Consumers Bayesian Learning and Misleading Advertising

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1 Consumers Bayesian Learning and Misleading Advertising VERY PRELIMINARY DRAFT Giuseppe Pignataro University of Bologna Alessandro Tampieri University of Florence Abstract This paper studies the effect of firms optimal advertising on consumers expected utility. Firms may induce consumers into a bad purchase through comparative deceptive advertising. The contribution of the paper is the characterization of a learning process that each consumer realizes in higher order beliefs. Information can be private or endogenously public and might be inferred from observing market. We show that the importance of higher order beliefs turns upon the type of information which agents face. The welfare effect of increasing the precision of public and private information may decrease the expected utility of each consumer. Different welfare analysis should be developed in the second part of the paper. Keywords: Bayesian Learning, Advertising, Higher Order beliefs, Information Externalities JEL codes: D8; D83; D6; L13; L15; M37. addresses: giuseppe.pignataro@unibo.it Giuseppe Pignataro, tamp79@gmail.com Alessandro Tampieri

2 1. Introduction Deceptive advertising is a typical business practice that violates the trust of consumers overstating the quality of the related product. As buyers cannot spot the value of the good as is often the case in the real society, then there is a clear incentive of the companies to offer a product touting the virtues of their characteristics. Indeed many consumers, if not impulse buying, confuse brand image with the quality of the product, causing them to place their interest in a product based on hype. Recent cases include Dannon sanctioned by Competition Authorities all over the world after it misled consumers about the health benefits of its Activia yoghurt products or Skechers which falsely stating that its toning shoes helped with weight loss or Kellogg s which wrongly claiming that its breakfast cereals helped immune systems of children. Such cases are just some in a long line of misleading behaviours adopted by companies. Information gaps are costly, as whoever has the least information can never be confident about what is being traded. But now customers can review products on shopping websites, talk to each other through social media and consult reviews websites like cnet.com and TripAdvisor. They can spend time and effort in acquiring information and verify the advertised claims on product quality. This is the reason to combine in the same framework the interaction effect between firms and consumers by evaluating firms advertisement choice and consumers effort under endogenous signals. Formally, the purpose of this paper is to present a tractable model of strategic competition in an oligopoly context with consumers Bayesian learning process. In particular, in the spirit of Vives 011 and Vives 017, we develop a setting in which consumers learn about the quality of the product by correcting the effect of firms advertising. Firms optimise their strategy on prices, advertising and quantities. Their profits depend upon demand conditions and the output choices of their competitors. If the quality is uncertain for consumers, then any information available to the consumers is used to form their beliefs about those conditions. Besides, such information can and should be used to evaluate the likely information available to others, and so to create higher order beliefs according to the efforts of the other consumers The model develops some intersting features in the next sections. Consumers utility is quadratic in the profile of differentiated products consumed, and so prices are linearly related to outputs and a demand shock. Consumers learn about that shock via a set of exogenous private and endogenous public normally distributed signals. A signal is characterised by its precision and by the conditional correlation of its realizations across the consumers. The novel aspect of the model is that the endogenous public signal sent by each firm is a function of its optimal process of advertising.

3 We study the effect of an endogenous public signal of misleading advertising. In particular, we observe how firms may influence consumers purchase and how they react differently to a private learning process. Moreover, the additional effect takes into account a process of higher-order beliefs. For instance, the increasing prevalence of online platforms and reviews sites and blogs emphasises the role of aggregating information across consumers with shared opinions. This implies that the effort of each consumer to learn about the quality of the product easily depends also on the learning effort of the other consumers in the market. Consumers reason in higher orders and consequently react to higher order beliefs they form. This becomes especially apparent in how consumers react to private and public signals, and it is true even if the precision of the private signal is larger than the public one, implicitly suggesting the importance of the interaction between private and public information for welfare analysis. Interestingly, the characterisation of linear equilibria involves a dependence of private and endogenous public signal for each firm with a noise correlation across consumers. Thus, some questions emerge on how consumers may use different sources of information and interact between them, the welfare properties of their interactions by taking into account the information externalities using a team solution see Radner 196 and Vives 008. The combination of the Bayesian equilibrium and the efficient solution determine a measure of welfare loss with some useful comparative statics. This influences the social value of information under the endogenous public signal as in Bayona 018. The remainder of the paper is organized as follows. Section reviews some contributions in the two strand of literature, learning and advertising. Section 3 introduces the model of competition between firms while Section 4 discusses the optimal firms strategy. Section 5 investigates consumers learning process and in section 6 we present the efficiency strategy with further evaluation and comparative statics. This section and the next welfare evaluations are still in progress. Conclusion follows in section 7.. Literature The paper is related to the literature that examines the relationship between imperfect information, learning and advertising. To the best of our knowledge, there is no contribution in the literature that tries to join a Bayesian learning process of the consumers with an optimal advertising choice of the firms under endogenous signals. First, our analysis refers to the strand of the literature on social value of information firstly popularized by Morris and Shin 00 and afterwards analyzed by several contributions, see among others Colombo et al. 014 and Myatt and Wallace 014, 015. The game has 3

4 a similar structure to the one proposed in the literature concerned with information sharing, see e.g. Vives 1993, Angeletos and Pavan 004, 007, where the Gaussian-quadratic model and the linear solutions are common hypothesis. According to the timing of our analysis, our utility function is derived from the optimal choice of the firm; thus we determine a coordination game into three components based on consumers effort and the quality of the product. The combination between private and public information in coordinating settings was investigated by Hellwig and Veldkamp 009 who shows that adopting strategic complementarities in actions may incentivise the role played by higher-order beliefs, for instance in the peer review process. Such equilibrium strategy à la Morris and Shin was recently proposed and enriched in many settings with asymmetric information including financial markets Allen et al., 006, business cycle models Angeletos and La O, 009, oligopolistic competition Myatt and Wallace, 015. The literature slightly distinguished between public and private information by looking at two signals. For the sake of simplicity, we intentionally make the same choice to build our structural model. Of course, multiple information sources may enrich the design and constitute a fertile ground for follow-up research. The literature slightly distinguished between public and private information by looking at two signals. For the sake of simplicity, we intentionally make the same choice to build our model. However, our analysis is based on the interaction between firms and consumers which explains the necessity to endogenise the public signal. Alternative studies that theoretically verified the effects of multiple information sources are among others, Angeletos and Pavan 004, 007, who consider an investment game evaluating welfare as aggregation of agents outcomes. 1 Our paper is conceptually related to Myatt and Wallace 01 where endogenous information appears through the relation between personal attention and the precision of the signal. In particular, agents have access to different information sources, and the importance of each source depends on both the accuracy of the source, i.e. sender noise, and the individual attention devoted to the information source, i.e. receiver noise. Our analysis differs however from theirs in some respects. First, we focus on the interaction between firms and consumers. Second, in our setting, the information acquisition arises endogenously by a linear strategy solution with exogenous private signal à la Burguet and Vives 000 and an endogenous public signal à la Bayona 018 and Vives 017, see Vives 008. The result we obtain involves a symmetric Bayesian equilibrium of learning effort as a function of public and private information in addition to the prior value. The second strand of literature in our case involves the recent literature on false adver- 1 A similar information structure is observed in a Lucas-Phelps island setting in Myatt and Wallace

5 tising. The closest paper to our contribution is the one proposed by Hattori and Higashida 01. They analyse endogenous misleading advertising in duopolistic markets with horizontal product differentiation. Unlike this paper, there are spillovers in firms advertising, so that what a consumer perceives from a firm s advertising is also based on the competitor s advertising. They find that firms engage in socially inefficient deceptive advertising. Nonetheless, the optimal amount of deceptive advertising is positive. The reason is that deceptive advertising mitigates the problem of underprovision of goods from the oligopolistic competition. In the same field, Barigozzi et al. 009 examine deceptive advertising in an entry game, where an entrant is not fully aware of its quality compared to the incumbent. Deceptive advertising occurs if the entrant claims that its quality is the same as the incumbent, and it is not true. The incumbent can verify the entrant s claim through the lawsuit. Rhodes and Wilson 018 analyse false advertising in an industry where the monopolist is privately aware of its product quality. It chooses price and advertises its product quality. On the other hand, the regulator chooses the fine level for deceptive advertising. Consumers beliefs on the product quality are then based on the firm and the regulator strategy. They determine the conditions under which the regulator policy optimally allows for a positive degree of deceptive advertising. Piccolo et al. 015 examine deceptive advertising in a quality-differentiated duopoly where consumers are not able to recognize quality. Firms may engage in advertising, which would be truthful in the case of the high-quality firm, and deceptive and costly, in terms of a regulatory fine in the case of the low quality firm. They show that consumers may be better off when the low-quality firm engages in deceptive advertising. The consumer s gain is due to the fact that, with deceptive advertising, the two firm are perceived are of same quality, so that market competition will be fiercer. The increase in consumer surplus driven by competition could be stronger than the fall in welfare of the deceived consumers. In our model, we consider an oligopoly rather than a duopoly, with symmetric firms. Also, consumers may exert learning effort to detect deceptive advertising. Compared to Piccolo et al. 015, Piccolo et al. 017 investigate deceptive advertising with credence goods: buyers cannot observe quality, even after consumption. Moreover, advertising is a continuous rather than a binary variable. Unlike their previous work, where deceptive advertising is desirable for the buyer only at the minimum price that sustains the pooling deceptive equilibrium, in this paper a deceptive equilibrium can be sustained also when consumers choose the maximal pooling price. Vives 017 show how the weight of private information may increase in principle being too much as prices are the main providers of endogenous public information. 5

6 Appropriate benchmark to measure the inefficiency of the market equilibria is the team soluion where agents The model Consider an industry for differentiated goods with M symmetric firms compete by choosing the price of their products and the level of advertising and N consumers with imperfect information about the quality of the m goods. Each consumer forms her demand based on quality expectation. Advertising is deceptive as it improves only the perceived quality of a good, without truly affecting the pleasure of consumption. As a consequence, advertising increases the demand and in turn prices of a good, but not an individual true utility, nor consumer surplus Hattori and Higashida 01. However, consumers can exert learning effort under a Bayesian process to infer extra information on a good s quality Expected utility Formally, each consumer i N = {1,..., n} has gross expected utility E[U i x;θ; ζ i ] = υ ij x j + γx x j ζi 1 where x x j j M R M and X = m k=1,k j x k. As in Häckner 000, utility is quadratic in the consumption of j-goods, j M {1,..., m}. The utility cost of effort is quadratic in learning effort, ζ i. Parameter γ [ 1, 1] indexes the extent of product differentiation of goods j M. It represents the degree of substitutability between products j, with γ = 1, γ = 0 and γ = 1 implying that goods are complement, independent and substitutes, respectively. The demand shifter υ ij identifies the willingness to pay of consumer i for each product j and it is equal to υ ij = θ+a j ζ i. The advertisement of the firm is a j, while the quality of each product θ is a random parameter that influences the market s demand conditions. Consumers are uncertain about the quality θ of the products and share a common prior, i.e., θ N θ, σ θ. Each consumer exerts effort ζ i to learn the quality of the product and overcome the impact that advertisement has on consumer s demand. For the sake of simplicity, we assume that exerting learning effort ζ i unravels information about the quality of all products in the industry. This is reasonable in particular if we assume a certain degree of substitutability among goods. Finally, since learning defies the effect ot false advertising, the amount of learning can at most offset the advertising level, i.e., 6

7 Assumption 1. ζ i a j. It is worthwhile to note that our framework adopts a discrete number of consumers, instead of the standard continuum of consumers, see e.g., Vives 011 for discrete firms. The contributions relying on this assumptions are, to the best of our knowledge, relatively scant, see Graba 1995 who employs a discrete number of consumers to highlight externalities among competing consumers in a model with excess demand. 3.. Demand Utility is also linear in the consumption of the composite good I, chosen as the numéraire. Consumers maximise their expected utility subject to the budget constraint, I + m p jx j R, where the price of the numéraire is normalised to one and R denotes income. Consumer s utility is expected as the consumer does not see the true quality of each good. Solving the budget constraint for the numéraire, and substituting the expression into eq. 1, each consumer i maximises max x 1,...,x m E[U i x;θ; ζ i ] + I from which we can determine good j s individual demand: p j x j, x ij = θ ζ i 1 γ p j a j [1 + γm ] + γ m 1 j p k a k. 1 γ [γm 1 + 1] Hence, j s total demand is the sum of N consumers demands: x j = n x ij = i=1 n θ ζ i 1 γ p j a j [1 + γm ] + γ m 1 j p k a k. 3 1 γ [γm 1 + 1] i=1 We call ζ = n i=1 ζ i /n, the average learning in the industry. Demand 3 can thus be rewritten as: θ ζ 1 γ pj a j [1 + γm ] + γ m 1 j p k a k x j = n. 4 1 γ [γm 1 + 1] Eq. 4 shows that there is a clear relationship between the quality of the product and the amount of product requested in the market. As explained above, consumers aggregate level of effort ζ helps in correcting the imperfect information of the market by reducing the effect of perceived quality. 7

8 3.3. Supply Consider a homogenous quality θ of the product for each firm j M {1,..., m}. The profit of each firm j is, π j x j = p j x j C a j 5 Firms simultaneously choose the optimal price p j and the level of advertising a j, with zero production costs, while expenditure in deceptive advertising entails a convex cost technology, C a j = λa j, 6 with λ > 0. Unlike classic papers such as Butters 1977 and Grossman and Shapiro 1984, in this model advertising reaches all consumers, so that cost convexity does not represent a penetration cost. The interpretation here is that it becomes increasingly costly to add up further advertising to each consumers, since there is a rising cost of consumer attention Teixeira Welfare Social welfare is determined by the sum of consumers true utilities and total industry profit. As stressed above, the true utility is not affected by the level of advertising a j, or the counter-effect of learning which in this case is zero, i.e., ζ i = 0. Hence, the demand shifter of the true utility contains only a good s quality, υ j = θ. Thus the true utility follows as: U i x;θ; 0 = θ x j + γx, 7 under the budget constraint I + m p jx j R. Total industry profit is instead, Π = p j x j C a j 8 where p j = θ + a j x j γx. Hence a general expression for social welfare under perfect information is given by: Wx;θ; 0 = n U i x;θ + Π 9 i=1 8

9 3.5. Information structure Before uncertainty is realized, all consumers face the same prospects. Each consumer i N receives a private signal about θ by each firm j M, s ij = θ+ɛ ij where ɛ ij N 0, σ ɛ, plus an endogenous public signal of each product j, z j = a j θ + ω j where ω j N 0, σ ω. where a j θ is the optimal level of advertisement simultaneously chosen by firms. The precision of a generic random variable y with a non-zero variance is given by τ y = σy. Conditional on θ, cov[ω j, ɛ ij ] = 0, while cov[θ, ɛ ij ] = 0 for i N. The correlation coefficient between signals s ij and s lj, i l N, is ρ j = σ ω/σ ω + σ ɛ with cov[ɛ ij, ɛ lj ] = ρ j σ ɛ, where ɛ ij are identical distributed with common precision τ ɛ, i N and j M. We obtain E[θ/s ij ] = ξs ij +1 ξ θ where ξ = τɛ τ ɛ+τ θ and E[s ij /s lj ] = E[θ/s lj ] = ξρ j s lj + 1 ξρ j θ. From the properties of the Normal distribution, the public posteriors y j = E [θ z j ] and E [θ s ij, z j ] should be computed as a function of the optimal choice of firms advertising. We postpone all these computations in Sections 5 and 6, see in particular the proofs of Propositions 1 and. The aggregate level of information for the product j M is equal to s j τ ɛ τ θ sj s ij, which is normally distributed as s j N θ, σ θ n 1ρ j σɛ and cov[ sj, s ij ] = var[ s j ] = σ θ n 1ρ j σɛ with precision τθ sj = τ θ +1+n 1ρ j τ ɛ and τ ɛ = 1+n 1ρ j τ ɛ. The precision-weighted signal average is thus a sufficient statistic for the signal s 1,..., s n so E[θ/ s j ] = θ + τ ɛ τ θ sj 1 n n s ij θ. i= Equilibrium Definition We restrict the attention to symmetric linear Bayesian equilibrium where each consumer i N chooses the level of effort ζ i to maximize E[U i x;sj ; ζ i ; ζ s j, a j ] conditional on the information set { s j, a j } for each product j M. Consumer i s strategy ζ i s j, a j, i N, is a mapping from the signal space to the effort space for each of them with some correlations accross signals. Given the optimal symmetric strategy of each firm j M, a j and for given realizations of signals, s ij, i N and j M, the conjectured strategy must be self-fulfilling Although the use of normal distributions is standard in the literature of Bayesian learning, it determines a positive probability that prices and quantities are negative. The problem is solved by a proper choice of variances of the distributions and the parameters. 9

10 as a fixed point of conjectured strategies to actual strategies. This is the definition of Rational Expectations Equilibrium REE introduced by Vives 011 and Vives It follows that, Definition 1. An equilibrium is a strategy ζ i such that, for all s j, a j, θ, E[ζ i s ij, s j, a j] = arg max E[Ui ζ i ζi, ζ, θ ] 10 The linearity of the equilibrium solution is ensured by any strategy that satisfies eq. 10 and is linear in s j and a j Timing The timing of the game is as follows. At t = 0, quality θ and all errors terms are drawn. At time t = 1, each consumer i receives a private signal ɛ ij and a public signal a j θ by each firm j. Each consumer i chooses her learning level to maximise her expected utility 1. At time t =, each firm j symmetrically chooses the optimal level of advertising a j, and price p j. We thus take into account the possibility in the real terms that firms make their decision on advertising after the review process conducted by consumers Market equilibrium In the last stage, firms simultaneously choose prices and deceptive advertising with the aim to maximize profits. For each firm j, the first order conditions with respect to p j and a j are, respectively: π j = n a j p j [γm + 1] γ m 1 k=1 p k + a k 1 γ θ ζ = 0, p j 1 γγm π j = a j λ + np j [γm + 1] a j 1 γ [γm 1 + 1] = 0, 3 For a general discussion on the use of REE in market games we refer to Vives 008, Section 3. 4 As a robustness check, we may observe that inverting the timing and postponing consumers learning process does not change the results substantially for both consumers Bayesian updating and firms investments. 10

11 while the second order conditions hold for λ > λ, where λ n [γm + 1] 41 γ [γm 1 + 1] see AppendixA for further details. Differentiation of profits of firm j with respect to price of j and competitors advertising yields: suggesting the following preliminary result. π j nγ m 1 = p j a j 1 γ [1 + m 1 γ] < 0, Lemma 1. Price and deceptive advertising are strategic substitutes. Lemma 1 implies that an increase in advertising may have a competitive effect and lower equilibrium prices. The result that competition can be stimulated by deceptive advertising is in line with other contributions, such as Piccolo et al. 017, Piccolo et al. 015 and Hattori and Higashida 01. Invoking symmetry and rearranging, optimal prices and advertising for any given learning level are: p j = λ θ ζ 1 γ [γm 1 + 1] λ [γ m 4m γ3m 5 + ] n [γm + 1] 11 a j = n θ ζ [γm + 1] λ [γ m 4m γ3m 5 + ] n [γm + 1] Assumption 1 requires a i ζ, which holds if and only if 1 λ < λ θnγm + 1 ζ [γ m 4m γ3m 5 + ]. 13 it is easy to show that condition 13 also ensures positive equilibrium values of price and advertising. Given 11-1, demand and profits are: x j = λn θ ζ [γm + 1] λ [γm 3 + ] [γm 1 + 1] n [γm + 1], 14 11

12 π j = λn θ ζ [γm + 1] [ 41 γλγm n γm + 1] [λ γm 3 + γm n γm + 1]. 15 Notice that average learning ζ has always an opposite effect to quality in all the equilibria components. This is due to the fact that, to contrast advertising, it has a negative effect on the demand shifter. We turn now on the comparative statics properties of the market equilibrium. We investigate how the equilibrium components are affected by variations in quality, the good s degree of substitutability, the market size i.e., the number of consumers, average learning in the population, the number of firms and the cost parameter of advertising. The following corollary examines the effect of variations on the equilibrium price. Corollary 1. Let λ λ, λ. The equilibrium prices are increasing in quality, in the degree of substitutability and in market size; decreasing in the average learning, the number of firms and in the cost parameter of advertising. Proof. See AppendixB Corollary 1 shows that the effect of variations on price is standard. Consider next the comparative statics properties of advertising and demand. Corollary. Let λ λ, λ. Equilibrium advertising and demand are: increasing in quality and in market size, decreasing in the degree of substitutability if m 3, or if m = and γ < 1/, and increasing if m = and γ > 1/, decreasing in the average learning, the number of firms and in the cost parameter of advertising. 1

13 Proof. See AppendixC A high quality product, as well as a large market size, provides and incentive to firms in advertising. By contrast, the increase of competition, due to a high number of firms or a high degree of substitutability, leads to lower gains from advertising and in turn lower investments. In addition, a decrease in advertising induces a decrease in equilibrium demand. The same applies if the cost parameter of advertising increases. Finally consider the effects of a variation of market parameters on profits. For convenience, define nγm + 1 λ γ 1γm 3 + γm 1 + 1, λ The results are summarised in the following corollary. γm + 1 γm γ m + 1. Corollary 3. Let λ λ, λ. Equilibrium profits are: increasing in quality, and decreasing in the average learning, the degree of substitutability and the number of firms, increasing in the cost parameter of advertising for γ 1+m, or γ < 1+m and λ λ, λ, and decreasing for γ < 1+m λ, and λ λ. increasing in the market size for λ λ, λ and decreasing for Proof. See AppendixD λ, λ. The effects on profits of a variation in product quality, in the degree of substitutability or in the number of firms are standard: a higher quality means a higher demand and prices. An increase in γ or in m implies an increase in competition, and thus lower profits. Things are less obvious, for changes in the cost parameter of advertising or the market size. Corollaries 1 and, equilibrium price, advertising and demand all fall with an increase in λ. It follows that firm j s total revenue falls: the rise in λ reduces the good s quantity purchased by each consumer. Moreover, total costs are decreasing in λ: C a j λ = θ n [γm + 1] [ λ γ m 4m γ3m nγm + 1 ] [nγm + 1 λ γ m 4m γ3m 5 + ] 3 < 0, 13 By 16

14 Since the cost function is quadratic in a, the negative effect of λ on advertising is stronger than the positive, linear effect of the cost parameter: the increase in λ makes advertising too costly and induces a lower equilibrium level. This explains the ambiguous effects on profits. With harsh competition γ > 1+m, the effects on costs is stronger than the effect on revenues, so that profits increase with the increase in λ. The same applies if competition is soft γ < 1+m but λ is sufficiently low. The effects of a variation of n are similar. Again by Corollaries 1 and, equilibrium price, advertising and demand all increase with an increase in n. The market size has the standard positive relationship with revenues. On the other hand, advertising costs increase with n, even if it is a fixed cost. The effects on costs is stronger when λ is small, which can be still explained by C a j / λ < Consumer s learning process Substituting p i, a i and x i in eq. yields, E[U i ζi, ζ, θ ] = m[n ζ i ζγm λγm 1 + 1ζ iγ + nγm θnγm γ n ζ ζ i ] ζi 17 where = n 1 + γm 1 γm n + n λγm 3 + γm This results in eq. 17 can be collected according to the personal and average learning effort and state, i.e., ζ i, ζ and θ, as: E[U i ζi, ζ, θ ] = ϑζ i + ξθζ i + ψ ζ + ϱθ ζ + ηζ i ζ + ιθ, 18 where the payoff function is quadratic in the action profile ζ ζ i i N R N and quality θ R, and it is symmetric with respect to the permutations of other consumers. parameters α, β, δ, ɛ, η, ι are constants and dependent of λ, m, γ such that, ϑ = The m γm λmγm + 1 ξ = λγm 3 + γm nγm

15 ψ = mγ 1λγm γm n + n γm λγm 3 + γm γm n + n 1 λmγm + 1γ 1λγm γm n + n ϱ = λγm 3 + γm γm n + n η = γ 1λmγm mnγm + 1 γm 1 + 1λγm 3 + γm nγm + 1 ι = 3 λ mγm + 1 γm λγm 3 + γm γm n + n 4 Constants ϑ, ξ, and η are coefficients of ζ i and contribute to defining consumer i s best response. This game may exhibit strategic complementarity or substitutability as η 0. The variation is possible according to the evolution of the cost parameter λ. For instance, provided λ > 0, it follows that λ < λ γm +1 γm 1+1 γm+1 and η > 0 and ξ > 0. Constants ψ, ϱ, ι are coefficients of terms not including ζ i, and they do not directly influence on consumer i s best response. However, the effect of higher order beliefs plays a role in the interplay between individual effort ζ i and the quality of the product θ. Such game is therefore strategically equivalently to a quadratic-payoff coordination game. The equivalence is possible by defining, κ 1 = mγ 1λγm γm n + n γm λγm 3 + γm γm n + n 1 κ = λ mγm + 1 γm λγm 3 + γm γm n + n 1 and rearranging the equilibrium condition, each consumer i seeks to maximize, E[U i ζ i, ζ, θ] = κ 1 ζ i θ + κ ζ i ζ + 1 κ 1 κ ζ θ 5 Eq. 5 refers to a loss function which is reminiscent of beauty contest Bayesian coordination game à la Morris and Shin 00. The novelty of this analysis is at least in two respects. First, eq. 5 is endogenously derived by the optimal firm choice, see subsection 3.7 for the timing of the analysis. Second, such endogeneity allows us to show an additional term involving the inner distance between the average effort of consumer and the uncertain quality. We can thus identify three motives that contribute to the quadratic payoff. The first term is the well-known fundamental motive and represents the action made by each consumer 15

16 to understand the quality of the product. The second term is the well-known coordination motive and is common in the literature of higher order beliefs as consumer tries to learn as a result of social interactions with other people. It contributes to explain what happened to a typical purchase in the market where user reviews are proven to be sales drivers for the majority of customers before making their decision. The last term instead is an informative externality motive motivated by the presence of higher order beliefs among consumers. It adds to the loss of each consumer the problem of uncertain quality compared to the average effort of individuals. We can define such information sharing effect as a fundamental motive at the aggregate level. This implicitly suggests a trade-off in the weights assigned to the three motives. In particular, both κ 1 and κ are functions of λ, γ, m and n. All components are strategic substitute and there is no possibility of complementarity here. For instance, if the weight of the fundamental motive κ 1 increases, then the importance of third component, i.e., information externality motive reduces. Similar results can be obtained in case of κ. This means that in case κ 1 + κ = 1 so that the importance is simply divided between fundamental and coordination motives of consumer i, then the information externality motive disappears. However, we point out that the third term does not influence directly the effort of each consumer i. Consequently the first order condition can be written as: n E[ζ i s ij ; s j ; z j ; θ] = κ 1 E [θ s ij, z j ] + κ E [ζ l. s lj, z j ] 6 l=1 Using properties of Normal distributions, we can compute conditional expectations and solve for the linear equilibrium by identifying coefficient with a conjecture on a candidate linear strategy as shown below. The following proposition characterises the linear equilibrium and the following corollaries its main properties: Proposition 1. There exists a unique symmetric linear Baeysian equilibrium of eq. 6 given by: ζ i s ij, z j = α + βs ij + δz j j M 7 16

17 where α = κ 1 θτ ɛ τ θ τ ωτ τ ɛ τ θ Λ3 n 1ρ j τ ɛ n1 κ τ ɛ τ θ τ ωτ ; 8 β = κ 1τ θ τ ωτ ɛ n 1ρ j + 1n 1ρ j + 1 κ τ ɛ n 1ρ j τ θ + τ ω τ ; δ = κ 1 Λτ θ τ ɛ n1 κ τ ; with Λm; n; γ; λ = n [γm + 1] λ [γ m 4m γ3m 5 + ] n [γm + 1] Λ Λ = 1 + Λn 9 τ = Λ + 1n 1ρ j Λτ ɛ 30 n 1ρ j + τ θ τ ɛ + τ ɛ τ θ τ ω Λ nρj + ρ j 1 + τ ɛ + Λτ ɛ τ ω τ = n 1ρ j + τ θ τ ɛ + τ ɛ Λ nρ j + ρ j + τ ω τ ɛ 1 + τ θ τ ɛ τ ω τ ɛ n 1ρ j Λ Λτ ɛ + Λ + 1 Proof. See AppendixE The results about α, β, δ in eq. 8 are extremely interesting. They provide the unique symmetric equilibrium with different characteristics. First, there cannot be asymmetric equilibrium as long as we restrict our attention to equilibria with bounded means and uniformly bounded variances accross consumers. The procedure implicitly involves an optimization process of a strictly concave function which delivers a unique solution. The optimal level of effort of each consumer i, ζ i s ij, z j, is thus a function of private and endogenous public signal. Endogenous public signal modifies the precision of the public signal compared to the traditional exogenous case and as agents respond more to private information, the more informative the public signal becomes. The public signal that consumers receive follows 17

18 the information structure z j = Λ θ ζ + ω j where the precisions τ θ, τ ω and τ ɛ influence the size of Bayesian weights of both private and public information, β and δ. According to Proposition 1, it is thus possible to capture the sensitivity of the equilibrium to private and public information. A complete snapshot of the relations results in comparative statics. Due to the cumbersome computations related to the variation of the precisions, some simplifying assumptions are necessary for the tractability of the problem. In particular, the following corollaries summarize the effect of the precisions τ θ, τ ω and τ ɛ and the correlation ρ j to Bayesian weights α, β, δ where m =, n =, ρ j = 1, γ = 1 and λ = 1. Corollary 4 thus describes the effect of variational precisions in a duopoly market with substitutes goods and unit cost of advertising and two consumers with perfect correlation of their signals. Corollary 4. Under simplifying assumptions on m, n, ρ j, γ and λ, the effects of variational precisions τ θ, τ ɛ, τ ω and the correlation ρ j on equilibrium parameters strategy α, β, δ are α τ θ < 0; β τ θ < 0; δ τ θ < 0; α > 0; τ ɛ β > 0; τ ɛ δ > 0; τ ɛ α < 0; τ ω β < 0; τ ω δ < 0; τ ω α < 0; ρ j β < 0; ρ j δ < 0; ρ j Proof. See AppendixF Interestingly, we can observe that the effect of variational precisions of both private, public and prior information have the same general sign, independently of the Bayesian weights. In particular, we observe that the higher precision of the prior τ θ reduces the weights assigned to all parameters α, β, δ. A similar effect of the precision of the public precision τ ω is obtained. Note that this noise term captures the exogenous impact of the public term independently by the endogenous effect of advertising. We can confirm instead that the higher the precision of private signal τ ɛ, the larger the values of all Bayesian weights. If instead we allow for the variation of the correlation ρ j, then we realize a negative impact in all terms. Similar results can be obtained by looking at the expected utility function in eq. 5. Results are proposed in the following corollary: 18

19 Corollary 5. The effect of variational precisions τ θ, τ ɛ, τ ω on consumer s expected utility U i ζ i, ζ, θ is: E[U i ] τ θ < 0; E[U i] τ ɛ > 0; E[U i] τ ω 0; Proof. See AppendixG Under the same simplifications above, Corollary 5 shows that the variational precisions have a clear sign in case of both prior and private information while the effect of public noise is ambiguous. This means that the higher the precision of prior and private signals, the larger the expected utility of consumer i. Intuitively, the result is justified by the fact that a higher precision implicitly requires a lower level of effort to compensate the effect of firms advertising. The impact is not evident in case of the public signal, endogenously taken. It can imply an overreaction or underreaction of the consumer and may involve an increase or decrease of her utility, the higher is the precision of the public signal. Corollary 6. Stronger substitutability between the fundamental and strategic components increases the dispersion of the learning distribution as V arζ i ζ κ 1 > 0 and V arζ i ζ κ > 0. Moreover, the larger the weight assigned to private information, the larger the dispersion as V arζ i ζ β > 0. Proof. See AppendixH Corollary 6 first shows that the relative weights κ 1 and κ of the fundamental and strategic components positively influence the effort distribution and increase the dispersion across the population such that the difference between individual effort and average one rises. Still, the higher the weight β assigned to the private signal, the larger the dispersion across the population. Interestingly the weights α and δ play no role as they have the same effect at the individual and aggregate level. See AppendixH for further details. 6. Efficiency equilibrium and welfare analysis As usual in the literature, welfare analysis requires a definition of the appropriate benchmark for this setting. Indeed the optimal linear strategy in case of perfect symmetric information suggests that the optimal level of effort is zero, i.e., ζ i = 0. Intuitively, this 19

20 is motivated by the fact that the level of effort is positive to counterbalance the effect of deceptive advertising and play no role when the uncertainty is solved and the quality is perfect knowledge. In this case the utility function of a consumer i as shown in eq. 7 is U i x;θ; 0 = m θ x j+γx x j, while E[U i x;s j ; ζ i ] = m υ ij x j+γx x j ζi identifies the expected utility as defined by eq. 1 where the quality is not observable and the willingness to pay υ ij is influenced by firm j s level of advertising and consumer i s effort. Then the social welfare of first-best full information allocation, W 0 x;θ; 0, follows as: 5 W 0 x;θ; 0 = mu i x;θ; 0 + a j x j C a j 31 Alternatively, the social welfare in case of imperfect information on quality, advertising and learning effort of consumers, i.e., υ ij = θ + a j ζ i, is: E [W x;υ ij ; ζ i ] = E[U i x;υ ij ; ζ i ] + ζ i C a j + a j x j 3 Using the expressions of eqs. 31 and 3, we observe that W 0 x;θ; 0 > E [W x;υ ij ; ζ i ], and in particular, E[W x;υ ij ; ζ i ] = W 0 x;θ; 0 WL 33 where WL is the welfare loss by looking at n = m 1-replica market, WL = m a j ζ i x j + ζ i ζ i 34 See AppendixI for a formal derivation of social welfare functions and welfare loss of eqs. 31, 3 and 34. The first term in eq. 34 corresponds to the aggregate ineffiency in the production of each firm due to the difference in the demand shifter where the effort requirement ζ i is proportional to a j and measures how distorted is the distribution of production due to imperfect information of quality θ. The second term captures instead the distributive inefficiency in consumers learning due to a convex cost of information and measures how imperfect information is costly in terms of the required effort. The advantage of the welfare representation of eq. 34 is that both inefficiencies in prodution and learning caused by quality uncertainty can be directly related to the social value of information and the precisions 5 The procedure conceptually follows Lemma 1 in Vives 00 by looking at a linear quadratic model and n = m 1 replica market. It is no necessary to take a Taylor series expansion in our case. 0

21 of the signals under Bayesian updating. According to Radner 196, Vives 1988, Vives 00 and Angeletos and Pavan 007, the first best case of full information allocation is considered to be too stringent for the comparison of market outcomes. The welfare benchmark tradionally used in the literature is the team solution strategy. An allocation is defined as team-efficient if it maximises expected social welfare subject to the use of linear decentralised strategies. This implies that agents use strategies which are measurable in their information. Information cannot be transferred from one individual to another and each agent would set his optimal choice conditioning only on her available signals. In our model, the novel aspect is that the effort strategy of each consumer is contingent only on her private information and not on the information of other agents. Consumers do not perceive to affect the aggregate information among individuals. Their strategies still depend on the endogenous public signal of advertising as in Proposition 1, but without the effect of higher order beliefs in their optimization process. The following proposition states the result for general efficiency under decentralized strategies. Proposition. The efficient linear strategy under asymmetric information is given by: ζ i s ij, z j = α + β s ij + δ z j j M 35 where α = 1 µ θκ 1 1 ϑ + ϑκ ; 36 1 κ 1 ϑκ β κ 1 ϑ = 1 ϑκ ; δ = µ1 ϑκ 1 1 κ 1 ϑκ ; where µ = Λτ ω /τλ and τλ = τ θ + Λ τ ω and ϑ = τ ɛ /τ ɛ + τλ. Proof. See AppendixJ It is possible to observe that the efficient strategy is of the same form as in the market solution in Proposition 1, by now differently from the previous Bayesian equilibrium, the 1

22 weights of the symmetric decentralized strategy α, β and δ are conditioned on private and public endogenous information, without evaluating the average effort of other consumers. At a candidate team decentralized strategy, therefore, we obtain that the inefficiency in production and learning can be defined as m m a j ζi x j = θλλm α + θβ + δ 1Λ and m ζ i ζi = mα + θβ + δ Λ 1α + θβ + δ Λ and from the expression of welfare loss in eq. 34, we have that: WLα ; α ; δ = mα + θα + δ Λ 1α + θβ + δ Λ 37 θλλmα + θβ + δ 1Λ Instead in case of market solution, the welfare loss WLα; β; δ has identical structure of eq. 37 with α, β and δ of Proposition 1. Comparing the welfare loss in both cases, it is easily to obtain, WLα; β; δ > WL α ; β ; δ 38 Thus the team solution minimizes WL. over weights α; β; δ, yielding that, α T = 1 + θβ + Λδ λm + 1 4θλΛmθΛλm β T = α + θλδ λm + 4θλΛmθΛλm θ δ T = α + βθ θλλm + 4θλΛmθΛλm θλ The following proposition states the result and provides an explicit expression of the team solution. Proposition 3. The team solution strategy under Bayesian updating is the most efficient strategy that minimizes the welfare loss due to asymmetric information, min WL. α;β;δ

23 such that α T β T ; δ T = α 39 β T δ T = β δ T β T = δ Proof. See AppendixK This yields a strictly convex welfare loss as a function of the weights of the decentralized strategy. The team solution strategy minimizes WL and optimally tradeoff both sources of inefficiency in production and learning. This implies that when consumers capture endogenously a public signal on firms advertising by taking into account also the effect that the effort of others may obtain, then the effect of higher order beliefs that The team solution terms. As in the previous section, we can capture the effect of variational precision on the new Remark 1. The effects of variational precisions τ θ, τ ɛ, τ ω on equilibrium parameters strategy α, β, δ are α τ θ 0; β τ θ < 0; δ τ θ < 0; α 0; τ ɛ β > 0; τ ɛ δ < 0; τ ɛ α 0; τ ω β < 0; τ ω δ > 0; τ ω Proof. See AppendixJ By comparing Proposition 1 and Proposition and the related Corollaries, we observe how the precision of prior, private and public information have partially a different sign. In 3

24 particular, we observe that while the precision of the private signal τ ɛ positively influences all Bayesian weights in case of private information, this is not true for the team solution as We show that the sign is ambiguous for the weight α, while it is positive and negative, respectively for β and δ. IN PROGRESS to be continued 7. Concluding remarks In the first part of our analysis, we show how firms advertising through their optimisation process influences consumers learning process. The analysis consists of a Bayesian update under an endogenous public signal across consumers. This is motivated by the general need to explain the effect that misleading advertising may induce on consumers expected utility as the quality of the product is unknown. We consider an additional effect typically induced by online review process regarding higher order beliefs. We observe the effect of evaluating market news on consumers behaviour. The endogenous signal even provides information not only about the quality of the product but also on the average effort of consumers in terms of information externalities. We thus observe whether higher-order beliefs and consequently public information empirically have a crucial role when the optimisation process relies on both individual evaluation and coordination across consumers. Our model was simple in concept. Formally, we have assumed that each has a quadratic-payoff function endogenous derived. A unique symmetric Bayesian equilibrium is obtained and compared to the decentralised strategy team solution. In the second part still, work in progress we want to make a welfare analysis showing how the effect of advertising may overturn the sign of the social value of private information and how this may change the related equilibria. 4

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28 AppendixA. Second order conditions The Hessian matrix is: H = n[1+γm ] 1 γ[m 1γ+1] n[m γ+1] 1 γ[m 1γ+1] n[m γ+1] 1 γ[m 1γ+1] λ, whose first element is negative, and its determinant is det H = n [γm + 1] [4λ1 γγm n 1 + γm ] 1 γ [γm 1 + 1] > 0, for λ > λ, where λ Hence the solution is a maximum for λ > λ. n [γm + 1] 41 γ [γm 1 + 1]. AppendixB. Proof of Corollary 1 Let λ λ, λ. Differentiation of p i, with respect to θ, ζ, γ, n, m and λ yields, respectively: p j θ p i ζ p j γ = = = λ1 γγm λ [γ m 4m γ3m 5 + ] n [γm + 1] > 0, γ 1λγm λ γ m 4m γ3m 5 + nγm + 1 < 0, θ ζ λm 1 [ γnγm + λγm ] [nγm + 1 λ γ m 4m γ3m 5 + ] < 0, p j θ ζ λ 1 γ [γm + 1] [γm 1 + 1] = n [nγm + 1 λ γ m 4m γ3m 5 + ] > 0, p [ j m = γ θ ζ λ1 γ λγm γn ] [nγm + 1 λ γ m 4m γ3m 5 + ] < 0, p i λ θ ζ n 1 γ [γm + 1] [γm 1 + 1] = [nγm + 1 λ γ m 4m γ3m 5 + ] < 0. AppendixC. Proof of Corollary Differentiation of a i and x i with respect to θ yield, respectively: a j θ x j θ nγm + 1 = nγm + 1 λ γ m 4m γ3m 5 + > 0, λnγm + 1 = λγm 3 + γm γm n + n > 0, 8