SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS

Transcription

1 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS GUY ARIE AND PAUL GRIECO Abstract. We study a switching cost model that uses a continuum of consumers. Using discrete choice demand, the model becomes entirely deterministic from the point of view of the firms, whereas in the finite consumer model, the idiosyncratic shocks of consumers creates a stochastic optimization problem at the firm level. The existence of equilibria is established along with conditions for uniqueness and algorithms for computing equilibria. Compared to previous models, numeric analysis suggests the dynamic considerations by the firms intensify competition and reduce prices. Switching costs provide firms with an incentive to increase their steady state shares and this drives prices down. In an open loop equilibrium this effect dominates the firm s gains from squeezing its loyal customer base resulting in overall lower prices and a limited and possibly negative effect on profits. Preliminary results suggest the same qualitative effect is sustained under Markov Perfect Equilibria for low switching costs values. This result implies that as switching costs rise, the gains from preventing competition (i.e. colluding) rise as well and implicit collusion becomes more likely. 1. Introduction Many differentiated good markets are characterized by switching costs. In such markets, consumers make repeated purchases and it is possible for consumers to increase utility by repeatedly purchasing the exact same good. Evaluating the effects of switching costs on market outcomes is an important and difficult question. Farrell et al. (2007) provide a detailed and current review of the economic analysis of switching costs. For firms, switching costs provide the opportunity to extract additional rents from locked-in customers but may make attracting new customers harder. If switching costs generate too much market power, regulators may need to intervene to prevent the formation of artificial monopolies 1. A key characteristic Date: February 14, This is a preliminary draft of a work in progress. It has errors which are entirely our fault. This work was inspired by a paper on the same subject by Dube et al. (2008). We are grateful to Uli Doraszelski and especially Mark Satterthwaite for their advice and encouragement. Financial support from the General Motors Research Center for Strategy in Management at Kellogg School of Management and the Robert Eisner Memorial Fellowship at Northwestern University is greatlfully acknowledged. 1 Such intervention, however, requires a conviction that welfare is decreased as a result of switching costs, and this may not always be the case. Switching costs may be due to brand loyalty or other 1

2 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 2 of the switching cost problem is its dynamic nature - decisions today affect future consumer preferences and by extension future profits. This paper characterizes steady state equilibrium market outcomes as a function of switching costs in an infinite horizon dynamic setting with myopic customers. The market is assumed to be a differentiated consumer goods mass market. Firms cannot price discriminate between customers on any dimension (and in particular by brand loyalty) and market shares are determined by a discrete choice model. An important implication of our setup is that some consumers will switch products in every period, so firms compete for both loyal and non-loyal customers. In a competitive market, switching costs have two strategic aspects: (Third degree) price discrimination against loyal customers and bargains to attract unloyal customers by partially compensating them for switching costs. The former is apparent if one considers very high switching costs. In such cases, consumers are extremely reluctant to switch products and each firm acts as a monopoly with respect to its consumers. Therefore, at the higher end, prices should increase with switching costs. In contrast, at very low switching costs, a significant share of the consumers switch between firms. Prices decrease because firms cannot price discriminate and need to compensate consumers for switching, giving rise to bargains. These two contradicting effects imply that steady state market price as a function of the switching cost intensity is undetermined for low values of switching costs and positive for high enough values however, there is on theoretical indication of how high is high enough. In this version we provide numerical evidence that the range in which the relationship between switching costs and prices is negative is quite significant. 2 extra consumer utility from repurchasing rather than extra costs of switching. Antitrust intervention in such cases may reduce welfare through the reduction in firm incentives for generating brand loyalty utility. 2 The comparative static analysis is effected by a third, direct effect of switching costs. This effect has to do with whether switching costs are assumed to be actual switching costs or forgone brand loyalty. If the switching cost is in fact forgone brand loyalty an increase in switching cost implies an increase to consumer utility and therefore has a direct positive effect on price.

3 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 3 While the effects of switching costs remain when the firms in the market collude via a weak cartel, their relative weights change in a way that relates switching costs to collusive behavior. Reduced competition means that the business stealing incentive to offer bargains is driven only by competition with the outside good. Therefore, the price discrimination effect, which is associated with an increase in prices (and firm profits) as a response to higher switching costs is stronger. As a result, higher switching costs may mean larger gains from collusive behavior and therefore may increase the likelihood of implicit collusion 3. A surprising result is that even for the most protected cartel the marginal effect of switching costs on prices is negative for extremely low switching cost values. The intuition for this is that the extra brand loyalty utility that switching costs provide increase the steady state shares in response to a price decrease more than in the static case. Therefore, the marginal benefit for the firm from reducing prices initially increases with switching costs more than the marginal cost of the price increase (i.e. the lost revenue). The industry is modeled using a dynamic framework. In many dynamic models of industrial organization, the state transition function may depend on the outcome of the stage game (total sales, profits, etc.), assuming discrete choice based demand (see Doraszelski and Pakes (2007) for a recent survey). Most recent studies interpret the predicted shares as the probability that a single (representative) consumer will choose the firm s product. The stage game outcome is therefore a random variable with few possible values (e.g. firm i made the sale). While the single representative consumer may be the best model for some problems (e.g., defense procurement), assuming a large number of consumers better suits consumer goods markets. In the latter case, the theory of discrete choice demand removes all uncertainty regarding the stage game shares. As a result, state transitions are deterministic 4 simplifying both the analytic and numeric analysis. 3 Farrell et al. (2007) note that high switching costs may also increase the firms capabilities to monitor each other s prices as demand will be less sensitive to price changes. 4 It is possible to assume some stochastic noise around the predicted shares without changing qualitatively any of the results of this paper. We leave this for future versions.

4 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 4 Our switching cost model follows Dube et al. (2008). At every period two firms compete in a market modeled using discrete choice demand. The mass of consumers is separated to three segments defined by loyalty to one of the two firms or to the outside good. 5 Consumers are myopic and derive a higher utility by purchasing the good they to which they are loyal. Consumers can only be loyal to one good at a time and become immediately loyal to the last good purchased. We show that this model always admits a symmetric pure strategy open loop Nash equilibrium that has a steady state in shares. In this steady state, firms name a constant price and firm shares never change, although consumers may switch. A symmetric Markov Perfect Equilibrium (MPE) in continuous pure strategies is also shown to exist. This MPE is shown to be unique 6 in the space of continuous Markov policies and to have a steady state. These results contribute to the question of the existence of a staggering orbit equilibrium in which at each time a different firm offers a sale and attracts customers. As Farrell et al. (2007) note, the current literature is ambivalent regarding this question. Finally, we extend the model to allow for weak cartel behavior and show that for sufficiently patient firms, discontinuous strategies can be used to sustain a collusive equilibrium in which firms enjoy cartel profits. The difference between each firm s profits under cartel and competition are positively correlated with switching costs, implying that high switching costs may encourage implicit collusion. Dube et al. (2008) use a single representative consumer to model period profits and state transitions. 7 This assumption has two effects. First, as noted in Farrell et al. (2007), the single consumer assumption removes the firm s price discrimination motive. At each period, each firm names a price for either a loyal or a disloyal 5 The model can be extended to allow for new and retiring consumers as well in each period, such as in Beggs and Klemperer (1992). We intend to do this in a future version. We expect all the theoretic results of section 2 (existence and uniqueness of equilibrium) to remain unchanged. We conjecture that the main effect strategic effect of this extension will be to flatten the effect of switching costs as the value of loyal customers decreases. 6 This may require some conditions on the parameter set. 7 This is indeed a widely used assumption both in dynamic IO models and in switching costs models. See Doraszelski and Pakes (2007) for a survey of the former and Farrell et al. (2007) for the latter.

5 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 5 customer, but is never required to name a single price for both. Second, in the dynamic context, a single consumer assumption increases the ante at each period - each firm plays an all-or-nothing game. While this has no implication in the static sense as firms are risk neutral, in the dynamic sense, if the real continuation value is concave in shares (which we expect is the case here) it is as if the firms become risk averse and the value for being at the better continuation state is artificially increases. state forcing the firms to take. Technically, using a single representative consumer is tantamount to approximating a continuous function (the continuation value) using the two extreme points of its domain, and thus generating a significant bias in the approximation. Our theoretic results are therefore substantially different from the analytic results of Dube et al. (2008). Nevertheless, the empiric results provided there - the negative effect of switching costs on prices may be interpreted using the analytic results of this paper. The next section presents the model to be analyzed. Section three considers three different equilibria concepts - Nash (open loop), Markov Perfect Equilibrium (MPE) and collusive equilibria (weak cartel). Existence of equilibrium is proved for each case along with sufficient conditions for uniqueness of the MPE. Section 4 presents the main intuitive results via a numeric example. 2. A Switching Costs Model The basic model closely resembles that of Dube et al. (2008). An example may assist the exposition, consider the market for apartment rents. A consumer (i) that rents an apartment in complex A has some complex specific capital - she already knows her neighbors, where to spend her leisure time (or where to avoid), where things are found, etc. We will say that the consumer is loyal to complex A. Let the utility value of this good specific capital be γ. Each year, our consumer needs to decide whether she stays in her complex or switches to a new one. If her utility from the current complex absent property specific capital is δ i,a, then her utility from staying is now δ i,a + γ. By choosing another complex (say B), her utility will be

6 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 6 only δ i,b. In other words, the switching cost is the loss of γ. Our industry has only J (two) possible goods (complexes) plus an outside good - she can move to a new area altogether. At each period, all potential consumers must choose between one of the industry s goods or the outside good. This choice determines the consumer s current period utility as well as her loyalty at the start of next period. Note that this is the case also for purchases of the outside good. We let S j,t, j {0, 1, 2} be the set of consumers who purchased good j in the previous period. 8 Under the standard assumptions, this means that at period t, the utility for customer i from product j {1, 2} is given by u ijt = δ j + αp jt + γ [i S j,t ] + ɛ ijt while the utility of the outside option is defined as, 9 u it0 = γ 0 [i S 0,t ] + ɛ it0 The mean utility of each of the two goods is captured by δ j, which is constant across time and consumers. p jt is product j s price at time t. Consumer specific idiosyncratic utility shocks are captured by ɛ ijt. As is typical of discrete choice models of differentiated products, ɛ ijt is assumed to be a type-1 extreme value consumer specific error that is i.i.d across products and time. γ 0 determines the level of switching costs (or brand loyalty). We allow for a separate loyalty benefit for the outside good γ 0. When γ = γ 0 = 0, preferences are not state dependent and the game reduces to an infinitely repeated pricing game. As γ consumers become increasingly captive. 8 Two alternative definitions of the loyalty set Sj,t should be noted. First, as in Dube et al. (2008), consumers may retain their last loyalty by choosing the outside good. This eliminates the possibility to be loyal to the outside good but fits better many consumer markets where the outside good is essentially forgo purchase at this period (e.g. soft drink purchases). Another alternative, used in Beggs and Klemperer (1992) is to allow a measure of consumers to retire each period and be replaced with unloyal new customers. The next version of the paper will allow for both of these cases as well. 9 We will explicitly consider a version of the model where consumers exhibit no loyalty to the outside good (i.e., γ 0 = 0).

7 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 7 Firms compete by setting prices in each period. From the firm perspective, the state of the game is the measures of the loyalty sets. To save on notation, let s i,t = µ(s i,t ). Clearly, the state of the game lies within the unit simplex. Consumers are assumed to be myopic - maximizing per period utility. 10 Using s to denote the loyalty shares of the following period and suppressing the time subscript, the shares in the next period are determined by the purchase shares in the current period, s = ψ (p, s) where 2 ψ j (p, s) = s k D j,k (p) k=0 Where D j,k (p) is the aggregate demand for j by consumers who are loyal to good k when firms select prices p. Note that consumers in a particular loyalty set have the same preferences up to ɛ ijt. The loyalty set shares can be found by integrating out the consumer level shock to get the usual logit demand formulation, D j,k (p) = eūj,k(p) 2 l=0 eūl,k(p) Where ū j,k (p) is the mean utility of good j for consumers in S k when the price vector is p. Note that D j,k (p) and ψ j (p, s) are deterministic. Each firm is seeking to maximize its profits over an infinite horizon with discount factor β. We assume that the firms have identical marginal cost c. Firm j profits are therefore β t (p j (t) c)ψ j (p(t), s t ) t=0 We are interested in how prices, market shares and firm profits in this model vary as we vary γ. We restrict our analysis to the case where δ 1 = δ 2 and consider only symmetric equilibria. 10 This assumption is common to the related switching costs models, though it is often criticized. See Farrell et al. (2007) for a survey of models with forward looking consumers.

8 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 8 3. Equilibrium Concepts 3.1. An Open Loop Steady State Equilibrium. The first solution concept we consider is the most tractable, open loop steady state equilibrium. In this approach each firm chooses a price function p (t) that is independent of state. A symmetric equilibrium is obtained whenever firm 1 s choice p 1 (t) solves the single firm problem given the plan of its opponent (p 2 (t)) and p 1 (t) = p 2 (t). Open loop equilibria are Nash equilibria but may not be sub-game perfect. Given p 2 (t), firm 1 s problem is (we omit the t argument from pricing and shares in the firm s problem): (3.1) V (s (0), p 2 ) = max {p 1(t)} t=0 β t (p 1 (t) c) ψ 1 (p 1 (t), p 2 (t), s(t)) subject to: s(t + 1) = ψ(p 1 (t), p 2 (t), s(t)) Existence of V and p 1 (t) for any p 2 (t) follows from the fact that per period profit is bounded and the domain of prices compact. The construction of ψ guarantees sj (t) = 1 for all t. The existence of a symmetric pure strategy open loop equilibrium is proven by construction of the equilibrium and its steady state. Definition 1. An open loop steady state equilibrium is a tuple p 1, p 2, s such that Firm j plays p j (t) = p j for all t. The steady state shares are s = s 1, s 2, s 3, s = ψ(p 1, p 2, s). p j (t) solves firm j s problem given his opponent is playing given p j (t). The equilibrium is symmetric if p 1 = p 2, which also implies s 1, s Existence of an open loop steady state equilibrium. Suppose that firm 2 s policy is constant: p 2 (t) = p 2. For every period, let p 1 (p 2 ; s(0)) denote firm 1 s strategy when the game starts at s(0) and firm 2 plays the constant strategy p 2. Let ψ (p 2 ; s(t)) ψ (p 1 (p 2 ; s(t)), p 2, s(t)) denote the continuation shares at time t and ψ 1 firm 1 s share. For every tuple p, s let BR O (p 2, s) = p 1 (p 2 ; s, s), ψ (p 2 ; s, s)

9 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 9 denote firm 1 s best response at time zero and the resulting continuation shares at time 1. Note that p 1 (p 2 ; s, s) is the first element of the sequence that solves firm 1 s problem (3.1). Suppose that in period t the market is at state s, since firm 2 s strategy is the constant p 2 and that BR O (p 2, s) = p 1, s. In period t + 1, firm 1 s optimization problem is identical to the one at period t and therefore it will continue to price at p 1. Because the shares will never change, p 1 (t) will also be constant at p 1. Now suppose that there is a pair p 2, s such that BR O (p 2, s) = p 2, s. That is, firm 1 s optimal pricing strategy p 1 (t) when firm 2 constantly prices at p 2 is to exactly price at p 2 as well. By symmetry, it is clear that p 2 is also optimal for firm 2 and therefore this is a steady state equilibrium. The discussion is summarized in the following lemma. Lemma 2. A fixed point of BR O defines a symmetric open loop equilibrium with prices p and steady state s. Proof. See the discussion above. The only requirement to prove existence of an open loop equilibrium and its steady state is that BR O has a fixed point. Proposition 3. There is always an open loop equilibrium with a steady state. Proof. We wish to apply Brouwer s fixed point theorem. The main obstacle is that BR O may be a correspondence. To overcome this problem, we restrict attention to the lowest price in BR O (p, s) 11 { } BR ˆ O (p, s) = p, s : min p, s BR O (p, s) p As the domain is compact (a lower and upper bound for optimal prices exists), all that is required is that ˆ BR O () is continuous in both its arguments. This follows from the construction of the problem above. 11 As price uniquely determines resulting shares there can be no multiplicity in s given p.

10 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 10 With the existence of an open loop equilibrium with a steady state at hand, the next subsection details how such a steady state may be calculated Using optimal control to find the open loop steady state. The single firm problem can be solved using the theory of optimal control. For convenience, we use a continuous time approach where the period length goes to zero to solve this problem, and will offer a discrete time version in future versions. In the continuous time case we introduce a parameter a which governs the speed of adjustment to price changes. The discrete time model is closely approximated by the continuous time model when a is high. 12 The current value Hamiltonian for the firm s problem is: H = (p 1 c) s m i a (ψ i (p, s) s i ) i=0 Where s = (s 1, s 2, s 3 ) is the share of consumers purchasing each good at the current instant and a is the speed of consumers adjustment to price changes. The optimality constraint for firm 1 is H p 1 = 0 or (3.2) s 1 + a 3 i=1 m i ψ i (p, s) p 1 = 0 The current value multipliers for the state variables are determined by 13 (3.3) m = rm s H Which is: ( ( ( m ψ1 1 = rm 1 (p 1 c) + a m 1 ( m ψ 1 2 = rm 2 a m 1 ) 1 s 1 ) ( ψ2 s m 2 s 2 ( m ψ 1 ψ 2 3 = rm 3 a m 1 + m 2 + m 0 s 2 s 2 )) ψ 2 ψ 0 + m 2 + m 0 s 1 s 1 ) ψ 0 + m 0 s 2 ( )) ψ0 1 s 2 12 In the discrete time model all consumers make a purchase decision in each period. In the continuous time model consumers reconsider their purchase at rate a. 13 The discount rate r for the continuous time problem is set to (1 β)/β.

11 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 11 By definition of the steady state, s = m = 0 at the solution. We must numerically verify that the second order condition is satisfied at the solution. For every p j, s we now have the optimality equation 3.2, the three multiplier equations 3.3 with m = 0 and the three steady state equations s = 0. The steady state is found by solving the problem for both firms simultaneously which requires a system of eleven equations: Each firm s optimality equation (total 2), three multiplier equations for each firm (total six) and the three steady state equations which are not firm specific (total 3). 14 Preliminary numeric results are presented in Section Markov Perfect Equilibrium. A natural refinement of the open loop solution above is to allow firms to condition their prices on shares. In this section we limit firms to Markov strategies - strategies that may depend only on current shares (Maskin and Tirole, 1988). We show that a Markov perfect equilibrium (MPE) exists. We conjecture and provide initial support for uniqueness of the MPE if firms are limited to continuous strategies. Let B : S R denote the space of bounded and continuous functions from states to firm net present values. A stationary Markov strategy σ (s) : S R defines the firm s price as a function of the state, which determines demand D (σ, s) Firm 1 s value is given by V (s; σ). 15 V (s; σ) = t=0 βt π i (σ (s), s) subject to s t+1 = ψ (σ (s), s) By our assumption on demand symmetry, V 1 ((s 1, s 2 ) ; σ) = V 2 ((s 2, s 1 ) ; σ). 16 To arrive at the MPE, let T σ : B B be firm 1 s Bellman operator when firm 2 prices 14 In practice, we replace one share equation with the adding up constraint s1 + s 2 + s 3 = 1 to improve numerical performance and exploit the symmetry of the problem to reduce the number of non-linear equations. If δ 1 = δ 2, we would need to solve the full problem. 15 For ease of notation we let πj (p, s) (p j c) ψ j (p, s) 16 The symmetry restriction used here is for notational conveinience only.

12 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 12 according to σ. (T σ V ) (s 1, s 2 ) = sup p 1 π 1 (p 1, σ (s 2, s 1 ), s 1, s 2 ) + βv (ψ (p 1, σ (s 2, s 1 ), s 1, s 2 ) ; σ) Lemma 4. For every continuous σ, there is a V σ B (S)such that V σ = T σ V σ. Proof. Apply Schauder s Theorem 17 (Stokey and Lucas, 1989, Chapter 17): 18 (1) There is some C (S) B (S) that is nonempty, closed, bounded and convex. This follows from the definition of B (S). (2) T σ : C (S) C (S) is continuous and that the family T σ (C) is equicontinuous: (a) X = S, π (x) defines the degenerate distribution such that P (x x) = 1 so the Feller condition trivially holds 19 (b) There is no uncertainty so the expected value at state s: (T σ V ) (s) equals the conditional value G (s, s ) (c) Therefore, one only needs to prove that T σ V is uniformly continuous in s given that V is continuous in s to show that the family T σ (C) is equicontinuous. (3) T σ V is uniformly continuous in s given that V is continuous. This part of the proof is omitted for this version. (4) Then T σ has a fixed point in B (S) in the space of uniformly cts V s by Schauder s fixed point theorem. The next step is to prove that V σ is attainable. This is straightforward as the space of relevant prices is compact. 17 We use the notation in Stokey and Lucas (1989) p Notation that is used only for this Lemma is not explained and the reader is referred to Stokey and Lucas (1989) for definitions. 18 An alternative and simpler proof may be to use the fact that Tσ is a monotone operator. We expect this approach to prove fruitful but have not finished exploring it yet. 19 This will hold also if we let ψ be stochastic in S as long as there are no jumps in probabilities

13 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 13 Lemma 5. Let V σ be a fixed point of T σ defined as above. Then there is a strategy for firm 1 : σ 1 : S R that achieves V σ. Furthermore, if σ is continuous then there is a strategy σ 1 that achieves σ and is continuous in s. Proof. Let Σ : S R be the space of pricing policies for the firm. Σ is complete so we only need to show it is bounded to show that the sup is always achieved. Consider the problem of a monopolist playing a repeated game where all consumers exhibit switching costs regardless of whether they purchased the good in the previous period. This monopolist simply solves the same period game every period. Let p be the optimal price for this monopolist given and π be his the per period profits. Both π and p exist because the per period profit function is concave. Because sup s V (s; σ) π 1 β and competitive prices will always be lower than this monopolist s price, the range of Σ (prices) can be bounded from below by π 1 β and bounded from above by p. Continuity of σ 1 is proven in steps: (1) π () is continuous in all of its parameters by construction. (2) ψ (p, s) is continuous in all of its parameters by construction. (3) σ is continuous by assumption (4) V (s; σ) is continuous given (1)-(3) (5) ˆπ (p 1, s) π 1 (p 1, σ (s 2, s 1 ), s 1, s 2 ) is continuous in p 1 and s. (6) ˆψ (p 1, s) ψ (p 1, σ (s 2, s 1 ), s 1, s 2 ) is continuous in p 1 and s. ( (7) σ (s) = arg sup pi ˆπ (p i, s) + βv ˆψ (pi, s)) is continuous in s because it is a sup on the sum of two continuous functions and the sup is obtained because domain is the bounded as proven above. Let P (V σ, σ) = {σ 1 : V σ is achieved by σ 1 } and BR (σ) = {σ 1 : σ 1 (s) P (V σ, σ), σ 1 (s) is continuous}

14 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 14 BR (σ)is a correspondence from an opponent s strategy σ to the possible best response policies. An MPE exists if BR (σ) has a fixed point. Lemma 6. BR (σ) has a fixed point. Proof. We present the outline of the proof. It is sufficient to show that BR (σ) is monotone and continuous and then apply Theorem 17.7 in Stokey and Lucas (1989). Monotonicity follows immediately from prices being strategic complements in discrete choice models. Continuity follows from V σ being continuous in σ and noting that BR (σ) is the pointwise maximizer of T σ V σ. This function is continuous in σ and V σ which proves the required result. This proves the existence of the MPE. Proposition 7. An MPE with continuous strategies always exists. Proof. Follows from Lemma 6. We conjecture that the BR (σ) operator has a unique fixed point. It is likely that BR (σ) is a contraction in a large subset of the relevant parameter set. 20 BR (σ) was already proved to be monotone and it is bounded by construction. Therefore, if the optimal response to a change in the opponent s price is never to fully mimic the change, BR (σ) is a contraction and has a unique fixed point. This is in fact a common feature of discrete choice models and therefore one would expect it to hold for this model as well. As we are unable yet to derive the conditions for this in our model, we leave this point as a conjecture at this version. Conjecture 8. For a well defined subset of the parameter set the symmetric Markov perfect equilibrium is unique in the space of continuous strategies Proof. Incomplete. See the discussion above. We now move to proving the existence of a steady state in shares for the MPE 21 : 20 An alternative path is to use recent results on monotone operators (see e.g. Wu and Liang (2006)) 21 Note that the proof does not require uniqueness.

15 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 15 Proposition 9. Every continuous MPE gives rise to a steady state in shares Proof. First, we show that every continuous Markov strategy has a steady state. Because the MPE is in continuous strategies, this implies it has a steady state. To see the first claim, define π σ (s) π (σ (s), s) and ξ σ (s) ψ (σ, s). It suffices to show that ξ σ : S S is continuous for every continuous σ. If σ (s) is continuous then ψ (σ, s) is continuous by construction and therefore ξ σ is continuous. S is compact so by Brouwer s theorem there is a fixed point s = ξ σ (s ) for every continuous σ (s). To summarize, this section had shown that an MPE in symmetric pure strategies always exists and that any such equilibrium must give rise to a steady state. 22 Sufficient conditions for uniqueness have been identified as well. In the next subsection we present an approach to numerically approximate the MPE Solving for Markov Perfect Equilibrium. In this section we detail the computational algorithm we use to compute a Markov Perfect Equilibrium and find its steady state. To search for the MPE, we employ value function iteration, because our state space is a continuum, projection methods must be used. Our computational approach to finding the MPE is similar to that employed in Doraszelski (2003) on a much different application. The numerical methods employed are discussed in further detail in Judd (1998). The value function for firm 1 is, V (s s 1, s 2 ) = max p 1(s) ψ 1 (p 1 (s) ; p 2 (s), s) (p 1 (s) c) + βv (ψ (p 1 (s) ; p 2 (s), s)) (3.4) The first order condition for firm 1 s optimal price in every state s is, [ ψ 1 V ψ 1 (p 1 (s) c) + ψ 1 (p 1 (s) ; p 2 (s), s) + β + V ] ψ 2 = 0 p 1 s 1 p 1 s 2 p 1 Where the derivatives are evaluated at (p 1 (s), p 2 (s), s). Evaluating (3.4) requires that we know the derivatives of the value function with respect to the state at an 22 We have not shown that repeated play of the equilibrium strategies will converge to the steady state, though we plan to investigate this possibility in future versions.

16 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 16 arbitrary point in the state space. We will approximate V with a two dimensional Chebyshev polynomial, ˆV. 23 V (s 1, s 2 ) ˆV K K (Z 1 (s 1, s 2 ), Z 2 (s 1, s 2 ); θ) = θ k1,k 2 T k1 (z 1 ) T k2 (z 2 ) k 1=0 k 2=0 Where T k (z) = cos(n cos 1 z) is the Chebyshev polynomial of order k evaluated at z. The function Z : S [ 1, 1] 2 is a continuous invertible mapping of our state space (the interior of the unit simplex) to the space on which Chebyshev polynomials are mutually orthogonal. We employ the mapping, 24 Z(s 1, s 2 ) = 2s 1 1 2s 2 1 s 1 1 The Chebyshev polynomial is fully described by the finite dimensional parameter θ. Furthermore, for every θ, ˆV z i are well defined and known. Therefore, for an approximation of the value function we can solve the approximation of (3.4) using ˆV z i and ẑi s i to determine player 1 s optimal price. This is done on a finite set of interpolation nodes which are chosen to be the Cartesian product of the roots of the order m Chebyshev polynomial where m > K. We then use these optimal prices to update the value function (Judd, 1998, p. 223). The procedure for an iteration, where r is the iteration count, is: (1) We begin with {V r 1, p r 1 1, p 2 r 1 } which the current approximations of firm 1 s value function and pricing strategies for both firms at our interpolation nodes. First, project V r 1 onto the space of Chebyshev polynomials to find θ r, θ r k 1,k 2 = m m i=1 j V r 1 (z i, z j )T k1 (z i )T k2 (z j ) ( m i=1 T k 1 (z i ) 2 )( m j T k 2 (z j ) 2 ) 23 Chebyshev polynomials are well suited to approximating arbitrary smooth functions. However, we have the additional information that V is monotone in shares. In future versions, we plan to investigate the use of shape preserving approximations of V. 24 Many other mappings are possible. The hope is that the results will be robust to the choice of mappings. We have not experimented with other mappings yet.

17 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 17 (2) Numerically solve the following for p r i (z) for every z = (z i, z j ) using p r 1 i as the opponents strategy and θ r to approximate the value function and its derivatives. (3) Use p r 1 computed above to generate V r on the interpolation points according to, V r (z) = ψ 1 ( p r 1 (z) ; p r 1 2 (z), Z 1 (z) ) (p r 1 (z) c)+β ˆV ( Z ( ψ ( p r 1 (z) ; p r 1 2 (z), Z 1 (z) )) ; θ r) (4) Check the stopping criteria. 25 The algorithm is finished if ˆV r ˆV r 1 ε otherwise increment r and return to step 1. Note that we must solve for p 2 separately, because of the transformed state space is not symmetric, i.e, p 2 (z 1, z 2 ) p 1 (z 2, z 1 ). Instead, p 2 (z 1, z 2 ) = p 1 ((1 z 1 ) z 2, ) z 1 1 z 2 (1 z 1 ) which we find using the transformation Z. Therefore, to find p 2 we solve firm 1 s first order condition at these additional points. While this algorithm is not guaranteed to converge 26, we have used it to find several MPEs at different parameter values Discontinuous Strategies and Cartels. McAfee and McMillan (1992) show that repeated duopoly games allow tacit collusion using discontinuous trigger punishments. The same analysis holds true in the dynamic case with deterministic transitions. In this section we first solve the cartel s optimal strategy (assuming no defections) and then characterize the requirements for sustaining the cartel as a competitive equilibrium. This version focuses on weak cartels - cartels that cannot 25 More work on the accuracy of the stopping criterion will be discussed in future versions. 26 Convergence is guaranteed if the operator Tσ is monotone operator with some additional properties that typically imply uniqueness of the MPE (e.g. a contraction)

18 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 18 make side payments between the firms. This is primarily because the weak cartel assumption is more in-line with the tacit collusion intuition Maximal Weak Cartel Profits. The optimal weak cartel strategy will involve both firms playing the same price over time. Thus we can write the cartel s problem under collusion as choosing a single price policy: max p t=0 βt (ψ 1 (p, p, s 1, s 2, s 0 ) + ψ 2 (p, p, s 1, s 2, s 0 )) (p c) subject to s t+1 = ψ (p, p, s t ) This program is a single agent problem and can be solved for a steady state using standard dynamic programming techniques. The result is a steady state price and share which maximizes the cartel s profits Sustaining a Weak Cartel Using Grim-Trigger Strategies. The cartel solution defines a pricing strategy p C (t) which converges to the weak cartel steady state. 27 Using these policies we can find the evolution of shares s C (t) and firm values for members of the cartel V C (t). In contrast, the continuous MPE defines p M (s), s M and V M (s). As V C (t) V M (s (t)), the weak cartel outcome can be supported as an equilibrium when we use the MPE strategies as the punishments strategies to be used in the event of a defection. As shares are deterministic, defecting from the cartel agreement is identified with probability one in the period following defection. Consider the strategy profile where both firms is to play the cartel strategy p C as long as a defection is not detected and revert to the least profitable (or unique) MPE strategy upon detection. The value of a defection when the at time t when the state is s t is therefore: 28 V D ( (t) = max ψ 1 p1, p C 2 (t), s C (t) ) (p 1 c) + βv M (ψ(p 1, p C 2 (t), s C (t))) p 1 The cartel is sustainable at state s C (t) if the solution to the defector s problem V D < V C. 27 p C (t) is a function of the initial shares. 28 We assume without loss of generality that player 1 is the defecting player.

19 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 19 Lemma 10. Let V C be the cartel s steady state value per firm. If for all s, V M (s) < V C then there exists r > 0 such that for r > r if the steady state shares of the cartel is reached then the cartel is always sustained. Proof. Trivial (following M&M) and omitted. Theorem 11. There exists β > 0 such that for β < β the weak cartel is sustained Proof. We showed above that the Markov perfect equilibrium converges to a steady state. Let the value for a firm under the MPE in the steady state be V M. Consider a sequence of r n converging to zero, then V M (s) V M. Similarly, V C (s) V C. V M V C. Therefore there is a sequence β n converging to zero such that Lemma 12 applies Sustaining a Weak Cartel using Discontinuous Markov Strategies. To be completed (maybe). The previous subsection used a non-markov strategy - the state space of the strategy automata was enlarged to account for the occurrence of a defection. The question is, if firms condition their strategies only on the current period shares, can they know they are on a punishment path and not on their way to the cartel steady state? If we start at the cartel steady state, then the problem disappears. Any change in the state would imply a defection and so the cartelsupporting MPE is well defined. However, if the game starts at an arbitrary state the problem is more involved. 4. Numerical Example In this version we compare the results for the weak cartel and the open loop equilibrium. These results are for a continuous time version of the model and are solved using continuous time optimal control. Results on Markov Perfect Equilibria will be provided in a future version. We have verified the key finding that equilibrium prices are decreasing in γ for low values of γ in the MPE s that we have computed so far. However, because the MPE uses a discrete time formulation and

20 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 20 the model presented here uses a continuous time formulation the two models are not directly comparable. Therefore we defer discussion of the MPE results. We solve the model numerically for a range of γ = [0, 3.8]. We are interested in how prices, market shares and firm profits in this model vary as we vary γ. We use a baseline parametrization with δ 1 = δ 2 = 1, c = 0, β = 0.95 and α = Since we assume that the firms are symmetric we restrict our analysis to symmetric equilibria. A useful feature of this parametrization is that γ is also the ratio of switching cost to baseline value of the two goods, so we plot switching costs that range from zero to 3.8 times the baseline utility of the good, we believe that the upper limit of our range can be interpreted as a very large switching cost. Our numerical results are presented in the following figures, these show the open loop equilibrium and cartel solution for two specifications. The only difference between the two models is consumers preferences towards the outside good. The red curves represent the results of the model explained above in which consumers have no loyalty to the outside good (γ 0 = 0). Consumers in the blue model experience a loyalty γ 0 = γ when they are in the set S 0. The solid lines are the values at the open loop Nash equilibrium steady state. Dashed lines correspond to the weak cartel. First we consider prices over this range for γ, which are plotted in Figure 4.1. Figure 4.2 plots the derivative of prices with respect switching costs. These figures show that prices are decreasing in the switching cost for low levels of γ but flatten out as γ increases and eventually become increasing. However, prices become increasing much more quickly for the cartel problem when the outside good is not insulated by switching costs (i.e., γ 0 = 0). When there are no switching costs (γ = 0), firms are playing a repeated game as they have no forward-looking interest in maintaining a high share of purchases. Once we increase γ slightly, firms balance the effect of a marginal increase in their share with the cost of a price reduction to current period profits. The extra brand 29 In the continuous time formulation β =.95 equates roughly to a discount factor r =.05.

21 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS Figure 4.1. Effect of the switching cost parameter on steady state prices under weak cartel (dashed) and open loop equilibrium (solid) for γ 0 = 0 (red) and γ 0 = γ (blue) Figure 4.2. Marginal effect of the switching cost parameter on steady state prices under weak cartel (dashed) and open loop equilibrium (solid) for γ 0 = 0 (red) and γ 0 = γ (blue). loyalty utility that switching costs provide increase the steady state shares in response to a price decrease more than in the static case. Therefore, the marginal benefit for the firm from reducing prices initially increases with switching costs more than the marginal cost of the price increase (i.e. the lost revenue). Thus the marginal effect of switching costs at γ = 0 on prices is negative, even in the most

22 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 22 protected cartel setting. 30 As γ becomes large, the steady state moves away from that of the repeated game steady state, and the sign becomes ambiguous. For high switching cost values, it becomes increasingly difficult to affect consumer purchases by lowering price, but the value of maintaining a large share increases. A firm s loyal customers consider the firms product to be far superior to the alternatives. Therefore the firm will tend to sell more to its loyal customers and it is less worthwhile to offer bargains to potential switchers. However, since some loyal customers are going to leave the pool, it is important to have prices low enough that they will be replaced by newcomers. Next, we compare the effect of whether consumers pay a cost to switch away from the outside good (the blue model) or not (the red model). Which assumption is appropriate will depend on the institutional setting. 31 For example, if we believe that the outside good is not using the product at all, it would seem unlikely that it would have its own switching costs. On the other hand, when the outside good is a substitute product or service, it may have its own switching cost. For example, if firms are offering maid service and the outside good is to clean yourself, cleaning on your own may exhibit switching costs due to knowledge gained about the cleaning job by actually carrying out the task. 32 We see that the question of whether the outside good exhibits switching costs or not has only a mild effect on the open loop equilibrium, but is extremely important in the case of the cartel. In the cartel, the only competition is with the outside good, so a weaker outside good quickly leads to higher prices. In the competitive equilibrium, firm pricing is disciplined by both the outside good and the potential for the opponent firm to undercut, the latter 30 It is worth noting that Beggs and Klemperer (1992) find their result that prices are increasing in switching costs for a model where agents are never able to switch between firm products, so their intuition would not apply to our case where γ is low. Instead their model assumes a persistent flow of new consumers who have never before purchased either good, and pools of locked in consumers who may repurchase their chosen good or not purchase at all. Their equilibrium is constructed such that all consumers always purchase one of the two goods. 31 In a future version we plan to examine a model where consumers purchasing the outside good maintain loyalties for their previous purchase, which is the framework considered by Dube et al. (2008). 32 Another example: if the two firms are airlines and the outside good is taking the car switching costs may arise due to increased familiarity with the road route. However, if the outside good is not traveling, than it may not exhibit switching costs.

23 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 23 restrains the effect of the outside good switching cost. As we will see below, this effect makes collusive equilibria both more attractive and easier to sustain in the case where the outside good is not insulated by switching costs Figure 4.3. Effect of the switching cost parameter on firms shares in the steady state under weak cartel (dashed) and open loop equilibrium (solid) for γ 0 = 0 (red) and γ 0 = γ (blue). Next we consider the effect of switching cost on firm shares. Figure 4.3 plots firm share as a function of γ for both models. The intuition from these plots is clear, as switching costs grow, firms react in a way that increases their share of loyal customers. When non-purchasers are loyal to the outside good, it is more difficult to maintain a high market share. We can also breakdown the probability that each type of consumer purchases firm 1 s product. This graph is presented for the competitive equilibrium where the outside good has no switching cost in Figure 4.4. This plot re-enforces our intuition about the third-degree price discrimination trade-off between loyal and non-loyal consumers. For low switching costs, consumers are only slightly more likely to purchase their loyal good, and firms wish to market effectively to all three types. For high switching costs, the firms attract primarily their loyal customers, and have little hope attracting customers who are loyal to the opposing firm When the outside good has switching costs, these consumers act in a manner similar to those who are loyal to good 2.

24 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS Figure 4.4. Effect of the switching cost parameter on the probability that each consumer type purchases good 1 in competitive open loop equilibrium where γ 0 = 0. Consumers who previously purchased good 1 are plotted in red; consumers previously purchasing good 2 in blue, and previously purchasing the outside good in green Figure 4.5. Effect of the switching cost parameter on firms profits in the steady state Figure 4.5 examines the effect of switching costs on firm profits. Here we see that switching costs do not necessarily lead to higher profits even for cartels. Cartel profits grow rapidly when the outside good is not insulated by a switching cost, but actually decline when the outside good is insulated with its own switching

25 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 25 costs. In other words, firms have little incentive to artificially generate industry wide switching costs unless they can effectively cartel to avoid price competition. Duopolistic competition (in the open loop sense at least) is sufficient to force the firms to transfer the extra welfare generated by brand loyalty to the consumers Conclusion We have examined a discrete choice switching cost model that uses a continuum of consumers. Our model is entirely deterministic from the point of view of the firms, whereas in the finite consumer model, the idiosyncratic shocks of consumers creates a stochastic optimization problem at the firm level. The existence of equilibria is established along with conditions for uniqueness. Compared to previous models, numeric analysis suggests the dynamic considerations by the firms intensify competition and reduce prices. Switching costs provide firms with an incentive to increase their steady state shares and this drives prices down. In an open loop equilibrium this effect dominated the firm s gains from squeezing its loyal customer base. Preliminary results indicate that the same qualitative effect is sustained under Markov Perfect Equilibria for low switching costs values. It is still possible that in the MPE solution concept prices will rise as a function of switching costs at moderate to high levels of switching costs. This would be the case if fierce competition is not be sub-game perfect. Nevertheless, this results imply that as switching costs rise, the gains from preventing competition (i.e. colluding) rise as well and tacit collusion becomes easier to support. References Beggs, A. and P. Klemperer (1992, May). Multi-period competition with switching costs. Econometrica 60 (3), Doraszelski, U. (2003, Spring). An R&D race with knowledge accumulation. RAND Journal of Economics 34 (1), This of course says nothing about the motivation for firm specific innovations that increase only the firm s switching cost (γ).

26 SWITCHING COSTS WITH A CONTINUUM OF CONSUMERS 26 Doraszelski, U. and A. Pakes (2007). A framework for applied dynamic analysis in io. In M. Armstrong and R. Porter (Eds.), Handbook of Industrial Organization, Volume 3, Chapter 30, pp Amsterdam: Elsevier. Dube, J.-P., G. J. Hitsch, and P. E. Rossi (2008, June). Do switching costs make markets less competitive. University of Chicago. Farrell, J., P. Klemperer, M. Armstrong, and R. Porter (2007). Chapter 31 Coordination and Lock-In: Competition with Switching Costs and Network Effects, Volume Volume 3, pp Elsevier. Judd, K. L. (1998). Numerical Methods in Economics. Cambridge, MA: MIT Press. Maskin, E. and J. Tirole (1988, May). A theory of dynamic oligopoly, I: Overview and quantity competition with large fixed costs. Econometrica 56 (3), McAfee, R. and J. McMillan (1992). Bidding rings. American Economic Review 82 (3), Stokey, N. L. and R. E. Lucas (1989). Recursive Methods in Economic Dynamics. Harvard University Press. Wu, Y. and Z. Liang (2006, November). Existence and uniqueness of fixed point for mixed monotone operators with applications. Nonlinear Analysis 65 (10),