Word of Mass: The Relationship between Mass Media and Word-of-Mouth

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1 Word of Mass: The Relationship between Mass Media and Word-of-Mouth Roman Chuhay Preliminary version Marh 6, 015 Abstrat This paper studies the optimal priing and advertising strategies of a firm in the presene of word-of-mouth ommuniation. In the model, a monopolist produes new produt and hooses the prie and amount of advertising. Consumers an learn about the produt diretly from advertising or from their neighbors, who have aquired it. We show that the optimal advertising level is a non-monotone funtion in the network onnetivity. An inrease in onnetivity failitates diffusion and onsequently raises the payoff to advertising. However, when onnetivity beomes suffiiently high, its further inrease leads to a ongestion effet and the optimal advertising level falls. We also show that an inrease in the advertising ost may atually lead to a higher onsumer surplus. JEL Classifiation numbers: D4, D83, D85 Keywords: word-of-mouth, advertising, diffusion, networks, priing 1 Introdution This paper studies the relationship between advertising and word-of-mouth ommuniation. Both advertising and word-of-mouth inrease onsumers awareness of the produt, but at quite differently. Although, an advertising is ostly ativity, a produer an freely hoose its amount. In ontrast, word-of-mouth depends on the onsumers behavior and network struture and an be affeted only in the indiret way. In the paper we show that Higher Shool of Eonomis, Department of Eonomis, Myasnitskaya 4, Mosow, Russia. rhuhay@gmail.om. I am partiularly indebted to Maarten Janssen. I also would like to thank Alexei Parakhonyak and partiipants of HSE seminars and funding from HSE Researh Laboratory for Strategi Behaviour and Institutional Design. 1

2 depending on the parameters of the model these two phenomena may ompliment eah other or serve as substitutes. In the model a firm reates new produt and sells it to a ontinuum of onsumers. We assume that the produt quality is revealed after the development proess and produer treats it as given exogenously. Consumers are embedded into a soial network, whih is represented by a generalized random graph 1. Eah onsumer has an outside option distributed aording to a uniform distribution and buys the produt if the utility of purhase is higher than the outside option. Initially, onsumers are not aware of the produt and to indue sales the innovator advertises the produt diretly to some proportion of onsumers. Advertising is ostly and has diminishing returns to sale. The rest of the population an learn about the produt and its quality from their neighbors. We assume that onsumers tell their friends about the produt only if they buy it and thus find information worth to spread. In the model there is no asymmetry of information and everyone who beomes aware of the produt immediately knows its quality. One of the explanations may be that nowadays onsumers find all relevant information in the internet. The produer knows statistial properties of onsumer network and hooses the amount of advertising and priing strategy to maximize profits. We show that for moderate levels of onnetivity, produt sales are non-monotone funtion in the ost of advertising. More preisely, as ost inreases sales of the produt first derease, but after some threshold level sales inrease. An inrease in the ost of advertising lowers the optimal advertising level. The produer partially offsets this by reduing the prie that failitates word-of-mouth ommuniation. These two effets work in the opposite diretions. In the beginning when ost of advertising is suffiiently low, the majority of onsumers beome aware of the produt through mass media. The diffusion in this ase looks like a great number of small interonneted islands emerged around onsumers who got an advertisement. The amoun of word-of-mouth is limited and the first effet dominates. However, when advertising ost beomes suffiiently the produer partially offsets the drop in advertising by lowering the prie. The produt diffusion looks like few big islands and a small derease in the prie generates a sizeable asade of sales. This happens sine the prie is already low and majority of onsumer neighbors are unaware of the produt. We show that in this ase the indiret effet dominates and sales inrease. Considering welfare impliations, we show that both onsumer surplus and soial welfare first derease in advertising ost, but then inrease. As we know, for a suffiiently low advertising ost its inrease lowers sales. At the same time the produer lowers the prie to stimulate word-of-mouth. However, the prie drop is not enough to offset derease in sales, sine for a suffiiently low ost word-of-mouth plays minor role in information 1 Generalized random graph is a graph seleted with a uniform probability from a set of graphs that obey given statistial properties. In our ase all graphs in the set have some speified degree distribution.

3 diffusion. As ost of advertising grows further at some point sales start to inrease. In this ase both effets work in the same diretion, and more onsumers buy the produt at a lower prie. Thus for a suffiiently high advertising ost onsumer surplus inreases. Moreover, for a suffiiently large ost, an inrease in the onsumer surplus beomes large enough to ompensate the derease in the produer surplus and total welfare inreases. We also show that the advertising level is a non-monotone funtion in the network onnetivity. More preisely, first the amount of advertising inreases in the onnetivity, but after a threshold value dereases. When the onnetivity is suffiiently low, eah advertisement generates small asade of sales as there are few hannels for the information to spread on. As a result produer hooses low advertising level. A growth in the onnetivity inreases effiieny of advertising and the optimal amount of advertising inreases. As the onnetivity grows further, both the amount of advertising and the average size of sales asade generated by an advertisement inrease. In this ase a further inrease in the onnetivity leads to a ongestion effet, when some part of reommendations are made to onsumers who already are aware of the produt. The diffusion slows down and the payoff on the advertising dereases. The produer lowers advertising and substitutes it with a higher word-of-mouth by lowering the prie. To our best knowledge this is the first paper that studies the interation of wordof-mouth and advertising in the expliit form, assuming non-trivial network struture. The previous papers that study diffusion of word-of-mouth suh as Lopez-Pintado 008, Chuhay 013 assume that only an infinitesimal part of population reeives advertising. The rest of population an find out about the produt only by means of word-of-mouth. In ontrast, in our paper the advertising level is firm s hoie variable, whih is affeted by the network struture. The most related paper to our work is Campbell 01, whih onsiders the optimal priing strategy of a monopolist in the presene of word-of-mouth ommuniation. The main result of this paper regarding advertising is that the prie elastiity of demand dereases in the advertising level. In our paper, we make the advertising level endogenous variable. We show that word-of-mouth and advertising behave as substitutes if we onsider a hange in the advertising ost, but may show omplementarity when we vary network onnetivity. Using numerial analysis Campbell 01 shows that onsumer surplus may inrease in the advertising ost. In our paper, we identify onditions under whih onsumer surplus is inreasing in advertising ost. Moreover, we show that an inrease in the advertising ost may be benefiial for the soiety as whole. Galeotti and Goyal 009 studies the model of strategi diffusion of information, where authors allow for network externalities in adoption deision. In the paper the authors limit diffusion proess only to immediate neighbors of a onsumer who reeives information. This assumption may play a ruial role, espeially in the ase when individual s adoption deision depends on the adoption ratio. In ontrast, we model diffusion proess in the As authors note, an inrease in the number of ontats negatively affets the probability of produt 3

4 expliit way, whih allows us to study the effets of average onnetivity on the optimal priing and advertising strategies. The rest of the paper is organized as follows. Setion desribes the model and demand funtion is derived. Setion 3 presents the main results regarding the impat of advertising ost and network onnetivity on the optimal prie, sales, advertising level and soial welfare. Setion 4 onludes. Model There is a ontinuum of onsumers that are embedded into a soial network represented by a lassial random graph with a given onnetivity. A firm reates a new produt, for whih there are no lose substitutes and ats as a monopolist on the market. We assume that produt quality v [0, 1] is realized after a prodution proess took plae and the firm treats it as given exogenously. Observing quality of the produt and knowing network onnetivity, the firm hooses prie P and the amount of advertising s to maximize profits. Initially, onsumers are not aware of the produt. To start sales the ompany advertises the produt to the population. With probability s eah onsumer reeives an advertising. With omplementary probability 1 s a onsumer does not get the advertising and may reeive information about the produt only from her neighbors, who already have aquired the produt. The advertising is ostly and produer pays for advertising the produt to proportion s of onsumers. The ost funtion is onvex in s, whih represents the idea that it is impossible to ontrol who gets an advertisement. Thus to reah an inreasing part of onsumers the amount of advertising should grow exponentially. All onsumers have outside option γ i, whih is distributed aording to uniform distribution U[0, 1]. A onsumer i buys the produt if the valuation of the produt purhase v P is higher than her outside option γ i. Thus a randomly seleted onsumer buys the produt with probability q = v P. One a onsumer buys the produt, all her neighbors beome aware of it and may buy it too. In the model we onsider an equilibrium state where diffusion already has taken plae and the demand is given by the number of purhases that onsumers made. The diffusion stops when all onsumers who learn about the produt do not buy it or do not have neighbors..1 Demand funtion With probability s a randomly hosen onsumer gets an advertisement diretly from the produer and buys the produt with probability v P. With probability 1 s the onsumer does not get advertising and the only way for her to find out about the produt is to hear adoption and impede diffusion. However, there is also additional effet. One a produt is adopted by a onsumer more neighbors beome aware of it. 1 s 4

5 about it from neighbors. Let s assume that a randomly seleted neighbor of a onsumer buys the produt with probability ŵ. Thus a onsumer with k links does not hear about the produt if no one of her neighbors buys it, whih happens with probability 1 ŵ k. With omplementary probability 1 1 ŵ k at least one of onsumer s neighbors buys the produt and onsumer learns about the produt. Sine, a randomly seleted onsumer has k links with probability pk, in expeted terms she hears about the produt and buy it with probability v P k=0 pk1 1 ŵk. Thus the demand funtion is the following expression: Ds, v, P = sv P + 1 sv P = v P 1 1 s pk1 1 ŵ k k=0 pk1 ŵ k To lose the model we should formulate a self-onsisteny ondition for ŵ. In general, degree distribution of neighbor is different from the one of a randomly seleted onsumer. The more links a onsumer has the greater is the probability that she is someone s neighbor. Thus a onsumer with k links has k-times higher probability to be a neighbor of randomly seleted onsumer than a onsumer with just one link. Therefore, the probability to have a neighbor with k links is proportional to kpk. After normalization we obtain a degree distribution of neighboring onsumer ξk, whih is the following: ξk = k=0 kpk kpk =. j=1 jpj z 1 A neighboring onsumer an be reahed through one of her links. Thus the probability that a neighbor hears about the produt from someone else and buys it is given by v P k=1 ξk1 1 ŵk 1. Thus the probability that a neighbor buys the produt is the following: ŵ = sv P + 1 sv P = v P 1 1 s ξk1 1 ŵ k 1 k=1 ξk1 ŵ k 1 The produer maximizes profits hoosing amount of advertisement s and prie P by solving the following problem: max s,p P v P 1 1 s k=0 k=1 pk1 ŵs, v, P k 1 s 5

6 s.t. ŵ = v P 1 1 s ξk1 ŵ k 1 In the following analysis we assume that the network is represented by a lassial random graph and thus the degree distribution is Poisson pk = k e k!. In the ase of Poisson degree distribution the distribution of links of neighboring node ξk equals to pk 1. In partiular, it implies that probabilities to buy the produt for a randomly hosen onsumer w and a randomly seleted neighbor ŵ are the same. The substitution of Poisson degree distribution redues the problem to the following form: k=1 max P v P 1 1 se w s,p 1 s s.t. w = v P 1 1 se w 1 The following Lemma presents the solution to the maximization problem in the impliit form. Lemma 1 Given that max{1, { } } < e w < min 1 v +, e there is a unique interior solution to the maximization problem, whih is given by the following equations: s = 1 e w ; P = v 1 v e w, where w is the solution to w = v. Proof See Appendix e w v e w Lemma 1 gives a solution for the optimal prie and advertising as a funtion of w. Note that w is the solution to the transendental equation, whih does not have losed form solution. The immediate orollary from the previous lemma is a ondition on the advertising ost, suh that there is non-zero advertising in the equilibrium. Corollary 1 If advertising ost is higher than v then there is no advertising in the equilibrium. Corollary 1 states that there is upper limit for the advertising ost, suh that for higher than there is no interior equilibrium for any parameters of the model. 3 Main results In this setion we onsider the effet of advertising ost and network onnetivity on the sales, optimal prie and awareness of the produt. In the ase of full information all 6

7 onsumers are aware of the produt and profit funtion has the following form P v P. The optimal prie in this ase is PF I = v and in equilibrium sales are v. Despite the fat that onsumers are not aware of the produt sales in the ase of limited awareness of onsumers may be higher than in the ase of full information. Proposition 1 For suffiiently high onnetivity sales in the ase of inomplete information are higher than in the ase of omplete information. The statement is also true for suffiiently high ost and v >. Moreover, the optimal prie is always lower than PF I, the prie in the ase of full information. Proof See Appendix When onsumers are not aware of the produt the firm uses ostly advertising and then relies on further word-of-mouth diffusion to inform onsumers about the produt. Perhaps surprisingly, Proposition 1 implies that when onsumers are not aware of the produt sales may be higher than in the ase of full information. This is the ase when ost of advertising or network onnetivity are suffiiently high. In the first ase when advertising is quite ostly, the firm relies mostly on the wordof-mouth diffusion, whih ruially depends on the produt prie. Reall, that onsumers spread information about the produt further only if they buy it. Thus to offset low level of advertising the firm lowers the prie. When the onnetivity is suffiiently high, a prie redution has substantial impat on word-of-mouth diffusion. Thus in the ase of high advertising ost the produer sets low prie and atual sales beome higher than in the ase of full information. The same logi applies in the ase of suffiiently high network onnetivity. To use high spreading effiieny of the network the monopolist sets low prie. 3.1 The impat of the advertising ost One of the important harateristis of the diffusion proess is a share of onsumers who are aware of the produt. Eah onsumer that beomes aware of the produt buys it with probability v P. Sine the share of onsumers that buy the produt is given by w the proportion of onsumers who know about the produt is simply ŵ v P. The following proposition formulates the results regarding the optimal prie, the amount of advertising, awareness of the produt and diffusion perimeter. Proposition The optimal prie P and amount of advertising s derease in the ost of advertising. The same is true about awareness of the produt and diffusion perimeter. Proof See Appendix The first result is quite straightforward. An inrease in the advertising ost leads to a lower level of advertising. The seond part of the result states that the optimal prie 7

8 falls in the advertising ost. We already know that the advertising level dereases in the ost. As advertising beomes ostlier the produer substitutes it with word-of-mouth ommuniation by lowering the prie. Indeed, a prie derease makes the produt attrative to a longer hains of buying onsumers. An inrease in word-of-mouth ommuniation offsets the derease in the advertising level. However, as the result implies word-of-mouth substitutes advertising only partially and overall awareness of the produt falls. Proposition 3 In general, sales of the produt is non-monotone funtion in advertising ost. More preisely, if 1 < v < 4 then sales of the produt first derease, but after some level inrease in. If v < 1 sales are dereasing in on the whole range, while if v > 4 sales always inrease in. Proof See Appendix Aording to Proposition an inrease in the advertising ost lowers both advertising level and the prie. If we onsider sales these two effets work in the opposite diretions. Proposition 3 states that for the intermediate values of v sales are non-monotone in the ost. In the beginning when the ost is suffiiently small most onsumers beome aware of the produt through the mass media. In this ase the diffusion looks like a great number of interonneted small islands emerged around onsumers who got the advertising. When the ost of advertising is suffiiently high, the advertising level and total awareness of the produt are quite low. Aording to Proposition the prie is low too. The diffusion now looks like few big islands. A purhase of the produt by a onsumer on the perimeter generates a sizeable asade of sales. This happens sine the prie is quite low and majority of onsumer s neighbors are not aware of the produt. In this ase, the indiret prie effet dominates the diret effet of advertising osts and sales inrease. The inreasing part appears only when v is higher than 1. If the opposite is true, whatever small is the prie, the diffusion is limited and sales always derease in the advertising ost. An interesting question is how the advertising ost affets onsumers welfare. If onsumer i buys the produt instead of the outside option she gains v P γ i. We know that a onsumer buys the produt only if γ i is lower than v P. Thus the hange in the onsumer surplus an be represented as the following: v P w v P 1 v P γ dγ = w v P 0 The following proposition relates onsumer surplus to the advertising ost. Proposition 4 If v > 1 the onsumer surplus is non-monotone funtions in advertising ost. More preisely, first onsumer surplus falls, but after some level onsumers beome better-off as the ost inreases. When v < 1 onsumer surplus dereases in the ost on the whole range. 8

9 Proof See Appendix We have seen that when advertising ost is suffiiently low, an inrease in the ost lowers sales. At the same time the produer lowers the prie to stimulate word-of-mouth ommuniation. Initially, when advertising ost is suffiiently low, the prie drop is not enough to offset derease in sales, sine word-of-mouth ommuniation plays minor role in the information diffusion. Thus onsumer surplus dereases in the advertising ost. However, for suffiiently high levels of advertising ost its further growth inreases sales. In this ase both effets work in the same diretion as more onsumers buy the produt at a heaper prie. Thus when the advertising ost is suffiiently high onsumer surplus inreases in it. The important point here is that onsumer surplus inreases only if v > 1 and thus even infinitesimal advertisement may lead to a purhase of the produt by some non-zero share of the population. When v < 1 the derease in the prie has limited impat of sales and that is why sales and onsumer surplus do not inrease. We have seen that onsumers may benefit from an inrease in the advertising ost. This happens beause the produer by lowering the prie substitutes word-of-mouth for advertising. The important question is whether taxation of advertising an be benefiial for the soiety as a whole. Total welfare onsists of three parts. Produer surplus P w +t 1 s, onsumer surplus w v P t and gains from taxation 1 s. Summing up we obtain: SW = w + t v + P + t 1 s Proposition 5 If < v < 6 then soial welfare first dereases in up to the point where = 1 v+ 1+4v and then inreases. If v < then soial welfare always dereases while for v > 6 soial welfare always inreases in. Proof See Appendix Thus if v is suffiiently high an inrease in the onsumer surplus due to a lower prie ompensates fall in the profits of produer and total welfare inreases. The result ruially relies on the suffiiently high onnetivity of the onsumer network whih failitates wordof-mouth spreading. 3. The impat of onnetivity In the previous setion we have seen substitution effet between advertising and word-ofmouth. More preisely, when advertising ost inreases the produer turns to a heaper word-of-mouth ommuniation by dereasing the prie. In this setion we study the effet of network onnetivity on the optimal amount of advertising and prie. In ontrast, to previous result we show that advertising and word-of-mouth may ompliment eah other. At the beginning, when onnetivity grows, word-of-mouth ompliments advertising 9

10 and both inrease. However, as onnetivity grows further, the relationship is reversed and word-of-mouth beomes a substitute for the advertising. The following propositions formalize the result. Proposition 6 When advertising ost is suffiiently lose to the upper limit, the amount of advertising is a non-monotone funtion in onnetivity. More preisely, for suffiiently small the amount of advertising inreases in, while for suffiiently high dereases. If the advertising ost is suffiiently low then the advertising level always dereases in. When onnetivity is suffiiently low, eah advertising generates small asade of sales as there are few hannels for the information to spread on. A growth in the onnetivity inreases spreading effiieny of the network and hene inreases the payoff on advertising. Thus for a suffiiently low levels of the onnetivity, word-of-mouth serves as a omplement to the advertising and both move in the same diretion with the onnetivity. However, at some point a further inrease in the onnetivity leads to a ongestion effet, when some advertisements are reeived by onsumers who already know about the produt from their neighbors. The payoff on the advertising dereases and the produer swithes from advertising to relatively more effiient word-of-mouth. One of the immediate orollaries is that the advertising level reahes its maximum for intermediate values of onnetivity, when advertising ost is suffiiently lose to the upper limit. Proposition 7 The optimal prie is a non-monotone funtion in onnetivity. More preisely, first the prie dereases in the onnetivity, but after some point inreases. The intuition behind the result of Proposition?? is the same as in the ase of relationship between advertising and. In both ases, an inrease in the onnetivity makes word-of-mouth and advertising more effiient, whih leads to a derease in the optimal prie. However, at some point the ongestion effet omes into play and rowds out both advertising and word-of-mouth, whih leads to an inrease in the optimal prie. Proposition 8 If network onnetivity is suffiiently small then both onsumer surplus and sales inrease in. When network onnetivity is suffiiently high and advertising ost is suffiiently low both onsumer surplus and sales derease in. Probably surprisingly Proposition?? states that a higher onnetivity and thus higher spreading effiieny of the network is not always better for onsumers. When the onnetivity is suffiiently low information about the produt is sare. In this ase an inrease in the onnetivity leads to a lower prie and higher advertising level, whih inreases onsumer surplus. However, when the onnetivity is suffiiently high and advertising ost is suffiiently low, a further inrease in the onnetivity leads to a higher prie. This happens sine a onsumer in one way or another beomes aware of the produt that is why it does not pain to ut some hannels by inreasing the prie. 10

11 4 Conlusion This paper studies the relationship between advertising and word-of-mouth ommuniation. Both advertising and word-of-mouth inrease onsumers awareness of the produt, but at differently. In the paper we show that depending on the parameters of the model these two phenomena may ompliment eah other or at as substitutes. In partiular, we show that for a suffiiently low onnetivity levels both word-of-mouth and advertising are omplements and grow with the onnetivity. However, when onnetivity beomes suffiiently high the produer substitutes advertising with relatively more effiient wordof-mouth. We show that an inrease in the advertising ost may atually inrease onsumer surplus. When advertising ost and network onnetivity are suffiiently high, wordof-mouth is relatively more effiient than advertising and produer redues the prie to inrease word-of-mouth ommuniation. This prie drop turns out to be suffiient to offset lower advertising and as a result lower produt awareness. We find that the optimal advertising level is a non-monotone funtion in the network onnetivity. An inrease in the onnetivity failitates word-of-mouth ommuniation and onsequently raises the payoff on advertising. However, for a suffiiently high onnetivity levels its further inrease leads to a ongestion effet, when some onsumers reeive multiple reommendations and the optimal advertising level falls. In the further researh we plan to study the effet of the produt quality on the optimal advertising and priing strategies. This will allow us to differentiate the optimal marketing strategies for goods of high and low quality and to onfront our findings with the data. Referenes [1] Campbell, A., 01, Word-of-mouth and Perolation in Soial Networks, Amerian Eonomi Review, forthoming. [] Chuhay, R., 013, Priing Innovation in the Presene of word-of-mouth Communiation, mimeo, Higher Shool of Eonomis. [3] Galeotti, A. and Goyal, S., 008, A Theory of Strategi Diffusion, Rand Journal of Eonomis, 009, pp [4] Lopez-Pintado, D., 008, Diffusion in omplex soial networks, Games and Eonomi Behavior, 6, APPENDIX Proof of Lemma 1 11

12 Taking the derivative with respet to s and P and expressing the derivative of w from the onstraint we get: P v P e w 1 sv P 1 s = 0 e w e w 1 s v P e w + 1 sv P e w 1 sv P = 0 3 Equations??,?? and onstraint 1 give us solution to the maximization problem. Solving?? for s we get two solutions. The first solution s = 1 e w implies zero diffusion. The seond is the following: v Substituting?? to?? we get: s = 1 v P ew v P 4 v P e w v P e w v P v P = 0 Solving the last equation for P, we get two possible andidates for the solution: P 1 = e w and P = v e w. The first one implies advertisement over 100%, the seond gives us interior solution. Substituting P into equation for s and then substituting both to the onstraint we get: s = 1 e w v P = v w = v 1 e w e w e w v 7 Note that by the definition 0 < s < 1, 0 < P < v and 0 < w < 1. The first ondition gives us 0 < e w < v. The seond ondition implies < e w. Finally, the third ondition implies 1 < e w < e. Combining all three onditions and rearranging we get: max{, } < e w { } v < min, e 8 1

13 Or we an get: Or we an get: max{1, { } } < e w 1 v < min +, e max{ 1, } < e w < min { } 1 v +, e 9 10 Uniqueness of the solution for w Lets denote by fw the right hand side of??. We know that f0 = v = v, whih is positive whenever 0 < s 0 and P 0 < v. Taking the derivative of fw we get: 1 e w 1 v e w Note that by?? for s to be lower than 1, prie P should be lower than v. Taking into aount?? we an onlude that e w < for any w. Thus the last expression hanges sign just one and in general fw first dereases and then inreases. Taking into aount that 0 < s < 1 and onstraint from 1 we an onlude that for any 0 < w < 1, funtion fw is lower then v, whih in turn is lower than 1. We an onlude that fw rosses 45 degree line only one from above and thus there is a unique solution to??. Proof of Corollary 1 First lets identify the threshold ost, s.t. there is advertising in the equilibrium. Substituting s = 0 into?? and solving for e w we get e w = +v. Substituting it into?? we get w = 1 4 +v + v. Substituting it bak to?? we get the following equation: 1 + e v +v = 0, v We an express threshold as a share of v. Substituting = α v we get: If we reexpress v as γ we get 1 + α v αe 1 4 v 1+α v 4α 1+α v 1 + α γ α e γ 1+α γ 4α 41+α γ 13

14 One an show that the funtion is negative for any γ > 1 only if α 1. Thus threshold level equals v. Lemma A variable q = e w dereases in. Rearranging the last expression and substituting q = e w w = v 1 q v q 1 Taking the derivative of q with respet to we get: we get: 11 e w w 1 3/ Thus if w < 0 then q dereases in. To onsider the ase when w > 0 lets alulate the derivative w using impliit funtion theorem. We get the following: w + 4 e w + v 4e w = 6 3/ e w 3 + v 1e w + 8 e 3w Substituting it to the the expression for q we get: 1 1 e w v e w e w 1 1 Substituting q we get that the previous expression is negative whenever the following holds: q q v q > 0 We know that q > and thus q is always negative. is true only for suffi- Lemma 3 Variable q always inreases in. Moreover, q < iently high. Proof 4 v 14

15 The full derivative of q = e w with respet to is e w 4w ve w + w + 4we w 6 3/ e w 3 + v 1e w + 8 e 3w The denominator is positive. Substituting bak q = e w into the numerator we get q qv + 4qwq + w, whih is higher than zero sine q >. Thus q inreases in. In partiular, this implies that w is inreasing in. Let s show that there is just one point of intersetion of urves represented by q and 4 v. Taking the derivative of q with respet to and substituting the derivative of w we get: e w 4w ve w + w + 4we w 6 3/ e w 3 + v 1e w + 8 e 3w Substituting q = e w we get: q 4q w q4w v + w 8q q v q = q 4q w q4w v + w q 3 + v q The denominator is positive, sine q >. We know that at the point of intersetion q = substituting it to the expression we get 4 v 4 v8 v w 4 v v 4 v One an show that the last expression for any w is lower than the derivative of 8 with respet to, whih is Thus we an onlude that whenever we have an 4 v. intersetion of two urves it should be that 4 v rosses q from below. Hene, there is at most one point of intersetion, s.t. q intersets 4 v from below. 1 When = 0, q equals and thus for suffiiently small, q is higher than 4 v 4 v. When approahes 4 v, expression approahes infinity, while q is always limited for any finite. Thus there is a unique intersetion point of q and 4 v, s.t. q intersets 4 v from below, and thus q < is true for suffiiently high. 4 v Proof of Proposition 1 Sales in the ase of omplete and inomplete information w = v e w v e w 15

16 Sales in the ase of inomplete information are higher than sales in the ase of full information if the seond term is positive. Substituting q = e w get e w + ve w 4e w e w = + qv 4q qq to the seond term we The denominator is always positive. The numerator is positive if v > 4 or v < 4 and q < 4 v. We know that q dereases in, while 4 v does not depend on. Thus sales will be higher than v for suffiiently high if there is q, suh that < q < 4 v. Rearranging we get that ondition always holds for v 4. We also know that q inreases in. By Lemma?? we know that there is a unique intersetion point of q and 4 v, s.t. 4 v intersets q from below. Thus we an onlude that for suffiiently high sales are higher than v, sine there always be suh that v. The fat that the optimal prie is always lower than in the ase of full information immediately follows from Lemma 1. Proof of Proposition s dereases in Taking the derivative of?? and assuming that w is a funtion of we get: Substituting?? we get: v e w w + 1 v 1 v 1 e w + ew v e w + e w 3 e w e w 1 We know that e w > 0 and thus the derivative is negative if the seond term in the brakets is positive. Lets denote by q = e w. From the previous analysis we know that s and P are meaningful when < q < v and q is higher than 1, sine w 0. Thus the. ondition beomes max{, 1} q 1 + v Using q we an rewrite the ondition in the following form fq = q q + q v + q 3 > 0. Taking the derivative of the last expression with respet to q we get f q = 3 + qv q. The expression represents upward 16

17 sloping parabola. Taking the derivative we an find the minimum q m = 10 v 1. Sine q m <, the range of admissible values for q is on the upward sloping part of the parabola. Thus if > 1 and f > 0 or 1 and f 1 > 0 we an onlude that f q is positive on the whole range. Substituting to f we get v 1, whih is positive if v > 1. Note that > 1 and thus > 1 and thus 1 < < v < v. Assume now that < 1, substituting q = 1 we get 3 + v , whih is positive, sine < < v. Going bak to f we should hek that if > 1 then f = 3/ 3 v 1 is higher than zero. We already have seen that if > 1 then v > 1 and thus f > 0. Substituting q = 1 we get v , whih is greater than zero. P dereases in Taking into aount that q always dereases in it is easy to show that the optimal prie represented by?? always dereases in too. w v P Number of informed onsumers dereases in We know that in the equilibrium w onsumers buy the produt. Eah of the informed onsumers buys the produt with probability v P and thus the number of informed onsumers is the following: w + qqv = v P q v Taking the derivative with respet to q we get q q 3 v whih is positive, sine by?? q is always greater than. Thus the number of informed onsumers inreases in q and onsequently always falls in. Proof of Proposition 3 In general w first dereases and then inreases in Taking the derivative of RHS of?? with respet to q we get: 1 q v q 13 The expression is positive if 4 vq 4q + > 0. If v < 4 the last expression is upward sloping parabola and it is positive if q < q 1 = + or q > q v =. v Substituting q into?? we get < q < 1 + v. Thus q is always greater than q1. We know that q dereases in and if q > q then there are values of suh that w dereases in. This happens if < 1 v + v, whih redues to v > v +. Sine we are interested in the existene we substitute = 0 and obtain v < 4. 17

18 When q is lower than q expression?? is negative and w inreases in. Atually, the inreasing behavior appears when <, whih holds if v > 1. v When v > 4 we have downwards sloping parabola. In this ase q beomes negative and is higher than +. Thus expression?? is negative and w inreases on the whole v range in. Proof of Proposition 4 Consumer surplus w v P is non monotone in Substituting expressions for w and P in terms of q to CS = w v P we get: Deriving it with respet to q we get: v + qqv q v + qv q 3 We know that the denominator is positive and thus the whole expression is negative if v > or q < v. By ondition?? we know that < q < 1 v +. Thus there is a region where the derivative is negative if v > 1. Taking into aount that the derivative of q with respet to is negative we an onlude that if v > 1 then there is a region where CS inreases in. Proof of Proposition?? In general soial welfare first dereases in and then inreases. Taking the derivative of the expression for soial welfare with respet to q we get: vq v 6q + 5q q q 3 The sign of the derivative depends on sign of the following expression v 6q +5q, sine we know that q >. If v > 6 it is always positive and thus SW dereases in q and onsequently inreases in. If however v < 6 we have downward sloping parabola with roots q 1 = 5 1+4v 6 v and q = v 6 v. We an show that if v < 6 the first root is always less than and thus does not belong to admissible parameter range. The seond root is higher than if v > and is lower than 1 v + if < 1 v+ 1+4v. Summing up if v < then SW always dereases in. If < v < 6 then SW first dereases in up to the point where = 1 v+ 1+4v and then inreases. If v > 6 then SW always inreases in. 18

19 Proof of Proposition?? The amount of advertisement s first inreases, but then dereases in Taking the full derivative of solution for s we get e 1 w w +w v. Substituting the derivative of w with respet to we get the following expression: + 3/ we w + 4w ve 3w 8 we w 8e 3w 3 3/ + v 1e w + 6 e w Substituting q = e w we get: 1 q 8q w q4w v + w q 3 + v q Expressing equation for w in terms of q we get: w = 1 v q v q Substituting last equation to the derivative we get: 1 + q q 3 v + 4q 4q q 3 + v q We know that < q < 1 v +. When equals 0, q = 1. Substituting it into the derivative we get v v v, whih is positive if > 16. Thus if > 16 the optimal advertising first inreases in and otherwise dereases in. Expression in the brakets in?? is positive if q 3 v q v 8 6q > 0 and is negative otherwise. The expression dereases in for all values. Taking the derivative with respet to q we get quadrati expression in q, 3 + 3q v + qv 8. One an show that if v 5 4 then the expression is negative and thus the derivative dereases in q. We know that q inreases in. Thus when 5 4v we an onlude that the amount of advertising dereases in Lemma 4 If there is no advertising in the equilibrium, the optimal prie P always inreases in Proof Substituting into equation for the optimal prie s = 0 we get: ˆP = v + e w e w v e w = e w e w e w v v 16 19

20 And equation for sales beomes: w = e w e w 1 e w e w e w v 17 Note that for existene of the solution e w should be higher than v, thus the smallest possible value of w is lnv. Substituting it to?? we get v < e v 1 whih is always true. Thus the urve on the left hand side is above 45 degree line at w = lnv. The maximal value for w is v. Substituting y = w = v we get ye y + 1 e e y y e y e y y. Subtrating y and taking out ommon fator we get y e y e y e y y, whih is lower than 0, and thus the urve represented by the RHS of?? is below 45 degree line. Substituting x = w and y = v into?? we get x = e x e x 1 e x e x e x y. The derivative of the right hand side of the equation with respet to x is: e x e x e x y e x + y + y e x e x y The denominator is positive sine 1 y e x. Solving equation?? for y we get y = ex x+e x x e x 1. Substituting it to the numerator we get: xex 1 e x + x1 + e x 1 e x, whih is negative for any x > 0. This implies that if the urve rosses 45 degree line it should ross it from above. Thus the solution exists and it is unique. Substituting z = ew into P and taking the derivative with respet to z we get: z 1 v zz v 1 The denominator is positive sine z = e w > v. It is easy to see that the expression is positive too. Thus we proved that P inreases in z = ew. Taking the full derivative of ew with respet to we get: e w e w v w e w e w v v + e w v + e w e w v e w e w e w v Substituting w from?? into the expression in the brakets in the numerator we get: e w v e w e w 1 e w e w e w v e w e w v = 0

21 = e w v e w e w v e w 1 e w e w e w v e w In the brakets we get a quadrati expression in e w whih is negative if v or e w < 1 v. Substituting ew = 1 v to the right hand side of?? we get w = v 1 or equivalently e w = e v 1. Sine e v 1 1 v, w is s.t. e w < 1 v, whih implies that the optimal w 1 is lower than v. Hene, the derivative of the prie with respet to is positive. Proof of Proposition?? Prie P dereases in Lets onsider term q = e w. Taking full derivative of it with respet to we get: e w e w + e w w v 1e w + 6 e w + 8e 3w 3 3/ As we know the denominator is positive. The numerator is positive whenever + e w w > 0 18 When is suffiiently lose to 0, the expression is negative and thus e w falls in, while by Lemma??, w inreases in. Assume that there is ˆ, suh that the expression beomes positive. Assume further that exists > ˆ, suh that the expression beomes zero again. In this ase we know that e w does not grow in, while w ontinues to grow and thus for any suffiiently small ɛ > 0 term e +ɛw +ɛ should inrease in ɛ. Taking into aount the ontinuity of all involved funtions we an onlude that for any > ˆ the term is non dereasing in, whih implies that the optimal prie is also non dereasing in. Thus we have shown that in general the optimal prie first dereases, but than inreases. Now let s hek the onditions under whih we have eah part of the urve. For suffiiently small we know that?? is negative and thus prie dereases in. Thus the prie first dereases and then may inrease. We also know that advertising first inreases and then dereases. Thus to hek whether there is inreasing part of the urve we substitute solution for s = 0 to??. Thus substituting?? we obtain: v + v

22 The expression is positive whenever v + > 4. Taking the full derivative of solution for P we get w we get: ve w + e w w v 1e w + 6 e w w ve w +w e w + 8e 3w 3 3/. Substituting Substituting q = e w we obtain: qv qv w q 3 + v q 19 The denominator is positive. Substituting?? into the numerator of?? we get: qv + q v 4 q The numerator of?? is negative iff v < 4 and q > 4 v. Note that for suffiiently lose to 0, q is higher or equal to 1. Thus for a suffiiently small the optimal prie dereases in. As grows further two things may happen. It may approah 4 v, in whih ase we know that the optimal prie starts to inrease in. Another possibility is that at some point the optimal s beomes zero and we get the orner solution with s = 0. However, by Lemma?? we know that when there is no advertising P inreases in if v > 1. Thus if v > 1 the prie is non-monotone in. 0 Proof of Proposition?? For suffiiently small sales inrease in, but when beomes suffiiently high sales derease in The derivative of w with respet to is 3/ w + e w v w + 4w 4we w v 1e w 3 3/ + 6 e w + 8e 3w The denominator is positive. Substituting = 0 we get 1 8 v+4w, whih is positive and thus for suffiiently small, sales inrease in. Substituting q = e w we get w + q v w + 4qwq q 3 + v q Substituting maximal value for q, q = 1 v + we get

23 4vw + + v w v + 4v + v Note that w inreases in and we an hoose arbitrarily large for suffiiently small. Thus the expression is negative for suffiiently high, sine w is limited from above by 1. For suffiiently small onsumer surplus inreases in, but when beomes suffiiently high onsumer surplus dereases in The onsumer surplus is given by w v P. We know that for suffiiently small sales inrease in and by Propositions?? we know that prie dereases in. Thus for suffiiently small onsumer surplus inreases in. We also know that when beomes suffiiently high sales derease in, while prie inreases in. Thus for suffiiently high onsumer surplus dereases in. 3