On two-part tariff competition in a homogeneous product duopoly

International Journal of Industrial Organization 41 (2015) 30–41

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International Journal of Industrial Organization journal homepage: www.elsevier.com/locate/ijio

On two-part tariff competition in a homogeneous product duopoly☆ Krina Griva a, Nikolaos Vettas b,c,⁎ a b c

Department of Economics, University of Ioannina, University Campus, Ioannina 451-10, Greece Department of Economics, Athens University of Economics and Business, Patision 76, Athens 104-34, Greece CEPR, UK

a r t i c l e

i n f o

Article history: Received 31 May 2012 Received in revised form 9 April 2015 Accepted 12 May 2015 Available online 19 May 2015 JEL classification: L13 D43

a b s t r a c t We explore aspects of two-part tariff competition between duopolists providing a homogeneous service when consumers differ with respect to their usage levels. Competition in only one price component (the fee or the rate) may allow both firms to enjoy positive profits if the other price component has been set at levels different enough between firms. Fixing one price component alters the nature of competition, indirectly introducing an element of product differentiation. Endogenous market segmentation emerges, with the heavier users choosing the lower rate firm and the lighter users choosing the lower fee firm. When no price component can be negative, competition becomes softer, profits tend to be higher but there is also a disadvantage for the firm that starts with a higher fee than that of its rival. © 2015 Elsevier B.V. All rights reserved.

Keywords: Two-part tariffs Non-linear pricing Market segmentation

1. Introduction A central feature of many markets is that consumers differ significantly with respect to their usage levels for a good or a service. The set of relevant examples is wide. When studying credit cards, telephone services, car rentals, club memberships, equipment leasing, amusement parks, tv subscriptions and in many other cases, some of the customers are heavier users while others are lighter users. Pricing in such cases is both of theoretical and of practical interest when oligopolists can price nonlinearly. In this paper we set up a simple homogeneous product duopoly model that allows us to explore some aspects of the nature of competition when pricing takes the form of a two-part tariff. Indeed, in many markets, pricing involves the use of a fee and of a per unit rate. Further, in such markets we often observe segmentation, with some firms attracting the heavier users (by charging a low per unit rate) and other firms attracting the lighter users (by charging a

☆ We are grateful to the Editor and the two anonymous referees for their extremely helpful and insightful comments, as well as to the participants at the EARIE and ASSET conferences and seminars at various universities. ⁎ Corresponding author at: Department of Economics, Athens University of Economics and Business, Patision 76, Athens 104-34, Greece. E-mail addresses: [email protected] (K. Griva), [email protected] (N. Vettas).

http://dx.doi.org/10.1016/j.ijindorg.2015.05.002 0167-7187/© 2015 Elsevier B.V. All rights reserved.

low fee). 1 While the relevant literature is growing, important aspects of the problem are not fully understood. Here we explore the nature of two-part tariff competition under alternative assumptions about which price components can be chosen by the firms and we show how these may affect the firms' profits. We study a simple duopoly model where all consumers view the products sold by the two firms as perfectly homogeneous. Importantly, consumers differ with respect to their usage levels. Naturally, if there are no restrictions on how pricing takes place, equilibrium can only be at zero profit.2 We explore when, how and why certain restrictions may lead to positive equilibrium profits. The first restriction is when one of 1 When the British household retail electricity market was liberalized in 1998, each supplier offered a single menu of two-part tariffs. These menus were different across firms, with entrants having a higher fee and a lower rate compared to the incumbent. Without cost asymmetries or variation in brand royalty, the firms segmented the market by targeting consumers of different usage rates (see Davies et al., 2014). There are many other related examples. Some restaurants offer an “all you can eat buffet”, while other price à la cart. Some hotels operate as “all inclusive” and other charge extra for every item of service required. Traditional airlines tend to include everything in the ticket price and others, positioning themselves as “low cost”, charge extra for each additional service item requested. Some insurance programs have a low premium but charge extra when some hospital service is used, while others allow for essentially unlimited use. 2 We abstract away from product differentiation, uncertainty, repeated interactions, capacity constraints, dynamic competition with switching costs and other considerations that may also imply the presence of some profits even when the products are homogeneous.

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the components of a two-part tariff can be taken as fixed and competition only takes place via the other component. This may lead to positive equilibrium profit, but only if there is enough differentiation between firms with respect to the component that is taken as given. Moreover, the nature of competition differs significantly when it is in fees or in rates. A second restriction that we study is how the problem is shaped when no price component can be set below zero. We find that, in this case, equilibrium behavior is significantly modified, implying a different level of industry profit and allocation between firms. Restricting attention to non-negative prices also allows us to endogenize the sequential choice of price components, with either the fees or the rates chosen first. In addition, this analysis allows us to directly compare our results to models of vertical product differentiation and to highlight not only similarities but also important differences. We should note upfront that much of our analysis examines the consequences of restrictions on setting some price component. There may be a variety of reasons that make the restrictions that we study relevant. In particular, studying a problem where one of the components of a twopart tariff has to be viewed as given makes sense in different settings. First, it may be that in certain markets the rates (or the fees) represent longer term choices which are, therefore, made at different levels of a business hierarchy, while competition in the other component takes place as a different and essentially independent element of a firm's strategy, or such choices are sometimes discussed centrally in business associations or are even suggested or imposed by some large customers. Second, a price component may be considered fixed if it is part of an existing contract and cannot be easily changed even though the contract allows some flexibility regarding other parts of pricing.3 Third, there may be regulatory restrictions that, for political or other noneconomic reasons, limit the choices with respect to one price component. In any such setting, it becomes important to understand the nature of competition in one of the components of a two-part tariff. Likewise, non-negativity restrictions on price components may also emerge for a variety of strategy or institutional reasons like the ones just mentioned and also when arbitrage possibilities emerge. More specifically, in our analysis we characterize how, when the perunit rates for the firms are set at levels that are not too close to each other, competition via fees leads to an equilibrium where both firms make positive profits. Also, within this range, both firms' profits increase as the difference between the rates increases. Essentially, even though the products are homogeneous, fixing one of the price components, along with differentiation in the usage levels across the consumers, implicitly introduces some differentiation in the market since consumers are no longer indifferent between the two products. This effect, in turn, may allow firms to segment the market and enjoy positive profits. We emphasize that for this result it is required not only that the rates are different, but also that their difference exceeds a certain threshold. When the rates are different enough, the high rate firm would have to lower its fee very significantly to capture the entire market. Instead (and as long as the rival fee is not too high), this firm finds it more profitable to lower its own fee by relatively less and to sell only to the low usage consumers, since this is the part of the demand that cares relatively less about the rates. A similar result is obtained when the fees are taken as given and firms compete via per-unit rates. Still, there are also significant differences between the two cases. These are due to the fact that not all consumers care equally about a difference in rates; in fact consumers with low usage levels make their product selection essentially only on the basis of 3 For example, there is often a fee between gas stations and their suppliers, which is set over a relatively long period (of one year or even for a longer period, possibly signifying an exclusive contract). Then a rate per unit of product ordered is also charged and may vary daily. Similar relations exist in a number of franchise settings. In credit card markets there may be restrictions (designed to protect the users who already carry some debt) on how fast interest rates can be increased. Therefore, these may be considered to some extent fixed over a certain time period, or at least not fully flexible. Alternatively, some card issuers make to their clients annual fee offers with a guarantee that the fee will never be increased. Thus, in subsequent periods, such banks may be viewed as competing only in interest rates.

31

the fee. This means that, when competing in rates, in equilibrium, the firm with the low fee can always guarantee for itself at least part of the demand. In general, when firms compete in rates competition is more intense than when firms compete in fees. We also examine the case with only non-negative prices. When it is not possible for firms to set either price component below zero, competition becomes softer in general, but again, the result is different depending on how competition takes place. When the rates are given, the firm with the low rate captures a larger market share and makes a higher profit than that of its rival. When the fees are given, the firm with the low fee captures the entire market and makes a positive profit. When the rates are chosen first, there is no equilibrium in pure strategies (because one firm prefers maximum differentiation but the other minimum), while when the fees are chosen first we have an equilibrium at zero profit (since both firms have a strong undercutting incentive). In our analysis, when one of the price components can be considered fixed, an element of vertical differentiation is indirectly introduced in a market where, otherwise, the consumers view the products as homogeneous. A comparison to models where the products differ with respect to their qualities is therefore in order. In particular, we show that there are important similarities, but also key differences, between our model and the classic analysis of Shaked and Sutton (1982), where vertical product differentiation enters directly. Like in their work, in our analysis when firms consider their rates (or fees) as given at different levels, all consumers would view one of the products as more attractive, but the intensity of this preference will be different (since the usage levels are different). Also, under certain conditions, we show that both firms' profits increase when the difference in rates increases, as would also be the case in Shaked and Sutton (1982) where profits increase as the quality difference increases. However, heterogeneity enters the two problems differently. A difference in qualities only affects the profit functions in Shaked and Sutton (1982) indirectly, through the demand and the consumers' choices. In our model a difference in rates also affects the profit functions directly, since the firms receive the relevant payment. As a result, in contrast to Shaked and Sutton (1982), we find that we do not have a pure strategy equilibrium unless the difference in rates is large enough. The two models become directly comparable when we restrict prices to be non-negative and we endogenize all choices. We then do not find a maximum differentiation result, contrary to what Shaked and Sutton (1982) find in qualities, again highlighting the differences between the models. An implication of our analysis is that, in a setting where one of the two pricing components can be fixed and firms subsequently compete via the other component, they can obtain positive profits. Price competition in only one dimension may not be enough to guarantee the perfectly competitive (Bertrand) outcome. As mentioned above, this case may be relevant when one of the price components has been set for the competing firms by a regulator, by some industry committee or association, or via some other institutional procedure. Setting fees or rates at different levels across firms serves to indirectly introduce heterogeneity to an otherwise homogeneous product market. Regulators should therefore be aware that fixing one price component, or allowing that it is fixed, may in turn allow firms to make positive profits even though they compete in another component and the products are not otherwise differentiated. Likewise, a competition authority should not view competition (no matter how intense) in one price component as sufficient, if the firms have been able to coordinate and set another price component at some level for each firm. There is growing literature on two-part tariffs following the classic work of Oi (1971) (see Armstrong, 2006; Stole, 2007; Vettas, 2011 for reviews and references). Much of this literature has focused on the monopoly case. 4 Despite its importance, the study of oligopoly 4 In addition, two-part tariffs can be used for commitment reasons in vertical relations, see e.g., Rey and Tirole (1986) for a classic analysis. Saggi and Vettas (2002) study how the use of two-part tariffs is related with the number of downstream firms.

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competition with two-part tariffs has received relatively less attention, especially when compared to the volume of work on oligopoly with linear pricing. Important papers on aspects of nonlinear competition in oligopolies include Spulber (1981), Oren et al. (1983), Hayes (1987), Holmes (1989), Mandy (1992), Stole (1995), Corts (1998), Harrison and Kline (2001), Rochet and Stole (2002) and more recently, Armstrong and Vickers (2010) where consumers differ in their preferences for a supplier and also in the quantity they want to purchase, Calzolari and Denicolò (2011) on quantity discounts and Reisinger (2014) on two-part tariff competition between two-sided platforms.5 Other work has examined the relation between horizontal product differentiation and two-part tariffs. In Armstrong and Vickers (2001) when competitive price discrimination is studied in a Hotelling context, two-part tariffs emerge in equilibrium with marginal cost pricing. Extending the analysis to more general horizontal preferences Yin (2004) finds that equilibrium prices equal marginal cost only if the marginal consumer's demand equals the average demand — two-part tariffs imply lower prices but higher profits than uniform pricing. Hoernig and Valletti (2007) model Hotelling competition when consumers could buy from more than one firm. The remainder of the paper is as follows. In Section 2 we set up the basic model. In Section 3 we study firms setting two-part tariffs simultaneously and setting their prices sequentially. This serves as a benchmark for the main analysis. In Section 4, we consider one of the two price components as given and examine the equilibrium in fees (for given rates) and in rates (for given fees). We discuss the details of the results and compare them with vertical differentiation. In Section 5, we study the problem when firms cannot set negative prices. We also examine subgame-perfect equilibrium behavior over the two-stage game. Section 6 concludes. With the exception of that of Proposition 3, all other proofs are relegated to Appendix A.

We shall consider various aspects of competition between the firms, depending on which price components are the strategic variables. Independently of which choices are endogenous, we first describe the firms' profits as functions of the fees and rates charged. Given the lack of product differentiation, for both firms to have a positive market share it should be either that the two firms charge identical prices or, if they charge different prices, that one firm charges a higher fee and the other charges a higher rate. We distinguish three cases: Case 1. If a firm has both a higher fee and a higher rate than its rival, no consumer uses its product and that firm ends up with zero profit. In that case, the firm with the lower fee and lower rate has demand equal to the entire market. The profit of such a firm, say firm i, that captures 1

~θ ≡ f B −f A : r A −r B

We consider competition between two firms, A and B, selling products viewed by all consumers as perfect substitutes — we say that firm i sells product i, i = A, B. Pricing by each firm i has two parts, a fee fi, to be paid by each consumer that chooses product i, and a rate ri, that is to be applied to the usage of the product. Demand is represented by a continuum of consumers with total mass normalized to 1. Consumers differ with respect to their usage levels. In particular, each consumer is represented by his level of usage, θ, where we further assume that θ is uniformly distributed on [θL, θH].6 We set θL = 0 and θH = 1. The goal of each firm is profit maximization — as we assume, for simplicity, zero costs, this amounts to maximization of the revenue from its product.7 Each consumer chooses the product that leads to the lowest total payment: a consumer with usage level θ will choose the product of firm A (“product A”) if fA + θrA b fB + θrB and product B otherwise. Consumers choose to purchase one of the two products.8 There is no uncertainty.9

5 Some papers examine how restrictions in parts of the strategy choices of firms affect the equilibrium outcome. For example, in Brekke et al. (2006) firms compete in locations and qualities, while their (linear) prices have been set exogenously. In our analysis, the restrictions are on dimensions of two-part tariffs. 6 To gain some additional insights about the role of the distribution of usage levels, we also examine in Section 4.2 the case of only two usage levels. 7 It is straightforward to introduce a constant unit cost, without altering the main results. 8 This formulation involves two implicit assumptions. First, that the utility of having a product is high enough so that no consumer would choose not to have either of the products. Second, that no consumer purchases both products. As long as prices are positive, this assumption follows from the fact that the products are perfect substitutes. When some prices are negative, in principle, consumers may wish to obtain both products in the case that the total payment for them is negative. In some markets such behavior may be possible (e.g., in some segments of the credit/payment card market), while in other cases it is not (e.g., a traveler may drive only one rental car at a time). 9 Introducing uncertainty into the analysis would generate additional insights, as also would the analysis of dynamics, see e.g., Griva and Vettas (2003).

ð1Þ

Both firms have positive market shares if ~θ ∈ ð0; 1Þ. In such a case, h i firm A sells to consumers with θ ∈ 0; ~θ , firm B sells to these with h i θ ∈ ~θ; 1 and the profit functions become Z

2. The basic model

1

the entire market by charging fi, ri is ∫ 0 ð f i þ θr i Þdθ ¼ f i þ r i ∫ 0 θdθ ¼ f i þ ri 2 . Case 2. If both firms have equal fees and equal rates (say f and r) then we will assume that they split the market equally and have equal profits
 f þ 2r =2. Case 3. Suppose now that one firm charges a higher fee and the other a higher rate. Without loss of generality, let firm A be the one that has the higher rate, that is, rA N rB. When rA N rB and fA b fB, a consumer with usage level θ is indifferent between product A and product B if fA + θrA = fB + θrB. We denote this indifference usage level by

~θ

~θ2 ð f A þ θr A Þdθ ¼ f A ~θ þ rA ¼ 2 0  2 f B− f A r A −r B f B −f A ¼ fA þ rA r A −r B 2

πA ¼

ð2Þ

and Z

  1−~θ2 ð f B þ θr B Þdθ ¼ f B 1−~θ þ r B ¼ 2  2 fA   1− frBA − −r B f −f A : þ rB ¼ f B 1− B r A −r B 2

πB ¼

1 ~θ

ð3Þ

If the indifference (Eq. (1)) implies ~θ ≥1, firm A captures the entire 1

1

market with profit equal to ∫ 0 ð f A þ θr A Þdθ ¼ f A þ rA ∫ 0 θdθ ¼ f A þ r2A , while firm B's market share and profit are zero. Regarding ~θ ≤ 0, we note that if a firm has a lower fee than its rival, it guarantees itself a positive market share. If, in addition, its rate is higher enough than that of its rival's, the rival will also have a positive market share, otherwise the low fee firm captures the entire market. The reason that a lower fee always implies a positive market share, regardless of the rates, is that there are always some consumers with usage level θ close to zero. Consumers with θ equal to zero (and, by continuity, also those near zero) make their selection strictly on the basis of which product has the lower fee. Since their usage level is very limited, such consumers do not pay attention to the per unit rates. We conclude that the profit functions when rA N rB and fA b fB are 8 > > > <



f −f A fA B πA ¼ r A −r B > > > : f A þ rA 2



 þ rA

f B− f A rA −rB

2

2 if ~θ ∈ ð0; 1Þ if ~θ ≥ 1

K. Griva, N. Vettas / International Journal of Industrial Organization 41 (2015) 30–41

33

and 8 > > <

 2 fA   1− frBA − −r B f B− f A þ rB πB ¼ f B 1− r A −rB 2 > > : 0

if ~θ ∈ ð0; 1Þ if ~θ ≥1:

3. Equilibrium: preliminaries Before proceeding to the main analysis, we provide some initial insights into the nature of competition in our model. First, it should be clear that when there is only a single price component, either a fee or a rate, the only equilibrium of the game is with pricing at (zero) cost and with profits equal to zero. With only one pricing component, we have essentially one-dimensional price competition with homogeneous goods and, therefore, the standard Bertrand arguments apply. What happens now when both fees and rates can be used? When both components are chosen at the same time, we have: Proposition 1. When firms are competing by setting fees and rates simultaneously, the only equilibrium is that firms set r A ⁎ = r B ⁎ = 0 and fA⁎ = fB⁎ = 0 and make zero profit. A sketch of the proof is as follows. First, we can never have an equilibrium where one or both firms have positive profit since, in such a case, at least one of the firms would have an incentive to deviate by undercutting its rival's prices and attract the entire market, thus obtaining a higher profit. So we can only have an equilibrium where both firms make zero profit. The next step is to find all the fees and rates that may form such a zero profit equilibrium. With the help of Fig. 1 we see that for r A N r B N 0 and fA b f B b 0, at least one firm enjoys a positive profit, therefore we cannot have this combination of prices in equilibrium. The two lines represent the total payment that a consumer would make to obtain each of the two products. Depending on their usage level, θ, consumers choose the product corresponding to the lower of the two lines. Indifference occurs where the two lines cross. From the viewpoint of the firms, for each customer they serve, the distance between the price line and the zero axis represents the profit (or loss). We see that, for r A N r B N 0 and fA b f B b 0 at least one firm makes necessarily nonzero profit. With the help of Fig. 2, we see that for r A N 0 N r B and fA b 0 b fB, firm A has an incentive to deviate by increasing its fee to fA = 0 and enjoy positive profit. Therefore, in equilibrium, we can only have prices and profits equal to zero. So, with simultaneous moves equilibrium prices have to be equal to cost. Next, we find that sequential moves are not a sufficient condition for both firms to enjoy positive profit. As in the case of simultaneous

Fig. 2. Profits for rA N 0 N rB and fA b 0 b fB.

moves, a price undercutting incentive exists, although the arguments now have to be modified. We have10: Proposition 2. Suppose that firms set their prices sequentially, so that firm A (the “leader”) sets its fee and rate, and then firm B (the “follower”) sets its fee and rate. There is a continuum of subgame perfect equilibria where the follower always matches the leader's price (as long as this is non-negative; and prices at zero otherwise) and the leader sets some non-negative price. The follower always captures the entire demand and the leader makes zero profit. The follower always seeks to attract the profitable consumers. If all consumers are profitable, the follower undercuts one of the rival's (strictly positive) prices and attracts the entire market. If only some of the consumers are profitable, then the follower undercuts the total payment that the leader charges to them. Fig. 3 illustrates this argument. When the leader, firm A, charges fA + θrA N 0 for some θ ∈ (0, 1), firm B wishes to capture the consumers that have θ ≤ θ˜ (because these are the profitable ones), so it charges a fee slightly lower than fA and a rate slightly higher than rA. 4. Market segmentation with competition in one price component Our analysis thus far does not lead to very surprising results. With homogeneous goods and no restrictions on the price strategies that can be used, equilibrium implies zero profit. We now explore the game where one price component is given at some predetermined level for each firm and competition takes place via the other component. 4.1. Competition in fees We proceed first to analyze the case with the fees as the strategic variables and taking the rates as exogenously given.11 We obtain the following result. Proposition 3. Suppose that the rates, rA ≥ rB ≥ 0, are given and the firms compete by choosing their fees simultaneously. Then: (i) if rA = rB = r, the equilibrium fees are fA⁎ = fB⁎ = − r / 2, the two firms share the market equally and make zero profit, (ii) if rB b rA b 2rB, there is no equilibrium in pure strategies and (iii) if rA ≥ 2rB, total demand is divided equally between the two firms, the equilibrium fees are 

fA ¼ −

rB 1  b 0 and f B ¼ ðr A −2r B Þ N 0 2 2

and firm B makes a higher profit than firm A.

Fig. 1. Profits for rA b rB b 0 and fA b fB b 0.

10 To simplify the statement of the following result we assume that, when prices are equal, consumers choose to purchase from the follower. 11 We analyze the case when the given rates are non-negative. Similarly, we can also show that we have an equilibrium where the firms' fees are as in case (iii) of Proposition 3 for any rA N 0 ≥ rB and also for 0 N rA N rB with 2rA N rB.

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K. Griva, N. Vettas / International Journal of Industrial Organization 41 (2015) 30–41

This best response correspondence is derived as follows. For firm B to capture the entire market, the highest fee it can charge is the one that, given fA, makes the consumer with the lowest usage level (θ = 0) choose firm B: fB = fA. When firm B shares the market with firm A, its reaction function, which is derived by setting ∂πB ∂fB

¼ 0 and solving for fB, is f B ¼ RB ð f A Þ ¼

f A r A þðr A −r B Þ2 . 2r A −rB

This is the

fee that maximizes firm B's profit when both firms have positive sales, therefore

Fig. 3. Profits when firms move sequentially and fA + θrA N 0 for some θ ∈ (0, 1) When the leader, firm A, charges fA + θrA N 0 for some θ ∈ [0, 1], firm B wishes to capture the consumers that have θ≤ ~θ (because these are the profitable ones), so it charges a fee slightly lower than fA and a rate slightly higher than rA.

Proof. (i) Suppose that rA = rB = r and one firm has a higher fee than the other. This firm would attract no clientele and would have an incentive to undercut its rival. Likewise, for any fees that are equal to each other at a level strictly above − r/2, each firm would slightly undercut its rival and capture the entire market. This motive will lead both firms to set fees such that profits are equal to zero and the market is divided equally; in other words, we have Bertrand competition. In equilibrium, the fees are fA⁎ =
 fB⁎ = − r/2 and the profit for each firm is πi ¼ 12 f i þ 2r . No firm wishes to lower its fee further because it would capture the entire market but make losses. Also, either firm will be indifferent to increasing its fee since it would lose all clientele and again make zero profit. (ii) To analyze the case when the exogenously given rates are different, we first examine the second-order conditions of the profit maximization problem for each firm. We have

2

∂ πA ∂ f 2A

¼ −rA þ2r2B , ðr A −r B Þ

which is positive for rB b rA b 2rB and then firm A's profit function is strictly convex and is negative for rA N 2rB and then firm A's profit function is strictly concave. For firm B, we have

2

∂ πB 2 ∂fB

¼ −2rA þr2B , ðr A −r B Þ

which is negative for all rA N rB and then firm B's profit function is strictly concave. We examine each firm's maximization problem in turn. For 0 b rB b rA b 2rB, the best response correspondence of firm A, given firm B's fee, is 8 1 > < f B −r A þ r B if f B ≥ ðr A −2rB Þ 2 f A ¼ RA ð f B Þ ¼ 1 > : any fee≥ f if f B b ðr A −2r B Þ B 2 and can be derived as follows. The highest fee that firm A can charge and still attract all customers is the one that, given fB, makes the consumer with the highest usage level (θ = 1) choose firm A: f A ¼ f B −r A þ r B :

ð4Þ

Since firm A has a convex profit function, it obtains a higher profit when it serves the entire market, compared to when it shares it with the rival firm. Firm A has a positive profit when it attracts all clientele only if f B ≥ 12 ðrA −2r B Þ , otherwise it prefers to set fA ≥ fB and have no clientele and zero profit. Next, we derive the best response correspondence of firm B, given firm A's fee: 8 f if f A N rA −r B > > < A f A r A þ ðrA −r B Þ2 f B ¼ RB ð f A Þ ¼ if −r A ≤ f A ≤ r A −r B ð5Þ > 2r A −r B > : any fee ≥ f A þ r A −r B if f A b −r A :

f A ≤ f B ≤ f A þ r A −r B ⇒ f A ≤

f A r A þðr A −r B Þ2 2r A −r B

≤fA þ

rA −r B , which implies that −rA ≤ fA ≤ rA − rB. If fA ≥ rA − rB, firm B maximizes its profit by charging fB = fA and capturing the entire market. If fA b − rA, firm B maximizes its profit by charging a fee that guarantees that no consumer would ever choose that firm, that is by choosing any fB N ( fA + rA − rB). Finally, we find that the two best response correspondences for firms A and B never intersect. Firm A finds it profitable to capture the entire market, otherwise it makes zero sales. It never aims at sharing the market. At the same time, as long as firm A has some customers, firm B finds it profitable to share the market with firm A. As a result, firm A can never capture the entire market in equilibrium. (iii) For rA N 2rB N 0, the best response correspondence for firm A is 8 f −r þ r B > > < B A −f B r B f A ¼ RA ð f B Þ ¼ > > : r A −2r B any fee ≥ f B

if f B ≥ r A −2r B if 0 b f B b r A −2r B if f B ≤ 0

and is derived as follows. The highest fee that firm A can charge and still capture all customers is fA = fB − rA + rB (derived as in Eq. (4)). When firm A shares the market with firm B, its A reaction function, which is derived by setting ∂π ¼ 0 and solving ∂f A

with respect to fA, is f A ¼ RA ð f B Þ ¼

−f B r B : rA −2rB

ð6Þ

This is the fee that maximizes firm A's profit when both firms operate in the market, that is when ~θ ∈ ð0; 1Þ, therefore f −r A þ B

rB b f A b f B ⇒ f B −r A þ rB b

− f B rB r A −2r B

b f B , which gives 0 b fB b rA −

2rB. If fB ≥ rA − 2rB, firm A makes a higher profit by charging fA = fB − rA + rB and capturing the entire market. If fB b 0, firm A maximizes its profit by charging a fee that guarantees that no consumer would choose that firm, and this fee is any fA ≥ fB. Firm B's best response correspondence, given the fee of firm A, is the same as in case (ii) (Eq. (5)). The two best response corre  spondences intersect at f A ¼ − r2B and f B ¼ 12 ðr A −2r B Þ. By direct  ⁎ ⁎ substitution of fA and fB into Eq. (1) we obtain ~θ ¼ 12. Finally, by substituting fA⁎ and fB⁎ into expressions (2) and (3), we obtain πA ¼ 18 ðr A −2r B Þ and πB ¼ 18 ð2r A −r B Þ. We then easily see that πB⁎ N πA⁎ if rA + rB N 0. ■ Some initial remarks on the properties of the equilibrium, for rA N 2rB, are useful here. First, the firm with the higher rate (firm A) chooses, in equilibrium, a fee lower than that of its rival. Second, the equilibrium fee of firm B is non-negative. In contrast, the equilibrium fee of firm A is negative.12 Third, the firm that obtains the higher equilibrium profit is the one that charges the relatively lower rate and higher fee, in 12 If the firms had a per unit cost equal to c, a similar analysis would imply that the equilibrium fee for firm A would be f A ¼ c− r2B and for c N r2B this fee would be positive. The fees in our formulation appear to be negative because we have assumed unit cost equal to zero. Also, if we had a more general uniform distribution with θ distributed uniformly on some  interval [θL, 1 + θL], we would have f A ¼ 12 ð−r B −2r A θL Þ. If θL could take negative values, then the equilibrium fee of firm A could be positive.

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other words, the one that attracts the consumers who tend to have a higher usage level. We discuss in greater detail the equilibrium properties just below. 4.2. Equilibrium properties Proposition 3 is central in our analysis. When one pricing component (the rates) is given, there are conditions that imply an equilibrium in the other pricing component (the fees) with positive profits for both firms. We examine various aspects of this result and provide some intuition. 4.2.1. Equilibrium existence and the difference in rates We have shown that when the rates satisfy rA N 2rB N 0, the two firms' best response correspondences intersect and there exists a   unique equilibrium where f A ¼ − r2B and f B ¼ 12 ðr A −2r B Þ . Fig. 4a illustrates this behavior. In contrast, when the rates are positive but rB b rA b 2rB, the two firms' best response correspondences do not intersect. This case is illustrated in Fig. 4b. We observe that when the rates are not different enough, firm A, the firm with the higher rate, aims at capturing the entire market, as long as the rival strategy makes this possible. Since firm A never finds it optimal to share the market with firm B, there exists a discontinuity in firm A's best response correspondence that prevents establishing a pure strategy equilibrium. In contrast, when the rates are different enough, firm A finds it profitable to share the market with its rival, for a given range of values for fB. This qualitative change in firm A's strategy implies that its best response correspondence is continuous and allows the existence of an equilibrium. The difference in rates plays a crucial role. When the difference is large, the high rate firm finds it more profitable to only serve part of the demand, the one with low usage levels, rather than to fight for all the demand by lowering its fee further. When the difference in rates is small, capturing all the demand would only require a relatively small decrease in this firm's fee, and hence this emerges as a more profitable strategy than serving only the low usage customers. It is also important to observe that for the range of fees where firm A maximizes its profit by sharing the market with firm B, fA is decreasing in fB (see Fig. 4a). The reason why this happens is as follows. When sharing the market, firm A, which is the high rate firm, attracts the consumers with a lower usage level, that is, the consumers located at the left of the indifferent consumer. Since price discrimination is infeasible, in order to attract additional consumers and to gain from their usage levels, firm A would have to lower its fee further and to lose some profit from the entire pool of its customers. Now, when the rates are different enough and for 0 b fB b rA − 2rB, firm A prefers not to serve some customers and to charge a relatively high fee, serving primarily the customers with low usage levels. In this range, an increase in the rival fee fB, makes firm A's profit function more elastic (with respect to its fee), as it now moves towards the high usage consumers. While its demand may be higher, it is locally more elastic. The relative importance of the low usage consumers in firm A's profit function now decreases. The higher the usage level, the more important the difference in the rates. Therefore, when a firm considers mainly low usage customers, it can attract them as long as its fee remains lower than its rival's. But when also considering high usage customers, these care more about the difference in rates, and so firm A, which now has a greater disadvantage in terms of its rate, has to become more competitive in terms of its fee. As a result, within this range, when the rival fee increases, firm A finds it optimal to respond by decreasing its own fee. The fact that both firms' best response correspondences are continuous and have opposite slopes allows us, in this range, to find a pure strategy equilibrium in fees. 4.2.2. Continuity in usage levels — comparison to a two-type model We have shown that, when the exogenously given rates are at levels close to each other, there is no pure strategy equilibrium in fees. In

35

contrast, there exists an equilibrium when the rates are at levels different enough. In addition to establishing the role of the difference between rates, it is useful to explore some alternative assumptions about the nature of demand. We show here that continuity in the usage levels is also essential for the existence of equilibrium. To demonstrate this point, we analyze a variant of our model with only two usage levels. We obtain the following result. Remark 1. Consider a version of the model where some (λ) of the consumers have low usage levels θL and some (1 − λ) have high usage levels θH, with θH N θL N 0. The rates are given, and firms compete by choosing their fees simultaneously. If the rates are equal, at some level r, there is an equilibrium with fA⁎ = fB⁎ = −[λθL + (1 − λ)θH]r, total demand divided equally between the firms and zero profits. If the rates are different (rA N rB), there exists no pure strategy equilibrium. In contrast to our main model where the usage levels follow a continuous distribution, we cannot have here an equilibrium in fees regardless of how large the difference in the rates is. In a two-type model, competition is more intense compared to a model with a continuum of types and this holds independent of the proportion of each type in the population. If we had an equilibrium where each firm attracted one of the types, firm A would have to attract the low type and firm B the high type. In order to capture the entire market, each firm would only need to attract one more type. We find that with discrete types, and starting from a candidate equilibrium where both firms have positive sales, there is always either an incentive for a firm to lower its fee and sell to both types, or an incentive for a firm to increase its fee and its profit. In our main model, with a continuum of usage levels, and when rates are not too close, the profit functions have the right curvature (they become concave) so that firms do not (always) find it profitable to deviate. Thus, the distribution of the usage levels should have certain properties to allow for an equilibrium with market segmentation to emerge, and having rates different enough is not sufficient. 4.2.3. Comparative statics Starting with rates such that an equilibrium in fees exists, we now study what happens when these rates change exogenously. Remark 2. When rA N 2rB, both firms' equilibrium fees and profits increase as the distance between the two rates increases. The result follows by differentiating the relevant functions with respect to the rates:  ∂πA 1 ∂f A and ¼ ¼ 0; ∂r A 8 ∂r A   ∂πA 1 ∂f A 1 and ¼− ¼− ; 4 2 ∂r B ∂r B  ∂πB 1 ∂f B 1 and ¼ ¼ ∂r A 4 ∂r A 2

and  ∂πB 1 ∂f B and ¼− ¼ −1: 8 ∂r B ∂r B

ð7Þ

It follows directly that both firms' profits and fees increase as rA increases or as rB decreases. This result is reminiscent of that of Shaked and Sutton (1982) concerning maximal quality differentiation. In our model, the firm that attracts the consumers with a higher usage level has a higher profit when its rate is reduced because this reduction softens the competition via fees. In Shaked and Sutton (1982), the low quality firm gains from reducing its quality because this softens price competition. We proceed to a more general comparison of the two models just below.

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(a)

(b)

Fig. 4. The firms' best response correspondences when (a) rA = 4 and rB = 1 and (b) rA = 4 and rB = 3.

4.2.4. Comparison to vertical differentiation We can understand our analysis better by drawing some analogies from the work of Shaked and Sutton (1982). In their well-known model, consumers are heterogeneous with respect to their tastes for quality (some would be willing to pay more for higher quality and others to pay less), while in our model consumers differ with respect to their usage levels (some are heavy users and some are light users). Also, in their model, each product is priced linearly, while in our model, a consumer has to compare both fees and rates. The sense in which our analysis is related to Shaked and Sutton (1982) is as follows. We deal with homogeneous products, but when one price component is fixed, an element that operates like vertical differentiation enters the analysis. Suppose that one firm's rate is fixed at a higher level than its rival's. Ignoring any effect from the fees, this would mean that all consumers would view the low rate firm as more attractive, but the intensity of this preference would be different across the population, with heavy users caring relatively more and light users caring less. However, there are also very important differences between the models. In Shaked and Sutton (1982), the heterogeneity of consumers determines which product each consumer will choose, but then each consumer always consumes only one unit. In our model, the heterogeneity of consumers reflects not only which product each consumer will choose, but also how much he will consume. As a result, the analysis is based on different profit functions. In their model, each firm only collects a linear payment from each consumer and only cares about how many consumers it will attract. In our model, each profit reflects the two-part tariff that each consumer pays. As a result, a firm cares not only about how many customers it attracts, but also who these customers are. A heavy user will generate higher profit for a firm than a light user. As a result, the heterogeneity of consumers with respect to their usage levels enters the profits at two levels. First, consumers, depending on their usage level, choose one of the two firms and obtain the product by paying the corresponding fee fi. Second, consumers use it at a different level θ by paying ri per unit of use. Thus, given the distribution θ, each consumer minimizes the total payment fi + θri, which is also exactly the amount that the firm collects. In Shaked and Sutton (1982) a firm does not collect a different amount from different users, nor does it collect a payment per unit of quality. Since the profit functions are different between the two models, it is not surprising that the analysis leads to different results. A comparison of the main results between the two models is as follows. When firms take product quality as given and compete via linear prices, Shaked and Sutton (1982) show that the firm selling the higher quality product charges a higher price than its rival, captures a larger market share (specifically twice as large assuming that preferences are uniformly distributed), and enjoys higher profit. In our model, when

rates are given (and are different enough so that rA N 2rB), in equilibrium, the firm that has been assigned the lower rate and therefore attracts the consumers with the higher usage levels, charges a higher fee than its rival, splits the market equally with its rival, and enjoys higher profit. In both cases, equilibrium profit increases for both firms the more differentiated they become. The difference in the equilibrium market shares is due to the different role that heterogeneity plays in the two models. Moving to the main differences, in our analysis there is a pure strategy equilibrium in fees only when the firms differ enough with respect to their rates. In Shaked and Sutton (1982) there is always a pure strategy equilibrium in prices, even for very similar qualities and there is also a maximum differentiation equilibrium in qualities. When a rival firm has high quality, a firm gains by having low quality because in this way it relaxes price competition — importantly, quality enters the profit function only via the demand function, and not directly. In our model, if one firm charges a high rate, the other firm may wish to undercut its rival's rate. This is because the rates do not only affect the consumers' choice of a product, but also represent direct revenue. While undercutting is not profitable in Shaked and Sutton (1982), it is profitable under certain conditions in our setting. Only when the rates are different enough, competition via the fees is relaxed and both firms are willing to share the market, in particular, with the high rate firm charging a negative fee. A final difference between the models has to do with negative prices. In Shaked and Sutton (1982), pricing is linear and a negative price would never make sense. With two-part tariffs, however, in our analysis, a price component may be set at a negative level in equilibrium. In Section 5 we examine our model when no price component can be negative, and then some important differences between our models and product differentiation become even clearer. 4.3. Competition in rates Now, we reverse the roles and examine the case where the fees are exogenously given (at some non-negative levels) and the rates are the strategic variables that the firms choose simultaneously. This case is not simply the mirror image of the one we have already examined. Since consumers differ in their usage levels, starting the analysis with different fees shapes the model differently than starting with different rates. When firms compete in rates, competition is more intense than when firms compete in fees. This happens for the following reason. Starting our analysis with different fees, fB N fA, we implicitly impose the restriction that firm B can never capture the entire market, no matter what the difference is between the rates. This advantage that firm A has stems from the fact that the consumers differ with respect to their usage levels and there are consumers in the neighborhood of θ = 0 (with very low usage levels) that essentially care only about the

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difference in fees and not about the difference in rates. This advantage of one firm was not present when we conducted our analysis starting with different rates, since all consumers equally care about the difference in fees and in such a case, it would be possible for the high rate firm to capture the entire market by setting a fee lower enough than that of its rival. We have13: Proposition 4. Assume that the fees are given, with fB ≥ fA ≥ 0, and firms choose their rates simultaneously. (i) If fA = fB = f, the equilibrium rates are rA⁎ = rB⁎ = − 2f, total demand is divided equally between the two firms and both firms have zero profit, (ii) if 12 f B ≤ f A b f B , there is no equilibrium in rates in pure strategies and (iii) if f A ≤ 12 f B , the equilibrium rates are r A

−3f A þ f B f þf pffiffiffi ¼ and r B ¼ − Apffiffiffi B b 0: 2 2

ð8Þ

Firm A has a larger market share than firm B, since θ ¼ p1ffiffi2, and makes a higher profit. When the fees are given at levels that are not very different between firms, firm B's best response is discontinuous, while firm A's rate is increasing in its rival's rate. As long as firm A seeks to capture the entire market, firm B never chooses to make no sales, and as long as firm B seeks to share the market with firm A, firm A makes a higher profit by capturing all the market. As a result, there is no pure strategy equilibrium. When the fees are set at different enough levels, firm B's best response becomes continuous. Moreover, for 3fA b fB firm A's best response becomes decreasing in rB within a specific range of rates. In this case, both firms have a positive market share. Importantly, it is too costly for firm A to maintain or increase its market share when firm B decreases its rate. Instead, it prefers to increase its rate and it accepts to lose some customers since the customers it serves are the ones that are located at the left of the indifferent consumer, that is, the customers with a lower usage level. In total, demand that comes from lighter users is less responsive to an increase in rA compared to demand from heavier users. Therefore, when the fees are very different and firms share the market, firm A chooses to increase its profit by increasing rA as a response to a decrease in rB. Finally, it is interesting to examine how a change in the fees affects equilibrium profits. We can directly differentiate the relevant equilibrium expressions (see the Proof of Proposition 4) with respect to fA and fB and obtain:

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take non-negative values. This may be relevant in different markets, for institutional or competition policy reasons. This analysis would also apply when consumers could in principle purchase more than one product from either firm. Then, and with a negative price component, some consumers would choose, in principle, to purchase two or more products not in order to use them, but simply to benefit from a negative payment.14 Examining this case also helps shed some light on the problem when we compare it with the case of unrestricted prices and also with vertical product differentiation. We first analyze a two stage game where firms simultaneously first choose their (non-negative) rates and then, given these rates, they simultaneously choose their (non-negative) fees. We obtain the following result: Proposition 5. Suppose that no price component can be negative and consider a two-stage game where rates are chosen first and fees are chosen  second. We have: (a) For any rA ≥ rB ≥ 0, in the second stage fA⁎ = 0, f B ¼ ðr A −r B Þ2 2r A −r B and r A −r B 2r A −r B ≤1=2.

firm B captures a larger market share than firm A, θ ¼ Given the rates, both firms enjoy a higher profit compared to the case where firms can also charge negative fees. (b) In the first stage, where firms choose their rates, there is no equilibrium in pure strategies.

With two-part tariffs, firms may choose one price component to be negative (this can only make sense if the other component is positive, of course). Here we examine the case where fees and rates can only

We find that both firms enjoy a higher profit when they can only charge non-negative fees compared to when they can charge negative fees — when prices are bounded below at zero, competition is less intense relative to when they are not. Consider firm A, the one with the higher rate, and which attracts the consumers with relatively low usage levels. In principle, this firm would like to increase its market share in order to attract a larger number of the more profitable consumers, those with relatively higher usage levels. According to Proposition 3, to do so it would have to charge a negative fee (when this is possible) and it would choose to attract the consumers that have a usage level up to 1/2. When firms are limited to non-negative prices, firm A cannot charge such a low fee, instead it charges a zero fee. As a result, firm A, the one that enters the problem having the higher rate, always has a smaller equilibrium market share than its rival. For a given difference in rates, the difference in the equilibrium fees is smaller when firms can charge only non-negative fees, competition is softer, and each firm enjoys higher profit. It is also useful to compare how firms behave in this case with Shaked and Sutton (1982). As already discussed in Section 4.2, the different first stage rates here can be paralleled to different quality levels under vertical differentiation. Competition then takes place in prices. Since all these components are non-negative, the comparison with the analysis here is direct. We have some similarities between the two models. Regardless of what has happened at the first stage, there is a second stage pure-strategy equilibrium. At this equilibrium, the firm that enters the second stage with a relative advantage (either because it has a lower rate in our analysis or a higher quality in Shaked and Sutton) charges a higher price, captures a higher market share and enjoys higher profit than its rival. However, the two models are different. In our model, in the first stage there is no equilibrium in rates (since firms have an undercutting incentive), while in Shaked and Sutton there is an equilibrium with maximum quality differentiation. Note that in this case, while there is no pure-strategy equilibrium in the first stage, there might exist mixed-strategy equilibria. Characterizing the set of these, however, would not be trivial. Part of the challenge is that the strategy sets are not bounded, when at the same time the payoff functions are not continuous. A mixed strategy equilibrium, when it exists, would involve a positive probability over a full support of prices, since otherwise there would be an undercutting incentive.

13 We present the analysis here when the exogenously given fees are non-negative. We can also show that we have an equilibrium where the firms' rates are as in case (iii) of Proposition 4 for fB N 0 and −fB b fA b 0.

14 Of course, such an incentive to some consumers to purchase only in order to enjoy negative prices would discipline the sellers to not offer such prices. See Section 6 in Griva and Vettas (2012).

∂πA ∂πA ∂πB ∂πB 1 1 −1 5 ¼ pffiffiffi and ¼ pffiffiffi ; ¼ pffiffiffi and ¼ 1− pffiffiffi : ∂f A 4 2 ∂f B 4 2 ∂f A 4 2 ∂f B 4 2 It follows that firm A's profit increases as each of the fees increases. For 0 ≤ f A ≤ 12 f B , firm A's profit is maximized when f A ¼ 12 f B . Firm B's profit increases as fA decreases and as fB increases. Therefore, firm B's profit is maximized when fA = 0. We have: Remark 3. For 0 ≤ f A ≤ 12 f B , firm A's equilibrium profit increases as both fees increase, while firm B's equilibrium profit increases as the distance between the fees increases. 5. Non-negative fees and rates

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Finally we consider the reverse case, a two-stage game where firms first simultaneously choose their fees and then, given these fees, they simultaneously choose their rates. Proposition 6. In a two-stage game of non-negative prices where fees are chosen first and rates second we have: (a) For any fB ≥ fA ≥ 0, in the second stage rA⁎ = fB − fA and rB⁎ = 0. For fB N fA firm A captures the entire market and enjoys a higher profit compared to the case where firms could charge negative rates, while firm B now has a lower profit, zero. For fB = fA the market is shared equally. (b) In the first stage, fees are chosen in equilibrium to be zero. This also implies that rates will also be zero at the second stage. In this case, where firms first choose their (non-negative) fees and then their (non-negative) rates, both firms have, in equilibrium, zero profit since they each seek to undercut their rival's fee at the first stage. This result is due to the fact that the firm with the higher fee always ends up with zero equilibrium profit in the second stage, a feature that implies a very strong undercutting incentive when setting fees at the first stage. Again, we note some similarity with Shaked and Sutton (1982) since the firm that enters the second stage with an advantage can price less aggressively than its rival, but especially some differences since the entire market is always captured in the second stage by the more favored firm. As a result, there is no maximum first-stage differentiation in our setting, instead the firms compete with each other towards zero profit.

6. Conclusion We have studied a homogeneous-product duopoly where pricing takes the form of two-part tariffs and consumers differ with respect to how much they use the product. When one price component is fixed at levels that differ enough between the two firms, both firms make positive profits and the market is segmented, with low usage consumers choosing the low fee firm, while high usage consumers choose the low rate firm. In a way, competition takes place as if the products were differentiated, when in fact they are not. The nature of competition is substantially different when it is in fees relative to rates. Importantly, the firm that has a lower given fee can guarantee for itself at least some part of the demand, since the consumers with very low usage levels essentially make their selection only on the basis of the lower fee. An important case in our analysis is when firms can only set nonnegative prices. First, when there is a zero low bound for all strategic choices, no matter at what level the first price component has been set, there is always a pure strategy equilibrium in the other price component. This allows us to fully study the two-stage game where the fees (or rates) are chosen first and the rates (or fees) are chosen subsequently. Second, restricting each price component to be non-negative does not allow firms to compete as aggressively as they otherwise would and, naturally, this property affects equilibrium profits. In comparison to the case where the firms can set a price component below zero, we now find that when competition is in fees (for given rates) both firms enjoy higher equilibrium profit. When competition is in rates (for given fees), the effect is different. The high fee firm cannot attract in equilibrium any customers and its profit is reduced to zero; its rival's profit is then higher. Thus, removing the ability of firms to price below zero affects the problem differently depending on whether the rates or the fees are chosen endogenously. We have kept our model as simple as possible, though the analysis is not trivial. Our main goal is to shed light on the distinct role that each price component plays in two-part tariff competition, in relation to the differentiation of the consumers along their usage levels. We demonstrate that fixing one price component indirectly introduces features of product differentiation, an important aspect of the problem and not yet explored in the literature. We also highlight features of the problem that should be kept in mind by competing firms, as well as by regulators, when components of two-part tariffs can be fixed or when non-

negativity constraints are important. Hopefully our analysis contributes to the growing literature on duopoly competition via two-part tariffs. Naturally it has its limitations and can be extended in various directions. A central theme in the analysis is that positive profits can be obtained only if one price component is set at levels that for the two rivals are different enough. More generally, it is restrictions on aspects of price setting that take us away from the Bertrand outcome and while this is a feature of our analysis, the limitations that these assumptions represent should be clearly acknowledged. In some applications these assumptions may be reasonable, but in others they may not be. In terms of the equilibrium construction, extending the analysis to mixed strategies may be of interest and generate additional insights. The main result, however, is expected to hold, that for a given distribution of usage levels, only if some price component differentiates enough the profit maximization problem between the two firms we could avoid zero profit at equilibrium. An interesting but nontrivial extension of the model would also be when there are additional dimensions of differentiation among the customers, in addition to their usage levels, perhaps along some horizontal characteristic of the product. We have also assumed, for simplicity, that the usage level for each consumer is fixed. The logic of the analysis could be extended to cases where the usage levels respond to price changes but only by a little relatively to how much they vary across the population. Yet, in settings where the usage levels may respond significantly to the rates, the problem becomes more complicated. Each firm would know that, by lowering its rate, it would not only attract some additional customers away from the rival, but it would also make all of its customers consume more. A first effect may be that each firm would unilaterally prefer to set a lower rate. However, since the rates would also depend on the fee levels, the nature of equilibrium will have to be studied carefully and will crucially depend on the details of the demand specification. Finally, throughout the analysis, we have also compared our analysis to vertical product differentiation, and in particular to Shaked and Sutton (1982), and we have noted similarities but also important differences. Additional work on the nature and meaning of product differentiation, vertical or horizontal, when pricing is not linear appears promising. Such extensions are left for future research.

Appendix A Proof of Proposition 1. First, we observe that any firm can achieve zero profit either by charging zero prices and attracting some customers, or by charging very high prices compared to its rival's and having no customers. So we investigate whether one or both firms can have a positive profit in equilibrium. If πj N πi ≥ 0, firm i prefers to undercut its rival's prices and attract the entire market, thus enjoying a higher profit. If πi = πj N 0 any firm has the incentive to undercut its rival's fee or both prices and attract the entire market instead of sharing it. We conclude that the only possible equilibrium is with πi = πj = 0. Now we need to investigate what fees and rates can support such a zero profit equilibrium. We start with the case where one firm charges a higher fee and the other a higher rate. (a) Suppose ri N rj N 0 and fi b fj b 0. Could such prices constitute an equilibrium with zero profits? The answer is negative, since at least one firm makes non-zero profit due to the heterogeneity of consumers with respect to their usage levels. (b) Suppose now that ri N 0 N rj and fi b 0 b fj. Could we have an equilibrium with zero profits? Again the answer is negative, because if both firms make zero profit, then firm i could increase its profit by increasing its fee to fi = 0. With a fee lower than its rival's, firm i still captures part of the market (it attracts the consumers with low usage levels) and since ri N 0 firm i earns strictly positive profit, thus this deviation is profitable. Let us now examine the case where all prices are equal to zero. Clearly, for rj = fj = 0, there is no (rj, fj) combination that allows firm i to enjoy a strictly positive profit. The reason is that with the rj = fj = 0

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combination available, no consumer would instead prefer the (fi, ri) combination unless it represented a negative net payment (i.e., a subsidy) to him (equal to (fi + riθ)). But this payment is exactly equal to the profit for firm i. Thus, by integrating over all the consumers that would accept such a combination, we find that any (fi, ri) combination that is accepted by some consumers would lead to a loss (or, at best to zero profit) for firm i. Thus, we reach the conclusion that ri = fi = 0 is the best response to rj = fj = 0, and therefore the unique equilibrium is for all prices to be equal to zero. ■ Proof of Proposition 2. Firm B wishes to attract only the profitable consumers. We distinguish three cases. Case (1) If fA + θrA ≥ 0 for all θ ∈ [0, 1], firm B charges ( fB, rB) = ( fA, rA) to attract the entire market. Case (2) If fA + θrA N 0 for the consumers with θ≤ ~θ (respectively θ ≥ ~θ), firm B charges a fee slightly lower than fA (respectively, higher) and a rate slightly higher than rA (respectively, lower) and attracts only the profitable consumers. Case (3) If fA + θrA ≤ 0 for all θ ∈ [0, 1], firm B charges (fB, rB) N ( fA, rA) so as to have no customers. Firm A's equilibrium behavior is derived as follows. Firm A, which sets its rate and fee first, takes the best response correspondence of firm B as given and tries to maximize its profit. Firm A knows that if it charges fA + θrA ≥ 0 for every θ ∈ [0, 1], firm B will leave firm A with no clientele and zero profit. If it sets fA + θrA ≥ 0 for some θ ∈ [0, 1], firm B will attract all the profitable consumers and leave firm A with losses. Finally, if firm A sets fA + θrA ≤ 0 for all θ ∈ [0, 1], firm B will leave firm A to attract all clientele and make losses. As a result, in equilibrium, firm A charges any pair ( fA, rA) such that ( fA + θrA ≥ 0) for every θ ∈ [0, 1] and has no customers, and firm B charges ( fB, rB) = ( fA, rA), attracts the entire market and makes πB ≥ 0. ■ Proof of Remark 1. Suppose that rA = rB = r. Then each firm has the incentive to slightly undercut its rival's fee in order to capture the entire market. This leads both firms to set fA⁎ = fB⁎ = −[λθL + (1 − λ)θH]r, with zero profits. Firms have customers from both types with the market being equally divided. This case obeys the standard logic of Bertrand competition. At this point, no firm has an incentive to deviate. If rA N rB, in order for each firm to attract one of the types we must have fA b fB. Then if firm A attracts one type, this has to be the low usage type by charging a fee such that fA + θLrA ≤ fB + θLrB which implies that fA = θL(rA − rB). Equality follows since we are interested in the highest fee that firm A can charge in order to attract the low usage type. Then πA = ( fA + θLrA)λ = ( fB + )θLrB)λ. As long as πA ≥ 0, that is fB ≥ − θLrB, firm A prefers to attract the low usage type than to have no customers. To attract both types firm A must charge a lower fee, denoted as c f , such that c f þ θ r ≤ f þ θ r which implies A

A

H A

B

H B

c cA ¼ c fA þ f A ¼ f B −θH ðr A −rB Þ . In this case, firm A's profit is π ½λθL þ ð1−λÞθH rA ¼ f B −λðθH −θL Þr A þ θH r B . Firm A prefers to attract cA and πA ≥ 0. Solving for fB we have only the low type if πA N π B þλðθH −θL Þr A . This inequality holds only if λrA N rB. −θL r B ≤ f B b −ðθH −λθLðÞr1−λ Þ

If λr A ≤ r B , firm A prefers to attract both types as long as fB ≥ λ(θH − θL)rA − θHrB, otherwise it prefers to have no customers. To summarize, if λrA N rB, firm A's best response correspondence is

Firm B's best response correspondence for rA N rB is

f B ¼ RB ð f A Þ ¼

8 > > > < f A þ θL ðrA −r B Þ

f A ¼ RA ð f B Þ ¼

> f B −θL ðr A −r B Þ > > > : any fee N f B −θL ðr A −rB Þ

−ðθH −λθL Þr B þ λðθH −θL Þr A ð1−λÞ −ðθH −λθL Þr B þ λðθH −θL Þr A if −θL r B ≤ f B b ð1−λÞ if f B b −θL r B ; if f B ≥

while if λrA ≤ rB ≤ rA, firm A's best response correspondence is f A ¼ RA ð f B Þ ¼

f B −θH ðr A −r B Þ any fee N f B −θL ðr A −r B Þ

if f B ≥ λðθH −θL Þr A −θH r B if f B bλðθH −θL Þr A −θH r B :

θH ð1−λÞðr A −rB Þ−θL ðr A −ð1−λÞr B Þ λ θH ð1−λÞðrA −r B Þ−θL ðr A −ð1−λÞr B Þ if −θH r A ≤ f A b λ if f A b −θH r A ;

if f A ≥

> f A þ θH ðr A −r B Þ > > : any fee ≥ f B þ θH ðr A −rB Þ

and is derived by following the same logic as when deriving firm A's best response correspondence, appropriately modified. The two best response correspondences never intersect. ■ Proof of Proposition 4. (i) Suppose fA = fB = f. This case follows partially the standard Bertrand case since, in equilibrium, profits are zero but the rate is not equal to cost (which is zero in our case). Each firm has an incentive to slightly undercut its rival's rate in order to capture the entire market. This leads both firms to set rates that make their profits equal to zero and the market is equally split between the   r two products. The equilibrium profit for each firm is πi ¼ 12 f þ 2i ¼ 0 and therefore, the equilibrium rates are rA⁎ = rB⁎ = − 2f. At this point neither firm wants to decrease its rate because it will capture the entire market but will make losses. Also both firms are indifferent towards increasing their rates as that would result in losing all clientele and again making zero profit. Suppose that fA b fB . First note that, in this case, firm A always attracts the consumers with usage level close to zero, no matter what the rates are. Differentiating the profit functions we find that they can be concave or convex depending on the level of the fees and of the rates. Shorting all the different possibilities, we end up with two distinct cases: (ii) For fB ≥ 0 and 12 f B ≤ fA b fB, the best response correspondence of firm A is

r A ¼ RA ðr B Þ ¼

8 > > < r B þ f B −f A ð3 f A −f B Þr B > > : fA þ fB

fA þ fB 2 f þ fB if r B ≤ − A 2 if r B N −

and is derived as follows. The highest possible rate that firm A can charge in order to attract all clientele is the one that, given r B , makes the consumer with the highest usage level (θ = 1) choose fA firm A, therefore the indifference usage level ~θ ¼ frBA − −r B should equal 1 and solving for rA we obtain rA = rB + fB − fA. If firm A shares the market with firm B, its reaction function, which is derived by setting ¼ 0 and solving it for rA, is r A ¼ ð3 ffA −þ ff B ÞrB. In order for both firms to A B operate in the market, ~θ ∈ ð0; 1Þ. Since ~θ ¼ f B − f A the following must ∂πA ∂r A

r A −r B

hold: f B N f A (which is exogenously given) and r A N f B −f A þ r B ⇒ ð3 f A − f B Þr B f Aþ f B

N f B −f A þ r B and thus r B b− ð f A þ2 f B Þ. For r B N− ð f A þ2 f B Þ, firm A

prefers to capture the entire market while for r B ≤− ð f A þ2 f B Þ firm A makes a higher profit by sharing the market with firm B. Next, we derive the best response correspondence of firm B, given the rate of firm A: rB ¼ RB ðr A Þ ¼

8 > > > > f B −θH ðr A −rB Þ <

39

8
pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if r A ≥ f A −3f B þ 2 2 f B ð− f A þ f B Þ pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if r A ≤ f A −3f B þ 2 2 f B ð− f A þ f B Þ;

B where W is defined uniquely as the root of the cubic equation ∂π ¼0 ∂r B that satisfies the relevant constraints for the fees (see below). As we have already noted, firm B can never capture the entire market (~θ can never be equal to zero). Therefore we need to investigate when sharing the market is more profitable than having no clientele and zero profit. We set the profit function equal to zero, solve with respect to its own rate, and obtain

r B ¼ r A −f B þ f A ;

ðA:1Þ

40

K. Griva, N. Vettas / International Journal of Industrial Organization 41 (2015) 30–41

rB ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 −f A − f B þ r A þ ð f A þ f B −r A Þ þ 8f B r A 2

ðA:2Þ

and rB ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 −f A − f B þ r A − ð f A þ f B −r A Þ þ 8f B r A : 2

As we observe, for certain values of rA, the profit function of firm B crosses the axis of rB more than two times. When firm B charges rB ≥ rA − fB + fA, it is left with no clientele (~θ ¼ 1) and zero profit. In order to share the market, firm B must charge rB b rA − fB + fA. We set ∂πB ∂r B

¼ 0, solve it with respect to rB and observe that only one of the

three roots satisfies the limitation for the specific range of given fees. We denote this root by W. In W the two cube roots are complex conjugates of each other and therefore the two imaginary parts cancel out. As qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð f A þ f B −r A Þ þ 8f B r A Þ≥0 , which implies r A ≥ f A −3f B þ long as pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 f B ð−f A þ f B Þ, firm B makes a higher profit when it shares the market than when it has no clientele, otherwise it prefers to have no clientele. When we combine these two best response correspondences, we see that they never intersect. As long as firm A tries to capture the entire market, firm B never chooses to have no clientele, and as long as firm B tries to share the market with firm A, firm A makes a higher profit by capturing all clientele. (iii) For 0≤ f A ≤ 12 f B , firm A's best response correspondence is the same as in case (ii). Next, we derive the best response correspondence of firm B, given the rate of firm A: r B ¼ RB ðr A Þ ¼

W any rate ≥ r A −f B þ f A

 pffiffiffi  − f A þ 4 2−5 f B fA þ fB  pffiffiffi ¼ pffiffiffi and πB ¼ 4 2 4 2

f A ¼ RA ð f B Þ ¼

ðA:3Þ

and conclude that πA⁎ ≥ πB⁎. ■ Proof of Proposition 5. (a) Start from the second stage. For any given rA ≥ rB derived in the first stage, each firm maximizes its profit with respect to its fee. For rA = rB = r, each firm has an incentive to undercut its rival's fee and to attract the entire market. In equilibrium, both firms charge zero fees, share the market by attracting half of the customers from the entire range of θ and enjoy πA ¼ π B ¼ 4r . For any given rA N rB derived in the first stage, firm A maximizes its profit by capturing the entire market and this is achieved by charging fA = fB − rA + rB as long as this fee is non-negative, that is, as long as fB ≥ rA − rB. If fB b rA − rB firm A charges fA =

f B −r A þ r B 0

if f B ≥ r A −r B if f B b r A −r B

Firm B's best response correspondence is 8 < fA f B ¼ RB ð f A Þ ¼ f A r A þ ðrA −r B Þ2 : 2rA −r B

if f A ≥ r A −r B if 0 ≤ f A b rA −r B ;

and it is derived in the same way as in Eq. (5), without the possibility of a negative fA. The two best response correspondences intersect when 



f A ðr A ; r B Þ ¼ 0 and f B ðr A ; r B Þ ¼

ðr A −r B Þ2 : 2r A −r B

Substituting the equilibrium fees into Eq. (1), we obtain θ ¼  1 r A −r B 2r A −r B . For rA N rB, we have θ b 2 and therefore firm B has a larger

market share than firm A. By substituting the equilibrium fees into the profit functions (Eqs. (2) and (3)), we obtain that πA ðr A ; r B Þ ¼

if r A ≥ − f A if r A ≤ − f A

The best response correspondence of firm B is derived as in case (ii) with one difference: for 0 ≤ f A ≤ 12 f B , we have that expression (A.1) is lower than expression (A.2) and therefore the profit function of firm B crosses the axes of rB, at most, twice. In order for both firms to operate fA in the market, ~ θ ∈ ð0; 1Þ. Since ~θ ¼ frBA − −r B and solving for rB, we find that rB b rA − fB + fA ⇒ W b rA − fB + fA which implies that rA N − fA. For rA b − fA firm B prefers to have no clientele and make zero profit by charging any rB ≥ rA − fB + fA, than to have some clientele and make losses. The two best response correspondences intersect when both firms share the market and the equilibrium rates are given by Eq. (8). Substituting Eq. (8) into Eq. (1) we observe that, in equilibrium, the indifference usage level is θ ¼ p1ffiffi2 ≈0:7071, thus firm A has a larger market share than firm B. Substituting Eq. (8) into Eqs. (2) and (3) we obtain the equilibrium profits:

πA

0 either because it would rather capture the entire market but it cannot charge a negative fee (this is the case when rA b 2rB or when rA ≥ 2rB and rA − 2rB ≤ fB ≤ rA − rB), or because it would rather share the market but its optimal response, given by Eq. (6), is negative (this is the case when rA ≥ 2rB and fB b rA − 2rB). Therefore, given that all prices must be non-negative and for rA N rB, firm A's best response correspondence is

r A ðr A −r B Þ2 2ð2r A −r B Þ2

r2

A and πB ðr A ; rB Þ ¼ 4rA −2r . B

In order to compare the case where firms can charge any price with the case where firms can charge only non-negative prices, we consider rates given with rA N 2rB ≥ 0. We compare each firm's profit when firms can charge only non-negative fees, denoted as πi⁎nf, with when they can charge any fee, denoted as πi⁎af. We have 4rA rB −r 2B 8ð2r A −r B Þ

f a f πp A −π A ¼

4r 2A r B −5r A r 2B þ2r 3B 8ð2r A −r B Þ2

N0

and

f a f πp ¼ B −π B

N 0 for rA ≥ rB. We observe that both firms enjoy a higher

profit when they can only charge non-negative fees. (b) Moving on to the first stage, we have to compare each firm's profit across three alternatives: when it charges i) the same rate as its rival, ii) a higher rate and iii) a lower rate than its rival. If in the first stage rA + rB = r, from the analysis of the second stage we know that π A ¼ πB ¼ 4r. If firm i charges a higher rate than its rival, from the analysis of the second stage we know
 r ðr −r Þ2 that πi r i ; r j ¼ i i j 2. If there is a limit on the level of the rate, 2ð2r i −r j Þ r, this function is maximized when firm i charges the maximum possible rate r. If firm i charges a lower rate than its rival, from the
 r 2j and analysis of the second stage we know that π i r i ; r j ¼ 4r j −2r i therefore it maximizes its profit by slightly undercutting. In this r case, we have πi → 2j . We conclude that for rj N 0 undercutting is always more profitable than charging the same rate. For r N 5:0489r j, firm i makes a higher profit when charging a higher rate than when undercutting. Summing up, firm i prefers to charge the highest possible rate when r N 5:0489r j , otherwise it prefers to undercut its rival. It follows that the two correspondences never intersect. ■ Proof of Proposition 6. (a) Start from the second stage. For any given fB ≥ fA, each firm maximizes its profit with respect to its rate. For fA = fB = f,

K. Griva, N. Vettas / International Journal of Industrial Organization 41 (2015) 30–41

both firms have an incentive to undercut their rival's rate in order to attract the entire market. In equilibrium, both firms charge zero rates, share the market equally, and enjoy πA⁎ = πB⁎ = f/2. For any fB N fA derived in the first stage, firm A has a convex profit function with respect to its rate and thus maximizes its profit when capturing the entire market with rA = rB + fB = fA. Firm B, which can never attract the entire market, has a concave profit function with respect to its rate. To derive firm B's best response B function we set ∂π ¼ 0 and solve with respect to rB, to find that ∂r B

the only acceptable solution is rB = W, where W is defined in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 the Proof of Proposition 4. For r A ≤ f A −4 f A f B þ 3f B , we have W ≤ 0 and so firm B charges a zero rate. Moreover, in order for firm B to share the market, the indifferent consumer's usage level must be less than 1 ( ~θb1 ), which implies that rA − fB + fA N rB. For rA ≤ fB − fA, firm B cannot charge a negative rate and as a result it is left with no customers, regardless of the non-negative rate it charges. To sum up, firm B's best response correspondence, given the rate of firm A, is:

r B ¼ RB ðr A Þ ¼

8 > > > : any rate N0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 f A −4 f A f B þ 3f B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 if f B −f A ≤ r A b f A −4 f A f B þ 3f B if 0 ≤r A bf B −f A :

if r A ≥

The two best response correspondences intersect at rA⁎( fA, fB) = fB − fA and rB⁎ = 0. By substituting the equilibrium rates into Eq. (1) we obtain θ⁎ = 1, thus firm A captures the entire market. We also substitute the equilibrium rates into the profit functions and obtain πA⁎( fA, fB) = ( fA + fB)/2 and πB⁎ = 0. Comparing these profits with the profits that firms make when they can charge negative rates (Eq. (A.3)), we observe that firm A now enjoys a higher profit. In contrast, firm B has a lower profit since it now has no customers and therefore has zero profit. (b) Proceeding to the first stage, we compare each firm's profit when it charges i) the same fee as its rival, ii) a higher fee, and iii) a fee lower than that of its rival. If fA = fB = f, it follows from the analysis of the second stage that πA⁎ = πB⁎ = f/2. If firm i charges a fee higher than that of its rival, from the analysis of the second stage we know that it is left with no customers and zero profit. If firm i charges a fee lower than that of its rival, then we know that πi⁎(fi, fj) = (fi + fj)/2 and therefore firm i maximizes its profit by undercutting its rival's fee. In this case, πi⁎ → fj. We conclude that undercutting is always more profitable than charging the same or a higher fee, as long as fj N 0, otherwise firm i charges any f N fj. In this case, the best response correspondences intersect

41

at fA⁎ = fB⁎ = 0. Subsequently, in the second stage we have rA⁎ = rB⁎ = 0. The two firms share the market equally, by attracting customers from the entire range of θ, and make zero profit. ■

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