Conformity based behavior and the dynamics of price competition: a new rational for fashion shifts

This paper deals with dynamic price competition in markets in which the perception of consumers regarding the value of goods depends on the choices of other consumers in the market. In particular, we consider the case in which consumers tend to imitate their peers, generating a conformity effect. In the context of a finite horizon model, we show that conformity based behavior creates new channels of dynamic interaction between firms, changing the nature of price competition. As time evolves, both price strategic complementarity and substitutability may arise along the equilibrium trajectory. This leads to V-shaped equilibrium price paths and oscillating trajectories of market shares. We provide also a new rational for the inversion of fashion trends.


INTRODUCTION
The emergence of trends, fads and fashion is a very common phenomenon.Often, agents view some of these trends as a path to social interaction and adopt them as means of social recognition by their peers.Accordingly, agents who are not willing to stand out imitate their peers in their consumption choices, generating conformity based behavior. 2There is a wide range of goods for which this behavior can arise, for example: alcoholic drinks, beverages, entertainment goods, garments, cars, restaurants, clubs, touristic destinations etc.In general, these are goods whose value has two components: the intrinsic value of the good and its social value.For example, the intrinsic utility of being a costumer of a certain restaurant is related to the meals served in that restaurant, whereas its social utility is related to the social status attributed to the people who frequent that particular place.
Conformity based behavior has implications on …rms'strategies both in a static environment, as studied by Grilo et al. (2001) and in a dynamic framework, as the one we study here.We consider the dynamic competition of two …rms that interact on prices during three periods.Each …rm produces a di¤erentiated variant of the good and there exist conformity e¤ects in consumption.We assume that agents take one period of time to observe the consumption trend and to conform with it. 3Accordingly, the number of past consumers is one of the main determinants of consumption patterns.In addition, we assume that the conformity e¤ect is noncumulative in time, in the sense that the current value of a good only depends on the number of consumers using a variant in the preceding period.In a context of fast shifting trends, consumers will have no interest in imitating the choices made by their peers in a distant past.
The hypothesis of non-cumulative and delayed conformity behavior is particularly suitable for some types of goods, as it is the case of fashion goods (garments, restaurants, dancing clubs, touristic destinations).On the hand, the conformity e¤ect in the consumption of these goods tend to be delayed as a consequence of some learning or word of mouth process (very often there is a time lag to get information on the most recent fashion tendency). 4On the other hand, the conformity e¤ect in the consumption of fashion goods tends to be non-cumulative since the number of consumers who was using the good far-o¤ in the past can hardly in ‡uence the current value of a fashion good.Some examples of goods in which this type of externalities arises quite often include clothes, shoes, restaurants, touristic destinations.
In this paper, we study to which extent the existence of conformity behavior in consumption creates new (dynamic) channels of interaction between …rms, a¤ecting the nature of price competition.From the point of view of …rms, the conformity based behavior of consumers generates inter-dependent demands across periods: the higher …rm's demand in the preceding period, the higher the social value provided by its good.This amounts to saying that, in each period, …rms'demands depend on the size of the base of users created in the preceding period.Hence, by lowering the price, …rms enhance both present and future consumption.In this context, we expect conformity based behavior to stimulate tougher competition between …rms in the …rst periods of interaction.
However, this paper reveals that there are conditions in which this initial competition boost does not arise in equilibrium.Our analysis unveils interesting results in what concerns the behavior of the duopolists.First, we observe that, for the levels of conformity considered in this paper, both …rms adopt V-shaped equilibrium price trajectories: prices decrease from period 1 to period 2, increasing afterwards.
The fact that …rms increase their prices as the game moves towards the end is not surprising.In the last period of interaction, …rms relax price competition in order to pro…t from the higher social value of theirs goods.On the contrary, …rms initial price reduction is not so intuitive.One could be led to think that, in the beginning of the game, …rms have maximal incentives to adopt price-cutting strategies since the base of users accumulated in the …rst period is the one with longer-lasting e¤ects.However, this result does not hold in this paper due to the …erceness of competition in the intermediary period.Indeed, we observe that both …rms quote negative prices, privileging a strategy of accumulating a larger basis of consumers for the last period.The higher …rms'demand in the intermediary period, the higher the cost of the former strategy since negative prices would be charged on a larger number of consumers.This situation refrains …rms from adopting aggressive demand-enhancing strategies in the …rst period.
We also analyse the evolution of the goods'social value along the equilibrium path.For the level of conformity e¤ects addressed in this paper, the …rm holding the good that initially yields a higher social value ends up reverting its position.In other words, the good à la mode becomes the unfashionable good!The reasons for this fashion reversion are twofold.First, the initially fashionable …rm has some slack and it is able to increase the price of the good à la mode, maintaining a high level of sales.Second, for the reasons explained above, in the intermediary period, this …rm has incentives to refrain from adopting demand-enhancing strategies, even if this makes its good less fashionable.
These results described above are related the speci…c nature of dynamic price competition in the presence of conformity based behavior.In each period of time, we have investigated the existence of strategic complementarity or substitutability on prices.The actions of two or more players are called strategic complements if they mutually reinforce one another, i.e. if an increase in the action of one player increases the marginal payo¤ of the other player, leading him to also increase his strategy.Likewise, they are called strategic substitutes if they mutually o¤set one another.In other words, in the case of strategic complements the reaction functions of the …rms are increasing in the rival's strategy, whereas in the case of strategic substitutes, they are decreasing.These terms were originally coined by Bulow et al. (1985). 5Our results can be summarized as follows.When the conformity e¤ect is relatively less important than di¤erentiation between products, …rms'prices are strategic substitutes in period one and two.Only in the last period of interaction, there is a switch to strategic complementarity.For higher intensity of the conformity e¤ect there is a change in the nature of strategic interaction: prices are strategic complements in periods one and three, and strategic substitutes in the second period.These results follow from the dynamic nature of our model together with the delayed non-cumulative conformity e¤ect that is being analyzed here.In a static setting (as well as in the last period of interaction), the only incentives that …rms have to compete for consumers stem from the instantaneous pro…ts that they get.This necessarily leads to price strategic complementarity.However, in a dynamic duopoly with non-cumulative conformity e¤ects, the inter-temporal dependence of demands creates additional incentives to compete for consumers.When quoting their prices, …rms will take into account that their attractiveness in the forthcoming periods depend on goods' social value, which, in turn depends on the size of the user base of the preceding period.
This paper belongs to the literature on conformity based behavior and conspic-uous consumption (Veblen (1922), Bikhchandani et al. (1992), Bernheim (1994), Corneo andJeanne (1997,1998) and Grilo et al. (2001)).As we deal with price competition when consumer behavior is characterized by conformity, the paper that is more closely related to ours is Grilo et al. (2001).However, while the latter brings a static perspective on this issue, we shed light on the dynamic aspects of …rm interaction.
Our paper also adds to the literature on dynamic price competition in network industries. 6A number of recent works have studied the problem of dynamic competition in network industries (Doganoglu (2003), Mitchell and Skrzypacz (2006), Laussel et al. (2004), Markovich (2008), Markovich and Moenius (2009), Doraszelski, Chen, and Harrington (2009) and Cabral (2007), among others).While some models predict monotonic price trajectories (e.g.Doganoglu (2003), or Laussel et al. ( 2004)), Cabral (2007) proposes a computational framework, in which prices do not evolve linearly with market shares, a prediction which is in line with our results.
The remainder of the paper is organized as follows.Section 2 presents the basic ingredients of the model.Section 3 characterizes the equilibrium in which both …rms survive in every period and provides conditions under which this equilibrium exists and it is unique.Section 4 analyzes strategic interaction along the equilibrium path and, …nally, Section 5 concludes.

THE MODEL
We consider a model in which two …rms, indexed by 1 and 2, produce di¤erentiated variants of a good, whose lifetime is equal to one period.Variants are di¤erentiated à la Hotelling and …rms are assumed to be located in the extrema of the Hotelling line [0; 1]: Firms interact for three periods choosing noncooperatively their prices.
The consumers (with mass equal to 1) are uniformly distributed in the Hotelling line.Every period t, they buy one of the two existing variants.Notice that, in our setup, consumers are considered to be atomistic and, therefore, there is no such thing as trendsetters.
Consumers'utility when buying good i is where (i) V represents the intrinsic value of consumers'ideal variant of the good; (ii) > 0; the unit travel cost; (iii) x 2 [0; 1], consumers' location along the Hotelling line; and (iv) x i , the location of …rm i, with x 1 = 0 and x 2 = 1: 7 The price of good i in period t is denoted by p i;t : The utility function speci…ed in (1) also takes into consideration the social value of goods stemming from consumers' desire for conformity.For the variant i; the social value is determined by the installed base of users for this variant in the preceding period (D i;t 1 ) : The intensity of the conformity e¤ect is measured by a parameter > 0: To better understand the spirit of our model, consider the following example regarding the choice of restaurants.Each period, consumers may opt for one of two restaurants: a sushi bar and a pizzeria.Each consumer has a preference for the type of food o¤ered by each restaurant.In our model, such preference is measured by V (x x i ) 2 : However, when choosing which restaurant to go to, consumers also take into consideration which is the restaurant à la mode, i.e., the restaurant with a larger number of patrons in the preceding period. 8In our model, this corresponds to the conformity e¤ect and it is measured by D i;t 1 : This type of conformity e¤ect is applicable to fashion goods, capturing the idea that consumers'choices do not depend on trends established in a distant past.In the restaurant example, this amounts to say that consumers'perception of which restaurant is à la mode in the last period does not depend on consumers'choices in period one.
According to (1), in period t; the consumer who is indi¤erent between the two existing variants of the good is located at xt (p 1;t ; p 2;t ; D 1;t 1 ; D 2;t 1 ) such that The demand for good 1 in period t as a function of the prices quoted by …rms at t (resp.p 1;t and p 2;t ) and …rms' market shares at the preceding period (resp.D 1;t 1 and D 2;t 1 ) is given by: with jp 2;t p 1;t + D t 1 j < to guarantee that both …rms have strictly positive market shares.The demand for good 1 in period t is simply D 2;t = 1 D 1;t . 9valuating (2) in period t 1; one obtains D 1;t 1 (p 1;t 1 ; p 2;t 1 ; D t 2 ) : Replacing this expression in (2) it is possible to de…ne market shares of period t conditional on the size of the installed base of …rms in period t 2 and the prices quoted by …rms at periods t and t 1. Repeating this exercise sequentially, we obtain the demand of …rm i in period t conditional on …rms'initial market shares (i.e.D 0 = D 1;0 D 2;0 ) and the sequence of prices charged by …rms from period 1 to period t (i.e.(p 1 ; :::; p t ); where p t = (p 1;t ; p 2;t ) and 1 t 3): Assuming that …rms produce goods at a zero cost, the instantaneous pro…ts of …rm i = 1; 2 in period t are equal to i;t (p 1;t ; p 2;t ; D t 1 ) = D i;t (p 1;t ; p 2;t ; D t 1 ) p i;t : In our dynamic game, the players are the two …rms and the strategies correspond to the three dimensional vectors of prices P i 2 R 3 ; with P i = fp i;t g 3 t=1 and i = 1; 2: In each period t, …rms' payo¤s are given by the value of the discounted pro…ts accumulated from t to T = 3: To characterize optimal price strategies in the context of this dynamic game, we rely on the equilibrium notion of Subgame Perfect Nash Equilibrium (SPNE): at equilibrium, for each period t = 1; :::; 3; each …rm i = 1; 2 quotes the price p i;t that maximizes the discounted pro…ts accumulated from t to T = 3 when …rm i takes as given the price of the rival in period t (p j;t ) as well as the differential on the size of the past demands faced by …rms in the preceding period D i t 1 = D i;t 1 D j;t 1 : Formally, at each period t = 1; ::3; each …rm solves the following optimization problem: where, for the sake of simplicity, we denote the vector (p 1;t ; p 2;t ; D t 1 ) by ( ) : Note that, for given p j;t and D t 1 ; the price p i;t (p j;t ; D t 1 ) that solves problem (3) takes into consideration the impact of p i;t on contemporaneous pro…ts of …rm i as well as the impact of this strategic decision on the future pro…ts of this …rm (dynamic e¤ect).This dynamic e¤ect arises because the choice of p i;t has an e¤ect on the size on the social value of the good o¤ered by …rm i in period t.Accordingly, price decisions in period t will in ‡uence the attractiveness of each variant for future consumers, a¤ecting …rms'future pro…ts through the conformity e¤ect.
In the following section, we focus on the domain of parameters ( ; ) and initial conditions ( D 0 ) ; for which the SPNE corresponds to interior solutions with both …rms having strictly positive market shares in every period. 10Under this restriction, we use backward induction techniques to derive the explicit path of P = (P 1 ; P 2 ) ; corresponding to a SPNE of the multi-stage game.

EQUILIBRIUM PATH
In this section, we derive the SPNE candidate in which both …rms are active in every period of interaction.Afterwards, we provide conditions under which the duopolistic SPNE candidate corresponds to an e¤ective equilibrium.
We start by investigating strategic interaction in the last period.At t = 3, the vector of equilibrium prices conditional on …rms' previous market shares p 1;3 ( D 2 ) ; p 2;3 ( D 2 ) solves max pi;3 with i = 1; 2 and D i;3 (p i;3 ; p j;3 ; D 2 ) obtained from (2) when t = 3: For the domain of parameters ( ; ) and initial conditions ( D 0 ) that leads to duopolistic equilibrium outcomes in period t = 3; the candidate equilibrium price strategies conditional on D 2 are given by the solution to the system of …rst order conditions, namely: In the light of ( 5) and ( 6), when both …rms are active in the market, equilibrium market shares conditional on D 2 are equal to: Equilibrium pro…ts at T = 3 conditional on the di¤erential D 2 are given by: In period 2, …rms choose p 1;2 ( D 1 ) ; p 2;2 ( D 1 ) that simultaneously solve the following optimization problems: where, each …rm i takes as given the price quoted by the rival at t = 2 as well as D 1 : For the sake of simplicity, the discount rate is assumed to be equal to 1.
The previous optimization problem can be re-written as follows: max pi;2 p i;2 D i;2 (p i;2 ; p j;2; D 1 ) + i;3 (p i;2 ; p j;2; D 1 ) ; i = 1; 2; since D 2 is a function of (p i;2 ; p j;2; D 1 ) :The candidate price equilibrium price can be obtained from the system of …rst order conditions, yielding: where = 2 9 2 :Accordingly, the equilibrium market shares of …rms in period and equilibrium instantaneous pro…ts of t = 2 conditional on D 1 are given by: : and it follows that: Introducing the equilibrium di¤erential D 2 ( D 1 ) in the expressions of equilibrium price strategies at t = 3 conditional on D 2 (respectively given by ( 5) and ( 6)), we obtain: To get a full description of equilibrium price paths, we move now to the analysis of …rms' price decisions at t = 1: Considering that …rms take as given the price quoted by the rival at t = 1 as well as initial market shares ( D 0 ), the duopolistic equilibrium candidate at t = 1 corresponds to the vector of prices conditional on initial market shares p 1;1 ( D 0 ) ; p 2;1 ( D 0 ) that simultaneously solves the following optimization problems: The candidate equilibrium prices at t = 1 are given by: and …rms'market shares at t = 1 conditional on D 0 are equal to: Introducing ( 18) in ( 9); (10); ( 13) and ( 14), it is possible to describe the vector P i ; corresponding to the path of prices that constitutes the SPNE candidate as function of the initial market shares and the parameters of the model: where D i 0 = D i;0 D j;0 ; i; j = 1; 2 and i 6 = j: From (19) follows that the vector of equilibrium prices is non monotonic in D i 0 : For b ( ) < < 3 ; the sign of is always positive, while the sign of is always negative.The sign of At the SPNE candidate, market shares evolve according to the following trajectory: 8 > > > < > > > : We proceed to investigate under which conditions the equilibrium candidate corresponds to an e¤ective SPNE of the multi-stage game.These conditions are identi…ed in Proposition 1 below: Proposition 1. (Existence and uniqueness) When the intensity of the conformity e¤ ect is such that 2 (b ( ) ; 3 ) ; the three dimensional vectors of prices P i constitute a unique SPNE of the multi-stage game if and only if the di¤ erential of …rms' initial installed basis of customers is not too large: Proof.See the Appendix.
In the next section, we investigate further properties SPNE of the multi-stage game when 2 (b ( ) ; 3 ) and, in addition, condition (21) holds.

Strategic substitutability and complementarity
Concentrating on the case of intermediate conformity e¤ects, 2 (b ( ) ; 3 ) ; and assuming that the initial degree of asymmetry between the size of …rms' installed basis of consumers is not too large (condition (21) holds), we investigate the properties of P i with respect to the strategic complementarity and substitutability between …rms'contemporaneous price strategies.As already mentioned in the introduction, the actions of two or more players are called strategic complements if they mutually reinforce one another, i.e. if the best reply of a …rm to an increase in the action of the rival is to increase her action.Likewise, they are called strategic substitutes if the reverse holds.
For the sake of exposition, within 2 (b ( ) ; 3 ) ; we distinguish a range of strong conformity e¤ects, b b ( ) < < 3 ; and a range of weak conformity e¤ects, b ( ) < < b b ( ).Di¤erent results are obtained for these two ranges of parameters.

Lemma 1. Strategic price complementarity and substitutability
In period 1, for weak conformity e¤ ects, …rms' prices are strategic substitutes, whereas for strong conformity e¤ ects; …rms' prices are strategic complements.In period 2, …rm's prices are strategic substitutes while in period 3, …rms' prices are strategic complements.
Proof.See the appendix.
In a context of price competition, it is well established in the literature that prices are often strategic complements: a price-cutting strategy tends to induce price-cutting strategies by the rival …rms.However, according to the previous Lemma, along the equilibrium path, both price strategic complementarity and price strategic substitutability arise.To clarify to which extent the existence of conformity e¤ects may a¤ect the nature of price competition, we investigate the characteristics of …rms'best reply functions in each period of interaction, starting with T = 3: In the last period of interaction, …rms'best reply functions are given by: We observe that, @pi;3(pj;3; D2) @pj;3 = 1 2 ; which yields strategic complementarity.This result is not surprising since, in the last period of interaction, …rms do not have any incentives to adopt strategies seeking to increase the social value of their goods in subsequent periods..For a given D 2 ; an increase in p j;3 leads to an increase in p i;3 : In period t = 2; the best reply functions are It is easy to observe that @pi;2 @pj;2 < 0, resulting in price strategic substitutability that sharply contrasts with the strategic complementarity result obtained for period t = 3.
The fact that prices are strategic substitutes in period t = 2 results from the speci…cs associated with the dynamic e¤ects created by conformity based behavior.If …rms only considered the impact of p j;2 on instantaneous pro…ts i;2 (p i;2 ; p j;2 ; D 1 ) ; we would get @ 2 i;2(pi;2;pj;2; D1) @pj;2@pi;2 = 1 2 > 0 (price strategic complementarity): However, along the SPNE …rms take into account the impact of their current decisions on future pro…ts.This dynamic e¤ect more than o¤sets the direct (static e¤ect), yielding strategic substitutability between p i;2 and p j;2 : In period t = 2; …rms have strong incentives to invest in price-cutting strategies in order to induce favorable variations in D 2 ;which increase the willingness to pay of consumers in the following period.In the light of these incentives, at t = 2; competition is quite tough and both …rms charge negative prices at equilibrium (see Figure 2).Finally, in period t = 1; depending on the intensity of conformity e¤ects, prices can be either strategic substitutes or strategic complements.The best reply functions are given by: with @pi;1 @pi;j < 0 when b ( ) < < b b ( ) and @pi;1 @pi;j > 0 when b b ( ) < < 3 : In this period; prices are not necessarily strategic complements because the static e¤ect coexists with dynamic e¤ects caused by the intertemporal linkages of demands.However, notice that, in period t = 1; the dynamic e¤ect is more complex than in period t = 2.The former can be decomposed in two distinct components: (i) the e¤ect of (p i;1 ; p j;1 ) on i;2 via D 1 (that directly depends on p i;1 and p j;1 ); and (ii) the e¤ect of (p i;1 ; p j;1 ) on i;3 via D 2 ( D 1 ) : To be more precise, the static and the two dynamic e¤ects are given by: @ 2 i;1(pi;1;pj;1; D0) @pj;1@pi;1 = 1 2 > 0; @ 2 i;2 ( D1(pi;1;pj;1; D0)) @pj;1@pi;1 = 9 2 2 +9 2 (4 +9 2 ) 2 > 0; @ 2 i;3 ( D 2 ( D1(pi;1;pj;1; D0))) @pj;1@pi;1 = 9

4
(4 +9 2 ) 2 < 0: Not surprisingly the static e¤ect favors price strategic complementarity, while the dynamic e¤ects have opposite signs.The full dynamic e¤ect is equal to 9 2 1 4 + 9 2 2 < 0; favoring strategic substitutability.It is also worth noting that the sign of the to-tal e¤ect (static and dynamic) depends on the intensity of the conformity e¤ect. 12 In the case of a weak conformity e¤ect, the dynamic e¤ect dominates, inducing tougher price competition.In contrast, in the case of a strong conformity e¤ect, the static e¤ect dominates and prices are strategic complements.

Non-monotonic equilibrium trajectories and fashion shifts
In the context of a dynamic duopoly with conformity based behavior, we observe that the nature of price strategic interaction changes substantially as time evolves.For this reason, along the equilibrium path, …rms' prices (as well as the corresponding market shares) will follow a non-monotonic equilibrium path.

Proposition 2. (Non monotonic price path)
Along the equilibrium price trajectory P i ; equilibrium prices evolve non-monotonically: decreasing between period t = 1 and t = 2 but increasing afterwards.
Proof.See the appendix.
From Proposition 4.2; equilibrium price trajectories are not monotonic, corresponding instead to a V-shaped trajectory: The fact that p i;3 D i 0 > p i;2 D i 0 8i = 1; 2 is associated with the lack of dynamic incentives to adopt price-cutting strategies in period t = 3.In period t = 2, di¤erently from period t = 3; …rms enjoy dynamic bene…ts from the accumulation of a larger installed base of users.For this reason, in period t = 2; The intensity of the conformity e¤ect does not in ‡uence the static e¤ect 1 2 but it positively in ‡uences the magnitude of the negative full dynamic e¤ect 9 2 1 4 + 9 2 2 .
price competition tends to be very tough.This is not the case in period t = 3; in which the accumulation of a larger user base does not yield any dynamic bene…t, relaxing competition between …rms and inducing an increase of prices: The fact that p i;2 D i 0 < p i;1 D i 0 is a priori less intuitive, since one could think that the dynamic interplay of the conformity e¤ects could lead …rms to quote lower prices in period 1 (as it it is the case in a monopoly setting, see Gabszewicz andGarcia (2007, 2008)).However, this is not the case.In our setting, competition in period 2 is very intense, leading both …rms to charge negative prices.Accordingly, in period 2;a strategy of creating a larger user base to increase the social value of its good in the subsequent period implies that …rms incur in pro…t losses in period 2 13 : Clearly, the larger are the …rms'market shares in period 2; the larger is the cost of adopting demand enhancing strategies (since …rms are charging a negative price on a larger basis of customers).As such, the …erceness of competition in period 2 ends up creating a dynamic e¤ect that dampens competition in period 1; since …rms may not bene…t so much from a a large installed base generated in this period. 14 In line with the equilibrium price paths, the equilibrium trajectories of market shares are not monotonic and, in addition, there is a phenomenon of fashion inversion in period t = 2: Proposition 3. (Fashion inversion) When existence conditions are met, the …rm bene…ting from a larger initial basis of customers (the …rm à la mode) loses this dominance after one period of interaction, becoming démodé.
Proof.See the appendix.
The phenomenon of fashion inversion is caused by the …erceness of competition in period 2: As mentioned before, in period 2; both …rms have strong incentives to invest in price cutting strategies in order to boost the social values of their goods for the last period.In fact prices are actually negative.Nevertheless, the magnitude of such incentives is not the same for both …rms, with the …rm inheriting a smaller social value from the …rst period (the former démodé …rm ) having more powerful incentives to invest in price cutting strategies 15 .As a consequence, the price charged by the previously "dominated …rm" is substantially more negative than the price charged by its rival, causing the phenomenon of fashion inversion.
In the following …gure, we depict the equilibrium trajectory of the …rms'market shares that exhibit an oscillating behavior.
This …gure re ‡ects the evolution of the di¤erential of equilibrium market shares inherent to (20) where i = 1; 2 denotes the …rm à la mode in the initial period.(the reverse inequalities hold in the case of the démodé …rm).

CONCLUSION
In this paper, we investigate dynamic price competition in a duopoly where consumers are conformist.We consider the conformity e¤ect to be delayed and non cumulative, in the sense that it takes one period of time to observe the consumption trends and to follow them and that distant past trends become irrelevant.To our knowledge, previous papers investigating dynamic price competition in the presence of consumption externalities have been mostly focused on dynamic models of in…nite time horizon, relying on equilibrium concepts of Markov Perfect Equilibrium, in certain cases involving linear strategies.Also, most results are obtained through numeric simulation.Our paper departs from this literature in two ways: …rst, we consider a …nite time horizon model and, second, we privilege SPNE in detriment of the Markov Perfect Equilibrium, which implies that we do not need to assume linear pricing strategies.Under these two assumptions, we show that equilibrium outcomes may substantially di¤er from the predictions of the existing models dealing with dynamic price competition in industries with positive externalities in consumption.
We have characterized the properties of the SPNE in which both …rms have positive market shares, identifying a range of parameters in which such equilibrium exists and it is unique (under the assumption of strong exit barriers or sunk costs associated with no-production).
We concluded that …rms change their pricing behavior as time evolves, with both price strategic complementarity and price strategic substitutability arising along the equilibrium path.We observed that, for the range of parameters in which the interior SPNE exists, the equilibrium path of prices is V-shaped, with …rms decreasing their prices in initial periods of interaction and increasing them afterwards.In the intermediary period (corresponding to the valley of the V trajectory), competition is quite tough, with both …rms investing in price-cutting strategies that lead to negative equilibrium prices.
With respect to the evolution of market shares, we observed that equilibrium paths are oscillating.Also, we unveiled a phenomenon of fashion inversion, with the …rm initially à la mode loosing its dominance after one period of interaction.In the context of our model, this fashion inversion phenomenon is explained by the …erceness of competition arising in the intermediary period.
Several questions naturally arise from our work.In our future research, we aim to investigate the possibility of eviction outcomes, studying under which conditions …rms …nd it pro…table to evict the rival.We also aim to investigate how the length of the interaction period a¤ects our results.To this end, we intend to develop a dynamic duopoly model with T periods.In the context of this model, we intend to investigate whether the properties of fashion inversion and V-shaped equilibrium path prices still hold when the time horizon is extended.

APPENDIX
Proof of Proposition 4.1 First, one must take into consideration that the SPNE candidate can only constitute an interior SPNE in which both …rms are active in the market, if the price strategies that compose P i , i = 1; 2; actually lead to duopolistic equilibrium outcomes.This amounts to say that the following conditions must hold: with t = 1; 2; 3: Substituting p 1;1 ( D 0 ) and p 2;1 ( D 0 ) in (22); we obtain, that in period t = 1 : Similarly, in period t = 2 : and, …nally, in period t = 3 : For 2 (b ( ) ; 3 ) ; it follows that condition (23) is more restrictive than the other consistency conditions and therefore this guarantees that P i is consistent with an interior equilibrium in which both …rms have strictly positive market shares.Under the assumption that there are exit barriers or signi…cant costs associated to no-production (i.e.we impose that both …rms are active in the market in every period), to demonstrate that the SPNE exists and is unique, it is su¢ cient to show that the objective functions in every period of interaction are concave.In period t = 3; we have @ 2 ( i;3 (p i;3 ; p j;3 ; D 2 )) @ 2 p i;3 = 1 < 0; therefore the objective function is concave.In period t = 2; we have @ 2 i;2 (p i;2 ; p j;2 ; D 1 ) + i;3 (p i;2 ; p j;2 ; D 1 ) @ 2 p i;2 = 1 9 3 ; which is negative for < 3 : Finally, in period t = 1; we obtain which is negative when > b ( ) : In the previous expression, for the sake of simplicity, the vector (p i;1 ; p j;1 ; D 0 ) is denoted by ( ) : Accordingly, the objective functions are concave in every period and, therefore, the equilibrium exists and it is unique.Proof of Lemma 4.1 Firms'contemporaneous prices are said to be strategic complements when @ 2 i;t (p i;t ; p j;t ; D t 1 ) @p i;t @p j;t > 0; or equivalently @p i;t (p j;t ; D t 1 ) @p j;t > 0; with i;t (p i ; p j ; D t 1 ) standing for the objective function of …rm i in period t and i; j = 1; 2; i 6 = j: Conversely, when the previous inequalities hold with the reverse sign, i.e. @ 2 i;t (p i ; p j ; D t 1 ) @p i @p j < 0; or equivalently, @p j (p i;t ; D t 1 ) @p j > 0; …rms'contemporaneous prices are said to be strategic substitutes.These derivatives can be directly obtained from ( 4), ( 8) and (15).In period t = 3; we obtain @ 2 1;3 (p 1;3 ; p 2;3 ; D 2 ) @p 2;3 @p 1;3 = @ 2 2;3 (p 1;3 ; p 2;3 ; D 2 ) @p 1;3 @p 2;3 = 1 2 ; which is always positive, yielding strategic complementarity on prices.In period t = 2; we get @ 2 1;2 (p 1;2 ; p 2;2 ; D 1 ) @p 2;2 @p 1;2 = @ 2 2;2 (p 1;2 ; p 2;2 ; D 1 ) @p 1;2 @p 2;2 = 1 18 2 + 9 2 3 ; which is negative for any such that b ( ) < < 3 : Accordingly, in period t = 2; prices are strategic substitutes.Finally, in period t = 1; we have @ 2 1;1 (p 1;1 ; p 2;1 ; D 0 ) @p 2;1 @p 1;1 = @ 2 2;1 ((p 1;1 ; p 2;1 ; D 0 )) @p 1;1 @p 2;1 = 4 + 9 The …rst term is negative when 2 (b ; 3 ).Likewise, the second term is positive, hence, the price di¤erence is increasing in D i 0 .If it is negative for the highest possible D i 0 ; it is negative everywhere.We evaluate the di¤erence upper limit of the relevant domain for D i 0 .The di¤erence is decreasing in D i 0 and hence, if it is positive for the highest value of D i 0 , it is positive everywhere.The negative sign of @D i;2 ( D0) @ D i 0 implies that, the …rm that accumulates a larger user base in period t = 2 corresponds to the …rm endowed with a narrower initial basis of customers.Furthermore, it is also the case that @D i;3 ( D0) @ D i 0 < @D i;2 ( D0) @ D i 0 < 0. Accordingly, in period t = 3;the initially démodé …rm has a larger market share than the …rm that was initially à la mode, and, furthermore, the equilibrium price di¤erential (in favour of the initially démodé …rm) is larger than in period t = 2:

2 ;
which is negative when b ( ) < < b b ( ); and positive otherwise.Hence, when b ( ) < < b b ( ); prices in period t = 1 are strategic substitutes.In contrast, when b b ( ) < < 3 ); prices in period t = 1 are strategic complements.Proof of Proposition 4.2 First we show that p i;2

3
The trajectory of market shares corresponding to the SPNE P i is described in(19), yielding:@D i;1 ( D 0 )