Two Part pricing, Consumer Heterogeneity and Cournot Competition

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1 Two Part pricing, Consumer Heterogeneity and Cournot Competition Sissel JENSEN Lars SØRGARD Discussion paper 20/2001 Revised February 2004 Abstract We analyze two part tariffs in oligopoly, where each firm in addition to the standard quantity commitment also commits to a fixed fee prior to determination of unit prices. In the case of homogeneous consumers, Harrison and Kline (2001) show the the equilibrium involve marginal cost pricing and positive fixed fees, whatever the number of firms. We show that these results are reversed when we allow for heterogeneous consumers. In particular, we find that price per unit can exceed marginal costs, and the fixed fee can be below costs if the firms are forced to serve all consumers. We also show that two part tariffs may collapse, because each firm would rather commit to a traditional Cournot price system (zero fixed fee). Finally, some numerical examples illustrate that both firms serving both types of consumers can be an equilibrium outcome in duopoly in cases where the monopolist would serve only one type of consumers. JEL.class.no: L11, L13, D82 Keywords: Oligopoly, Two part tariff, Cournot We are grateful for comments from Petter Osmundsen and Patrick Rey. We are also indebted to participants at the Nordic Workshop on ICT-related research at Norwegian School of Economics and Business Administration in June 2001, EARIE meeting in Madrid in September 2002 and to participants at the doctoral seminar series at the Norwegian School of Economics and Business Administration for helpful comments. Institute for Research in Economics and Business Administration, Mailing address: Breiviksveien 40, N-5045 Bergen, Norway, sissel.jensen@snf.no Norwegian School of Economics and Business Administration. Mailing address: Helleveien 30, N-5045 Bergen, Norway, lars.sorgard@nhh.no

2 1 Introduction Nonlinear prices are common in many industries, and have been studied extensively in the economic literature. One motive for nonlinear prices is rent extraction. 1 As shown in Oi (1971), in a monopoly a single two part tariff is more effective when it comes to extracting consumer surplus than a uniform price. Harrison and Kline (2001) analyze two part tariffs in an oligopoly with strategic interaction and homogeneous consumers. In their paper, increased competition affects industry profit and tariff structure solely by reducing the fixed fee, and the tariffs are merely a modification of the tariffs described by Oi (1971) in a monopoly. It is not obvious that the same kind of tariff modifications applies to a setting with heterogeneous demand. Hence, the present model builds a more realistic set-up for a the study of two part tariffs by assuming two instead of one type of consumers. It turns out that our results differ in many respects both from the results in the Harrison and Kline (2001) oligopoly model with homogenous consumers as well as the results in the Oi (1971) monopoly model. Moreover, the set-up enables us to offer an explanation to the the introduction of two part tariffs with inclusive consumption alongside increased competition in the telecommunications industry. Our starting point is the framework developed in Harrison and Kline (2001). Each firm commits to a certain quantity, as in a traditional Cournot model. In addition, each firm sets its fixed fee while the unit prices are determined endogenously by market forces. Without the first assumption it not possible to escape a situation where a competing firm charges a uniform price and captures all consumers. The latter assumption is analogous to what is the case in a traditional Cournot model. Harrison and Kline (2001) extend the basic problem of charging a group of N identical consumers according to a two part tariff instead of a uniform unit price to a setting with competition, i.e., they extend the first part of the model in Oi (1971) to oligopoly. It is found that in equilibrium the unit price is set equal to marginal costs, and the fixed fee is positive for a given number of firms. We derive conclusions based on two different regulatory frameworks. First, no regulation is imposed on the firms. In the unregulated case firms may choose to serve only high demand consumers. In such a case the equilibrium outcome replicates the one shown in Harrison and Kline (2001), except there are now some consumers not being served. If both types of consumers are served, we find that the price per unit is above marginal costs. This is in line with Oi (1971), but departs from Harrison and Kline (2001). By using a numerical example we 1 Another motive for price discrimination is screening. The firm takes into account the unobserved heterogeneity of individual demand by using a menu of two part tariffs and thereby introducing self-selection. See Rochet and Stole (2000) for a guide to the screening literature and Stole (2001) for a comprehensive guide to the literature on price discrimination in models with competition. 1

3 show that there can be multiple equilibria. Moreover, it is shown that both firms serving both types of consumers can be an equilibrium duopoly outcome in cases where the monopolist would prefer to serve only one type of consumers. The driving force is that the rival, non-deviating firm supplies a given quantity which it is committed to sell, acting as a constraint on the deviating firm s price setting. Second, we assume that the firms operate under a regulatory contract forcing them to provide a universal service, i.e., they have to serve both consumer types. We then find that the fixed fees can be zero or even negative for a finite number of firms. This result departs from both Oi (1971) and Harrison and Kline (2001). In Oi (1971), if a monopoly is forced to cover the entire market, the fixed fee is strictly positive since it extract low demand consumers surpluses, which is positive if they consume a positive amount. Harrison and Kline (2001) obtains strictly positive fixed fees when the number of firms is finite. In fact, we show that firms can be better off committing to traditional Cournot competition where they can only charge a unit price. This contrasts the conclusions obtained in monopoly models on two part tariffs. An interpretation of a negative fixed fee can be that the fixed fee is positive, but below costs. This is what we observe in the cellular telephony industry where mobile phones are given away almost for free. 2 Otherwise, without any fixed costs, tariffs with a negative fixed fee may be difficult to implement and one must look for ways to get around the problem. Figure 1 below shows a pricing practice commonly used by telephony companies. Under a two part tariff with inclusive consumption consumers pay a fixed fee and are given a free consumption allowance, for subsequent consumption they pay a strictly positive unit price. If we instead let consumers pay for the included consumption according to the same unit price and coerce a two part tariff through the kink, the corresponding fixed fee would be negative. Hence, two part tariffs with inclusive consumption are very interesting in light of our result since they easily can, and frequently do, cover a negative fixed fee in a two part tariff. 3 Telecommunications is of interest to our model because many firms operates subject to a universal service obligation, and because it focuses on subscriber penetration rates as an indicator of market coverage. Hence, a regulatory mandate to serve all consumers is as plausible as a mandate to meet all demand. Although there are many difficulties surrounding empirical models with non- 2 In Norway for example, mobile phones have been sold at a price of NOK 1 each, while some retailers have received a payment of approximately NOK 2000 from the producer. The producer then incurs a loss of approximately NOK 2000 for each consumer it captures, and earns revenues on the same consumer from what he pays for the use of the mobile phone. Competition between Nintendo and Sony Playstation is another example. Buyers pay a fixed fee when they buy the game terminal, and the price on each game depends on the number of games that are developed. 3 To see this, note the dotted lines in Figure 1. If there were no inclusive consumption allowance, the fixed fee would have been negative. 2

4 GBP 80 Vodafone Anytime 200 (UK) BT Mobile Home Plan (UK) Orange Viva (DK) 20 PSfrag replacements Call minutes 20 Figure 1: A selection of tariffs with inclusive minutes (solid lines) linear pricing under competition, both Ivaldi and Martimort (1994) and?) support the notion that oligopoly firms implement a nonlinear outlay schedule. Nevertheless, the economic literature shows that nonlinear pricing may not be sustainable in oligopoly. For example, Mandy (1992) finds that in a traditional Bertrand oligopoly with homogeneous products where we allow the firms to set nonlinear prices all prices may collapse to a uniform price. The finding illustrates that, except for some special cases which he explores, some of the assumptions in the traditional oligopoly model have to be relaxed in order to make nonlinear prices sustainable in oligopoly. This has been done in the emerging literature on competitive nonlinear pricing. One extension is to introduce capacity constraints, as is done in Harrison and Kline (2001), Oren, Smith and Wilson (1983), Scotchmer (1985a) (1985a, 1985b) and Wilson (1993). Scotchmer (1985b) (1985a, 1985b) only considers existence when the number of firms becomes large, while both Oren et al. (1983) and Wilson (1993) assume that the firms predict the market shares of their rivals. Another extension of the traditional Bertrand model is to introduce product differentiation, see Calem and Spulber (1984), Castelli and Leporelli (1993), Economides and Wildman (1995), Shmanske (1991), and Young (1991). If future demand is to some extent stochastic, Hayes (1987) manage to sustain nonlinear pricing under competition by assuming that consumers are riskaverse and, hence, have a preference for two part tariffs. In the case of telephony, Miravete (2002) rules out risk aversion as a possible source of consumers biased 3

5 taste for flat rate tariffs. The article is organized as follows. In Section 2 we formulate our model using identical firms and characterize the outcome in each of the two subgames where either both types of consumers are served or only one. In Section 3 we explore the equilibrium outcomes with respect to pricing and market coverage of the game without regulatory intervention. We do so by asking whether one firm would profit if it deviates and serve high demand types only in the subgame where every other firm serve both types, and whether it would profit if it deviates and cover the entire market in the subgame where every other firm serve high demand consumers only. Section 4 focusses on the case when firms are mandated to provide universal service, i.e., when the firms by assumption serve both types of consumers. We characterize the pricing strategy, and analyze whether firms would rather commit to traditional Cournot price system (zero fixed fee). Lastly, our results are summarized in Figure 2 and Figure 4. Finally, Section 5 offers some concluding remarks. 2 Cournot competition with two part tariffs Consider a setup with k identical firms, k 2, supplying a homogeneous product to N consumers, firm i {1, 2,..., k} serving n i N consumers. The cost function is characterized by constant returns to scale, C (Q) = cq where c > 0 is the marginal cost and Q is output. For simplicity fixed costs are omitted from all measures. Let us just consider the case when the number of firms is exogenous and leave the question of entry outside the model. 4 Consumers preferences are defined by a quasi-linear utility function { u (q) T if they pay T and consume q units V = 0 if they do not buy, (1) The utility function is assumed to be increasing and strictly concave in q, u (0) = 0, lim q 0 u q (q) c, lim q u q (q) 0. For any tariff T = A + pq, where A is a fixed fee that is paid up-front and p is a unit price, utility maximization yields a downward sloping demand curve for each individual which is independent of income and therefore also of the fixed fee. Indirect utility gross of the fixed fee is V (p) = u(q(p)) pq(p), q(p) max q u(q, ) pq A, (2) V p = q(p). With quasilinear utility we can measure the indirect utility in monetary terms. Consumers choose to buy if they obtain a nonnegative net surplus at some firm i, 4 See Harrison and Kline (2001) on entry in this model. 4

6 that is, iff V (p i ) A i 0, i {1, 2,..., k}. They buy from the firm providing them with the highest surplus, i.e., they buy from firm i if V (p i ) A i > V (p j ) A j, (i, j {1, 2,..., k}, i j). Firms act to maximize profit by choosing a strategy s i = (Q i, A i ), with Q i > 0 for all i = 1, 2,..., k, and we assume that firms are able to commit to this strategy. The firm cannot exclude any consumer from buying. 2.1 Homogeneous consumers Consider first the case when the firms serve N identical consumers as a reference. This model is presented in Harrison and Kline (2001) and we only give a brief presentation of the set-up. 5 Assume that all firms charge the same fixed fee and the same unit price. When rival firms charge their consumers according to the tariff T i = A i + p i q a consumer is indifferent between buying from firm i and one of the other firms when the participation constraint is binding. If the firm leaves the consumer with additional surplus, it sacrifices profit. If there is at least one additional active firm where consumers would buy a strictly positive quantity, we expect that the participation constraint is binding, hence V (p i ) A i = V (p i ) A i. (3) Profit for firm i is given by Π i = n i A i + (p i c) Q i. (4) Since the fixed fee is a lump sum transfer from consumers to the firm, the unit price in firm i s tariff is adjusted in such a way that aggregate demand for firm i s product is equal to firm i s supply. Hence, the unit price is independent of the fixed fee. Whenever the fixed fee is positive, consumers will make all or nothing purchases at firm i. When firm i serves a total of n i consumers, the unit price is adjusted to satisfy the following market clearing condition Q i = n i q (p i ). (5) For a given strategy combination there exists a consumer equilibrium defining a consumer-price profile ((n 1,..., n k ), (p 1,..., p k )). This is formally defined in Harrison and Kline (2001). Although we define a firm s strategy in capacity and the fixed fee, from a consumer s point of view he chooses the quantity that maximizes his utility for a given A i and p i. The notion behind this reasoning is that it is a competitive equilibrium where a large number of consumers without market power trade, given the fixed fees and quantities from each firm. 5 It would be helpful for the reader to examine the paper by Harrison and Kline, which is the seminal paper that explores this model setup. They give a thorough treatment of the model and the assumptions that are needed in order to ensure that equilibrium exist. 5

7 Firm i maximizes profit subject to the condition that the unit prices charged by rival firms are adjusted to satisfy the market clearing condition and subject to voluntary participation. Let the aggregate supply by firms competing with firm i be given by Q J j i Q j. (6) The unit price p i charged by every other firm must satisfy the condition Q J = (N n i ) [q (p i )]. (7) The following Lemma states the equilibrium pricing strategies in a k-firm oligopoly with homogeneous demand. Lemma 1 (Harrison and Kline) Equilibrium pricing in a k-firm oligopoly with N identical consumers is given by { } cq(c) A i A = min V (c),, (k 1) ε (q(c)) p i p = c, (8) Q i Q = N k q(c). Lemma 1 is the result in Harrison and Kline (2001). They give a thorough treatment of Cournot competition with two part tariffs and a single consumer type, and they also guide the reader through all proofs. They show that the pricing described in Lemma 1 is a unique Nash equilibrium in pricing strategies for the game. In addition to the equilibrium with symmetric market shares, there also exist equilibria that are asymmetric in market shares. Calculations leading to Lemma 1 is given in appendix A. According to this Lemma, the fixed fee in the single-type case converges toward zero as the number of firms approaches infinity. The optimal tariff is a cost-plus-fixed-fee tariff. If firm i takes the number of consumers it serves as given, for any tariff charged by rival firms the reservation utility is defined as a constant and will not affect the optimization with respect to unit price. The problem then resembles the monopoly problem, and the marginal price is identical to that in a monopoly. To attract additional consumers from rival firms, firm i has to adjust the fixed fee. Hence, a marginal increase in market share affects firm i s profit via the fixed fee. Finding the profit maximizing strategy reduces to finding the optimal number of consumers to serve. 2.2 Heterogeneous consumers We now extend the model by assuming that there are two groups of consumers; low and high demand consumers. Given a single two part tariff, there are two 6

8 alternatives: (i) both types of consumers are served, or (ii) only the high demand type of consumers. In this Section we will characterize each of those two subgames, and discuss the equilibrium outcome of the overall game by checking for possible deviations Characterizing the subgames We assume that there are N consumers. Preferences are defined as before and heterogeneity is introduced by adding a type parameter, θ, into the utility function. Consumers with taste parameter θ 1 are in proportion λ and consumers with taste parameter θ 2 are in proportion (1 λ), θ 1 < θ 2. We will refer to the first group as type 1 consumers or low demand consumers and to the other group as type 2 consumers or high demand consumers. Hence { u (q, θl ) T if they pay T and consume q units, V = 0 if they do not buy, θ l = {θ 1, θ 2 }, u (q, θ 2 ) u (q, θ 1 ) and u q (q, θ 2 ) u q (q, θ 1 ) q (9) Consumers indirect utility and the two consumer types demand curves are then given by q l (p) q (p, θ l ) = max q u (q, θ l ) pq A, V (p, θ l ) = u (q l (p), θ l ) pq l (p), V p = q l (p), l = 1, 2. (10) Again, a consumer will buy from the firm leaving him largest surplus, and he will buy form firm i iff V (p i, θ l ) A i V (p j, θ l ) A j, (i, j {1, 2,..., k}, i j, l = 1, 2). When the two consumer types are charged the same tariff, a type 2 consumer obtains a surplus that is at least as large as the surplus a type 1 consumer obtains. Thus, if type 1 is able to obtain a nonnegative surplus, type 2 obtains a strictly positive surplus. If all firms charge the same tariff (apply identical strategies) we assume that all firms serve an equal share of each consumer type. Hence, if both types are served, firm i serves a number of consumers given by n i = N/k, and if low demand types are excluded the number is n i = (1 λ)n/k. This will be sufficient to ensure that there exists a consumer equilibrium defining a consumer-price profile ((n 1,..., n k ), (p 1,..., p k )) for this specific strategy combination when there is demand side heterogeneity. If there is at least two active firms, the relevant participation constraint in firm i s optimization problem is given by V (p i, θ l ) A i V (p j, θ l ) A j, l = 1, 2, j {1, 2..i 1, i + 1,...k}. (11) 7

9 Profit for firm i is given by Π i = n i A i + (p i c) Q i, (12) and the unit price is adjusted to satisfy the market clearing condition Q i = n i [λq 1 (p i ) + (1 λ) q 2 (p i )], (13) if both types are served. If only high demand consumers are served the market clearing condition is given by Q i = n i (1 λ) q 2 (p i ). (14) Assume now that all firms but firm i, charge the same fixed fee A i and choose the same quantity level Q i. Firm i maximizes profit subject to the condition that the unit prices charged by rival firms are adjusted to satisfy the market clearing condition and subject to voluntary participation. As is well known from monopoly, in some cases it is beneficial for a firm to exclude the type with low willingness to pay in order to exploit high demand types larger willingness to pay, and in other cases it is preferable to serve both types of consumers. Let either all firms but i serve only high demand consumers or let all other firms but i serve both consumer types. When the strategy by other firms is such that only the high demand consumer is able to obtain positive utility, the unit price p i charged by the firms must satisfy the market clearing condition Q J = (N n i ) (1 λ) q 2 (p i ). (15) If both types are served, the market clearing condition is Q J = (N n i ) [λq 1 (p i ) + (1 λ) q 2 (p i )]. (16) If all firms adopt a symmetric strategy, the resulting two part tariff, ((A 2 T T, p2 T T ) or (A12 T T, p12 T T )), is defined by adjusting for demand side heterogeneity in Lemma 1. This give us the following two lemmas Lemma 2 (Serve only type 2) In the subgame where low demand consumers are excluded, and where all firms adopt a symmetric strategy, the optimal corresponding two part tariff, (A 2 T T, p2 T T ), is given by } A 2 T T {V = min cq 2 (c, θ 2 ), (k 1) ε (q 2 ) p 2 T T = c (17) Q 2 T T = N k (1 λ)q 2 q 2 = q 2 (c) 8

10 Lemma 3 (Both types served) In the subgame where the entire market is covered, and where all firms adopt a symmetric strategy, the optimal corresponding two part tariff, (A 12 T T, p12 T T ), is given by { ( ) } A 12 T T = min V (p 12 T T, θ 1 1), (1 λ) q 2 q 1 k 1 q 1 p 12 T T q 1 ε(q 12 T T ) p 12 T T = c + (1 λ) [q 2 q 1 ] (λq 1 + (1 λ) q 2 ) Q 12 T T = N k (λq 1 + (1 λ) q 2 ) q l = q l (p 12 T T ), q l = q l (p12 T T ), l = 1, 2 (18) Lemma 2 and 3 are proved in appendix B. In order to check whether one of these two strategies also constitutes equilibrium strategies in this game we examine firm i s incentives to deviate from a proposed symmetric strategy when low demand types are excluded and served respectively. It turns out to be hard to obtain closed form solutions when we assess the firm s incentive to deviate from an equilibrium with symmetric tariffs given by Lemma 2 or Lemma 3, and with symmetric market shares. We have therefore chosen to present some numerical examples to illustrate possible equilibrium outcomes in the overall game, i.e. whether the subgame characterized in Lemma 2 or the subgame characterized in 3 is the equilibrium outcome. To secure the existence of a unique equilibrium we use a numerical example with a quadratic utility function as well as constant marginal costs. 3 The unregulated case To simplify the problem, let us consider a duopoly with firm a and firm b, instead of an oligopoly setting. Also, from now and onwards we focus on a specific case when consumers preferences are represented by a quadratic utility function. We let the reservation utility be zero for both consumers. V = θ l q 1 2 q2 T, l = 1, 2, if they pay T and consume q units, otherwise they obtain zero utility. Each consumer has a linear demand function q l = θ l p, l = 1, 2. Letting θ λθ 1 + (1 λ) θ 2 θ 1, expected demand is λq 1 + (1 λ) q 2 = θ p. The indirect utility exclusive of the fixed fee for a consumer paying a unit price of p is V (p, θ l ) = 1 2 (θ l p) 2, l = 1, 2. Because we are interested in how equilibrium strategies are affected by heterogeneity in demand, the example is simplified by letting θ 1 = 1 and c = 1 2. Increased demand side heterogeneity is captured by variations in λ and θ 2. Large heterogeneity can then come about either by an increase in the number of type 2 consumers (λ decreases), or because a type 2 consumer has larger willingness to pay relative to a type 1 consumer (θ 2 increases). Hence, increased demand side heterogeneity is captured by an increase in θ. 9

11 Consider now the two equilibrium candidates given if firms announces identical tariffs and serve the same customer base. In the first equilibrium candidate, both consumer types are served with a tariff (A 12 T T, p12 T T ) and each firm earns a profit per consumer π 12 T T. In the second equilibrium candidate, low demand consumers are excluded from making purchases and type 2 is served with a tariff (A 2 T T, c). We assume that the firms have equal market shares, i.e., n a = n b = 1 N 2 or n a = n b = 1(1 λ)n. Each firm earns a profit per consumer given by 2 π2 T T or. Expected profit in each of the two cases is π 12 T T and Π 12 T T = N 2 π12 T T (19) Π 2 T T = N 2 (1 λ) π2 T T. (20) They would generate the same profit in each of the symmetric cases when Π 12 T T = Π2 T T, i.e., if λ λ = θ 2+ (4θ 2 2 3)(4θ2 2 8θ 2+5) 8(θ 2 1) 2 (21) λ is also the monopolist cutoff value, if λ < λ it serves only type 2 consumers, while if λ > λ it serves both types of consumers. If demand side heterogeneity is not too large, a duopoly is able to extract all surplus from type 1 when both consumer types are served, given that λ (2θ 2 2 1)/(2θ 2 2). Since λ > (2θ 2 2 1)/(2θ 2 2), a duopoly that is maximizing joint profit would prefer to exclude low demand types even if it were able to extract all surplus from low demand types. However, as we will demonstrate it is difficult to sustain a duopoly outcome which excludes low demand types from the market. Let us use λ as a reference point for our numerical examples. If λ < λ, demand side heterogeneity is large and we conjecture that the firms would tend to exclude type 1. Conversely, we conjecture that each firm would tend to serve both types of consumers if λ > λ. Note, however, that it is not at all obvious that the cutoff point is the same in duopoly as in monopoly. A monopoly firm can extract surplus from high demand segments by deliberately excluding low demand segments, because a monopoly can choose to design a tariff low demand segments would never accept. A duopoly firm, however, can not prevent that a competing firm applies a tariff that low demand segments will accept. Such a tariff will be strictly preferred by high demand segments. 3.1 Low demand types excluded First, let us consider the symmetric pricing strategy where both firms serve only type 2, and the firms tariffs are given by (A 2 T T, c). Type 1 is excluded and the 10

12 firms extract the entire surplus from type 2 via the fixed fee, and A 2 T T = V (c, θ 2). The two firms split the base of type 2 consumers equally, n a = n b = (1 λ) N. 2 Would a unilateral deviation from an outcome where both firms serve only type 2 be profitable? One firm, say firm a, could deviate by setting a tariff that type 1 is just willing to accept and capture all type 1 consumers, λn. However, since low demand consumers derive nonnegative surplus, high demand consumers will derive strictly positive surplus by switching to low demand types tariff. The deviating firm will then serve a mix of type 1 and type 2 consumers. It will serve all type 1 consumers and more than half of all type 2, but not all since the other firm holds a capacity which it will sell. Since firm a captures some of the high demand types as well, this tends to make such a deviation profitable. However, profit per consumer is reduced. Let the deviating firm choose a strategy 12 ( Q T T, Ã12 T T ), or equivalently charge a tariff (Ã12 T T, p12 T T ) in order to maximize profit subject to individual rationality and firm b s strategy (Q 2 T T, A2 T T ). The problem is to maximize Π 12 subject to V T T = [N n b ] Ã12 T T + (22) 12 ( p T T c ) [Nλ q 1 + (N (1 λ) n b ) q 2 ] V ( p 12 T T, 1) Ã12 T T (23) ) 12 ( p T T, θ 2 V 12 ( p T T, 1 ) = V ( p ) 2 T T, θ 2 V (c, θ2 ) (24) N (1 λ) q 2 2 n b q 2 (25) where q i = q i ( p 12 T T ), l = 1, 2, q 2 = q 2 ( p 2 T T ), and q 2 = q 2 (c). Firm b, the nondeviating firm, will then lose type 2 consumers. This leads to a price reduction at firm b in order to restore individual rationality, the unit price falls to p 2 T T < c. Since a unit price reduction in turn leads to an increase in a type 2 consumer s demand, q 2 ( p 2 T T ) > q 2(c), the capacity supplied by firm b becomes insufficient to serve all type 2 consumers, and n b < (1 λ)n/2 is adjusted to restore market clearing at firm b. Formally, the individual rationality constraint (24) and the market clearing condition (25) jointly determine firm b s share of type 2 consumers as a function of firm a s strategy, n b = n b ( p 2 ( p 12 T T )). Although firm a obtains lower profit per consumer when it deviates, it expands its market. When λ is low, or θ 2 is high, the market expansion effect is less likely to cover the per-consumer-loss in profit. In that case there are few type 1 consumers to serve and expected profit per consumer is significantly lower when firm a deviates. Conversely, we expect that a deviation is profitable when demand side heterogeneity is low. For λ close to λ the expected revenue per consumer is identical and we therefore conjecture that it is profitable to deviate. In Table 1 we have reported some numerical examples for N = 100 and c = 1. Hence, 2 p2 T T = 1 and 2 A2 T T = V (c, θ 2). The results in Table 1 confirm 11

13 our conjecture. Note that when λ < λ, the monopolist would serve only type 2 consumers. This particular case therefore suggests that a Nash equilibrium in a duopoly where both firms serve only one type of consumers to a large extent coincides with the case where a monopolist prefers to serve only one type of consumers. However, there are parameter values for which low demand types would be served in a duopoly, whereas they would be excluded in a monopoly. Table 1: Deviation from a symmetric equilibrium where type 1 is excluded. θ 2 λ λ p 12 T T à 12 T T p 2 n T T n b b N(1 λ) Π 2 T T Π 12 T T Both types served Second, let us consider the equilibrium candidate where both firms serve both types of consumers, and the firms tariffs are given by (A 12 T T, p12 T T ). Then, type 2 consumers enjoy positive surplus, and type 1 consumers receive their reservation utility, which is non-negative. Firms have equal market shares and serve N/2 each, and they serve low types and high types in proportion λ and (1 λ). Consider a unilateral deviation by firm a, (Ã2 T T, p2 T T ), and keep the strategy for firm b fixed (Q 12 T T, A12 T T ). In this case firm a can deviate by using one of two strategies: (i) aim for all type two consumers n a = N(1 λ), but leave them a positive surplus. In that case it sets Ã2 T T < V (c, θ 2); (ii) knowing that firm b has a limited capacity, firm a could act as a monopoly on any residual demand. He will then serve less than the entire pool of type 2 consumers, n a < N(1 λ), but extract all surplus à 2 T T = V (c, θ 2). 12

14 Consider the first strategy. Firm a announces a tariff (Ã2 T T, c) that is strictly preferred by type 2 consumers. It will extract as much as possible from type 2 consumers via the fixed fee and will maximize Π 2 T T = N (1 λ) Ã2 T T (26) subject to ) V (c, θ 2 ) Ã2 12 T T V ( p T T, θ 2 A 12 T T (27) N (1 λ) q 2 ) Nλ q 1 (28) where q i = q i (p 12 T T ), (l = 1, 2), and q 1 = q 1 ( p 12 T T ). The unit price p12 T T is adjusted to account for the fact that firm b is now left with only type 1 consumers instead of a mix of type 1 and type 2. Given that type 1 consumers receive exactly their reservation utility, the unit price that clears the market at firm b cannot exceed p 12 T T, instead, type 1 consumers are rationed at firm b. Hence, 0 < p12 T T < p12 T T. is low as well in order to restore market clearing at firm is also low in this case. The more intensely firms compete, either via a low fixed fee or a low unit price, the more binding is the restriction on Ã2 T T. This suggests that in duopoly an outcome where both firms serve both types of consumers can be an equilibrium outcome in situations where a monopolist would have preferred to serve only one type of consumers. In our numerical example, the second effect always dominates the first and a deviation is never profitable. In Table 2 we have reported some numerical examples, again using N = 100, and This restricts the fixed fee in (27), which in turn will restrict the profitability earned on type 2 consumers. From (26) it would seem that a deviation is profitable when λ is small. However, when λ is small, p 12 T T b. Hence, Ã 2 T T c = 1, hence p12 2 T T > c, A12 T T = V (p12 T T, 1). Note that symmetric two part tariffs as given in Lemma 3 may imply a negative fixed fee. In Proposition 2 in section 4, we show that uniform pricing dominates a two part tariff with a negative fixed fee, hence we apply the result in Proposition 2 in the following table. The other possible deviation strategy in this situation was for firm a to act as a monopoly on any residual demand from type 2. This time, consider a deviation where firm a announces a tariff that extracts all surplus from type 2, (V (c, θ 2 ), c). Type 2 enjoys positive surplus by switching to firm b s tariff. Hence, type 2 consumers will crowd out type 1 consumers at firm b since capacity at firm 1 is insufficient to meet all demand. Firm a earns monopoly profit on each type 2 consumer it serves and aggregate profit is given by Π 2 T T = [N (1 λ) n b] V (c, θ 2 ) (29) where n b is the number of type 2 consumers that can be served by firm b. Type 2 is indifferent between the two firms tariffs when he receives zero surplus. Hence, 13

15 Table 2: Deviation from a symmetric equilibrium (Q 12 T T, A12 T T ), the fixed fee in type 2 s tariff is restricted. θ 2 λ λ p 12 T T A 12 T T p 2 T T Ã 2 T T V (c, θ 2 ) p 12 T T Π 12 T T Π 2 T T the unit price in firm b s tariff must be adjusted in order to restore individual rationality for type 2, p 2 T T. V ( p 2 T T, θ ) 2 A 12 T T 0 (30) N (λq (1 λ) q 2 ) n b q 2 (31) N (1 λ) n b (32) This time, firm b is left with type 2 consumers only, instead of with a mix of type 1 and type 2. Again, we would have thought it is profitable to deviate when λ is small. But now, when λ is small, the fixed fee A 12 T T is low. And therefore, type 2 consumers will gain considerably if they switch to firm b. Hence, the unit price p 2 T T is high and demand from type 2 is restricted. This means that q2 T T is low and that n b is large in order to restore market clearing. In Table 3 we report some numerical examples, still using N = 100 and c = 1. Again, we apply Proposition 2 and use uniform Cournot pricing instead 2 of a negative fixed fee. As shown, we find no examples where such a deviation is profitable. Again, the fact that the non-deviating firm has committed itself to sell a certain quantity acts as a constraint on the deviating firm s behavior. If there are few type 2 consumers, the non-deviating firm would serve them all and the deviating firm would have no residual demand. If there are many type 2 consumers, the price per unit would be close to marginal costs. If so, there is a limited scope for the deviating firm to generate additional consumer surplus from type 2 by setting price per unit equal to marginal costs. 14

16 Table 3: Deviation from a symmetric equilibrium (Q 12 T T, A12 T T ), acting as a monopoly on the residual demand from type 2. θ 2 λ λ p 12 T T A 12 T T p 2 T T Â 2 T T n b N (1 λ) Π 12 T T ˆΠ 2 T T Figure 2 summarizes our conclusions in this section. The north west part of the diagram characterizes a case with low heterogeneity, while the south east part characterizes a case with large heterogeneity. In region A and B the unique equilibrium will be that the entire market is covered and the firms apply a two part pricing strategy. A monopoly would exclude low demand consumers and serve only high demand consumers if the parameter values lies in region B, whereas a monopoly would serve both consumers in region A. In region C and D we have multiple equilibria. If only high demand types are served the optimal strategy is to apply a two part pricing arrangement. If the entire market is covered firms will revert to uniform pricing under large heterogeneity, in region D, and apply a two part pricing arrangement in region C. Note that in region C and D a monopoly firm would always choose to exclude the low demand types. 15

17 1 0.8 A B 0.6 λ 0.4 D PSfrag replacements 0.2 C θ 2 Figure 2: Equilibrium outcomes under free market coverage 4 Pricing under a universal service obligation Let us keep the description of the demand side from the previous section, but say that there are k active firms, k 2. Assume now that all firms are obliged to serve both types, i.e., to provide universal service. This could be due to some institutional restrictions, forcing them to provide a universal service. Given such a restriction, which combination of fixed fee and price per unit would each firm choose? Proposition 1 In a regulatory regime in which all firms are obliged to serve both types of consumers, and in which each firm sets a two part tariff, for any pair of parameter values such that (i) 0 λ λ 4θ 2 5 4θ 2 4, or (ii) k > k 1 then A 12 T T < 0 and p12 = λ + (1 λ)θ p12 T T p 12 = p 12 T T > c. 2(θ 2 1)(1 λ), > c. Otherwise, A12 T T > 0 and The critical values λ and k are derived in the appendix. First, we see that each firm would set a price per unit that exceeds marginal costs (λ+(1 λ)θ 2 1). In contrast, Harrison and Kline (2001) found that each firm would set a price per unit equal to marginal costs. Obviously, the extension of the model from one to two types of consumers explains the change in the result. It is well known from a monopoly model that a firm serving two types of consumers with one two part tariff should let the unit price exceed marginal costs, see Oi (1971). By doing so it is able to extract more profits from the high demand consumer, and this 16

18 outweighs the loss in profit extraction from the low demand consumer as long as the price-cost margin is not above a certain threshold level. The price-cost margin is higher the larger the difference between the consumer types (θ 1 versus θ 2 ), and the larger the proportion of the high demand consumers (λ approaches zero). This is natural, since a large difference between those two groups of consumers would lead to a relatively high price-cost margin to extract profits from the larger group. Second, note that the price-cost margin is not influenced by the number of firms. At first glance, this may come as a surprise. Why do they not compete on prices? The reason is that they compete on access prices, not prices per unit. The prices per unit are set to balance the revenues from the two consumer groups, after they have competed on fixed fees to attract consumers. Note that our result is in line with the result in Harrison and Kline (2001), where the price per unit is always equal to marginal costs since the unit price in both cases just replicates the monopoly price. Third, we see that each firm s fixed fee can be set below the fixed cost of serving a consumer (which is normalized to zero in our setting). In contrast, Harrison and Kline (2001) found that the fixed fee is always above costs, but approaches costs when the number of firms approaches infinity. In their setting, as well as in ours, profits approach zero when the number of firms approaches infinity. But the fact that we have a positive price-cost margin, implies that the fixed fee is competed away even for a finite number of firms. In fact, if the demand side heterogeneity is sufficiently large, the fixed fee is competed away even in a duopoly. Obviously, the existence of many firms would lead to fierce competition on fixed fees. But even with two firms, fixed fees can be negative if the fraction of the high type consumer is large or when the difference in consumers type is large. In such a case the price per unit is high, to extract profits from the large consumer. Then the fixed fee is low even in a monopoly setting, and competed away in a duopoly setting. Hence, an obvious question is whether the firm would have been better off constraining its tariff policy to uniform pricing. What, then, if the firm sets a fixed fee equal to zero rather than a negative fixed fee? It can then be shown that the following would emerge as equilibrium outcomes Proposition 2 In a regulatory regime in which all firms are obliged to serve both types of consumers, and in which each firm can choose between a two part tariff and uniform pricing (fixed fee equal to zero), each firm chooses a uniform price if the fixed fee in a two part tariff would be negative (see the previous Proposition), where p 12 UP p 12 UP > p12 < p12 T T. T T. Otherwise, it chooses a two part tariff with A12 T T > 0 and First, we see that as long as the fixed fee is above costs in a setting with a two part tariff, the firm would set a two part tariff rather than restrict its pricing 17

19 policy to a uniform price. A uniform price, which equals the traditional Cournot price, would in that case be higher than the unit price in a two part tariff. This suggests that a firm would find it profitable to deviate from an outcome where both firms set a uniform price. It could deviate by setting a lower price per unit, and extract the gross consumer surplus it generates through a positive fixed fee. Therefore, we would expect that the firms would end up with a two part tariff with a positive fixed fee. Second, we see that each firm would choose a uniform price if the alternative is that both firms set a two part tariff with a negative fixed fee. To understand this, note that in such a case the price per unit in a two part tariff is higher than the traditional Cournot price (a uniform price). In our model, the firms compete in utility levels. Then if other firms hold a high unit price and generate consumer surplus via a negative fixed fee, it will be profitable to match other firms offer by restricting the fixed fee to zero and lowering the price per unit, thereby increasing consumer surplus. Unit price Cournot price Two-part tariff Unit cost PSfrag replacements Number of firms Figure 3: Price per unit Note that competition between the firms leads to a low price per unit: The equilibrium outcome is a uniform (Cournot) price if that price per unit is lower than the price per unit in a two part tariff, and vice versa. This is illustrated in Figure 3, where the solid lines show the price per unit in equilibrium. As explained above, in some instances the institutional setting is such that the firms are forced to set a fixed fee. In other instances, though, firms are more flexible. If the choice is either to set a negative fixed fee and a relative high price per unit or a low uniform price, each firm may end up choosing the 18

20 latter price system because that would generate a larger sale and thereby a larger profit. This suggests that there is no conflict between public policy and private incentives concerning the choice of tariff structure. Each firm has incentive to choose the tariff structure with the lowest price per unit, which is beneficial for consumers and leads to only a limited dead weight loss. Figure 4 below summarizes the conclusions provided in this section. If consumer heterogeneity is sufficiently low the firms will apply two part tariffs, in region D and E. In region D the fixed fee will extract a low demand type s entire surplus, while the fixed fee is positive but lower than a low demand type s surplus if heterogeneity is more moderate, in region E. In region F the firms prefer uniform Cournot pricing over two part tariffs D E 0.6 λ 0.4 F PSfrag replacements θ 2 Figure 4: Tariffs under mandatory market coverage 5 Concluding remarks Harrison and Kline (2001) have shown how we can extend the traditional Cournot model to a setting with not only a unit price, but also a fixed fee. They found that each firm sets a price per unit equal to marginal costs, and a positive fixed fee that approaches zero only when the number of firms becomes large. Thus, we extend their model from one to two types of consumers, i.e., from homogeneous to heterogeneous demand. It turns out that the conclusions in Harrison and Kline (2001) are not robust to such an extension. We show that there might be multiple Nash equilibria. First, both firms serving only one type of consumers can be an 19

21 equilibrium outcome. Numerical examples suggest that this equilibrium outcome coincides with the cases where the monopolist chooses to serve only one type of consumers. Second, we find that both firms serving both types of consumers can be an equilibrium outcome. In fact, we find no numerical examples where the firms would deviate from such an outcome. The intuition is that the rival, non-deviating firm s given quantity acts as a constraint on the deviating firm s behavior. If we assume that firms are imposed a universal service obligation, we find that the price per unit exceeds marginal costs and that the fixed fee can be negative. The industry profit amounts to the revenues made through the price-cost margin and the fixed fee. Competition tends to reduce profit when the number of firms increases. Hence, if the price-cost margin is positive, the fixed fees will become negative at some point. When the mark-up is large, the fixed fee can become negative even with a limited number of firms. The mark-up tends to be large when the demand side heterogeneity is large, hence, the model may help us explaining the practice of subsidizing the the price of mobile phones or below cost prices of game-stations. If the firms can choose between a traditional Cournot pricing (a uniform price) and a two part tariff, we show that they may choose a uniform price. The analysis show that competition can lead firms to serve consumers with smaller willingness to pay, and thus, competition increases the share of the market that is served. Under competition, firms serve both small and large consumers, whereas a monopoly might choose to concentrate solely on high demand segments. This happens for two reasons. First, since competition tends to reduce the fixed fee, it is more likely that low demand consumers finds it individual rational to buy. Second, given that other firms choose to serve the high demand segment only, it may be profitable for a single firm to deviate and serve both types. On the other hand, given that other firms choose to serve both segments, it is difficult for a single firm to deviate and target high demand consumers only. As long as competition is not too fierce in terms of the number of firms, and the demand side is homogeneous, Harrison and Kline (2001) demonstrates that the firms will apply two part tariffs in a similar way as a monopoly would. Especially, as long as there is some scope for positive industry profit, it is sufficient to use the fixed fee to extract consumer surplus. Our paper shows that a single two part tariff might not be a sufficient instrument with heterogeneous consumers; a positive fixed fee may not be sustainable, and it may be difficult to serve high demand segments only. 20

22 Appendix A Calculations leading to Lemma 1 Assuming N identical consumers. Given that firm i serves a number n i when it charges a tariff (p i, A i ), and that all other firms charge identical tariffs (p i, A i ), firm i maximization problem reduces to maximizing profit subject to voluntary participation and market clearing. Π i = n i A i + (p i c)q i (A.1) A i = V (p i ) V (p i ) + A i Q J = (N n i )q(p i ) Q i = n i q(p i ) (A.2) Choosing Q i knowing that the market clearing condition must be satisfied amounts to maximizing profit wrt p i, holding A i, Q J and A i constant. Hence (p i c) dq(p i) dp = 0 p i = c (A.3) Now, given that p i = c is optimal, the firm determines the size of the fixed fee A i by determining the optimal number of consumers to serve. Hence, under the constraints of voluntary participation and market clearing, firm i maximizes profit wrt n i, which yields dπ i dn i = V (p i ) V (p i ) + A i + n i [ V p ] dp i = 0 (A.4) dn i Substituting from the participation constraint in (A.2), and differentiating the market-clearing condition Q J = (N n i )q(p i ) holding Q J fixed, enable us to state the following condition A i = n [ / i p i q p q(p i )p (p ] i) i N n i q(p i ) (A.5) Further, let firms have equal market shares, n i = N/k. When all firms charge the same unit price, p i = p i = c, and the participation constraint binds we get A i = A i A. Evaluating the condition in (A.5) for p = c, keeping in mind that consumer surplus must be nonnegative, gives the following expression for the fixed fee A = min { V (c), } cq(c) (k 1) ε(q(c)) This verify the statements in Lemma (A.6)