Exposure Order Effects and Advertising Competition

Transcription

1 Exposue Ode Effects and dvetising Competition Oksana Loginova Univesity of Missoui-Columbia Depatment of Economics 118 Pofessional ldg Columbia, MO May 10, 2008 bstact This pape applies the theoies of exposue ode effects, developed in the psychology liteatue, to an industial oganization model to exploe thei ole in advetising competition. Thee ae two fims and infinitely many identical consumes. The fims poduce a homogeneous poduct and distibute thei bands though a common etaile. Consumes andomly aive at the etaile and buy thei most pefeed bands. The ode in which a consume sees the advetising messages affects his band pefeences. Unde the pimacy effect the consume pefes the band he fist saw advetised, unde the ecency the last encounteed band. The equilibium of the advetising game is chaacteized sepaately unde the pimacy and the ecency effects. In the fist setting all consumes ae initially unawae of the poduct existence. The equilibium advetising intensities, emakably, do not depend on the type of exposue ode effect. In the othe two settings some consumes have aleady fomed thei band pefeences. The pimacy and the ecency effects give ise to diffeent equilibium outcomes. JEL Classifications: C73, D11, D43, L13, M37. Keywods: advetising ode effects, pimacy, ecency. 1

2 I Intoduction Taditionally, advetising has been studied by two vey diffeent subfields of social science, economics and psychology. 1 The economic liteatue on advetising has been concened with such impotant questions as the infomational content of ads, the socially optimal level of advetising, and fim ivaly. The psychology liteatue, among othe things, has exploed the connection between infomation pocessing and advetising effectiveness. The goal of this pape is to develop a model that incopoates economics and psychology to examine how infomation pocessing biases affect advetising competition between fims. Past eseach in psychology has shown that the ode in which consumes ae exposed to bands influence consume pefeences. Two advetising ode effects have eceived attention. The pimacy effect is chaacteized by geate pesuasion consequence of the fist advetising message. On the othe hand, the ecency effect occus when the last encounteed band is pefeed. Whethe exposue ode esults in the pimacy effect o in the ecency, depends on diffeent factos, such as oveall involvement, motivation, attitude stength, and the time between infomation exposue and pefeence constuction. Lana 1963) showed that subjects geatly inteested in the topic exhibited the pimacy effect, wheeas subjects with minimal inteest exhibited the ecency effect. The pimacy effect occus when paticipants ae motivated to elaboate the initial message and show citical thinking towad late infomation, wheeas the ecency effect occus when motivation is low Haugtvedt and Wegne 1994). unel and Nelson 2003) examined how advetising ode effects connect to gende diffeences. Thei esults suggested that unde conditions of low involvement, females exhibited the pimacy effect, males the ecency effect. Unde conditions of highe involvement, females continued to exhibit the pimacy effect, the ecency effect with male disappeaed when the advetisement matched thei values. Neidich and Swain 2007) found the pimacy effect when the delay between attibute encoding and pefeence constuction is long a few days). On the othe hand, the ecency effect is moe likely to occu when the delay is shot a few minutes). 1 Of couse, advetising is one of the pimay subjects studied in the field of maketing. Most papes in maketing concening advetising, howeve, can be classified as having eithe economic o psychologic foundations. 2

3 The model pesented hee applies the pimacy and the ecency effects to advetising competition. The agents ae two fims and infinitely many identical consumes of total mass one. The fims poduce a homogeneous poduct; fim poduces band and fim poduces band. The fims distibute thei bands though a common etaile. Each consume andomly accoding to a Poisson pocess) aives at the etaile and buys his most pefeed band. Consumes lean about the poduct existence and fom thei band pefeences though the fims advetising. The ode in which a consume sees the advetising messages affects his band pefeences. Unde the pimacy effect the consume puchases the band he fist saw advetised, wheeas unde the ecency effect whicheve band he most ecently saw advetised. The fims compete in advetising intensities the ate paametes of Poisson pocesses that advetising messages follow. The equilibium of the advetising game is chaacteized sepaately unde the pimacy and the ecency effects. Thee setting ae consideed. In Setting 1 New Maket) consumes initially ae unawae of the poduct existence. In this setup, the equilibium advetising intensities ae symmetic and, emakably, do not depend on the type of exposue ode effect. In Setting 2 Gowing Maket) it is assumed that at the beginning of the game a numbe of consumes believe band is bette, the equal numbe pefe band, and the est ae unawae of the poduct existence. to diffeent equilibium outcomes. In this setup the two exposue ode effects give ise Results show that the fims choose highe advetising intensities unde the ecency effect than unde the pimacy. In Setting 3 fim has advantage ove fim. Specifically, at the beginning of the game a numbe of consumes pefe band and the est ae unawae of the poduct existence. Unde the pimacy effect the equilibium advetising intensities ae symmetic, wheeas unde the ecency effect fim chooses highe advetising intensity than fim. Vaious welfae implications ae investigated. This pape contibutes to the small but gowing body of economic liteatue on advetising gounded in psychological eseach. Kähme 2004) developed a monopolistic model in which the buye may not ecall coectly his past expeience with the poduct. dvetising 3

4 can activate memoy and help consumes to ecollect thei past expeiences, o distot actual consumption expeiences. Shapio 2006) exploed two altenative advetising mechanisms that yield vey diffeent pedictions. In the fist, advetising convets memoies of bad expeiences into memoies of good ones. In the second, advetising makes favoable expeiences moe likely to be emembeed. ekke and Rege 2007) captued availability heuistic phenomenon, accoding to which consumes cannot distinguish between the ecommendations o eal people and fictitious chaactes in advetisements. Thei analysis showed that even if a peson knows that his obsevations of othes may be distoted by advetising, it is still ational fo him to choose whicheve poduct he has obseved most often. In the next section, the fomal model is pesented. The thee settings ae analyzed in sections III though VI. Concluding emaks appea in Section VII. ll poofs ae elegated to the ppendix. 4

5 II The Model Fims and Consumes Two fims, and, poduce a homogeneous poduct using zeo maginal cost technology. The fims distibute thei bands though a common etaile. The demand side consists of infinitely many identical consumes of total mass 1. Consumes ae isk-neutal, possess continuous-time discount ate > 0, and have a sequence of unit demands fo the poduct. In paticula, a consume eceives a goss payoff of 1 wheneve he aives at the etaile and buys his most pefeed band. The numbe of times the consume visits the etaile follows a homogenous Poisson pocess with ate paamete σ > 0. That is, in a small time inteval t the consume visits the etaile with infinitesimal pobability σ t + o t). 2 Consumes lean about the poduct existence and fom thei band pefeences though advetising. The numbe of ads descibing band a consume eceives is a homogeneous Poisson pocess with ate paamete, the numbe of ads descibing band ads he eceives follows a Poisson pocess with ate paamete. It is assumed that consumes eceive advetising messages independently fom each othe they watch/ listen to diffeent TV/ adio pogams at diffeent times). dvetising intensities, and, ae chosen by the fims. The discounted cost needed to achieve intensity 0 is c)/. Function c ) is convex and inceasing, c > 0 and c > 0. The standad bounday assumptions, c0) = c 0) = 0 and lim c ) =, guaantee the existence of an inteio solution. dvetising Ode Effects The ode in which a consume sees the advetising messages affects his band pefeences. Two main effects ae consideed: pimacy and ecency. The pimacy effect is chaacteized by geate pesuasion consequence of the fist advetisement. In the context of the pesent 2 Fomally, let N t denote the numbe of times a consume visits the etaile by time t. N t is a Poisson pocess with ate paamete σ > 0 if and only if i) the numbe of visits duing one time inteval is independent of the numbe of visits duing a diffeent non-ovelapping time inteval; ii) in a small time inteval t the consume visits the etaile with pobability P{N t+ t = N t + 1} = σ t + o t), and he visits the etaile moe than one time with pobability o t). On Poisson pocesses see, fo example, Doob 1953). 5

6 model, the pimacy effect implies that the consume puchases whicheve band he fist saw advetised. The ecency effect, on the othe hand, occus when the last encounteed band is pefeed. That is, the consume buys whicheve band he most ecently saw advetised. Let x t) denote the numbe of consumes that at time t believe band is bette. Similaly, x t) is the numbe of consumes that pefe band. The est of the consumes, x t) = 1 x t) x t), ae unawae of the poduct existence. These consumes will be efeed to as ignoant consumes, fo the lack of a bette tem. How do x t), x t), and x t) evolve unde the pimacy and the ecency effects given advetising intensities and? Conside the pimacy effect. The pobability that a consume sees fim s advetisement in a small time inteval t is t + o t). Hence, the numbe of ignoant consumes who see fim s ad is x t) t + o t). Similaly, the numbe of ignoant consumes who see fim s ad is x t) t + o t). Theefoe, x t) inceases at instantaneous ate x t), x t) inceases at ate x t), and the numbe of ignoant consumes deceases at ate + )x t). lgebaically, d dt x t) = + )x t), d dt x t) = x t), d dt x t) = x t). 1) The ecency effect diffes fom the pimacy effect in that consumes switch fom one band to the othe when they see an advetisement fo the latte. Thus, the numbe of consumes who switch fom band to band is x t) t + o t). 6

7 Similaly, the numbe of consumes who switch fom band to band is x t) t + o t). Theefoe, x t) inceases at ate x t) + x t)) and deceases at ate x t), x t) inceases at ate x t) + x t)) and deceases at ate x t). lgebaically, d dt x t) = + )x t), d dt x t) = x t) + x t)) x t), d dt x t) = x t) + x t)) x t). 2) The tansition matices infinitesimal geneatos) fo the pimacy and the ecency effects ae, espectively, and + ) 0 0 P = ) 0 0 R =. Note that the column sums of P and R ae equal to zeo, the nondiagonal enties ae nonnegative, and the diagonal enties ae nonpositive. Let x t) xt) = x t). x t) Then the systems of diffeential equations 1) and 2) can be ewitten in a matix fom, d xt) = P xt) dt 7

8 and d xt) = R xt). dt The solutions ae given by xt) = e tp x0) 3) and xt) = e tr x0), 4) whee x0) is the vecto of initial conditions. 3 Pofit Functions of the Fims The diagonalization 4 is used to compute the exponentials e tp and e tr, e + )t 0 0 e tp = e + )t e + )t ) and e tr = e + )t 0 0 e + )t + + e + )t e + )t + + e + )t e + )t + e + )t ) Staightfowad but tedious calculations ae elegated to the ppendix. Substituting the above expessions into 3) and 4) allows to compute xt) unde the pimacy and the ecency effects, hence the numbe of consumes that pefe band, x t), and the numbe of consumes that 3 The exponential e t is defined as e t = X n=0 t) n. n! 4 The fist step is to diagonalize the matix, = QDQ 1, whee D = diagd 1, d 2,...). Once is diagonalized it is easy to compute exponential e t, e t = X QtD) n Q 1 n=0 n! = Q diag e d1t, e d2t,... Q 1. 8

9 believe band is bette, x t). Thee is no pice competition between the fims, as, by assumption, each consume buys whicheve band he believes is bette. The fims set thei pices equal to 1, consume valuation fo the poduct. Hence, the pofit functions of the fims ae Π, ) = 0 e t x t)σ dt c ) and Π, ) = 0 e t x t)σ dt c ). Thee Diffeent Settings The fims choose thei advetising intensities and simultaneously at the beginning of the game. The equilibium concept employed is Nash equilibium. Thee diffeent settings ae consideed. In Setting 1 it is assumed that at time t = 0 consumes ae unawae of the poduct existence, 1 x0) = 0. 0 In Setting 2, consumes ae ignoant, 1 )/2 believe band is bette, and 1 )/2 pefe band. That is, x0) = ). 1 ) In Setting 3 fim has advantage ove fim. In paticula, it is assumed that at the beginning of the game consumes ae unawae of the poduct existence, the est believe band is bette, x0) =

10 In the next thee sections each setting is analyzed sepaately unde the pimacy and the ecency effects. III Setting 1 New Maket) Conside the setting in which initially consumes ae unawae of the poduct existence. Unde the pimacy effect, xt) = e + )t e + )t = e + )t e + )t e + )t + e + )t +. Hence, and x t) = e + )t + 7) x t) = e + )t +. 8) Obseve that the fist column of matix 5) coincides with the fist column of matix 6). Hence, functions x t) and x t) ae the same unde the ecency effect as unde the pimacy. This appaently supising esult can be easily explained. Indeed, the numbe of consumes who switch fom band to band in a small time inteval t is x t) t + o t). Similaly, x t) t + o t) consumes switch in the opposite diection fom band to band. It follows fom 7) and 8) that fo any given and x t) = x t) 10

11 holds fo all values of t. Theefoe, consume switching does not affect x t) and x t) obtained unde the pimacy effect. Substituting 7) and 8) into the fims pofit functions and computing the integals yield Π, ) = σ + + ) c ) and Π, ) = σ + + ) c ). Let and denote the advetising intensities chosen by the fims in equilibium. Poposition 1 Equilibium in Setting 1). Unde both the pimacy and the ecency effects = =, whee is implicitly defined by σ + ) 2 + ) 2 = c ). How do the equilibium levels of advetising intensities compae with the social optimum? The social welfae is given by W, ) 0 e t x t) + x t))σ dt 1 c ) + c )). In the pesent model it coincides with the sum of the fims pofits. Hence, in Setting 1 the social welfae equals W, ) = σ + ) + + ) 1 c ) + c )). Let and denote the socially optimal advetising intensities. The next poposition shows that the fims ove-advetise in equilibium. Intuitively, each fim caes only about the numbe of consumes that pefe its own band. The social planne, on the othe hand, aims to incease the numbe of consumes that lean about the poduct existence though the fims advetising messages, x t) + x t). 11

12 Poposition 2 Social Optimum in Setting 1). The socially optimal levels of advetising intensities ae = =, whee is implicitly defined by σ 2 + ) 2 = c ). dvetising is socially excessive, >. In the simple setting with x 0) = 1, the equilibium outcome is the same unde the pimacy and the ecency effects. Howeve, if at t = 0 a numbe of consumes have aleady fomed thei pefeences about the two bands, as in Setting 2, o if one of the fims has competitive advantage, as in Setting 3, the two advetising ode effects have diffeent stategic implications. IV Setting 2 Gowing Maket) Conside the setting in which at the beginning of the game 1 )/2 consumes believe band is bette, 1 )/2 consumes pefe band, and the est ae ignoant consumes. Unde the pimacy effect, xt) = e + )t 0 0 e + )t e + )t ). 1 ) Hence, and x P t) = e + )t ) 9) + 2 x P t) = e + )t ). 10)

13 Supescipt P stands fo pimacy.) Substituting 9) and 10) into the fims pofit functions and computing the integals yield Π P σ, ) = + + ) )σ c ) 11) and Π P σ, ) = + + ) )σ c ). 12) These fomulas deseve some discussion. t the beginning of the game, 1 )/2 consumes pefe band and unde the pimacy effect will neve switch to band they ae locked up by fim ). Fim eans )σ fom these consumes. Similaly, 1 )/2 consumes ae locked up by fim. Fim eans )σ fom these consumes. Finally, consumes ae unawae of the poduct existence. If an ignoant consume fist sees an ad descibing band, he becomes fim s custome. If the consume fist sees fim s ad, he becomes fim s custome. Fims and ean, espectively, and σ + + ) σ + + ) fom initially ignoant consumes. In contast to Setting 1, the ecency effect in Setting 2 gives ise to a diffeent vecto xt). 5 5 Recall that in Setting 1, x t) = x t) holds fo any values of and. Hee, x P t) = x P t) holds only fo =. 13

14 The pofits of the fims ae given by Π R σ, ) = + + ) )σ ) σ ) + + ) c ) 13) and Π R σ, ) = + + ) )σ ) σ ) + + ) c ). 14) Supescipt R stands fo ecency.) It follows that fim s pofit function unde the ecency effect 13) diffes fom that unde the pimacy effect 11) by ) σ ) + + ). 15) This tem eflects consume switching between the bands. It entes with negative sign fim s pofit function 14). Let P and P unde the pimacy effect. Similaly, let denote the advetising intensities chosen by the fims in equilibium and denote the equilibium intensities chosen unde the ecency effect. Poposition 3 shows that the equilibium advetising intensities ae highe unde the ecency effect. This esult is diven by 15), which ceates exta incentives fo the fims to advetise. Poposition 3 Equilibium in Setting 2). i) Unde the pimacy effect P = P = P, whee P is implicitly defined by σ P + ) 2 P + ) 2 = c P ). ii) Unde the ecency effect = =, whee is implicitly defined by σ + ) 2 + ) ) σ 2 + = c ). iii) Thee is moe advetising unde the ecency effect, > P. 14

15 The social welfae is the same unde the pimacy and the ecency effects, W, ) = σ + ) + + ) + 1 )σ 1 c ) + c )). Indeed, it does not matte fo the social planne whethe a consume switches fom one band to the othe afte he obseves an advetising message fo the latte the ecency effect), o not the pimacy effect). The next poposition shows that the fims ove-advetise unde the pimacy effect, and, hence, unde the ecency effect. Poposition 4 Social Optimum in Setting 2). The socially optimal levels of advetising intensities ae = =, whee is implicitly defined by σ 2 + ) 2 = c ). dvetising is socially excessive, > P >. V Setting 3 One Fim Has dvantage) Conside the setting in which at the beginning of the game consumes ae ignoant and the est believe band is bette. s in Setting 2, the pimacy and the ecency effects give ise to diffeent equilibium outcomes. Unde the pimacy effect, the pofit functions of the fims ae given by Π P σ, ) = + + ) + 1 )σ c ) 16) and Π P σ, ) = + + ) c ). 17) Unde the ecency effect, Π R σ, ) = + + ) + 1 ) σ + ) + + ) c ) 18) 15

16 and Π R, ) = σ + + ) c ). 19) Fomulas 16) though 19) deseve some discussion. Conside initially ignoant consumes. The pofits that fim and fim ean fom these consumes ae the same unde the ecency effect as unde the pimacy the esult obtained in Setting 1). In paticula, fim eans σ + + ) and fim eans σ + + ). Conside the est of the consumes that at time t = 0 pefe band. These consumes ae locked up by fim unde the pimacy effect, hence fim eans 1 ) σ 20) and fim eans 0. Unde the ecency effect fim loses pat of 20) to fim. In paticula, fim eans and fim eans σ + ) 1 ) + + ) σ 1 ) + + ). Poposition 5 Equilibium in Setting 3). i) Unde the pimacy effect P = P = P, whee P is implicitly defined by σ P + ) 2 P + ) 2 = c P ). 16

17 ii) Unde the ecency effect lso, < < and ae implicitly defined by + 1 ). σ +) + +)2 σ +) + +)2 = c ), = c ). iii) Fim chooses highe advetising intensity unde the ecency effect than unde the pimacy, > P. The fist esult of Poposition 5 is easily undestood. Unde the pimacy effect 1 ) consumes ae locked up by fim. Fim and fim ae symmetic with espect to initially ignoant customes, fo which they compete. Hence, the fims choose the same advetising intensities in equilibium, P = P. The esult that fim chooses highe advetising intensity than fim unde the ecency effect is also intuitive. t the beginning of the game, fim s advetising messages affect only ignoant consumes, as the est of the consumes aleady believe band is bette. Fim s messages, on the othe hand, affect all consumes ignoant as well as those who initially pefeed band. Thus, fim s incentives to advetise ae stonge than fim s, especially at the beginning. This esults in >. Finally, Poposition 5 shows that fim chooses highe advetising intensity unde the ecency effect than unde the pimacy, intensities > P = P. Fim s equilibium advetising and P = P, howeve, cannot be anked, because Fim s best eply unde the ecency effect is non-monotonic in. The social welfae function in the cuent setting coincides with the one obtained in the pevious setting, W, ) = σ + ) + + ) + 1 )σ 1 c ) + c )). The next poposition shows that unde the pimacy effect both fims ove-advetise, and, hence, unde the ecency effect fim ove-advetises and 17 cannot be anked).

18 Poposition 6 Social Optimum in Setting 3). The socially optimal levels of advetising intensities ae = =, whee is implicitly defined by Moeove, > P >. σ 2 + ) 2 = c ). VI Setting 2 vesus Setting 3 The social welfae function is the same in Setting 2 and Setting 3. This makes compaison between the equilibia in the two settings legitimate and, theefoe, valuable. Poposition 7 Equilibium in Setting 2 vs. Equilibium in Setting 3). i) Unde the pimacy effect, the fims equilibium advetising intensities in Setting 2 ae the same as in Setting 3, P [S2] = P [S2] = P [S3] = P [S3]. ii) Unde the ecency effect, fim s equilibium advetising intensity is highe in Setting 3 than in Setting 2, [S3] > [S2]. The coesponding settings ae specified in squae backets.) The fist esult of Poposition 7 follows diectly fom Poposition 3i) and Poposition 5i). Intuitively, in both settings unde the pimacy effect the fims compete fo initially ignoant consumes. Whethe the est of the consumes ae locked up by both fims, as in Setting 2, o by fim, as in Setting 3, does not change the competition between the fims. Hence, the equilibium advetising intensities in Setting 2 ae the same as in Setting 3. Next, conside the ecency effect. In Setting 3 at the beginning of the game, fim s advetising messages affect all consumes, ignoant and 1 ) that pefe band. Howeve, 18

19 in Setting 2 fim s messages have actual effect only on + 1 )/2 consumes ignoant and those who initially pefeed band. Thus, fim s incentives to advetise ae stonge in Setting 3. s a esult, fim chooses highe advetising intensity in Setting 3 than in Setting 2. VII Conclusion This pape has applied the theoies of exposue ode effects, developed in psychology liteatue, to an industial oganization model to exploe thei ole in advetising competition. The equilibium advetising intensities wee chaacteized sepaately unde the pimacy and the ecency effects in thee diffeent settings. It was shown that in a new maket Setting 1) the advetising equilibium does not depend on the type of exposue ode effect. Howeve, in a gowing maket whee a numbe of consumes have aleady fomed thei pefeences about the two bands Setting 2), o if one of the fims has moe customes than the othe Setting 3), the pimacy and the ecency effects give ise to diffeent equilibium outcomes. Unde the pimacy effect the consumes that pefe a paticula band fom the outset will neve switch to the othe band. The fims compete fo initially ignoant consumes, and, theefoe, choose the same advetising intensities acoss both settings. Unde the ecency effect, on the othe hand, each fim s advetising messages affect ignoant consumes as well as those who pefe the ival band. In Setting 2 the fims choose highe advetising intensities unde the ecency effect than unde the pimacy. In Setting 3 fim chooses highe advetising intensity than fim. It was also shown that in all thee settings the fims ove-advetise in equilibium. One of the limitations of the pesent model is that the fims advetising intensities ae fixed thoughout the game. dynamic vesion of the advetising game, in which the fims choose thei advetising intensities continuously, does not have an analytical solution. This is because the diffeential equations that descibe the pimacy and the ecency effects ae not linea in the advetising intensities. 6 nalyzing the static game unde diffeent initial 6 On diffeential games see, fo example, Zaccou 2002) and Eickson 2003). 19

20 conditions, howeve, uncoves some featues of the dynamic setup. The numbe of ignoant consumes,, deceases with the passage of time. Theefoe, unde the pimacy effect the equilibium advetising intensities gadually convege to zeo. Unde the ecency effect the fims advetise moe at the beginning, but thei equilibium advetising intensities will always be bounded away fom zeo. Refeences [1] ekke, Kjell ne, and Mai Rege, 2007, dvetising as a Distotion of Social Leaning,.E. Jounal of Theoetical Economics, vol. 7, iss. 1, aticle 38. [2] unel, Fédéic F., and Michelle R. Nelson, 2003, Message Ode Effects and Gende Diffeences in dvetising Pesuasion, Jounal of dvetising Reseach, 43, [3] Doob, Joseph L., 1953/1990, Stochastic Pocesses, 1990 edition, Wiley Classics Libay, New Yok. [4] Eickson, Gay M., 2003, Dynamic Models of dvetising Competition, 2nd edition, Intenational Seies in Quantitative Maketing, Kluwe cademic Publishes, Nowell, M. [5] Haugtvedt, Cutis P., and Duane T. Wegene, 1994, Message ode Effects in Pesuasion: n ttitude Stength Pespective, Jounal of Consume Reseach, 21, [6] Kähme, Daniel, 2004, dvetising and Consume Memoy, Univesity College London Mimeogaph. [7] Lana, Robet E., 1963, Inteest, Media, and Ode Effects in Pesuasive Communications, Jounal of Psychology, 56, [8] Niedich, Ronald W., and Scott D. Swain, 2007, The Effects of Exposue-Ode and Maket Enty-Infomation on and Pefeence: Dual Pocess Model, Jounal of the cademy of Maketing Science, DOI /s x. [9] Shapio, Jesse M., 2006, Memoy-Jamming Theoy of dvetising, Univesity of Chicago Mimeogaph. [10] Zaccou, Geoge ed.), 2002, Optimal Contol and Diffeential Games, dvances in Computational Management Science, vol. 5, Kluwe cademic Publishes, Nowell, M. 20

21 ppendix Deivations of 5) and 6). Conside matix P. This matix has eigenvalues 0, + ), 0. Hence, P = Q P Q 1, whee P = 0 + ) 0, Q = , Q 1 = The columns of Q ae ight eigenvectos of P and the ows of Q 1 ae left eigenvectos. The eigenvectos ae unique up to a multiplicative constant. The constant in the ight eigenvecto fo eigenvalue 1 is chosen so that it is the invaiant pobability distibution fo P, Next, conside matix R. This matix has eigenvalues 0, + ), + ). Hence,. R = Q RQ 1, whee R = 0 + ) 0, ) 21

22 Q is the same as above. Once P and R ae diagonalized, it is easy to compute exponentials e tp and e tr, e tp = n=0 Q t P ) n Q 1 n! = Q 0 e + )t 0 Q 1 = e + )t 0 0 e + )t e + )t and e tr = n=0 = R) n Q t Q 1 n! = Q 0 e + )t e + )t e + )t 0 0 e + )t + + e + )t Q 1 e + )t + + e + )t e + )t + e + )t Poof of Poposition 1. The pofit function of fim i =, is Π i, ) = 0 = σ i + e t x i t)σ dt c i) = σ i ) c i) = ) e t e + +)t dt c i) σ + + ) c i). Fim i s best eply is implicitly defined by the fist ode condition fo the pofit maximization poblem, o i j. ecause i Π i, ) = 2 2 i σ j + ) + + ) 2 c i ) = 0, σ j + ) + + ) 2 = c i ), Π i, ) = 2σ j + ) + + ) 3 c i ) < 0, 22

23 the fist ode condition is sufficient. Theefoe, equilibium advetising intensities satisfy σ +) + +)2 = c ), σ +) + +)2 = c ). The equilibium is unique and symmetic. Indeed, it follows fom above that + ) c ) = + ) c ). ecause + )c ) is an inceasing function of, it must be the case that = =, whee satisfies σ + ) 2 + ) 2 = c ). 21) Next, obseve that the left-hand-side of 21) is deceasing and the ight-hand-side is inceasing in. Hence, the solution is unique. Poof of Poposition 2 The fist ode conditions fo the social planne s optimization poblem ae σ 2 = c + + σ 2 = c + + ), ). The matix of second deivatives 2σ + c ) 2σ +) 3 + +) 3 2σ 2σ + +) 3 + c ) +) 3 23

24 is negative semidefinite, which implies the sufficiency of the fist-ode conditions. y symmety, = =, whee satisfies σ 2 + ) 2 = c ). 22) It follows fom 21), 22), and the convexity of c) that >. Poof of Poposition 3 Each pat is poven in tun. i) Conside the pimacy effect. Fim s best eply is given by the fist ode condition σ + ) + + ) 2 = c ), fim s by σ + ) + + ) 2 = c ), s in the poof of Poposition 1, it can be shown that the fist-ode conditions ae sufficient and that the equilibium is unique and symmetic, P P satisfies = P = P, whee σ P + ) 2 P + ) 2 = c P ). 23) ii) Conside the ecency effect. Fim s best eply is given by the fist ode condition σ + ) + + ) ) σ2 + ) + + ) 2 = c ), fim s by σ + ) + + ) ) σ2 + ) + + ) 2 = c ). It is staightfowad to show that the fist-ode conditions ae sufficient and that the 24

25 equilibium is unique and symmetic, = =, whee satisfies σ + ) 2 + ) ) σ 2 + = c ). 24) iii) It follows fom 23), 24), and the convexity of c) that > P. Poof of Poposition 4 The fist ode conditions fo the social planne s optimization poblem ae σ 2 = c + + σ 2 = c + + ), ). y symmety, = =, whee satisfies σ 2 + ) 2 = c ). 25) It follows fom 23), 24), 25), and the convexity of c) that > P >. Poof of Poposition 5 Each pat is poven in tun. i) Conside the pimacy effect. Fim s best eply is given by the fist ode condition σ + ) + + ) 2 = c ), fim s by σ + ) + + ) 2 = c ), The equilibium is unique and symmetic, P = P = P, whee P satisfies σ P + ) 2 P + ) 2 = c P ). 26) 25

26 ii) Conside the ecency effect. Fim s best eply is given by the fist ode condition σ + ) + + ) 2 = c ), fim s by σ + ) + + ) 2 = c ). Theefoe, and satisfy σ +) + +)2 σ +) + +)2 = c ), = c ). 27) It is left to show that < < + 1 ). y contay, suppose. Then c ) c ). It follows fom 27) that o σ + ) + + ) 2 σ + 1 ). + ) + + ) 2, This contadicts the supposition. Hence, Next, suppose o <. + 1 ). Then c ) > c σ + ) + + ) 2 > σ + ) + + ) 2, < + 1 ). ). It follows fom 27) that This contadicts the supposition. Hence, < + 1 ). iii) y contay, suppose P. Then c P ) c ). It follows fom 26) and the 26

27 second equation in 27) that σ P + ) 2 P + ) 2 σ + ) The ight-hand-side of the above inequality is geate than + + ) 2. 28) σ + ) 2 + ) 2. Indeed, can be ewitten as σ + ) + + ) 2 σ + ) 2 + ) 2 σ ) + + ) ) + )2 2 + ) 2 > 0. Theefoe, 28) implies σ P + ) 2 P + ) 2 σ + ) 2 + ) 2. ecause + )/2 + ) 2 is a deceasing function of, it follows that contadicts the supposition. Hence, > P. > P. This Poof of Poposition 6 The esult follows diectly fom Poposition 3i), Poposition 4, and Poposition 5i,iii). Poof of Poposition 7 Each pat is poven in tun. i) The esult follows diectly fom Poposition 3i) and Poposition 5i). ii) y contay, suppose [S2] [S3]. Then c [S2]) c [S3]). It follows 27

28 fom 24) and the second equation in 27) that σ [S2] + ) 2 [S2] + ) ) σ 2 [S2] + σ [S3] + ) [S3] + [S3] + ) 2. 29) The ight-hand-side of the above inequality is geate than σ [S3] + ) 2 [S3] + ) 2, as was shown in the poof of Poposition 5. Theefoe, 29) implies σ [S2] + ) 2 [S2] + ) ) σ 2 [S2] + σ [S3] + ) 2 [S3] + ) 2, o σ [S2] + ) 2 [S2] + ) ) σ 2 [S2] + ) 2 σ [S3] + ) 2 [S3] + ) 2. ecause +)/2+) 2 is a deceasing function of, it follows that This contadicts the supposition. Hence, [S3] > [S2]. [S3] > [S2]. 28