Inert Consumers in Markets with Switching Costs and Price Discrimination

Transcription

1 Inert Consumers in Markets with Switching Costs Price Discrimination Marielle C. Non March 15, 2011 Abstract This paper analyzes an infinite-period oligopoly model where consumers incur costs when switching supplier. Every period only a fraction α of consumers considers switching. Firms can price-discriminate between their current customers consumers who did not buy from the firm in the past period. In this setting, firms set relatively low introductory prices to attract new consumers. Contrary to previous findings in switching costs models, the full price can both increase decrease in switching costs, while the introductory price always decreases. Moreover, the introductory price increases in α while the full price again can both increase decrease. Keywords: switching costs, pricing, price discrimination, inert consumers JEL codes: C73, D43, L13 Contact:University of Groningen, DUI 314, P.O. Box 800, 9700 AV Groningen, The Netherls. M.C.Non@rug.nl. Phone: 0031 (0)

2 1 Introduction There exist quite some consumer markets where switching costs play an important role. Some examples that are often mentioned are telecom, insurance energy. Starting with Klemperer (1987), several researchers have investigated markets with switching costs, see Farrell Klemperer (2007) for an overview. One of the main themes in this previous research is the question whether switching costs will increase prices. There are however other interesting dimensions to switching, that have received less or no attention in previous research. In the current paper three of those elements are included in a switching model. First of all, in many markets with switching costs introductory offers are quite common. This can also be interpreted as price discrimination between current future customers. Most previous research on switching costs however assumes that firms can not discriminate between consumers. In the current paper it is shown that the introduction of introductory offers in a switching costs model has an important effect on the relation between switching costs prices. Second, previous research either assumes that consumers are perfectly rational in their switching behavior, or that consumers are completely inert in the sense that they always stay with their first choice (for a model with inert consumers see e.g Beggs Klemperer (1992) Padilla (1992)). This paper assumes that every period a fraction of consumers considers switching, but that the remaining consumers do not consider switching stay with their old firm. This is in line with observed behavior, see Wieringa Verhoef (2007). The common intuition is that an increase in the fraction of consumers who consider switching will increase competition lead to lower prices. The analysis in this paper shows that in a market with introductory offers this is not necessarily true. Third, most of the previous literature examines duopoly models, mainly because an extension to the oligopoly case is not computationally feasible. The only oligopoly model I am aware of is the one by Dube et al. (2009). The model in this paper also allows a general number of firms, which gives the possibility to examine the effect of an increase in the number of competitors on prices, switching behavior welfare. The model in this paper is an extension of the basic model of Cabral (2009). Cabral proposes a duopoly model with infinitely lived consumers firms. Each firm can price discriminate between consumers who have bought from the firm in the previous period, consumers who have not. Every period consumers assess whether they want to stay with their current firm, or switch to the competitor. Switching however comes at a cost. This model is extended by assuming that each period only a fraction α of the consumers considers switching. The parameter α can be a constant, but could also be a 1

3 function of the expected benefits of switching. Moreover, a general number of firms, n, is allowed. In equilibrium, the possibility of price discrimination indeed leads to introductory pricing. Consumers who return to their current firm pay the full price, while switching consumers pay a reduced price. This result is not very surprising. More interesting is the pricing behavior when the switching costs increase. The introductory price decreases in switching costs, for a relevant part of the parameter space the full price also decreases in switching costs. Although not mentioned by Cabral, this result also holds in the basic model with two firms α = 1. The reason why the introductory price decreases in switching costs is fairly simple. The introductory price is meant to attract switchers, when switching costs increase firms try to offset this by lowering the introductory price. For the full price, there are two effects. On the one h, an increase in the switching costs implies the firm has more monopoly power over its current customers. This should lead to an increase in prices. But on the other h the gap between the introductory full price becomes larger, which induces more consumers to switch. To prevent this, a firm should lower the full price as well. As a result of those two effects the full price can both increase decrease. As a total effect, the gap between the introductory full price always increases in the switching costs. This increase is however less than the increase in switching costs, implying that the net gains from switching decrease. Next to analyzing the effect of switching costs on prices, the model in this paper also allows to analyze the effect of an increase in the number of consumers who consider switching. The common idea is that an increase in the fraction of those consumers will lead to more competition lower prices. This issue is also interesting from a government perspective. In many recently liberalized markets only very few consumers switch supplier often the government sees it as its task to stimulate competition by raising awareness among consumers. The analysis in this paper shows that this might have an opposite effect. When the fraction of consumers who consider switching increases, the introductory prices tend to increase, while the full prices can both increase decrease. The intuition again is not very complicated. When setting the introductory price the firm weighs the price itself against the number of consumers it will attract. This number of consumers decreases in the introductory price, but increases in the fraction of consumers who consider switching. Therefore, when more consumers consider switching, it gets more attractive for firms to increase the introductory price. For the full price, on one h firms want to decrease it to keep aware customers in the firm. On the other h, the increase in the introductory price gives room to increase the full price as well. Firms balance those two effects, as shown by a full price that can both increase decrease. 2

4 The prices seem to increase in the number of firms... TO BE INVESTI- GATED FURTHER. The next section present the model that will be used. Section 3 will analyze the duopoly case of the model, presents some very clear results on the duopoly model. Section 4 will analyze the general oligopoly case. In section 5 an example will be investigated section 6 concludes. 2 Model Consider a market with n heterogenous firms, that each produce a single variety of the product. In the next section I will assume that n = 2. Later, I will generalize to any n 2. The firms have identical constant marginal costs of production, that will be normalized to 0. The firms choose their prices (or their price path) to maximize discounted profits with an infinite time horizon. The discount factor is δ. Firms are able to discriminate between consumers, based on whether the consumer has bought from the firm in the previous period or not. A consumer who has bought from firm j in period t 1, again buys from firm j in period t will have to pay the old consumer s price p oj,t. If the consumer would instead buy from firm k in period t (s)he will pay the new consumer s price p nk,t. Consumers have unit dem. They are infinitely lived buy in every period (alternatively, they have an infinite reservation price). If consumer i buys from firm j (s)he gets utility ɛ ij,t ˆp, where ˆp is either p oj,t or p nj,t, dependent on where the consumer bought in the previous period. ɛ ij,t is an iid rom variable drawn from F (ɛ) describes the match between consumer i firm j in period t. Note that every period ɛ ij,t is redrawn, this draw is independent from the draw in the previous period. When a consumer switches to another firm (s)he incurs switching costs s > 0. Consumers try to maximize their discounted utility, again with discount factor δ. At the same time, however, consumers are boundedly rational. With probability 0 < α < 1 a consumer considers switching to another firm, but with probability 1 α the consumer will not consider switching at all, will stay at his/her current firm. For a consumer who bought at firm j in period t 1 the probability α depends on P j will be denoted by α(p j ). In a duopoly market P j = p oj,t p nk,t s. In an oligopoly market with n firms P j = p oj,t 1 k j n 1 p nk,t s. That is, P j gives the expected first-period gain from switching to another firm. This gain increases in the old price, decreases in the new price switching costs. To ensure that the consumer more often considers switching when the expected gains from switching are higher, assume that α(p j) j = α (P j ) 0. Note that this allows α(p j ) to be a constant between

5 Whenever a consumer considers switching, (s)he compares utilities across firms, taking the switching costs into account. The consumer expects that in future periods (s)he will not switch anymore. Also, the consumer expects that in the future firms are symmetric, so that p oj,t+h = p ok,t+h p nj,t+h = p nk,t+h (which in equilibrium holds true). 3 Duopoly In this section I will analyze the model when there are two firms in the market. I will focus on a symmetric equilibrium. 3.1 Consumer behavior Consider a consumer at the start of period t who bought at firm j in period t 1. It is clear that with probability 1 α(p j ) the consumer does not consider switching will stay at firm j. With probability α(p j ) the consumer considers switching to firm k. In this case the consumer compares the utility of buying at j with the utility of buying at k. The (expected) utility of buying at j is ɛ ij,t p oj,t + E δ h (ɛ ij,t+h p oj,t+h ). h=1 Note that at the start of period t the consumers knows the realization of ɛ ij,t, but does not yet know the realization of ɛ ij,t+h with h 1. Also, the consumer expects that in future periods the old price is p oj,t+h. This could be the equilibrium old price, but this is not necessary as long as the consumer expects the same old price in firms j k. The old price in period t is observed at the start of period t. The (expected) utility of buying at k is ɛ ik,t p nk,t s + E δ h (ɛ ik,t+h p ok,t+h ). h=1 It is assumed that the consumer expects the firms to be symmetric, so expects p oj,t+h = p ok,t+h for h 1. Also, Eɛ ij,t+h = Eɛ ik,t+h. This implies that E h=1 δh (ɛ ij,t+h p oj,t+h ) = E h=1 δh (ɛ ik,t+h p ok,t+h ). Therefore, a consumer will decide to switch if only if ɛ ik,t p nk,t s > ɛ ij,t p oj,t, or ɛ ij,t ɛ ik,t < p oj,t p nk,t s = P j. In the remainder of this section, let G(x) be the CDF of ɛ ij,t ɛ ik,t, with g(x) the corresponding PDF. Then the probability that a consumer who considers switching indeed switches is G(P j ). 4

6 3.2 Pricing decision In each period each firm has to set two prices: p o.,t p n.,t. I will focus on a symmetric equilibrium where prices are constant over time, so p o.,t = p o, p n.,t = p n P = p o p n s. To derive the equilibrium prices, consider the pricing decision of firm j in period t. Let π o denote the (expected) discounted profits a firm can make on a single consumer who bought from the firm in period t 1 let π n denote the (expected) discounted profits a firm can make on a single consumer who did not buy from the firm in period t 1. Then π o = max p oj,t α(p j )G(P j )δπ n + (1 α(p j )G(P j ))(p oj,t + δπ o ) (1) π n = max p nj,t α(p k )G(P k )(p nj,t + δπ o ) + (1 α(p k )G(P k ))δπ n. (2) Note that π o depends only on p oj,t p nk,t π n depends only on p nj,t p ok,t. When setting its old price, firm j therefore only has to consider its old consumers when setting its new price firm j only has to consider its new consumers. Furthermore, the optimal prices do not depend on market share. This is a general feature of duopoly switching costs models with price discrimination, see also... The first-order conditions give the following equilibrium condition. Proposition 3.1 Let Q 1 (x) be defined by Q 1 (x) = (α(x)g(x)) = α(x)g(x)+ α (x)g(x). In any symmetric equilibrium, P is implicitly defined by P Equilibrium prices are given by 1 2α(P )G(P ) = s. (3) p o = (1 δ)(1 α(p )G(P )) + δα(p )G(P ) p n = Equilibrium profits are given by δ + (2δ + 1)α(P )G(P ). π n = α(p )2 G(P ) 2 (1 δ) 5

7 π o = δα(p )2 G(P ) 2 + (1 δ)(1 α(p )G(P )) 2. (1 δ) The proof of this proposition is in the Appendix. The first order conditions alone do not guarantee existence or uniqueness of the equilibrium. For this, the following assumption is needed. Assumption A1 with 2Q 1 (x) 1 α(x)g(x) < Q 2(x) Q 1 (x) < 2Q 1(x) α(x)g(x) Q 1 (x) = (α(x)g(x)) = α(x)g(x) + α (x)g(x) Q 2 (x) = (α(x)g(x)) = α (x)g(x) + 2α (x)g(x) + α(x)g (x). When assumption A1 holds, the second-order conditions are satisfied. Moreover, it can be shown that under this assumption (3) has a unique solution. This gives the following Proposition. Proposition 3.2 When A1 holds, the equilibrium defined by proposition 3.1 exists is the unique symmetric equilibrium. Whether assumption A1 holds depends strongly on α(x). Assume for the moment that α(x) is constant. If α(x) = 1, using that g(x) is symmetric around 0, assumption A1 simplifies to g (x) g(x) < 2g(x) G(x). This holds for every log-concave density. If on the other h α(x) = 0 assumption A1 simplifies to 0 < g (x) g(x) < 2g(x) G(x). Since g(x) is symmetric around 0, g (x) g(x) > 0 can never hold for all x in the support. More general, if α(x) is a constant, assumption A1 will hold for large values of α(x), while assumptions A1 will not hold for small values of α(x). Intuitively, this is clear from the profit functions. When only a few fixed number of consumers consider switching, a firm that has to set a price for its current customers wants to focus on the part that is loyal by definition, since those consumers have no maximum willingness to pay, the firm wants to set an infinite price. When α(x) is not constant, this reasoning will not hold anymore. Increasing the price will then lead to an increase in α(p ) a decrease in customers. this will give a moderate price, even when α(p ) is relatively small. It is fairly easy to find combinations of α(x) G(x) for which assumption A1 holds. One example with α(x) logistic will be discussed later. 6

8 3.3 Equilibrium properties In this section we will assume A1 holds. The proof of proposition 3.2 shows that the unique symmetric solution has P > s. Since P = p o p n s, P > s implies p o > p n. This is a quite intuitive result that mirrors the classic bargain-then-ripoff result of the switching literature. Firms compete fiercely for new consumers, trying to persuade them to switch by setting a low introductory price. But once a consumer is in the firm, the firm tries to rip him off by setting a higher price. At the same time, the proof of proposition 3.2 shows that in equilibrium firms make sure that less than 50% of their old customers switch: α(p )G(P ) < 1 2. If α(p ) is high enough, α(p )G(P ) < 1 2 implies that p o p n is bounded above. Even though firms have some power over old consumers, they do not have monopoly power. The higher the price for old consumers, the more those consumers consider switching (α(p ) does not decrease in P ), the more those consumers indeed switch (G(P ) increases in P ). This makes firms cautious not to raise their old price too much lose most of their customer base. Proposition 3.1 gives expressions for p o p n in terms of P. The price for old customers, p o, is always above marginal costs. To see this, note that both the numerator the denominator are strictly positive. The price for new customers can be below marginal costs. Again, the denominator of the expression for p n is always strictly positive, but the numerator can be both positive negative. For example, suppose δ = 1, α(p ) = ep 1+e P F (P ) Exp(1). Then for s = 1 2 the numerator ( consequently p n) is positive, but for s = 1 the numerator ( consequently p n ) is negative. The expected profits π o π n are both strictly positive, even though p n can be negative. A switching consumer can give a short term loss, but once this customer is old, the firm expects to make a profit on him/her. Note that because α(p )G(P ) < 1 2, π o > π n. So, as expected, a customer who bought from the firm in period t 1 is more profitable than a customer who did not buy from the firm in t 1. The next proposition summarizes the discussion above. Proposition 3.3 In equilibrium, the following properties hold p o > p n p o > 0 p n can be both positive negative π o > π n > 0 α(p )G(P ) < 1 2 7

9 An interesting issue is how the equilibrium changes when the switching costs increase. Intuitively, when the switching costs increase, consumers switch less. This increases the power firms have over their old consumers. At the same time, firms need to put in more effort to attract new consumers, which lowers the prices for new consumers. The next proposition formally states the effects of a change in switching costs s. Proposition 3.4 A change in s has the following effects 1. 1 < < 0 0 < po pn < 1 2. α(p ) 0, G(P ) < 0 3. po 4. For δ πo 6. πn < 0 α(p )G(P ) < 0 can be both positive negative. p n < 0 can be both positive negative. The proof of this proposition is in the Appendix. Proposition 3.4 confirms part of the intuition above. First, 2. shows that indeed consumers switch less when the switching costs increase. This is not surprising, given that the expected gains from switching, P, decrease in the switching costs. Part 1. also claims that the difference between the old new price increases in s. This reflects that indeed firms have more power over their old consumers compete more for new consumers. The competition for new consumers is also reflected in 4. The price for new consumers decreases when the switching costs increase, at least for δ 1 2. For δ < 1 2 it is not clear what happens to p n. Theoretically, the derivative could be positive, but I have not been able to find an example where this indeed happens. Perhaps a bit surprising is that the price for old consumers can both increase decrease in s. Intuitively, the firm should have more power over its old consumers, allowing it to increase prices. Still, p o can also decrease the key to understing this result is that the switching probability, α(p )G(P ), not only decreases in s but also increases in p o decreases in p n. If the decrease in p n is strong, many consumers will consider switching, which lowers π o. A firm might therefore want to counter the effect of p n on α(p )G(P ) by decreasing p o. If the decrease in p n is low (or if p n increases), the change in p n will be offset by the change in s a firm might want to increase p o. The increase in p o will however always be limited, since a too large increase will induce too much switching. This is also visible in 1. The 8

10 difference between p o p n will increase, but the derivative is bounded above. Given that the probability a consumer switches is decreasing in s, given that p n also tends to decrease in s, it is intuitive that the expected profits from new consumers decrease in s. After all, profits on those consumers can only be made when the consumer switches, those profits depend on p n. The profits on old consumers, π o can both increase decrease in s, reflecting that p o can both increase decrease in s. Next, consider the effect of a change in the discount factor δ. From (3) it is clear that P does not depend on δ. This greatly simplifies the analysis. The next proposition summarizes the effect of δ. Proposition 3.5 A change in δ has the following effects 1. δ 2. po δ = 0, po pn δ = 0, α(p ) δ = 0 G(P ) δ = 0 = pn δ = 1 2α(P )G(P ) < 0 3. (1 δ)πo 1 2α(P )G(P ) δ = < 0 4. (1 δ)πn δ = 0 This proposition can easily be derived from proposition 3.1 therefore the proof is omitted. The effect of δ on the prices is fairly intuitive. When the discount factor increases, the future gets more weight. Since the largest part of the profits that are made on switching consumers (π n ) are made in future periods, the future gets more weight, firms lower p n in order to induce more switching. More switching decreases the profits on old consumers. Therefore, firms lower p o to match the decrease in p n, thereby make sure that the switching rate α(p )G(P ) does not change. The analysis of profits needs some care. The profits do not only depend on prices market shares, but also on the level of δ. When δ increases, profits tend to increase as well, just because all future periods get more weight. Therefore, proposition 3.5 corrects the profits by dividing them by 1 1 δ = 1 + δ + δ Once the profits are corrected, the expected profits on old consumers decrease in δ. This can be explained by the decrease in prices. Surprisingly, the (corrected) profits on switching consumers do not change in δ. The reason for this is that firms make more profits on switching consumers in future periods. Even with the proposed correction, the future periods get more weight compared to the current period, which tends to increase total expected profits. On the other h, the decreasing prices tend to decrease profits. These two effects cancel out, leading to no net change in profits. 9

11 4 Oligopoly In this section I will analyze the model when there are n 2 firms in the market. I will again focus on a symmetric equilibrium. 4.1 Consumer behavior Consider a consumer at the start of period t who bought at firm j in period t 1. Recall that in an oligopoly P j = p oj,t 1 k j n 1 p nk,t s. With probability 1 α(p j ) the consumer does not consider switching will stay at firm j. With probability α(p j ) the consumer considers switching to some other firm. In this case the consumer compares the utility of buying at j with the utility of buying at some other firm k. The (expected) utility of buying at j is ɛ ij,t p oj,t + E δ h (ɛ ij,t+h p oj,t+h ). h=1 The (expected) utility of buying at k j is ɛ ik,t p nk,t s + E δ h (ɛ ik,t+h p ok,t+h ). h=1 Recall that it is assumed that the consumer expects the firms to be symmetric. Therefore, a consumer will decide to stay at firm j if only if for all k j ɛ ij,t p oj,t ɛ ik,t p nk,t s. Since ɛ i.,t is iid with cdf F, the probability that the consumer stays at firm j can be written as ( F (ɛ (p oj,t p nk,t s)) ) f(ɛ)dɛ. (4) k j Now consider the probability that a consumer switches to firm l. A consumer will switch to l if only if ɛ il,t p nl,t s ɛ ij,t p oj,t, for all k j, l, ɛ il,t p nl,t ɛ ik,t p nk,t. The probability that this occurs equals F (ɛ + p oj,t p nl,t s) ( k j,l F (ɛ p nl,t + p nk,t ) ) f(ɛ)dɛ. (5) As in the previous section, I will focus on a symmetric equilibrium where prices are constant over time. This implies that in equilibrium p o.,t = p o, p n.,t = p n P = p o 1 k j n 1 p n s = p o p n s. In such an equilibrium, the probability that a consumer stays at firm j can be written as F (ɛ P ) n 1 f(ɛ)dɛ 10

12 the probability that a consumer switches from firm j to firm l can be written as Note that by definition F (ɛ + P )F (ɛ) n 2 f(ɛ)dɛ. (n 1) F (ɛ + P )F (ɛ) n 2 f(ɛ)dɛ = 1 F (ɛ P ) n 1 f(ɛ)dɛ. (6) In this section, define G(P ) by G(P ) = (n 1) F (ɛ + P )F (ɛ)n 2 f(ɛ)dɛ. Note that G(P ) is a different expression than in the previous section, but the interpretation is the same. G(P ) denotes the probability that a consumer who considers switching indeed switches. Also define g(p ) = (n 1) f(ɛ + P )F (ɛ)n 2 f(ɛ)dɛ, the derivative of G(P ) w.r.t. P. Note that because of (6) g(p ) can also be written as g(p ) = (n 1) f(ɛ P )F (ɛ P ) n 2 f(ɛ)dɛ. 4.2 Pricing decision To derive the optimal pricing decision of firm j, consider π o π n. As before, π o gives the expected current discounted future profits from a single consumer who has bought from firm j in the previous period, π n gives the expected current discounted future profits from a single consumer who has not bought from firm j in the previous period. The expression for π o is a simple generalization of (1). ( π o = max 1 α(p j )+α(p j ) p oj,t ( α(p j ) 1 ( k j ( k j F (ɛ (p oj,t p nk,t s)) ) ) f(ɛ)dɛ (p oj,t +δπ o )+ F (ɛ (p oj,t p nk,t s)) ) ) f(ɛ)dɛ δπ n. The expression for π n is a bit more complicated, since the probability that a consumer who is currently in firm k considers switching, α(p k ) depends on k. Therefore, the expected profits that firm j makes on a consumer who is not currently in j depend on the firm where the consumer currently is. Define ρ k as the current market share of firm k define π nk by π nk = α(p k ) F (ɛ+p ok,t p nj,t s) ( F (ɛ+p nl,t p nj,t ) ) f(ɛ)dɛ(p nj,t +δπ o )+ l j,k 11

13 ( 1 α(p k ) F (ɛ + p ok,t p nj,t s) ( F (ɛ + p nl,t p nj,t ) ) ) f(ɛ)dɛ δπ n. l j,k Then ρ k π n = max π nk. p nj,t 1 ρ j k j Note that as before π o depends only on p oj,t p nk,t with k j. Also, π n depends only on p nj,t p ok,t p nk,t with k j. Therefore, when setting its old price, firm j therefore only has to consider its old consumers when setting its new price firm j only has to consider its new consumers. It is immediately clear that the optimal p oj,t does not depend on market shares. From the expression for π n it seems that the optimal p nj,t does depend on market shares, but in a symmetric equilibrium π nk = π nl for k l. Therefore in a symmetric equilibrium for π n. ρ k 1 ρ j will drop out of the expression The first-order conditions give the following equilibrium condition. Proposition 4.1 Let Q 1 (x) R 1 (x) be defined as Q 1 (x) = α(x)g(x) + α (x)g(x) R 1 (x) = 1 n 1 α (x)g(x)+α(x)g(x)+(n 1)α(x) In any symmetric equilibrium, P is implicitly defined by (n 2)F (ɛ+x)f (ɛ) n 3 f(ɛ) 2 dɛ. P 1 α(p )G(P ) Equilibrium prices are given by + α(p )G(P ) R 1 (P ) = s. (7) p o = δ α(p ) 2 G(P ) 2 (1 δ(1 α(p )G(P )))(1 α(p )G(P )) + n 1 R 1 (P ) p n = α(p )G(P ) (1 + δ (1 α(p )G(P ))2 α(p )G(P )) δ. R 1 (P ) n 1 Equilibrium profits are given by 12

14 π n = α(p ) 2 G(P ) 2 (1 δ)(n 1)R 1 (P ) π o = δα(p ) 2 G(P ) 2 (1 δ)(n 1)R 1 (P ) (1 α(p )G(P ))2 +. The proof of this proposition is in the appendix. Note that when R 1 (x) = Q 1 (x) (which holds for n = 2) the equilibrium in Proposition 4.1 is equal to the equilibrium in Proposition 3.1. In general, however, R 1 (x) Q 1 (x). To underst why this is the case, first consider the interpretation of Q 1 (x) R 1 (x). Q 1 (x) can be interpreted as the derivative w.r.t. p oj,t of the probability that a consumer who is currently at firm j stays at firm j. R 1 (x) can be interpreted as the derivative w.r.t. p nj,t of the probability that a consumer who is currently at firm k switches to firm j. These derivatives differ for two reasons. First, the effect of p oj,t on P j differs from the effect of p nj,t on P k : j p oj,t = 1 k p nj,t = 1 n 1. This explains the 1 n 1 in the first term of R 1 (x). Second, a consumer who decides to stay at firm j compares p oj,t p nk,t +s for all k j. For every one of those comparisons the effect of a change in p oj,t is the same. But a consumer who switches from k to j compares p ok,t with p nj,t + s compares p nj,t with p nl,t for all l j, k. The effect of a change in p nj,t on the first comparison is different from the effect on the second set of comparisons. This is reflected in R 1 (x) by the last term. The second-order conditions can be derived by twice differentiating the profit functions. For p o this gives the condition Q 2(x) Q 1 (x) > 2Q 1(x) 1 α(x)g(x), with Q 2(x) = α (x)g(x) + 2α (x)g(x) + α(x)g (x). For p n, deriving the second derivative gives a lengthy expression that cannot be expressed in terms of α(x), G(x) /or R 1 (x). For sake of brevity, this expression is left out of the paper, but there are combinations of α(x) F (x) for which both second order conditions hold, for which the lhs of (7) is increasing in P. 4.3 Equilibrium properties Assuming that the second-order conditions are satisfied, the equilibrium defined in Proposition 4.1 has some properties that are easy to derive. First note that Q 1 (x) > 0 R 1 (x) > 0 for any x. Also using that by definition 0 < α(x) < 1 0 < G(x) < 1 it is easy to see that p o > 0, π o > 0 π n > 0. Moreover, p n can be both positive negative. For example, suppose F (x) Exp(1), α(p ) = ep, δ = 1 n = 5. Then for s = 1 p 1+e P n is negative, but for s = 0, 5 p n is positive. The next proposition summarizes. Proposition 4.2 Assume the second-order conditions are satisfied. In equilibrium, the following properties hold 13

15 p o > 0 p n can be both positive negative π o > 0 π n > 0 In the duopoly case it was also derived that p o > p n, π o > π n α(p )G(P ) < 1 2. These properties do not necessarily generalize to the case n > 2. The problem here is that Q 1 (x) R 1 (x), assumptions on Q 1 (x) R 1 (x) are necessary to evaluate the prices profits. For example, it can be shown that when Q 1 (x) > R 1 (x) α(0) 1 2 all the properties mentioned above hold. The condition Q 1 (x) > R 1 (x) is however quite restrictive. Assuming that (n 1)Q 1 (x) > R 1 (x), which is a less restrictive condition, it can be shown that α(p )G(P ) < n 1 n. The restriction (n 1)Q 1(x) > R 1 (x) is however not sufficient to show that p o > p n or π o > π n. The fact that the equilibrium depends on both Q 1 (x) R 1 (x) also complicates the derivation of the effects of a change in s or δ. In the next section some results will be derived for one specific example. 5 Results for a specific example In this section it will be assumed that α(p ) follows a logistic distribution that F (x) follows an exponential distribution. That is, α(x) = Then it can be derived that ex+a 1 + e x+a F (x) = 1 e x. Also, G(x) = { 1 1 n e x + 1 n e x (1 e x ) n if x < n e x if x 0 { 1 g(x) = n e x 1 n e x (1 e x ) n (1 e x ) n 1 if x < 0 1 n e x if x 0 α (x) = e x+a (1 + e x+a = α(x)(1 α(x)), ) 2 Q 1 (x) = α(x)g(x) + α(x)(1 α(x))g(x) for n > 2 14

16 For n = 2 P is defined by R 1 (x) = 1 α(x)(1 α(x))g(x) + α(x)g(x). n 1 P P 1+eP +a 1+e P +a e2p +a e P +a e 2P +a ( ep +a ) 1+eP +a 1+e a e P +a e P +a e P +a ( ea ) = s if P < 0 = s if P 0 For n = 2 it can be checked that assumption A1 holds, therefore all the results in Section?? hold. For n > 2 P is defined by h(p ) = s with h(p ) defined by h(p ) = P 1 e P +a 1 (n 1)(1+e P +a )+1 (1 e P ) n 1 (1+e P +a ) 2 e P +a (1+ 1 n ea 1 n ea (1 e P ) n ) e P +a (1+e P +a )(1 e P ) n 1 if P < 0 P 1 e P +a 1 (n 1)(1+e P +a )+1 if P 0 Note that the denominator of the last term of h(p ) for P < 0 can be written as e P +a (1 + 1 n ea 1 n ea (1 e P ) n (1 + e P +a )(1 e P ) n 1 ) = e P +a (1 + e P +a f(p ) (1 e P ) n 1 ) > 0. For P 0 it is clear that h(p ) is increasing in P. For P < 0 it is shown in the appendix that when a ln 75 7 h(p ) is also increasing in P.1 Note that for P h(p ) that for P h(p ). Therefore, for a ln 75 7 there is a unique solution to h(p ) = s. The second order conditions can be checked, show that this solution indeed defines a maximum. In the duopoly case, Proposition 3.3 gives some properties of the equilibrium. Except for the last property, all properties also hold in the oligopoly case of the example in the current section. Proposition 5.1 Assume that α(x) = ex+a, F (x) = 1 e x a 1+e x+a. Then in equilibrium, the following properties hold ln 75 7 p o > p n p o > 0 1 a ln 75 is a sufficient condition for 7 h (P ) to be positive. It is likely that h(p ) is also increasing in P for a > ln 75 75, but this is hard to prove. a ln is not a very restrictive 7 7 assumption. For a = ln 75 consumers who expect no gain from switching (P = 0) will 7 consider switching with probability

17 p n can be both positive negative π o > π n > 0 The proof of this proposition is in the appendix. The interpretation of the results is the same as before. Most of the effects of a change in s that were specified in Proposition 3.4 also generalize to the oligopoly case of the example of the current section. Proposition 5.2 Assume that α(x) = ex+a, F (x) = 1 e x a 1+e x+a. Then in equilibrium, a change in s has the following effects ln < 0 2. α(p ) < 0, G(P ) < 0 3. po 4. pn < 0 5. πo 6. πn < 0 α(p )G(P ) < 0 can be both positive negative. can be both positive negative. The next table gives some simulation results for different value of n. TO BE CONTINUED 6 Conclusion TO BE WRITTEN References [1] Beggs, A. P. Klemperer, Multi-period competition with switching costs, Econometrica, [2] Cabral, L., Small switching costs lead to lower prices, Journal of Marketing Research, [3] Dube, J., G.J. Hitsch P.E. Rossi, Do switching costs make markets less competitive?, Journal of Marketing, [4] Farrell, J. P. Klemperer, Coordination lock-in: competition with swiching costs network effects, Hbook of Industrial Organization, North Holl Publishing,

18 α(p ) G(P ) α(p )G(P ) po pn πo πn

19 [5] Klemperer, P., The competitiveness of markets with switching costs, The RAND Journal of Economics, [6] Padilla, A.J., Mixed pricing in oligopoly with consumer switching costs, International Journal of Industrial Organization, [7] Wieringa, J.E. P.C. Verhoef, Understing customer switching behavior in a liberalizing service market: an exploratory study, Journal of Service Research,

20 A Proof of Proposition 3.1 First, it can be derived that α(p j ) = α(p j) j = α (P j ) p oj,t j p oj,t α(p k ) = α(p k) k = α (P k ). p nj,t k p nj,t This gives first order conditions (α (P j )G(P j )+α(p j )g(p j ))δπ n (α (P j )G(P j )+α(p j )g(p j ))(p oj,t +δπ o )+1 α(p j )G(P j ) = 0 (α (P k )G(P k )+α(p k )g(p k ))(p nj,t +δπ o )+α(p k )G(P k )+(α (P k )G(P k )+α(p k )g(p k ))δπ n = 0. In a symmetric equilibrium with p o.,t = p o, p n.,t = p n P = p o p n s this translates to (α (P )G(P )+α(p )g(p ))δπ n (α (P )G(P )+α(p )g(p ))(p o +δπ o )+1 α(p )G(P ) = 0 (α (P )G(P )+α(p )g(p ))(p n +δπ o )+α(p )G(P )+(α (P )G(P )+α(p )g(p ))δπ n = 0. Rewriting gives p o = δ(π n π o ) + 1 α(p )G(P ) p n = δ(π n π o ) + α(p )G(P ). Plugging this back into the profit functions gives π o = α(p )G(P )δπ n + (1 α(p )G(P )) ( δπ n + ) 1 α(p )G(P ) 19

21 This gives ( π n = α(p )G(P ) δπ n + ) α(p )G(P ) + (1 α(p )G(P ))δπ n. π n = α(p )2 G(P ) 2 (1 δ) π o = δα(p )2 G(P ) 2 + (1 δ)(1 α(p )G(P )) 2. (1 δ) Finally, plugging those expressions for π o π n back into the expressions for p o p n gives p o = 1 δ + (2δ 1)α(P )G(P ) p n = δ + (2δ + 1)α(P )G(P ). Note that all prices profits depend on P, which is defined as P = p o p n s. P is therefore implicitly defined by or P = 1 2α(P )G(P ) s P 1 2α(P )G(P ) = s B Proof of Proposition 3.2 To show that under A1 (3) has a unique solution, first note that any solution should have 1 2α(P )G(P ) > 0. Suppose to the contrary that 1 2α(P )G(P ) 0. Then it should be that G(P ) > 0.5, since g(x) is symmetric around 0, 2 P > 0. But if P > 0 1 2α(P )G(P ) 0, then the lhs of (3) is strictly positive, which can never hold. Therefore, any solution should have 1 2α(P )G(P ) > α(P )G(P ) Next, for 1 2α(P )G(P ) > 0 under A1, α(p )g(p )+α (P )G(P ) is decreasing in P. To show this, it is sufficient to show that the derivative of is negative: 1 2α(P )G(P ) α(p )g(p )+α (P )G(P ) 2 Recall the G(x) is the CDF of ε ij,t ε ik,t, with g(x) the corresponding CDF. Since ε ij,t ε ik,t are iid, g(x) should be symmetric around 0. 20

22 2 2 (1 2α(P )G(P ))Q 2 (P ) < 0 Note that the denominator of the derivative is left out, since the denominator is strictly positive. Since 1 2α(P )G(P ) > 0, one can use A1 to show that the derivative is indeed negative. This implies that for 1 2α(P )G(P ) > 0 the lhs of (3) is strictly increasing in P. Note that 1 2α(P )G(P ) is decreasing in P approaches 1 when p. Therefore, there does exist a region where 1 2α(P )G(P ) > 0, this region has the form (, P ), with P > 0 with P = if lim P 1 2α(P ) > 0. In this region the lhs of (3) is strictly increasing in P. For P = s, 1 2α(P )G(P ) > 0, the lhs of (3) is strictly below s. For P = P, the lhs of (3) is strictly positive. This implies that there is a unique solution to (3), this unique solution has s < P < P. To show that the unique solution to (3) defines a maximum, consider the second derivatives δπ n [ α (P )G(P ) + 2α (P )g(p ) + α(p )g (P ) ] + (p o + δπ o ) [ α (P )G(P ) 2α (P )g(p ) α(p )g (P ) ] 2α(P )g(p ) 2α (P )G(P ) (p n + δπ o ) [ α (P )G(P ) + 2α (P )g(p ) + α(p )g (P ) ] + δπ n [ α (P )G(P ) 2α (P )g(p ) α(p )g (P ) ] 2α(P )g(p ) 2α (P )G(P ) To ensure that the first order conditions maximize profits, both second derivatives should be negative at the value of P defined by (3). Using the values of p o, p n, π o π n defined by Proposition 3.1 the second derivatives can be written as 1 α(p )G(P ) Q 2 (P ) 2 (8) α(p )G(P ) Q 2 (P ) 2. (9) Under assumption A1 both second derivatives are negative. 21

23 C Proof of Proposition 3.4 First note that p o, p n, π o π n only depend on s via P. Therefore, it is sufficient to know the derivatives of prices profits w.r.t. P to know the derivative of P w.r.t. s. 1. Using (3), it can be derived that 2Q 1(P ) 2 (1 2α(P )G(P ))Q 2 (P ) 2 with Q 2 (P ) as defined in A1. Rewriting gives = 1 or (3 + (1 2α(P )G(P ))Q 2(P ) 2 ) = 1 = (1 2α(P )G(P ))Q 2 (P ). Using A1, using that 1 2α(P )G(P ) > 0 gives 1 α(p )G(P ) 1 < 1 + α(p )G(P ) < Since = po pn 1, 1 < 2. Since α(p ) G(P ) 0, α(p ) G(P ) 0 = α(p ) G(P ) > 0, < α(p )G(P ) 2 α(p )G(P ) < 0. < 0 implies 0 < po pn < 1. > 0 = G(P ) < 0 it is immediately clear that α(p )G(P ) = α(p ) G(P ) + α(p ) G(P ) < 0. < 0. Also, since α(p ) > 0 3. First note that the derivative of (1 δ)(1 α(p )G(P )) + δα(p )G(P ) w.r.t. P is (2δ 1). This gives p o = (2δ 1)Q 1(P ) 2 ((1 δ)(1 α(p )G(P )) + δα(p )G(P ))Q 2 (P ) 2. Rewriting gives p o = 2δ 1 ((1 δ)(1 α(p )G(P )) + δα(p )G(P ))Q 2(P ) 2. Since po = po po po < 0, the sign of is -1 times the sign of. Now suppose that f(x) Exp(1), α(p ) = ep, δ = 0.95 s = 2. Then 1+e P it can be shown that po < 0. But when G(x) Exp(1), α(p ) = ep, 1+e P δ = 0.95 s = 1, po po > 0. This shows that can be both positive 22

24 negative. 4. First note that the derivative of δ + (2δ + 1)α(P )G(P ) w.r.t. P is (2δ + 1). This gives or p n = (2δ + 1)Q 1(P ) 2 ( δ + (2δ + 1)α(P )G(P ))Q 2 (P ) 2 p n = 2δ + 1 ( δ + (2δ + 1)α(P )G(P ))Q 2(P ) 2. Now suppose δ + (2δ + 1)α(P )G(P ) > 0. Using A1 gives 2δ(1 α(p )G(P )) α(p )G(P ) α(p )G(P ) < p n 1 + α(p )G(P ) + 2δα(P )G(P ) <. 1 α(p )G(P ) When δ 1 2, 2δ(1 α(p )G(P )) α(p )G(P ) 1 2α(P )G(P ) > 0, so p n pn > 0 < 0. Now suppose δ + (2δ + 1)α(P )G(P ) < 0. Again using A1 gives 1 + α(p )G(P ) + 2δα(P )G(P ) 1 α(p )G(P ) < p n 2δ(1 α(p )G(P )) α(p )G(P ) <. α(p )G(P ) Since 1+α(P )G(P )+2δα(P )G(P ) 1 α(p )G(P ) is clearly positive, pn pn > 0 < First note that the derivative of δα(p ) 2 G(P ) 2 + (1 δ)(1 α(p )G(P )) 2 w.r.t. P equals 2δα(P )G(P ) 2(1 δ)(1 α(p )G(P )), or 2(δ + α(p )G(P ) 1). This gives π o = 1 ( 2(δ + α(p )G(P ) 1) 1 δ (δα(p ) 2 G(P ) 2 + (1 δ)(1 α(p )G(P )) 2 )Q 2 (P ) 2 When f(x) Exp(1), α(p ) = ep, s = 1 δ = 0.95 πo 1+e P ). > 0, since πo < 0, < 0. But when f(x) Exp(1), α(p ) = ep, s = 2, 5 1+e P πo πo < 0, since < 0, > 0. This shows that can be δ = 0.95 πo both positive negative. 6. First note that the derivative of α(p ) 2 G(P ) 2 w.r.t. P equals 2α(P )G(P ). This gives 23

25 or Using A1 gives Since π n = 1 2α(P )G(P ) 2 α(p ) 2 G(P ) 2 Q 2 (P ) 1 δ 2 π n = 1 ( 2α(P )G(P ) α(p )2 G(P ) 2 ) Q 2 (P ) 1 δ 2. 0 < π n < 1 2α(P )G(P ) 1 δ 1 α(p )G(P ). < 0 this gives πn < 0. D Proof of Proposition 7 First, it can be derived that α(p j ) = α(p j) j = α (P j ) p oj,t j p oj,t α(p k ) = α(p k) k = 1 p nj,t k p nj,t n 1 α (P k ). This gives first order conditions ( α (P j ) + α ( (P j ) F (ɛ (p oj,t p nk,t s)) ) f(ɛ)dɛ α(p j ) α(p j ) k j f(ɛ (p oj,t p nl,t s)) ( l j k j,l F (ɛ (p oj,t p nk,t s)) ) ) f(ɛ)dɛ (p oj,t +δπ o )+ ( 1 α(p j ) + α(p j ) F (ɛ (p oj,t p nk,t s)) ) f(ɛ)dɛ+ k j ( α (P j )(1 ( F (ɛ (p oj,t p nk,t s)))f(ɛ)dɛ)+ k j f(ɛ (p oj,t p nl,t s)) ( l j k j,l F (ɛ (p oj,t p nk,t s)) ) ) f(ɛ)dɛ δπ n = 0 24

26 k j ρ k 1 ρ j (( 1 n 1 α (P k ) F (ɛ+p ok,t p nj,t s) ( F (ɛ+p nl,t p nj,t ) ) f(ɛ)dɛ+ l j,k α(p k ) f(ɛ + p ok,t p nj,t s) ( F (ɛ + p nl,t p nj,t ) ) f(ɛ)dɛ+ l j,k α(p k ) F (ɛ+p ok,t p nj,t s) h j,k f(ɛ+p nh,t p nj,t ) ( l j,k,h F (ɛ+p nl,t p nj,t ) ) ) f(ɛ)dɛ (p nj,t +δπ o )+ α(p k ) F (ɛ + p ok,t p nj,t s) ( F (ɛ + p nl,t p nj,t ) ) f(ɛ)dɛ+ l j,k ( 1 n 1 α (P k ) F (ɛ + p ok,t p nj,t s) ( F (ɛ + p nl,t p nj,t ) ) f(ɛ)dɛ l j,k α(p k ) f(ɛ + p ok,t p nj,t s) ( F (ɛ + p nl,t p nj,t ) ) f(ɛ)dɛ l j,k α(p k ) F (ɛ+p ok,t p nj,t s) h j,k f(ɛ+p nh,t p nj,t ) ( l j,k,h In a symmetric equilibrium with p o.,t = p o, p n.,t = p n P = p o p n s this translates to ( α (P )+α (P ) F (ɛ P ) n 1 f(ɛ)dɛ α(p ) 1 α(p ) + α(p ) F (ɛ P ) n 1 f(ɛ)dɛ+ ( α (P )(1 F (ɛ P ) n 1 f(ɛ)dɛ)+α(p ) F (ɛ+p nl,t p nj,t ) ) ) f(ɛ)dɛ )δπ n = 0. ) (n 1)f(ɛ P )F (ɛ P ) n 2 f(ɛ)dɛ (p o +δπ o )+ ) (n 1)f(ɛ P )F (ɛ P ) n 2 f(ɛ)dɛ δπ n = 0 25

27 ( 1 n 1 α (P ) F (ɛ+p )F (ɛ) n 2 f(ɛ)dɛ+α(p ) f(ɛ+p )F (ɛ) n 2 f(ɛ)dɛ+ ) α(p ) F (ɛ+p )(n 2)F (ɛ) n 3 f(ɛ) 2 dɛ (p n +δπ o )+α(p ) F (ɛ+p )F (ɛ) n 2 f(ɛ)dɛ+ ( 1 n 1 α (P ) F (ɛ+p )F (ɛ) n 2 f(ɛ)dɛ α(p ) f(ɛ+p )F (ɛ) n 2 f(ɛ)dɛ ) α(p ) F (ɛ + P )(n 2)f(ɛ)F (ɛ) n 3 f(ɛ)dɛ δπ n = 0. Using the definition of G(x), this can be written as ( α (P )G(P ) α(p )g(p ))(p o +δπ o )+1 α(p )G(P )+(α (P )G(P )+α(p )g(p ))δπ n = 0 ( (p n + δπ o ) 1 1 n 1 α (P ) n 1 G(P ) α(p ) 1 g(p )+ n 1 ) α(p ) F (ɛ+p )(n 2)F (ɛ) n 3 f(ɛ) 2 dɛ +α(p ) 1 n 1 ( 1 G(P )+ 1 ) α(p ) g(p ) + α(p ) F (ɛ + P )(n 2)f(ɛ)F (ɛ) n 3 f(ɛ)dɛ δπ n = 0. n 1 Isolating p o p n gives 1 n 1 α (P ) G(P )+ n 1 p o = δ(π n π o ) + 1 α(p )G(P ) p n = δ(π n π o ) + α(p )G(P ). R 1 (P ) Plugging this back into the profit functions gives π o = α(p )G(P )δπ n + (1 α(p )G(P )) 26 ( δπ n + ) 1 α(p )G(P )

28 1 π n = α(p )G(P ) n 1 This gives ( δπ n + ) α(p )G(P ) R 1 (P ) 1 + (1 α(p )G(P ) n 1 )δπ n. π n = α(p ) 2 G(P ) 2 (1 δ)(n 1)R 1 (P ) π o = δα(p ) 2 G(P ) 2 (1 δ)(n 1)R 1 (P ) (1 α(p )G(P ))2 +. Finally, plugging those expressions for π o π n back into the expressions for p o p n gives p o = δα(p )2 G(P ) 2 (1 α(p )G(P ))(1 δ(1 α(p )G(P ))) + (n 1)R 1 (P ) p n = α(p )G(P ) (1 + R 1 (P ) δα(p )G(P ) ) n 1 δ(1 α(p )G(P ))2. All prices profits depend on P, which is defined as P = p o p n s. P is therefore implicitly defined by or P = 1 α(p )G(P ) α(p )G(P ) R 1 (P ) s P 1 α(p )G(P ) + α(p )G(P ) R 1 (P ) = s. E Proof that the example has a unique solution Derivative wrt P P 1 α(p )G(P ) Q(P ) + n 1 n α 1+1+ (1 α(p )G(P ))(2α(P )(1 α(p ))g(p ) + α(p )g(p ) + α(p )(1 α(p ))(1 2α(P ))G(P )) (α(p )g(p ) + α(p )(1 α(p ))G(P )) 2 + (n P < 0 27

29 2+ (1 α(p )G(P ))(2(1 α(p ))g(p ) g(p ) + (n 1)(1 ep ) n 2 e P + (1 3α(P ) + 2α(P ) 2 )G(P )) α(p )(g(p ) + (1 α(p ))G(P )) 2 + Define X = 1 α(p )+α(p )g(p )+α(p )(1 e P ) n 1 = g +(1 α(p ))G(P )+ (1 e P ) n 1 2α(X (1 e P ) n 1 ) 2 +X((X (1 e P ) n 1 )(1 2α(P ))+(n 1)(1 e P ) n 2 e P X) (X (1 e P ) n 1 )(X 2α(P )(1 e P ) n 1 ) + (n 1)(1 e P ) n 2 e P X 2 X (1 e P ) n 1 = (1 α(p ))(1 (1 e P ) n 1 ) + α(p )g(p ) > 0 X 2α(P )(1 e P ) n 1 = 1 α(p ) + α(p )g(p ) α(p )(1 e P ) n 1 1 2e P e a (1 e P ) n n ea 1 n ea (1 e p ) n First deceasing then increasing in P. Minimum at e P = 1 minimum is n ea (1 2( 2n 2 2n 1 )n 1 ) 2n 1. Value at For n = 3 1 2( 2n 2 2n 1 )n 1 = n ea (1 2( 2n 2 2n 1 )n 1 ) > 0 for e a < n 2 2n 1 )n 1 is decreasing in n. For n > 3 but 1 2( 2n 2 2n 1 )n 1 < n ea (1 2( 2n 2 2n 1 )n 1 ) is increasing in n so > 0. When 1 2( 2n 2 2n 1 )n 1 > n ea (1 2( 2n 2 2n 1 )n 1 ) > 0 by definition. So minimum is always positive. F Proof of Proposition When P = s, h(p ) < s. Since h(p ) is increasing in P this implies that P should be larger than s, or p o p n > It is immediately clear from the expression for p o in Proposition 4.1 that p o > Suppose δ = 1, a = 0 n = 5. Then for s = 1 p n < 0 for s = 0.5 p n > 0. 28

30 4. It is immediately clear from the expressions for π o π n in Proposition 4.1 that π o > 0 π n > 0. Also, it has already been shown that π o π n > 0 for n = 2. Therefore, in this proof it will be assumed that n 3. α(p )G(P ) Using that R 1 (P ) = n 1 n α(p ), note that π o π n > 0 if only if α(p )G(P ) + (1 α(p )G(P )) 2 (n α(p )) > 0. This can be rewritten as α(p ) 2 G(P )g(p ) α(p ) 2 G(P ) 2 +n(1 α(p )G(P )) 2 α(p )+2α(P ) 2 G(P ) > 0. When P < 0, g(p ) = 1 G(P ) (1 e P ) n 1. Then π o π n > 0 if only if α(p ) 2 G(P )(1 e P ) n 1 + n(1 α(p )G(P )) 2 α(p )(1 α(p )G(P )) > 0. It is clear that for P < 0 α(p ) 2 G(P )(1 e P ) n 1 > 0. It will be shown now that n nα(p )G(P ) α(p ) > 0, which implies that n(1 α(p )G(P )) 2 α(p )(1 α(p )G(P )) > 0. Plugging α(p ) = ep +a 1+e P +a G(P ) = 1 1 n e P + 1 n e P (1 e P ) n in n nα(p )G(P ) α(p ) multiplying by 1 + e P +a leaves to show that n + e a e a (1 e p ) n P +a > 0. This expression is increasing in n, so n 3 gives n+e a e a (1 e p ) n P +a 3+2e P +a 3e P e P +a +e P +a e 2P. This expression in turn is increasing in a, so 3 + 2e P +a 3e P e P +a + e P +a e 2P lim a 3 + 2e P +a 3e P e P +a +e P +a e 2P = 3. Therefore n+e a e a (1 e p ) n P +a 3 > 0. Now assume P 0 Then g(p ) = 1 G(P ) π o π n > 0 if only if n(1 α(p )G(P )) 2 α(p )(1 α(p )G(P )) > 0, or, equivalently, n ep +a nα(p )G(P )) α(p ) > 0. Plugging in α(p ) = G(P ) = e P +a n e P, multiplying by 1 + e P +a leaves to show that n + e a e P +a > 0. Suppose that this is not true, that is, suppose that e P +a n+e a. Then the definition of P gives P 1 e P +a 1 (n 1)(1 + e P +a ) + 1 ln(n+ea ) a 1 n + e a 1 (n 1)(n e a ) + 1. The rhs of this expression is increasing in n, so because n 3 ln(n+e a ) a 1 n + e a 1 (n 1)(n e a ) + 1 ln(3+ea ) a e a e a. The rhs of this expression is decreasing in a, so since a ln

31 ln(3 + e a ) a e a 1 96 ln 9 + 2ea 7 ln > 0. Therefore, if e P +a n + e a P 1 e P +a 1 (n 1)(1 + e P +a ) + 1 > 0 which can never hold for s 0. So, in equilibrium e P +a < n + e a, or π o π n > 0. G Proof of Proposition 5.2 For n = 2 all statements have been proved in Proposition 3.4, except for < 0. To prove this, it has to be shown that p n p n = 2δ+1 ( δ + (2δ + 1)α(P )G(P ))(2(1 α(p ))g(p ) + g (P ) + (1 α(p ))(1 2α(P ))G(P )) > g(p ) + (1 α(p ))G(P ) When P < 0 g (P ) = g(p ) = G(P ), so it it left to show that (2δ + 1)(2 α) ( δ + (2δ + 1)α(P )g(p ))(4 5α(P ) + 2α(P ) 2 ) > 0. The derivative of this expression wrt δ is 8 8α(P )g(p ) 7α(P ) + 10α(P ) 2 g(p ) + 2α(P ) 2 4α(P ) 3 g(p ). This expression in turn is decreasing in g(p ), since P < 0 gives G(P ) < 1 2 the derivative wrt δ is larger than 8 11α(P ) + 7α(P )2 2α(P ) 3 > 0. Since the derivative wrt δ is positive, (2δ+1)(2 α) ( δ+(2δ+1)α(p )g(p ))(4 5α(P )+2α(P ) 2 ) > 2 α α(p )g(p )(4 5α(P )+2α(P ) 2 ). The rhs of this expression is again decreasing in g(p ), so (2δ+1)(2 α) ( δ+(2δ+1)α(p )g(p ))(4 5α(P )+2α(P ) 2 ) > 2 α α(p )(2 5 2 α(p )+α(p )2 ) > 0. When P 0 g (P ) = g(p ) G(P ) = 1 g(p ), so it it left to show that 2δ (1 2α(P ))(δ 2δα α + (2δ + 1)α(P )g(p )) > 0. The derivative of this expression wrt δ equals 30

32 3 4α(P ) + 2α(P )g(p ) + 4α(P ) 2 4α(P ) 2 g(p ). Since P 0 g(p ) 1 2, for any possible value of 0 < g(p ) 1 2 the derivative wrt δ is positive. This gives 2δ+1+(1 2α(P ))(δ 2δα α+(2δ+1)α(p )g(p )) > 1 (1 2α(P ))(α α(p )g(p )) > 0. in the remainder of this proof consider the case n The definition of P gives h (P ) = 1. Since h (P ) > 0, < Using that α(p ) < 0 that > 0, G(P ) > 0 is immediately clear. 3. First note that α(p )G(P ) > 0 this so p n = n 1 α(p )G(P ) (1 α(p )G(P ))2 + δ δ n α(p ) n α(p ) p n = (n 1)α(P )(1 α(p )) (n α(p )) 2 +δ Q 1(P ) n α(p ) +δ α(p )2 G(P )(1 α(p )) (n α(p )) 2 +2δ(1 α(p )G(P ))+δ(1 α(p )G where Q 1 (P ) is the derivative of Q 1(P ) wrt P. When P < 0, g(p ) = 1 G(P ) (1 e P ) n 1 g (P ) = 1 + G(P ) + (1 e P ) n 1 + (n 1)(1 e P ) n 2 e P. This gives = α(p )(1 α(p )G(P ) (1 e P ) n ) Q 1 (P ) = α(p )(1 2α(P ))(1 α(p )G(P ) (1 ep ) n 1 ) + α(p )(n 1)(1 e P ) n 2 e P. This in turn gives p n (n 1)α(P )(1 α(p )) = (n α(p )) 2 +δ α(p )(1 α(p )G(P ) (1 ep ) n ) +δ α(p )2 G(P )(1 α(p )) n α(p ) (n α(p )) 2 +2δ(1 It is clear that the first three terms are positive. The last two terms together give δ(1 α(p )G(P )) 2α(P )(1 α(p )G(P ) (1 ep ) n ) 2 + (1 α(p )G(P ))((1 2α(P ))(1 α(p )G(P ) α(p )(1 α(p )G(P ) (1 e P ) n I will show that these two terms together also are positive, so consequently pn > 0. The sign of the last two terms is determined by 2α(P )(1 α(p )G(P ) (1 e P ) n ) 2 +(1 α(p )G(P ))(1 2α(P ))(1 α(p )G(P ) (1 e P ) n 1 )+(1 α(p )G 31

33 Rewriting gives that it has to be shown that (1 α(p )G(P ) (1 e P ) n 1 )α(p )(1 G(P ) (1 e P ) n 1 )+(1 α(p )G(P ) (1 e P ) n 1 )(1 α(p ) α(p ) Since 1 G(P ) (1 e P ) n 1 = g(p ) > 0 consequently 1 α(p )G(P ) (1 e P ) n 1 > 0, the first term of this expression is positive. Suppose 1 α(p ) α(p )(1 e P ) n 1 0, then the second term is positive, the third term is positive the total expression is positive. QED When 1 α(p ) α(p )(1 e P ) n 1 < 0 the second third term together are larger than (1 α(p )G(P ))(1 α(p ) α(p )(1 e P ) n 1 + (n 1)(1 e P ) n 2 e P ). Plugging in α(p ) = ep +a 1+e P +a gives 1 α(p ) α(p )(1 e P ) n 1 +(n 1)(1 e P ) n 2 e P = e P +a (1 ep +a (1 e P ) n 1 +(1+e P +a )(n 1)(1 e When e P > 1 n, 1 ep +a (1 e P ) n 1 + (1 + e P +a )(n 1)(1 e P ) n 2 e P is increasing in a, so 1 e P +a (1 e P ) n 1 + (1 + e P +a )(n 1)(1 e P ) n 2 e P > 1 + (n 1)(1 e P ) n 2 e P > 0. QED When e P < 1 n, 1 ep +a (1 e P ) n 1 + (1 + e P +a )(n 1)(1 e P ) n 2 e P is decreasing in a, so 1 e P +a (1 e P ) n 1 + (1 + e P +a )(n 1)(1 e P ) n 2 e P > 1 e P 75 7 (1 ep ) n 1 + (1 + e P 75 7 )(n 1)(1 ep ) n 2 e P When P 0, g(p ) = 1 G(P ), g (P ) = g(p ), = α(p )(1 α(p )G(P )) Q 1 (P ) = α(p )(1 2α(P ))(1 α(p )G(P )). Plugging this rewriting gives in pn p n (n 1)α(P )(1 α(p )) α(p )(1 α(p )G(P )) = (n α(p )) 2 +δ +δ α(p )2 G(P )(1 α(p )) n α(p ) (n α(p )) 2 +2δ(1 α(p )G(P ))+ The first three terms of this expression are positive. The remaining two expressions can be written as δ(1 α(p )G(P ))frac1α(p ) > 0 so pn > 0 pn < < α(p ) < 0, G(P ) < 0 p o α(p )G(P ) < 0 can be both positive negative. 32