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Network externalities and the Coase conjecture
European Economic Review 44 (2000) 1981}1992
Network externalities and the Coase conjecture Robin Mason* Department of Economics, University of Southampton, Highxeld, Southampton SO17 1BJ, UK Received 1 December 1998; accepted 2 December 1999
Abstract This paper addresses two general questions. First, what is the e!ect of market structure on the development of a network in a dynamic model with rational expectations? Secondly, is the intuition that network externalities are &economies of scale on the demand side' correct? These questions are examined in a model of durable good production in the presence of network externalities. Two results are presented. First, the Coase conjecture fails in its strongest sense when network bene"ts are increasing in the current network size. Secondly, a committed monopolist may be socially preferable to a time consistent producer when network externalities are su$ciently large. The analysis indicates an analogy between network externalities and learning-by-doing. 2000 Elsevier Science B.V. All rights reserved. JEL classixcation: C73; C78; D42; L12 Keywords: Coase conjecture; Network externalities
1. Introduction This paper examines the production of a durable good in the presence of network externalities in order to address two general questions. First, what is the e!ect of market structure on the development of a network in a dynamic
* Tel.: #44-1703-593268; fax: #44-1703-593858. E-mail address: [email protected] (R. Mason). 0014-2921/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 4 - 2 9 2 1 ( 9 9 ) 0 0 0 6 7 - 7
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model with rational expectations? Secondly, is the intuition that network externalities are &economies of scale on the demand side' correct? The paper brings together two sets of literature. The "rst details the conditions under which the Coase conjecture (Coase, 1972) can be expected to hold. Central to this work is a careful treatment of the time consistency issues that arise when a good is durable and consumers anticipate future price changes. See Bulow (1982) and Gul et al. (1986). The second considers the e!ect that network externalities have on market outcomes. This research examines (amongst other issues) the tendency for concentrated industry structures when there are network e!ects. See e.g. Katz and Shapiro (1985) and Farrell and Saloner (1985). While a few authors have suggested the importance of combining the two approaches, there has not yet been a satisfactory treatment of the problem of durable good sales with network externalities. Katz and Shapiro (1986) "rst raised the possibility that network externalities might lead to the monopoly price of a durable good increasing over time. The intuition is straightforward. Network size is equivalent to a parameter of vertical di!erentiation: the larger the network, the higher the value of the good to consumers. Therefore, a lower initial price which increases the number of initial users raises the value of the good to future consumers; this may allow the monopolist to charge a higher price later. Bensaid and Lesne (1996) provide an explicit analysis of this possibility in a discrete time model. They show for network externalities of a certain type that the monopolist prices above marginal cost at all times; and (if the externalities are su$ciently large) price may rise over time and the monopolist's pro"ts are una!ected by its inability to commit to a production plan. This paper develops a model of production of a durable good in the presence of network externalities. It asks: what is the (strong) Markov perfect equilibrium of the model when producers have an in"nitesimal period of commitment? The use of a discrete time model implicitly allows the monopolist a period of commitment. Stokey (1981) makes clear that this is a crucial assumption. The continuous time framework used here means that the implicit commitment period is zero. Xie and Sirbu (1995) also analyse a continuous time model of durable good sales; but they allow commitment on the part of the monopolist, who chooses open-loop rather than feedback strategies. Two results are presented. First, the Coase conjecture may fail in its strongest form when network externalities are present. For a broad class of network bene"t functions (those that are increasing over some range of the network size), the Markov perfect equilibrium is the same for monopolistic and perfect competitive market structures. Socially optimal pricing (price equal to marginal cost) prevails in both cases. But the time-consistent producers grow the network more slowly than is socially optimal. Secondly, a committed monopolist may be socially preferable to a time consistent producer when network externalities are su$ciently large. The former grows the network at the socially optimal rate, but
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restricts the long-run size of the network. The latter grows the network too slowly, but attains the socially e$cient network size in the long-run. These results indicate that the model of network externalities can be viewed as a demand-side counterpart of Olsen's (1992) learning-by-doing model.
2. The model The model is based on Stokey (1981) and Karp (1996). A monopolist chooses output to maximise the present value of a discounted stream of pro"ts from production of a durable good. There is no depreciation of the good; there are no capacity constraints; the monopolist must sell, rather than rent its output; and the monopolist uses an in"nite time horizon. There is a continuum of nonatomic, in"nitely-lived consumers, each with a demand for one unit of the durable good. They do not consider the e!ect of their decisions on others, and hold rational expectations about the monopolist's production plans. The notation is as follows: output at time t is q(t); the stock of the durable good (the state variable) is Q(t); the selling price is P( . ); constant unit production costs are c; and the common continuous time discount rate is r. There is common knowledge of all aspects of the model. Consumers' utility from purchasing the good has two components. The "rst is intrinsic, derived from the #ow of services from the good itself; this is denoted b. It is assumed throughout the paper that intrinsic valuations are private information, but it is common knowledge that they are uniformly distributed on the interval [b, 1]. In addition, there is a network e!ect, so that the gross surplus derived byM a consumer with intrinsic valuation b buying at time t is v "b#kn , (1) R R where n is the size of the network externality at time t and k50 is a scaling R parameter. The network externality can be interpreted quite generally, and represents any increase in the gross valuation of the good when the total quantity demanded increases. One speci"c and simple example is n "Q : that is, R R the current network externality is equal to the current network size. Bensaid and Lesne (1996) call this type of externality &excluded', and cite computer software as an example } early buyers of software do not bene"t from the externalities that they generate, such as discovery of bugs, unless they buy updated versions. The only type of externality ruled out at this stage is one which arises from sales of an associated non-durable good; this case is considered by KuK hn and Padilla (1996), who show that the Coase conjecture fails.
To be precise: a unilateral deviation by a consumer from its equilibrium strategy does not change the actions of other consumers or the monopolist.
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The following assumptions are made about the network function n( . ): (i) it is non-negative: n 50 ∀t; (ii) it is Markovian and stationary: n ,n(Q(t)); (iii) it is R R piecewise continuous; and (iv) there exists at least one value QK such that 1!QK #kn(QK )"c. Assumption (i) con"nes attention to positive network externalities. The Markovian assumption in part (ii) is less restrictive than it appears. Consumers' expectations of future output are assumed to depend only on the current level of the durable good stock i.e. Q(t#s)"Q(Q(t), s) for some continuously di!erentiable function Q( . ) (where Q(s) denotes the stock of the good expected at time s). It is assumed further that consumers' expectations are ful"lled in equilibrium: Q(s)"Q(s) ∀s. Consequently, although network bene"ts may depend on future sizes of the network, they can be written as a function of the current network size. Assumption (iii) ensures (local) existence of a solution to the problem that is analysed; assumption (iv) ensures that a steady state exists. Finally, the following parametric restrictions are made: (v) kn (Q)(1 ∀Q; and (vi) b#kn(1!b)4c. Part (v) ensures that the inverse demand function for the M services of the M good is downward sloping (see below); it requires k to be su$ciently small. The size of the network in period t is therefore Q(t)"1!b , R where b is the intrinsic valuation of the marginal consumer who is indi!erent R about buying at time t. Part (vi) means that price eventually is driven to marginal cost and the market covered. In other words, this paper examines only the &no gap' case identi"ed by Gul et al. (1986). It is straightforward to modify the paper's results for the &gap' case (which Bensaid and Lesne, 1996 analyse); the main conclusions are unchanged. Note that the possibility of equilibrium multiplicity in the &no gap' case identi"ed by Gul et al. (1986) is not an issue here, since the continuity assumptions impose the regularity required for uniqueness. In order to derive the correct continuous time expression for the equilibrium price function, the model is written "rst in discrete time (as in Gul et al., 1986); the time between periods is then reduced to zero. The extensive form of the discrete time game is: in each period, the monopolist "rst names a price at which he is willing to trade. Consumers then either accept the o!er, or reject. Once a consumer has accepted, she drops out; those that reject continue to receive
Stokey (1981) shows that multiple equilibria exist when expectations are discontinuous in the state; see also the discussion in Karp (1996). A stronger assumption is that the network function is stationary. Ausubel and Deneckere (1989) show that non-stationarity (of preferences and strategies) leads to a folk theorem: there is a continuum of subgame perfect reputational equilibria with monopoly pro"ts ranging from zero to the open-loop, full commitment level. There is little to be gained from simply repeating this result in a model with network externalities } hence, the focus on the stationary case. That expectations are Markovian is consistent with the assumptions in the discrete time version of the model that consumers' decision rules of whether to buy or not are Markovian and that the monopolist's strategy is Markovian.
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o!ers in the future. Each period is of length D. The monopolist's strategy speci"es the price P he will charge in each period t; a strategy for a consumer R speci"es whether to accept or reject the current price o!er of the monopolist. Strategies are assumed to be stationary and Markovian. The monopolist's beliefs about the types of consumers are revised according to Bayes' rule. Consider the acceptance/rejection decision of the marginal consumer in some arbitrary period t. Let this consumer have intrinsic valuation b ; and let the R network function at time t be n . Then, the consumer is indi!erent between R buying one unit of the durable good at time t, and deferring purchase until the next period t#D, if b #kn !P "d(b #kn D !P D ). (2) R R R R R> R> Here, the per-period discount rate is d"exp(!rD). Note that Eq. (2) considers only a unilateral deviation by the marginal consumer: if she does not purchase the good at time t, nevertheless, the monopolist sells so that the network externality is n D in the next period. R> Eq. (2) can be rearranged, d replaced with exp(!rD), and exp(!rD) approximated by 1!rD for small D, to give
k(1!rD) n D !n 1!rD P D !P R> R "P ! R> R . b #kn ! R R R r r D D
(3)
If the total quantity sold during the interval D is bounded for su$ciently small D (i.e. the stock of the durable good changes continuously), then the limit of Eq. (3) as DP0 can be taken, to give a di!erential equation for the price function dP dn R "rP !r(b #kn )#k R . R R R dt dt
(4)
Eq. (4) can be rewritten to make comparison with Stokey (1981) and Karp (1996) clearer. The inverse demand for services from the durable good, or rental rate, can be written as F ,r(b #kn ), i.e. the constant #ow of utility per period R R R which, when received in perpetuity from time t, yields a present discounted value of total utility of b #kn . Therefore, R R dP dn R "rP !F #k R . (5) R R dt dt When the quantity sold during the interval D is not bounded } i.e. when there is a discrete jump in the stock of the durable good sold, from Q to Q , say } then rationality requires that the price just before the jump must be equal to the price immediately after: P(Q )"P(Q ). In other words, consumers anticipate any discrete change; in the continuous time limit, any price di!erential would represent an arbitrage opportunity.
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The economic intuition for Eq. (5) can be understood by supposing that there is a perfectly functioning second-hand market for the durable good. The equality states that the marginal consumer should be indi!erent between borrowing to buy the good, deriving an instantaneous utility, and reselling to pay o! the loan. Therefore, utility F should equal the interest payment rP on the loan minus the price capital gain dP/dt plus the &network capital gain', kdn/dt. If this last term is positive, then the consumer has an added incentive (relative to the no network externality case) to delay purchase until the network is larger. In summary: the assumptions of continuous Markovian expectations and consumer rationality have two consequences for the price function in Eq. (5). The former implies that the price function is a continuous and almost everywhere di!erentiable function of the current state. When the stock of the durable good changes continuously, price therefore changes continuously. The latter means that consumers anticipate any discrete jump in the state; and so the price just before the jump must be equal to the price immediately after.
3. The Markov perfect equilibrium This section derives the strong Markov perfect equilibrium for the continuous time model. It is convenient to take the monopolist's strategy at time t as a choice of output, q . Taking the price function as given, the monopolist's R problem is
max exp(!r(s!t))q [P(Q)!c] ds Q OQ Y R dQ(t) "q(t), Q(0) given. dt
s.t.
(7)
The solution to problem (7) and the function P(Q) generate a price path which in equilibrium satis"es Eq. (5). The monopolist cannot commit to a production
In Stokey (1981) and Karp (1996), the di!erential equation for the price function is derived from the requirement that
P( . )"
exp(!r(s!t))F(Q(s)) ds (6) R i.e. the current price is the present discounted value of the future expected rents. This approach is not adopted here because it would mask the fact that only unilateral deviations by consumers are considered in Eq. (2). It is this that gives rise to the additional term kdn/dt in Eq. (5).
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path, but must choose a stationary Markovian decision rule for output. The necessary condition for pro"t maximisation is given by the Bellman equation
*J rJ"max P(Q)!c# q. (8) *Q OY J is a solution to the Bellman equation; it is veri"ed below that it is di!erentiable in Q (as assumed in Eq. (8)), and that it is also a value function of problem (7). The problem is linear in the control q, and so the optimal solution may involve discontinuities in output. q is therefore restricted to be a piecewise continuous function of time. Let F(QK )"rc. Consider any interval in [Q , QK ) over which output is non-zero and "nite. The linearity of the Bellman equation means that the singular solution *J P(Q)!c# "0 *Q
(9)
must hold over this interval. Therefore J"0; and hence *J/*Q"0 and P(Q)"c. Consumer rationality (i.e. Eq. (5)) then requires 0"rc!F(Q)#kn (Q)q.
(10)
If kn (Q)"0 on this interval, then Eq. (10) implies that F(Q)"rc i.e. Q"QK (a constant). This is inconsistent with q'0 over this interval. If kn (Q)(0, then Eq. (10) implies that q"(F(Q)!rc)/kn (Q). Since output must be non-negative, this requires that Q5QK . This is inconsistent with a stable steady state (with Q"QK ). Therefore, output can be non-zero and "nite only when kn (Q)'0. Consider now an interval in [Q , QK ] over which q"0. Clearly, the stock Q is constant during such an interval; and, due to stationarity, so is the price P(Q). Therefore P(Q)"F(Q)/r, from Eq. (5). If Q(QK , P(Q)'c, and the time-consistent monopolist sets output above zero. Therefore, q"0 is consistent with equilibrium only when Q"QK and P(Q)"c. Finally, consider an interval in [Q , QK ) over which q is in"nite, producing a discrete change in the durable good stock. Since the problem is linear in the control, the principle of the Most Rapid Approach Path applies (see Clark, 1990). The stock is increased to the start of a singular interval, or to the steady state if the former does not exist. In either case, price equals cost after the jump in the stock; and, by consumer rationality, therefore equals cost before the jump. The preceding argument shows that the unique Markov perfect equilibrium is as follows: P(Q)"c, F(Q)!rc q" , kn (Q)'0, Q(QK , kn (Q)
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Fig. 1. The steady state in the Markov perfect equilibrium: n '0.
"R, kn (Q)40, Q(QK , "0,
Q"QK .
(11)
In equilibrium, the monopolist prices at marginal cost and earns zero pro"ts (as predicted by the Coase conjecture). Over any interval in [Q , QK ) where kn (Q)'0, the rate of production is non-zero and "nite. Production continues until the stock of the durable good reaches the steady state level of QK , which is determined by the intersection of the network function n(Q) with the line (c!1#Q)/k; this is illustrated in Fig. 1 for the case when kn '0 ∀Q3[Q , QK ]. The time to reach the steady state may be greater than zero, and may even be in"nite (depending on the form of n( ) )). Consider next production of the durable good by a perfectly competitive industry. Irrespective of the degree of compatibility of the "rms' goods, price equals marginal cost. At the same time, the competitive price path must satisfy Eq. (5). The competitive equilibrium is therefore precisely the same as the monopolistic (Markov perfect) equilibrium: in both, price is set at marginal cost and production occurs according to Eq. (11).
The degree of compatibility may have an e!ect on whether perfect competition is sustainable; see e.g. Katz and Shapiro (1985). This issue is not considered here: the question is, conditional on there being perfect competition, what is the equilibrium price and stock of the durable good?
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Finally, consider socially optimal production of the durable good. The Bellman equation for the social planner is
*< F(Q) r<(Q)"max !c# q, r *Q OY where <(Q) is the planner's value function. For a non-zero but "nite production rate to be optimal, there must be an interval over which F(Q) *< !c# "0. r *Q
(12)
During such an interval, <(Q)"0 and F(Q)/r"c. But the latter equality implies that qF (Q)"0; since F (Q)(0, q must equal zero. The singular interval is, therefore, the point Q"QK . The linearity of the problem means that it is optimal to adjust the stock of the durable good instantaneously to the steady state level QK at the start of the planning period. Comparison of the programs of the time-consistent producers and the social planner gives the following proposition. Proposition 1. The Markov perfect equilibria of monopolistic and perfectly competitive industries are identical. In both, price equals the socially optimal level of marginal cost. If kn (Q)'0 over any interval in [Q , QK ), then monopoly and perfect competition exhibit delay: the network grows too slowly compared to the social optimum. Proposition 1 shows that there is no loss in welfare due to market power; nevertheless, the strong form of the Coase conjecture fails when network bene"ts are increasing in the current network size (over some interval). The intuition for the pricing part of the result is straightforward. The last consumer to buy has a total valuation equal to the marginal cost of production. Consumers anticipate, therefore, that price will equal cost in the long run. Price drops immediately to this level in the continuous time model. The network growth result is more surprising; the intuition is clearest using Coase's original insight that the time consistent monopolist is equivalent to a sequence of monopolists. Each "rm in the sequence does not gain the full bene"t from an increase in the current size of the network, since future "rms will act against its interest. Each "rm therefore has a lower incentive to produce at a high rate than a producer who can commit, and also (the proposition shows) the social planner. Three points are worth noting about the proposition. First, failure of the strong Coase conjecture occurs with a broad class of network bene"t functions. All that is required for ine$ciency is the function to be increasing over some interval. Secondly, the result is similar to those that emerge from learning-bydoing models in which marginal costs decrease with cumulative production. For
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example, Olsen (1992) "nds that a durable good monopolist produces more slowly than is socially optimal; and that the time-consistent monopoly and perfect competition equilibria are identical. Here, a similar e!ect is at work, but on the demand, rather than supply, side. Finally, Bensaid and Lesne (1996) argue that, when the network externality function is n "n(Q ) and k is R R\ su$ciently large: (i) price is always above marginal cost, (ii) price rises over time, and (iii) commitment is of no value to the monopolist. None of these features arise here, for three reasons: (i) the &no gap' case is analysed (so that price falls eventually to cost); (ii) a continuous time approach is used (so that price falls immediately to cost); and (iii) network externalities cannot be too large (so that the solution identi"ed exists and is stable). A further welfare comparison is of interest. The time consistent monopolist is socially ine$cient, since it grows the network too slowly; but it does not restrict the long-run size of the network. A &committed' monopolist (one who uses an open-loop production strategy) is socially ine$cient, not because it sells too slowly (all production occurs immediately, as Stokey, 1979 shows), but because it sets the long-run network size too low. Which monopolist is socially preferable depends on the size of the network externality (amongst other things). When the externality is small, the time consistent monopolist causes little delay. For a larger externality, the Markov perfect equilibrium involves considerable delay; the welfare loss of this delay may be greater than the loss due to output restriction by the committed monopolist. This general result is illustrated in Proposition 2 for the speci"c case of n(Q)"Q. Proposition 2. With n(Q)"Q and Q "0, social welfare is higher (lower) when the monopolist is unable to commit, relative to the commitment case, when k((') 0.4. Proof. In this case, QK "(1!c)/(1!k), Eq. (11) implies that production of the time consistent monopolist is q"(r/k)(QK !Q), and the committed monopolist produces so that the stock jumps instantaneously at t"0 to QR" (1!c)/2(1!k). Social welfare from these production programs is 3(1!c) (1!c) < " , < " . +.# -*# 2!k 8(1!k) The result is immediate. 䊐
This form is chosen for its tractability, and also because it is the continuous time version of the network externality function used by Bensaid and Lesne (1996). In fact, the proposition can be proved for any isoelastic network externality function.
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4. Conclusions This paper has examined the production of a durable good in the presence of network externalities. It has shown that network externalities can cause the strong form of the Coase conjecture to fail. In this model, market structure makes no di!erence to the price or production in the Markov perfect equilibrium. When network bene"ts are increasing in the network size over some interval, however, both industries grow the network too slowly compared to the social optimum. This leads to the possibility that it may be socially preferable to allow a monopolist the ability to commit (e.g. by renting rather than selling the durable good). The analysis suggests an analogy between network externalities and learning-by-doing. The key is that, in both, each "rm in the sequence which makes up the time-consistent monopolist does not consider the welfare of future versions of himself when making its current decision. With network externalities, this leads to delay in network growth; with learning-by-doing, costs decrease too slowly. This provides further support for the intuition that network externalities can be viewed as &economies of scale on the demand side'.
Acknowledgements I would like to thank Larry Karp for many comments on various drafts which have improved the paper substantially. Discussions with David Newbery and David Myatt, and comments from two referees, have also been very helpful. Any errors are my own. Funding from Alcatel Bell is gratefully acknowledged.
References Ausubel, L.M., Deneckere, R.J., 1989. Reputation in bargaining and durable goods monopoly. Econometrica 57 (3), 511}531. Bensaid, B., Lesne, J.P., 1996. Dynamic monopoly pricing with network externalities. International Journal of Industrial Organization 14 (6), 837}855. Bulow, J.I., 1982. Durable goods monopolists. Journal of Political Economy 90 (2), 314}332. Clark, C.W., 1990. Mathematical Bioeconomics, The Optimal Management of Renewable Resources, 2nd Edition. Pure and Applied Mathematics Series. Wiley, New York. Coase, R.H., 1972. Durability and monopoly. Journal of Law and Economics 15 (1), 143}149. Farrell, J., Saloner, G., 1985. Standardization, compatibility, and innovation. Rand Journal of Economics 16 (1), 70}83. Gul, F., Sonnenschein, H., Wilson, R., 1986. Foundations of dynamic monopoly and the Coase conjecture. Journal of Economic Theory 39 (1), 155}190. Karp, L., 1996. Depreciation erodes the Coase conjecture. European Economic Review 40 (2), 473}490.
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Katz, M.L., Shapiro, C., 1985. Network externalities, competition, and compatibility. American Economic Review 75 (3), 424}440. Katz, M.L., Shapiro, C., 1986. Technology adoption in the presence of network externalities. Journal of Political Economy 94 (4), 822}841. KuK hn, K.-U., Padilla, J., 1996. Product line decisions and the Coase conjecture. Rand Journal of Economics 27, 391}414. Olsen, T.E., 1992. Durable goods monopoly, learning by doing and the Coase conjecture. European Economic Review 36 (1), 157}177. Stokey, N.L., 1979. Intertemporal price discrimination. Quarterly Journal of Economics 93 (3), 355}371. Stokey, N.L., 1981. Rational expectations and durable goods pricing. Bell Journal of Economics 12 (1), 112}128. Xie, J.H., Sirbu, M., 1995. Price competition and compatibility in the presence of positive demand externalities. Management Science 41 (5), 909}926.
Network externalities and the Coase conjecture Robin Mason* Department of Economics, University of Southampton, Highxeld, Southampton SO17 1BJ, UK Received 1 December 1998; accepted 2 December 1999
Abstract This paper addresses two general questions. First, what is the e!ect of market structure on the development of a network in a dynamic model with rational expectations? Secondly, is the intuition that network externalities are &economies of scale on the demand side' correct? These questions are examined in a model of durable good production in the presence of network externalities. Two results are presented. First, the Coase conjecture fails in its strongest sense when network bene"ts are increasing in the current network size. Secondly, a committed monopolist may be socially preferable to a time consistent producer when network externalities are su$ciently large. The analysis indicates an analogy between network externalities and learning-by-doing. 2000 Elsevier Science B.V. All rights reserved. JEL classixcation: C73; C78; D42; L12 Keywords: Coase conjecture; Network externalities
1. Introduction This paper examines the production of a durable good in the presence of network externalities in order to address two general questions. First, what is the e!ect of market structure on the development of a network in a dynamic
* Tel.: #44-1703-593268; fax: #44-1703-593858. E-mail address: [email protected] (R. Mason). 0014-2921/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 4 - 2 9 2 1 ( 9 9 ) 0 0 0 6 7 - 7
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model with rational expectations? Secondly, is the intuition that network externalities are &economies of scale on the demand side' correct? The paper brings together two sets of literature. The "rst details the conditions under which the Coase conjecture (Coase, 1972) can be expected to hold. Central to this work is a careful treatment of the time consistency issues that arise when a good is durable and consumers anticipate future price changes. See Bulow (1982) and Gul et al. (1986). The second considers the e!ect that network externalities have on market outcomes. This research examines (amongst other issues) the tendency for concentrated industry structures when there are network e!ects. See e.g. Katz and Shapiro (1985) and Farrell and Saloner (1985). While a few authors have suggested the importance of combining the two approaches, there has not yet been a satisfactory treatment of the problem of durable good sales with network externalities. Katz and Shapiro (1986) "rst raised the possibility that network externalities might lead to the monopoly price of a durable good increasing over time. The intuition is straightforward. Network size is equivalent to a parameter of vertical di!erentiation: the larger the network, the higher the value of the good to consumers. Therefore, a lower initial price which increases the number of initial users raises the value of the good to future consumers; this may allow the monopolist to charge a higher price later. Bensaid and Lesne (1996) provide an explicit analysis of this possibility in a discrete time model. They show for network externalities of a certain type that the monopolist prices above marginal cost at all times; and (if the externalities are su$ciently large) price may rise over time and the monopolist's pro"ts are una!ected by its inability to commit to a production plan. This paper develops a model of production of a durable good in the presence of network externalities. It asks: what is the (strong) Markov perfect equilibrium of the model when producers have an in"nitesimal period of commitment? The use of a discrete time model implicitly allows the monopolist a period of commitment. Stokey (1981) makes clear that this is a crucial assumption. The continuous time framework used here means that the implicit commitment period is zero. Xie and Sirbu (1995) also analyse a continuous time model of durable good sales; but they allow commitment on the part of the monopolist, who chooses open-loop rather than feedback strategies. Two results are presented. First, the Coase conjecture may fail in its strongest form when network externalities are present. For a broad class of network bene"t functions (those that are increasing over some range of the network size), the Markov perfect equilibrium is the same for monopolistic and perfect competitive market structures. Socially optimal pricing (price equal to marginal cost) prevails in both cases. But the time-consistent producers grow the network more slowly than is socially optimal. Secondly, a committed monopolist may be socially preferable to a time consistent producer when network externalities are su$ciently large. The former grows the network at the socially optimal rate, but
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restricts the long-run size of the network. The latter grows the network too slowly, but attains the socially e$cient network size in the long-run. These results indicate that the model of network externalities can be viewed as a demand-side counterpart of Olsen's (1992) learning-by-doing model.
2. The model The model is based on Stokey (1981) and Karp (1996). A monopolist chooses output to maximise the present value of a discounted stream of pro"ts from production of a durable good. There is no depreciation of the good; there are no capacity constraints; the monopolist must sell, rather than rent its output; and the monopolist uses an in"nite time horizon. There is a continuum of nonatomic, in"nitely-lived consumers, each with a demand for one unit of the durable good. They do not consider the e!ect of their decisions on others, and hold rational expectations about the monopolist's production plans. The notation is as follows: output at time t is q(t); the stock of the durable good (the state variable) is Q(t); the selling price is P( . ); constant unit production costs are c; and the common continuous time discount rate is r. There is common knowledge of all aspects of the model. Consumers' utility from purchasing the good has two components. The "rst is intrinsic, derived from the #ow of services from the good itself; this is denoted b. It is assumed throughout the paper that intrinsic valuations are private information, but it is common knowledge that they are uniformly distributed on the interval [b, 1]. In addition, there is a network e!ect, so that the gross surplus derived byM a consumer with intrinsic valuation b buying at time t is v "b#kn , (1) R R where n is the size of the network externality at time t and k50 is a scaling R parameter. The network externality can be interpreted quite generally, and represents any increase in the gross valuation of the good when the total quantity demanded increases. One speci"c and simple example is n "Q : that is, R R the current network externality is equal to the current network size. Bensaid and Lesne (1996) call this type of externality &excluded', and cite computer software as an example } early buyers of software do not bene"t from the externalities that they generate, such as discovery of bugs, unless they buy updated versions. The only type of externality ruled out at this stage is one which arises from sales of an associated non-durable good; this case is considered by KuK hn and Padilla (1996), who show that the Coase conjecture fails.
To be precise: a unilateral deviation by a consumer from its equilibrium strategy does not change the actions of other consumers or the monopolist.
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The following assumptions are made about the network function n( . ): (i) it is non-negative: n 50 ∀t; (ii) it is Markovian and stationary: n ,n(Q(t)); (iii) it is R R piecewise continuous; and (iv) there exists at least one value QK such that 1!QK #kn(QK )"c. Assumption (i) con"nes attention to positive network externalities. The Markovian assumption in part (ii) is less restrictive than it appears. Consumers' expectations of future output are assumed to depend only on the current level of the durable good stock i.e. Q(t#s)"Q(Q(t), s) for some continuously di!erentiable function Q( . ) (where Q(s) denotes the stock of the good expected at time s). It is assumed further that consumers' expectations are ful"lled in equilibrium: Q(s)"Q(s) ∀s. Consequently, although network bene"ts may depend on future sizes of the network, they can be written as a function of the current network size. Assumption (iii) ensures (local) existence of a solution to the problem that is analysed; assumption (iv) ensures that a steady state exists. Finally, the following parametric restrictions are made: (v) kn (Q)(1 ∀Q; and (vi) b#kn(1!b)4c. Part (v) ensures that the inverse demand function for the M services of the M good is downward sloping (see below); it requires k to be su$ciently small. The size of the network in period t is therefore Q(t)"1!b , R where b is the intrinsic valuation of the marginal consumer who is indi!erent R about buying at time t. Part (vi) means that price eventually is driven to marginal cost and the market covered. In other words, this paper examines only the &no gap' case identi"ed by Gul et al. (1986). It is straightforward to modify the paper's results for the &gap' case (which Bensaid and Lesne, 1996 analyse); the main conclusions are unchanged. Note that the possibility of equilibrium multiplicity in the &no gap' case identi"ed by Gul et al. (1986) is not an issue here, since the continuity assumptions impose the regularity required for uniqueness. In order to derive the correct continuous time expression for the equilibrium price function, the model is written "rst in discrete time (as in Gul et al., 1986); the time between periods is then reduced to zero. The extensive form of the discrete time game is: in each period, the monopolist "rst names a price at which he is willing to trade. Consumers then either accept the o!er, or reject. Once a consumer has accepted, she drops out; those that reject continue to receive
Stokey (1981) shows that multiple equilibria exist when expectations are discontinuous in the state; see also the discussion in Karp (1996). A stronger assumption is that the network function is stationary. Ausubel and Deneckere (1989) show that non-stationarity (of preferences and strategies) leads to a folk theorem: there is a continuum of subgame perfect reputational equilibria with monopoly pro"ts ranging from zero to the open-loop, full commitment level. There is little to be gained from simply repeating this result in a model with network externalities } hence, the focus on the stationary case. That expectations are Markovian is consistent with the assumptions in the discrete time version of the model that consumers' decision rules of whether to buy or not are Markovian and that the monopolist's strategy is Markovian.
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o!ers in the future. Each period is of length D. The monopolist's strategy speci"es the price P he will charge in each period t; a strategy for a consumer R speci"es whether to accept or reject the current price o!er of the monopolist. Strategies are assumed to be stationary and Markovian. The monopolist's beliefs about the types of consumers are revised according to Bayes' rule. Consider the acceptance/rejection decision of the marginal consumer in some arbitrary period t. Let this consumer have intrinsic valuation b ; and let the R network function at time t be n . Then, the consumer is indi!erent between R buying one unit of the durable good at time t, and deferring purchase until the next period t#D, if b #kn !P "d(b #kn D !P D ). (2) R R R R R> R> Here, the per-period discount rate is d"exp(!rD). Note that Eq. (2) considers only a unilateral deviation by the marginal consumer: if she does not purchase the good at time t, nevertheless, the monopolist sells so that the network externality is n D in the next period. R> Eq. (2) can be rearranged, d replaced with exp(!rD), and exp(!rD) approximated by 1!rD for small D, to give
k(1!rD) n D !n 1!rD P D !P R> R "P ! R> R . b #kn ! R R R r r D D
(3)
If the total quantity sold during the interval D is bounded for su$ciently small D (i.e. the stock of the durable good changes continuously), then the limit of Eq. (3) as DP0 can be taken, to give a di!erential equation for the price function dP dn R "rP !r(b #kn )#k R . R R R dt dt
(4)
Eq. (4) can be rewritten to make comparison with Stokey (1981) and Karp (1996) clearer. The inverse demand for services from the durable good, or rental rate, can be written as F ,r(b #kn ), i.e. the constant #ow of utility per period R R R which, when received in perpetuity from time t, yields a present discounted value of total utility of b #kn . Therefore, R R dP dn R "rP !F #k R . (5) R R dt dt When the quantity sold during the interval D is not bounded } i.e. when there is a discrete jump in the stock of the durable good sold, from Q to Q , say } then rationality requires that the price just before the jump must be equal to the price immediately after: P(Q )"P(Q ). In other words, consumers anticipate any discrete change; in the continuous time limit, any price di!erential would represent an arbitrage opportunity.
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The economic intuition for Eq. (5) can be understood by supposing that there is a perfectly functioning second-hand market for the durable good. The equality states that the marginal consumer should be indi!erent between borrowing to buy the good, deriving an instantaneous utility, and reselling to pay o! the loan. Therefore, utility F should equal the interest payment rP on the loan minus the price capital gain dP/dt plus the &network capital gain', kdn/dt. If this last term is positive, then the consumer has an added incentive (relative to the no network externality case) to delay purchase until the network is larger. In summary: the assumptions of continuous Markovian expectations and consumer rationality have two consequences for the price function in Eq. (5). The former implies that the price function is a continuous and almost everywhere di!erentiable function of the current state. When the stock of the durable good changes continuously, price therefore changes continuously. The latter means that consumers anticipate any discrete jump in the state; and so the price just before the jump must be equal to the price immediately after.
3. The Markov perfect equilibrium This section derives the strong Markov perfect equilibrium for the continuous time model. It is convenient to take the monopolist's strategy at time t as a choice of output, q . Taking the price function as given, the monopolist's R problem is
max exp(!r(s!t))q [P(Q)!c] ds Q OQ Y R dQ(t) "q(t), Q(0) given. dt
s.t.
(7)
The solution to problem (7) and the function P(Q) generate a price path which in equilibrium satis"es Eq. (5). The monopolist cannot commit to a production
In Stokey (1981) and Karp (1996), the di!erential equation for the price function is derived from the requirement that
P( . )"
exp(!r(s!t))F(Q(s)) ds (6) R i.e. the current price is the present discounted value of the future expected rents. This approach is not adopted here because it would mask the fact that only unilateral deviations by consumers are considered in Eq. (2). It is this that gives rise to the additional term kdn/dt in Eq. (5).
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path, but must choose a stationary Markovian decision rule for output. The necessary condition for pro"t maximisation is given by the Bellman equation
*J rJ"max P(Q)!c# q. (8) *Q OY J is a solution to the Bellman equation; it is veri"ed below that it is di!erentiable in Q (as assumed in Eq. (8)), and that it is also a value function of problem (7). The problem is linear in the control q, and so the optimal solution may involve discontinuities in output. q is therefore restricted to be a piecewise continuous function of time. Let F(QK )"rc. Consider any interval in [Q , QK ) over which output is non-zero and "nite. The linearity of the Bellman equation means that the singular solution *J P(Q)!c# "0 *Q
(9)
must hold over this interval. Therefore J"0; and hence *J/*Q"0 and P(Q)"c. Consumer rationality (i.e. Eq. (5)) then requires 0"rc!F(Q)#kn (Q)q.
(10)
If kn (Q)"0 on this interval, then Eq. (10) implies that F(Q)"rc i.e. Q"QK (a constant). This is inconsistent with q'0 over this interval. If kn (Q)(0, then Eq. (10) implies that q"(F(Q)!rc)/kn (Q). Since output must be non-negative, this requires that Q5QK . This is inconsistent with a stable steady state (with Q"QK ). Therefore, output can be non-zero and "nite only when kn (Q)'0. Consider now an interval in [Q , QK ] over which q"0. Clearly, the stock Q is constant during such an interval; and, due to stationarity, so is the price P(Q). Therefore P(Q)"F(Q)/r, from Eq. (5). If Q(QK , P(Q)'c, and the time-consistent monopolist sets output above zero. Therefore, q"0 is consistent with equilibrium only when Q"QK and P(Q)"c. Finally, consider an interval in [Q , QK ) over which q is in"nite, producing a discrete change in the durable good stock. Since the problem is linear in the control, the principle of the Most Rapid Approach Path applies (see Clark, 1990). The stock is increased to the start of a singular interval, or to the steady state if the former does not exist. In either case, price equals cost after the jump in the stock; and, by consumer rationality, therefore equals cost before the jump. The preceding argument shows that the unique Markov perfect equilibrium is as follows: P(Q)"c, F(Q)!rc q" , kn (Q)'0, Q(QK , kn (Q)
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Fig. 1. The steady state in the Markov perfect equilibrium: n '0.
"R, kn (Q)40, Q(QK , "0,
Q"QK .
(11)
In equilibrium, the monopolist prices at marginal cost and earns zero pro"ts (as predicted by the Coase conjecture). Over any interval in [Q , QK ) where kn (Q)'0, the rate of production is non-zero and "nite. Production continues until the stock of the durable good reaches the steady state level of QK , which is determined by the intersection of the network function n(Q) with the line (c!1#Q)/k; this is illustrated in Fig. 1 for the case when kn '0 ∀Q3[Q , QK ]. The time to reach the steady state may be greater than zero, and may even be in"nite (depending on the form of n( ) )). Consider next production of the durable good by a perfectly competitive industry. Irrespective of the degree of compatibility of the "rms' goods, price equals marginal cost. At the same time, the competitive price path must satisfy Eq. (5). The competitive equilibrium is therefore precisely the same as the monopolistic (Markov perfect) equilibrium: in both, price is set at marginal cost and production occurs according to Eq. (11).
The degree of compatibility may have an e!ect on whether perfect competition is sustainable; see e.g. Katz and Shapiro (1985). This issue is not considered here: the question is, conditional on there being perfect competition, what is the equilibrium price and stock of the durable good?
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Finally, consider socially optimal production of the durable good. The Bellman equation for the social planner is
*< F(Q) r<(Q)"max !c# q, r *Q OY where <(Q) is the planner's value function. For a non-zero but "nite production rate to be optimal, there must be an interval over which F(Q) *< !c# "0. r *Q
(12)
During such an interval, <(Q)"0 and F(Q)/r"c. But the latter equality implies that qF (Q)"0; since F (Q)(0, q must equal zero. The singular interval is, therefore, the point Q"QK . The linearity of the problem means that it is optimal to adjust the stock of the durable good instantaneously to the steady state level QK at the start of the planning period. Comparison of the programs of the time-consistent producers and the social planner gives the following proposition. Proposition 1. The Markov perfect equilibria of monopolistic and perfectly competitive industries are identical. In both, price equals the socially optimal level of marginal cost. If kn (Q)'0 over any interval in [Q , QK ), then monopoly and perfect competition exhibit delay: the network grows too slowly compared to the social optimum. Proposition 1 shows that there is no loss in welfare due to market power; nevertheless, the strong form of the Coase conjecture fails when network bene"ts are increasing in the current network size (over some interval). The intuition for the pricing part of the result is straightforward. The last consumer to buy has a total valuation equal to the marginal cost of production. Consumers anticipate, therefore, that price will equal cost in the long run. Price drops immediately to this level in the continuous time model. The network growth result is more surprising; the intuition is clearest using Coase's original insight that the time consistent monopolist is equivalent to a sequence of monopolists. Each "rm in the sequence does not gain the full bene"t from an increase in the current size of the network, since future "rms will act against its interest. Each "rm therefore has a lower incentive to produce at a high rate than a producer who can commit, and also (the proposition shows) the social planner. Three points are worth noting about the proposition. First, failure of the strong Coase conjecture occurs with a broad class of network bene"t functions. All that is required for ine$ciency is the function to be increasing over some interval. Secondly, the result is similar to those that emerge from learning-bydoing models in which marginal costs decrease with cumulative production. For
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example, Olsen (1992) "nds that a durable good monopolist produces more slowly than is socially optimal; and that the time-consistent monopoly and perfect competition equilibria are identical. Here, a similar e!ect is at work, but on the demand, rather than supply, side. Finally, Bensaid and Lesne (1996) argue that, when the network externality function is n "n(Q ) and k is R R\ su$ciently large: (i) price is always above marginal cost, (ii) price rises over time, and (iii) commitment is of no value to the monopolist. None of these features arise here, for three reasons: (i) the &no gap' case is analysed (so that price falls eventually to cost); (ii) a continuous time approach is used (so that price falls immediately to cost); and (iii) network externalities cannot be too large (so that the solution identi"ed exists and is stable). A further welfare comparison is of interest. The time consistent monopolist is socially ine$cient, since it grows the network too slowly; but it does not restrict the long-run size of the network. A &committed' monopolist (one who uses an open-loop production strategy) is socially ine$cient, not because it sells too slowly (all production occurs immediately, as Stokey, 1979 shows), but because it sets the long-run network size too low. Which monopolist is socially preferable depends on the size of the network externality (amongst other things). When the externality is small, the time consistent monopolist causes little delay. For a larger externality, the Markov perfect equilibrium involves considerable delay; the welfare loss of this delay may be greater than the loss due to output restriction by the committed monopolist. This general result is illustrated in Proposition 2 for the speci"c case of n(Q)"Q. Proposition 2. With n(Q)"Q and Q "0, social welfare is higher (lower) when the monopolist is unable to commit, relative to the commitment case, when k((') 0.4. Proof. In this case, QK "(1!c)/(1!k), Eq. (11) implies that production of the time consistent monopolist is q"(r/k)(QK !Q), and the committed monopolist produces so that the stock jumps instantaneously at t"0 to QR" (1!c)/2(1!k). Social welfare from these production programs is 3(1!c) (1!c) < " , < " . +.# -*# 2!k 8(1!k) The result is immediate. 䊐
This form is chosen for its tractability, and also because it is the continuous time version of the network externality function used by Bensaid and Lesne (1996). In fact, the proposition can be proved for any isoelastic network externality function.
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4. Conclusions This paper has examined the production of a durable good in the presence of network externalities. It has shown that network externalities can cause the strong form of the Coase conjecture to fail. In this model, market structure makes no di!erence to the price or production in the Markov perfect equilibrium. When network bene"ts are increasing in the network size over some interval, however, both industries grow the network too slowly compared to the social optimum. This leads to the possibility that it may be socially preferable to allow a monopolist the ability to commit (e.g. by renting rather than selling the durable good). The analysis suggests an analogy between network externalities and learning-by-doing. The key is that, in both, each "rm in the sequence which makes up the time-consistent monopolist does not consider the welfare of future versions of himself when making its current decision. With network externalities, this leads to delay in network growth; with learning-by-doing, costs decrease too slowly. This provides further support for the intuition that network externalities can be viewed as &economies of scale on the demand side'.
Acknowledgements I would like to thank Larry Karp for many comments on various drafts which have improved the paper substantially. Discussions with David Newbery and David Myatt, and comments from two referees, have also been very helpful. Any errors are my own. Funding from Alcatel Bell is gratefully acknowledged.
References Ausubel, L.M., Deneckere, R.J., 1989. Reputation in bargaining and durable goods monopoly. Econometrica 57 (3), 511}531. Bensaid, B., Lesne, J.P., 1996. Dynamic monopoly pricing with network externalities. International Journal of Industrial Organization 14 (6), 837}855. Bulow, J.I., 1982. Durable goods monopolists. Journal of Political Economy 90 (2), 314}332. Clark, C.W., 1990. Mathematical Bioeconomics, The Optimal Management of Renewable Resources, 2nd Edition. Pure and Applied Mathematics Series. Wiley, New York. Coase, R.H., 1972. Durability and monopoly. Journal of Law and Economics 15 (1), 143}149. Farrell, J., Saloner, G., 1985. Standardization, compatibility, and innovation. Rand Journal of Economics 16 (1), 70}83. Gul, F., Sonnenschein, H., Wilson, R., 1986. Foundations of dynamic monopoly and the Coase conjecture. Journal of Economic Theory 39 (1), 155}190. Karp, L., 1996. Depreciation erodes the Coase conjecture. European Economic Review 40 (2), 473}490.
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Katz, M.L., Shapiro, C., 1985. Network externalities, competition, and compatibility. American Economic Review 75 (3), 424}440. Katz, M.L., Shapiro, C., 1986. Technology adoption in the presence of network externalities. Journal of Political Economy 94 (4), 822}841. KuK hn, K.-U., Padilla, J., 1996. Product line decisions and the Coase conjecture. Rand Journal of Economics 27, 391}414. Olsen, T.E., 1992. Durable goods monopoly, learning by doing and the Coase conjecture. European Economic Review 36 (1), 157}177. Stokey, N.L., 1979. Intertemporal price discrimination. Quarterly Journal of Economics 93 (3), 355}371. Stokey, N.L., 1981. Rational expectations and durable goods pricing. Bell Journal of Economics 12 (1), 112}128. Xie, J.H., Sirbu, M., 1995. Price competition and compatibility in the presence of positive demand externalities. Management Science 41 (5), 909}926.