Entry-Deterring Nonlinear Pricing with Bounded Rationality

Transcription

1 Entry-Deterring Nonlinear Pricing with Bounded Rationality Dawen Meng Guoqiang Tian May 7, 2014 Abstract This paper considers an entry-deterring nonlinear pricing problem aced by an incumbent irm o a network good. The analysis recognizes that the installed user base/network o incumbent monopolist has preemptive power in deterring entry i the entrant s good is incompatible with the incumbent s network. This power is, however, dramatically weakened by the bounded rationality o consumers in the sense that it is vulnerable to small pessimistic orecasting error when the marginal cost o entrants alls in some medium range. These indings provide a ormal analysis that helps reconcile two seemingly contrasting phenomena: on one hand, it is very diicult or a new, incompatible technology to gain a ooting when the product is subject to network externalities; on the other hand, new technologies may requently escape rom ineicient lock-in and supersede the old technologies even in the absence o backward incompatibility. Our results thereore shed light on how the market makes transition between incompatible technology regimes. Keywords: Nonlinear pricing, Entry deterrence, Network Externalities, Bounded rationality. JEL Codes: D42, D62, D82 1 Introduction New entrants challenge the monopoly power and depress proits o incumbent irms. Incumbents are thereore strongly motivated to deter entry o newcomers. In act, entry deterrence is among a irm s most important strategic decisions and has long been a central issue in industrial organization theory. Institute or Advanced Research, Shanghai University o Finance and Economics; Key Laboratory o Mathematical Economics (SHUFE), Ministry o Education, Shanghai , China, devinmeng@hotmail.com Department o Economics, Texas A&M University, tian@econmail.tamu.edu 1

2 When acing threat o entry, incumbent irms must decide how to respond. Early theoretical research, notably by Bain (1956), Modigliani (1958), and Sylos-Labini (1962), emphasizes limit pricing: an incumbent irm setting its pre-entry price low enough to make entry appear unproitable. Various researches have oered explanations or limit pricing. Milgrom and Roberts (1982a) show that when the entrant is uncertain about the incumbent s cost, the incumbent tries to signal its low cost and then discourage entry by setting a low price. Kreps and Wilson (1982) and Milgrom and Roberts (1982b) show that when the entrant is uncertain about the incumbent s payos, the incumbent may have an incentive to cut prices even ater entry as a way to build a tough reputation or ighting uture entry. Firms may use other types o signals instead o or in addition to price to deter entry. Bagwell and Ramey (1988), Linnemer (1998), and Bagwell (2007), among many others, show that advertising also has an entry-deterring eect. Espínola-Arredondo et al. (2014) investigate the signaling role o tax policy in promoting, or hindering, the ability o a monopolist to practice entry deterrence. In complete inormation environment, however, the limit pricing theory is criticized on the grounds o credibility o commitment. Game-theoretic research has considered a variety o s- trategies and conditions that might provide credible deterrents to entry. Some literature (Spence (1977), Dixit (1980), Bulow et al. (1985), Maskin (1999), Allen et al. (2000), etc.) suggests that, irms might hold excess capacity in order to deter entrants. Other devices such as R&D expenditures, increased advertising, capital structure, exclusive contract, may also be used as commitments to deter entry. (See Fudenberg and Tirole (1984), Aghion and Bolton (1987), Tirole (1988), Segal and Whinston (2000) and Tarzijan (2007), among many others, or detailed discussion.) In multiproduct environment, it is shown that strategies such as product prolieration, bundling, and diversiication, are all eective entry-deterrent measures. (See Schmalensee (1978), Omori and Yarrow (1982), Whinston (1990), Choi and Steanadis (2001, 2006), Carlton and Waldman (2002), and Nalebu (2004), etc., or detailed discussion.) In many cases, network externalities may serve as barriers to market entry even when the newcomers have superior technologies and oer lower prices. An obvious example is a telephone network. In a world without interconnection, a user will not switch to a new telephone network eaturing better technologies at lower prices as long as there are no subscribers on that network to communicate with. I all consumers postpone purchase o a product with network externalities until the critical mass is reached, then new entrants will not be able to establish themselves. A cheaper product or a product o better quality would not be suicient to gain a user base in the ace o strong network eects which guide users to previously established networks. 2

3 In industrial practice, incumbent irms oten purposely create network externalities to impede entry. Examples are numerous, such as IBM s amous practice o requiring purchasers o its tabulating machines to also purchase tabulating cards rom IBM; Microsot s attempts to bundle Internet Explorer and Oice with Windows operating system to keep out a rival product (e.g., Linux) 1 ; China Mobile s ighting against China Unicom by releasing Fetion, an instant messenger sotware used only between China Mobile s subscribers. It is o great importance to determine how the incumbent s strategies are aected by threat o entry; how the installed user base o a network good can serve as an incumbent s preemptive power to deter potential entry; and what actors inluence an incumbent s entry-deterring ability. We answer these questions in the present paper by analyzing a nonlinear pricing problem aced by an entry-deterring monopolist under asymmetric inormation and network externalities. 2 Our model diers rom the existing literature along several dimensions. First, this paper investigates the inluence o the consumers bounded rationality on the incumbent irm s entry-deterring ability. Even though entry deterrence has long been a central issue in industrial organization theory, not much attention has been given to consumers. In most existing literature, irms (usually an incumbent monopolist and a potential entrant) compete with various orms o strategic weapons, while consumers do not play an active role. In our model, however, the market power o incumbent monopolist depends crucially on consumers belie. We provide an entry-deterrence model where consumers choose whether or not to bypass the present network based on their expectations. Equilibria at which consumers initial belie on network size is ulilled in the sense that it is consistent with the actual outcome o the entry game are characterized. We also show that, under certain conditions, these equilibria are unstable in the sense that they are vulnerable to perturbation o initial expectation. Thereore, the incumbent irm s entry-deterring ability is weakened when acing boundedly rational consumers. This result rationalizes a substantial number o stylized acts that new technologies/brands which have not yet built their own networks could successully supersede the old and mature technologies/brands with installed networks. It thereore throws light on how the market system escapes rom ineicient lock-in due to network eects. Second, this paper draws rom and adds to the literature on mechanism design under bounded 1 This practice stiles competition by reducing the desirability o entry o competing irms into the market o operating systems, and it is the main allegation in the antitrust case against Microsot in In the context o network goods, Fudenberg and Tirole (2000) have shown that an incumbent monopolist may have an incentive to charge a low price to build a large installed user base in order to deter entry with an incompatible product. Their model, however, assumes away the possibility o perorming price discrimination across dierent groups o consumers, which is the ocus o our analysis. 3

4 rationality. The traditional mechanism design theory assumes that agents are able to orecast correctly the key parameters in the economy and then respond rationally to the principal s oer, so the correct design o mechanisms is decisive or achieving economic systems with good welare properties. Yet as agents usually lack ull rationality, the designer s desirable outcome may not be obtained any more. Intuitively, in a society populated by boundedly rational agents, the central planner s policy objective may oten ail or even converge to an undesirable outcome. A great number o studies, both experimentally and theoretically, ocus on the evolutionary properties o mechanism under bounded rationality (See Chen and Gazzale (2004), Healy (2006), Chen (2005, 2008), De Trenqualye (1988), Walker (1984), Cabrales (1999), Sandholm (2002, 2005), Cabrales and Serrano (2011), Healy and Mathevet (2012), etc., or detailed discussion.) In these existing studies, agents bounded rationality is mainly modeled as their non-optimal reports to the mechanism designer due to behavioral biases or orecasting error. We ocus instead on a setup where the agents are assumed to update gradually their participation decisions rather than reports. This makes a main dierence between the previous works and ours: the equilibrium in our model is at least semi-stable rom above since the agents consumption is controlled by the principal as long as they accept the contract. Third, this paper challenges the no distortion at the top convention by integrating t- wo classes o literatures: network externalities and countervailing incentives. The canonical principal-agent model makes two simpliying assumptions. First, there is no externality among agents; second, an agent is assumed to obtain a minimal type-independent level o utility i he rejects the oer made by the principal. Under these assumptions, the optimal contract exhibits no distortion or the best type agent and downward distortions or all other types. (See or example Mussa and Rosen (1978), Maskin and Riley (1984), Baron and Myerson (1982), Myerson (1981), Hellwig (2010), Chade and Schlee (2012), among many others, or detailed discussion.) Many studies challenge this result by relaxing the above two assumptions. Lockwood (2000) shows that when the cost o any agent depends positively on his own eort and that o his co-workers, the optimal contract oered by the principal exhibits a two-way distortion in output. Hahn (2003) builds a model o telecommunication to examine the role o call and network externalities in nonlinear pricing model. It is shown that the presence o call externalities leads to downward distortions o the outgoing call quantity or all types o subscribers including the highest type. Also in a nonlinear pricing setup, Csorba (2008) shows that the joint presence o asymmetric inormation and positive network eects leads to a strict downward distortion or all types o consumers in the quantities provided. The ailure o no distortion at the top result 4

5 makes an appearance also in the countervailing incentives problem where an agent s reservation utility is type-dependent. Lewis and Sappington (1989) show in a regulation model that, when the ixed cost o a regulated irm is type-dependent, the output can be distorted above the eicient level or some types and below the eicient level or others. Maggi and Rodriguee-Clare (1995) give a complete analysis o the principal-agent problem with countervailing incentives. They show that the pattern o the two-way distortion depends crucially on whether the reservation utility o the agent is convex or concave in his private inormation. In our model, the monopolist must provide each customer with surplus at least equal to the maximum utility they could obtain rom an outside market. As a result, the agent s participation constraint may be type-dependent and countervailing incentives could arise. We ully characterize the optimal pure monopoly and entry-deterring nonlinear pricing contracts or dierent kinds o network eects. It is shown that the nonlinear pricing contract exhibits either one-way or two-way distortion in quantities depending on the curvature properties o network. Thereore, our work incorporates as special cases many existing studies, such as Lockwood (2000) and Csorba (2008). It is also shown that the distortionary pattern o entry-deterring contract depends on both the properties o network and eiciency o outside competitors. The remainder o this paper is organized as ollows. Section 2 sets up the basic model. Section 3 gives the benchmark results without entry threats. Section 4 analyzes the entrydeterring model with, respectively, ully and boundedly rational agents. Finally, concluding remarks are oered in Section 5. 2 The model Consider a monopolist who produces a good exhibiting network eects at a constant marginal cost c. There exists a continuum o consumers with heterogenous preerences or the good, and this heterogeneity is captured by the one-dimensional parameter with distribution unction F () and density () over [, ]. As usual in the standard adverse selection model, we assume the ollowing monotone hazard rate properties. ] ASSUMPTION 1 d d 0 d d [ 1 F () () [ F () () The preerence o a customer o type is represented by the linearly separable utility unction ]. U(, q,, t) = v(q) + Ω() t, (1) where q is the amount he consumes (individual consumption), is the total quantity o the product used by all customers in the market (oten reerred to as gross consumption or network 5

6 size) and t is the tari charged by the seller. v(q) and Ω() are oten reerred to, respectively, as intrinsic value and network value. The above speciication shows that the network eect is homogeneous, i.e., network value depends only on gross consumption rather than individual consumption. We assume v > 0, v < 0, and the Inada conditions v (q) or q 0 and v (q) 0 or q. We also assume sup [0,+ ) Ω () < c, the monopolist may otherwise wish to produce unbounded output levels. In the presence o network externalities, each agent s utility depends not only on his own trade with the principal, but also on others trades. The ollowing properties o network externalities are deined according to the inluences o an agent s trade on the other agents total and marginal utilities. DEFINITION 1 (The sign o network externalities) Externalities are positive (negative, absent) i or all, U(, q(),, t()) is increasing (decreasing, constant) in q( ), or all. DEFINITION 2 (The curvature properties o network externalities) Externalities are (strictly) increasing (decreasing, constant) i or all [, ], U q (, q(),, t()) is (strictly) increasing (decreasing, constant) in q( ), or all. In network having positive externalities, a buyer beneits rom an increase in the consumption o other buyers. In the situation with negative externalities, however, an increase o one consumer s consumption hurts the others. This may arise due to congestion or internal competition. Various kinds o networks, such as transportation, communication, and computer networks exhibit congestion eects, whereby increased demand or certain network elements (e.g., roads, telecommunication lines, and servers) tends to downgrade their perormance or increase the cost o using them. The curvature properties o externalities relect the change o marginal utilities o an agent with the consumption o others. With increasing (decreasing) externalities, an agent is more (less) eager to trade more when other agents trade more because the externalities at higher trades are larger (smaller) than at lower trades. I the utility unction is twice dierentiable, the property o positive (negative, absent) externalities involves the sign o irst order derivative U q( ), while the property o increasing (decreasing, constant) externalities involves the sign o cross partial derivative 2 U q() q( ). With the special utility structure in this paper, the externalities are positive (negative, absent) i and only i Ω () > 0, (< 0, = 0) or all ; while the externalities are increasing (decreasing, constant) i and only i Ω () > 0, (< 0, = 0) or all. Suppose there exists an outside market composed o many identical potential entrants. Let U 0 () represent the reservation utility available to an agent o type when he rejects the contract 6

7 and switches to the outside product. The seller cannot explicitly distinguish between customer types prior to contracting. Thus, the entire menu o quantity-price pairs must be available to all customers. The revelation principle ensures that the irm can restrict its attention to a direct mechanism, i.e., a menu o quantity-price pairs {q(ˆ), t(ˆ)}, where ˆ [, ]. A contract {q(ˆ), t(ˆ)} is said to be incentive easible or i, or each, it satisies incentive compatibility and individual rationality constraints 3 : v(q()) + Ω() t() v(q(ˆ)) + Ω() t(ˆ), ˆ [, ] (2) v(q()) + Ω() t() U 0 (). (3) The incumbent s objective is to choose an optimal contract to maximize his expected payo and successully deter entry under constraints (2) and (3). 3 Nonlinear pricing without entry threats In order to give a benchmark result against which entry-deterring contract is compared, we suppose in this section that the incumbent does not ace the threat o entry. In this case, the agents reservation utility would be zero, i.e., U 0 () = 0 or all [, ]. We now proceed to analyze the incumbent s problem by considering in turn ully and boundedly rational agents. 3.1 Fully rational agents In this subsection, the consumers are assumed to be ully rational in the sense that their common expectation on the network size will be actually ulilled, i.e., = q()()d. Let U() = v(q()) t() + Ω() be the rent obtained by type, then it ollows rom the well-known nonlinear pricing result o Maskin and Riley (1984) that the optimal ully rational (ulilled expectation) contract, denoted by {q (), t ()}, solves the ollowing program. [P] : max {q(),u()} [v(q()) cq() U()] ()d + Ω() s.t. : U () = v(q()), U() 0, q () 0, = q()()d Neglecting momentarily the monotonicity condition q () 0 and applying integration by parts technique, the proit unction can be rewritten as a unctional o q(), ( Π[q()] 1 F () ) ( ) [v(q()) cq()] ()d + Ω q()()d. (5) () 3 Since every agent is o measure zero, each agent takes as ixed when deciding on his own announcement ˆ. (4) 7

8 Applying the calculus o variations, we can reormulate the principal s optimization problem as 4 where [P ε ] : maxπ(ε), (6) ε R [( Π(ε) 1 F () ) ] v(q () + εq()) c(q () + εq()) ()d () ( ). + Ω (q () + εq())()d I ε = 0 maximizes P ε (and thus q () maximizes P), then, or any ixed admissible unction q(), we must have Π (0) = [( 1 F () ) ] v (q ()) c + Ω ( ) q()()d = 0. (7) () Since this is true or any variation q(), it ollows that the continuous unction which is the coeicient o q() under the integral sign must vanish identically on [, ]. We thus have the Euler s equation: [ 1 F () ] v (q ()) + Ω ( ) = c. (8) () Note that (8) equates the consumer s marginal revenue taking into account inormational rent and the externality to the marginal cost to the principal o an increase in q(). The irst term on the let-hand side o (8) is the marginal intrinsic value in terms o virtual valuation o raising q() incrementally, and the second term is the marginal network value o raising q(). Letting ( )] q (, ) (v ) [(c 1 Ω ())/ and φ () q (, )()d, we have that 1 F () () the optimal network size is a ixed point o φ (), i.e., = φ( ) and q () = q(, ). The existence o is guaranteed by the boundedness o Ω () and the Inada conditions. 5 I the externalities are nonincreasing, i.e., Ω () 0 or all, then φ () is nonincreasing, and is thus determined uniquely. However, the uniqueness o ixed points is not guaranteed in the case o increasing externalities, wherein an increasing curve φ () might cross the 45 line several times. The monopolist thus simply picks one rom the set = { R + φ () = } to maximize Π () [( 1 F () ) ] ( ) v(q (, )) cq (, ) ()d + Ω q (, )()d. () 4 With a slight abuse o notation, Π still represents the proit o the principal. 5 It ollows rom condition sup x [0,+ ) Ω (x) < c and Inada conditions that φ(0) = q(, 0)()d > 0, φ(+ ) = lim x + q(, x)()d < +. Since q (, ) is continuous, so is φ (). Thereore, there exists a satisying φ( ) =. 8

9 Consider the ollowing parametrized orm [( Π(, ϵ) ϵ 1 F () ) ] ( v(q(,, ϵ)) cq(,, ϵ) ()d + Ω () q(,, ϵ)()d [ ]} where q(,, ϵ) = (v ) {[c 1 Ω 1 F () ()] / ϵ (), ϵ = 0 and ϵ = 1 correspond respectively to the irst-best and second-best cases. Note that Π = [ Ω (φ(, ϵ)) Ω () ] q (,, ϵ)()d = [Ω (φ(, ϵ)) Ω ()] φ, where φ(, ϵ) = q(,, ϵ)()d, so in the case o strictly increasing externalities, any interior maximizer o Π(, ϵ) must be a ixed point o φ(, ϵ). This allows us to search the optimal network size (ϵ) over the whole real line, i.e., arg max Π(, ϵ) = arg max R Π(, ϵ). ), LEMMA 1 I Ω () is o constant sign, given condition sup [0,+ ) Ω () < c, a ixed point is optimal only i φ 1. PROOF. See appendix. It could be seen rom the expression o dπ/d that, at the optimum φ(, ϵ) must cross the 45 line rom above. The points at which φ(, ϵ) is tangent to or crosses the 45 line rom below are not optimal. It ollows directly rom the irst order condition φ(, ϵ) = 0 that d/dϵ = φ ϵ (, ϵ)/[φ (, ϵ) 1] < 0. We thus obtain the ollowing result. LEMMA 2 I Ω () is o constant sign, the network size with incomplete inormation shrinks relative to the irst-best case, i.e., (1) < (0). We can now turn to analyze the distortions induced by the presence o both asymmetric inormation and externalities at the second-best consumption q () q(, (1), 1) relative to the irst-best consumption q () q(, (0), 0). In the standard model without externalities, quantity is distorted downward or all values o except the highest. In the case with externalities, however, this outcome can no longer be sustained. THEOREM 1 The pattern o distortion in the second-best contract may depend on the curvature properties o externalities: I network externalities are increasing, i.e., Ω () > 0 or all, there is one-way distortion: q () < q (), or all [, ]; I externalities are decreasing, i.e., Ω () < 0 or all, there exists two-way distortion: (, ), q () > q () or (, ] and q () < q () or [, ); 9

10 I externalities are constant, i.e., Ω () = 0 or all, then the standard result still holds: q () < q (), [, ), q () = q (). PROOF. See appendix. In the standard nonlinear pricing setup, the principal s rent extraction purpose motivates her to reduce the consumptions or all but the most eicient types. This eect may still exist in our model. Moreover, in our model, the principal also has an incentive to increase every type s consumption and thus make the network more attractive. I the network is increasing, the rent-extraction eect dominates the network-expansion eect or all types. As a result, every type s consumption is lower than that in the irst-best case. I the network is decreasing, however, or the higher types the network-expansion eect dominates, while or the lower types the rent-extraction eect dominates, so two-way distortion is the outcome. With decreasing externalities, an agent would be even more eager to trade more i he expects the other agents trade less with the principal. Thereore, whenever an outcome is ulilled, any initial expectation smaller (larger) than will induce an actual aggregate consumption larger (smaller) than, thus the optimal contract must be uniquely implemented. With increasing externalities, however, an agent would be more eager to trade more i he expects the other agents trade more. In this case, multiple equilibria are more o a norm than an exception. This is due to the positive eedback associated with expectations: i all agents in the economy believe something will not succeed, it will usually ail; on the contrary, i they expect it to succeed, it usually will. Now we need some conditions ensuring the uniqueness o the ixed point o φ (). Let ϕ be deined as ϕ() φ (). Then a ixed point o φ () is a solution o equation ϕ() = 0. I is unique, then ϕ( ) = 0 and ϕ(0) > 0 implies that ϕ is nonnegative everywhere on the line rom to 0. The ollowing deinition ormalizes this property. DEFINITION 3 The unction g : R R is radially quasiconcave ( R concave ) i g(x) = 0 or some x > 0 implies g(λx) 0 or all λ [0, 1]. I the strict inequality holds or all λ (0, 1), then g is strictly R concave. Armed with this concept, we are now able to give a uniqueness result. PROPOSITION 1 (The Uniqueness Result) Suppose that sup [0, ) Ω () < c, then φ () has a unique positive ixed point i and only i ϕ() = φ () is strictly R concave. PROOF. See appendix. Remarks. (i) Since quasiconcavity (quasiconvexity) means convexity o upper (lower) contour sets, a unction ϕ() is (strictly) R-concave i it is either (strictly) quasiconcave or (strictly) 10

11 quasiconvex provided that ϕ(0) > 0 and lim + ϕ() =, but not vice versa. Thereore, any one o the ollowing assumptions on ϕ() is suicient (but not necessary) or equilibrium to be unique: (a) it is strictly concave; (b) it is strictly convex; (c) it is decreasing; (d) it increases slower than. Conditions (c) and (d) hold or decreasing or mildly increasing externalities. (ii) Note that quasiconcavity (quasiconvexity) o φ () does not necessarily imply the quasiconcavity (quasiconvexity) o ϕ(). (iii) It is clear that the strict version o R-concavity cannot be replaced by the weak version. For example, unction ϕ() = ( 1) 2 ( 2), [0, ) satisies ϕ(0) > 0, lim ϕ() =. Also, it is R-concave but not strictly R-concave. O course the uniqueness result ails or this unction. With increasing externalities, φ () is increasing, but ϕ() is not necessarily R-concave. Thereore, multiple ixed points may arise when unction φ () crosses the 45 line more than once. Throughout the rest o the paper, we assume ϕ() to be strictly R-concave to avoid multiple equilibria. 3.2 Boundedly rational agents In the preceding subsection, the second-best contract {q (), t ()} is ulilled i agents could orm rational expectations without any systematic orecasting errors about the real network size. This is in line with the traditional neoclassical view propagated by Muth (1961) and Lucas (1972). This result is in act attributing too much inormation and rationality on the part o agents. It is more reasonable to assume that agents are boundedly rational in their ability o orecasting and decision-making. They modiy adaptively their expectations over time on the basis o observations o past perormance. In this subsection, we will discuss whether or not the ully rational equilibria may emerge as outcomes rom repeated adaptive learning process o boundedly rational agents. Agents must orm an expectation on network size one period ahead and make their participation decisions based on this expectation. Their adaptive learning process works as ollows. At time t, all the agents orm a common expectation e t o the network size, then consumers with { nonnegative expected utilities, i.e., [, ] } v(q ())d Ω( ) Ω( e t ), accept the contract {q (), t ()}. 6 The rest o the agents will reject it and quit the market. This orms 6 In this paper, the agents belies are assumed to be homogeneous, i.e., all the agents have identical expectations e t. Please see Hommes (2006), LeBaron (2006), Hommes and Wagener (2009) and Chiarella et al. (2009), among many others, or detailed discussion o models with heterogeneous expectations. 11

12 an actual network size t = ρ( e t ), where ρ(x) = i Ω(x) [Ω( ), + ) [ (x) q ()()d i Ω(x) max{0, Ω( ) ) v(q ())d}, Ω( ) [ 0 i Ω(x) 0, max{0, Ω( ) ) v(q ())d}, (x) is given implicitly by v(q ())d + Ω(x) = Ω( ) (see FIGURE 1). I t = e t, the ( ) ( ) ( ) v(q ())d e ( t ) ( ) v(q ())d * e ( t ) Consumers rejecting the contract Consumers accepting the contract FIGURE 1. rational expectation outcome is reached; otherwise, the expectation is updated according to an adaptive learning rule e t+1 = α t + (1 α) e t, where α (0, 1] is the expectations weight actor. The expected network size is a weighted average o yesterday s expected and realized values, or equivalently, the expected network size is adapted by a actor α in the direction o the most recent realization. This process is repeated until an expectation is sel-ulilled, i.e., e t = t. is obviously a ixed point o the eedback unction ρ() αρ() + (1 α). In what ollows, we will discuss whether or not the adaptive learning procedures will lead the economy to converge to ully rational equilibrium (FRE). 7 We irst introduce a ew preliminary deinitions and lemmas. 7 Throughout this paper, it is assumed implicitly that there is a new independent draw o the agents types every period, so there is no dynamic contracting issue arising rom the gradual elimination o the inormational asymmetry over time. Moreover, the principal is assumed to believe that the agents are or will eventually be rational. That is, the principal is irrational on the rationality o agents. Thereore, she will always provide the ully rational contract {q (), t ()} in every period. 12

13 DEFINITION 4 A ixed point x o (x) is called attractive i there exists a neighborhood U(x ) such that the iterated unction sequence { (n) (x)} converges to x or all x U(x ). DEFINITION 5 A ixed point x o (x) is called Lyapunov stable i, or each ϵ > 0, there is a δ > 0 such that or all x in the domain, i x x < δ, (n) (x) (n) (x ) < ϵ or all n N. DEFINITION 6 A ixed point x o (x) is called asymptotically stable (an attractor) i it is Lyapunov stable and attractive; it is called neutrally stable i it is Lyapunov stable but not attractive 8 ; it is called unstable (a repeller) i it is not Lyapunov stable. Lyapunov stability o an equilibrium means that solutions starting close enough to the equilibrium remain close enough orever. I x is an unstable ixed point, then there always exists a starting value x very near to it so that the system moves ar away rom x upon iteration: an open interval I containing x, x I\{x }, n > 0 such that (n) (x) / I. Asymptotic stability means that solutions that start close enough not only remain close enough but also converge to the equilibrium eventually. Neutral stability means or all initial values x near x, the solution stays near but does not converge to x. 9 LEMMA 3 Let x be a ixed point o the discrete time dynamical system x n+1 = (x n ), i 0 +(x ) < 1(0 (x ) < 1), then x is asymptotically stable rom above (below); i +(x ) > 1( (x ) > 1), then x is unstable rom above (below). PROOF. See appendix. The above lemma leaves out the case with neutral ixed point, i.e., (x ) = 1, wherein the stability o x could not be determined until urther inormation regarding higher-order terms o Taylor expansion are available. 10 Armed with Lemma 3, we are now able to characterize the stability o. PROPOSITION 2 I Ω ( ) > v(q ())/()q (), then is unstable rom below and asymptotically stable rom above; i Ω ( ) < v(q ())/()q (), then is asymptotically stable rom both sides. 8 It is also possible or a ixed point to be attractive but not Lyapunov stable. For example, trajectories starting at any point x 0 may always go to a circle o radius r beore converging to x. 9 The center o a linear homogeneous system with purely imaginary eigenvalues is an example o a neutrally stable ixed point. 10 When x is neutral, nothing deinitive can be said about the behavior o points near x (Holmgren (1991), p.53), and several situations are possible: x may be stable, unstable, semistable or neutrally stable. 13

14 PROOF. See appendix. This theorem shows that when the externalities are negative or weakly positive at optimum, the equilibrium is robust to small initial orecasting errors. Adaptive learning processes o consumers lead the economy to converge toward the unique rational expectation equilibrium. When the externalities are strongly positive, however, the equilibrium is only semi-stable rom above. The basic intuition behind this result is that the principal is capable o preventing the overheating but not necessarily the overcooling o economy. The overoptimistic expectations o consumers boost their participation in contracting but the good is not overconsumed since the principal controls their total consumptions. However, overpessimistic expectations will decrease the real network size since a set o consumers with positive measure will choose to quit the market. 11 In the case with negative externalities, i all consumers initially orm a shared overpessimistic prior on the network size, then a raction o consumers will quit the market and the good is actually underconsumed. The adaptive learning process will lead the consumers gradually to an overoptimistic expectation. Ater that, all the consumers participate in contracting and the expectation converges increasingly to the ully rational equilibrium. Things are dierent or the case with positive externalities. I externalities are weakly positive and consumers orm an overpessimistic expectation, the realized network size will outperorm their expectation, then the consumers over-pessimism disappears eventually. In contrast, with strongly positive externalities, the low initial expectation reduces the real aggregate consumptions, which in turn conirms the expectations. The eects that pessimism and depression reinorce each other lead the network size to a very low level. Proposition 2 encompasses several special cases o interest, which are depicted in the ollowing FIGUREs 2 to 5 with α = 1. Case a. Ω () > 0 or all [0, + ), Ω( ) > v(q ())d, Ω ( ) < v(q ())/()q () ( ρ() is strictly concave in [Ω 1 Ω( ) ) ] v(q ())d,. In this case, such that ρ(), [0, ) and < ρ(), (, ). When the curve ρ() lies below the 45 line, we have downward pressure on the consumption o the good: the realized network size outcome will underperorm the consumers expectation, and there will be a downward spiral in consumption. Correspondingly, when the curve ρ() lies above the 45 line, we have upward pressure on the consumption o the good. is not just an unstable equilibrium, but is actually a critical point, or a tipping point, in the success o the good. I the irm producing the good can get the consumer s expectations or the total 11 In the case with positive externalities, agents with expectations larger than the equilibrium value are called optimistic, and those who have expectations smaller than the equilibrium value are called pessimistic. It is opposite or negative externalities. 14

15 consumption above, then it can use the upward pressure o demand to get its market share to the stable equilibrium at. On the contrary, i the consumer s expectations are even slightly below, then the downward pressure will tend to drive the market to shut down. This result suggests that the success o a product depends crucially on consumers initial conidence on it. Case b. Ω () > 0 or all [0, + ), Ω( ) > v(q ())d, and unction σ() = ρ() is strictly R-concave on (0, + ). In this case, unction ρ() has two ixed points 0 and. Since the curve ρ() lies below the 45 line, there is always downward pressure on the consumption o the good. Any initial overoptimistic expectation will disappear gradually. However, even a very small perturbation in the let will lead the market to shut down. Case c. Ω () > 0 or all [0, + ), Ω( ) v(q ())d, σ() = ρ() is strictly R-concave on (0, + ). In this case ρ() has a unique ixed point, where ρ() crosses the 45 line rom above. Thereore, is globally stable. It is robust to perturbation in any amount and either direction. Case d. Ω () 0 or all [0, + ). In this case is also globally stable due to the negative eedback eect. Realized network size t 45 e ( t ) ' 1 [ ( ) v( q ( )) d] Shared expectation e t FIGURE 2. Case a 15

16 Realized network size t 45 e ( t ) 1 [ ( ) v( q ( )) d] Shared expectation e t FIGURE 3. Case b Realized network size t 45 e ( t ) Shared expectation e t FIGURE 4. Case c 4 Nonlinear pricing with entry threats The monopolist in the above model may ace a threat o entry rom rival irms, whose product is intrinsically a perect substitute or the monopolist s product, but is incompatible with the existing network. By virtue o being the incumbent, the monopolist s product generates network value or all customers. The entrant s product, on the other hand, is assumed to provide only its intrinsic value to the customers. This section discusses the optimal entry-deterring nonlinear pricing contracts when consumers are, respectively, ully and boundedly rational. 16

17 Realized network size t 45 e ( t ) 1 [ ( ) v( q ( )) d] Shared expectation e t FIGURE 5. Case d 4.1 Fully rational agents We assume that the competitive outside rivals set their price equal to their marginal cost ω. Thereore, in order to deter entry, the monopolist s pricing scheme must provide customers o type with a surplus o at least U 0 () = max q [0, ) [v(q) ωq]. Theoretically, this is a principal-agent model with type-dependent individual rationality constraints. q 0 () arg max q [0, ) [v(q) ωq], U() v(q()) t() + Ω() U 0 (). Facing ully rational consumers, the monopolist s entry-deterring problem can be written as: [ [P e ] : max v(q()) cq() U 0 () U() ] ()d + Ω() q(),u() subject to: U () = v(q()) v(q 0 ()), q()is nondecreasing, U() 0, = q()()d. Note that the consumer s rents may either increase or decrease with depending on the comparison between q() and q 0 (). I the quantity a consumer purchases rom the incumbent irm exceeds that rom the outside rivals, rents rise with ; otherwise they all with. Neglecting momentarily the monotonicity condition o q() and checking it ex post, we can write the principal s problem as a standard control problem with U as state variable and q as control variable. The Hamiltonian unction or this control problem would take the orm H(U, q, µ, ) = [ v(q()) cq() + U 0 () U() ] () ( ) + Ω q()()d + µ()[v(q()) v(q 0 ())] 17 Let

18 where µ is the costate variable. The Lagrangian unction is L = H + γ()u(), where γ() is the multiplier o constraint U() 0. The irst-order condition or the maximization o the Hamiltonian with respect to q is [ + µ() ] v (q()) + Ω () = c. (9) () For a ixed network size, all variables (q, µ, U, γ) being represented as unctions o and, the ollowing conditions must be satisied: costate equation: µ = L U = () γ() (10) state equation: U = v(q(, )) v(q 0 ()) (11) complementary slackness: γ(, )U(, ) = 0, γ(, ) 0, U(, ) 0 (12) transversality conditions: µ(, )U(, ) = µ(, )U(, ) = 0, (13) µ(, ) 0, µ(, ) 0. Let ˆq(µ,, ) denote the value o q that maximizes the Hamiltonian given µ, and deined implicitly by (9), and let ˆµ(, ) be the solution in µ to the ollowing equation ˆq(µ,, ) = q 0 (). It can be easily obtained ˆµ(, ) = [(c Ω ()) /ω 1] (), ˆµ(, ) is the value o costate variable such that the agent s utility is constant (U = 0). From (10) and (12), we have γ(, ) = () µ (, ) 0, then there must be µ (). I the IR constraint is binding on a nondegenerate interval, then µ(, ) must be equal to ˆµ(, ) and µ = ˆµ < () on that interval. To get an optimal costate unction µ(), we impose the ollowing assumption. ASSUMPTION 2 Function F () () is nondecreasing. This assumption is obviously a bit stronger than the usual monotone hazard rate condition which only requires F ()/() to be nondecreasing. In order to solve the problem, we conjecture a solution and then veriy whether or not it satisies conditions (10) to (13). The critical work is to construct the right solution or µ(, ). Consider the ollowing unction, F () i ˆµ(, ) F () µ (, ) = ˆµ(, ) i F () 1 < ˆµ(, ) < F () F () 1 i ˆµ(, ) F () 1. (14) For simplicity, we deine Θ 1,Θ 2, and Θ 3 as the subintervals o Θ where µ (, ) is equal to F (), ˆµ(, ) and F () 1, respectively. In order or µ (, ) to satisy the conditions (10) to (13), we 18

19 need to check the condition µ (). It is obviously satisied on Θ 1 and Θ 3. So we need only to check µ = ˆµ (, ) () or Θ 2. Notice that ˆµ (, ) = [() + ()]ˆµ(, )/(). I + 0, then we have ˆµ (, ) < F ()[() + ()] () (), where the last inequality ollows directly rom Assumption 2. I + < 0, then we also have ˆµ (, ) < [F () 1][ () + ()] () 2 () [1 F ()] () () = () 1 F () (). The second inequality ollows rom Assumption 1. Thereore, the optimal quantity q + c Ω (, ) i () ω 1 + F () () q (, ) ˆq(µ (, ),, ) = q 0 1 F () () i 1 () < c Ω () ω < 1 + F () () q (, ) i 1 c Ω () ω is obviously nondecreasing in, where q (, ) = (v ) [(c 1 Ω ())/ ( (v ) [(c 1 Ω ())/ + F () () 1 F () () ( (15) )] 1 F () (), q + (, ) = )]. We then need to distinguish the ollowing cases, depicted in FIGURE 6 to FIGURE 10, depending on the values o and ω: CASE 1 c Ω () ω < 1 1 (). In this case ˆµ(, ) < F () 1 and thus q (, ) = q (, ) or all [, ]; CASE () c Ω () ω < 1. In this case F () 1 ˆµ(, ) < 0, [, ()) and ˆµ(, ) < F () 1, ( (), ]. Thereore, we have q q 0 () i [, ()) (, ) = q (, ) i [ (), ], where () is the critical type given implicitly by c Ω () ω = 1 1 F () () ; CASE 3 c Ω () ω or all [, ]; = 1. In this case ˆµ(, ) = 0, and q (, ) = q (, ) (v ) 1 [(c Ω ())/] CASE 4 1 < c Ω () ω < (). In this case ˆµ(, ) > F () or [, + ()); 0 < ˆµ(, ) < F () or [ + (), ). Thereore, we have q q + (, ) i [, + ()) (, ) = q 0 () i [ + (), ) 19

20 where + () is the critical type given implicitly by c Ω () ω = 1 + F () () ; CASE 5 c Ω () ω (). In this case, ˆµ(, ) F () and thus q (, ) = q + (, ) or all [, ]. 1 F( ) F( ) 1 ˆ, -1 FIGURE 6. Case 1: c Ω () ω < 1 1 () 1 F( ) ˆ, F( ) 1-1 FIGURE 7. Case 2: 1 1 () c Ω () ω < 1 20

21 1 F( ) ˆ, F( ) 1-1 FIGURE 8. Case 3: c Ω () ω = 1 1 ˆ, F( ) F( ) 1-1 FIGURE 9. Case 4: 1 < c Ω () ω < () φ () q (, )()d is consequently a piecewise unction given by φ + c Ω () i () ω () + () q + (, )()d + + () q0 ()()d i 1 < c Ω () ω < φ () = φ c Ω () i () ω = 1 () q 0 ()()d + () q (, )()d i 1 > c Ω () ω 1 1 φ c Ω () i () ω < 1 1 () () (), (16) where φ i () = qi (, )()d, i {,, +}. Let i be the ixed point o φ i (), and q i () q i (, i ), i {+,, }. We assume throughout this section that φ i (), i {+,, } are all 21

22 ˆ, F( ) F( ) 1 FIGURE 10. Case 5: c Ω () ω () strictly R-concave, then i are uniquely determined. It is easy to ind that < < + and [ ] [c Ω ( )] / 1 1 () = v (q ()) > v (q ()) = c Ω ( ) = v (q ()) > v (q + ()) = [ ] [c Ω ( + )] / We now proceed to determine the ixed point o φ () and () then characterize the optimal contract {U (), q ()}. In particular, it is interesting to compare the optimal allocation and network size with those in the ull-inormation case. THEOREM 2 Given assumptions (1) and (2), i the sign o Ω () is constant on [0, + ), sup 0 Ω () < c, and φ i (), i {,, +} are all strictly R-concave, then the optimal nonlinear pricing contract o incumbent irm changes with the marginal cost o potential entrants: I ω [, ]; ( I ω [ c Ω ( ) ) 1 1/(), +, then =, q () = q (), U () = [ v(q ()) v(q 0 ()) ] d, c Ω ( ), c Ω ( ) 1 1/() U () = ), then <, q q 0 () i [, ( )) () = q (, ) i [ ( ), ] ( ) 0 i [, ( )) [ v(q (, )) v(q 0 ()) ] d i [ ( ), ] 12 Analogous to THEOREM 1, we have the ollowing result: i Ω () > 0,, then q + () > q (), [, ]; i Ω () < 0,, then [, ] such that q + () < q (), [, ), q + () > q (), (, ]; i Ω () = 0,, then q + () = q (), q + () > q (), (, ]. Thereore, i Ω () has constant sign, we always have q () < q + ()., ; 22

23 I ω = c Ω ( ), then =, q () = q 0 () = q (), U () 0, [, ]; I ω [ I ω [, ]. ( ) c Ω ( + ), c 1+1/() Ω ( ), then >, U () = 0, c Ω ( + ) 1+1/() q q 0 () i [ + ( ), ] () = q + (, ) i [, + ( )) [ + ( ) v(q + (, )) v(q 0 ()) ] d i [, + ( )) 0 i [ + ( ), ] ], then = +, q () = q + (), U () = [ v(q + ()) v(q 0 ()) ] d,, ; PROOF. See appendix. Facing potential entrants, consumers have an opportunity o using an alternative, incompatible but competitively supplied good and the incumbent irm s proit may be depressed, so it is in the interest o incumbents to deter entry i possible. Whether and to what extent would the incumbent irm s pricing strategy be aected by entry threat has constituted a major theme in the marketing and industrial organization literature. In the standard nonlinear pricing model, where the reservation utility o consumers is independent o their types and normalized to zero, a consumer has incentive to underreport his type to earn inormation rents. This will also be the case in our model i the potential entrants are very ineicient (ω [c Ω ( )]/[1 1/()]). In this case, or all types o consumers, the incentive to understate their valuations to get inormation rents always outweighs the incentive to bypass the present network and get their reservation utilities. Thereore, the pure monopoly pricing contract remains optimal. As the potential entrants become more eicient (ω (c Ω ( ), [c Ω ( )]/1 ())), the low demand consumers preer bypassing to staying in the network and understating their types. Thereore, the principal has to oer them less distorted quantities to prevent them rom quitting the present network. I ω = c Ω ( ), all types have incentive to quit the market rather than to misreport their types. So the principal inds that it is no longer optimal to distort quantities away rom the irst-best level. The remaining tool available to the principal to retain all consumers in the market is the transer t. In this case, all consumption levels are the eicient ones. As ω continues to decrease, the outside market becomes attractive to the high demand consumers, so the principal has to oer them quantities higher than the irst-best level to prevent them rom quitting. The low demand types are now attracted by the allocation given to the high demand types. That is, they have incentive to overstate their types and thereby 23

24 convince the seller that greater reward is required to prevent them rom switching to the outside market. I the potential entrants are highly eicient with ω [c Ω ( + )] / [ 1 + 1/() ], then or all types o consumers, the incentive to overstate always dominates their incentive to quit the market, and the agent s participation constraint binds or the highest realization o. For each type, i ω > c Ω ( ) (ω < c Ω ( )), decreasing (increasing) his consumption may (1) reduce the inormation rents obtained by the higher (lower) types; (2) decrease (increase) the whole network value; (3) reduce (intensiy) the congestion o network i the externalities are decreasing. The joint presence o these three eects leads to complex patterns o contract distortions, which are depicted in the ollowing FIGUREs 11 to 15. In each igure, the red curve represents the benchmark case with complete inormation, the blue curve represents the pure monopoly case, while the green curve represents the case with entry threat. The distortionary way o quantities is described in TABLE 1. q () * - - q () q() q(, ) q () * - - q () q() q(, ) * - - q () q() q(, ) q () q (, ) (a) increasing externalities (b) constant externalities (c) decreasing externalities FIGURE 11. ω c Ω ( ) 1 1 () q( ) - - q() q(, ) q q (, ) q(,) q - * q(,) q() q(, ) q 0 q q q(, - * q(, ) 0 q () * q() q(, ) q(,) - * () (a) increasing externalities - * () (b) constant externalities - * ( ) (c) decreasing externalities FIGURE 12. c Ω ( ) < ω < c Ω ( ) 1 1 () The above results show that when network externalities are positive, subtracting a term Ω () rom the incumbent irm s marginal cost c makes him more competitive in ighting against 24

25 * 0 q () q () q () FIGURE 13. ω = c Ω ( ) 0 + q + () q (, ) + * q (,) q q () q (, ) q() q(,) * q () q (, ) q (,) 0 q q q(,) q q (, ) 0 q + * q (,) - - q() q(, ) + + q () q (, ) q q (, ) + * () (a) increasing externalities + * () (b) constant externalities + * () (c) decreasing externalities FIGURE 14. c Ω ( + ) 1+ 1 () < ω < c Ω ( ) his rivals. That implies a network with positive externalities may serve as a powerul anticompetitive weapon. It allows an ineicient incumbent to successully keep out an eicient rival. It is very diicult or newcomers with more advanced but incompatible technology to establish a market position. As a result, the incumbent irm may be reluctant to explore a superior but incompatible technology in the presence o network externalities. In this sense, our indings are consistent with the amous lock-in argument 13 : the diiculty o gaining a ooting by a new, incompatible technology when a product is subject to network externalities. Despite this argument s popularity, ample historical evidence suggests that many new, incompatible technologies have been successully introduced. For example, Microsot Word was introduced ater WordPerect dominated the market; MS-DOS was introduced and subsequently cornered the market ater CP/M-80 became established as the industry standard operating system. Cases o lock-in to inerior technologies are rare in the long history o technological change (Liebowitz 13 See Farrell and Saloner (1985, 1986), Katz and Shapiro (1985), Farrell and Klemperer (2004), among many others, or detailed discussion. 25

26 * + + q ()=q () q (, ) * + + q () q () q (, ) * + + q ()=q () q (, ) q q q - - q() q(, ) - - q() q(, ) - - q() q(, ) (a) increasing externalities (b) constant externalities (c) decreasing externalities FIGURE ω c Ω ( + ) 1+ 1 () Curvature Properties IE CE DE Marginal Cost ω [ ) c Ω ( ) 1 1/(), + OWD OWD+NDT TWD ( ) c Ω ( ), c Ω ( ) 1 1/() OWD OWD+NDT TWD c Ω ( ) ) ND ND ND, c 1+1/() Ω ( ) [ ] OWU OWU+NDB TWD OWU OWU+NDB TWD ( c Ω ( + ) 0, c Ω ( + ) 1+1/() TABLE 1. IE: increasing externalities; CE: constant externalities; DE: decreasing externalities; OWD: one-way downward distortion; OWU: one-way upward distortion; NDT: no distortion at the top; ND: no distortion; NDB: no distortion at the bottom; TWD: two-way distortion. 26

27 and Margolis (1995)). These observations raise a question: under what circumstances and to what extent will the entry-deterring power o network externalities be limited? In the sequel, we are trying to answer these questions rom the perspective o bounded rationality. 4.2 Boundedly rational agents Analogous to the pure monopoly case described in Section 3, the optimal contract {q (), U ()} depends heavily on the rationality o agents. I agents are boundedly rational, this contract may not necessarily be ulilled. Given a common prior expectation e t, consumers belonging to Θ( e t ) { [, ] U () Ω( ) Ω( e t) } will accept the contract proposed by the principal, then a network size o t = ρ( e t) Θ( e t ) q ()d is actually realized. The eedback unction ρ( ) is given as ollows: I ω > c Ω ( ), then i Ω(x) [Ω( ), + ) [ ρ(x) = (x) q ()()d i Ω(x) max{0, Ω( ) ) [v(q ()) v(q 0 ())]d}, Ω( ) [ 0 i Ω(x) 0, max{0, Ω( ) ) [v(q ()) v(q 0 ())]d}, where (x) is given implicitly by [v(q ()) v(q 0 ())]d + Ω(x) = Ω( ). I ω = c Ω ( ), then i Ω(x) [Ω( ), + ) ρ(x) = 0 i Ω(x) [0, Ω( )), I ω < c Ω ( ), then i Ω(x) [Ω( ), + ) [ ρ(x) = (x) q ()()d i Ω(x) max{0, Ω( ) ) [v(q0 ()) v(q ())]d}, Ω( ) [ 0 i Ω(x) 0, max{0, Ω( ) ) [v(q0 ()) v(q ())]d}, where (x) is given implicitly by [v(q 0 ()) v(q ())]d + Ω(x) = Ω( ). As shown in the preceding section, the consumers then update their expectation according to e t+1 = (1 α)e t + α t, and this adaptive learning process will not end until an equilibrium is reached or the market is totally shut down. We can easily obtain that i Ω ( ) < 0, then ρ ( ) = 0, ρ +( ) = Ω ( )q ( ( ))( ( ))/ v(q ( ( ))) v(q 0 ( ( ))) < 0; i Ω ( ) = 0, then ρ +( ) = ρ ( ) = 0; i Ω ( ) > 0, then ρ +( ) = 0, ρ ( ) = Ω ( )q ( ( ))( ( ))/ v(q ( ( ))) v(q 0 ( ( ))) > 0. Following the same logic as 27