Market Share Dynamics in a Duopoly Model with Word-of-Mouth Communication. Preliminary and Incomplete

Transcription

1 Market Share Dynamics in a Duopoly Model with Word-of-Mouth Communication Preliminary and Incomplete Robert C. Schmidt Eugen Kovac February 1, 01 Abstract This paper analyzes price competition in an infinitely repeated duopoly game with homogeneous goods. Information about suppliers spreads via word-of-mouth communication among consumers. A fraction of consumers remains locked-in at their previous supplier in each period. Market outcomes are, therefore, historydependent. We derive Markov perfect equilibria in which firms use random pricing strategies. The endogenous price dispersion drives market share dynamics. We show that market shares tend to be more asymmetric when future profits are important. In an attempt to secure its dominant position in the market, it is often the firm with the larger customer base that prices aggressively. Keywords: dynamic duopoly; consumer lock-in; mixed pricing; Markov perfect equilibrium JEL classification: D43, D83, L11 The authors would like to thank Roland Strausz for helpful comments. Department of Economics, Humboldt University, Spandauer Str. 1, Berlin, Germany; robert.schmidt.1@staff.hu-berlin.de Department of Economics, University of Bonn, Adenauerallee 4 4, Bonn, Germany; eugen.kovac@uni-bonn.de; URL: 1

2 1 Introduction This paper analyzes dynamic price competition in a homogeneous goods duopoly, where consumers exchange information about suppliers via word-of-mouth communication. 1 In each period, a fraction of consumers exits the market and is replaced by the same mass of new consumers who are fully informed about both suppliers. The remaining consumers are repeat-purchasers. Unless they receive information about the alternative supplier via word-of-mouth, they remain locked-in at the supplier they visited in the previous period. Similarly as in a market with switching costs, this creates inertia on the demand side. Our assumption of homogeneous goods rules out the existence of pure-strategy equilibria. We derive Markov perfect equilibria in mixed strategies. The payoff-relevant state variable is the size of firm 1 s customer base, given by firm 1 s market share in the previous period. The firms usage of randomized pricing strategies is a feature well-known also from models of consumer search. However, most of the models from this strand of literature adopt a static modelling framework. 3 In our model, the endogenous price dispersion is what drives market shares dynamics. 4 The goal of our analysis is to characterize these dynamics, and to reveal the driving forces behind them. The idea that mixed pricing strategies can generate market share dynamics is not entirely novel. E.g., Chen and Rosenthal (1996) introduce an infinitely repeated pricing game where a firm that currently offers the higher price, loses market shares. However, these authors assume that a loss (a gain) in market shares is always associated with the same number of consumers switching the supplier. The resulting uniform step size in the market share space is an assumption that seems poorly justified. Our results are closely related to a strand of literature that seeks to reveal conditions under which market shares in duopolies tend to become more or less skewed over time. Similarly as in Budd, Harris and Vickers (1993) and Cabral (011), we show that a rise in the discount factor for future profits can induce a tendency towards more skewed market splits (relative to the myopic case where future profits are fully discounted away). However, these authors assume that there is an external source of uncertainty, while in our model, market share dynamics are generated endogenously due to the firms usage of randomized pricing strategies. In Cabral (011) and Mitchell and Skrzypacz (006), a 1 This information may e.g. concern the existence and location of a supplier. An alternative interpretation is to assume that suppliers send ads of their current price at the beginning of each new period, to those consumers who visited them in the previous period. In this case, the information is about current prices. For an overview over the literature on switching costs, see Klemperer (1995). An earlier concept of demand inertia was introduced by Selten (1965). 3 See e.g. Stahl (1989), Janssen and Moraga-Gonzalez (004). Salop and Stiglitz (1977) and Varian (1980) also introduce models in which some consumers learn only one supplier s price. Unlike these authors, we do not assume that the identity of this supplier is determined randomly. See also Fishman and Rob (1995). 4 The prevalence and persistence of price dispersion is demonstrated empirically by Lach (00).

3 tendency towards skewed market splits results from network effects, while Budd, Harris and Vickers (1993) show that asymmetric market splits can emerge also due to selfreinforcing cost effects under strategic interaction. An important feature that our approach shares with several models from this stand of literature is the assumption of a one-dimensional state space. E.g., Budd, Harris and Vickers (1993) assume that firms current revenues depend only on the gap between firms, that can be interpreted as technological differences, differences in quality or goodwill. For simplicity, the authors interpret the state variable in terms of firm 1 s market share, and assume that it can take on values between 0 and 1. It evolves according to a stochastic process with an exogenous random drift. The mean is given by the difference in firms efforts. In our model, there is no exogenous random drift. 5 However, there is a close relation between the effort costs in Budd, Harris and Vickers (1993), and prices in our model. Namely, low prices are associated with lower (expected) revenues and can, therefore, be associated with higher efforts to gain customers (or to avoid a loss of market shares). One of the main differences is that in our model, the step sizes in the market share space are related to the state itself, based on our assumptions about the process of information transmission. There is no equivalent to this feature in the model by Budd, Harris and Vickers (1993). In our model, the tendency to produce skewed market splits, becomes more pronounced when many consumers rely on word-of-mouth communication. Intuitively, the popularity weighting property of word-of-mouth communication favors skewed market splits. As a result, when future profits are sufficiently important, a firm that has reached a dominant position in the market starts to defend this position against the smaller competitor whenever it is at stake. This is the case when market shares are skewed but not extreme. The firm with the larger customer base, then, prices aggressively and tends to gain market shares. When market shares are closer to one of the extremes of the market share space, an opposite effect dominates: the firm with the larger customer base now prefers to exploit the locked-in consumers in its customer base by charging higher prices. The combination of these two effects often induces a zig-zag pattern near one of the extremes of the market share space. A firm with a dominant position in the market can usually maintain this position for many periods. The word-of-mouth process underlying our results is a simplified version of a communication process introduced by Ellison and Fudenberg (1995). In a non-market environment, these authors assume that each agent can ask a sample of N other agents about the payoff received from choosing one (out of two) alternatives. In our model, the sample 5 Budd, Harris, and Vickers (1993) point out that in their model, an equilibrium typically fails to exist when the variance of the random drift is zero. The reason for this is that the authors focus on Markov perfect equilibria in pure strategies, and neglect the possibility of mixed strategies. In this sense, our analysis is complementary to their analysis. 3

4 size is restricted to 1. This is sufficient to capture the popularity weighting property characterizing word-of-mouth communication. It implies that when most consumers rely on word-of-mouth, a firm with a small customer base can attract few additional consumers when charging the lower price in the market, because few people discover this firm s offer. As a result, market shares are more sticky near the extremes of the market share space than in the center, where information spreads more efficiently. We also show that when few consumers rely on word-of-mouth and most consumers are fully informed, prices converge to the competitive level. Under (almost) full information, market shares are very volatile and fluctuate between the extremes of the market share space. In this case, the discount factor has little effect upon the dynamics. Rob and Fishman (005) analyze a market with quality investments and word-ofmouth communication. New consumers who enter the market meet with a certain probability old consumers and find out about the tenure of the firm that was patronized by the old consumer last period. The tenure is the number of periods that have elapsed since the firm produced a low-quality product. However, these authors abstract from price competition by assuming that each supplier has monopoly power over the consumers visiting it this period. Caminal and Vives (1996) introduce a two-period duopoly model with differentiated products in which consumers in the second period observe market shares realized in the first period. Furthermore, consumers receive an imperfect signal about an unobservable quality differential between products (the authors point out, that this can also be interpreted as word-of-mouth communication). However, unlike in our model, there is no repeat purchasing, as consumers are assumed to stay in the market for just one period. The authors show that - since today s market shares are an informative signal for future consumers about the quality difference - firms have an incentive to over-invest in market shares in order to interfere with consumers beliefs. The remainder of this paper is organized as follows. In Section, the model is introduced and equilibrium conditions are derived. Section 3 derives analytical results under a discretization of the state space. Section 4 uses numerical simulations that serve as a robustness check. Section 5 concludes. All proofs are relegated to the Appendix. The model There is a market for a homogeneous good with two firms (i 1, ). Time is discrete, the time horizon is infinite. The firms choose simultaneously prices p i,t in every period t = 1,,... Both firms marginal costs are constant and normalized to zero. On the demand side, there is a continuum of consumers with mass 1. Each consumer purchases either zero or one unit of the good in each period, and all consumers have the same 4

5 reservation price, normalized to 1. Prices above 1 (the monopoly price) are eliminated from the strategy space without loss of generality. Hence, the market demand in each period equals 1. We consider the situation where market shares in period t depend on prices in period t, and on market shares in period t 1. This inertia reflects consumers imperfect information acquisition. Firm 1 s demand (equal to its market share) in period t 1 represents its customer base in period t; we denote it n t D 1,t 1. The size of firm s customer base in period t is then D,t 1 = 1 n t. In particular, we will assume that firm 1 s market share increases to the value h(n t ) if it charges a lower price in period t, and drops to the value l(n t ) if it charges a higher price. If both firms charge identical prices, their market shares remain constant. Hence, the market share dynamics is guided by the following transition function h(n t ) if p 1,t < p,t, n t+1 = D 1,t (p 1,t, p,t, n t ) = n t if p 1,t = p,t, (1) l(n t ) if p 1,t > p,t. We assume that this process is the same for both firms. Thus, the functions h( ) and l( ) are required to satisfy the following consistency condition l(n) + h(1 n) = 1, for all n [0, 1]. () This condition says that the aggregate demand equals 1 in each period, for any given market split (captured by the state variable n t ). In addition, we assume that the function h : [0, 1] [0, 1] is continuous, strictly increasing, and that n < h(n) < 1 holds for all n [0, 1). Hence, a firm that charges the lower price in the current period gains market shares, but it does not serve the entire market unless its market share was equal to 1 in the previous period. In the next section we present an information transmission technology that gives rise to the functions h( ) and l( ). Note that the change in the demand (or market share) depends, besides the customer base, only on whether the firm charges a higher or a lower price; the size of the price difference is irrelevant. It would be indeed realistic to consider also this dependence, which could be modelled by introducing product differentiation. However, we will see later, even a model with homogeneous products is able to generate interesting price dynamics. In this respect, our model is represents only a minor departure from the Bertrand model, which could be actually interpreted by setting h(n) = 1 and l(n) = 0. Note also that given a specification of the functions h( ) and l( ), the history of market shares up to period t is fully described by the initial size of firm 1 s customer base in period 1 (which is n 1 ), and a trinary sequence that indicates whether firm 1 gains, loses, or retains the 5

6 market share in each period..1 Information transmission The specification of the functions h( ) and l( ) depends on the micro-structure underlying the technology of information transmission. In the following, we introduce two specifications, based on two technologies how consumers exchange information via word-of-mouth. The first specification of the information technologies yields a linear-quadratic functional form, the other one yields an exponential form. Under both specifications, consumers remember the relevant information about the supplier they visited in the previous period. In addition, at the beginning of the period, they may learn the other supplier s price via communication with other consumers. Under the first specification, consider a consumer from a firm 1 with customer base n. We assume that the consumer learns the other firm s price with an exogenous probability 1 φ, where φ (0, 1). In this case, the consumer becomes informed and buys from the firm with a lower price (by equal prices, she visits the same firm as in the previous period). With a complementary probability φ, the consumer does not learn the price automatically and tries to learn it from a fellow consumer. In particular, with a probability µ she can meet a fellow and ask her about her price. Should the fellow be from another firm, which occurs with probability 1 n, the consumer learns the other firm s price and again chooses the firm with a lower price. For simplicity, the communication occurs only in one direction, i.e., the fellow does not learn anything from the agent who met him. Should the consumer meet a fellow from the same firm, which occurs with probability n, she does not learn anything and visits the same firm as in the previous period. 6 In this specification we assume that each agent meets at most one fellow. This then yields the following functional for l( ): l(n) = φn(1 µ + µn). (3) Recall that l(n) is the mass of consumers out of n consumers who visited firm 1 and remain ignorant about firm s offer because they don t learn the price of firm directly (which occurs with probability φ), and also the don t meet a fellow (with probability 1 µ) or meet a fellow who purchased from firm 1 (with probability µn). Note, that the corresponding specification of the function h( ) follows immediately from (). The second specification differs from the first one only in the number of fellows a consumer can meet and ask. Each consumer can meet several other fellows and ask each of them about the supplier they visited in the previous period. If the consumer meets at 6 An alternative interpretation of this meeting technology would be that a fraction 1 φ of consumers (randomly chosen from the population) exits, and an equal mass of new consumers enters the market. Upon entry, the new consumers become informed about the offers of both suppliers (e.g., via active search). 6

7 least one fellow who visited the alternative supplier, the consumer learns the other firm s price. Otherwise, she remains locked-in at her own previous supplier. Here we assume that the number of fellows a consumer meets is drawn from a Poisson distribution with an arrival rate of λ.in that cases, firm 1 s demand in the current period, given that it loses customers, is given by: l(n) = φne λ(1 n) (4) As before, firm 1 s demand h(n) when it gains customers, follows from (). It turns out that the dynamics of the model are qualitatively similar under the two technologies of information transmission. In this respect, we will focus more on the first, linear-quadratic, specification. Moreover, we will be mostly interested in settings where the market share dynamics is mainly driven by the word-of-mouth communication. This means that φ as well as µ are large enough. 7 This is exactly the case, where the share of informed consumers is rather small and meeting a fellow is likely.. Profit maximization and equilibrium In the following, we characterize the firms profit maximization problem for a general specification of the functions h( ) and l( ). We derive equilibrium conditions that can be used to solve the model. Given the prices p 1,t and p,t, firm 1 s profit in period t is p 1,t D 1,t (p 1,t, p,t, n t ). Let us also denote π E 1,t(n t, p 1,t ) = p 1,t E p,t [D 1,t (p 1,t, p,t, n t )] firm 1 s expected profit, if firm uses a (potentially) mixed strategy, where the expectation is taken over firm s price. As we will show later, in equilibrium the firms indeed use only mixed strategies. Firm s profit is obtained in an analogous way (we replace n by 1 n, after swap the indices of the firms). Let δ [0, 1) be the discount factor for future profits. In period t, firm i maximizes the present discounted value of future expected profits over an infinite horizon: V i (H t ) max {p i,t } δ τ t πi,τ(n E τ, p i,τ H τ ), (5) τ=t where H t denotes the history up to period t. The history H t contains all prices chosen up to period t 1, as well as the initial state n 1. The expectation in (5) is taken over firm j s prices (j i), conditional on the history. Note that there is no external source of uncertainty in the model. As we show below, in equilibrium, uncertainty arises only due the firms usage of mixed pricing strategies. If firms can condition their choice in period t on the entire history H t, the set of equilibria may potentially be overwhelming. To narrow the set of equilibria, the Markov perfection equilibrium (MPE) refinement is used. The payoff-relevant state variable in 7 In the second specification we would assume that φ as well as λ are large enough. 7

8 period t is n t. Hence, we are looking for equilibrium, where the firms condition their price only on the current state n t. This is a natural requirement, as the profit in each period depends only the prices in this period and market shares in the previous period. We start by considering all Markov perfect equilibria and derive some their properties (Propositions 1 and ). equilibria. However, we later limit our attention only to the symmetric The solution of firm i s optimization problem then satisfies the following Bellmanequation V i (n) = max p π E i (n, p) + δe[v i (n ) n, p], (6) where n and n stand for customer base n t and current demand n t+1, respectively. 8 As a first simple observation, note that V i (n) is non-negative (as optimum). This follows from the fact that the firm can always choose to set price p i,t = 0 in every period, guaranteeing it zero profit. Second, observe that V i (n) is bounded from above by 1/(1 δ), as the maximal profit in each period is at most 1, which is the monopoly profit with full market. Thus, V i (n) 1 + δ + δ + = 1/(1 δ). The following propositions summarize further properties of Markov perfect equilibria. Proposition 1. If δ is sufficiently small, then in any Markov perfect equilibrium (MPE), the firms use only mixed strategies for any n [0, 1]. Proposition. Consider MPE that involve only mixed strategies. Then the following statements hold: (i) Both equilibrium price distribution functions F 1 (n, p) and F (1 n, p) have the same support of prices (for any given n). 9 (ii) If δ is sufficiently small, the support of the price distribution is an interval of the form [p(n), 1]. (iii) If δ is sufficiently small, at most one firm attaches a positive probability mass to some price. If so, then it is the firm with a smaller market share and the mass point is located at p = 1 (the monopoly price). We require δ to be be sufficiently small in most of the results. The difficulty that arises for δ large, is that an increase in the size of a firm s customer base may lead to reduced profits in the future when the intensity of future price competition increases. Hence, the value function V (n) may not be generally monotone. If δ is large, a firm may be reluctant to undercut the competitor s price if this price can be predicted with certainty. However, 8 We replace the full history H t in the argument of V by the payoff relevant state n = n t. 9 Let the distribution function be defined as F (n, p) Pr(P < p n). Under this convention, 1 F i (n, p(n)) is the probability mass at the maximal price p(n). 8

9 the logic of the mixed pricing strategies requires that, within the support of F i ( ), a firm would always undercut the competitor s price if it were known. From now on, let us consider only symmetric Markov perfect equilibria when δ is sufficiently small (as required in Propositions 1 and ). We thus omit the index i identifying the firm. The expected profit of a firm with market share n (let it be firm 1) in the current period can be written as π E (n, p) = p [l(n)f (1 n, p) + h(n)(1 F (1 n, p))] (7) where l(n) and h(n) are firm 1 s demands if it ends up having a higher and a lower price, and F (1 n, p) and 1 F (1 n, p) are probabilities of these events, respectively. Firm 1 s expected value in the next period is E[V (n ) n, p] = V (l(n))f (1 n, p) + V (h(n))(1 F (1 n, p)) (8) where V (l(n)) and V (h(n)) are the firm s values after losing and market shares, respectively. In a mixed strategy equilibrium, the maximum in (6) is attained over a range of prices, and the support of F (n, p) contains only prices within this set. For all prices p outside the support of F (n, p), it must hold that: π E (n, p) + δe[v (n ) n, p] V (n), i.e., there is no profitable deviation. Using (7) and (8) in (6), we obtain the following expression for the value function V (n): V (n) = ph(n) p[h(n) l(n)]f (1 n, p) + δv (h(n)) δ[v (h(n)) V (l(n))]f (1 n, p), (9) for all prices within firm 1 s support. The max-operator has been omitted because, for prices within the support of F ( ), the right-hand side must be independent of p. Equation (9) can be solved for F (1 n, p): F (1 n, p) = δv (h(n)) V (n) + ph(n) δ[v (h(n)) V (l(n))] + p[h(n) l(n)], (10) for all n [0, 1]. Note that the cumulative distribution function F ( ) is a ratio of two linear functions of the price p. If the function V ( ) were known, (10) can be used to evaluate F ( ) for all n. In the following we derive equilibrium conditions that can be used to determine V ( ). It follows from Proposition that F (n, p(n)) = F (1 n, p(n)) = 0 for all n [0, 1]. Using 9

10 (10), we obtain after eliminating p(n) the first equilibrium condition: h(n)[v (1 n) δv (h(1 n))] = h(1 n)[v (n) δv (h(n))]. (11) We obtain further conditions by considering the maximal price p = 1, as at most one of the firms chooses the monopoly price with positive probability (see Proposition prop1- mixed). Hence, it must either hold that either (A) F (n, 1) = 1 and F (1 n, 1) 1, or (B) F (n, 1) < 1 and F (1 n, 1) = 1 We will refer to the former as case (A) and to the latter as case (B). Suppose that for a given value of the state n, case (A) is relevant. 10 The condition F (n, 1) = 1 can, thus, be used to derive an equilibrium condition for the given value of n. Using (10), we obtain V (1 n) δv (l(1 n)) = l(1 n) (1) If case (B) is relevant (for a given n), then it similarly follows from (10) that V (n) δv (l(n)) = l(n) (13) Together with an (until now) unknown rule that states whether case (A) or case (B) is relevant for every given state n, conditions (11), (1), and (13) implicitly define the value function V ( ). Note that this is a continuum of equations because these conditions must be fulfilled for all n [0, 1]..3 Benchmark: Myopic case Similarly as in Budd, Harris and Vickers (1993), we start by analyzing the myopic benchmark case when δ = 0. This corresponds to a repetition of a static game, as the future profits are fully discounted As in the full dynamic game, the dynamics are governed by each firm s probability of gaining or losing market shares, depending on value of the current state n. Results for the myopic case can be obtained analytically, as the current behaviour is not influenced by future value functions. First of all, note that Propositions 1 and also hold when δ = 0. In order to derive 10 Until now, it is not clear under what conditions case (A) or case (B) is relevant. Intuitively, we would expect that case (A) is relevant (firm chooses the monopoly price with positive probability) whenever firm 1 s customer base n is not greater than 1, because it should, then, have a stronger incentive to compete for new customers, while firm plays less aggressively and charges the monopoly price with positive probability. As shown below, this intuition is generally correct for sufficiently small values of the discount factor δ. 10

11 the price distribution, let us first consider the price p = p(n). Then F (1 n, p(n)) = 0 and it follows from (9) that V (n) = p(n)h(n). Substituting this into (10), we obtain that h(n) F (1 n, p) = (1 p(n) ) h(n) l(n) p (14) for p [p(n), 1]. Now recall that given n, both firms price distributions have the same support (Proposition ), i.e., p(1 n) = p(n). Comparing the cumulative distribution functions of firms with market shares n and 1 n, we obtain the following proposition. Proposition 3. In the myopic case (δ = 0), firm s price distribution function F (1 n, p) first-order stochastically dominates firm 1 s price distribution function F (n, p) if and only if n < 1. The proposition reveals that in the myopic case, it is always the firm with the smaller customer base that prices more aggressively than its larger rival. 11 It also suggests that there is a tendency towards even splits of the market. In Section 4, we illustrate this result using numerical simulations. As the last step we specify the lower bound of the price distribution p(n). It follows from Proposition 3 that for the monopoly price p = 1 and for n < 1, we obtain F (1 n, 1) < F (n, 1) = 1, which means that the case (A) applies. Clearly, for n > 1, then case (B) applies. Substituting this into (14), we obtain p(n) = { l(1 n)/h(1 n), if n 1, l(n)/h(n), if n > 1. This together with (14) give a complete description of firms equilibrium strategies. 3 Discretizing the state space Departing from the myopic case, computation of the equilibrium becomes much complex. The derivation of a closed-form solution for such infinitely repeated pricing game with non-trivial market share dynamics is often not feasible. Various authors have developed different approaches in order to characterize the outcome of a dynamic game in light of this caveat. Maskin and Tirole (1988) discretize the action space (by introducing a price grid) in order to derive tractable results. Budd, Harris and Vickers (1993) conduct an asymptotic expansion (similar to a Taylor expansion) around a discount factor of zero in 11 Proposition 3 can be seen as an extension of Varian s (1980) model, for an asymmetric distribution of uninformed consumers. For a more rigorous analysis of this model, see also Baye, Kovenock, de Vries (199). 11

12 order to obtain analytical results. Cabral (010) derives analytical results for his model only for the case with three consumers. 1 As shown in Section., the conditions that can be used to determine Markov perfect equilibria for the game introduced in this paper, are linear in the value function V ( ). Therefore, a useful starting point for the derivation of analytical results for this model is to discretize the state space. More precisely, we will consider a finite grid with N points G = {a 1, a 1,..., a N }, where a 1 = 0 < a < < a N 1 < a N = 1 so that a raise (a loss) in market shares is represented by a single step to the right (to the left) along the grid. This can be interpreted by considering functions l( ) and h( ) defined only on the set G so that l(0) = 0, l(a k ) = a k 1 and h(a k ) = a k+1, h(1) = 1 where the second equality holds for k and the third one for k N 1. An important feature of such discretization is h(l(n)) = n (15) for any position n > 0 on the grid. The use of a above grid not only enables us to derive a closed-form solution (see below). It also allows us to drop the restriction to sufficiently small δ in Propositions 1 and, as highlighted in the following proposition. Proposition 4. For any finite market share grid, the result of Propositions 1 and also hold for all values of the discount factor δ (0, 1). The restriction to sufficiently small values of δ was used in the proof of Proposition 1 in order to rule out the possibility that a firm may be reluctant to undercut the competitor s price if the competitor s price were known (and positive) because this may imply tougher price competition in the next period. The problem is that for an arbitrary specification of the functions h( ) and l( ), we cannot compare the values V (n) and V (h(l(n))) without knowing the shape of the value function V ( ), as (15) may in general not hold. Given a market share grid where changes in market shares correspond to a single step along the grid, such knowledge is not necessary since h(l(n)) = n, and thus V (h(l(n))) = V (n) for all n > 0. The introduction of a market share grid also transforms the continuum of equilibrium conditions (11), (1), resp. (13) into a discrete finite set of linear equations. The difficulty that remains is that a rule to determine when case (A) and case (B) is relevant, that is, whether (1) or (13) must be used for a given position n on the market share grid, is a priori not known. One possibility is to make a guess, and then use the set of equilibrium conditions to compute a candidate solution. 13 Then it remains to verify whether the value 1 See also Cabral (011). 13 The most obvious guess is that case (A) is relevant if n 1, and case (B) otherwise. 1

13 function fulfills F (n, 1) 1 (using (10)) for all values of n on the market share grid. If not, the decision rule must have been wrong. In principle, via trial and error, one can go through all possible combinations to determine whether case (A) or case (B) should be used for each position on the market share grid. Since the number of combinations grows exponentially in the size of the grid, a procedure via trial and error is viable only for small grids. However, for some specifications of the functions h( ) and l( ), very fine grid may be needed to accurately describe the transitions between states. The reason lies again in the fact that (15) may in general not hold. A simple computational exercise illustrates the underlying difficulty. Suppose, the parameters in the linear-quadratic specification (3) are given by φ = 0.4 and µ = 0, and firm 1 starts with a customer base of size 1 in stage 1. Then, it loses market shares in two consecutive periods, and subsequently gains market shares in two periods. The resulting sequence of firm 1 s market shares is: 1, l(1) = 0.4, l(0.4) = 0.16, h(0.16) = 0.664, h(0.664) = Hence, the step sizes downwards in the market share space (reflected by the function l( )), are generally of different size than the reverse steps upwards. Therefore, a grid that is not sufficiently fine, may not adequately describe the dynamics of market shares for a given sequence of events (describing which firm gains or loses market shares in which period). Fortunately, there exists a remedy to this problem for some parameter values it is possible to define a market share grid that contains a small number of positions N, but that nevertheless describes the market share dynamics reasonably accurately. To give an example, consider once more specification (3), but now with values φ = 0.8 and µ = 1. If firm 1 starts with a customer base of size 1 in stage 1, then loses market shares twice, and subsequently gains market shares in two periods, the sequence of market shares is: 1, l(1) = 0.8, l(0.8) = 0.51, h(0.51) = , and h(0.8094) = This suggests that a grid with positions {0, 0., 0.5, 0.8, 1} might sufficient to describe any sequence of events quite accurately (for the given parameter values). 3.1 Market share grid with 5 positions Let us generalize the above idea by considering the following grid with five positions G 5 = { 0, s, 1, 1 s, 1}, (16) where s (0, 1 ) is a parameter. Given this grid, we have l(0) = l(s) = 0, l( 1) = s, l(1 s) = 1, l(1) = 1 s, (17) h(0) = s, h(s) = 1, h( 1 ) = 1 s, h(1 s) = h(1) = 1. (18) 13

14 l n n Figure 1: Approximation by a grid with 5 positions (φ = 0.8, µ = 1, s = 0.) By varying the new parameter s, we can adjust the grid to reflect different choices of the original parameters µ and φ. Not all combinations can be adequately reflected. However, it turns out that for large values of φ (around 0.8) and large values of µ, this grid can be used to approximate the true market share dynamics sufficiently well. This is also the range of parameter values that we are especially interested in, because high values of φ imply a large amount of word-of-mouth communication among consumers. In order to find the appropriate value of s, we could simply calibrate the function l( ) by matching its values. In the linear-quadratic specification (3), we have l(1) = φ, so s should be close to φ. Moreover, our approximation using the grid G 5 makes sense when (i) l(l(1)) is close to 1 ; (ii) l(l(l(1))) is close to 1 l(1); and (iii) l(l(l(l(1)))) is sufficiently close to zero. This is indeed the case for φ close to 0.8 and µ large. 14 Setting s = 1 φ will then serve as a pretty good approximation. 15,16 Figure 1.1 illustrates the accuracy of our approximation for φ = 0.8 and µ = 1, using the grid G 5 for s = 0.. The curve shows the true function l(n). The dots represent the discrete approximation as given by (17); the vertical coordinate of each dot is given by the horizontal location of the next step on the grid to the left (because a loss of market shares is reflected by a motion to the left along the grid). The figure shows that for the 14 For the values φ = 0.8 and µ = 1, we have l(1) = 0.8, l(l(1)) = 0.51, l(l(l(1))) = 0.097, and l(l(l(l(1)))) = Setting s = 1 φ matches the condition l(1) = 1 s. The equality l(l(1)) = 1, is for φ [1/, 1/ 3 ] [0.7071, ], satisfied by µ = (φ 1)/[φ (1 φ)]. This expression is increasing in φ. Thus, we need higher values of in order to match the condition (i). Moreover, as l(n) is increasing in φ and decreasing in µ, we need sufficiently low µ in order to match the condition (iii). Note also that we cannot match all these conditions, as l(n) > 0 for any n > Alternatively, we can also minimize the sum of square errors implied by imposing the grid. For example, for φ = 0.8 and µ = 1, the the sum of squared errors LS = N k= [a k 1 l(a k )] is minimized by s = 0.013, for which LS <

15 given parameter values, the function l(n) is reflected quite accurately. Smaller values of s can be used to match larger values of φ and µ. Intuitively, when most consumers communicate via word-of-mouth (large φ and µ), a firm with a small customer base gains only a moderate amount of market shares when it currently charges the lower price in the market, because few consumers discover this firm s offer via wordof-mouth. Hence, market shares tend to me more volatile near the center of the market share space than near the extremes. Such a situation is well described by small values of s. Larger values of s (closer to 1 ) can be interpreted as market shares that are more volatile near the extremes than near the center of the market share space. This corresponds more closely to a situation where few (or no) consumers communicate via word-of-mouth, but in this range of parameters the grid yields a less accurate representation of our microfoundation. 17 We now derive a closed-form solution for the value function given the market share grid G 5 introduced above. We apply the equilibrium conditions (11), (1), resp. (13) to compute values V (0), V (s), V ( 1 ), V (1 s), and V (1). We proceed as follows. We first compute a candidate equilibrium using the decision rule: Apply case (A) if n 1, and case (B) otherwise. We show that this decision rule is correct only when the discount factor δ is sufficiently small. Then we derive a critical value for δ above which condition case (A) must be replaced by case (B) also at the position n = s in the derivation of an MPE. This modification allows us to derive an analytical solution of the model also for large values of the discount factor. Applying the condition (1) to the positions n = 0, n = s, and n = 1 along the grid we obtain the following three linear equations: V (1) δv (1 s) = 1 s, V (1 s) δv (1/) = 1/, V (1/) δv (s) = s. (19) For the positions n = 1 s and n = 1 on the grid, case (B) is relevant. 18 However, as can easily be verified, applying condition (13) that corresponds to case (B) to these positions, yields the same equations as (19). The equilibrium conditions for the remaining positions n = 1 s and n = 1 can be derived by from (11): V (s) δv (1/) = V (1 s)/ δv (1)/, V (0) δv (s) = s(1 δ)v (1). (0) Note that condition (11) can be applied to any position on the grid (it holds for all 17 For any given set of parameter values, an individual grid can be designed that contains a limited number of positions but nevertheless reflects market share dynamics reasonably well. We focus on the grid introduced above because it is simple, yields a good approximation for the parameter range we are interested in, and still delivers interesting pricing patterns, as can be seen later. 18 For n = 1, the conditions (1) and (13) coincide. Thus, either case (A) or case (B) can be applied. 15

16 n [0, 1]). However, as the condition is symmetric, replacing n by 1 n yields an identical equation. It is also trivially fulfilled for n = 1. Therefore, again an application of this condition to the other positions along the grid does not yield any additional information. Equations (19) and (0) form a system of five linear equations with five unknowns. A solution can easily be obtained. For example, the value function at position s is given by V (s) = 1 (1 4s)δ δ sδ 3 ( 3δ + δ 4 ) The complete solution can be found in the proof of Lemma 1 in the Appendix. Lemma 1. Consider the grid G 5. If s < 1 3, there exists a critical value δ crit (0, 1) such that the firm with a smaller customer base charges the monopoly price with positive probability in the positions s (resp. 1 s) on the grid if and only if δ > δ crit. In addition, if s 1 3, then δ crit 1. The lemma implies that (in an MPE) for sufficiently large values of δ, we must apply case (B) (rather than case (A)) at position n = s of the grid. The critical value δ crit marks the turning point between an MPE where it is always the firm with the larger customer base that charges the monopoly price with positive probability, towards a solution in which also the firm with the smaller customer base sometimes charges the monopoly price with positive probability. For δ > δ crit, this occurs in the positions s and 1 s of the market share grid. This indicates that when future profits are sufficiently important, in these positions, the firm with the larger customer base prices very aggressively in order to maintain the dominant position in the market. For smaller values of δ, this effect does not occur. In the following, we analyze the resulting dynamics of market shares. Market share dynamics in a MPE that uses mixed strategies is governed by the probability that firm 1 charges the higher (the lower) price in the current period. If a firm with a small customer base prices aggressively (chooses a lower price in the current period with a high probability), then the resulting dynamics exhibit a tendency towards the center of the market share space. Conversely, if the firm with the larger customer base prices aggressively when market shares are skewed but not extreme (positions s and 1 s on the grid), then a tendency towards the extremes of the market share space emerges. Note that in any symmetric equilibrium, for each firm, the probability of gaining or losing market shares when n = 1 (center of the market share space) is always equal to 1. Proposition 5. Given the grid G 5, for δ sufficiently small, F (1 n, p) first-order stochastically dominates F (n, p) in the positions n = 0 and n = s of the grid. The proposition implies that for n < 1, the firm with the smaller customer base (firm 1) gains market shares with probability greater than 1, and a tendency towards 16

17 the center of the market share space results. Intuitively, for small values of δ, a firm with a large customer base has a stronger incentive to exploit the locked-in consumers in its customer base than a firm with a smaller customer base. Therefore, the smaller firm competes more vigorously for new customers than the larger firm, and a tendency towards even market splits emerges. The stochastic dominance result of Proposition 5 confirms earlier findings obtained in the myopic case (Proposition 3). The effects are especially strong in situations where many consumers communicate via word-of-mouth (see also Figure in Section 4). For larger values of δ, an additional effect can occur. Since future profits are valuable, the firm with the larger customer base may start to defend its dominant position in the market whenever it is threatened, by pricing very aggressively. Hence, for larger values of δ, we might expect tougher price competition around the center of the market share space than at the extremes. If this holds true, then a tendency towards skewed market splits emerges. If price competition is very intense in the center of the market share space, a firm with a smaller customer base may shy away from this region, which allows the larger firm to maintain its dominant position in the market. This is highlighted by the following proposition. Proposition 6. Consider the grid G 5. If δ < δ crit and δ is sufficiently close to δ crit, then F (n, p) first order stochastically dominates F (1 n, p) in the position n = s. Moreover, as δ δ crit, the probability that the firm with the larger customer base (firm ) gains market share converges to 1. If δ δ crit, firm gains market share when n = s with probability 1. If follows from the proposition that, for sufficiently large values of δ, a tendency towards the extremes of the market share space emerges. If δ becomes very large (above δ crit ), market shares never cross the center of the market share space in equilibrium, and a firm that has reached a dominant position in the market, maintains this position forever. Note, that δ δ crit can be fulfilled only if s is sufficiently small (namely if s < 1 3, as follows from Lemma 1). Therefore, Proposition 6 is relevant only when s is not too large. As indicated earlier in this section, small values of s correspond to large values of the parameter φ and µ, hence, to situations where most consumers communicate via word-of-mouth. It is intuitive that in such markets, a high market share is particularly valuable, since firms with a small customer base can attract few additional customers when gaining market shares. Therefore, the finding that market shares tend to become more skewed when s is small and δ is raised, confirms our intuition. Conversely, in markets where few consumers communicate via word-of-mouth, we would expect this effect to be less pronounced. An increasing value of s can be interpreted as a smaller fraction of consumers communicating via word-of-mouth, while a larger fraction of consumers remain locked-in at their previous supplier. Therefore, the finding 17

18 that for larger values of s, the critical value δ crit increases, and that for s > 1 3, δ crit exceeds 1, confirms this intuition. When δ crit 1, even for large values of δ, a dominant firm in the market will not maintain this position indefinitely, and market shares continue to cross the center of the market share space. Unfortunately, larger values of s yield less accurate representation of the microfoundation given in Section than smaller values of s. Therefore, the interpretation of the results derived using this grid for larger values of s (in particular s > 1 ), must 3 be seen in light of this caveat. Nevertheless, it is interesting to note that there are fundamental differences both in the dynamics and in the shape of the value function when comparing situations where s is small (s < 1) to situations where s is large (s > 1 ), given 3 3 the above grid. The following result highlights these differences. Proposition 7. Consider the grid G 5. For δ 1, the aggregated payoff (1 δ)[v (n) + V (1 n)] converges to the half of the monopoly payoff (i.e., to 1 ) when s 0 or s 1, while the aggregated payoff is zero for s = 1 3. It is interesting to note that the same aggregated payoff is realized in the two opposite polar cases s 0 and s 1 (for δ 1). Nevertheless, the dynamics underlying this apparent similarity are markedly different. In the case s 0, the firm with the larger customer base (say, firm ) earns the entire market payoff, while firm 1 s payoff is zero. When s is small and δ is large, then firm prices very aggressively in the position n = s (Proposition 5). Furthermore, in position n = 1, both firms compete vigorously to become the dominant firm, and the expected prices can even fall below zero. 19 Therefore, for n = s, firm 1 (the smaller firm) charges the monopoly price with a high probability, rather than to enter into a fierce price competition, and thus, loses market share with a high probability (Proposition 6). When s is small, the profit earned by firm 1 is also small because the mass of informed consumers it can attract when charging the lower price, is approximately given by s. 0 between n = 0 and n = s. 1 Hence, for s close to 0, market shares mostly fluctuate However, due to intense competition when n = 1, and a demand of zero in the first two positions of the grid (for s 0), it is impossible for firm 1 to earn any positive profits. Firm achieves half of the monopoly payoff because (on average) it reaches a market share of 1 n = 1 in every second period and charges the monopoly price. In all other periods, its market share is 1 s. In these periods, it conducts limit pricing and, thus, does not earn any profits. 19 We do not rule out negative prices, since prices can be interpreted as net of marginal production costs. 0 Note, that firm 1 s demand in position n = s is zero if firm 1 charges the higher price (since l(s) = 0). The reason why it nevertheless charges the monopoly price, is to go back to n = 0 as soon as possible, because the main source of profit is the transition from n = 0 to n = s. 1 It can be shown that when s 0 and n = 0, the probability that firm loses market shares is 1. When n = s, firm gains market shares with probability 1. A zig-zag pattern of market shares, thus, emerges. 18

19 In contrast, when s 1 (for δ 1), market shares repeatedly cross the center of the market share space. In this case, price competition is not particularly intense when n = 1. In fact, in the limit s 1, we obtain uniform distribution of prices, i.e., F ( 1, p) = p for p [0, 1], so negative prices do not occur. Whether a firm starts off with a large or a small customer base does not affect the discounted profit in the long run, because firms alternate in their role as the dominant firm. Nevertheless, it can be shown that price competition intensifies when δ increases, but the effect is less pronounced than for smaller values of s. The finding that for s = 1 (given δ 1), market payoffs are equal to zero, while they 3 are positive for all other values off s, is intriguing. It marks a turning point between two different classes of dynamics. For δ 1 and s < 1, market shares are always skewed. 3 For s > 0, this permits both firms to earn a positive payoff, because firms often alternate in charging very high and very low prices. For δ 1 and s > 1, market shares are often 3 evenly distributed, and firms alternate in the role of the dominant firm. This permits both firms to earn positive (expected) profits in all periods. For n = 1, we are neither in 3 the one nor in the other case. At each position of the market share grid, firms compete vigorously for the dominant position in the market. However, once they have reached this position, they vigorously defend this position. This leads to intense price competition in all positions of the grid, to an extent that eliminates all payoffs in the long run. Formally, it can be shown that for δ 1 and s = 1, the probability that firm 3 charges a price of zero in the position n = 0 converges to 1, while firm 1 continues to randomize over the entire interval p [0, 1]. Hence, firm conducts limit pricing even when it already has a customer base of size 1. Therefore, when δ 1 and s = 1 3, there are no real dynamics. Firm (or firm 1) serves the entire market demand in each period (possibly after one or two initial periods, unless n = 0 or n = 1 already holds in the first period), and in each of these infinitely many periods, the competitive price p = 0 is charged with probability 1. 3 For δ 1 and s = 1, the firms desire to reach a dominant 3 market position is, thus, so strong that all payoffs are dissipated. Unlike for s < 1, there 3 is no niche at the extremes of the market share grid where price competition is less intense, allowing firms to earn some positive profits. This explains why for s > 1 3 and δ 1, the payoffs are independent of the current state n (see the proof of Proposition 7). 3 In the position n = s, firm also conducts limit pricing, charging a negative price, while firm 1 continues to randomize over an interval [p(n), 1]. In position n = 1, both firms randomize over an interval [p(n), 1]. In both cases p(n) < 0. 19