Informative advertising under duopoly

Transcription

1 Informatie adertising under duopoly Scott McCracken June 6, 2011 Abstract We consider a two-stage duopoly model of costless adertising: in the first stage each firm simultaneously chooses the accuracy of signals informing consumers about how much they alue its product; in the second stage firms compete in prices. We find that when the distributions of consumers aluations for each product are identical, independent and symmetric, and the market is coered for any combination of first-stage choices, the subgame-perfect equilibrium inoles both firms perfectly informing consumers in order to increase ex post differentiation and reduce price competition. When the market is not coered for all first-stage choices, one firm may no longer choose perfectly accurate signals in equilibrium, as an increase in the informatieness of a signal further lowers the ex post aluation of those consumers indifferent between purchasing or not, causing them to drop out of the market. Numerical results indicate that if the symmetric distribution of aluations is sufficiently peaked, then both firms inform perfectly; otherwise one firm informs perfectly and the other chooses the highest leel of informatieness such that the market is coered. Thanks to Martin Richardson, José Rodrigues-Neto, Kieron Meagher, Hodaka Morita, and seminar participants at the Australasian Economic Theory Workshop All remaining errors are of course my own. Research School of Economics, College of Business and Economics, HW Arndt Building 25a, Australian National Uniersity, ACT 0200, Australia. scott.mccracken@anu.edu.au

2 1 Introduction Firms often hae some control oer the degree of information that consumers hae about their products, and choose to exercise this control through a ariety of methods such as adertising, product demonstration, trial ersions, and preiews. In an industry with more than one firm, one function of informing consumers is to differentiate one s product from that of rials. If an increase in the amount of information that consumers hae about a product results in consumers haing quite different expected aluations for that product compared to other products, then consumers become less price-sensitie and price competition is reduced. In this paper we consider the incenties of firms in a duopoly to proide information. Each firm has the ability to send costless signals through adertisements or otherwise which inform consumers about their aluation of the good the firm is selling. The informatieness of the signals can be interpreted as the information content of a firm s adertisements. Although firms may hae a complete knowledge of the horizontal attributes of their product, they do not know how any indiidual consumer alues these attributes. After obsering the adertising, consumers are able to form a posterior belief about how much the product is worth to them. The higher the information content, the more precise is this posterior aluation of the product. When no information is proided, information is symmetric, with neither firms nor consumers haing any priate information. In this case, all that consumers and firms know is the distribution of aluations for each good and so consumers alue both products identically at the mean. We consider a two-stage game between firms. In the first stage firms simultaneously choose the accuracy of the informatie signals which each consumer receies; in the second stage firms compete in prices. The aluations for each good are distributed independently according to the same continuous distribution. 1 We isolate two effects of an increase in the informatieness of a firm s signals: the direct effect is the change in the firm s profits as a result 1 Allowing for correlation of aluations across goods would complicate the model while adding little. In section 5 we comment on the robustness of our results to correlation. 2

3 of the change in demand while keeping prices fixed; the strategic effect is the change in profits as a result of the shift in demand caused by the rial firm s adjustment to its new equilibrium price. In general, the balance of these effects will determine the equilibrium outcome of the game. We find that when the distributions of consumers aluations for each product are symmetric, and the market is coered for any combination of first-stage choices, the subgame-perfect equilibrium inoles both firms perfectly informing consumers in order to increase ex post differentiation and reduce price competition. The intuition is as follows. Gien the assumptions of a symmetric distribution of consumer aluations and coered market, the price equilibrium is symmetric for any first-stage choice of information proision. An increase in the informatieness of a firm s signal produces a meanpresering spread of the distribution of differences in posterior aluations. This has no direct effect on demand, with those consumers receiing a signal from that firm which is aboe the mean reising their aluations for that firm s good further upwards, while those with signals below the mean reise further downwards. The firm gains some consumers with high signals and loses some with low signals. Symmetry of the distribution implies that the mass of consumers gained is the same as the mass of consumers lost and that each firm shares the market equally in eery price equilibrium. There is howeer a strategic effect: The spread in the distribution causes a clockwise rotation of demand. The resulting decrease in the elasticity of demand at current equilibrium prices reduces the gain in market share from a cut in price, so that an increase in information raises equilibrium prices and profits. Welfare is maximized in the subgame-perfect equilibrium: All consumers hae aluations aboe marginal cost for each good and purchase in equilibrium; perfect information enables consumers to choose the good giing them the highest aluation net of price; equilibrium prices are identical, so that consumers purchase the good which gies them the highest aluation. For the case where the market is not coered in the price equilibrium of eery subgame, we are unable to sole the model analytically. Instead, we sole the model numerically for a number of distributions. We find that when the density is symmetric and sufficiently peaked, the subgame-perfect equilibrium 3

4 inoles both firms proiding perfectly informatie signals. For such distributions, there are few consumers with aluations at the lower end of the support. Also, since many consumers hae similar aluations for each good, price competition is igorous enough that equilibrium prices are low. Together these properties imply that the direct effect on demand of increased information is small relatie to the strategic effect, and so the subgame-perfect equilibrium inoles firms perfectly informing. For less peaked symmetric distributions, one firm perfectly informs while the other firm may proide an intermediate leel or no information. In section 2 we reiew the related literature. In sections 3 and 4 we present and sole the model, and then inestigate its welfare properties. In section 5 we consider the robustness of the results to changes in some of the model s assumptions. Section 6 concludes. All proofs are in the appendix. Details of the numerical methods are aailable upon request. 2 Related Literature A number of papers hae inestigated the monopoly incenties for information proision under the assumption that the monopoly has no priate information about its product. Lewis and Sappington (1994) explore the incenties for a monopoly to allow potential buyers to acquire priate information about their idiosyncratic aluations for the monopoly s product. In the model, consumers hae a taste parameter which may not be known to consumers. The monopoly is able to send to each consumer (or allow each consumer to acquire) a costless signal of their taste parameter, and to choose its informatieness. The signal is assumed to hae the following properties: with some gien probability the signal is informatie (i.e. higher signals are associated with higher taste parameters in the sense of first-order stochastic dominance); with complementary probability the signal is uninformatie (i.e. the signal is simply a draw from the same marginal distribution as the informatie signals). The monopoly s information choice is then simply a choice of the probability that the signal is informatie. As well as choosing the informatieness of the signals, the monopoly also specifies a menu of quantities 4

5 and corresponding payments. Once each consumer has obsered their priate signal they then choose their preferred consumption-payment pair. Although improed priate information for consumers allows the monopoly to charge higher prices to those with higher aluations, it also proides rents to these consumers. It is shown that profit is conex in the informatieness of the signal and so the trade off is such that it is optimal to proide either as much information as possible, or no information. Johnson and Myatt (2006) use a more general information structure than Lewis and Sappington to analyze the incentie to inform. Specifically, they look at a single-unit monopoly s preferences oer a family of distributions of consumer aluations ordered by dispersion. In particular, they define a family of distributions indexed by s and ordered by rotation. 2 Their definition orders, by clockwise rotation, cumulatie distribution functions that cross only once. The idea is similar to an ordering by second-order stochastic dominance, but the spread need not be mean presering. The main result is for a family of distributions indexed by s [s L, s H ] and ordered by a sequence of rotations. It says that if the point of rotation, θ s, is decreasing in s, then monopoly profits are quasi-conex in s and so maximized at an extreme s {s L, s H }. They apply this result to informatie adertising and product choice. One application of their paper is to a particular type of informatie signal which they call truth or noise. We use this type of signal in our paper. It is a special case of that used by Lewis and Sappington (1994): with some gien probability the signal is the consumer s true aluation; with complementary probability the signal is a random draw from the true distribution of consumer aluations. This produces a family of distributions of ex post consumer willingness to pay indexed by the accuracy of the signal (i.e. the probability that the signal is the consumer s true aluation). In this case a single-unit monopoly s profits satisfy a stronger condition: they are conex 2 For a distribution F s parameterized by s, with support [θ s, θ s ], a local change in s leads to a rotation of F s if, for some θ s (θ s, θ s ) θ θ s F s(θ) s 0. 5

6 in the accuracy of the signal. Saak, in a series of papers (2006a, 2006b, 2008, 2009), also examines the monopoly incenties for information proision. In (2006a) it is shown that in a single-unit monopoly (with risk neutral consumers) it is optimal for the monopoly to inform consumers only about whether their aluation is greater than marginal cost, and then price at the mean aluation of those consumers with aluations aboe marginal cost. This allows the monopolist to achiee the first-best outcome: only consumers with aluations aboe marginal cost purchase the good, and the monopolist extracts all the consumer surplus from these consumers, i.e. it is able to practise ex ante perfect price discrimination. The result shows that the monopolist may prefer buyers with an intermediate leel of information for sufficiently high production cost. This generalizes an example in Lewis and Sappington (1994) which showed that for sufficiently high marginal costs, perfect information was preferred to no information as it allows the monopoly to screen out consumers with aluations below marginal cost. In our model we assume zero marginal cost, so that the optimal leel of information proision for a monopoly is zero. Howeer, as soon as we introduce another firm, it is clear that this can no longer be optimal for both firms. If the firms are ex ante identical and neither firm proides any information, then Bertrand competition results in both firms earning zero profit. If one of the firms proides a small amount of information to consumers, and this leads to some consumers haing different willingness to pay across firms, then price competition is reduced and the Bertrand outcome is aoided. We explore the effects of a nonzero marginal cost in section 5. Moscarini and Ottaiani (2001) inestigate price competition in a model with an informed buyer and two symmetric sellers of different arieties of a good. Before purchasing, the consumer receies a priate binary signal about the relatie quality or match alue of each of the goods. In contrast to our model, the informatieness of the priate signals is not a strategic ariable, but a fixed constant. Comparatie statics are performed on the equilibrium outcome of the pricing game with respect to the consumer s prior belief and the precision of the priate information. We can directly compare the two models if we interpret our model as haing one consumer, and profits as 6

7 expected profits. Then the only other differences are that we allow for a continuous distribution of aluations for each good, no correlation in aluations across firms, an independent signal from each firm, and assume the consumer has a common prior belief of her aluation for each good equal to the mean of the distribution. The continuous distribution in the setup of our model allows us to aoid many of the technical complications (i.e. mixed strategies) introduced by the two point distribution in Moscarini and Ottaiani s model. This comes at the price of losing the ability to sole the model explicitly for asymmetric distributions. Whereas in our model, the firms products are ex ante identical, the prior belief in the Moscarini and Ottaiani model may faour one firm so that firms can be ex ante differentiated. Howeer, when the prior belief on the relatie quality is such that sellers are only weakly ex ante differentiated, and the priate signal is sufficiently precise (so that the sellers are strongly ex post differentiated), there is little competitie pressure on prices just as in our model. Damiano and Hao (2007) alter Moscarini and Ottaiani s model by allowing each firm s good to be of low or high alue (with equal probability) independently of the other firm. Furthermore, before the firms choose prices, each firm is able to send an independent binary signal and choose the quality of that signal (which is obsered by both firms). Because the assumption of a discrete distribution is maintained, the model retains the complication of mixed strategy price equilibria. Although this feature precludes a complete analysis of the model, two equilibria in signal informatieness are identified: one is a symmetric equilibrium with both firms informing completely; the other is an asymmetric equilibrium with one firm informing completely and the other firm proiding no information. In proing these results, they show that informing completely and proiding no information are both best responses to perfect reelation of information by the other firm. This result for a discrete uniform distribution is identical to the result for the continuous uniform distribution in our model when the market is coered. Howeer, we show that when we moe away from the uniform distribution to any other symmetric distribution, the unique best response to any choice of informatieness of the rial firm is to choose perfectly informatie signals. 7

8 Iano (2009), in a model ery similar to ours, inestigates the role the number of firms plays in the choice of how much to inform consumers. The two important departures from our model are that firms make pricing and information choices simultaneously, and distributions are restricted to hae support on the unit interal. These assumptions together lead to quite different results. Iano shows in his model that for any duopoly with symmetric distribution of aluations on the unit interal, it is not an equilibrium for both firms to perfectly inform consumers. Howeer, our numerical results indicate that, for our model, it is an equilibrium for both firms to fully reeal consumers aluations for sufficiently peaked symmetric beta distributions. The reason for the contrasting results is the difference in timing. Suppose that firms A and B hae chosen to perfectly inform consumers. First consider the case where prices and signal accuracy are chosen simultaneously. The appropriate equilibrium concept is that of Nash equilibrium. Gien the choices of informatieness, the only candidate for a pure-strategy Nash equilibrium inoles both firms choosing the same positie price and obtaining half the total demand. Since firms choose positie prices, total demand must be less than unity. But then firm B, say, would be strictly better off deiating and choosing to send no signals, as then total demand would increase to unity and firm B s demand would increase to half. 3 Now consider the model in which signal accuracies are chosen in the first stage (and obsered by both firms) and then prices are chosen in the second stage. In this case we look for a subgame-perfect equilibrium. Firms choose Nash equilibrium prices for any gien first-stage choices of informatieness. When both firms choose to perfectly inform in the first stage, the equilibrium prices are equal, and firms split demand equally. As in the first case, demand is less than unity. In contrast to the first case, a deiation by firm B to sending no signals will be accompanied by a change in the second-stage equilibrium prices. Although total demand will increase to unity, there will at the same time be a decrease 3 When firm B sends no signals, consumers hae a common aluation for firm B s good equal to the mean of the distribution. Thus, when firms A and B choose the same prices, those consumers who are informed that they alue good A aboe the mean purchase from A, while the remainder purchase from B. Since the distribution is symmetric, half the consumers alue good A aboe the mean. 8

9 in prices. It is no longer true that firm B must be better off from such a deiation. We beliee our choice of timing is more natural, and closer to the obsered actions of firms. It also aoids the complications of mixed strategy equilibria that would result if prices and adertising were chosen simultaneously. The restriction to distributions on the unit interal is not innocuous in itself. It rules out cases where the distribution is symmetric and firms choose to perfectly inform in Iano s simultaneous-moe model. Suppose both firms perfectly inform. If the distribution is bounded away from zero, it may be that at the candidate Nash equilibrium prices total demand is unity and each firm has demand of one half. But then the incentie to deiate to another leel of informatieness (because it increases total demand) is absent. 4 In their study of comparatie adertising Anderson and Renault (2009) derie some results on information disclosure in a two-stage model, under the assumption that the market is always coered, for the case where the consumer aluations are independent across firms and continuously and symmetrically distributed. In contrast to our model, firms are assumed to make a simple binary choice of no information or full information. Our results under the coered market assumption, show that this assumption of a binary choice is not restrictie for the case where the distribution of aluations for each firm is identical because firms choose full information in equilibrium. Howeer, when the market is not coered in the price equilibrium of eery subgame, it may be that in equilibrium a firm chooses intermediately informatie signals. Allowing for asymmetric (but still identical) distributions of aluations also generates a new result: For sufficiently right-skewed distributions, the best response of a firm to no information proision by its rial is intermediate information proision. In Anderson and Renault s model, when comparatie adertising is allowed, firms also hae the option of adertising the consumer s match alues for its rial. There is no role for comparatie adertising when both products hae identical distributions of aluations, as each firm wishes to inform in equilibrium een if comparatie adertising is banned. Thus Anderson and Renault also consider cases where the mean al- 4 Note, howeer, that there may be still be an incentie to deiate to a different leel of informatieness and price. 9

10 uation (or quality) for each firm is different while the symmetric zero mean distribution of deiations (or match alues) is identical across firms. They show that if the quality difference is small enough, both firms inform in equilibrium. On the other hand (when comparatie adertising is banned), if the quality difference is large enough, neither firm informs if the density of match alues is strictly positie at the lower bound of the support; otherwise only the high quality firm informs. The interpretation of consumer aluations as being made up of the sum of a quality and match component raises the following question. If consumers did not initially know each firm s quality, would the firms hae an incentie to certifiably reeal quality and/or match information in equilibrium? Anderson and Renault discuss this possibility. They find that if the consumers prior is that qualities for each firm are drawn independently from the same distribution, and firms initially know their own and their rial s quality, then both firms would fully inform consumers of their quality and the match alues proided the quality difference between firms is not too large. This leads us to a strand of literature which assumes firms hae priate information about the quality of their product, and looks at when firms hae an incentie to reeal that information. The standard result for monopoly (see Milgrom 1981; Grossman 1981) is that a firm that can costlessly and certifiably (i.e. lying is impossible) disclose information about its product quality will do so in equilibrium. 5 The basic intuition can most easily be seen in the case where there are two possible qualities, high and low. A firm producing the high quality is always weakly better off reealing its quality (and strictly better off if the low quality firm chooses not to reeal with some positie probability), as not reealing can only lead to a rational consumer haing a lower expectation of that firm s quality. On the other hand, a low quality firm cannot be strictly better off reealing its quality, as this can only reduce a consumer s expectation of its quality. Thus if a firm fails to disclose its quality, a consumer rationally infers that it must be a low quality firm. It follows that it is an equilibrium for firms of either quality to reeal (with the 5 Of course, when quality disclosure is costly, this result can change. See, for example, the analysis of the monopoly case in Guo and Zhao (2009). 10

11 low quality firm being indifferent between reealing and not reealing). This argument extends readily to cases with a range of possible quality leels. The results under duopoly (with no horizontal differentiation) are similar. Guo and Zhao (2009) present a model in which both firms and consumer hae the same prior that firms qualities are distributed independently, identically, and uniformly on the unit interal. Each firm only knows its own quality. 6 Two cases are inestigated. In the first case firms make their quality disclosure choice simultaenously. In the second case, firms choices are sequential. When disclosure is costless, the results are obious. 7 In the first case both firms disclose in equilibrium no matter their quality, just as under monopoly. In the second case, the first firm always discloses, while the second firm discloses if its quality is higher than the first firm s (when the disclosure cost is zero, the second firm is indifferent between disclosing and note if its quality is less than the first firm s). Eaton and Grossman (1986) use the Hotelling model of product differentiation with two firms, quadratic transportation costs and no outside option to inestigate the incenties for firms to reeal the characteristics of their product. The choice of information reelation for each firm is restricted to the extremes of either no information or perfect information. 8 The other main differences between their model and ours are: consumers and firms in their model hae some ex ante priate information, 9 the true aluations of each of the goods are perfectly correlated, and information proision changes the mean willingness to pay. Ex ante, consumers beliee that firms are equally likely to be located at any point on the line so that as in our model at least one firm will reeal its location in equilibrium in order to aoid 6 If each firm were to know its rial s quality, then the higher quality firm would reeal. The lower quality firm would be indifferent between reealing and not because it would hae zero demand in either case. 7 When disclosure is costly, the equilibrium leel of disclosure under simultenous reelation is less than under monopoly, i.e. the cutoff leel aboe which firms in duopoly reeal is higher than under monopoly. 8 A natural extension would be to allow for firms to reeal that they lie in a particular sub-interal of the Hotelling line. 9 That is, consumers know their location on the line, while firms know their location and their rial s. 11

12 the Bertrand outcome of pricing at marginal cost. It is shown that when marginal costs are equal and firm locations exogenous, reelation by both firms will be the equilibrium outcome when competitors hae ery dissimilar products, or when one of the firms has a product close to the centre of the taste distribution. In the former case competition after reelation will not be too intense as the firms are sufficiently horizontally differentiated. In the second case the firm away from the centre (the niche market firm) will reeal in order to obtain the customers at its end of the market at a higher price while the firm at the centre (the mass market firm) will reeal in order to use its centre position as an adantage to gain a large portion of the demand. When the products are similar, there are two equilibria in which exactly one firm reeals in order to aoid Bertrand price competition reducing profits to zero. The results in our paper are also related to those obtained in the context of location or product choice in the Salop (1979) circle model. Consider the duopoly ersion of the Salop circle model where firms first choose location and price. In any price subgame, the equilibrium prices are symmetric and each firm sells to half the market, just as in our model. This implies that the only effect of changing location is through the strategic effect on prices. An increase in the distance of one firm from the other leads to a mean-presering spread in the distribution of differences in willingness to pay. This reduces price competition, just as increasing the informatieness of signals does in our model. Thus in some sense, our model can be though of as an information analogue of the Salop circle model with location choice. 3 The Model There are two identical firms, indexed by i {A, B} and a continuum of consumers of mass 1. All agents are risk neutral. The firms each produce a good at zero marginal cost. Consumers choose to purchase a unit from exactly one firm or make no purchase. The utility of the outside option of no purchase is normalized to zero. The distribution of consumer aluations for each of the firms goods is common knowledge among consumers and firms. 12

13 The aluations for each good are independently and identically distributed according to F on the interal [, ], where 0 <. We assume that there is a corresponding probability density function f with support [, ]. Furthermore, we assume that f is differentiable on (, ) and log-concae (i.e. ln f is assumed to be concae). This assumption is used to ensure the existence (and in some cases uniqueness) of a price equilibrium. Although consumers know the distribution of aluations, they may not be perfectly informed of their own aluations of each good. Each firm can choose how informed consumers are about their own product. We assume each firm i can send costless signals x i [, ] of accuracy r i [0, 1] to consumers. The parameter r i can be thought of as a measure of the intensity or information content of firm i s adertising. 10 Since we assume consumers are aware of both products and their prices, we can think of a ery small alue of r i as representing adertising which informs consumers of little more than the product s existence and its selling price. If firm i chooses to also include descriptions of the product, proide product demonstrations, or allow consumers to try the product before buying, the alue of r i would be large. We consider the two-stage game where and r B are chosen simultaneously in the first stage (and obsered by consumers), and then prices p A and p B are chosen simultaneously in the second stage. Information structure The following simple truth or noise information structure is assumed: with probability r i the signal is the consumer s true aluation; with probability (1 r i ) the signal is a random draw from F. A consumer receiing signal x i of accuracy r i is willing to pay i (r i, x i ) = r i x i + (1 r i )µ for good i, where µ is the expected alue of the distribution F. We will denote the median and mode of F by m and M respectiely. Consumers 10 We assume that firms cannot mislead consumers, i.e. they must adertise truthfully. The legal system often has an important role in ensuring this is the case. 13

14 make their purchase decision based on i. Let F ri and f ri respectiely denote the distribution and density of consumers ex post aluations i, where ( F ri (z) = F µ + z µ ) and f ri (z) = 1 ( r i r f µ + z µ ). i r i An increase in r i induces a special type of mean-presering spread of f ri illustrated in figure 1. The shape of the density remains the same, with an increase in signal accuracy producing a new density that is a horizontally stretched and ertically contracted copy of the old one. The effect of an increase in r i on the distribution F ri is to rotate it clockwise about the mean. 11 Consumers will purchase from firm i if i p i max{ j p j, 0} for i j {A, B}. We will denote demand and profit for each firm i by D i (p A, p B,, r B ) and Π i (p A, p B,, r B ) respectiely. Subscripts will denote partial deriaties. We will denote the second stage Nash equilibrium prices, demand and profits by p i (, r B ), Di (, r B ), and Π i (, r B ). as f r F r f r F r µ (a) Spread of the density. µ (b) Rotation of the distribution. Figure 1 Effect of an increase in informatieness from r to r on the density and distribution of ex post aluations. 11 In Johnson and Myatt s (2006) terminology, the family of distributions {F r } is ordered by a sequence of (clockwise) rotations. Here the point of rotation is fixed at µ. In fact, the family of distributions {F r } also satisfies Johnson and Myatt s definition of a arianceordered family of distributions, with the added restriction that the mean is constant. 14

15 Incentie to inform We are interested in the incenties of a firm to increase information proision to consumers. Here we decompose, quite generally, the effect of an increase in the informatieness of a firm s signal on its profits. Differentiating equilibrium profits with respect to and using firm A s first-order condition we obtain the following expression Π A 1 (, r B ) = p A [D3 A ( p A, p B,, r B ) + D2 A ( p A, p B,, r B ) p B 1 ]. (1) The first term is the direct effect on A s demand of an increase in. It represents the marginal increase in A s profit keeping firm B s price fixed. The second term is the strategic effect. It represents the change in demand as a result of firm B s adjustment to its new equilibrium price. In the following section we will be able to say more about the direction and balance of these effects. 4 Soling the Model In the following three subsections we consider the price equilibrium and incenties to increase product information. We split the analysis up into three cases: neither firm sends a signal, exactly one firm sends a signal, and both firms send a signal. We then sole for the subgame-perfect equilibrium. No signals When neither firm sends a signal, all consumers hae a common willingness to pay of µ for each product. Through the Bertrand process, firms will compete prices down to zero. Proposition 1. When = r B = 0 the unique price equilibrium is p A (0, 0) = p B (0, 0) = 0. Equilibrium profits are zero and consumer surplus is µ, so that total welfare is also µ. 15

16 One firm sends a signal We now consider the case where > 0 but r B = 0. The other case is symmetric. A consumer receiing signal x A prefers to buy from firm A oer firm B if A p A B p B, i.e. if x A µ + pa p B := x A (p A p B, ). We will refer to x A as the marginal signal. It is easy to see that both firms must hae positie demands in equilibrium. Een if firm A were to choose the lowest possible price p A = 0, firm B can choose a positie price such that those consumers receiing low signals from firm A purchase from firm B. 12 Furthermore, firm B will not choose a price larger than µ, as then its demand is zero. When p B [0, µ], the inequality x A x A implies that a consumer with signal x A prefers buying from A oer buying from B and oer buying nothing and so will purchase from firm A. It follows, that when only one firm sends a signal, the market is always coered in equilibrium and demand for each firm is D A (p A, p B,, 0) = 1 F ( x A (p A p B, )) D B (p A, p B,, 0) = F ( x A (p A p B, )). (2) Here F ( x A ( A B, )) is simply the distribution of differences in consumers ex post aluations A B. The existence and uniqueness of price equilibrium when only one firm sends a signal is a straightforward consequence of the log-concaity of f. Lemma 1. For > 0 and r B = 0, a unique price equilibrium in pure strategies exists. Since firm B will neer choose a price larger than the mean, there are two possible types of equilibria: firm B chooses an interior price and both firms set marginal reenue equal to marginal cost (i.e. zero); firm B chooses to price exactly at the mean because marginal reenue is positie for all prices 12 Any price p B such that x A > when p A = 0, or equialently p B < (µ ), ensures positie demand for firm B for all p A 0. 16

17 in [0, µ]. Furthermore, it can be shown that for less informatie signals of firm A, the equilibrium will be of the first type. For certain distributions the equilibrium may be of the second type when signals become sufficiently informatie. The following proposition characterizes the price equilibrium. Proposition 2. (i) If > 0 and r B = 0, the price equilibrium ( p A, p B ) satisfies one of 13 where x A ( ) = x A ( p A p B, ). 1 F ( x A ( )) = pa f( xa ( )), p B < µ, F ( x A ( )) = pb f( xa ( )); 1 F ( x A ( )) = pa f( xa ( )), p B = µ, F ( x A ( )) > µ f( xa ( )); (3) (4) (ii) There exists a unique > 0 such that for < the price equilibrium is defined by (3) while for > the price equilibrium is defined by (4). Note that the of the proposition may be larger than 1, in which case only the first type of price equilibrium is possible. We expect that will be smaller than 1 only for distributions which are heaily right skewed, i.e. when a large proportion of consumers are below the mean. In this case firm B is able to capture this large section of consumers unsatisfied with firm A, and so has an incentie to choose a high price. Incentie to inform Here we take a closer look at the incenties for firm A to increase the informatieness of its signal. Using our demand expressions (2) in the decomposition (1) we obtain the following expression Π A 1 (, 0) = pa f( x A ) [ p A p B + p B 1 ]. (5) 13 There is also a borderline case where p B = µ and F ( x A ( )) = µ f( x A ( )). 17

18 The first term represents the direct effect of raising. Suppose that firm A charges the higher equilibrium price. In this case the marginal signal is aboe the mean. An increase in the accuracy of such a positie signal causes the posterior expected alue of a consumer receiing it to increase. It follows that the new marginal signal is lower and demand for firm A increases. The second term represents the strategic effect. If firm B responds to a marginal increase in by raising its price, then firm A receies more demand at its current price. These effects are illustrated in figure 2. The first panel shows the direct effect on demand induced by an increase in informatieness of firm A s signals, i.e. the clockwise rotation of demand (from D A to D A ) about the point where p A = p B. In this example p A > p B, and the shaded area indicates the resulting increase in profits. The second panel shows the strategic effect. As increased informatieness of A s signal causes demand for each firm to become more inelastic, equilibrium prices rise and demand shifts out (from D A to D A ). The positie strategic effect is gien by the shaded area. p A p A D A D A D A p A p A D A (a) Direct effect (demand rotation) (b) Strategic effect (demand shift) Figure 2 Decomposition of the effects of a marginal increase in informatieness from to. Here D A = D A (p A, p B,, 0), D A = D A (p A, p B,, 0), and D A = D A (p A, p B,, 0) where p B = p B (, 0) and p B = p B (, 0). Flatter lines represent the corresponding own-price deriaties of profit, Π A 1. We now consider the balance and direction of these effects when r < 18

19 and the equilibrium is of the first type. Although the direct effect may be negatie, we can show that the strategic effect is always positie and large enough that the total effect on profits for both firms is positie. The reasoning is as follows. An increase in has no effect on the equilibrium marginal signal x A. 14 This result stems from the information structure we hae assumed. An increase in informatieness of the signal has no effect on the shape of the distribution. For any, the density f ra of willingness to pay for firm A is just a rescaling of the density f horizontally by the factor, and ertically by the factor 1/. It follows that if we think of firms as choosing prices normalized by (i.e. p i / ) then the firms profit maximization problems are independent of, and normalized equilibrium prices are constant in. We can then conclude that the equilibrium marginal signal is constant and equilibrium prices linear in. This leads to the result that the increase in profits for each firm is equal to their (constant) equilibrium demand multiplied by the increase in their own equilibrium prices. The next lemma summarizes these findings. Lemma 2. If <, then: (i) Equilibrium prices are increasing in : p A 1 (, 0) = pa > 0 p B 1 (, 0) = pb > 0. (ii) Equilibrium profits are increasing in : Π A 1 (, 0) = pa [1 F ( xa ( ))] > 0 Π B 1 (, 0) = pb F ( xa ( )) > Taking the difference in the first-order conditions (3), we hae x A ( ) µ = 1 2F ( xa ( )) f( x A (. )) If the left-hand side were to increase with a change in, then because of the log concaity of f, the right-hand side would decrease, a contradiction. 19

20 We can also say something about the incenties for firm A to further inform consumers about its product when r > and the price equilibrium is of the second type. When r >, p B = µ and firm B has no incentie to increase its price. Thus the only incentie left for firm A is the direct effect identified in equation (5). It follows that firm A has the same incenties as a monopoly to inform consumers further about its product: if its equilibrium price is aboe the mean, informing is profitable; if its equilibrium price is below the mean, informing reduces profits. Lemma 3. If >, then: (i) Equilibrium prices satisfy: ( ) p p A 1 ( A µ, 0) r, pa A p B 1 (, 0) = 0. (6) (ii) Equilibrium profits satisfy: ( ) p Π A 1 ( A µ, 0) = [1 F ( x A )] (7) Π B 1 (, 0) > 0. We can now derie firm A s best response to firm B proiding no information. Lemma 2 tells us that profit is at a maximum on [0, max{, 1}] when = max{, 1}. The behaiour of profit for depends on whether the mean is larger than the median. We can show that when the median is at least as large as the mean, the informing firm charges the higher price in equilibrium. 15 It follows that the direct effect on firm A s profits of informing is always positie. Since the strategic effect is zero, it must be that firm A s best response is to perfectly inform. On the other hand, when the median is less than the mean, the direct effect may be negatie or positie (while again the strategic effect is zero). The informing firm is effectiely a monopolist when. As Johnson and Myatt (2006) show, a monopolist s profits 15 See Lemma 8 in the appendix. 20

21 are conex in the degree of information under the truth or noise information structure. Thus firm A maximizes its profit by choosing a signal accuracy of either or 1. Proposition 3. Suppose r B = 0. If m µ, then firm A s best response is = 1. If m < µ, then firm A s best response is either = or = 1. We now present an example in which the distribution is sufficiently right skewed that < 1 and firm A s best response to no information disclosure by firm B is to choose to inform consumers partially. Example 1. Consider the triangle distribution on [0, 1] with mode M = 0, i.e. the distribution with density f(z) = 2(1 z) for all z in its support. The mean is µ = 1/3 and the median is m = M < m < µ. Profits are 0.29, so that we hae Π A (p A, p B,, 0) = p A [2rA + 3(p B p A )] 2 9( ) ( 2 ) Π B (p A, p B,, 0) = p B 1 [2rA + 3(p B p A )] 2. 9( ) 2 Simple calculations yield the equilibrium prices { ( 10+1) p A (, [0, ], 0) = , [, 1] 9 { (3 10 5) p B (, [0, ], 0) = 12 1, 3 ra [, 1] and equilibrium profits Π A (, 0) = Π B (, 0) = { ( 10+1) 3, [0, ] 432 4(1+2 ) 3, [, 1] 729( ) { 2 (3 10 5)( ), [0, ] 432 (2+13 )(5 2), [, 1], 243( ) 2 where =

22 In figure 3 we plot the equilibrium prices and profits of each firm. The dashed lines indicate the profit maximizing price and maximum profits for a singlegood monopoly as a function of. Note that monopoly profits at the optimal price are decreasing in the degree of informatieness. For >, the equilibrium prices satisfy (4). As stated in Lemma 3 firm B s profits are increasing, while firm A s profits are decreasing in because firm A s equilibrium price is below the mean. We also see that firm A s profits coincide with the monopoly profits oer this range. The intuition for decreasing profits is that the strategic incentie is absent. Because the median is below the mean, a majority of consumers are receiing signals below the mean. Further increasing the accuracy of information simply causes these consumers to further reduce their willingness to pay for firm A s product. In the absence of any offsetting rise in equilibrium prices, A s profits fall. 1 3 p B p A 1 3 Π B Π A (a) Prices. (b) Profits Figure 3 Equilibrium prices and profits in example 1. When the distribution is symmetric we are able to sole explicitly for the equilibrium prices. Definition 1. A distribution F is symmetric about its mean µ if F (µ z) = 1 F (µ + z) for all z 0. 22

23 A density f is symmetric about its mean µ if f (µ z) = f (µ + z) for all z Since a symmetric density has the property that m = µ, we can conclude immediately from Lemma 8 (see appendix) that if the price equilibrium is of the first type, then equilibrium prices are equal. That the price equilibrium must be of the first type follows from fact that the unimodality of f implies the alue of the density at the mean is large enough for the prices defined by (3) to be below the mean. This equality of prices allows us to compute explicitly the equilibrium prices in any subgame in which r B = 0. Proposition 4. If > 0, r B = 0 and F is symmetric, then the equilibrium prices are p A (, 0) = p B (, 0) = ra 2f(µ), where µ = + 2. When the distribution is symmetric, half of the consumers receie signals aboe the mean. Without any information about firm B s product, all consumers alue its product at the mean. It follows that when both firms choose the same price, they split the demand equally. Furthermore, since demand depends only on the difference in prices, p A p B, we hae D1 A = D2 B. It must then be that both firms first-order conditions are satisfied when p A = p B. It is easy to see that the equilibrium prices are increasing in. Corollary 1. If > 0, r B = 0 and F is symmetric, then the equilibrium profits of both firms are increasing in. The intuition for the result is straightforward. When the distribution is symmetric, demand is independent of and only the strategic effect of raising in expression (5) remains. Since prices are equal, there is no direct effect on demand as illustrated in panel (a) of figure 4. The strategic effect increases profits and is indicated in panel (b). 16 Equialently the distribution F is symmetric if F (+z) = 1 F ( z) and the density f is symmetric if f( + z) = f( z) for all z 0. 23

24 Example 2. Suppose f is uniform on [, ] with density gien by f(z) = 1 δ for all z in its support, where δ =. The equilibrium prices are and the equilibrium profits are p A (, 0) = p B (, 0) = δra 2 Π A (, 0) = Π B (, 0) = δra 4. Note that the uniform distribution is the least peaked of all symmetric distributions and so the aboe equilibrium prices and profits are an upper bound for those obtained for any other symmetric distribution. p A p A D A D A p A D A p A D A (a) Direct effect (demand rotation) (b) Strategic effect (demand shift) Figure 4 Decomposition of the effects of a marginal increase in informatieness from to for example 2. The direct effect is zero. Here D A = D A (p A, p B,, 0), D A = D A (p A, p B,, 0), and D A = D A (p A, p B,, 0) where p B = p B (, 0) and p B = p B (, 0). Flatter lines represent the corresponding own-price deriaties of profit, Π A 1. Both firms send a signal We now suppose that both firms are proiding some information to consumers, i.e. that, r B > 0. We will begin by soling for the price equilibrium under the assumption that the market is coered in all subgames. Once the equilibrium prices are found, we will proide a condition to ensure that 24

25 all consumers do indeed prefer purchasing one of the goods to the outside option. Since we hae made the assumption that all consumers prefer purchasing one of the products to non-purchase, a consumer will buy from firm A if A p A B p B, i.e. if x A µ + rb (x B µ) + p A p B. Then for any gien realization of x B, the probability of purchasing good 1 is ) 1 F (µ + rb (x B µ) + p A p B. Integrating oer all possible realizations x B gies the demand for firm A. Since the market is coered, total demand is one, and each firm s demand is gien by D A (p A, p B,, r B ) = 1 H(p A p B,, r B ) D B (p A, p B,, r B ) = H(p A p B,, r B ), where H is the distribution of A B and is gien by ( ) H(z,, r B ) = f(x)f µ + rb (x µ) + z dx. The corresponding density is h(z,, r B ) = 1 f(x)f ( ) µ + rb (x µ) + z dx. The demand functions can be shown to inherit log concaity from f, so that a price equilibrium exists. Although it is not possible to show that in general this price equilibrium must be unique, we can show that there can be only one equilibrium where all consumers purchase. Lemma 4. For, r B (8) > 0, a price equilibrium in pure strategies exists. There can only be one price equilibrium in which the market is coered That is, we can show that a price equilibrium exists in which the market may or may not be coered. If we assume no outside option, we can show further that this price equilibrium is unique. Thus if the distribution and, r B > 0 are such that there is a price equilibrium in which the market is coered, then it is the unique such equilibrium. We cannot rule out that there may coexist other equilibria where the market is not coered. 25

26 The mean of the distribution of differeneces in willingess to pay, H, is zero for any distribution F. When H also has the property that H(0,, r B ) = 1/2 for all, r B > 0 as is the case when H is symmetric about zero a simple explicit solution for the price equilibrium of any subgame is aailable. It is easy to show that if F is symmetric then H will also be symmetric. Lemma 5. If F is symmetric about µ, then H is symmetric about zero for all, r B > 0. The symmetry of H implies that if prices are equal then each firm obtains the same demand. Furthermore, since each firms demand is a function only of the difference in prices p A p B, the own-price deriaties of demand are equal (i.e. D1 A = D2 B ). Together, these properties of demand imply that each firm charges the same price in equilibrium, with the equilibrium prices being directly proportional to the inerse of the magnitude of the peak of the density of differences in willingness to pay. Proposition 5. If, r B > 0, the distribution F is symmetric and the market is coered, then the equilibrium prices are p A (, r B ) = p B (, r B ) = 1 2h(0,, r B ). A necessary and sufficient condition for the market to be coered at the prices gien aboe is h(0,, r B ) 1 2 max{ + (1 )µ, r B + (1 r B )µ}. The condition requires that the peak of the distribution of differences in willingness to pay is sufficiently large at the gien choices of signal accuracy. This ensures that price competition is igorous enough that equilibrium prices are below the willingness to pay of those consumers receiing the lowest signals. Incentie to inform As before, we can decompose the effect of a change in into a direct effect and strategic effect. Using the demand expressions (8) and the general 26

27 decomposition (1) we hae Π A 1 (, r B ) = p [ ] A H 2 ( p A p B,, r B ) + h( p A p B,, r B ) p B 1. For the special case where F is symmetric, an increase in causes a clockwise rotation of H about its mean. Since the median of H is zero for any, r B, and the equilibrium prices of each firm are equal in any subgame, the direct effect is zero. Only the strategic effect remains. The strategic effect is positie if firm B s price rises with increases in firm A s signal accuracy. We later show that this is the case for any symmetric distribution. This is because an increase in causes demand to rotate clockwise and become more inelastic at the preious equilibrium price, decreasing the incentie for firm B to reduce price to gain market share. In the following example the equilibrium prices and profits are computed for for the case where the distribution of aluations is uniform. Example 3. Suppose f is uniform on [, ], with density gien by f(z) = 1 δ for all z in its support, where δ =. When r B, the density of differences in willingness to pay is gien by [ ] 1 r A +r B + z z δ δ r B 2 δ 2 (ra r B ) h(z,, r B 1 ) = z [ δ δr A ] 2 (ra r B ), δ 2 (ra r B )] 1 z δ 2 (ra r B ). δ r B [ +r B 2 z δ By Proposition 5, when r B the equilibrium prices are and profits are p A (, r B ) = p B (, r B ) = Π A (, r B ) = Π B (, r B ) = 1 2h(0,, r B ) = δra 2 1 4h(0,, r B ) = δra 4. We can see that for r B both prices are strictly increasing in and constant in r B. The effect of an increase in is illustrated in figure 5. For the special case of the uniform distribution, an increase in the proision of information by firm B, when it is proiding less information than 27