Competing for Customers Attention: Advertising When Consumers Have Imperfect Memory

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1 Competing fo Customes Attention: Advetising When Consumes Have Impefect Memoy Oksana Loginova Univesity of Missoui-Columbia Depatment of Economics 118 Pofessional Bldg Columbia, MO Decembe 1, 2006 Abstact This pape applies the theoy of memoy fo advetising, developed in the consume behavio liteatue, to an industial oganization setting to povide insight into advetising stategies in impefectly competitive makets. 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 and ae willing to buy one unit of the poduct. They ae unawae of the existence of a paticula band unless they emembe an ad descibing it. Unde etoactive intefeence consumes emembe ecently seen ads and foget about ads they saw in the past. Unde poactive intefeence the ability of consumes to ecall new ads is hampeed by past ad exposue. The equilibium of the advetising game is chaacteized fo both poactive and etoactive intefeences acoss thee stategic settings. In the Simultaneous Move setting, the fims equilibium advetising fequencies, emakably, do not depend on the type of intefeence. In the Sequential Move and Dynamic settings, poactive and etoactive intefeences do give ise to diffeent equilibium outcomes. JEL Classifications: C73, D11, D43, L13, M37. Keywods: advetising, memoy, fogetting, competitive intefeence. 1

2 1 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 has among othe things) dealt with memoy. In this pape, a model is constucted that incopoates economics and psychology to examine the inteplay between advetising and memoy with espect to fims advetising decisions. The banch of economic liteatue to which the model pesented in this pape most closely concens is infomative advetising in new makets, whee consumes ae a pioi uninfomed about the existence of goods 2. In a pioneeing study, Buttes 1977) investigated the setting in which fims sell a homogeneous poduct, and consumes lean about poduct availability and pices though advetising. He showed that the esulting equilibium involves pice dispesion, with high pices advetised moe intensively than low pices. Moe stiking was his finding that fee makets poduce the socially optimal level of advetising. Stegman 1991) modified Buttes model to allow buyes to have heteogeneous esevation values. He showed that each fim unde-advetises in equilibium. Gossman and Shapio 1984) examined infomative advetising with spatially diffeentiated poducts, finding that the maket-detemined levels of advetising ae excessive. The cuent pape takes a vey diffeent tack fom the extant liteatue on infomative advetising. Specifically, in this pape a setting is studied in which consumes occasionally foget about the availability of goods and fims advetise in ode to emind them. The psychology liteatue on advetising is also vast and ich. Of paticula inteest fom the pespective of this study ae the theoies concening why consumes might foget the ads they have seen. Most widely known among them ae decay theoy, intefeence theoy, 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 Fo infomative advetising in established makets, whee consumes ae well infomed about availability and popeties of goods, see, fo example, Beste and Petakis 1995), LeBlanc 1998). 2

3 and situational fogetting Tulving and Pealston 1966) 3. Accoding to the decay theoy, memoy taces ae fomed when a peson leans infomation, and when no longe in use, these taces gadually fade with the passage of time Ebbinghaus 1885, Woodwoth 1938). The main agument against this intuitively undestandable theoy is that it does not account fo pocesses occuing between initial leaning and etieval. Altenatively, fogetting may occu because in the limited stoage capacity ealie leaning competes, o intefees, with late leaning Jenkins and Dallenbach 1924, Undewood 1957, Munane and Shiffin 1991). The intefeence theoy can wok eithe way. Unde poactive intefeence, ealie leaning makes it moe difficult to encode new infomation. Unde etoactive intefeence, late leaning hampes the ability to emembe peviously leaned infomation. Anothe theoy fom the infomation pocessing appoach was intoduced by Tulving 1974), who suggested that people foget cetain infomation because the featues of cuent infomation do not match those they ty to etieve. The theoy of situational fogetting implies that the taces of old infomation ae not lost but equie adequate stimulation fo thei activation and etieval. These theoies of fogetting, the competitive intefeence phenomenon in paticula, have been caied ove into consume behavio liteatue in analysis of consume memoy fo advetising. Buke and Sull 1988) wee the fist to conduct a seies of expeiments that examine intefeence-induced fogetting in an advetising context. The fist expeiment tested how consume memoy fo magazine ads is affected by subsequent exposue to competitos ads, finding that ecall is significantly diminished by highe level of etoactive intefeence. The second expeiment examined how the leaning of new ads is affected by past ad exposue, demonstating that poactive intefeence has a negative effect on consume ability to lean and etain new infomation pesented in futue advetising. The thid expeiment investigated how consume memoy fo a paticula ad is affected by the ad s level of epetition in vaious competitive envionments. The esults suggested that epetition of the same ad only has a positive effect on ecall of the advetised poduct when thee is little o no advetising fo 3 See Solso 1988) fo a textbook eview fo these theoies. 3

4 competitive poducts. Theefoe, highe levels of ad epetition may not ovecome detimental effects of ad intefeence 4. In subsequent studies, eseaches exploed the ole of band familiaity in competitive advetising. When consumes ae exposed to an ad fo a familia and matue band, the ad is easily ecognized and stoed in memoy in the well-established schema fo the band. Kent and Allen 1994) showed expeimentally that, in compaison to unfamilia bands, advetising infomation elated to familia bands is less sensitive to competitive intefeence 5. Summaizing the above, economists have dealt with advetising in competitive makets without taking into account the ole of memoy. Psychologists have studied consume memoy fo advetising but not, of couse, equilibium advetising stategies employed by fims. This pape bings the two togethe. Its pupose is to constuct a model of consume memoy fo advetising, based on the decay and intefeence theoies of fogetting, and to embed this model in a setting of oligopolistic fim ivaly. The agents of the model pesented in this pape ae two fims and infinitely many identical consumes of total mass one. The fims poduce a homogeneous good; fim A poduces band A and fim B poduces band B. The fims distibute thei bands though a common etaile. Consumes andomly aive at the etaile and ae willing to buy one unit of the good. To povide a ole fo advetising, it is assumed that a consume is unawae of the existence of a paticula band unless she saw and emembe an ad descibing it. All the events aivals to the etaile, advetising messages, and fogetting ae Poisson pocesses. Consume memoy fo advetising is modelled as a continuous-time Makov chain that takes values in the state space consisting of thee elements: emembe an ad descibing band A, emembe an ad descibing band B, and unawae of the poduct existence 6. Initially, all consumes ae in state unawae of the poduct existence. A consume s memoy jumps to state emembe an ad descibing band A o emembe an ad descibing band B once she sees an advetising message of the espective fim. Suppose the consume saw fim A s 4 See also Kelle 1987, 1991). 5 See also Jewell and Unnava 2003), Kuma and Kishnan 2004). 6 In Wilson 2002), a decision-make is also esticted to a finite set of memoy states. The pape exploes how the decision-make with a bounded memoy updates he beliefs upon eceiving new infomation, finding that the agent displays a confimatoy and oveconfidence/undeconfidence biases. 4

5 ad fist. If the consume subsequently sees fim B s ad, then, unde etoactive intefeence, the memoy pocess jumps to state emembe an ad descibing band B, as new infomation facilitates fogetting of what was leaned peviously. Unde poactive intefeence, the memoy pocess stays in state emembe an ad descibing band A, as pevious leaning hampes the ability to encode new infomation. Finally, the consume may simply foget about the band she is awae of, and etun to state unawae of the poduct existence, a manifestation of the decay theoy. The fims set thei pices equal to the consume valuation fo the good and compete in advetising fequencies. The equilibium outcome of the advetising game is chaacteized fo both poactive and etoactive intefeences acoss thee settings: simultaneous, sequential, and dynamic moves. In the Simultaneous Move setting, the fims make thei advetising decisions simultaneously at the beginning of the game. In this setup, the equilibium advetising fequencies, emakably, do not depend on the type of intefeence. It is shown that in this setting each fim ove-advetises in equilibium elative to the social optimum. In the Sequential Move setting, whee one fim makes its advetising decision at the beginning of the game, and the othe fim entes the maket and chooses its advetising fequency at some exogenously given) time T, poactive and etoactive intefeences give ise to diffeent equilibium outcomes. Results show that, in ode to undemine the advetising incentives of the second fim, the fist fim chooses highe advetising fequency unde etoactive intefeence than unde poactive. In the thid setting studied, the fims choose thei advetising fequencies twice, at the beginning of the game, and then at time T. As in the Sequential Move setting, the equilibium outcome in the Dynamic setting depends on the intefeence type. At the beginning of the game, when competing fo consumes, the fims choose highe advetising fequencies unde poactive intefeence than unde etoactive intefeence. In the next section, the fomal model is pesented. The Simultaneous Move setting is analyzed in Section 3. Some technical lemmas ae contained in Section 4. In Sections 5 and 6, the Sequential Move and Dynamic settings ae exploed espectively. Concluding emaks appea in Section 7. All poofs ae elegated to an Appendix. 5

6 2 The Model Supply Side On the supply side, thee ae two fims, A and B, that poduce a homogeneous poduct at zeo maginal cost. The fims distibute thei bands though a common etaile. Demand Side 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 v = 1 fom consuming the poduct wheneve she i) visits the etaile i.e., goes shopping) and ii) emembes to buy the poduct. 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. That is, 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 about σ t, P{N t+ t = N t + 1} = σ t + o t). Hee o t) is much smalle than t, i.e., o t) lim = 0 7. t 0 t Advetising Technology To povide the ole fo advetising, it is assumed that consumes ae unawae of the existence of a paticula band unless they emembe an ad descibing it. The fims decide on advetising 7 Fo the definition of a Poisson Pocess see, fo example, Doob 1953), pp , o Lawle 1995), pp

7 fequencies α A and α B the ate paametes of Poisson pocesses that advetising messages follow. Each consume eceives messages independently of othe consumes. The discounted) cost needed to achieve advetising fequency α 0 is cα)/, whee c ) is a convex and inceasing function, c > 0, c > 0. The bounday assumptions, c0) = c 0) = 0 and lim α c α) =, guaantee existence of an inteio solution. Consume Memoy fo Advetising Consume memoy fo advetising is a continuous-time Makov pocess M t taking values in the state space consisting of thee elements: s, s A and s B. A consume emembes an ad descibing band A band B) if he memoy is in state s A state s B ). The consume is unawae of the poduct existence if he memoy is in state s. It follows that consumes neve emembe two ads descibing each band at a time. In othe wods, memoy has a limited stoage capacity. The pobability that the memoy pocess in state s jumps to state s A state s B ) in a small time inteval of length t is equal to the pobability that the consume sees fim A s fim B s) advetising message: P{M t+ t = s A M t = s } = α A t + o t), P{M t+ t = s B M t = s } = α B t + o t). Convesely, the pobability that the pocess in state s A state s B ) jumps to state s is popotional to the ate of fogetting φ: P{M t+ t = s M t = s A } = φ t + o t), P{M t+ t = s M t = s B } = φ t + o t). Paamete φ 0 eflects the geneal obsevation that people foget as time passes i.e., the decay theoy). Next, the pobability that the pocess in state s A jumps to state s B is negligibly small 7

8 Figue 1: Tansition Diagams Poactive Intefeence s Retoactive Intefeence s φ t + o t) φ t + o t) φ t + o t) φ t + o t) α A t + o t) α B t + o t) α A t + o t) α B t + o t) 0 + o t) α B t + o t) s A s B s A s B 0 + o t) α A t + o t) in case of poactive memoy: old infomation hampes the ability to lean new infomation. This pobability is popotional to α B in case of etoactive memoy: new infomation pushes out old infomation. That is, o t), P{M t+ t = s B M t = s A } = α B t + o t), if memoy is poactive, if memoy is etoactive. Similaly, o t), P{M t+ t = s A M t = s B } = α A t + o t), if memoy is poactive, if memoy is etoactive. It is assumed M 0 = s. That is, at t = 0 consumes ae unawae about the existence of the poduct. Figue 1 shows gaphically the pobabilities of tansition fom one state to anothe fo the two memoy types. 8

9 Equilibium Concept Thee is no pice competition between the fims, as, by assumption, consumes neve emembe two ads descibing each band at a time. The fims set thei pices equal to 1, the consumes valuation fo the good. The stategic vaiables of the fims ae advetising fequencies. The equilibium concept employed is subgame pefect Nash equilibium. In the Simultaneous Move setting, the fims choose thei advetising fequencies simultaneously. An equilibium is a pai α A, α B ). In the Sequential Move setting, fim A chooses its advetising fequency fist; fim B obseves fim A s fequency, then chooses its own at exogenous time T > 0. In this case, an equilibium is a pai α A, α B )). In the Dynamic setting, the fims choose thei advetising fequencies twice: at t = 0, and then at t = T. Theefoe, an equilibium is a quaduple α A0, α AT, ); α B0, α BT, )). 3 Simultaneous Move Setting In this section the setting in which fim A and fim B make thei advetising decisions simultaneously at the beginning of the game is analyzed. Let p s t) denote the pobability that the memoy pocess is in state s at time t: p s t) P{M t = s}, whee the dependence on α A and α B is suppessed fo the moment. Fom the aggegate point of view, p sa t) and p sb t) ae the factions of consumes that at time t emembe ads descibing bands A and B espectively, while p s t) is the faction of consumes that ae unawae of the poduct existence. When memoy is poactive, functions p s t), p sa t) and p sb t) satisfy the following system 9

10 of linea diffeential equations 8, p s = α A + α B )p s + φp sa + p sb ), p s A = φp sa + α A p s, p s B = φp sb + α B p s. The fist diffeential equation coesponds to state s. The change in numbe of consumes that ae unawae of the poduct existence is on the left hand side of the equation. The minus tem on the ight hand side is the ate at which consumes leave state s, α A + α B, multiplied by the numbe of consumes in this state. A consume leaves state s when she obseves an ad of fim A o B.) The plus tem on the ight hand side is the ate at which consumes aive at state s, φ, multiplied by the numbe of consumes in states s A and s B. A consume aives at state s fom state s A o s B when she fogets about the existence of band A o B, espectively.) The second equation coesponds to state s A. The change in numbe of consumes that know about band A s existence is on the left hand side of the equation. The minus tem on the ight hand side is the ate at which consumes leave state s A, φ, multiplied by the numbe of consumes in this state. A consume leaves state s A when she fogets about the band.) The plus tem on the ight hand side is the ate at which consumes aive at state s A, α A, multiplied by the numbe of consumes in state s. A consume aives at state s A fom state s when she obseves an ad of fim A.) Simila logic applies to the thid diffeential equation that coesponds to state s B. When memoy is etoactive, p s = α A + α B )p s + φp sa + p sb ), p s A = α B + φ)p sa + α A p sb + p s ), p s B = α A + φ)p sb + α B p sa + p s ). The fist diffeential equation, coesponding to state s, coincides with its poactive coun- 8 See, fo example, Lawle 1995, pp

11 tepat. The second and thid equations ae diffeent. On the left hand side of the second equation is the change in numbe of consumes that know about band A s existence. The minus tem on the ight hand side is the ate at which consumes leave state s A, φ + α B, multiplied by the numbe of consumes in this state. Unde etoactive intefeence, a consume leave state s A when she eithe foget about the band o when she obseves an ad descibing band B.) The plus tem on the ight hand side is the ate at which consumes aive at state s A, α A, multiplied by the numbe of consumes in states s and s B. A consume aives at state s A fom state s o s B when she obseves an ad message of fim A.) Simila logic applies to the thid diffeential equation. It tuns out that unde initial conditions p s 0) = 1, p sa 0) = p sb 0) = 0 the pocess is in state s at t = 0), the two systems geneate the same solution. Lemma 1. Pobabilities p s t), s = s, s A and s B, do not depend on the type of consume memoy and ae given by p s t; α A, α B ) = φ + α A + α B )e α A+α B +φ)t, α A + α B + φ p si t; α A, α B ) = α i 1 e α A +α B +φ)t ), α A + α B + φ whee i = A, B. Limiting pobabilities p s lim t p s t), s = s, s A, s B, epesent long-un behavio of the memoy pocess. It is easy to see that these pobabilities ae popotional to φ, α A and α B, espectively: p s = p si = φ α A + α B + φ, α i α A + α B + φ, whee i = A, B. Lemma 1 allows to compute the discounted) pofits of the fims, as functions of α A and α B. In a small time peiod [t, t + t], a consume buys band i with pobability σ t the 11

12 pobability that she visits the etaile) multiplied by p si t; α A, α B ) the pobability that she emembes band i). Fim i s pofit function is, theefoe, Π i α A, α B ) = 0 e t p si t; α A, α B )σ dt cα i). Pefoming simple algeba yields the following poposition. Poposition 1 Pofit Functions). In the Simultaneous Move setting, the pofit functions of the fims do not depend on the type of consume memoy and ae given by Π i α A, α B ) = σα i α A + α B + + φ) cα i), whee i = A, B. In line with economic intuition, the goss pofit of fim i the fist tem of Π i ) is inceasing in its own advetising fequency, deceasing in the advetising fequency of its competito, and is popotional to the shopping ate. Since pobabilities p s t; α A, α B ), s = s A, s B, ae the same fo the two types of consume memoy, the pofit functions ae the same. This esult is due to symmety in the initial conditions at t = 0 consumes do not know band A, no band B. Poposition 2 Ove-Advetising in Competitive Equilibium). In the Simultaneous Move setting: i) The social optimum is α A = α B = α, whee α is implicitly defined by σ + φ) 2α + + φ) 2 = c α ). ii) Equilibium is unique and symmetic, α A = α B = α, whee α is implicitly defined by σα + + φ) 2α + + φ) 2 = c α ). iii) Advetising is socially excessive, α > α. 12

13 While a social planne would aim to incease the faction of consumes that know about the poduct availability though the fims advetising messages, each fim caes only about the faction of consumes that emembe its own band. As a esult, the two fims ove-advetise in equilibium. In the simple setting whee the fims choose thei advetising fequencies simultaneously at the beginning of the game, the equilibium outcome is the same independent of whethe consume memoy is poactive o etoactive. Howeve, if some sot of asymmety is intoduced fo example, one fim entes the maket and chooses its advetising fequency fist, as in the Sequential Move setting), o if the fims make thei advetising decisions dynamically the Dynamic setting), the equilibium outcome does depend on the type of memoy pocess. Fo the est of the pape φ = 0 is assumed. That is, no diect fogetting occus. This assumption geatly simplifies the analysis and makes the diffeence between poactive and etoactive intefeence as stak as possible. 4 Technical Lemmas Let p m ss τ) denote the pobability that type m memoy will be in state s in a time inteval of length τ > 0, given that the pocess is cuently in state s: p m ss τ) P{M t+τ = s M t = s}, whee the dependence on α A and α B is suppessed fo the moment. This pobability is independent of t, since the memoy pocess has the Makov popety. Lemma 2. i) Suppose the memoy pocess is cuently in state s. Then, p m s s i τ; α A, α B ) = α i 1 e α A +α B )τ ), α A + α B whee m = P, R, and i = A, B. 13

14 ii) Suppose the memoy pocess is cuently in state s i. Then, p m 1, if m = P, s i s i τ; α A, α B ) = α i +α j e α A +α B )τ α A +α B, if m = R, 0, if m = P, p m s i s j τ; α A, α B ) = whee i, j = A, B, and i j. α j1 e α A +α B )τ ) α A +α B, if m = R, This technical lemma deseves some discussion. Fist, suppose the memoy pocess is cuently in state s. Pobabilities p m s s A τ) and p m s s B τ) ae obtained by applying φ = 0 to the fomulas of Lemma 1. Next, suppose the pocess is in state s i, i = A, B. Conside poactive memoy: the only way the pocess can leave this state is though fogetting, which neve happens as φ = 0. Theefoe, fo any τ > 0, p m s i s i τ) = 1 and p m s i s j τ) = 0. Conside etoactive memoy. It is easy to see that p m s i s i τ) is always geate than the coesponding long-un pobability p si, but conveges to it as τ. In contast, p m s i s j τ) conveges to p sj Define fom below. V m i α A, α B ) s = 0 e τ p m ss i τ; α A, α B )σ dτ. In othe wods, Vi m α A, α B ) s is the goss i.e., excluding the advetising costs) discounted pofit function of fim i given that the memoy pocess is cuently in state s. Lemma 3. i) Suppose that the memoy pocess is cuently in state s. Then, V m i α A, α B ) s = σα i α A + α B + ), whee m = P, R, and i = A, B. 14

15 ii) Suppose that the memoy pocess is cuently in state s i. Then, whee i, j = A, B, and i j. V m i α A, α B ) si = V m j α A, α B ) si = σ, if m = P, σα i +) α A +α B +), if m = R, 0, if m = P, σα j α A +α B +), if m = R, Conside poactive memoy. If the memoy pocess is cuently in state s i, fim i s goss pofit is equal to σ/ maximum possible). The goss pofit of its competito is zeo in this case. When consume memoy is etoactive, the compaative advantage that fim i has is less ponounced: V R i α A, α B ) si = σα i + ) α A + α B + ) < σ. 5 Sequential Move Setting In this section the setting in which the fims make thei advetising decisions sequentially is analyzed. Fim A chooses its advetising fequency at the beginning of the game. Fim B entes the maket at some exogenous time T > 0, obseves fim A s advetising decision, and then chooses its own advetising fequency. When fim B entes the maket, consumes of total mass p sa T ; α A, 0) = 1 e α AT know about band A, the est faction p s T ; α A, 0) = e α AT ) ae not awae of the poduct existence. Lemmas 1 and 3 allow computation of the fims pofit functions. Fo m = P, R, Π m A α A, α B ) = V m A α A, 0) s e T s=s,s A p s T ; α A, 0)V m A α A, 0) s + e T p s T ; α A, 0)VA m α A, α B ) s cα A), s=s,s A Π m B α A, α B ) = p s T ; α A, 0)VB m α A, α B ) s cα B). s=s,s A 15

16 The fist two tems of Π m A constitute the goss discounted pofit of fim A befoe time T. It is equal to σα A α A + ) e T e α σα AT A α A + ) + 1 e α AT ) ), and is the same fo both types of consume memoy. The thid tem of Π m A is the goss pofit of fim A fom time T on, discounted to t = 0. It is equal to e T e α σα AT A α A + α B + ) + 1 e α AT ) ) when memoy is poactive, and to e T e α σα AT A α A + α B + ) + 1 e α AT ) ) σα A + ) α A + α B + ) when memoy is etoactive. The fist tem of Π m B to t = T. It is equal to is the goss discounted pofit of fim B fom time T on, discounted e α AT σα B α A + α B + ) when memoy is poactive, and to σα B α A + α B + ) when memoy is etoactive. This seves as a poof of the following poposition. Poposition 3 Pofit Functions). In the Sequential Move setting, the pofit functions of the fims do depend on the type of consume memoy. i) When consume memoy is poactive, Π P σα A Aα A, α B ) = α A + ) cα A) e T e α σα AT A α B α A + )α A + α B + ), Π P Bα A, α B ) = e α σα AT B α A + α B + ) cα B). 16

17 ii) When consume memoy is etoactive, Π R Aα A, α B ) = σα A α A + ) cα A) e T σα B α A + α B + ) + e T e α σα AT B α A + )α A + α B + ), Π R σα B Bα A, α B ) = α A + α B + ) cα B). It follows that fo any given values of α A and α B, fim A s pofit is highe when memoy is poactive than when it is etoactive, while fim B s pofit is highe when memoy is etoactive than when it is poactive. Specifically, Π P Aα A, α B ) Π R Aα A, α B ) = e T 1 e α AT ) α B α A + α B +, which is exactly e T Π R Bα A, α B ) Π P Bα A, α B ) ). This is not a coincidence. Obseve that the faction of consumes that ae awae of the poduct availability tough the fims ads is independent of whethe consume memoy is poactive o etoactive. Theefoe, the sum of the fims pofit functions, Π m A α A, α B ) + e T Π m B α A, α B ), is the same fo m = P, R. Poposition 4 Fim B s Stategy). In the Sequential Move setting: i) When consume memoy is poactive, fim B s stategy, α P B α A), is implicitly defined by e α AT σα A + ) αa + α P B + ) 2 = c α P B). ii) When consume memoy is etoactive, fim B s stategy, α R B α A), is implicitly defined 17

18 by σα A + ) αa + α R B + ) 2 = c α R B). iii) Fo each value of α A, fim B chooses lowe advetising fequency when memoy is poactive: α P B α A) < α R B α A). Fim B has less incentive to invest in advetising when memoy is poactive, since at time T the faction of its potential customes i.e., those not aleady locked up by fim A) is only e αat. When memoy is etoactive, by contast, fim A cannot foeclose any of the maket to fim B because consumes will puchase accoding to the last ad they obseve. Poposition 5 Fim A s Equilibium Advetising Fequency). In the Sequential Move setting, if T is lage enough, fim A chooses highe advetising fequency when consume memoy is etoactive: α R A > αp A. Sketch of the Poof. Fist, let T =. Fim A maximizes monopoly payoff Π A α A ) σα A α A + ) cα A), which is independent of whethe consume memoy is poactive o etoactive. Let ᾱ A denote ag max αa ΠA α A ). Next, conside lage but finite T. Fim A maximizes Π P A αa, α P Bα A ) ) = Π A α A ) + O e +α A)T ) when memoy is poactive, and Π R A αa, αbα R A ) ) = Π A α A ) + e T σαb Rα A) ) α A + αb Rα A) + ) + O e +α A)T when memoy is etoactive. Hee O e +α A)T ) is of the same ode as e +α A)T. That is, O e +α A )T ) 0 < lim T e +α A)T 18 <.

19 The maginal effect of inceasing advetising fequency fom monopoly level ᾱ A is negligibly small of ode e +ᾱ A)T ) when memoy is poactive. The effect is sizable when memoy is etoactive. Staightfowad algeba see the Appendix) shows that dπ R A αa, αb Rα A) ) ) = e T Hᾱ A ) + O e +ᾱ A)T, dα A ᾱa whee Hᾱ A ) > 0. It follows that in equilibium fim A chooses highe advetising fequency when memoy is etoactive. 6 Dynamic Setting In this section, the setting in which the fims choose thei advetising fequencies twice, at t = 0 and at t = T, is analyzed. Let α A0 and α B0 denote the fims advetising fequencies chosen at time 0; α AT and α BT denote the fequencies chosen at time T. Lemmas 1 and 3 allow computation of the fims pofit functions. Π m i = Vi m α A0, α B0 ) s e T p s T ; α A0, α B0 )Vi m α A0, α B0 ) s s=s,s A,s B 1 e T ) cα A0 ) + e T Π m it α A0, α B0, α AT, α BT ), whee m = P, R, and i = A, B. The fist two tems constitute the goss discounted pofit of fim i fom t = 0 to T. It is equal to σα i0 α A0 + α B0 + ) e T σα i0 e α A0+α B0 )T α A0 + α B0 + ) + σα i0 1 e α A0 +α B0 )T ) ) α A0 + α B0 ), and is the same fo both types of consume memoy. The last tem, e T Π m it, is fim i s pofit 19

20 fom time T on, discounted to t = 0. Π P it = p s T ; α A0, α B0 )Vi P α AT, α BT ) s cα AT ) s=s,s A,s B = σα it e α A0+α B0 )T α AT + α BT + ) + σα i0 1 e α A0 +α B0 )T ) cα it ), α A0 + α B0 ) Π R it = σα it e α A0+α B0 )T α AT + α BT + ) + σα it + )α i0 1 e α A0 +α B0 )T ) α AT + α BT + )α A0 + α B0 ) + σα it α j0 1 e α A0 +α B0 )T ) α AT + α BT + )α A0 + α B0 ) cα it ). This seves as a poof of the following poposition. Poposition 6 Pofit Functions). In the Dynamic setting, the pofit functions of the fims do depend on the type of consume memoy. i) When consume memoy is poactive, Π P σα i0 i = α A0 + α B0 + ) 1 e T ) cα i0 ) e T cα it ) )) e T e α A0+α B0 )T α i0 α A0 + α B0 + α it α AT + α BT + whee i = A, B. ii) When consume memoy is etoactive, Π R i = Π P i e T 1 e α A0+α B0 )T ) σα i0 α jt α j0 α it ) α AT + α BT + )α A0 + α B0 ), whee i, j = A, B, and i j. It follows that fim i s pofit is highe when memoy is poactive if and only if α i0 α jt α j0 α it > 0, o α i0 /α j0 > α it /α jt. 20

21 This is vey intuitive. Fix α it and α jt fo a moment. If α i0 is high elative to α j0 and memoy is poactive, then, fom time T on, fim i gets most of its pofit fom selling the poduct to lage faction p i T ; α A0, α B0 ) of consumes having eceived an ad descibing fim i s poduct, these consumes will neve switch to band j). If α i0 is low elative to α j0, then fim i gets highe pofit when memoy is etoactive, since even consumes that ae awae of band j at time T faction p j T ; α A0, α B0 ), which is lage in this case) ae among its potential buyes. Also, obseve that the sum of the fims pofit functions, Π m A α A, α B ) + e T Π m B α A, α B ), is the same fo m = P, R at any point in time, the faction of consumes that know about the poduct is independent of whethe consume memoy is poactive o etoactive). Poposition 7 Equilibium Stategies at t = T ). In the Dynamic setting: i) When consume memoy is poactive, the fims equilibium stategies at time T, α P AT α A0, α B0 ) and α P BT α A0, α B0 ), ae implicitly defined by the system of equations e α A0+α B0 )T e α A0+α B0 )T σα P +) BT = c α P ) α P AT +αp BT +)2 AT, σα P +) AT = c α P ) α P AT +αp BT +)2 BT. ii) When consume memoy is etoactive, the fims equilibium stategies at time T, α R AT α A0, α B0 ) and α R BT α A0, α B0 ), ae implicitly defined by the system of equations σα A0 +α B0 )α R BT +e α A0 +α B0 )T α A0 +α B0 ) = c α R ) α A0 +α B0 )α R AT +αr BT +)2 AT, σα A0 +α B0 )α R AT +e α A0 +α B0 )T α B0 +α A0 ) = c α R ) α A0 +α B0 )α R AT +αr BT +)2 BT. Poposition 8 Equilibium Stategies at t = 0). In the Dynamic setting, if T is lage enough, the fims choose highe advetising fequencies when consume memoy is poactive: α P A0 = αp B0 > αr A0 = αr B0. 21

22 Note that in the limit as T the Dynamic setting conveges to the Simultaneous Move setting, in which case advetising fequencies chosen by the fims at t = 0 do not depend on the type of memoy. Sketch of the Poof. Conside lage but finite T. Incopoating time T stategies into the pofit functions yields Π P i = σα i0 α A0 + α B0 + ) 1 e T ) cα i0 ) e T c αit P ) + O e ) α A0+α B0 +)T when memoy is poactive, and Π R σα i0 i = α A0 + α B0 + ) 1 e T ) cα i0 ) e T c α R ) it ) σ α i0 α R e T jt α j0αit R ) α A0 + α B0 ) αat R + αr BT + ) + O e α A0+α B0 +)T when memoy is etoactive. In the symmetic equilibium the fims choose advetising fequencies α m A0 = αm B0 = αm 0, whee αm 0 satisfies dπ m i dα i0 αi0 =α j0 =α m 0 ) Π m i + Πm i dα m α i0 αit m it + Πm dα m i jt dα i0 αjt m dα i0 αi0 =α j0 =α m 0 = 0. The second tem in backets is zeo, as α m it is chosen optimally. Pefoming simple but, in the case of etoactive memoy, tedious algeba yields the following equations fo α P 0 and αr 0 : σ α0 P + ) 2α P 0 + ) 2 1 e T ) c α0 P ) + O e 2αP+)T) 0 = 0 and σ α0 R + ) 2α R 0 + ) 2 1 e T ) c α0 R ) e T G α0 R ) + O e 2αR+)T) 0 = 0, whee Gα0 R ) > 0 see the Appendix). It follows that in equilibium the fims choose highe advetising fequencies when memoy is poactive: α P 0 > αr 0. 22

23 7 Conclusion This pape has applied the theoies of poactive and etoactive intefeence developed in the psychology liteatue to exploe how consume memoy affects advetising competition between fims. The equilibium advetising fequencies wee chaacteized unde thee diffeent stategic settings: simultaneous, sequential, and dynamic moves. It was shown that when fims choose thei advetising fequencies simultaneously, the equilibium outcome emakably does not depend on whethe memoy coesponds to poactive o etoactive intefeence. The equilibium outcome in this setting exhibits excessive advetising due to a type of band-stealing effect. In the Sequential Move and Dynamic settings, poactive intefeence was shown to have simila competitive effects as consume switching costs 9. Indeed, unde poactive intefeence a consume emembes the band that was fist to captue he attention though an advetisement. Moeove, this exposue hampes he ability to lean subsequently about competing bands. In the Sequential Move setting, the second fim chooses lowe advetising fequency when memoy is poactive than when it is etoactive, since by the time it entes the maket, many consumes have aleady been locked up by the fist fim. In the Dynamic setting, the fims compete fo consumes vey aggessively at the beginning of the game when memoy is poactive. By the time the fims econside thei advetising stategies, the faction of consumes that have not been locked up is small, and the fims consequently choose elatively low levels of advetising fo the est of the game. It was assumed that consumes could neve emembe moe than a single band at a time which implied no scope fo pice competition. Specifically, each fim chages the monopoly pice to the consumes who emembe its ad. A natual extension of the model would be to intoduce additional memoy egistes and fims. Fo instance, suppose thee wee thee fims and consumes could emembe two bands. Then, the fims would compete not only in advetising, but also in pices. Anothe impotant extension would be to intoduce a ole fo band familiaity. Familia bands ae less sensitive to intefeence effects.) This could be 9 On switching costs see, fo example, Klempee s 1987) pioneeing aticle and his subsequent 1995) suvey. 23

24 modelled by supposing that consumes wee moe likely to emembe bands that they had used in the past. The study of the economic consequences of bounded memoy is still in its infancy 10. This pape has made a fist attempt at undestanding how the limited memoy of consumes might impact the advetising stategies of fims. While seveal impotant themes emeged fom this investigation, thee ae clealy many avenues fo futue eseach in which the insights gleaned hee may be efined and extended. 10 See, fo example, Wilson 2002), Mullainathan 2002), and Kähme 2004). 24

25 Appendix Poof of Poposition 1. Fo i = A, B, Π i α A, α B ) = e t p si t; α A, α B )σ dt cα i) 0 σα i = α A + α B + φ 0 σα i 1 = α A + α B + φ 1 α A + α B + + φ σα i = α A + α B + + φ) cα i). ) e t e α A+α B ++φ)t dt cα i) ) cα i) Poof of Poposition 2. Each pat is poven in tun. i) Social planne solves max Π A α A, α B ) + Π B α A, α B ), α A,α B o σα A + α B ) max α A,α B α A + α B + + φ) cα A) cα B), F.O.C.: σ+φ) α A +α B ++φ) 2 = c α A ), σ+φ) α A +α B ++φ) 2 = c α B ). It is easy to show that the matix of second deivatives 2σ+φ) α A +α B c α A ) ++φ) 3 2σ+φ) α A +α B ++φ) 3 2σ+φ) α A +α B ++φ) 3 2σ+φ) α A +α B c α B ) ++φ) 3 is negative semidefinite, which implies sufficiency of the fist-ode conditions. Theefoe, α A = α B = α, whee α satisfies σ + φ) 2α + + φ) 2 = c α ). 25

26 ii) Fim A s best-esponse function BR A α B ) is a solution to max α A Π A α A, α B ), o max α A σα A α A + α B + + φ) cα A), F.O.C.: σα B + + φ) α A + α B + + φ) 2 = c α A ). As 2 α 2 A Πα A, α B ) = 2σα B + + φ) α A + α B + + φ) 3 c α A ) < 0, the fist-ode conditions ae sufficient. Similaly, BR B α A ) can be found fom σα A + + φ) α A + α B + + φ) 2 = c α B ). A solution to σα B ++φ) α A +α B ++φ) 2 = c α A ), σα A ++φ) α A +α B ++φ) 2 = c α B ). yields equilibium advetising fequencies. Uniqueness and symmety of the solution is shown in two steps. Fom the system above, it follows α A + + φ)c α A ) = α B + + φ)c α B ). The LHS RHS) is inceasing in α A α B ). This obsevation implies the solution is symmetic, α A = α B = α, whee α satisfies σα + + φ) 2α + + φ) 2 = c α ). 26

27 Second, BR Aα B ) = BR Bα A ) = 2σα A α B φ) 2σα B + + φ) + α A + α B + + φ) 3 c α A ), 2σα B α A φ) 2σα A + + φ) + α A + α B + + φ) 3 c α B ). Obseve that BR A α B) αa =α B and BR B α A) αa =α B < 0. Theefoe, the solution is unique. iii) The LHS s of the equations that define α and α, 2α + + φ) 2 c α ) = σ + φ) and 2α + + φ) 2 c α ) = σα + + φ), ae inceasing functions of α and α, espectively. This obsevation implies α > α. Poof of Lemma 2. Each pat is poven in tun. i) See Lemma 1. ii) When memoy is poactive, p P s i s i τ) = 1, p P s i s j τ) = 0. When memoy is etoactive, pobabilities p R s i s A τ) and p R s i s A τ) satisfy the following system of linea diffeential equations: d dτ pr s i s A = α B p R s i s A + α A p R s i s B, d dτ pr s i s B = α A p R s i s B + α B p R s i s A, with initial conditions p R s i s i 0) = 1, p R s i s j 0) = 0. The deivation will appea in the next daft of this pape. 27

28 Poof of Lemma 3. Each pat is poven in tun. i) Fo i = A, B, V m i α A, α B ) s = 0 = σα i α A + α B = σα i α A + α B e τ p m s s i α A, α B )σ dτ 0 ) e τ e α A+α B +)τ dτ ) = 1 1 α A + α B + σα i α A + α B + ). ii) When memoy is poactive, V P i is etoactive, α A, α B ) si = σ/ and Vj P α A, α B ) si = 0. When memoy V R i α A, α B ) si = V m j α A, α B ) si = 0 σ = α A + α B σ = α A + α B 0 = σα j α A + α B = σα j α A + α B e τ p R s i s i α A, α B )σ dτ 0 αi + ) α i e τ + α j e α A+α B +)τ dτ ) α j = α A + α B + e τ p R s i s j α A, α B )σ dτ 0 ) e τ e α A+α B +)t dτ ) = 1 1 α A + α B + σα i + ) α A + α B + ), σα j α A + α B + ). Poof of Poposition 4. Each pat is poven in tun. i) When memoy is poactive, fim B solves max α B Π P Bα A, α B ), o max e α σα AT B α B α A + α B + ) cα B), 28

29 F.O.C.: e α AT σα A + ) αa + α P B + ) 2 = c α P B). As 2 α 2 B ) Π m B α A, α B ) = e α 2σα AT A + ) α A + αb P + ) 3 c α P B < 0, the fist-ode conditions fo α P B α A) ae sufficient. ii) Similaly, α R B α A) is a solution to σα A + ) αa + α R B + ) 2 = c α R B). iii) Rewite the fist-ode conditions fo α P B and αr B as αa + αb P + ) 2 c αb P ) = e α A T σα A + ) and αa + αb R + ) 2 c αb R ) = σαa + ). The LHS s of the above equations ae inceasing functions of α P B and αr B, espectively. This obsevation implies α P B α A) < α R B α A), fo each value of α A. Poof of Poposition 5. Conside lage but finite T. Evaluating d dα A Π m A α A, α m B α A)) at ᾱ A yields dπ P A = O e ) +ᾱ A)T dα A ᾱa and dπ R A = e T σαb Rᾱ A) dα A ᾱa ᾱ A + αb Rᾱ A) + ) 2 σᾱ A + ) ᾱ A + αb Rᾱ A) + ) 2 dα R B ᾱ A) dᾱ A ) ) + O e +ᾱ A)T. 29

30 Applying the Implicit function theoem to equation ii) of Poposition 4 yields dα R B ᾱ A) dᾱ A = ᾱa + α R B ᾱ A) 2ᾱ A + ) + hᾱ A ), whee hᾱ A ) 1 σ c α R Bᾱ A ) ) ᾱ A + α R Bᾱ A ) + ) 3 > 0. Thus, dπ R A = e T σαb Rᾱ A) dα A ᾱa ᾱ A + αb Rᾱ A) + ) 2 σᾱ A + ) ᾱ A + αb R + ᾱ A) ) ) ᾱ A + αb Rᾱ A) + ) 2 2ᾱA + ) + hᾱ A )) = e T ᾱ A + ) ᾱ A + α R B ᾱ A) + ) + α R B ᾱ A)hᾱ A ) ᾱ A + α R B ᾱ A) + ) 2 2ᾱA + ) + hᾱ A )) + O e ) +ᾱ A)T ) + O e +ᾱ A)T. It follows fom the above that dπ R A > dπp A dα A dα ᾱa A ᾱa. Theefoe, fim A chooses highe advetising fequency when memoy is etoactive. Poof of Poposition 7. Each pat is poven in tun. i) Conside poactive memoy, t = T. Fim i s best-esponse function is a solution to max α it Π P it α A0, α B0, α B0, α BT ), F.O.C.: e α A0+α B0 )T σα jt + ) α AT + α BT + ) 2 = c α AT ). As 2 α 2 it Π P it = e α A0+α B0 )T 2σα jt + ) α AT + α BT + ) 3 c α AT ) < 0, 30

31 the fist-ode conditions ae sufficient. A solution to e α A0+α B0 )T e α A0+α B0 )T σα P +) BT = c α P ) α P AT +αp BT +)2 AT, σα P +) AT = c α P ) α P AT +αp BT +)2 BT. yields time T equilibium stategies, α P AT α A0, α B0 ) and α P BT α A0, α B0 ). ii) Conside etoactive memoy, t = T. Fim i s best-esponse function is a solution to max α it Π R it α A0, α B0, α B0, α BT ), F.O.C.: σα jt + )e α A0+α B0 )T α AT + α BT + ) 2 + σα jt α i0 1 e α A0 +α B0 )T ) α AT + α BT + ) 2 α A0 + α B0 ) + σα jt + )α j0 1 e α A0 +α B0 )T ) α AT + α BT + ) 2 = c α AT ). α A0 + α B0 ) As 2 Π R it / α2 AT < 0, the fist-ode conditions ae sufficient. A solution to σα A0 +α B0 )α R BT +e α A0 +α B0 )T α A0 +α B0 ) = c α R ) α A0 +α B0 )α R AT +αr BT +)2 AT, σα A0 +α B0 )α R AT +e α A0 +α B0 )T α B0 +α A0 ) = c α R ) α A0 +α B0 )α R AT +αr BT +)2 BT. yields time T equilibium stategies, α R AT α A0, α B0 ) and α R BT α A0, α B0 ). Poof of Poposition 8. Diffeentiating the system of equations that jointly defines α R AT and αr BT pat ii) of Poposition 7) with espect to α i0 yields dαjt R dα i0 = αi0 =α j0 =α 0 c αt R ) 2 2α 0 c ) αt R + 2α0 αt R + α 0 ) c )) αt R + O e 2α 0T ), whee α R T αr AT α 0, α 0 ) = α R BT α 0, α 0 ). 31

32 Next, dπ R i dα i0 = σ α 0 + ) αi0 =α j0 =α 0 2α 0 + ) 2 1 e T ) c α 0 ) σα R T e T 2α 0 2α R T + ) + σ 2 dαr jt 2αT R + ) dα i0 ) + O e 2α 0+)T. αi0 =α j0 =α 0 So, α R 0 is defined by the following equation: σ α0 R + ) 2α R 0 + ) 2 1 e T ) c α0 R ) e T G α0 R ) + O e 2αR+)T) 0 = 0, whee Gα R 0 ) 2α R 0 σα R T 2α R T + ) + σ 2 2α R T + ) dαr jt dα i0 αi0 =α j0 =α R 0 > 0. 32

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