Web Appendix or The Value o Switching Costs Gary Biglaiser University o North Carolina, Chapel Hill Jacques Crémer Toulouse School o Economics (GREMAQ, CNRS and IDEI) Gergely Dobos Gazdasági Versenyhivatal (GVH) February 3, 2010 In this Appendix, we provide the ormal deinitions and proos or the ininite horizon models o the text. First, we describe the elements that are exactly common to all the models or are small modiications o each other. Next, we ormally deine equilibria. For compactness, we reer to the uniorm model when discussing the model when all consumers have the same switching cost. 1 Elements common to all the games studied in the paper We introduce the common element or all the games studied in this paper. 1.1 The players The players are: The incumbent I; Potential entrants E = {(i, t)} i N,t N, where N is the set o strictly positive natural numbers {1, 2,... }. We will use the terminology entrant when there is no possible ambiguity. Entrant (i, t) is the i th entrant in period t; we will denote by E t the set {(i, t)} i N o potential entrants in period t. The set o all irms is F = {I} E. Consumers are represented by a number rom c [0, 1]. In the uniorm switching cost model, all consumers have a switching cost σ > 0. In the 1
Stackelberg and Bertrand models, consumers c [0, α] have switching cost σ, while consumers c (α, 1] have a switching cost equal to 0. 1.2 First period moves Let us denote by R = R {+ } the set o real numbers to which we have appended the element +, which will be interpreted as not making an oer. In the uniorm and Bertrand switching cost models in period 1, irms {I} E 1 oer their price in R, while in the Stackelberg model in period 1, the incumbent {I} chooses its prices irst, then all entrants in E 1 oer their price in R. It will be notationally convenient to assume that the other irms (i.e., the potential entrants in subsequent periods) also make an oer, which is restricted to +. The} set o active irms, those who make an oer is F 1 = { F p 1 +. O course, F 1 {I} E 1. I F 1 =, the game ends. Consumers The consumers choose to purchase rom one irm or another. We will designate by φ 1 (c) the irm rom which consumer c buys in period 1, and call λ 1 () the set o consumers who buy rom in period 1: λ 1 () = { c = φ 1 (c) }. 1.3 First period payos Firms The period 1 proit o irms which do not belong to F 1 is equal to 0. I λ 1 () is measurable, let Q 1 = µ(γ1 ()) be the total quantity sold by irm F 1. The period 1 proit o irm, Π 1 () is equal to p 1 Q1. I λ 1 () is not measurable, the proit o irm is 0, and the game ends. Consumers I F 1, the disutility o consumer c is p 1 φ 1 (c) i φ1 (c) = I or i the consumer is a lsc consumer (c > α in the Stackelberg and Bertrand models), while it is p 1 φ 1 (c) + σ i φ1 (c) I and the consumer is a hsc consumer (c α in the Stackelberg and Bertrand models and all consumers in the uniorm model). I F 1 = or i or some irm λ 1 () is not measurable, the payo o all consumers is, by harmless abuse o language, +. 1.4 Moves or periods t > 1 We assume that in period t 1, a non-empty set F t 1 o active irms; unctions φ t 1 (c) and λ t 1 () which describe the allocation o customers to irms in the previous period. The total sales o the irms and such quantities are also deined as in the case o period 1. Firms We assume that i Q t 1 = 0 or some irm active in period t 1, then it drops out o the game. So that the set o potential irms in period t is F { t = F t 1 Q t 1 > 0} E t. 2
The moves that the irms can make are R or F t ; the others are restricted to playing +. The active irms are the} irms which make an oer; they belong to the set F t = { p t +. In the Stackelberg model, in every period, incumbent irms choose prices irst and, ater observing these prices, entrants choose their prices. I F t =, the game ends. Consumers I F t consumer c must choose a irm φ t (c) rom which to buy rom. The set λ t () is the set o consumers such that φ t (c) =. 1.5 Period payos or t > 1 Firms The proit o a irm which does not belong to F t is equal to 0; or the irms in F t the proit is p t Qt where Qt = µ(γt ()), when λ t () is measurable and 0 otherwise. Consumers I F t, the disutility o consumer c is p t φ t (c) i φt (c) = φ t 1 (c) or i the consumer is an lsc consumer (c > α in the Stackelberg and Bertrand models) and p t φ t (c) +σ otherwise. I F t = or i or some irm λ t () is not measurable, the payo o all consumers is + and the game stops. 1.6 Intertemporal Payos For both irms and consumers the intertemporal payos is computed by adding each period payo weighted by the discount actor δ (0, 1) over t N (except that in all cases where the game ends in inite time, the disutility o the consumers is equal to + ). 1.7 Strategies Histories are well deined. Deine h 0 = { } and or t 1, h t is constructed in the usual way by the concatenation o h t 1 and the moves in period t. The strategies o irms in the Stackelberg Model For the Stackelberg model, a strategy or the incumbent irm in period 1, ρ 1 I, is a period 1 move, with the constraints described above. In period 1, a strategy or an entrant is a unction rom the incumbent s irst period price, p I, to a price p 1, with the constraints described in 1.3. A strategy in period t > 1 or an incumbent, is a unction rom h t 1 into moves at period t with the constraints described above, while a strategy or period t entrants is a unction rom h t 1 and the prices set by the incumbents in period t, p t I. A strategy ρ is a strategy or { } every period ρ t t N. 3
Firm Strategies or Bertrand and Uniorm Models As we will see, in the Bertrand model there exist no pure strategy stationary equilibrium which satisies our assumptions. Thereore, we need to expand our deinitions to mixed strategies or irms. For all irms, a strategy in period 1, ρ 1 is a probability distribution over period 1 prices, with the constraints described above. In period t > 1, a strategy is a unction rom h t 1 into a distribution over prices at period t with the constraints described above. A strategy ρ is a strategy or every { } period ρ t t N. Strategies or Consumers For consumers, a strategy ρ t c in period t is a unction rom the concatenation o h t 1 and the moves o the irms in that period t into moves in period t, with the constraints imposed above. A strategy ρ c is a strategy or every period { ρ t } c.(because we will assume that t N consumers coordinate to purchase rom the same irm, we need to assume that they have a pure strategy.) The outcomes are deined as usual rom the strategies, and the payos associated with strategies are the discounted payos o the outcomes. 2 Equilibria in the ininite horizon models: deinitions In the games with ininite horizons studied in Sections?? and??, we identiy subgame perect equilibria, which satisy the ollowing added constraints: inite entry, consumers have mass, no play o undominated strategies and stationarity. { { } } Subgame perection A set o strategies ρ, F {ρ c} c [0,1] is a subgame perect equilibrium i or every irm and every consumer c, there exist no other strategy and no history such that a deviation to that other strategy ater that history would increase the payo and i in any period γ t () is measurable. Finite number o active entrants We will only consider equilibria where in every period there is only a inite number o entrants, that is irms which announce a price smaller than + : or this it is suicient to require that F t be inite in every period (there are only a inite number o active irms in each period). Consumers have mass We want to prevent the ollowing type o situation: a irm makes a much better oer than the incumbent, taking 4
into account the act that the hsc consumers have to pay the switching cost σ. However, all hsc consumers think that the others will reuse the oer. Thereore, every consumer eels that i he accepts the oer, he will be the only one, which implies that will not make an oer in the ollowing period, and he will have to pay the switching cost once again in that period. Thereore, it is an equilibrium or all consumers not to accept the oer. Thereore, we assume that consumers have mass : ormally, we will allow small groups o consumers { } to coordinate on a strategy. For all t N, all h t 1 and all (easible) p t (that is or all possible histories at the F time the consumers choose their moves) it is not true that or all ε > 0 there exists a set C ε [0, a] o consumers, with 0 < µ(c ε ) ε; φ t 1 (c) is invariant over C ε ; ρ t c(h t 1, p t ) is independent o c; there exists c t (h t 1, p t ) such that all the consumers in C ε would be strictly better o purchasing rom than rom ρ t c(h t 1, p t ) (in the sense that their discounted disutility would lower ater this deviation rom the equilibrium). Stationarity We will make two stationarity assumptions. 1. Incumbents use the same mixed strategy, i they have proitable (hsc) customers, whatever the history. Formally, there exists ρ such or all t, whatever h t 1, i a strictly positive mass o hsc consumers purchased rom irm in period t 1, then ρ t (ht 1 ) = ρ. (In the Uniorm and Stackelberg models, there are pure strategy equilibria; F I will be a mass point at the equilibrium price.) 2. In or out o equilibrium, the distribution o the minimum o the prices charged by entrants (and, when relevant, the irms which sold only to lsc customers in the previous period), is independent o history. We will call it F E. Firms do not play undominated strategies As in one period Bertrand models with dierent costs or the dierent irms, we need an assumption that irms play undominated strategies: there does not exist, t, h t 1 and a easible move ˆp t such that a) the proits o irm generated by choosing ˆp t ater history ht 1 is equal to its proits when it chooses ρ t (ht 1) when the other irms play their equilibrium strategies; b) the proits o irm generated by choosing ˆp t ater history ht 1 are at least as large as its proits when it chooses ρ t (ht 1) whatever the moves chosen by the other irms in period t and strictly greater or (at least) one set o moves, i we assume that the irms play their equilibrium strategies in subsequent periods. This assumption only plays a role in our proos in the Bertrand model when all consumers have the same switching costs. 5