Second-degree Price Discrimination in the Presence of Positive Network Effects

Transcription

1 Second-degree Price Discrimination in the Presence of Positive Network Effects Gergely Csorba Central European University February 14, 2003 Abstract This paper uses tools provided by lattice theory to describe and solve the second-degree price discrimination problem faced by a monopolist seller of a network good. We build a general model to demonstrate that positive network effects and asymmetric information together will lead to a downward distortion for all consumers in the quantities provided. Despite the overall downward distortion result, positive network effects lead to higher consumption levels than in the standard model without network effects. Keywords: network effects, price discrimination, contracting with externalities, nonlinear pricing, supermodularity JEL Classification: D42, D62, D82, L12 The author thanks Attila Ambrus and Peter Benczur for useful discussions and comments. Postal address: Central European University, Economics Department, Nador u. 9, 1051 Budapest, Hungary. address: cphcsg01@phd.ceu.hu 1

2 1 Introduction This paper derives a general model of second-degree price discrimination in the presence of positive network effects, and examines how their presence changes the classical results of the literature. 1 Positive network effects are present if an economic agent s utility derived from the consumption of the good is positively affected by the consumption level (or number) of other agents consuming the same or compatible products. These effects are common to arise in various modern industries, like telecommunications, hardware and software, and these industries are similar in the properties that they are highly concentrated and firms are charging different nonlinear tariffs. Our goal in this paper is to use the tools provided by lattice theory to describe the problem faced by a monopolist seller of a network good, and to give a complete characterization of the optimal contracts it can use. We build a unifying framework to combine two well-known results. The first is one of the main conclusions in second-degree price discrimination models, namely that the incentive problem due to information asymmetry makes the monopoly distort the quantity supplied to consumers with smaller willingness to pay for the good, but makes no distortion at the top : consumers with the largest willingness to pay are provided with the first-best optimal quantities. The second result is due to the externality literature: 2 once a consumer s utility is not only the function of his own consumption level, but of the others consumption levels as well, all economic agents end up with socially suboptimal quantities. Network effects, which are generally assumed to be positive, result then in underconsumption of the network good for all consumers. Our main aim is to show that if asymmetric information and positive network effects are both present, these two results reinforce each other, so there will be a downward distortion for all consumers in the quantities provided. Despite the overall downward distortion result, we find that the presence of positive network effects leads to higher consumption levels for all consumers than in the standard models without network effects. To characterize the optimal contracts, we build heavily on the theory of optimizing supermodular functions, pioneered by Topkis (1978), Vives (1990) 1 When I mention classical second-degree price discrimination results, I refer to the works of Mussa and Rosen (1978) and Maskin and Riley (1984), summarized for example in Fudenberg and Tirole (1991) and Laffont and Martimort (2002). 2 For a general overview on externalities, see for example Laffont (1988). 2

3 and Milgrom and Shannon (1994). 3 The supermodularity assumption plays a key role in our analysis for two reasons. First, the supermodularity of a utility (or payoff) function is equivalent of the goods (or actions) being strategic complements, which is exactly the case in the presence of positive network effects. Second, with supermodular functions the optimal solution is a monotone function of the model parameters, which property will be very useful in ordering the resulting allocations. The old literature on network effects, which was basically on telecommunication pricing, focused on the question whether the network size would be the one that maximizes social welfare. 4 It is found that in perfect competition the network effects cannot be fully internalized, and a monopoly can perform better, since it controls both the price and the quantity and may use crosssubsidization policies more effectively. 5 Incentive-compatible non-linear tariffs were first studied by Oren et al. (1982), but in this seminal model the choice of the usage level of the network (how much quantity is supplied to the consumers) and of network size (how many consumers will join the network) were still separated. The new literature on network effects considered mainly homogeneous types of consumers and concentrated on the multiple equilibria problem created by different (rational) expectations. 6 Models with heterogeneous types of consumers were only recently used by Fudenberg and Tirole (2000) and Ellison and Fudenberg (2001). However, these models analyze the effectivity of dynamic strategies, like entry deterrence or software upgrades, in the presence of network effects, hence they concentrate on intertemporal discrimination aspects. Recently, Hahn (2003) uses tools provided by standard incentive theory to study the role of call and network externalities in a telecommunication example of nonlinear pricing. He establishes the result that in equilibrium all types end up with suboptimal quantities, so the no distortion on the top result does not hold. Nevertheless, since he works with a special utility structure, which is separable in call and network externalities, he attributes this result to the 3 Summarizing works of these seminal results are Topkis (1998) and Vives (1999). 4 This classification is based on Liebowitz and Margolis (2002). 5 Similar conclusions have been derived in the macroeconomics literature of imperfect competition, for example in Cooper and John (1988, p. 454): a demand externality may arise, though, in market structures where agent require information on both prices and quantities in making choices [...] In these cases quantities matter to individual decision makers, and prices do not completely decentralize allocations. 6 Farrell and Saloner (1985) and Katz and Shapiro (1985) were the first to raise this problem. 3

4 existence of call externalities. Two works closely related to ours are Segal (1999, 2001), in which he studies a general model of contracting with externalities and characterizes the nature of the arising inefficiencies. When externalities are positive, he finds that the resulting equilibrium allocation is smaller than the socially efficient one. However, his results rely on two restrictive assumptions: first, consumers are identical and second, total welfare depends only on aggregate trade, and not on its allocation across consumers. A contribution of this paper is to show that the assumption about homogeneous types (and hence symmetric information) is not needed to achieve this underconsumption result if externalities are positive. The rest of this paper is organized as follows. Section 2 introduces the main setup of the model. In Section 3 we analyze a model of two types to demonstrate the main characteristics of the incentive contracts in the presence of positive network effects. Section 4 demonstrates that the results of the previous chapter generalize for the continuous type case. Finally, Section 5 concludes and discusses further extensions of the model. 2 The model Consider a monopoly selling a good exhibiting positive network effects, which can be produced at a constant marginal cost c. We assume that the network goods sold to the consumers are perfectly compatible with each other. Consumers have heterogenous preferences for the good, and this heterogeneity is captured by the one-dimensional parameter. Suppose that each consumer of type i has a utility function U( i ) = i V (q i, q) t i, where q i is the amount of the network good he consumes, q is the total amount of network good in the economy (network size), and t i is the tariff charged for q i by the monopoly. Alternatively, we could give another interpretation for this problem: the monopoly is selling at most one network good to each consumer, and the goods differ in their quality q i. If in this case we normalize the mass of consumers to 1, q can be seen as average quality level in the network. We assume that consumer types are independently distributed by a given cumulative distribution function F (), and we require a continuum of consumers in each type. The latter assumption ensures that a single consumer s contribu- 4

5 tion to the network is negligible, so a single deviation does not affect network size. Suppose that there are no externalities on non-traders: for all q, V (0, q) = 0, so the outside option is zero for all consumers. our analysis to pure network goods, i.e. differ from 0. However, we do not restrict the stand-alone utility V (q, 0) may We assume that V ( ) is twice continuously differentiable and that V 1 > 0, V 11 0, so the marginal utility of individual consumption level is positive and increasing. 7 The positivity of network effects is captured by following two key assumptions. First, V 2 > 0, so the marginal utility of network size is always positive. Second, the individual consumption levels of each consumer are strategic complements, which is equivalent of stating that V (q i, q i ) has increasing differences on (q i, q i ). 8 This means that each consumer s marginal utility of individual consumption increases as the consumption level of an other consumer increases. 9 These assumptions imply that individual and total consumption levels are strategic complements as well, so V The monopoly does not observe the consumers type, or even if he does, it is prohibited from directly discriminating among consumers. Thus it has to offer a menu of contracts {(q, t)}, among which consumers self-select. We assume that there is no shutdown of inefficient types, i.e. the optimal tariff structure is determined such that all consumers have incentives to buy some positive amount of the good. However, we do not restrict the tariff charged by the monopoly to be positive, so cross-subsidization policies are possible. Naturally, as many models with network effects show, consumers expectations about others choices crucially influence their behavior, which may result in multiple equilibria. However, since consumers individual consumption levels are strategic complements, which induces a so called supermodular game, there will be a (rationally expected) equilibrium that Pareto-dominates the others. 10 We assume that once the monopoly offers the menu of contracts {(q, t)}, consumers coordinate on this equilibrium (Pareto-criterion), or alternatively we concentrate on non-unique implementation. 7 In all of this paper, lower indexes refer to partial derivatives of the function V ( ) in its respective argument. Second, the word increasing (decreasing) and bigger (smaller) will be used in the weak sense. 8 A function g(x, t) has increasing differences in (x, t) if g(x, t ) g(x, t) is increasing in x for all t > t. 9 Segal (1999, 2001) defines positive effects (he calls them externalities) in a similar way. 10 See Milgrom and Roberts (1990). 5

6 3 A model of discrete types Suppose that { 1, 2 } so that 1 < 2, and the probability for a consumer being of a type 1 and 2 is p and (1 p), respectively. To facilitate notations, let us denote ( 2 1 ) by. The monopoly will offer two contracts, (q 1, t 1 ) and (q 2, t 2 ). In all of this paper we restrict ourselves to deterministic mechanisms, since by the Revelation Principle the monopoly cannot do better by offering more complex contracts. If every consumer selects the menu devoted to him, as it will be the case in equilibrium, total quantity purchased will be q = pq 1 + (1 p)q 2. As we will see later, the optimal quantity schedule determines the optimal tariff schedule, so to key decision variable for the monopoly is q = (q 1, q 2 ) Q = R 2, the latter being a lattice. 11 The function V ( ) is defined on this lattice Q, and since we have assumed that q 1 and q 2 are strategic complements, this implies that V ( ) is supermodular in (q 1, q 2 ) on Q The first-best case As a benchmark case, suppose that the monopoly knows each consumer s type, and based on this information it can discriminate among them. Then the contracts will be designed such that in equilibrium each type realizes non-negative utility: Monopoly profit is given by 1 V (q 1, q) t 1 0, and (1) 2 V (q 2, q) t 2 0. (2) Π = p(t 1 cq 1 ) + (1 p)(t 2 cq 2 ), and it has to be maximized such that participation constraints (1) and (2) are satisfied. Naturally, both of these constraints will be binding in optimum, so 11 A lattice is a partially ordered set, which contains the least upper bound (so called join) and greatest lower bound (so called meet) of each pair of its elements. The join (meet) of two elements x and x is denoted by x x (x x ). In R 2, x x = (max{x 1, x 1 }, max{x 2, x 2 }) and x x = (min{x 1, x 1 }, min{x 2, x 2 }). 12 This follows from Theorem in Topkis (1998). A function g(x) is supermodular on the lattice X, if g(x ) + g(x ) g(x x ) + g(x x ) for all (x, x ) X. 6

7 after substitution we have the following profit-maximization problem: max Π = p[ 1V (q 1, pq 1 +(1 p)q 2 ) cq 1 ]+(1 p)[ 2 V (q 2, pq 1 +(1 p)q 2 ) cq 2 ]. {q 1,q 2} Since there are no externalities on non-traders, the perfectly discriminating monopoly internalizes all network effects. (3) Thus the monopoly s problem is equivalent with the welfare-maximizing one, and the optimal allocation produces no social inefficiency. This result is in line with Segal (2001), since so far the heterogeneity of consumers played no role. Note that the profit function is the sum of supermodular functions multiplied by positive constants, so it will be supermodular on Q. 13 We do not impose any sufficient conditions to ensure either existence or boundedness of the maximum for this problem, since these would be very restrictive. 14 Instead, we restrict ourselves to the problems when such a maximum exists, since otherwise the problem does not make economic sense. On the other hand, we do not exclude multiple optima. The following first-order conditions characterize the first-best allocation q F B = (q F B 1, q F B 2 ): 15 1 V 1 (q 1, q) + p 1 V 2 (q 1, q) + (1 p) 2 V 2 (q 2, q) = c, and (4) 2 V 1 (q 2, q) + p 1 V 2 (q 1, q) + (1 p) 2 V 2 (q 2, q) = c. (5) In both equations, the first term measures the marginal utility of individual consumption, we will call it first-best individual effect. The second and the third term together will be called first-best network effect, and it is the same in both equations. If there are no network effects, the second and third terms vanish, so individual effect equals marginal cost, as in standard first-best implementation. By combining the two first-order conditions (4) and (5), we have that 1 V 1 (q 1, q) = 2 V 1 (q 2, q). Since V 11 0, this equation implies that q F B 1 < q F B 2, so in the first-best case high-type consumers get more than low type ones. 13 See Lemma in Topkis (1998). We use the fact that a one-dimensional function, like cq i, is both super- and submodular, and a supermodular minus a submodular function is still supermodular. 14 A set of sufficient conditions would be that V (q, q) is concave (quasi-concavity is not enough), and both V 1 and V 2 satisfy the Inada conditions. 15 Throughout the paper, the final forms of the first-order conditions are derived after dividing the equation by the density of the respective type, which is now p and (1 p). 7

8 3.2 The second-best case In the second-best case the optimal menus, (q 1, t 1 ) and (q 2, t 2 ), must satisfy participation constraints (1) and (2) and the following incentive constraints: 1 V (q 1, q) t 1 1 V (q 2, q) t 2, and (6) 2 V (q 2, q) t 2 2 V (q 1, q) t 1. (7) Note that in the incentive constraints we require only that no consumer has any incentive to deviate individually. Since we have assumed a continuum of consumers in each type, a single consumer s choice cannot have a significant effect on network size, so q remains unchanged if other consumers select the menu devoted to them. As standard in incentive theory literature, we try to reduce the number of relevant constraints. First, the participation constraint for high-type consumers will be automatically satisfied if constraints (1) and (7) are satisfied: 2 V (q 2, q) t 2 2 V (q 1, q) t 1 1 V (q 1, q) t 1 0. Second, we expect that the incentive constraint for the low-type consumers holds with an inequality in equilibrium, so we ignore it for the moment and check at the end whether it is satisfied. Then in optimum the two relevant constraints (1) and (7) must be binding and we can write down the tariffs charged by the monopoly as functions of the quantities: t 1 = 1 V (q 1, q), and (8) t 2 = 2 V (q 2, q) V (q 1, q). (9) These two equations demonstrate the standard intuition of second-degree discrimination: the surplus of the low-type consumers is fully grasped, while high-type consumers get their information rent in equilibrium. By using these results, we see that the low-type consumers incentive constraint is satisfied if which requires that q 2 q 1. [V (q 2, q) V (q 1, q)] 0, Now, if we substitute equations (8) and (9) into the profit function, we end up with the following profit-maximization problem: max Π = p[ 1V (q 1, q) cq 1 ] + (1 p)[ 2 V (q 2, q) V (q 1, q) cq 2 ] = {q 1,q 2} ( = 1 1 p ) p pv (q 1, q) + (1 p) 2 V (q 2, q) cq. (10) 8

9 The constant multiplier of the first term is always positive, 16 so the profit function is supermodular on Q. Again, we restrict ourselves to the relevant cases where this function has a maximum. 17 The optimal allocation q SB = (q SB 1, q SB 2 ) can be characterized by the firstorder conditions. First, it is more instructive to analyze the first-order condition in respect of q 2 : 2 V 1 (q 2, q) + p 1 V 2 (q 1, q) + (1 p) 2 V 2 (q 2, q) (1 p) V 2 (q 1, q) = c. (11) The first term is the first-best individual effect, the second and the third terms are the first-best network effects. However, there is an additional term, which will be called second-best network effect. The presence of a second-best term in the optimum condition for the high-type consumers is due to the fact that the information rent is affected by total network size, which contains q 2 as well, while in standard incentive theory the information rent depends only on q 1. As a consequence, the no distortion at the top result does not hold any more. The other first-order condition in respect of q 1 is the following: 1 V 1 (q 1, q) + p 1 V 2 (q 1, q) + (1 p) 2 V 2 (q 2, q) (1 p) V 2 (q 1, q) 1 p p V 1(q 1, q) = c. (12) The first three terms are the first-best effects, the fourth one is the secondbest network effect. The fifth term will be called standard second-best effect, and this is exactly the same effect, which creates the downward distortion for the low-type consumers in the standard discrete-type model without network effects. If we combine the two first-order conditions (11) and (12), we have that ( 1 1 p ) V 1 (q 1, q) = 2 V 1 (q 2, q). (13) p First, in order to have a positive solution for q 1, 1 1 p p must be positive. Second, since 1 1 p p < 2 and V 11 0, it follows that q 1 < q 2, so hightype consumers end up with higher quantities than low-types. This condition is sufficient to ignore the incentive constraint of low-type consumers, so we have given a full characterization of the equilibrium. 16 It follows from the assumption that the monopoly does not want to exclude low-type consumers from the network. See the discussion after equation (13). 17 If the first-best maximum exists and it is bounded, Proposition 1 (see below) will ensure that the seond-best problem has a bounded maximum as well. 9

10 3.3 Comparison of different outcomes In the preceding two subchapters we have derived the contracts offered by a monopolist producer of a network good for the perfect (first-degree) and incentive (second-best) discrimination case. In this subchapter we would like to compare the resulting allocations to each other and to two other benchmark case: perfect competition and monopolist discrimination in the absence of network effects. We have seen that in the second-best case the presence of network effects distorts the first-best allocations for all consumers. which builds on the Monotone Selection Thorem, 18 downward distortion for all consumers. In the next proposition, we prove that this is a Proposition 1 The second-best allocation is strictly smaller than the first best allocation, that is q SB < q F B. Proof. Consider the following parametrized form Π : Q T R, where T = R + is the parameter space: Π(q, α) = αpv (q 1, q) + (1 p) 2 V (q 2, q) cq. When α = 1, we have the first-best profit function given in (3), while for α = 1 1 p p (0, 1) we have the second-best profit function given in (10). We have seen that the function Π(q, α) is supermodular in q on Q. Moreover, Π(q, α) has strictly increasing differences in (q, α) on Q T, because both Π q 1 α and Π q are strictly positive. Then since 2 α αf B > α SB, for any optimal solution picked from the set of maximizers q(α) the Monotone Selection Theorem ensures that q F B q SB. But we have seen a distortion for both consumer types, so the inequality should be strict. Now we turn our attention to the case of perfect competition, where identical firms supply the network good at a price equal to the marginal cost c. Then each consumer of type i derives a utility of i V (q i, q e ) cq i, where q e is the expected network size, which must be fulfilled in equilibrium. For each i, maximizing in q i results in the first-order condition of i V 1 (q i, q) = c. 18 We use Theorem in Topkis (1998), which is a slightly modificated version of the Monotone Selection Theorem derived by Milgrom and Shannon (1994). 10

11 Let us denote the solution of this equation system for each i by qi P C, which results in the allocation q P C. By comparing two different first-order condition, we can easily see that if i > j then qi P C > qj P C. If we compare these first-order conditions with those of the first-best discrimination case (equations (4) and (5), we see that for all i i V 1 (qi F B, q F B ) < i V 1 (qi P C, q P C ). (14) Note that the network size in the first-best discrimination case cannot be smaller than in perfect competition, since otherwise the monopoly could offer contracts replicating the perfect competition network size and reap the increased surplus of the consumers, increasing thereby it s profit. Then since V 12 0, for all q F B i V 1 (qi F B, q P C ) V 1 (qi F B, q F B ). Combining this inequality with inequality (14), we have that i V 1 (qi F B, q P C ) < i V 1 (qi P C, q P C ), which yields that qi F B qi P C for all i, since V Note that this proof is valid for any number of types. So the perfectly competitive outcome is smaller than the first-best discrimination outcome. This is because perfectly competitive firms cannot internalize the network effects implied by higher allocations, since they cannot influence the quantity choice of the consumers. On the other hand, the perfectly discriminating monopoly can set the (higher) socially optimal allocation, and reap the increased surplus of each consumer. The comparison of second-best discrimination and perfect competition allocations does not give unambiguous results, since we are comparing two outcomes, which fail to be the first-best because of two different reasons: incentive problems due to information asymmetry and the incapability of internalizing network effects. Last, we show that the standard outcome is always smaller than the outcome in the presence of network effects, no matter which type of discrimination we consider. This result is natural in the case of first-best discrimination, but it also shows that despite the overall downward distortion in the second-best discrimination in the presence of network effects, the resulting allocation is not smaller than in second-best discrimination without network effects. 11

12 Proposition 2 The equilibrium allocation in the presence of network effects is smaller than the equilibrium allocation without network effects, both for firstand second-best discrimination. That is qne i qi ST for i = F B, SB. Proof. Let us consider the utility function U( i ) = i V (q i, βq) t i, where β measures the importance of network size. If β = 0, we have the standard discrimination case without positive network effects, while our original model refers to β = 1. In this case the profit function takes the form Π : Q T R: Π(q, β) = α i pv (q 1, βq) + (1 p) 2 V (q 2, βq) cq, where β T = R +. As before, α F B = 1, α SB = 1 1 p p. We have seen that the function Π(q, β) is supermodular in q on Q for each β. In this case Π(q, β) has strictly increasing differences in (q, β) only on Q T \ {0}, since Π q 2 β Π q and 1 β are strictly positive only for β > 0, and it has increasing differences in (q, β) on Q T. Then by the Monotone Selection Theorem for any β (0, 1) the greatest element of the optimal solutions q(β ) is smaller than the smallest element of q(1), and by the Monotone Comparative Statics Theorem the greatest element of q(β ) is bigger than q(0). 19 So q NE q ST for both α i. 4 The continuous type model Suppose now that is distributed on [, ] according to a distribution function F (), with a positive density f() at each point. We( assume) that F ( ) satisfies the monotone hazard rate property, namely that d 1 F () d f() 0. As in the discrete-type case, the relevant decision variables are the quantity choices of the monopoly, since the optimal tariff schedule t() will be determined by the optimal quantity schedule q(). Now let Q be the partially ordered set of bounded functions q() defined on [, ]. Let us define the join and meet of two elements q () and q () as the upper and lower envelope of the two functions: q () q () = max{q (), q ()} and q () q () = min{q (), q ()}. Then Q is a lattice, since it always contains the join and meet of any two elements. By applying the definitions it can be shown that if V ( ) has increasing differences in q() on Q then it is also supermodular in q() on Q. 19 For the Monotone Comparative Statics Theorem see Theorem in Topkis (1998). 12

13 4.1 The first-best case Again, we consider first the benchmark case of perfect discrimination. For each type, the monopoly offers a contract (q(), t()) that grasps all the surplus of the consumer: where q = q(u)f(u)du. Monopoly profit is given by Π = t() = V (q(), q), (15) [t() cq()]f()d, and after substitution of equation (15), the monopoly s profit-maximization problem is the following: [ max q() Π = V (q(), q(u)f(u)du) cq() ] f()d. (16) Note that in the square brackets we have a supermodular function in q(), since it is the sum of two supermodular functions, and the expected value of a supermodular function is supermodular, so the profit function is supermodular as well. Pointwise maximization in q( i ) gives the following first-order condition for each i : i V 1 (q( i ), q) + V 2 (q(), q)f()d = c. (17) The continuum of first-order conditions define the first-best quantity profile q F B (). Again, we see the intuition presented in the discrete-type case: the first term is the individual effect and the second term is the sum of network effects, which is the same in all first-order conditions. These two effects together should equal the marginal cost of the network good. By combining two first-order conditions for and, we have that V 1 [q( ), q] = V 1 [q(), q]. Since V 11 0, if > then q( ) > q(), so a consumer of higher type gets a higher quantity level in equilibrium. 13

14 4.2 The second-best case: implementability We now turn to the second-best case, and first we derive necessary conditions for implementability. Based on the Revelation Principle we may restrict ourselves to truthful direct revelation mechanisms {(q( ), t( ))}, where q( ) and t( ) are assumed to be piecewise differentiable functions. If every consumer truthfully reveals his type (as it will be the case in equilibrium), total quantity purchased will be q = q(u)f(u)du. Each consumer of type maximizes his utility in answer, while he is conjecturing that every other consumer truthfully reveals his type: maxu(, ) = V (q( ), q) t( ). { } Truthtelling is optimal for all if the following first-order condition holds: V 1 (q(), q) q() t() = 0. (18) If we differentiate this equation by and combine it with the local secondorder condition, we have V 1 (q(), q) q() 0, (19) and since V 1 > 0, it leads to the the necessary condition of q() 0. (20) So in order to have an implementable mechanism, the quantity scheme (and according to equation (18) the tariff scheme as well) has to be a non-decreasing function of the type. The reason why we end up with exactly the same implementability conditions as in the standard continuous type problem without network effects is the following: in the incentive constraints we are considering only individual deviations, and even if a consumer lies about his type and chooses a different menu, by the continuum of consumers in each type assumption his action leaves network size q unchanged. The very same reasoning can be used to demonstrate why the single-crossing condition (which is automatically satisfied with this utility structure) is sufficient to ensure that locally optimal truthful responses (characterized by equations (18) and (19)) are globally optimal as well The proof presented at the Appendix 3.2 of Laffont and Martimort (2002) can be easily modified to show this result. 14

15 4.3 The second-best optimal contract To solve the optimal program for the monopolist it is easier to consider the rent of the -type consumer: U() = V (q(), q) t(). (21) If we differentiate this equation by and use equation (18), we arrive to the differential equation which has a solution of U() = V (q(), q), U() = U() + V (q(v), q)dv. By substituting this result into equation (21) and using that U() = 0, we can express the tariff function for each type: t() = V (q(), q) V (q(v), q)dv. Now, we can write down the monopoly s profit-maximization problem as [ ] max q() Π = V (q(), q) s.t. q() 0. V (q(v), q)dv cq() f()d, First, we ignore the monotonicity constraint and check at the end whether it is satisfied in equilibrium. integrand by parts, we have [ ] V (q(v), q)dv f()d = Second, if we integrate the second part of the V (q(), q)[1 F ()]d. By using this result the reduced problem is [ ( max Π = 1 F () ) ] V (q(), q(u)f(u)du) cq() f()d. (22) q() f() The profit function is expected value of a supermodular function multiplied by a positive constant, therefore it will be supermodular Again, the constant is positive by the no shutdown of inefficient types assumption. 15

16 Pointwise maximization in q( i ) gives the following first-order condition for each i : i V 1 [q( i ), q] + 1 F ( i) V 1 (q( i ), q) f( i ) V 2 (q(), q)f()d [1 F ()]V 2 (q(), q)d = c. (23) If we compare this equation with its counterpart of the discrete-type case, equation (12), we see the same result on the left-hand side: the first two terms are the first-best effects, the third term is the standard second-best effect on information rent (this is exactly the same as in the model without network effects), and the fourth one is the sum of second-best network effects. These effects together should equal the marginal cost of the network good. The standard second-best effect vanishes if i =, as in equation (11) of the discrete-type case, but the second-best network effects are still creating a distortion for the highest type as well. If we examine two first-order conditions for and, we see that ( 1 F ) ( ( ) f( V 1 (q( ), q) = 1 F () ) V 1 (q(), q). ) f() ( ) ( If >, by monotone hazard rate property 1 F ( ) f( ) < holds, and since V 11 0, it follows that q( ) > q(). Consumers of higher types ) 1 F () f() end up with higher quantities, so the omitted necessary condition is fulfilled in equilibrium. 4.4 Comparison of different outcomes In Subsection 3.3 we have discussed the comparison of first-best and second best outcomes in the presence of network effects to each other and two other cases. The proof given about the comparison of the different discrimination outcomes to the perfect competition outcome is valid also for continuous types, while Proposition 2 about the ordering of discrimination outcomes in the presence / lack of network effects can be easily modified. We are proving here only that the strict downward distortion result generalizes to the case of continuous types. Proposition 3 The second-best allocation is strictly smaller than the first-best allocation, that is q SB < q F B. 16

17 Proof. If we compare the profit function for the first-best and the second-best case given in (16) and (22), they can be captured by the following general form Π : Q T R: Π(q(), γ) = = g(q(), γ)f()d cq = ( + γ 1 F () ) V (q(), q)f()d cq, f() where γ T = [ 1, 0]. If γ = 0, we have the first-best profit function, while γ = 1 refers to the second-best case. The function Π(q(), γ) is supermodular in q() on Q for each γ T (the multipliers are always positive for these values) and since g(q(), γ) has strictly increasing differences in (q, γ) on Q T for each, Π(q, γ) has strictly increasing differences in (q, γ) on Q T. Then since γ F B > γ SB, for any optimal solution picked from the set of maximizers q(γ) the Monotone Selection Theorem ensures that q F B q SB. But we have seen a distortion for all consumer types, so the inequality should be strict. 5 Concluding remarks In this paper we have derived a general model to analyze the second-degree price discrimination problem of a monopoly selling a network good with positive network effects. By using the theory of supermodular functions, we were able to give a full characterization of the contracts that successfully screen the consumers. Both in the discrete and continuous type case, we have seen that positive network effects and asymmetric information together lead to a downward distortion for all consumers; however, consumption levels were higher for all consumers than in the standard models without network effects. A crucial feature in our model was that the optimal contracts are designed such that it is individually not profitable for deviating from the truthtelling equilibrium. However, the natural question arises whether it could be advantageous for some consumers to form a coalition to coordinate their decisions and then reallocate the goods among themselves. Jeon and Menicucci (2002) show that when buyer coalitions are formed under asymmetric information, the monopoly s profit is not harmed by the presence of such coalitions, because the buyers fail to realize the gains from joint deviations due to the transaction costs of asymmetric information. Although network effects are not present in their model, the intuition seems to hold in our setting as well. 17

18 So far the network goods provided to different types of consumers were assumed to be compatible with each other. Let us briefly examine in the discrete model the case when the monopoly chooses the network good provided to lowtype consumers to be incompatible with the high-types goods. Now low-types benefit less from the network, so the monopoly cannot charge such a high tariff for them. However, a high-type consumer will now have less incentive to choose the menu devoted to low-type consumers, since then he excludes himself from using a part of the network, so information rent of high-type consumers should decrease as well, which is profitable for the monopoly. Our conjecture is that if the monopoly chooses to make its good partially incompatible, then it will choose to do so with the good devoted to low-type consumers, since high-types have a higher marginal utility of the network by the single-crossing property. If the good devoted to low-type consumers is incompatible with the good devoted to high-type consumers, then in equilibrium low-type consumers utilities depend only on low-type consumers choices. This is exactly the same case as if low-type consumers had the pessimistic expectation that high-type consumers will stay out of the market, so the monopoly has to design the contract devoted to low-type consumers such that they would accept it without the high-types as well. But if high-type consumers observe the contract devoted to low-type consumers, no matter how pessimistic prior expectations they had about lowtypes behavior, they will realize that low-types will accept that contract in any case. Then they will make their choices by expecting low-type ones in the network, thus the monopoly can design the menu devoted to high-types accordingly. 22 This reasoning is the same as the divide and conquer strategy presented in Jullien (2002) and Segal (2001), which may help to overcome the problem of multiple equilibria induced by different consumers expectations and to end up with unique implementation. 22 For this result to hold we should assume that consumers of the same type coordinate their acions or use the Pareto-criterion for equilibrium selection. If we do not, then the low-type contract should be designed such that it is accepted by any low-type consumer even in the case when he expects no one else to join. 18

19 References [1] Cooper, R., and John, A. (1988), Coordinating coordination failures in Keynesian models, Quarterly Journal of Economics 103, [2] Ellison, G. and Fudenberg, D. (2000), The Neo-Luddite s Lament: Too Many Upgrades in the Software Industry, Rand Journal of Economics 31, [3] Farrel, J. and Saloner, G. (1985), Standardization, Compatibility, and Innovation, Rand Journal of Economics 16, [4] Fudenberg, D., and Tirole, J. (1991), Game Theory, MIT Press. [5] Fudenberg, D. and Tirole, J. (2000), Pricing a Network Good to Deter Entry, Journal of Industrial Economics 48, [6] Hahn, J-H. (2003), Nonlinear Pricing of Telecommunications with Call and Network Externalities, forthcoming in International Journal of Industrial Organization. [7] Jeon, D-S and Menicucci D. (2002), Buyer Coalition Against Monopolistic Screening: On the Role of Asymmetric Information among Buyers, working paper, Universitat Pompeu Fabra. [8] Jullien, B. (2002), Competing in Network Industries: Divide and Conquer, working paper, IDEI, University of Toulouse. [9] Katz, M. and Shapiro, C. (1985), Network Externalities, Competition, and Compatibility, American Economic Review 75, [10] Laffont, J-J. (1988), Fundamentals of public economics, MIT Press. [11] Laffont, J-J., and Martimort, D. (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press. [12] Liebowitz, S. and Margolis, S. (2002), Network Effects, in Cave, M. et al. (editors), Handbook of Telecommunications Economics, Volume 1, 75-96, Elsevier Science. [13] Maskin, E. and Riley, J. (1984), Monopoly with Incomplete Information, Rand Journal of Economics 15,

20 [14] Milgrom, P., and Roberts, J. (1990), Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities, Econometrica 58, [15] Milgrom, P., and Shannon, C. (1994), Monotone Comparative Statistics, Econometrica 62, [16] Mussa, M. and Rosen, S., 1978, Monopoly and Product Quality, Journal of Economic Theory 18, [17] Oren, S., Smith, S., and Wilson, R. (1982), Nonlinear Pricing in Markets with Interdependent Demand, Marketing Science 1, [18] Segal, I., 1999, Contracting with Externalities, Quarterly Journal of Economics 114, [19] Segal, I. (2001), Coordination and Discrimination in Contracting with Externalities: Divide and Conquer?, working paper, Stanford University. [20] Topkis, D. (1978), Minimizing a Submodular Function on a Lattice, Operations Research 26, [21] Topkis, D. (1998), Supermodularity and Complementarity, Princeton University Press. [22] Vives, X. (1990).Nash Equilibrium with Strategic Complementarities, Journal of Mathematical Economics 19, [23] Vives, X. (1999), Oligopoly Pricing, MIT Press. 20