witching Costs, Relationship Marketing and Dynamic Price Competition Francisco Ruiz-Aliseda May 010 (Preliminary and Incomplete) Abstract This paper aims at analyzing how relationship marketing a ects dynamic price competition between two rms in an in nite horizon setting We consider an environment in which a new consumer arrives at the market at every period, with each consumer living for two periods All consumers initially perceive both rms products as equally valuable, but consuming a product randomly increases the value of consuming the same rm s product in the future (vis-à-vis that of the rival) Relationship marketing allows a rm to learn the random switching cost for each of its customers, thereby allowing the rm to take advantage of private information about its customer base when setting (nondiscriminatory) prices We show that a unique symmetric Markov Perfect Bayesian Equilibrium exists for this game, and we nd that higher (average) switching costs lead to: (i) less intense price competition for consumers; (ii) higher equilibrium payo s for rms; and (iii) shorter periods of market dominance by rms Key words: Market Dominance, Information Technology, Private Information, Markov Perfect Bayesian Equilibrium, trategic Complementarity JEL code: L13, M1 Universitat Pompeu Fabra and IEE P-P E-mail: franruiz@upfedu
1 Introduction In the light of recent developments in information (gathering) technologies, popular maxims such as "Get to know your customer" have become easier to ful ll Not incidentally, management practices such as relationship marketing have become widely used in the last years Relationship marketing involves attracting customers and improving their retention through the increase of their switching costs and the use of information gathered from their interaction with a rm However, it is still unclear what the e ect of relationship marketing on product market competition is, as recently emphasized by Musalem and Joshi (009) The objective of this paper is to shed a light on this issue We focus on the competitive implications of two of the leading aspects of relationship marketing: (i) the enhancement of switching costs for consumers who have already purchased a rm s product; and (ii) the acquisition of private information about these consumers preferences We show in this paper that higher switching costs increase rm payo s by relaxing price competition, and we also nd that higher switching costs lead to shorter periods of market dominance by a rm We show these results in an in nite horizon duopoly model in which a new consumer arrives at the market at every period, with each consumer living for two periods All consumers initially perceive both rms products as equally valuable, but consuming a product makes any consumer bear a random cost for switching to the competing product We assume that marketing relationship programs allow a rm to estimate this random switching cost borne by its customer base better than its rival This results in a rm enjoying private information about its own customer base s switching costs Viewed in this vein, the paper analyzes how private information about customers switching costs a ects the classical trade-o between harvesting and investing (as described by Klemperer 1995) In our dynamic (nondiscriminatory) price competition setting with privately observed switching costs, a rm must weigh whether to milk its current customer base or to attract new customers who can be milked later on What is novel and critical in our environment is that the attractiveness of these two options available at each time depends on the observed switching cost Thus, a rm that (privately) observes a customer with a low switching cost will be willing to forgo milking this customer and will price aggressively to capture a new consumer with higher switching costs (in expected terms) If a customer s switching cost is observed to be high enough, though, the rm will take advantage of its market power over this customer and will extract rents from her, thus forgoing capturing a newly arrived consumer Nevertheless, the competing rm perceives that not milking the customer base becomes less likely as switching cost grows on average (for a xed spread of the distribution from which switching costs are drawn) As a result, a spread-preserving increase in 1
average switching costs will lead the rm without any customer base to price higher, in the belief that the rival is more tempted to harvest Because of strategic complementarity, this will in turn imply that the rm with a customer base will have an incentive to price higher Therefore, dynamic price competition is relaxed as average switching costs increase, and the payo attained in equilibrium by rms will grow accordingly In addition, the fact that it is less likely for the rm that has a customer base to be willing to compete for a new consumer will imply that market dominance will be shorter (in a rst-order stochastic dominance sense) Our paper contributes to the literature on dynamic price competition with switching costs, as pioneered by the seminal paper by Beggs and Klemperer (199) In their paper, they assume that switching costs are in nitely large, and they nd that dynamic price competition is softer than in the absence of switching costs In their duopoly setting, they also nd that one rm increases its dominance over the competing rm as time goes by The latter result is reversed if consumers have nite lives, as shown by To (1996) Also, work by Padilla (1995) relaxes the in nite switching cost assumption made by these papers He shows when rms are allowed to use mixed strategies that switching costs lead to higher pro ts by relaxing price competition and that rms alternate in their market dominance 1 Our paper derives a related conclusion for any level of switching costs: switching costs relaxes price competition and enhances rm value when the level of a consumer s switching cost is information private to the rm that sold to this consumer in the past However, we are able to show additional results: higher switching costs lead to shorter market dominance The paper is organized as follows The model is introduced in ection, and the equilibrium of the dynamic game proposed in such section is characterized in ection 3 ection 4 concludes The model We consider an in nite horizon game in discrete time, with time t running from 0 to 1 There are two ex ante rms that can produce units of a homogeneous product at a constant marginal cost that is set equal to zero without loss of generality Firms simultaneously choose a single price for their products and discount pro ts at rate [0; 1) It is assumed that a di erent consumer arrives at the market at each period Each consumer lives for two periods, and any consumer is assumed to demand one unit of the product sold by rm i f1; g at each of 1 It is worth noting that recent work by Cabral (008) shows that low switching costs may enhance dynamic price competition if a rm can price discriminate between consumers who have never traded with the rm in the past and those who have In addition, products are nonstorable and nondurable, so that they can only be consumed immediately
the periods in which she is alive A unit of the good generates a gross utility of v > 0 to any consumer regardless of whether she is "young" or "old" In addition, a consumer who arrived at time t and bought the product sold by rm i f1; g at such date experiences a cost es if she switches to rm i s competitor once she becomes old in the next period We assume that the switching cost es is an independent random variable uniformly distributed between 0 and >, where we will typically think of as being close to zero (so that an equilibrium exists for esentially all parameter values) 3 The draw obtained by each consumer is known by the consumer only once she has made her rst purchase Hence, no consumer knows how locked-in she will be, although her expected switching cost equals ( + )= regardless of which rm she purchases from Consumers are forward-looking and have a discount factor that can be set equal to zero without any loss of generality, as we will discuss below We also assume that rm i gets a private signal of the realized value of es for a particular consumer, although this happens only if such consumer has purchased rm i s product in the past For simplicity, we assume that the private signal received by rm i is perfectly correlated with es This captures in an extreme way our idea that a rm that has traded with a particular consumer knows how to best serve her because of some relationship marketing program However, this extreme advantage is short-lived in principle because the consumer giving rise to such advantage is bound to disappear in the next period We shall look at the symmetric Markov Perfect Bayesian Equilibrium in which all consumers purchase one of the existing products The relevant (payo -relevant) state variables at time t 1 are the identity of the rm that made the sale to the young consumer at time t 1 and the realized value of the switching cost of the old consumer at time t This realized switching cost is information private to the rm that made the sale to the then-young consumer at time t 1 Because the consumer disappears at t + 1, we do not have to worry about o -the-equilibrium-path prices and the beliefs they would induce on the revealed type of this consumer This independence of private information across consumers and across time enormously simpli es the analysis Hence, rm i s strategy at any t 1 consists of fp i 0; p i (s)g, where p i 0 denotes the price that should be charged by rm i when it did not make a sale in the last period of play, and p i : [; ]! < denotes the price charged by rm i when it has made the last sale and the consumer s realized switching cost is s In addition, rm i s strategy at time t = 0 consists of just a price bp i Because we shall look at symmetric equilibria, we shall omit superscripts henceforth Lastly, we shall make the tie-breaking assumption that a consumer confronted with identical prices chooses to purchase from the rm with private information; if no rm has private information, then the consumer chooses randomly with equal probability for each rm 3 When = 0, the model can be easily generalized from a uniform distribution to a power distribution 3
3 Resolution of the model We start by analyzing the optimal behavior of consumers, recalling that we are looking at symmetric equilibria and that the expected value of es for a young consumer is ( + )= regardless of which rm captures such consumer As a result, it holds that the consumer who arrives at time t will expect earning the same payo when she grows old regardless of which product she purchases Consequently, she nds it optimal to act myopically, and thereby chooses to purchase the lowest priced product when she is young (as long as the price does not result in a negative utility) When she is old, she also acts myopically for obvious reasons o suppose now that rm i f1; g made the sale to the young consumer at time t 1 and consider the pricing subgame that takes place at t if this consumer s realized switching cost is s Recall that s is observed only by rm i and that a young consumer has just arrived at the market at time t Throughout, we let V 0 denote the value of a rm that did not make a sale to the young consumer in the last period of play and behaves optimally thereafter given its competitor s equilibrium strategy imilarly, let V (s) denote the value of a rm that made a sale in the last period of play to a consumer whose realized switching cost turns out to be s Given that rm i s competitor clearly charges some p 0 not exceeding v, we proceed to nd rm i s best response uppose that rm i has made the last sale to a consumer whose realized switching cost equals s ince rm i s rival charges some p 0 v, rm i knows that the old consumer compares v s p 0 with v p, where p denotes the price charged by rm i Hence, rm i nds it optimal either to charge p = min(v; s + p 0 ) and thus get just the old consumer or to charge p = p 0 and thus get both consumers Let s v p 0 denote the switching cost above which an old consumer would be completely locked-in by rm i, and suppose rst that s > s Then rm i compares the payo to selling just to the old consumer, v + V 0, and the payo to selling to both consumers, p 0 + E(V (s)) (where E() denotes the expectation operator) We focus on equilibria in which v + V 0 > p 0 + E(V (s)), so we will have to verify later that it indeed holds that p 0 < v (E(V (s)) V 0) (1) In short, we have when s > s that it holds that p(s) = v and V (s) = v + V 0 4
Consider now the cases in which s s, so that rm i compares s + p 0 + V 0 and p 0 + E(V (s)) Let s p 0 + (E(V (s)) V 0 ) denote the realized switching cost for which rm i is indi erent between pricing to get just the old consumer and pricing to get both consumers Then s [s; s] implies that p(s) = s+p 0 and V (s) = s + p 0 + V 0, whereas s [; s] implies that p(s) = p 0 and V (s) = p 0 + E(V (s)) Besides (1), we will also need to check that s < s, noting that s is smaller than s (by our working hypothesis in (1)) We have that a rm that captured the young consumer in the last period of play follows the following (weakly) increasing pricing strategy when the consumer grows old: 8 >< p 0 if s [; s] p(s) = s + p 0 if s [s; s] >: v if s [s; ] () Its value function is continuous and (weakly) increasing in the consumer s realized type: 8 >< p 0 + E(V (s)) if s [; s] V (s) = s + p 0 + V 0 if s [s; s] >: v + V 0 if s [s; ] o we now analyze the behavior of rm i s rival Given that it faces a competitor with private information and pricing strategy given by (), the following Bellman equation must hold: V 0 = maxf p 0 Z s 1 (V 0 ) dx + Z s 1 (p 0 + E(V (s))) dxg, since rm i is believed to sell to both consumers if s [; s] and just to the old consumer if s [s; ] Taking into account that s = p 0 + (E(V (s)) rewritten as: V 0 ), the Bellman equation can be V 0 = 1 max p 0 f[p 0 + E(V (s))] [p 0 + (E(V (s)) V 0 )] V 0 g, (3) so noting that the right hand side of the equation is strictly concave in p 0 yields that the optimal price charged by rm i s rival is p 0 = (E(V (s)) V 0 ) (4) 5
Plugging p 0 into the right hand side of (3) yields V 0 = 4(1 )( ) (5) In addition, we have that and s = v + (E(V (s)) V 0) s =, so s exceeds if > We proceed now to compute the expected value attained by the rm that captures a young consumer: E(V (s)) = Z s Integrating, we have Z (p 0 + E(V (s))) s dx + s E(V (s)) = (p 0 + E(V (s)))(s ) + (v + V 0)( s), + Z (x + p 0 + V 0 ) (v + V 0 ) dx + s dx (s s)(s + s) ( ) + (p 0 + V 0 )(s s) so the facts that s = = and s + s = v + (E(V (s)) V 0 ) yield after some algebra that E(V (s)) = (p 0 + 3v + (E(V (s)) + 3V 0 )) 4( ) s(v p 0 (E(V (s)) V 0 )) ( ) (p 0 + E(V (s))) Using that p 0 = (E(V (s)) V 0 ) and s = v +(E(V (s)) V 0), we arrive at the following expression after performing some manipulations: (E(V (s)) V 0 ) +[( )( )+(v ) ](E(V (s)) V 0 )+[v(v 3)++(1 )( )V 0 ] = 0 Hence, we have that = 0 implies that E(V (s)) = v(3 v), ( ) 6
whereas > 0 implies that ( )( ) (v ) + E(V (s)) = V 0 + q [( )( ) + (v ) ] 4 [v(v 3) + + (1 )( )V 0 ] When > 0, some manipulations using that V 0 = 4(1 )( ) yield (E(V (s)) V 0 ) = ( )( ) (v ) + q 8(1 + )v( ) + 4(1 )( ) 1( ) + (4 4 ) Because it must hold that v p 0 = (E(V (s)) V 0 ), (E(V (s)) V 0 ) = ( )( ) (v ) + q 8(1 + )v( ) + 4(1 )( ) 1( ) + (4 4 ) implies that q 0 ( (1+))+ 8(1 + )v( ) + 4(1 )( ) 1( ) + (4 4 ), which cannot be (since Hence, we are left with (1 + ) > 0, taking into account that s = = must exceed ) (E(V (s)) V 0 ) = ( )( ) (v ) + (6) q 8(1 + )v( ) + 4(1 )( ) 1( ) + (4 4 ) + In order for the term within the square root in condition (6) to be nonnegative, the following must hold too: v ( + 4 4) + 4( + + ) 4( + + 1) 8(1 + )( ) 7
We have the requirements that v p 0 = (E(V (s)) V 0 ) and v (E(V (s)) V 0 ) > p 0 = (E(V (s)) V 0 ), but satisfaction of the latter clearly implies satisfaction of the former Because v (E(V (s)) V 0 ) > (E(V (s)) V 0 ) if and only if v > ( + 4( )) 4(1 + )( ) (7) and we have that ( + 4( )) 4(1 + )( ) > ( + 4 4) + 4( + + ) 4( + + 1) 8(1 + )( ) for all >, it then follows that the most stringent condition is that in (7) We also need that s <, that is, v < ( + 4(3 + )( )) 8(1 + )( ), so overall, the following set of inequalities should hold: ( + 4( )) 4(1 + )( ) where note that it always holds that < v < ( + 4(3 + )( )), (8) 8(1 + )( ) ( + 4( )) 4(1 + )( ) < ( + 4(3 + )( )) 8(1 + )( ) It also holds that (E(V (s)) V 0 ) > 0 if and only if 3 p (7 8) < v < 3 + p (7 8), so the facts that these bounds on v are tighter than those required by (8) imply that indeed E(V (s)) > V 0, as was expected 8
We now study what happens with the relative size of the market dominance region (ie, (s )=( )) if switching costs increase on average, keeping the spread of the probability distribution xed Given our uniform distribution assumption, a spread-preserving increase in average switching costs can be accomplished by increasing and by the same amount Because Pr(es s) = 1 ( ), a spread-preserving increase in average switching costs implies that the rm that made the sale in the last period of play dominates the market for a shorter period (in the sense of rst-order stochastic dominance) Also, note that p 0 = ( ) + (v ) q 8(1 + )v( ) + 4(1 )( ) 1( ) + (4 4 ), so p 0 > 0 if and only if (v) 4 (v ) 8 ( )(v ) + (3 ) + 4(1 )( ) > 0 Because (v) is minimized at v = and () = [(4 3) 4(1 )] is positive even if = = (the maximum value that can take), it holds that p 0 > 0 for all feasible parameter values We can now summarize all the relevant results so far Proposition 1 uppose that it holds that > and ( + 4( )) 4(1 + )( ) < v < ( + 4(3 + )( )) 8(1 + )( ) Then it holds that a spread-preserving increase in average switching costs leads to shorter periods of market dominance by a rm (in the sense of rst-order stochastic dominance) o consider now the initial period, t = 0 If rm i prices lower than its competitor, then it gets p 0 + E(V (s)) and its rival gets V 0, so Bertrand competition yields that both rms 9
would charge the following price in the rst period: bp = (E(V (s)) V 0 ) < 0 Competition to lock-in the rst consumer arriving at the market would partially dissipate rms pro ts, and each rm would get an equilibrium payo b V that equals V 0, that is, bv = 4(1 )( ) Therefore, a rm s equilibrium increases with a spread-preserving increase in average switching costs (as well as with the discount factor) We then have the following Proposition uppose that it holds that > and ( + 4( )) 4(1 + )( ) < v < ( + 4(3 + )( )) 8(1 + )( ) Then it holds that a spread-preserving increase in average switching costs increases a rm s equilibrium payo, b V The results thus far have considered situations in which there exists some s [; ) such that v = s + p 0, so that s < We now deal with the situations in which v + p 0, which corresponds to the case above in which s = It is important to note in these situations that es could be equivalently interpreted as the extra value derived from purchasing repeatedly from the same rm Relative to our previous analysis, it is easy to see that the only aspect that changes is that E(V (s)) = Z s Z (p 0 + E(V (s))) dx + s so making use of s = = yields upon integrating that (x + p 0 + V 0 ) dx, E(V (s)) = [8p 0 + 4(E(V (s)) V 0 )]( ) + 3 + 4p 0 + 8V 0 ( ) 8( ) Using (4) and (5) yields after some manipulations that so it holds that E(V (s)) > V 0 E(V (s)) = (9 5) 8(1 ) 8(1, )( ) 10
We also have that p 0 = ( + 4(1 )( )) 8(1 + )( ) > 0 Because we need v + p 0, we have that this case requires that v ( + 4(3 + )( )) 8(1 + )( ) As can be seen by direct inspection, a spread-preserving increase in average switching costs increases both p 0 and E(p(s)) = p 0 + 3, and hence softens price competition Because 8( ) and V 0 = 4(1 )( ) E(V (s)) = (9 5)( ) + [(1 + 3) + (5 )( )] 8(1, )( ) an increase in average switching costs keeping the spread xed increases both V 0 and E(V (s)) as well, so rms payo s increases as well Furthermore, a spread-preserving increase in average switching costs implies that the rm that made the sale in the last period of play dominates the market for a shorter period (in the sense of rst-order stochastic dominance), since (s )=( ) decreases if and increase by the same amount In addition, the price charged in the rst period is bp = (E(V (s)) V 0 ), that is, bp = (7 8) 8(1 + )( ), which results in each rm earning an equilibrium payo equal to b V = V 0, so we have that bv = 4(1 )( ) increases when average switching costs increase while keeping the spread of the distribution xed All these results can be summed up as follows Proposition 3 uppose that it holds that > and v ( + 4(3 + )( )) 8(1 + )( ) Then it holds that a spread-preserving increase in average switching costs increases p 0, E(p(s)), 11
V 0, E(V (s)), and V b It also leads to shorter periods of market dominance by a rm (in the sense of rst-order stochastic dominance) This setting generalizes a pure Bertrand setting Indeed, for = 0, it holds as # 0 that p 0 # 0, V 0 # 0 and E(V (s)) # 0, whereas p(s) # 0 pointwise, so we get the Bertrand outcome in the limit as switching costs (and hence private information about them) vanish 4 Conclusion TO BE WRITTEN 1
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