effect, each (cid:222)rm competes with only its closest neighbors in the product space (see Eaton and Lipsey 1989 for a survey of the literature on product differentiation). Therefore, even if the number of (cid:222)rms in the market as a whole is large, product differentiation makes all but asmall number of them irrelevant to any one (cid:222)rm.5 As a consequence, product differentiation may offset the tendency toward efficiency when the number of (cid:222)rms increases. More generally, our model is one of product choice, and the economic forces at work are similar to those found in the product differentiation model introduced by Hotelling (1929).6 Hotelling derived the celebrated (cid:147)principle of minimum differentiation,(cid:148) which refers to the tendency of product choice to be bunched together in order to gain market share. The tendency in our model for producers to cater to the consumer and offer her favorite project even when it is inferior is an example of this principle. More generally, when making product choices, one incentive for (cid:222)rms is to (cid:147)be where the demand is(cid:148) (Tirole 1988, p. 286). If consumers know less than producers about the product choice, then this incentive, which is accentuated by competition, may pull (cid:222)rms away from choices that are objectively better. 5However, the larger is the number of dimensions in which the product is differentiated, the weaker is the tendency towardsa localization of competition (Eaton and Lipsey 1989). 6Thesharpdiscontinuityinthemodel(cid:146)sconclusionswhenthereisanoddnumberof(cid:222)rmsisalso reminiscent of a peculiarity of the Hotelling model: if prices are (cid:222)xed, then no pure-strategy Nash equilibrium in product characteristics exists when there are three (cid:222)rms in the market (Eaton and Lipsey 1975). 15

In Hotelling(cid:146)s model, there is also a tendency to move away from minimum differentiation. When (cid:222)rms locate close to each other, price competition tends to decrease the value of their product choice even if their market share is high. Our model also has a cost to minimum differentiation, but the economics is quite different: a product choice has a lower per-unit expected value not because of the presence of competitors, but rather because it is an intrinsically inferior choice in light of the information about the uncertain investment environment. 3 Discussion 3.1 Assumptions There are a number of assumptions of the model that warrant discussion. First, because the payoffs of all players are proportional to the expected payoff from investment, the interests of the (cid:222)rms and the consumer are perfectly aligned. This assumption eliminates any agency problems in order to focus on the informationaggregation aspect of competition, creating a benchmark of (cid:222)rst-best efficiency under monopoly. Obviously, in the presence of agency problems, competition would have incentive bene(cid:222)ts that could offset its informational costs. But the fact that our results emerge even when incentive problems are absent makes them especially striking. The inefficiencyfromcompetitionthatwe highlight hasnothingtodowith (cid:222)rms paying insufficient attention to the customer(cid:146)s true long-run interest. On the contrary,thepointthatwemakeisthatcompetitionendogenouslydistortsproducer incentives away from being concerned about long-run consumer value toward being concerned about short-run consumer opinions. When these opinions may be erroneous due to lack of information, competition may force (cid:222)rms to make inefficient choices. Second, we have not considered alternative mechanisms that might restore ef- (cid:222)ciency. In particular, efficiency could be restored if the consumer were able to conceal her prior. One way to do this is to let the consumer choose how to allocate her investment after the (cid:222)rms have made their offers. Then the (cid:222)rms would 16

ignore the consumer(cid:146)s opinion if she promised to split her dollar evenly across the (cid:222)rms regardless of their investment choice. Although this seems like an attractive arrangement at (cid:222)rst glance, we believe that we have given the game a more reasonable dynamic structure for two reasons. First, the consumer(cid:146)s promise is not credible: once the (cid:222)rms offered their investment strategy, it would be in the consumer(cid:146)s best interest to go back on her promise and choose instead the (cid:222)rm that offers the project that the consumer perceives has the highest expected payoff. The consumer may, of course, (cid:222)nd ways to precommit, for example through an explicit contract, but such contracts appear exceedingly rare. Second, a con(cid:223)ict of interest between the consumer and the (cid:222)rms is likely to be present as well, in which case the consumer would be reluctant to relinquish her ability to respond to actions taken by (cid:222)rms. Third, the (cid:222)rms can compete only through product choice. A natural second dimension of competition would be through pricing. This could be modeled if we allowed the (cid:222)rm to cede a larger share of the investment payoff. Although this extension may alter the analysis signi(cid:222)cantly in some applications, we conjecture thatinthecurrentmodelitwouldnotchangethe(cid:223)avoroftheresults: itseemslikely that Bertrand-like competition would shrink the (cid:222)rm(cid:146)s payoff share, but otherwise leave the analysis unchanged.7 Fourth, the investment projects are indivisible. In some applications, it would be sensible to allow for compromises between different alternatives. For example, we could certainly allow the (cid:222)rms to offer a portfolio consisting of both projects. However,no(cid:222)rmwouldtakethatopportunity. Obviously,theγ-type(cid:222)rmhasnouse for anything but the green project, which means that any portfolio containing the redprojectrevealsperfectlythatthe(cid:222)rmhasaρsignal. Butintheinefficientregion, 7Some caution is warranted, however, because an equilibrium does not exist in many games in which (cid:222)rms choose both price and product characteristics (see, for example, Caplin and Nalebuff 1991 and d(cid:146)Aspremont, Gabszewicz, and Thisse 1979). We conjecture that this is a non-issue in ourmodelbecausethereisno interaction between priceandproduct location: thecost oflocating close to another (cid:222)rm is not due to stiffer price competition, but rather due to that location being intrinsically inferior. 17

perfect separation makes the red project strictly inferior to the green one, even if the (cid:222)rm has a ρ signal, so a mixed portfolio would never be used in equilibrium. Fifth,theconsumerisforcedtorelyonthe(cid:222)rms(cid:146)expertinvestmentmanagement because she lacks a viable investment option that she can manage on her own. Without this assumption, the consumer herself might, in effect, put competitive pressure on the (cid:222)rms by partly acting as an investment manager in her own right. This couldmake the distortions inthe (cid:222)rms(cid:146) investment strategyappearevenunder monopoly. However,theessenceoftheeconomicargumentwouldnonethelessremain intactsincecompetitivepressure(cid:150)albeitfromtheconsumerratherthanfromother (cid:222)rms (cid:150) would still be what forces (cid:222)rms to cater to the consumer(cid:146)s uninformed preference. Finally, in our model, the (cid:222)rms supply their expertise by offering products. An alternative is that they sell their advice to the consumer, who then makes the investmentdecisionherself. However,aslongasthe(cid:222)rms(cid:146)havevaluableinformation and their compensation depends only on the ultimate payoff from the investment that the consumer herself makes, the analysis is unaffected. The important point is thatcompetitionallowstheconsumertochoosethe(cid:222)rmthatsayswhatshewantsto hear, be it in the form of advice about, or the execution of, her investment decision. 3.2 Applications Although we cannot hope to capture the full richness of real world problems using our very parsimonious model, there do appear to be many situations in which competitive pressure can impede the management of private information. Here are several examples in which our results may be an important element of the story: Savings and Loan Crisis (cid:149) Milgrom and Roberts (1992) say that competitive pressure was one of the key forces in the S&L crisis of the 1980s: (cid:147)The moral hazard problem in the S&L industry was actually intensi(cid:222)ed by the effects of competition. Normally we think of competition, which tends to drive out those executives who are 18

unwillingtotakethepro(cid:222)t-maximizingactions, aspromotingefficiency. Inthe contextoftheS&Lindustryinthe1980s, however, competitionhadaperverse effect. ManyconservativeS&Lexecutiveshadnochoicebuttogambleonrisky investments if they were to survive.(cid:148) (p. 175). Credence Goods (cid:149) With credence goods, an expert must act as the agent for the customer in making production decisions, and it is difficult for the customer to evaluate the quality of the expert(cid:146)s decision-making even ex post. Examples include medicalservices, legal services, andrepairservices. The literatureoncredence goodshasfocusedontheproblemofagentswhoexploitcustomers(cid:146)information disadvantage (see Dulleck and Kerschbamer 2001 for a survey). However, againstthebackdropoftheanalysispresentedhere,itisnaturaltoaskwhether competitioncanforceexpertstocatertothecustomer(cid:146)suninformed,short-run opinion rather than pursue her long-run welfare. Anecdotes suggest that this problem is especially important when it comes to prescription drugs. Patients demand certain widely advertised drugs from doctors (see, e.g., Nexium which is marketed as the (cid:147)purple pill(cid:148) so it(cid:146)s easy for consumers to remember). Doctors usually know better than their patients whether the (cid:147)purple pill(cid:148) and the like is appropriate. Nevertheless, doctors prescribe brandname drugs, rather than generics, for fear of upsetting the consumer, according to reports.8 Competition for Votes (cid:149) In many countries, there is a continuing constitutional debate over how often politicians should face the voters. The standard argument for holding elections with lower frequency is that it shields politicians from competitive pressure that may force them to make populist decisions that are not in the 8(cid:147)Prilosec(cid:146)sMakerSwitchesUserstoNexium,ThwartingGenerics,(cid:148)WallStreetJournal,p. A1. June 6, 2002. 19

best long-run interest of the country. In support of this view, Heidhues and Lagerl¤of (2002) argue that self-interested political candidates have an incentive tochoosepoliticalplatformsthat getthemelected, ratherthanthosethat implementefficientoutcomes. Ouranalysissuggeststhatthisconclusionisnot limited to when politicians are self-interested, but that it applies even when candidates have the electorate(cid:146)s best interest in mind. Internal Labor Markets (cid:149) Organizations are an important arena where management of information on behalf of others takes place. Holmstr¤om and Milgrom (1991) were the (cid:222)rst to point out that incentive provision is delicate when decision-making is complex. Prendergast (1993) applies this idea to information management, and establishesatrade-offbetweenincentiveprovisionandinformationaggregation when incentives are provided through compensation contracts. But incentives canalsobeprovidedbyinternallabormarketsinwhichemployeescompetefor promotion(see, e.g., Lazear andRosen1981). Ouranalysis suggeststhat even though competition can induce employees to supply the efficient amount of labor, it may also give them the incentive to use private information relevant to the (cid:222)rm(cid:146)s pro(cid:222)ts in an inefficient way.9 In other words, Prendergast(cid:146)s (cid:147)yes men(cid:148)may be theresult not onlyofincentive contracts, butofthe competition for promotion as well. 4 Conclusion Our model establishes that competitive pressure can lead to inefficient decisions when producers are better informed than consumers about the bene(cid:222)ts of different alternatives. Consumers typically have a favorite alternative, which gives producers the incentive to enhance their competitiveness by choosing that project, even when 9Thereare,ofcourse,otherimportantdisadvantagestoincentiveprovisionthroughcompetition aswell,mostnotablythatitdiscouragescooperationamongemployees(Lazear1989)andmaylead tocostlyin(cid:223)uenceactivitiesiftheevaluationofperformanceissubjective(Milgrom1988,Milgrom and Roberts 1988, and Tirole 1992). 20

their own expert opinion suggests otherwise. However, the efficiency cost of choosing the wrong alternative provides (cid:222)rms with a countervailing incentive to ignore the consumer(cid:146)spreferences, andtoact on theirsuperior private information. Therefore, the inefficiency from competition disappears when the producers(cid:146) information advantage becomes sufficiently large. This insight is only a (cid:222)rst step in our research, so the agenda for re(cid:222)nements and extensions is extensive. First, we want to extend the model so that competition has an incentive bene(cid:222)t as well as an information-aggregation cost. This would allow for an explicit analysis ofthe relationshipbetweenthe twoand how the tradeoff between them depends on the economic environment. Second, an intriguing extension of the model would be to allow for the customer to be a (cid:222)rm. To make this interpretation interesting, we would like to incorporate an additional layer of agency problems between management and shareholders in the demand-side (cid:222)rm. The market for auditing services could be one application of this formulation of the model. Third,itwouldbeinterestingtotrytotransplanttheeconomicsofourmodel totheLazear-Rosenframeworkofincentiveprovisionthroughrank-orderpromotion tournaments. Finally, wehave shownthat competitionprevents private information from being used, which begs the question of whether competitive pressure puts a damper on experimentation and the acquisition of private information. 21

A Proofs of Propositions Proof of Proposition 1: In a perfectly-separating equilibrium the (cid:222)rms invest efficiently, so the consumer can infer the (cid:222)rms(cid:146) signals perfectly. If the two (cid:222)rms have different signals and therefore offer different projects, then the consumer(cid:146)s payoff from choosing the green project looks as follows: E[Π(g) g,r]=E[Π(g) γ,ρ]=Pr(Gγ,ρ) 1+Pr(R γ,ρ) 0=(cid:181) (1 (cid:181)). | | | • | • ≥ − Suppose that (cid:181) = 1. The consumer now splits the dollar evenly between the two 2 projects if both are offered, so the (cid:222)rms receive the same payoff from both projects in case theyhave different signals. Whenthe (cid:222)rms receive the same signal, choosing the efficient investment project yields a strictly higher payoff. Hence, if (cid:181)= 1, then 2 efficient investment is an equilibrium. Suppose instead that (cid:181) > 1 > (1 (cid:181)). Now the consumer always chooses the 2 − greenproject ifthe (cid:222)rms offer different projects. Therefore, incase the (cid:222)rms receive differentsignals,the(cid:222)rmthatreceivesaρsignalstandstogainfromswitchingtothe inefficient investment project, which could make the efficient outcome strategically unstable. The (cid:222)rm with a γ-signal, on the other hand, clearly has no incentive to deviate). Consider (cid:222)rst the ρ-type (cid:222)rm(cid:146)s expected payoff from the efficient strategy. The probability of two ρ signals given a ρ signal is Pr(ρρ ρ) =Pr(ρR)Pr(R ρ)+Pr(ρG)Pr(Gρ)=σPr(R ρ)+(1 σ)Pr(Gρ) | | | | | | − | σ(1 (cid:181)) (1 σ)(cid:181) =σ − +(1 σ) − σ(1 (cid:181))+(1 σ)(cid:181) − (1 σ)(cid:181)+σ(1 (cid:181)) " # " # − − − − σ2(1 (cid:181))+(1 σ)2(cid:181) = − − . σ(1 (cid:181))+(1 σ)(cid:181) − − 22

In this case, the (cid:222)rms share the payoff, which has an expected value of Pr(R ρρ). | The probability of a γ signal and a ρ signal given a ρ signal is Pr(γρ ρ) =Pr(γ R)Pr(R ρ)+Pr(γ G)Pr(Gρ) =(1 σ)Pr(R ρ)+σPr(Gρ) | | | | | − | | σ(1 (cid:181)) (1 σ)(cid:181) =(1 σ) − +σ − − σ(1 (cid:181))+(1 σ)(cid:181) (1 σ)(cid:181)+σ(1 (cid:181)) " # " # − − − − σ(1 σ) = − . σ(1 (cid:181))+(1 σ)(cid:181) − − In this case, the payoff to the (cid:222)rm that receives the ρ signal is zero. Hence, the expected payoff from efficient investment for a (cid:222)rm that receives a ρ signal is 1 1 Πρ ∗ =Pr(ρρρ) Pr(R ρρ) +Pr(γρ ρ) 0= Pr(ρρρ)Pr(R ρρ) . (1) | 4 | | • 4 | | • ‚ • ‚ Theexpectedpayofffromamixedstrategyissimplyaconvexcombinationofthe payoffs from the two pure strategies. Therefore, to examine the strategic stability of the efficient outcome, we only need to consider a deviation from the other pure strategy βρ = 1. If the other (cid:222)rm receives a γ signal, then the two (cid:222)rms share the payoff, which has an expected value of Pr(Gγρ). If the other (cid:222)rm receives a ρ | signal, then by deviating and offering the green project the (cid:222)rm captures the entire payoff, which has an expected value of Pr(Gρρ). Hence, the expected payoff for a | (cid:222)rm that receives a ρ signal and offers the green project is 1 1 ΠρD =Pr(γρρ) Pr(Gγρ) +Pr(ρρρ) Pr(Gρρ) | 4 | | 2 | • ‚ • ‚ (2) 1 = Pr(γρρ)(cid:181)+2Pr(ρρρ)Pr(Gρρ) . 4 | | | • ‚ The expected net bene(cid:222)t of offering the green project instead of the red project when the (cid:222)rm receives a ρ signal is the difference between (2) and (1): 1 1 ΠρD Πρ ∗ = Pr(γρρ)(cid:181)+2Pr(ρρ ρ)Pr(Gρρ) Pr(ρρ ρ)Pr(R ρρ) − 4 | | | − 4 | | • ‚ • ‚ 1 = Pr(γρρ)(cid:181) Pr(ρρ ρ) Pr(R ρρ) 2Pr(Gρρ) (3) 4 | − | | − | ( ) • ‚ 1 1 1 2 = 2σ2 (cid:181) 3(cid:181) σ . 4 (1 σ)(cid:181)+σ(1 (cid:181)) − 2 − − 3 " #( ) − − (cid:181) ¶ (cid:181) ¶ 23

The bracketed expression is strictly positive, so the sign of the entire expression depends on the sign of the expression in braces. Let denote the expression in braces. This function is strictly increasing in (cid:181): H ∂ 2 H =2σ2 3 σ ∂(cid:181) − − 3 (cid:181) ¶ =2 σ(3 2σ) >2 (3 2σ) >2 (3 1)=0. − − − − − − It is strictly decreasing in σ: ∂ 1 1 H =4σ (cid:181) 3(cid:181)<4 (cid:181) 3(cid:181)=(cid:181) 2<0. ∂σ − 2 − − 2 − − (cid:181) ¶ (cid:181) ¶ Moreover, it is easy to verify that is strictly negative when (cid:181) = 1 and when H 2 σ = 1, and that it is strictly positive when (cid:181) = σ. Therefore, the Intermediate Value Theorem implies that for each value of (cid:181) (σ), there exists a unique threshold value of σ ((cid:181)), call it σ ((cid:181)), that makes = 0. When σ < σ ((cid:181) > (cid:181)) is strictly H H positive, and when σ >σ ((cid:181)<(cid:181)) is strictly negative. b b H b b The parameter space can thus be split into two regions, one region (high σ and b b low (cid:181)) where the efficient outcome is an equilibrium and one region (low σ and high (cid:181)) where it is not. The boundary between the two regions is implicitly de(cid:222)ned by the net bene(cid:222)t from deviation being equal to zero, which happens when 1 2 =2σ2 (cid:181) 3(cid:181) σ =0. H − 2 − − 3 (cid:181) ¶ (cid:181) ¶ It is easy to verify that the end points of the boundary are (cid:181)= 1,σ = 2 and 2 3 ((cid:181)=1,σ =1). It is also easy to verify that for any admissibl¡e σ and (cid:181), ¢ 2. H ≤ Finally, to analyze the slope of this boundary, notice that along it = 0. This H implies that when the partial derivatives of the net bene(cid:222)t of deviation with respect to σ and (cid:181) are evaluated at the boundary, the effects of the parameters on the (cid:222)rst multiplicative term, 1 vanish. Hence, the partial derivatives of the net (1 σ)(cid:181)+σ(1 (cid:181)) − − bene(cid:222)t with respect σ and (cid:181) take on the same sign as ∂ and ∂ , respectively. It ∂Hσ ∂H(cid:181) now follows from the Implicit Function Theorem that d dσ dH(cid:181) = >0. d(cid:181) −d dHσ fl fl =0 b flH fl b 2fl4 ¥

Proof of Proposition 2: Consider (cid:222)rst the part of the parameter space where the (cid:222)rst-best efficient outcome is not an equilibrium. To con(cid:222)rm that the proposed equilibrium is strategically stable, start from the end of the game with the consumer(cid:146)s decision when both projects are offered. Call the (cid:222)rm offering the green project i and the (cid:222)rm offering the red project j. The consumer(cid:146)s payoff from the greenprojectisproportional tothe probabilitythatthe state isgreen, andtherefore her best-response correspondence is 1 1 if Pr(Gg,r)> { } | 2  1 α ∗    (0,1) if Pr(Gg,r)= ∈  | 2   1 0 if Pr(Gg,r)< .   { } | 2    In the proposed equilibri um, the red project is chosen with positive probability, so Pr(Gg,r) is given by Bayes Rule, amended by the Intuitive Criterion if βρ =1: | j Pr(g G)Pr(r G)Pr(G) Pr(Gg,r) = | | | Pr(g G)Pr(r G)Pr(G)+Pr(g R)Pr(r R)Pr(R) | | | | (1 σ)(cid:181)[σ+(1 σ)βρ] = − − i (4) (1 σ)(cid:181)[σ+(1 σ)βρ]+σ(1 (cid:181))[(1 σ)+σβρ] − − i − − i =Pr(Gg,ρ). | The Pr(Gg,r) is strictly decreasing and strictly convex in βρ: | i ∂Pr(Gg,r) σ(1 σ)(cid:181)(1 (cid:181)) σ2 (1 σ)2 | = − − − − − < 0 ∂βρ 2 i ρ £ ρ⁄ [σ+(1 σ)β ](1 σ)(cid:181)+[(1 σ)+σβ ](1 (cid:181)) − i − − i − ‰ (cid:190) ∂2Pr(Gg,r) 2σ(1 σ)(cid:181)(1 (cid:181)) (1 σ)2(cid:181)+σ2(1 (cid:181)) | = − − − − > 0. ∂βρ2 3 i [σ+(1 σ)βρ](1 σ) £ (cid:181)+[(1 σ)+σβρ](1 ⁄ (cid:181)) − i − − i − ‰ (cid:190) Moreover, it is easy to verify that Pr(Gg,r) is strictly greater than 1 when | 2 βρ = 1 and is strictly less than 1 when βρ = 0. It follows from the Intermediate i 2 i ValueTheoremthat thereexistsauniqueβρ forwhichPr(Gg,r)= 1. Atthisvalue i | 2 25

of βρ, which we will denote by β ρ , the consumer switches from choosing the green i i project to choosing the red project. b ρ A closed-form expression for β can be derived based on equation (4) i (1 σ)(cid:181) σ+(1b σ)β ρ i 1 − − = (1 σ)(cid:181) σ+(1 σ)β hρ +σ(1 (cid:181)) i (1 σ)+σβ ρ 2 ⇔ i b i − − − − h i h i b ρ b ρ σ+(1 σ)β (1 σ)(cid:181)= (1 σ)+σβ σ(1 (cid:181)) i i − − − − ⇔ h i h i b ρ b σ(1 σ)(2(cid:181) 1) β = − − . i σ2(1 (cid:181)) (1 σ)2(cid:181) − − − b Since Pr(Gg,r) is strictly increasing in (cid:181) and strictly decreasing in σ, it follows | ρ from the Implicit Function Theorem that β is strictly increasing in (cid:181) and strictly i decreasing in σ. b To derive the ρ-type (cid:222)rm(cid:146)s expected net bene(cid:222)t from offering the green rather than the red project we need the following conditional probabilities: σ(1 σ)+β ρ σ2(1 (cid:181))+(1 σ)2(cid:181) Pr(g ρ )= − j − − j | i σ(1 (cid:181))+(1 σ)(cid:181) £ ⁄ − − σ2(1 (cid:181))+(1 σ)2(cid:181) Pr(r ρ ) = 1 βρ − − j | i − j σ(1 (cid:181))+(1 σ)(cid:181) ‡ · • − − ‚ β ρ (1 σ)2(cid:181)+(1 σ)σ(cid:181) Pr(Gρ ,g )= j − − | i j (1 σ)σ+βρ[(1 σ)2(cid:181)+σ2(1 (cid:181))] − j − − σ(1 σ)(1 (cid:181))+β ρ σ2(1 (cid:181)) Pr(R ρ ,g )= − − j − | i j σ(1 σ)+βρ[σ2(1 (cid:181))+(1 σ)2(cid:181)] − j − − (1 σ)2(cid:181) Pr(Gρ ,r ) = − ifβρ <1 | i j (1 σ)2(cid:181)+σ2(1 (cid:181)) j − − σ2(1 (cid:181)) Pr(R ρ ,r ) = − if βρ <1. | i j σ2(1 (cid:181))+(1 σ)2(cid:181) j − − 26

The ρ-type (cid:222)rm(cid:146)s net bene(cid:222)t from choosing the green project instead of the red project is 1 α Πρ = Pr(g ρ )Pr(Gg ,ρ )+ Pr(r ρ )Pr(Gr ,ρ ) i,(g r) 4 j | i | j i 2 j | i | j i − ‰ (cid:190) 1 (1 α) Pr(r ρ )Pr(R r ,ρ )+ − Pr(g ρ )Pr(Gg ,ρ ) − 4 j | i | j i 2 j | i | j i ‰ (cid:190) 1 = Pr(g ρ )Pr(Gg ,ρ ) [Pr(r ρ )Pr(R r ,ρ )+2Pr(g ρ )Pr(R g ,ρ )] 4 j | i | j i − j | i | j i j | i | j i ‰ +2α[Pr(r ρ )Pr(Gr ,ρ )+Pr(g ρ )Pr(R g ,ρ )] . j i j i j i j i | | | | (cid:190) This net bene(cid:222)t is continuous and linearly increasing in α. If βρ <β ρ , then α =1. Hence, we can express the payoff as i i ∗ b1 σ(1 σ)(cid:181)+βρ(1 σ)2(cid:181) Πρ = − j − i,(g r) 4 σ(1 (cid:181))+(1 σ)(cid:181) − ( • − − ‚ 1 β ρ (1 σ)2(cid:181) 1 β ρ σ2(1 (cid:181)) +2 − j − − j − σ‡(1 (cid:181))·+(1 σ)(cid:181) − σ‡(1 (cid:181))·+(1 σ)(cid:181) ) • − − ‚ • − − ‚ 1 1 1 2 = 2σ2((cid:181) ) 3(cid:181)(σ ) +βρ σ2(1 (cid:181)) (1 σ)2(cid:181) . 4 σ(1 (cid:181))+(1 σ)(cid:181) − 2 − − 3 j − − − " #( ) − − • ‚ £ ⁄ Because σ > (cid:181), this expression is linearly decreasing in βρ, and by the assumption j that the (cid:222)rst-best efficient outcome is not strategically stable, the expression is strictlypositivewhenβρ =0. Thisallowsustoconcludethat βρ <β ρ Πρ > j ∀ i i ⇒ i,(g r) − 0. Therefore, no such βρ can be a best response. j b If βρ >β ρ , then α =0. Hence, we can express the payoff as i i ∗ 1 Πρ = b Pr(g ρ )Pr(Gg ,ρ ) Pr(r ρ )Pr(R r ,ρ )+2Pr(g ρ )Pr(R g ,ρ ) i,(g r) 4 j | i | j i − j | i | j i j | i | j i − ( • ‚ ) 1 1 = σ[2(1 σ)(1 (cid:181))+(σ (cid:181))]+βρ[σ2(1 (cid:181)) (1 σ)2(cid:181)] . −4 σ(1 (cid:181))+(1 σ)(cid:181) − − − j − − − " #( ) − − Because σ > (cid:181), this expression is linearly decreasing in βρ and is strictly negative j when βρ = 0. This allows us to conclude that βρ > β ρ Πρ < 0. Therefore, j ∀ i i ⇒ i,(g r) − no such βρ can be a best response. i b 27

Thus, the only remaining candidate for an equilibrium strategy for the ρ-type ρ (cid:222)rm is β , which makes the consumerindifferent betweenthe greenand redproject. i Moreover,itfollowsfromtheIntermediateValueTheoremthatthereexistsaunique b α that makes the ρ-type (cid:222)rm indifferent between the green and the red project. ∗ For Πρ =0 we have i,(g r) − 1 Pr(r ρ )Pr(R r ,ρ )+2Pr(g ,ρ )Pr(R g ,ρ ) Pr(g ρ )Pr(Gg ,ρ ) α = j | i | j i j i | j i − j | i | j i ∗ 2 Pr(r ρ )Pr(Gr ,ρ )+Pr(g ρ )Pr(R g ,ρ ) ( j i j i j i j i ) | | | | 1 σ (cid:181) 1 = 1+ − > . 2 (1 σ)[(1 σ)(cid:181)+σ(1 (cid:181))]+βρ[σ2(1 (cid:181)) (1 σ)2(cid:181)] 2 ( − − − j − − − ) ρ Sinceβ isuniqueanddependsonlyontheinformationstructure, itmust bethe i equilibriumchoice ofboth (cid:222)rmsifthey receive a ρ signal. Therefore, inequilibrium, b βρ ∗ =βρ ∗ =β ρ β ρ and i j i ≡ 1 b b σ (cid:181) α ∗ = 1+ − 2 ( (1 σ)[(1 σ)(cid:181)+σ(1 (cid:181))]+βρ [σ2(1 (cid:181)) (1 σ)2(cid:181)] ) • i ‚ − − − − − − 1 σ 1 (cid:181) = − . 2 1 σ (cid:181) c (cid:181) − ¶(cid:181) ¶ By assumption, the condition for an inefficient outcome derived in Proposition 1 is satis(cid:222)ed. This in turn implies that σ 1 (cid:181) <2, so α (1,1). 1 σ −(cid:181) ∗ ∈ 2 − The expected efficiency loss can‡be de·r‡ived a·s follows E[Π(β ∗ )] =(cid:181) σβγ ∗ +(1 σ)βρ ∗ +(1 (cid:181)) (1 σ) 1 βγ ∗ +σ 1 βρ ∗ − − − − − h i h ‡ · ‡ ·i ρ ρ =(cid:181) σ+(1 σ)β +(1 (cid:181)) σ 1 β − − − h ρ i h ‡ ·i =σ β (σ (cid:181)).b b − − Hence, the welfare loss can be calculated as the difference between the (cid:222)rst-best b efficient equilibrium in which information is fully revealed and this last expression ρ ρ W E[Π(β )] E[Π(β )]=σ σ β (σ (cid:181)) =(σ (cid:181))β . ∗∗ ∗ ≡ − − − − − h i To argue that the above equilibrium is unbique, recall from Propbosition 1 that when it obtains, the perfectly-separating (cid:222)rst-best efficient outcome is not strategically stable. Furthermore, notice that for any given strategy pro(cid:222)le, the net payoff 28

from choosing the green rather than the red project is strictly higher for the γ-type (cid:222)rm than for the ρ-type (cid:222)rm. It therefore follows that βγ ∗ βρ ∗ and that at most ≥ one (cid:222)rm-type can play a mixed strategy. Hence, apart from the above equilibrium, there exists no other in which the ρ-type (cid:222)rm plays a mixed strategy. Moreover, the discussion above also ruled out the perfectly-pooling equilibrium in which both (cid:222)rm-types choose the green project. Next, consider the possibility that the γ-type (cid:222)rm plays a mixed strategy in equilibrium: βγ ∗ (0,1) βρ ∗ = 0. With βρ ∗ = 0, choosing the green project ∈ ⇒ perfectly reveals that the (cid:222)rm(cid:146)s signal is γ, so the consumer strictly prefers the green project to the red one. But this implies that the γ-type (cid:222)rm also strictly prefers the green project to the red one, i.e., βγ ∗ (0,1) cannot be sequentially ∈ rational. Moreover, the Intuitive Criterion ensures that the γ-type (cid:222)rm enjoys the same bene(cid:222)t from, and incentive to deviate to, the green project in the perfectly pooling outcome in which both (cid:222)rm-types invest in the red project. ¥ 29

B The Number of Competing Firms To get a sense of how the analysis changes when the number of (cid:222)rms increases, we outline the analysis of the problem when there are four (cid:222)rms.10 Intuitively, increasing the number of (cid:222)rms decreases the ρ-type (cid:222)rm(cid:146)s net bene(cid:222)t from choosing the green project. This shifts down the net-bene(cid:222)t curve in Figure 2, panel A, and, as a consequence, the inefficiency region in Figure 1 shrinks. However, it leaves ρ unaffected β , which is determined by a single deviating (cid:222)rm(cid:146)s in(cid:223)uence on the consumer(cid:146)s inference about the true state of the world. b We therefore wanttodemonstrate that the ρ-type (cid:222)rm(cid:146)s netbene(cid:222)t fromunilateral deviation from the efficient outcome is strictly larger with two (cid:222)rms than with four. When there are two (cid:222)rms, this net bene(cid:222)t looks as follows: 1 Pr(γρρ)Pr(Gγρ)+Pr(ρρρ)[2Pr(Gρρ) Pr(R ρρ)] . 4 { | | | | − | } ¡ ¢ Theleadingone-quarteristhe(cid:222)rm(cid:146)sshareoftheinvestmentpayoffwhenthemarket issharedevenly. The (cid:222)rstterminsidethebracketsisthebene(cid:222)tfromcapturinghalf the market with the green project rather than none of it with the red project when the other (cid:222)rm offers the green project. The second term represents the net bene(cid:222)t when both (cid:222)rms receive a ρ signal. In this state, the green project once again gives a larger market share than the red one, namely the entire market instead of only half of it, but also has a lower expected value since it contradicts two ρ-signals. 10Ifthereisanodd numberof(cid:222)rms, then theinefficiencydisappearscompletely. Thereason for thisisthatwithanoddnumberof(cid:222)rms,theaggregateprivateinformationofthe(cid:222)rmsalwaysfavors oneprojectovertheother. Thismeansthatthepivotal(cid:222)rmwhoseprivateinformation(cid:150)asinferred bytheconsumer(cid:150)determinestheconsumer(cid:146)schoicecanswaytheconsumernomatterwhichsignal shereveals. Therefore,thereisnoincentiveforthepivotal(cid:222)rmtomisrepresentherinformation. In contrast,when thenumberof(cid:222)rmsiseven, thepivotal(cid:222)rm(cid:146)strueprivateinformation makesfora cancellation of the (cid:222)rms(cid:146) signals taken together, leaving it up to the uninformed, but opinionated consumer to determinethe choice of investment project. 30

Consider next the same net bene(cid:222)t from deviation when there are four (cid:222)rms: 1 4 Pr(γγρρρ) Pr(Gγγρρ) 8 | 3 | ( (cid:181) ¶ (cid:181) ¶ 4 +Pr(γρρρρ) 2Pr(Gγρρρ) Pr(R γρρρ) | | − 3 | • (cid:181) ¶ ‚ +[Pr(γγγρρ)Pr(Gγγγρ) Pr(ρρρρρ)Pr(R ρρρρ)] . | | − | | ) Again, the leading multiplicative term scales the payoff to each (cid:222)rm(cid:146)s baseline market share when the market is shared equally. The (cid:222)rst two terms are similar to the ones when there are two (cid:222)rms. The (cid:222)rst term captures the net bene(cid:222)t from getting a part of the market rather than nothing when the aggregate private information is uninformative and two other (cid:222)rms are offering the green project. Notice that the marketshareinthiscaseishigherthanone-eighthsinceone(cid:222)rmfailstoshareinthe payoff by offering the red project. The second term represents the net bene(cid:222)t when the aggregate information sums to two ρ-signals. Just as in the two-(cid:222)rm case, the ρ-type (cid:222)rm(cid:146)s decision is critical for the consumer(cid:146)s preference over the two projects, so the green project, which shares the market with one other (cid:222)rm, captures a larger market share than the red one, which shares the market with two other (cid:222)rms. Finally, the last term does not appear when there are only two (cid:222)rms. This captures the bene(cid:222)t when the aggregate information is so unambiguously in favor of one of the projects that a single (cid:222)rm(cid:146)s decision is unable to change the consumer(cid:146)s beliefs enough to alter her ranking of the two projects. This happens when the other three (cid:222)rms have the same signal. In this case, the (cid:222)rm(cid:146)s market share is one-fourth if it offers the same project as the other three (cid:222)rms and is zero if it offers a different one. Therefore, the net bene(cid:222)t from offering the green project instead of the red one amounts to trading equal shares of a project with a high expected value for one with a low expected value. Compare term-by-term the net bene(cid:222)t from deviation with two and four (cid:222)rms. With two (cid:222)rms, the (cid:222)rst term looks as follows: 1 Pr(γρρ)Pr(Gγρ) = 1 σ(1 − σ) (cid:181). 4 | | 4 σ(1 (cid:181))+(1 σ)(cid:181) − − h i ¡ ¢ ¡ ¢ 31

With four (cid:222)rms, the (cid:222)rst term is 1 Pr(γγρρρ) 4 Pr(Gγγρρ) = 1 3σ2(1 − σ)2 4 (cid:181) 8 | 3 | 8 σ(1 (cid:181))+(1 σ)(cid:181) 3 − − h i ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ =2σ(1 σ) 1 σ(1 σ) (cid:181) . − − 4 σ(1 (cid:181))+(1 σ)(cid:181) − − n h i o ¡ ¢ Since 2σ(1 σ) < 1, this expression is strictly smaller than the corresponding − expression when there are two (cid:222)rms. The second term looks as follows when there are two (cid:222)rms 1 Pr(ρρ ρ) 2Pr(Gρρ) Pr(R ρρ) 4 | { | − | } = ¡ ¢ 1 4 σ σ 2( ( 1 1 − (cid:181) (cid:181) ) ) + + ( ( 1 1 − σ σ ) ) 2 (cid:181) (cid:181) 2 (1 σ ( ) 1 2(cid:181) − + σ σ )2 2 (cid:181) (1 (cid:181)) − σ2(1 σ (cid:181) 2( ) 1 + − (1 (cid:181)) σ)2(cid:181) − − − − − − h in h i h io ¡ ¢ = 1 1 2(1 σ)2(cid:181) σ2(1 (cid:181)) . 4 σ(1 (cid:181))+(1 σ)(cid:181) − − − − − h ih i ¡ ¢ With four (cid:222)rms, the second term is 1 Pr(γρρρ ρ) 2Pr(Gγρρρ) 4 Pr(R γρρρ) 8 | | − 3 | ¡ = ¢ 1 8 3σ(1 − σ σ ( ) 1 [σ ' 2 (cid:181) (1 )+ − ( (cid:181) 1 )+ σ (1 )(cid:181) − σ)2(cid:181)] ¡ 2 ¢ (1 σ ( ) 1 2(cid:181) − + σ σ )2 2 (cid:181) (1 “ (cid:181)) − 4 3 σ2(1 σ (cid:181) 2( ) 1 + − (1 (cid:181)) σ)2(cid:181) • − − ‚ − − − − n h i h io ¡ ¢ ¡ ¢ = 3 σ(1 σ) 1 1 2(1 σ)2(cid:181) 4 σ2(1 (cid:181)) 2 − 4 σ(1 (cid:181))+(1 σ)(cid:181) − − 3 − − − n h ih io ¡ ¢ ¡ ¢ ¡ ¢ < 3 σ(1 σ) 1 1 2(1 σ)2(cid:181) σ2(1 (cid:181)) . 2 − 4 σ(1 (cid:181))+(1 σ)(cid:181) − − − − − n h ih io ¡ ¢ ¡ ¢ Since 3 σ(1 σ) < 1, this expression is strictly smaller than the corresponding 2 − expres¡sio¢n with only two (cid:222)rms. Finally, it is easy to con(cid:222)rm the economic intuition that the third and last term of the net bene(cid:222)t with four (cid:222)rms is strictly negative 1 Pr(γγγρρ)Pr(Gγγγρ) Pr(ρρρρ ρ)Pr(R ρρρρ) 8 { | | − | | } ¡ = ¢ 1 8 σ(1 − σ σ ) ( [ 1 σ2(cid:181) (cid:181) + )+ (1 (1 − σ σ )2 )(cid:181) (1 − (cid:181))] σ2(cid:181)+(1 σ2 σ (cid:181) )2(1 (cid:181)) − (1 σ − (1 σ)4 (cid:181) (cid:181) ) + +( σ 1 4(1 σ − )(cid:181) (cid:181)) σ4(1 σ (cid:181) 4( ) 1 + − (1 (cid:181)) σ)4(cid:181) ‰• − − ‚ − − − − − − (cid:190) h i h ih i ¡ ¢ = 1 1 σ3(1 σ)(cid:181) σ4(1 (cid:181)) 8 σ(1 (cid:181))+(1 σ)(cid:181) − − − − − h i = ¡ 1 ¢ σ3 £ ((cid:181) σ)<1. ⁄ 8 σ(1 (cid:181))+(1 σ)(cid:181) − − − h i ¡ ¢ Therefore, the ρ-type (cid:222)rm(cid:146)s net bene(cid:222)t from unilateral deviation from the efficient outcome is strictly greater with two (cid:222)rms than with four (cid:222)rms. 32

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