Competition and Irreversible Investments under Uncertainty

Competition and Irreversible Investments under Uncertainty Michele Moretto NOTA DI LAVORO 32.2003 MARCH 2003 KNOW – Knowledge, Technology, Human Capital Michele Moretto, Department of Economics, University of Brescia This paper can be downloaded without charge at: The Fondazione Eni Enrico Mattei Note di Lavoro Series Index: http://www.feem.it/web/activ/_wp.html Social Science Research Network Electronic Paper Collection: http://papers.ssrn.com/abstract=XXXXXX The opinions expressed in this paper do not necessarily reflect the position of Fondazione Eni Enrico Mattei Competition and Irreversible Investments under Uncertainty Summary This paper examines the effect of competition on the irreversible investment decisions under uncertainty as a generalization of the “real option” approach. We examine this issue with reference to an industry where each firm has only one investment opportunity which is completely irreversible and the product market reveals an inverted U-shape relationship between firm profits and industry size. That is, there are positive externalities for low level of the market size and negative externalities at high level of the market size. In the latter case, which corresponds to the traditional competitive industries, firms invest sequentially as market profitability develops. In the former case, which corresponds to industries in which investments are mutually beneficial, firms invest simultaneously after profitability of the market has developed sufficiently to capture all network benefits and to recover the option value of waiting. Put together, these extensions of the “real option” analysis, with strategic interactions, may help to explain both the cases of rapid and sudden developments such as the recent internet investments and the cases of prolonged start-up problems while waiting for the market to develop as the story of fax machines shows. Keywords: Competition, network effect, real options JEL: D81, C73, G13, O31 This research was financed in part by MURST-2002 grant. The author would like to thank Fabio Manenti, Piero Tedeschi and Paola Valbonesi for helpful comments and discussions. Any errors remain the author’s responsibility. Address for correspondence: Michele Moretto Department of Economics University of Brescia Via S. Faustino 74b 25122 Brescia Italy E-mail: [email protected] 1 Introduction Investment is deÞned as the act of incurring an immediate cost in the expectation of future payoff. However, when the immediate cost is sunk (at least partially) and there is uncertainty over future rewards, the timing of the investment decision becomes crucial (Dixit and Pindyck, 1994, p.3). In particular it is shown that irreversibility and uncertainty induce the Þrm to invest optimally only when the value of the investment exceeds the value of the option of waiting before making the irreversible decision. This paper extends the above standard irreversible investment model, taking strategic interactions into account. We deal speciÞcally with the case where a large number of identical Þrms are engaged in an investment game to enter a new product market and analyse the effect of competition on the optimal investment strategy of the Þrms. We examine this issue with reference to an industry where each Þrm has only one investment opportunity which is completely irreversible and the product market reveals an inverted U-shape relationship between Þrm proÞts and industry size: that is, positive externalities tend to dominate for low initial market size levels, whereas negative externalities tend to dominate at higher market size levels. Although we do not refer in the paper to a particular product, there are many markets that show, at least for some dimensions, greater proÞtability when more then one Þrm has already invested. This situation could arise in the case of goods that exhibit “network externalities” so that the utility of each consumer increases as the total number of consumers purchasing the same or compatible brands increases.1 One of many examples concerns the decision by multiple rival Þrms to set up an interconnected network to satisfy an interdependent demand for telecommunication services by a signiÞcant number of potential customers (Rohlfs, 2001, p. 34). A different case is when a high degree of complementarity between different goods is present as for software and hardware. Generally, software packages are produced by a large number of Þrms so that they can be used by the same hardware. Thus the greater the variety of software supporting a certain hardware, the greater the value of this hardware and the greater the utility consumers derive directly from the variety of software supporting the speciÞc hardware. Some 1 Jeffrey H. Rohlfs coined the term bandwagon effect for the beneÞt that a person enjoys as a result of others’ doing the same thing that he or she does, and speciÞcally he used the term network externalities for the bandwagon effect that applies to the user set of a comunication network (Rholfs, 2001) 3 authors refer to this as “indirect network externalities” (Shy, 2001, p.52) or “complementary bandwagon effects” (Rohlfs, 2001, p. 47-48 ). In other cases, the utility of each consumer decreases as more consumers buy the good. This occurs because of congestion, as the communication and information-based industries are recently experiencing. If on one hand the introduction of a new Web site increases the value of Internet to every existing user, on the other hand the progressive increase of its use increases congestion measured in term of excessive delay of transmission (longer connection time spent to load a Web page) or loss of service altogether (Odlyzko, 1999). Congestion then reduces consumers’ utility of joining the Internet and passes this dis-beneÞt to the Þrms by reducing the demand of access.2 The negative externalities case, with or without congestion, corresponds to the traditional competitive industry in which the investment of one Þrm lowers the proÞtability of the others. In this case the introduction of competition has two opposing effects which annul each other. Firstly, competition reduces the expected proÞt ßow that derives from the investment which tends to delay investment. Secondly, competition introduces a strategic beneÞt in favour of the investment as it deters the investments by rivals. Leahy (1993) Þrst discovered this property showing that the optimal investment strategy of a competitive Þrm remains equal to that of a single Þrm in isolation. In this case, Þrms enter sequentially as market proÞtability increases. On the contrary, in the case where investments are mutually beneÞcial, the optimal investment policy is essentially a question of coordination. As the timing of a Þrm’s entry is inßuenced by the entry decisions of others, Leahy’s result cannot be applied. Two equilibriums can emerge: either the industry remains locked-in with no entry as long as very pessimistic expectations dominate the market, or a mass of Þrms simultaneously runs to enter, driven by the expected rents generated by the positive externalities.3 Excluding the former for the sake of subgame-perfectness, we show that the level of market proÞtability that triggers these Þrms’ “network run” is the same as the one that justiÞes the entry of the Þrst Þrm under negative externalities. In other words, the Þrms make their decision simultaneously when the proÞtability of the market has developed sufficiently to capture all bandwagon beneÞts and 2 See, for example, DaSilva (2000) and Falkner et al. (2000), for a sarvey on the literature on how to price congestible networks as Internet. 3 This is what Rohlfs (2001, p.16-17) deÞnes a chicken-egg problem: nobody joins the network because the size of the network is zero, but the size of the network is nul because no one has joined it. 4 to recover the option value of waiting due to the irreversibility. This also determines endogenously the optimal start-up size of the industry. The paper is organized as follows. Section 2 presents the model and states the main results of the paper, namely the optimal entry strategy in the presence of positive and negative externalities. Section 3 deals with the coordination equilibrium induced by positive externalities using a discretetime game. The approach of this section is left at a heuristic level to highlight the link between a single Þrm’s decision and the beneÞts of coordinating investment. The formal analysis for pure strategies is presented in section 4 showing the conditions according to which, given the other Þrms’ policy of entry, no individual Þrm Þnds it optimal to follow a different policy. By the continuous time representation we show that the optimal policy is also subgame perfect. Section 5 applies the main results to the decision of building up a competitive network for satisfying a demand for telecommunication services and section 6 places the paper in the context of the literature on irreversible investment and market structure. Finally section 7 concludes. 2 The model We consider the decision to enter a new market subject to uncertain returns by a large number of identical Þrms. Yet, in order to focus exclusively on the timing decisions we abstain from explicitly characterizing the product market decisions (price or quantity), the Þrm size and, in line with this approach, we assume that the entry costs required to initiate the technology projects are given. This is summarized by the following assumptions: Assumptions 1. At any time t an idle Þrm may decide to enter a new market. Firms are risk-neutral and discount the future returns at the riskless interest rate ρ.4 2. All Þrms are identical and their size dm is inÞnitesimally small with respect to the market. 4 Introducing risk aversion does not change the results since the analysis can be developed under a risk neutral probability measure (Cox and Ross, 1976; Harrison and Kreps, 1979). 5 3. Each Þrm can enter by committing forever to a ßow cost w or undertaking a single irreversible investment which requires an initial sunk cost K = w/ρ. 4. Firms are free to enter. That is, in the free-entry game the Þrms Þrst decide whether or not to enter (and pay the entry cost K) and then compete for the available rents (generated by the positive externalities). Since entry is irreversible the Þrms already in the market do not have other decisions to make. 5. Each Þrm has zero operating options.5 6. Indicating by mt = m the number of Þrms currently active at time t (incumbents), each of them yields a ßow of operating proÞts that we abbreviate as: π(m, θ) ≡ u(m)θ (1) where θ is an industry-speciÞc shock. Time is continuous, t ∈ [0, ∞), and suppressed if not necessary. 7. The function u(m) is twice continuously differentiable in m, and it is increasing over the interval [0, m̄) and decreasing thereafter (see Þgure 1). That is, there are positive externalities to investment which can be caused by “network externalities” or the fact that the Þrms produce complementary products, over [0, m̄). After m̄ it is better for any single Þrm that the others have not invested: competition and/or congestion occur. We also assume that at zero and at some Þnite number of Þrms M (M >> m̄), proÞts falls to zero, i.e. u(0) = 0, and u(M ) = 0, whatever the value of θ. As M could be arbitrarily large, this assumption is harmless in our setting. Figur e 1 about her e 8. Finally, the industry-speciÞc shock θ follows a geometric diffusion process: dθ = αθdt + σθdW with θ0 = θ and α, σ > 0. 5 (2) This assumption allows us to focus on when, rather than whether, the entry takes place. The most important operating option is the ability of the Þrm to reduce output or even shut down and thereby avoid variable costs. The presence of operating options raises the value of the Þrm, see MacDonald and Siegel (1985) and, for a thorough discussion, Dixit and Pindyck (1994, chs. 6 and 7). 6 Applying Itô’s Lemma to (1) and substituting (2) to eliminate dθ, an expression for the proÞt process in terms of the shock and the number of Þrms emerges as: dπ = µ(m)πdm + απdt + σπdW, with π0 ≡ u(m0 )θ0 = π (3) where µ(m) ≡ u0 (m)/u(m) captures the direct effect of entry. From (3), entry inßuences the level of proÞts through its effect on the market equilibrium depending on the initial size of industry. In particular, given any value of the shock θ, more Þrms in the market implies a higher or lower equilibrium level of proÞts depending on the presence of positive µ(m) > 0 or negative µ(m) < 0 externalities respectively. The rest of this section is devoted to summarising the main properties of the entry process driven by (3), emphasizing the economic intuition behind it; the rigorous analysis is given in Section 4. 2.1 Negative externalities Although the inverted U-shape of (1) implies an entry process that meets positive externalities Þrst, we solve the investment problem by working backward starting from the negative externalities interval. If the initial size of the industry is m ≥ m̄, we expect entry to work in the following way: for a Þxed number of Þrms, proÞts move according to the above stochastic process with µ(m)πdm = 0. If proÞts then climb to a level π ∗ ≡ u(m)θ∗ , entry will become feasible and at the moment of entry, proÞts will drop downward along the function u(m). In technical terms this means that the threshold π ∗ becomes an upper reßecting barrier on the proÞt process.6 ProÞts will then continue to move stochastically without the term µ(m)πdm until another entry episode occurs. Under this setting a (competitive) equilibrium can be deÞned as a symmetric Nash equilibrium in entry strategies which bound the proÞt process of the Þrms. Although, in general, it is difficult to construct such an equilibrium, fortunately it can be built much more simply from the entry policy of a single Þrm in isolation regardless of future entry decisions: “...., each Þrm can make its entry decision by Þnding the expected present value of its proÞts as if it were the last Þrm that would enter this industry, and then making the 6 The proÞt function follows a regulated Brownian motion in the sense of Harrison (1985). 7 standard option value calculation. While the Þrm should entertain rational expectations about the stochastic process θ, it can be totally myopic in the matter of other Þrm’s entry decisions” (Dixit and Pindyck, 1994, p.291). This remarkable property of the competitive equilibrium, Þrst discovered by Leahy (1993), has an important operative implication: the optimal competitive equilibrium policy need not take account of the effect of entry. The proÞt level, say π̂, that triggers entry by the single Þrm in isolation is identical to that of the Þrm that correctly anticipates the other Þrm’s strategies π∗ . That is, when a Þrm decides to enter claiming to be the last to enter the industry, it is ignoring two things. First, it is thinking that its proÞt ßow is given by u(m)θ with m hold Þxed forever. Thus, as u0 (m) < 0, it is ignoring that future entry by other Þrms, in response to higher value of θ, will reduce its proÞts. Other things being equal, this would make entry more attractive for the Þrm that behaves myopically. Second, it ignores the fact that the prospect of future entry by competitors reduces its option value of waiting. That is, pretending to be the last to enter the industry, the Þrm also thinks that it still has a valuable option to wait before making an irreversible decision. Other things being equal, this makes the decision to enter less attractive. The two effects offset each other, allowing the Þrm to act as if it were in isolation. This offsetting behavior can be summarized by the following result. Result 1 The candidate policy for optimal entry in a competitive industry, characterized by an initial mass of Þrms m ∈ [m̄, M ), is described by the following upper proÞt threshold: u(m)θ∗ (m) = β1 (ρ − α)K ≡ π ∗ (= π̂), β1 − 1 with β1 >1 β1 − 1 (4) where ρ > α and β 1 > 1 is the positive root of the auxiliary quadratic equation Ψ(β) = 21 σ 2 β(β − 1) + αβ − ρ = 0. Over the range [m̄, M ), new additional entry occurs every time the proÞts climb to the known threshold π ∗ ; if proÞts stay below this barrier no new investment is undertaken. Proof. See Leahy (1993) and Section 4. With m incumbents, an ∗idle additional Þrm will enter if the present value (m) of its proÞts at entry u(m)θ exceeds the cost of the investment K augρ−α mented by the option of waiting to invest β 1−1 K, i.e. by waiting a little the 1 8 Þrm obtains a new observation of the market proÞtability, reducing its downside risk.7 We can have a better intuition of the competitive equilibrium by writing the above threshold in terms of the shock θ. Since π ∗ ≡ u(m)θ∗ (m) and u(m) is decreasing in the region [m̄, M ], the optimal policy can be restated by the following upward-sloping curve (Þgure 2): θ∗ (m) ≡ K β1 (ρ − α) , β1 − 1 u(m) for m ∈ [m̄, M ) (5) In the region above the curve, it is optimal to enter. A discrete mass of Þrms will enter in a lump to move the proÞts level immediately to the threshold curve. In the region below the curve the optimal policy is inaction: Þrms wait until the stochastic process θ moves it vertically to θ∗ (m) and then again a mass of Þrms will jump into the market just enough to keep the proÞts from crossing the threshold. 2.2 Positive externalities Working backward towards the start-up of the industry, if the initial size m is less than m̄, any potential entrant is subject to positive externalities, that is the value of entering the industry depends on the number of Þrms who have already entered. Therefore, the timing of a Þrm’s entry is inßuenced by the entry decisions of others and intuition suggests that Leahy’s result cannot be extended to cover this case: a single Þrm cannot continue to claim to be the last to enter the industry in constructing its optimal entry policy. The gist of our argument relies on the presence of “network beneÞts” so the higher the number of Þrms in the industry, the greater the advantage in terms of proÞt ßow. However, although investing is proÞtable, it is “more expensive” to do it alone than to enter together with others or even later on when others have already done so. This makes the Nash equilibrium represented by the myopic trigger π̂ no longer subgame-perfect. By the Þrstmover disadvantage and the strategic nature of the timing decision, each Þrm can do better by delaying entry. Generally speaking, potentially conßicting preferences over appropriation of the positive “network beneÞts” make them face a choice between no entry and agreement. 7 In other words, the decision to enter entails the exercise of an option to delay, when the Þrm enters its loss of ßexibility is given by β 1−1 K. 1 9 However, as all Þrms are subject to the same (industry-wide) uncertainty shock, two equilibrium patterns are the only ones possible: either the industry remains locked-in at the initial size, sustained by self-fulÞlling pessimistic expectations8 , or a mass of Þrms simultaneously runs to enter, driven by the expected rents generated by the positive externalities. Excluding the former for the sake of subgame-perfectness (see section 3 for a discussion of this case), we are left with the latter. In this speciÞc case, we expect entry to work in the following way: for a Þxed number of Þrms, proÞts move according to (3) with µ(m)πdm = 0. If proÞts climb to π ∗∗ ≡ u(m)θ∗∗ , it will trigger an entry of discrete size that raises the dimension of the industry instantaneously by a jump. The exact form of the trigger π ∗∗ as well as the size of the mass of Þrms that jump into the industry upon reaching it is given in the following result. Result 2 The candidate policy for optimal entry in a competitive industry, characterized by positive externalities and initial mass of Þrms m ∈ [0, m̄), is described by the following upper proÞt threshold: π ∗∗ ≡ u(m)θ∗∗ (m) = u(m̄)θ∗ (m̄), for m ∈ [0, m̄) (6) Over the range [0, m̄), the optimal entry policy is to set the threshold π ∗∗ equal to the known threshold u(m̄)θ∗ (m̄) where the proÞt ßow is maximum. No Þrms enter if proÞts stay below this barrier, but a discrete mass of (m̄ − m) new Þrms “coordinate” entry the Þrst time that π ∗∗ is reached. Proof. See Section 4. An immediate corollary that follows from Results 1 and 2 is: Corollary 1 The proÞt threshold that triggers the “network run” of (m̄ − m) new Þrms is the same reßecting barrier that triggers the marginal competitive entry under negative externalities at m̄: u(m̄)θ∗ (m̄) = β1 (ρ − α)K ≡ π∗∗ (= π ∗ ). β1 − 1 8 The Þrms may delay entry till θ reaches, for the Þrst time, the upper level θ∗ (m) which indicates the “optimal” entry trigger for each idle Þrm in isolation. 10 Again, we can have a better intuition of the equilibrium by writing the above threshold in terms of the aggregate shock θ. Since π ∗∗ ≡ u(m)θ∗∗ (m) and u(m) is increasing in the region [0, m̄), the optimal policy is given by a ßat curve starting at θ∗∗ (0) = θ∗ (m̄) deÞned by: θ∗∗ (m) = θ∗ (m̄) ≡ K β1 (ρ − α) , β1 − 1 u(m̄) for all m ∈ [0, m̄) (7) Figure 2 summarizes the effect of positive externalities on entry. Thus starting at m, if the initial shock is below the known trigger at m̄, all the Þrms wait until the θ rises vertically to this level, and then “coordinate” their entry to bring the size to the optimal level m̄. Once the optimal size is reached and to the right of m̄, further decisions to enter proceed as explained in the previous section with negative externalities. Intuitively, starting at any m < m̄, (6) (or (7)) locates the optimal entry threshold so as to maximize the total proÞts of the incremental number of Þrms that enter (m̄ − m). The shock value θ∗ (m̄) that triggers these Þrms’ “network run” is the same threshold that justiÞes a further marginal entry under negative externalities. Section 4 conÞrms that this is in fact an equilibrium. No Þrm would ever invest at a lower entry trigger since this trigger is based on the most optimistic assessment with respect to the other Þrms, namely that they all invest at θ∗ (m̄). On the other hand no Þrm Þnds it convenient to delay its entry given that the other Þrms invest, since θ∗ (m̄) is also the investment trigger of the rivals. Figur e 2 about her e 2.3 Dynamics of Industry Investments By Results 1 and 2 and inverting (4), we are able to represent the properties of the industry’s dynamic entry pattern with positive and negative externalities. The optimal boundary function: m∗t = θ∗−1 (θt ; K, ρ, α, σ) determines the optimal industry size as a function of the state variable θ and the vector of parameters (K, ρ, α, σ). For movements of the shock to the right of the boundary, new Þrms enter; if the shock stays on the left of the boundary, no new investment is undertaken. Assuming as an example 11 u(m) = m(40−m), ρ = 0.04, α = 0, σ = 0.2 (at annual rate) and normalizing K = 10, 000 Þgure 3 below shows a possible entry pattern for this industry. Figur e 3 about her e The industry size process mt is singular: entry takes place only when θ = θ∗ (m), except for the initial jump to m̄ = 20 (M = 40) necessary to bring θ into the region [θ∗ (20), ∞). Formally, we get: Z t ∗ mt = 20 + J[θ=θ∗ ] dm∗s T1 1 ∗ where T = inf(t ≥ 0) | θ = θ (20)) and J[θ=θ∗ ] denotes the indicator function for all the instants in which the process θ hits the upward-sloping curve θ∗ (m). 1 From θ∗ (20) ≡ β β−1 = 2, it is evident that the Þrms invest when market 1 proÞtability is sufficient to guarantee that if they make their investment decision simultaneously all the “network beneÞts” will be captured and the option value of waiting, indicated by β 1−1 = 1, will be recovered. 1 3 Coordination and Pareto-dominant equilibria: a heuristic analysis This section is devoted to highlighting where the above Nash equilibrium in pure entry strategies comes from and its perfectness. The approach of this section, however, is more on a heuristic level; the formal analysis with pure strategies is performed in the next section. Moreover, although the heuristic analysis uses mixed strategies, we show that the optimal policy is outcome equivalent to that in which the Þrms employ pure strategies. This justiÞes formally proceeding in the next section as if each Þrm uses pure strategies. Finally, although we use discrete time rather than continuous time, the reference paper for analysing the timing of investment is Fudenberg and Tirole (1985). Let’s begin by assuming an industry of m ∈ [0, m̄) incumbent Þrms and a sufficiently high number of all equal outsider Þrms looking to enter. To focus on the basic question of the paper, we impose the following restrictions: 1. We start considering a one-shot-discrete-time game between a generic ith Þrm and a pool of (m̄ − m)−i < M other Þrms. Although, for 12 convenience, we can refer to the (m̄ − m)−i Þrms as the “other player”, we consider strategies and payoffs of each individual Þrm;9 2. Since entry (investment) is irreversible the incumbents do not have other decisions to make, while the outsiders choose a randomization over {Enter, Don’t Enter} . We can therefore simply use the term Þrm for the potential entrants. 3. The strategies (and payoffs) of all Þrms in the investment game are taken at Þxed (stochastic) times: T 1 for θ∗ (m̄), and T 2 for θ∗ (m). Since θ∗ (m̄) < θ∗ (m), then almost surely T 2 > T 1 > 0. When θ hits θ∗ (m̄) for the Þrst time, Þrm i ’s action set is {Enter, Don’t Enter} . If Þrm i decides to invest at T 1 its actions set becomes the null action “stay in” forever. Conversely, if Þrm i does not invest at T 1 it must wait for θ∗ (m) to be reached before entering, i.e. it waits until it is optimal to enter as a single Þrm in isolation.10 We shall relax some of these assumptions later. 3.1 Pure strategy equilibria As stated, for the above one-shot-discrete-time game, each Þrm has two strategies available at T 1 : Entry (E) or go alone (No Entry, NE). Since the deÞnition of T 1 implies that simultaneous entry is not optimal before T 1 , by convention we evaluate the payoffs by referring to time T 1 . Then T 2 is the “optimal” policy for each idle Þrm in isolation and payoffs are evaluated accordingly. Finally, K is normalized to one. Taking advantage of Fudenberg and Tirole’s notation, we deÞne the following four functions: the function M(T 1 ) is the expected discounted value of each Þrm if all invest together at T 1 . The function L(T 1 ; T 2 ) is the (leader’s) discounted value for the Þrm that invests at T 1 while all the rivals wait till T 2 .F (T 1 ; T 2 ) is the (follower’s) discounted value for the Þrm that waits till T 2 before investing while the rivals go at T 1 . Finally, as at T 2 it is always 9 To avoid complication we do not consider the possibility of coalitions among the m̄ Þrms. See Bernheim, Peleg and Whinston (1987) for coalition games and the related deÞnition of coalition-proof Nash equilibria. 10 The same result holds if we assume that T 2 corresponds to any trigger θt with θ∗ ( m̄) < θt ≤ θ∗ (m), see below. 13 optimal to enter, F F (T 2 ) ≡ M(T 2 ) = L(T 2 ; T 2 ) = F (T 2 ; T 2 ) is the payoff for joint-investment. From the above deÞniton, if at time T 1 all m̄ − m Þrms simultaneously enter, the net present value of the investment is :11 ¸ ·Z ∞ ∗ −ρt 1 (8) e u(m̄)θt dt | θT 1 = θ (m̄) − 1 M(T ) ≡ ET 1 T1 ¶ µ 1 u(m̄)θ∗ (m̄) −1 = = ρ−α β1 − 1 If none of them enter at T 1 , under our Þxed time assumption, all wait until T 2 > T 1 before entering. Then, by (25), their present value becomes: ½ ·Z −ρ(T 2 −T 1 ) F F (T ) ≡ ET 1 e ∞ 2 = µ −ρt e T2 u(m̄)θt dt | θT 2 ¸ ¾ = θ (m) − 1 ∗ (9) ¶β ¶β ¶µ ∗ ¸µ · θ (m̄) 1 u(m) 1 β 1 u(m̄) u(m̄)θ∗ (m) −1 −1 = ρ−α θ∗ (m) β 1 − 1 u(m) u(m̄) ´β 1 h i³ u(m) 1 u(m̄) As β 1 > 1 and u(m̄) > u(m), it follows that β 1−1 > β β−1 . − 1 u(m) u(m̄) 1 1 It is always convenient to coordinate. Although we stated that Þrms have complete information, they have imperfect information, i.e. they choose their strategies without knowledge of the other’s choice. Therefore, in our “two-player” game, we should also evaluate the payoff by a player who coordinates when the other fails to do so. In particular, if at time T 1 the ith Þrm invests but the rest do not, its net present value can be expressed as: 11 As for m ≥ m̄ the optimal competitive equilibrium policy need not consider strategically simultaneous entry of other Þrms (i.e. Result 1 holds), we simplify evaluating the payoffs at entry without considering the option value of future new entry. 14 L(T 1 ; T 2 ) ≡ ET 1 "Z T 2 −∆T e−ρt u(m+i )θt dt + Z ∞ e−ρt u(m̄)θt dt | θT 1 = θ∗ (m̄) − 1 T 2 −∆T T1 ¸µ ∗ ¶β 1 ¸ · θ (m̄) (u(m̄) − u(m+i ))θ∗ (m+i ) u(m+i )θ∗ (m̄) −1 + = ρ−α ρ−α θ∗ (m+i ) µ ¶β ¸ · β 1 u(m̄) − u(m+i ) u(m+i ) 1 β 1 u(m+i ) −1 + = β 1 − 1 u(m̄) β1 − 1 ρ−α u(m̄) · where T 2 − ∆T = inf(t > T 1 | θ = θ∗ (m+i )) is the Þrst time to which the rivals respond by entering and m+i indicates that m (old) Þrms plus the (new) ith are now present in the market. On the contrary, if at time T 1 the (m̄ − m)−i Þrms invest but the ith does not, the net present value of the ith Þrm is equal to: ·Z ½ −ρ∆T F (T ; T ) ≡ ET 1 e 1 ∞ 2 µ −ρt e u(m̄)θt dt | θT 1 T 1 +∆T ¶µ ∗ ¶β 1 u(m̄)θ∗ (m̄−i ) θ (m̄) = −1 ρ−α θ∗ (m̄−i ) · ¶β ¸µ u(m̄) β1 u(m̄−i ) 1 = −1 β 1 − 1 u(m̄−i ) u(m̄) ¸ ¾ = θ (m̄) − 1 (11) ∗ That is, as θ∗ (m) is decreasing for m < m̄, the “other player” who has not coordinated responds “almost” immediately at T 1 + ∆T = inf(t > 0 | θ = θ∗ (m̄−i )). By the properties of the above payoffs, we are able to conclude that the following disequality holds: Result 3 L(T 1 ; T 2 ) < F F (T 2 ) < F (T 1 ; T 2 ) < M(T 1 ) Proof. See Appendix The payoffs when a particular pair of strategies is chosen are given in the appropriate cell of the bi-matrix below: the payoff to the ith Þrm is the one at the top left of the cell. 15 (10) # (m̄ − m)−i E NE , .. i E M , M L NE F , .. F F , F F Referring to the above bi-matrix, as each Þrm within the m̄ − m can play the role of ith and F F > L, the one-shot-discrete-time game presents only two candidates for symmetric Nash equilibria in pure strategies: “all E” and “all NE”. Nevertheless, although “all E” is the Pareto-dominant equilibrium, it is not clear that it is the one that will be played. In fact, as Þrms are inÞnitesimally small, this makes F (T 1 ; T 2 ) ' M(T 1 ) and L(T 1 ; T 2 ) << F F (T 2 ), and the above game resembles a “one-sided coordination game” where one agent strictly prefers to match the action played by the other, with player i strictly preferring to match the “other player” if it plays NE. Putting some numbers in the cells of the above bi-matrix, the game can be illustrated by the following example: (m̄ − m)−i E NE i E 10 , 10 −5 , .. NE 9 , .. 4 , 4 While the Pareto outcome (10,10) may tend to make the strategy (E,E) a focal point of the game, playing NE is much safer for player i, as it guarantees 4 regardless of how the other “players” play. In this situation we are not certain what outcome to predict.12 The same uncertainty remains even if we extend the game to include mixed strategies. 3.2 Mixed strategy equilibria In this case we write: 12 Without entering into the details of coalition-proof equilibria, if the ith Þrm expects its rivals to form a coalition, “all E” remains the only candidate for symmetric Nash equilibria in pure strategies. To see that this is the case we have to complete the above bi-matrix considering the payoffs of the ( m̄ − m) −i Þrms if, at time T 1 , the ith Þrm coordinates but they do not: 16 • si (T 1 ) as the probability Þrm i enters (plays E) at time T 1 , if it has not previously entered, with i ∈ (m̄ − m). In pure strategies si (T 1 ) equals zero or one, that is it maps each Þrm’s information set θ(T 1 ) to one action: NE or E. In mixed strategies, si (T 1 ) maps each Þrm’s information set θ(T 1 ) to a probability distribution over action. Returning to the above one-shot-discrete-time game, if all the potential entrants are out of the market at time T 1 , Þrm i’s expected present discounted 1 2 A(T ; T ) ½ ·Z −ρ(T 2 −∆T −T 1 ) e ≡ E T1 = = ∞ e −ρt u( m̄)θt dt | θ T 2 −∆T ∗ · ¶β 1 ¸µ u( m̄)θ∗ (m+i ) θ ( m̄) −1 ρ−α θ∗ (m+i ) · ¶β ¸µ β1 u(m+i ) 1 u( m̄) −1 β 1 − 1 u(m+i ) u( m̄) ¾ = θ (m) − 1 ∗ T2 ¸ The value of the ( m̄ − m) −i if they coordinate but the ith does not: 1 2 B(T ; T ) ≡ ET 1 "Z T 1 +∆T −ρt e u( m̄−i )θt dt + T1 = = Z ∞ −ρt e ∗ # u( m̄)θt dt | θT 1 = θ ( m̄) − 1 T 1 +∆T ¸µ ∗ ¶β 1 ¸ · θ ( m̄) (u( m̄) − u( m̄−i ))θ∗ ( m̄−i ) u( m̄−i )θ∗ ( m̄) −1 + ρ−α ρ−α θ∗ ( m̄−i ) · µ ¶β ¸ β 1 u( m̄−i ) β 1 u( m̄) − u( m̄−i ) u( m̄−i ) 1 −1 + β 1 − 1 u( m̄) β1 − 1 ρ−α u( m̄) · Furthermore, L(T 1 ; T 2 ) < A(T 1 ; T 2 ) and B(T 1 ; T 2 ) < F (T 1 ; T 2 ). Adding these payoffs, the bi-matrix becomes: i E NE ( m̄ − m) −i E NE M , M L , A F , B FF , FF With F F (T 1 ; T 2 ) < A(T 1 ; T 2 ) (F F (T 2 ) ' A(T 1 ; T 2 )) and B(T 1 ; T 2 ) > F F (T 2 ). Strategy E strictly dominates NE for the coalition, which makes (E,E) the only Nash equilibrium. 17 value is: £ ¤ Pi (T 1 ) = si (T 1 ) s−i (T 1 )M + (1 − s−i (T 1 ))L (12) £ ¤ 1 1 1 +(1 − si (T )) s−i (T )F + (1 − s−i (T ))F F ¤ £ 1 = s−i (T )F + (1 − s−i (T 1 ))F F £ ¤ +si (T 1 ) s−i (T 1 )(M − F ) + (1 − s−i (T 1 ))(L − F F ) where s−i (T 1 ) ≡ s(m̄−m)−i (T 1 ), indicates the probability that all the (m̄−m)−i opponents play E. Following the usual procedure for solving a maximization problem, we differentiate (12) with respect to the choice variable si (T 1 ) to obtain the Þrst order condition: s−i (T 1 )(M − F ) + (1 − s−i (T 1 ))(L − F F ) = 0 or: L(T 1 ; T 2 ) − F F (T 2 ) ŝ−i (T ) = [L(T 1 ; T 2 ) − F F (T 2 )] − [M(T 1 ) − F (T 1 ; T 2 )] 1 (13) Since F F (T 2 ) > L(T 1 ; T 2 ) and M(T 1 ) > F (T 1 ; T 2 ), we get 0 ≤ ŝ−i (T 1 ) ≤ 1. Taking account of (13) we are able to rewrite (12) in a different way: Pi (T 1 ) = £ ¤ s−i (T 1 )F + (1 − s−i (T 1 ))F F (14) £ ¤ 1 1 1 +si (T ) [(M − F ) − (L − F F )] s−i (T ) − ŝ−i (T ) If the opponents’ probability of playing E is sufficiently small, s−i (T 1 ) < ŝ−i (T 1 ), Þrm i’s expected present discounted value is nonpositive, and Þrm i maximizes its payoff by playing NE with certainty, i.e. si (T 1 ) = 0. If the opponents’ probability of playing E is sufficiently high, s−i (T 1 ) > ŝ−i (T 1 ), Þrm i’s expected present discounted value is positive, and Þrm i maximizes its payoff by coordinating entry with certainty, i.e. si (T 1 ) = 1. Finally, if s−i (T 1 ) = ŝ−i (T 1 ), Þrm i’s expected present discounted value is zero, and independent of the probability of entering selected by i. Which equilibrium strategies are more plausible depends on the number of players. If each Þrm plays E Q with an equal probability independent of the 1 others, this implies ŝ−i (T ) = ŝi (T 1 ) = ŝi (T 1 )m̄−m−i . To exemplify, let’s assume an infant industry characterized by an initial mass of Þrms m = 0, and m̄ = 20. It is easy to show that ŝ−i (T 1 ) = 0.9, which requires: si (T 1 ) ≥ ŝi (T 1 ) = 0.91/19 = 0.994 47 18 With a mass of m̄ − m potential entrants, “all E” is the optimal strategy only if each individual Þrm assesses the probability of E greater than 0.994 47. In other words, going alone at T 2 “risk dominates” coordinate entry at T 1 in the sense of Harsanyi and Selten (1988).13 As the example suggests, when there are more players, each player relies more on someone else coordinating. The more Þrms that have to decide entry, the less likely the coordination. 3.3 Subgame perfect equilibria So far we have presented the entry process as a simultaneous game justifying it by assuming a fairly unrealistic situation in which the Þrms either decided immediately (at T 1 ) or the period they had to wait before being able to reconsider (observe) the possibility of entering was so long that it was as if they were choosing their strategies simultaneously. However, if the interval between the different decisions is shorter, even in continuous time, the hypothesis of sequential decisions seems more realistic. In this case the question is: can the Pareto superior coordinating outcome (E) be sustained in a dynamic game? The answer is positive. Before going on to the model in continuous time let’s go further with the discrete-time game, formally adding the probability that Þrms enter between T 1 and T 2 . If at T 2 no Þrm has entered, as θ∗ (m) is the “optimal” policy for each idle Þrm in isolation, it will not be expedient for any Þrm to wait further. This implies that si (T 2 ) = 1 for all i. Recalling that each Þrm in the mass m̄ − m can play the role of ith, proceeding inductively we can identify at most three subgame perfect equilibrium strategies:14 (1) Firms play si (t) = 1 for t = T 1 , for all i ∈ (m̄ − m) : the industry shows coordinated entry; (2) Firms play si (t) = 0 for t = T 1 and si (t) = 1 for t = T 2 , for all i ∈ (m̄ − m) : the industry shows lock-in; 13 A well-known example of a game with multiple equilibria is the one described in the stag-hunt game; see Fudenberg and Tirole (1991, ch.1) for a thorough discussion of games with multiple equilibria and Pareto optimality. 14 Only Markov perfect equilibria are examined. That is, the equilibrium concept applied is that of subgame perfect Nash equilibrium in Markov strategies for the exogenous variable θ at which Þrms decide to enter. 19 (3) Firms play si (t) = s̃i (t) for t = T 1 and si (t) = 1 for t = T 2 , for all i ∈ (m̄ − m) : the positive externalities result in equilibrium strategies in which all Þrms take a positive chance of making a mistake in order to get the highest payoff. Which of the three is the strategy proÞle that will be deÞnitely chosen by the Þrms is generally difficult to assess, and working backward from the last period does not help as it does not lead to uniqueness. However, if each Þrm i behaves optimally along any enter probability path that includes the mixed enter probabilities si (t) = s̃i (t) in T 1 , the above arguments suggest that the third subgame perfect equilibrium strategy will be payoff-equivalent and outcome-equivalent to the Þrst one of the pure strategy equilibria: Þrms enter at T 1 and the mixed probabilities are never implemented. This reduces the subgame perfect equilibrium strategies to only pure strategies. Maintaining the heuristic spirit of this section we proceed in arguing why the strategy proÞle (1) is the most reasonable outcome of the game. We do this checking that the strategy proÞle (1) yields a subgame perfect Nash equilibrium as it is unimprovable in a single step, that is it never pays to deviate from it in a single period while conforming to it thereafter.15 In particular, we know that no strategy that calls for stay out at T 2 can be a Nash strategy, because the same strategy with entry replacing stay-out dominates it. But if all the Þrms have strategies calling for entry in the last period, then a strategy calling for entry in the next-to-last period (i.e. at T 1 ) is Nash perfect only if it shows that it is not optimal to deviate by replacing entry with stay-out at T 1 . This should rule out any strategy that does not call for “all E” everywhere along the equilibrium path.16 Take (1) as a candidate strategy solution and suppose the ith Þrm deviates in period T 1 to return to the candidate solution at T 2 , i.e. it follows the strategy proÞle (2). In order to verify if the one-step deviation is optimal, 15 Essentially this is the one-step-deviation principle. This principle is an application of the fundamental dynamic programming principle of pointwise optimization, which says that a proÞle strategy is optimal if and only if it is optimal in each time period. For a proof of the one-step-deviation principle see Fudenberg and Tirole (1991, p.109). Although this principle applies, for both Þnite and inÞnite horizon game, provided that events in the distant future are made sufficiently insigniÞcant through discounting, its use in the above two-periods game can guide us as to how to come up with a candidate solution. 16 It is also worth noting that the strategy “always E” is not a “dominant” strategy, as it is in the one-shot game at T 2 , because it is not the best response to various suboptimal strategies at T 1 . 20 we evaluate, at time T 1 , the difference in the net present value between (1) and (2) as: s−i (T 1 )M + (1 − s−i (T 1 ))L − F F ≥ 0. This difference is positive if: s̃−i (T 1 ) ≡ F F (T 2 ) − L(T 1 ; T 2 ) < ŝ−i (T 1 ) < 1 M(T 1 ) − L(T 1 ; T 2 ) (15) By (15) if the opponents’ probability of playing E is s−i (T 1 ) > s̃−i (T 1 ), Þrm i’s expected present discounted value is positive, and it maximizes its payoff by coordinating entry with certainty, si (T 1 ) = 1. Simple application of the above example shows that s̃−i (T 1 ) = 0.6 much lower than ŝ−i (T 1 ) = 0.9. In other words, coordinating entry at T 1 becomes less risky. If we now allow the Þrms to change their actions at any point in the interval [T 1 , T 2 ] (i.e. in the interval [θ∗ (m̄), θ∗ (m)]), intuition suggests that there are an inÞnite number of symmetric equilibrium strategies like the one described above, characterized by its movement date t which calls for entry at T 1 . To understand how this can occur, there are two aspects of the entry game in continuous time that must be considered. First of all, if after reaching θ∗ (m̄) Þrms do not coordinate in the expectation that no-one will enter, they may still do so at any successive “instant”, say at t > T 1 with θt > θ∗ (m̄), at the same proÞts u(m̄). By the Markov property of the state variable θ, this game has inÞnite subgame equilibria which are Pareto ranked by their date of entry with earlier entry being more efficient from the Þrms’ point of view. In fact, deÞning with L(t; T 2 ), F F (T 2 − ∆T ) and M(t) the respective payoffs evaluated at t > T 1 ,the probability that the ith Þrm will play E decreases as t increases without entry, and increases as the optimal entry time by the single Þrm T 2 becomes more remote.17 That is: Proposition 1 The per-period probability s̃−i (t) ≡ following properties: 17 F F (T 2 −∆T )−L(t;T 2 ) M(t)−L(t;T 2 ) has the To simplify, we indicate the interval [T 1 , T 2 ] as a synonym of the interval [θ ( m̄), θ∗ (m)] of the state variable θ.Obviously this is not always the case. In fact, although the Þrms can make their entry decisions within an apparently Þnite time span [T 2 − T 1 ], it is as if they can do so indeÞnitely. Owing to uncertainty, no Þrm can perfectly predict θ at each date and since θ follows a random walk there is, for each time interval dt, a constant probability of moving up or down. Formally this mean that we must consider only the time interval for which θt > θ∗ ( m̄). Having speciÞed this, we continue to use the above synonym, conÞdent that it will not lead to confusion. ∗ 21 1) 2) 3) ∂s̃−i (t) >0 ∂t ∂s̃−i (t) <0 ∂T 2 ∂s̃−i (t) |T 2 →∞ < ∂σ2 with limt→T 2 −∆T s̃−i (t) = 1; with limT 2 →∞ s̃−i (t) = β 1 −1 ; β1 0 Proof. See Appendix In words, although for the ith Þrm delaying the decision to enter means an expected reduction in the beneÞts of coordination with respect to going alone, i.e. M(t) − L(t; T 2 ), there is also an equivalent reduction in the costs associated with the delay itself expressed in terms of an increase in the advantage of going alone with respect to waiting T 2 and entering together, i.e. F F (T 2 − ∆T ) − L(t; T 2 ). The two effects offset each other so that the opponents’ probability threshold s̃−i (t) that makes Þrm i’s expected present discounted value positive converges to one as t increases and, consequently, the probability of Þrm i entering if it has not previously entered si (t) tends to zero. The second part of the proposition says that the farther off the moment when it will not be expedient for any Þrm to wait any longer, the lesser the advantage of going alone and the greater the advantage of coordinating; s̃−i (t) decreases while si (t) increases. The intuition of this result relies on the deÞnition of T 2 . By (5), T 2 → ∞ as m → 0 : a smaller number of incumbents implies more externalities in the market which increase the degree of coordination among potential entrants. The greater the number of externalities to be exploited, the lower the probability of mistakes and the coordination problem becomes less severe. The probability of mistakes is reduced also as uncertainty increases (the third part of the proposition). The greater the uncertainty over future values of the shock θ, the larger the return the Þrms will demand before they will consider making the irreversible investment, which translates into an increase of θ∗ (m̄).However, a high level of θ∗ (m̄) if delays the moment at which it becomes advantageous to enter, in the same way it signals that the proÞtability of the market will be maintained even longer, which favours coordination among potential entrants. The second problem that must be considered is that in continuous time games there is no notion of last time before t. The real line is not well ordered and therefore induction cannot be applied. This denies the possibility of building up an expected value such as (14), from which to deduce the 22 optimal subgame perfect equilibrium strategies by working backward from the end using (longer) subgames. Fudenberg and Tirole (1985), and Simon and Stinchcombe (1989), to which we refer for further details, highlight the fact that there is a loss of information in the attempt to represent continuoustime equilibria as the limits of discrete time mixed strategy equilibria. They argue that in these kind of games a strategy cannot be represented by a single distribution function. To correct for this loss of information they extend the strategy space to specify not only the cumulative distribution that player i has entered by time t given that the others have not yet entered, but also the intensity of atoms on the interval between [t, t + dt].18 With this formalism these authors see continuous time as discrete-time with a length of reaction (or information lag) that becomes inÞnitely negligible to allow the Þrms to respond immediately to the rivals’ actions. A class of continuous strategies is then deÞned so that any increasingly narrow sequence of discrete-time grids generates a convergent sequence of game outcomes whose limit is independent of the grid sequence. In the limit when the period length converges at zero, an entry will occur immediately regardless of the value assumed by the perperiod probability. However, the probability of having simultaneous entry varies with this probability. In this speciÞc case, the Pareto superior joint moving outcome of the above “one-sided coordination game”, all moving at T 1 , seems to be the most reasonable outcome of the game. Furthermore, Simon and Stinchcombe (1989, p. 1198-1200) show that the Pareto superior joint moving equilibrium is the unique equilibrium that survives iterated elimination of weakly dominated strategies. 18 In this speciÞc case, it is worth noting how the per-period probability (15) coincides with the notion of “intensity of entry” introduced by Fudenberg and Tirole (1985). The function value s̃−i (t) should be interpreted as the probability that the ( m̄−m) −i opponents play E in the matrix game below: ( m̄ − m) −i i s̃i (t) 1 − s̃i (t) s̃−i (t) M (t) , M (t) F (t; T 2 ) , .. 23 1 − s̃−i (t) L(t; T 2 ) , .. repeat the game 4 A formal analysis This section is devoted to the proof of Results 1 and 2. The aim is to demonstrate that the candidate policies presented in (4) and (6) are indeed optimal. As the simultaneous investment scenario by letting the Þrms play mixed strategies is outcome equivalent to the one in which the Þrms employ pure strategies, we conduct the analysis as if each Þrm uses a stopping rule (a pure Markovian strategy) that speciÞes the critical value of the shock θ beyond which the Þrms invest.19 We refer to some dynamic optimization solutions extensively studied in the Operations Research literature where an Itô process is constrained never to leave an (optimal) region (see Harrison and Taksar, 1983, Karatzas and Shreve 1984, 1985; Harrison, 1985), and to some well-known applications to the case of a competitive economy (see Leahy, 1993; Bartolini, 1993; Dixit and Pindyck, 1994). The results presented by these authors can be applied with some modiÞcations to the problem at hand. In particular, the special structure of the industry considered here leads to some important new insights into the analysis. For the optimal entry policy, the Þrst thing to do is to Þnd the value of an established Þrm V (m, θ) as the expected discounted stream of proÞts π(m, θ) ≡ u(m)θ, given each Þrm’s optimal future entry policy: "Z # ∞ X V (m, θ) = max E0 J[t=τ i ] K | m0 = m, θ0 = θ e−ρt u(mt )θt dt − τi 0 τi (16) where J[t=τ ] is the indicator function that assumes the values one or zero depending on whether the argument is true or false, and the expectation is taken considering that the number of active Þrms may change over time by new entry. A solution of (16) can be obtained starting within a time interval where no new entry occurs. Over this interval the number of Þrms is Þxed and the Þrm is an asset which pays a ßow of proÞts u(m)θ per unit of time, and experiences a “capital” gain as θ evolves stochastically. The proÞts and the expected “capital” gain must add up to the risk-adjusted return ρ if the 19 Since Markovian strategies incorporate all the information relevant for the game, if a player uses a Markovian strategy, then the best response that his rivals can adopt is Markovian as well. This means that a Markovian equilibrium remains such even if the players are allowed to use history-dependent strategies (Fudenberg and Tirole, 1991, p. 501). 24 Þrm wishes to stay active (Bellman equation): ρV (m, θ)dt = u(m)θdt + E[dV (m, θ)] (17) Assuming V (m, θ) to be a twice-differentiable function with respect to θ and using Itô’s Lemma to expand dV (m, θ), the no-arbitrage condition (17) becomes a differential equation equal to: 1 2 σ Vθθ (m, θ) + α2 Vθ (m, θ) − ρV (m, θ) + u(m)θ = 0 2 (18) As long as the number of active Þrms m is Þxed, (18) is an ordinary differential equation familiar in the option pricing methodology (Dixit and Pindyck, 1994, p.179-180). Provided that ρ > α in order for the value of the Þrm to be bounded, the general solution of (18) can be written as: V (m, θ) = A(m)θβ 1 + B(m)θβ 2 + v(m, θ) where 1 < β 1 < ρ/α, β 2 < 0 are, respectively, the positive and the negative root of the characteristic equation Ψ(β) = 12 σ 2 β(β − 1) + αβ − ρ = 0, and A, B are two constants to be determined. To keep V (m, θ) Þnite as θ becomes small, i.e. limV (m, θ) = 0, we discard θ→0 the term in the negative power of θ setting B = 0. Moreover, the boundary conditions also require that limθ→∞ {V (m, θ) − v(m, θ)} = 0, where the second term in the limit represents the discounted present value of the proÞt ßows over an inÞnite horizon starting from θ (Harrison 1985, p.44): v(m, θ) ≡ E0 ·Z ¸ ∞ −ρt e u(m)θt dt | m0 = m, θ0 = θ = 0 u(m)θ ρ−α (19) The general solution then reduces to: u(m)θ (20) ρ−α Since the last term represents the value of the active Þrm in the absence of new entry, then A(m)θβ 1 is the correction of the Þrm’s value due to the new entry and A(m) must therefore be negative. To determine this coefficient for each m we need to impose some suitable boundary conditions. First of all, perfect competition (free entry) requires V (m, θ) = A(m)θβ 1 + 25 the idle Þrms to expect zero proÞts at entry. Then, indicating by θ∗ (m) the value of the shock θ at which the mth Þrm is indifferent between entry right away or waiting another instant, the matching value condition requires: u(m)θ∗ (m) =K (21) ρ−α The Þrm’s competitive behavior keeps the value of active Þrms below the level K, by increasing the number of Þrms in the market. Moreover, as we assumed that the Þrm’s size is inÞnitesimal, then the trigger level θ∗ (m) is also a continuous function in m. Secondly, it is worth noting that the number of Þrms m affects V (m, θ) depending on the sign of θ∗ (m). Since the term θβ 1 in (21) is always positive, any change in m either raises or lowers the whole function V (m, θ), depending on whether the coefficient A(m) increases or decreases. This simpliÞes the optimization of θ∗ (m);by totally differentiating (21) with respect to m we obtain: V (m, θ∗ (m)) ≡ A(m)θ∗ (m)β 1 + dV (m, θ∗ (m)) dθ∗ (m) = Vm (m, θ∗ (m)) + Vθ (m, θ∗ (m)) dm dm ¸ · ∗ 0 u(m) dθ∗ (m) u (m)θ ∗ ∗β 1 β 1 −1 0 + A(m)β 1 θ (m) + =0 = A (m)θ + ρ−α ρ−α dm Furthermore, since each Þrm rationally forecasts the future development of all the market and new entries by competitors, at the optimal entry threshu0 (m)θ∗ ∗ ∗β1 0 old we get Vm (m, θ (m)) ≡ A (m)θ + ρ−α = 0 (Bartolini, 1993; proposition 1).20 This reduces the above condition to: ¸ · u(m) dθ∗ (m) dθ∗ (m) ∗ β 1 −1 Vθ (m, θ (m)) ≡ A(m)β 1 θ (m) + =0 dm ρ−α dm ∗ (22) In conjunction with the matching value condition (21), the above extended smooth pasting condition says that either each Þrm exercises its entry option at the level of θ at which its value is tangent to the entry cost, i.e. Vθ (m, θ∗ (m)) = 0, or the optimal trigger θ∗ (m) does not change with m. 20 Note that this is a generalization of the condition in Dixit (1993, p. 35). If the Þrm claims to be unique or the last to enter the market, then u0 (m) = A0 (m) = 0 and the Þrst order (22) reduces to Vθ (m, θ∗ (m)) = 0. 26 While the former case means that the value function is smooth at entry and the trigger is a continuous function of m, the latter case says that if this condition is not satisÞed, a single Þrm would beneÞt from marginally anticipating or delaying its entry decision. In particular if Vθ (m, θ∗ (m)) < 0 it means that the value of a Þrm is expected to increase if θ drops (investing now will be expected to lead to almost sure proÞts); on the contrary if Vθ (m, θ∗ (m)) > 0 it means that an active Þrm would expect to make losses versus a future drop in θ. In both situations (22) is satisÞed by imposing dθ∗ (m) = 0, and therefore the same level of shock may either trigger entry by dm a positive mass of Þrms or lock-in the industry at the initial level of Þrms.21 The rest of the proof is devoted to showing that for the m ≥ m̄ the smooth pasting condition reduces to the traditional one, where Vθ (m, θ∗ (m)) = 0 and θ∗ (m) is increasing in m. For m < m̄, we get Vθ (m, θ∗ (m)) > 0 which requires dθ∗ (m) = 0. dm 4.1 Optimal trigger value with negative externalities In the case of m ≥ m̄ we show two things: (1) the smooth pasting condition (22) reduces to Vθ (m, θ∗ (m)) = 0; (2) the optimal competitive trigger θ∗ (m) is equivalent to the trigger of a Þrm in isolation, that is of a Þrm claiming to be the last to enter the industry. For (1), let’s consider the value of an active Þrm starting at the point (m, θ < θ∗ ), that would follow the optimal policy hereafter. Indicating by T the Þrst time that θ reaches the trigger θ∗ , the optimal policy must then satisfy: ·Z T ¸ Z ∞ −ρt −ρT E0 e u(m)θt dt + e u(mt )θt dt | m0 = m, θ0 = θ (23) V (m, θ) = max θ∗ 0 T ·Z T ¸ Z ∞ −ρT −ρt −ρT = max E0 e u(mt )θt dt | m0 = m, θ0 = θ e u(m)θt dt + e max θ∗ θ∗ T 0 ·Z T ¸ ∗ −ρt −ρT E0 e u(m)θt dt + e V (m, θ (m)) | m0 = m, θ0 = θ = max ∗ θ 0 where V (m, θ∗ (m)) represents the optimal continuation value of the Þrm. Since, by (21), the present value of proÞts at T is K, the above value can be 21 If this condition does not hold, the expected “capital” gain or loss at θ∗ (m) would be inÞnite due to the inÞnite variation property of the stochastic process θ. 27 written as: · Z u(m)E0 [ V (m, θ) = max ∗ θ T −ρt e −ρT θt dt | θ0 = θ] + KE0 [e 0 ¸ | θ0 = θ] (24) Moreover, the expected value that appears in the above expression can be found by using some standard results in the theory of the regulated stochastic processes22 . In particular we use the fact that: µ ¶β 1 Z T θ θ − θβ 1 (θ∗ )1−β 1 −ρT −ρt , and E0 [e | θ0 = θ] = E0 [ e θt dt | θ0 = θ] = ρ−α θ∗ 0 (25) Substituting these expressions in (24) and rearranging, we get: " µ ¶ µ ¶β 1 # ∗ u(m)θ u(m)θ θ V (m, θ) = max (26) − −K ∗ θ ρ−α ρ−α θ∗ Now, to choose optimally θ∗ , the Þrst order condition is: ¸ µ ¶β 1 · K u(m) θ ∂V − β1 ∗ =0 ∗ = (β 1 − 1) ∂θ ρ−α θ θ∗ (27) and the optimal threshold function takes the form: u(m)θ∗ (m) = β1 (ρ − α)K ≡ π ∗ , β1 − 1 with β1 >1 β1 − 1 (28) Since u(m) is decreasing in the interval [m̄, M), θ∗ (m) is increasing. Moreover, substituting (28) into (26) we can solve for A(m) which is negative as required by (20): (π∗ )1−β 1 u(m)β 1 Kθ∗ (m)−β 1 ≡− <0 A(m) = − β1 − 1 β 1 (ρ − α) (29) Finally, substituting (29) into (26) and rearranging we obtain (20): V (m, θ) = A(m)θβ 1 + u(m)θ (π ∗ )1−β 1 u(m)β 1 β 1 u(m)θ ≡− θ + ρ−α β 1 (ρ − α) ρ−α 22 (30) For these results see Karlin and Taylor (1974, ch.7); Harrison and Taksar (1983); Harrison (1985, ch.3), and for a non-technical review, Dixit and Pindyck (1994, 315-316) and Moretto (1995). 28 from which it is easy to verify that Vm (m, θ) 6= 0 within the interval θ < θ∗ (m) and zero at the boundary. For (2), let’s consider an idle Þrm pretending to be the last to enter the industry. With m Þrms already active, if the Þrm decides to enter when the shock is θ, it pays K and receives in return an asset that values v(m, θ) as in (19). Write F (m, θ) for the value of its option to enter; this takes the form: n o F (m, θ) = max E0 e−ρT [v(m, θ̂) − K] | m0 = m, θ0 = θ (31) θ̂ where T indicates the Þrst time that θ hits the trigger θ̂. Substituting (19) and rearranging, we get: ( ) u(m)θ̂ − K]E0 [e−ρT | θ0 = θ] F (m, θ) = max [ ρ − α θ̂ à !µ ¶ β θ 1 u(m)θ̂ = max −K ρ−α θ̂ θ̂ (32) Taking the derivative of the above expression with respect to θ̂ and solving it, it is easy to show that the optimal threshold is equivalent to (28). Although at Þrst glance this result seems surprising, it is not. It is consistent with the properties of the dynamic programming principle of optimality for a symmetric Nash equilibrium in entry strategies. The optimality principle says that an optimal path has the property that given the initial conditions and control values over an initial period, the control over the remaining period must be optimal for the remaining problem, with the state resulting from the early decisions considered in the initial condition. This principle matches with the deÞnition of subgame perfect Nash equilibrium where a strategy proÞle is a Nash equilibrium if no Þrm has the incentive to deviate from its strategy given that the other Þrms do not deviate (Fudenberg and Tirole, 1991, p. 108). Therefore, for the problem at hand, a perfect Nash equilibrium means that if all Þrms follow a policy of entry, no individual Þrm can Þnd it optimal to follow any other policy. Formally this implies Þnding a trigger level θ∗ such that a single Þrm Þnds it optimal to enter with the others. Suppose that all Þrms have decided to enter at θ∗ , with θ∗ > θ̂. This cannot be a Nash equilibrium since a single Þrm can do better by entering at θ̂. In fact, 29 since by (3) the myopic proÞt process and the competitive proÞt process are identical until θ∗ , the proÞt ßow that the Þrm is able to obtain following the policy θ̂ is the best that it can do, at least till T. However, by the principle of optimality this choice is also optimal for the rest of the period as (23) shows: if the optimal policy of the single Þrm calls for it to be active at θ∗ tomorrow, it is obvious that the optimal policy today is to enter at θ̂. Finally, as (23) is a continuous function in θ∗ , the limit as θ∗ → θ̂ shows that θ̂ is a Nash equilibrium (Leahy, 1993; proposition 1). Another way of considering the same result is to compare (26) with (32). The value of a competitive Þrm (26) that is active in the market is the difference between the value of an active myopic Þrm and the value of an inactive myopic Þrm as expressed by (32). Competition, therefore, not only does not alter the incentive to trade an idle Þrm for an active Þrm but also encourages both to have the same price at entry. Using (28) in equation (30) gives V (m, θ∗ (m)) − K = 0, i.e. in equilibrium Þrms expect zero proÞt at entry (Dixit and Pindyck, 1994, ch.8). 4.2 Optimal trigger value with positive externalities In the case of m < m̄ we have to show three things: (1) that a single Þrm can no longer claim to be the last to enter the industry and, therefore, the optimal competitive trigger is no longer equivalent to the trigger of a Þrm in isolation; (2) that the candidate policy described in Result 2 satisÞes the necessary and sufficient conditions of optimality; (3) that it is a subgame perfect equilibrium. For (1) and (2), let’s consider an (idle) Þrm that follows the optimal policy ∗ θ (m). As θ∗ (m) is decreasing in the interval m < m̄, the higher the number of Þrms in the industry, the greater the proÞt ßow at entry. The (idle) Þrm would then maximize its entry option by claiming to be always the last to enter the market expecting an inadmissible upward jump in proÞts. To see this more formally, consider a Þrm that claims to have been the last to enter at θ = θ∗ (m). By (19) its value is simply V (m, θ∗ (m)) ≡ v(m, θ∗ (m)) = u(m)θ∗ (m) . It is then easy to check that: ρ−α V (m, θ) = V (m, θ∗ (m)) − lim ∗ θ→θ (m) K >0 β1 − 1 (33) In (33) the inequality holds since it represents the correction due to the 30 new entry (i.e. A(m)θβ 1 in (20)). This contradicts the smooth pasting condition Vθ (m, θ∗ (m)) = 0 and then the optimality of θ∗ (m). As all (idle) Þrms are equal, all expect an upward jump in proÞts at θ = θ∗ (m) if no other Þrm enters afterwards. This may induce each of them to delay entry waiting for the others to enter Þrst. However, as θ∗ (m) is decreasing in the interval m < m̄, the upward jump in proÞts would decrease as more Þrms have already entered and it disappears at m = m̄ where the Þrm’s value function at entry is just the known function (30). This conÞrms ∗ (m) that: a) the candidate policy for the interval m < m̄ is to impose dθdm = 0; ∗ b) the optimal level of shock that triggers entry is θ (m̄) where the proÞt ßow is maximum for all the discrete sizes of investment (m̄ − m); c) at m̄ the necessary condition for optimality Vθ (m̄, θ∗ (m̄)) = 0 turns out to be satisÞed again. To verify that the necessary conditions are satisÞed, let’s calculate the value of an active Þrm starting at the point (m, θ), that would follow a policy deÞned by two parameters: wait until the Þrst instant T at which the process θ rises to a level c > θ, corresponding to an immediate increase of the industry size to b > m. Making use of (23) the expected payoff V (m, θ) from this policy is equal to: V (m, θ) = E0 ·Z T −ρt e −ρT u(m)θt dt + e ¸ V (b, c) | m0 = m, θ0 = θ (34) · ¸ Z T −ρt −ρT = E0 u(m) e θt dt + e V (b, c) | m0 = m, θ0 = θ 0 0 ¸ µ ¶β 1 u(m)θ u(m)c θ − − V (b, c) = ρ−α ρ−α c · If the Þrm were able to choose the best moment for the industry size’s jump as well as the dimension of the jump, the Þrst order conditions would be: ¸ µ ¶β 1 · V (b, c) ∂V (b, c) ∂V (m, θ) u(m) θ =0 = (β 1 − 1) − β1 + ∂c ρ−α c ∂c c ∂V (b, c) ∂V (m, θ) = ∂b ∂b 31 µ ¶β 1 θ =0 c When b and c are chosen according to the candidate policy so that b = m̄ and c = θ∗ (m̄) the value function reduces to (20) and the matching value condition requires V (b, c) = K. These properties verify that the candidate policy satisÞes the above Þrst order conditions. By processing (33) we can say more about the necessary conditions. Let the Þrm, as in (34), wait until the Þrst time the process θ rises to the myopic trigger level c ≡ θ∗ (b), corresponding to an immediate increase of the industry size to b > m, and assume also that the Þrm expects no more entry after b. Therefore its expected payoff V (b, θ) from this time onwards equals the discounted stream of proÞts Þxed at u(b), i.e. by (19): V (b, θ) = u(b)θ ρ−α (35) Comparing (35) with (20) gives A(b) = 0. Therefore to obtain the constant A(m), subject to the claim that beyond b no other Þrm will enter the market, we substitute (20) into the condition Vm (m, θ∗ (m)) = 0 to get A0 (m)θ∗ (m)β 1 + u0 (m)θ∗ (m) = 0 resulting in: ρ−α θ∗ (m)1−β 1 u0 (m) (π ∗ )1−β 1 u0 (m) ≡− A (m) = − ρ−α ρ − α u(m)1−β 1 0 (36) Integrating (36) between m and b gives: Z Z b (π ∗ )1−β 1 b u0 (x) 0 dx A (x)dx = − ρ − α m u(x)1−β 1 m Taking account of the fact that A(b) = 0, the above integral gives the constant A(m) as: A(m) = ¤ (π ∗ )1−β 1 £ u(b)β 1 − u(m)β 1 β 1 (ρ − α) (37) Substituting (37) into (20), which we rewrite to make explicit its dependence on the end size b, yields: V (m, θ; b) = ¤ (π ∗ )1−β 1 £ u(m)θ u(b)β 1 − u(m)β 1 θβ 1 + β 1 (ρ − α) ρ−α (38) As long as u(b) > u(m) the Þrst term in (38) is positive and it forecasts the advantage the Þrm would experience by the entry of b − m Þrms when 32 θ hits θ∗ (b). That is, if the Þrm were able to choose the optimal dimension of the jump, it would be b → m̄ which happens the Þrst time that θ reaches θ∗ (m̄). Thus, as opposed to before non-sequential investments are possibile, the necessary conditions would coordinate an optimal simultaneous entry by all the Þrms, i.e. θ∗ (m̄) is a (symmetric) Pareto-dominant Nash equilibrium for all m < m̄. Finally, if u00 (m) < 0 the necessary conditions are also sufficients. As the stochastic process θ is common knowledge, each Þrm can foresee the beneÞt from the entry of others and observing the realization of the state variable θ instantaneously considers when to enter by maximizing (38). In addition, as the reaction lags are literally nonexistent, no Þrm has the incentive to deviate from the entry strategy θ → θ∗ (m̄) and b → m̄ given that the other Þrms do not deviate. Finally, since θ is a Markov process in levels (Harrison, 1985, p.5-6), the conditional expectation (34) is in fact a function solely of the starting states so that, at each date t > 0, the Þrm’s values resemble those described in (38) which makes the equilibrium subgame perfect. 5 Positive externalities and the case of telecommunication services So far we consider the function u(m) as a reduced form of a more general proÞt function or, in a simpler setting without operating costs, as the inverse demand function of a network good. This section is devoted to developing this application a bit further and to analysing the implications of the above optimal entry policy for a network product. In this regard, we consider the Þrms’ decision to set up a network for satisfying a demand for telecommunication services. 5.1 Interconnection and competitive provider Following the pioneering approach of Rohlfs (1974)23 , we consider a group of a M continuum of potential telecommunication customers uniformly indexed by i ∈ [0, M] and ranked in decreasing order of willingness to pay. We interpret customers indexed by low i as those who place high valuation on 23 See also Shy (2001) ch.5. 33 the ability to communicate. The utility of a consumer indexed by i is deÞned as:  if s/he subscribes  (1 − Mi )q − u (39) Ui =  0 if s/he does not subscribe where q is the total number of consumers who actually subscribe and u is the connection fee.24 To derive the consumers’ aggregate demand for phone services we look at the consumer m who is, for a given price u, indifferent to subscribing or not subscribing the service. By (39) the indifferent consumer m )q − u = 0 and assuming fulÞlled-expectations about the is found by (1 − M number of subscribers, we get q = m. Substituting we obtain the inverse demand function for telecommunication services: u(m) = (1 − m )m M (40) The inverse aggregate demand function (40) exhibits a path similar to the one in Þgure 1. It is upward sloping at small demand levels (i.e. over the interval [0, m̄)) and becomes downward sloping at high demand levels. In particular u(0) = 0, u(M) = 0 and m̄ = M2 . For any given m, u(m) is therefore the reservation price of the marginal subscriber. On the supply side, we assume that there are many idle Þrms ready to provide telecommunication services with the following characteristics: • Each Þrm can serve one single customer with a Þxed coefficient technology, i.e. each Þrm provides one unit of service per period.25 Then m indicates the total number of consumers that subscribe as well as the size of the industry providing the phone system. • Each Þrm can enter by building its own network at cost K, but this cost is sunk and the investment is irreversible (K is the cost of connecting the house of a new customer to the total network). 24 Congestion can be easily adpated to this model introducing an utility function of type Ui = (1 − Mi )f (q) − u, where the network effect is given by the function f (q) with the properties that f (0) = 0, f 00 (q) < 0 and there exist a maximum at some positive level of subscribers (see Lee and Mason, 2001). 25 It is worth noting that the quality of our results would not change if we assume that each Þrm serves a single network with an equal number of costumers. 34 • Interconnection is provided. That is, each Þrm may use the infrastructure owned by other Þrms in the industry paying a Þxed access price per unit of time which is the same for all Þrms. Then m also indicates the total dimension of the network.26 With m > 0 incumbent Þrms currently active, if the interconnection fees are the only operative cost borne by the potential entrants, in view of (40), each provider will expect to yield a ßow of operating proÞts equal to: π(m, θ) ≡ u(m)θ = (1 − m )mθ M (41) where θ is a stochastic variable that summarises different kinds of randomness from variable inputs to shifts of technology. 5.2 Equilibrium network size Going on with the case of demand for telecommunication services m stands for the number of users that subscribe before the network grows and generates bandwagon beneÞts, while m̄ = M2 indicates the minimal demandbased equilibrium network achieved by rolling over the upward-sloping part of the inverse demand curve (40) by positive feedback. This positive feedback process starts as θ reaches an upper level. Proposition 2 The minimal demand-based size of the network M2 is reached by the connection of M2 − m new customers when θ hits for the Þrst time the upper level: 4K β1 (ρ − α) θ∗ ≡ β1 − 1 M Above this trigger more customers will be connected, following the rule: θ∗ (m) ≡ β1 K , (ρ − α) m )m β1 − 1 (1 − M for m ∈ [ M ,M) 2 In other words, M2 is the minimal number of customers needed to ensure that at least they will beneÞt from subscribing to the service at the fee u = M4 . 26 As the Þrms are inÞnitesimal and indistinguishable (as well as the customers) it seems reasonable to assume an equal access price. This is equivalent to assuming free access among Þrms. 35 The timing to build up this minimal network depends on the evolution of the exogenous shock θ. In the region below θ∗ the optimal policy is inaction, 1 (ρ − α) 4K a mass of M2 − m outsiders the Þrst time that θ hits the level β β−1 M 1 coordinate their entry subscribing. Once the network has reached its minimal size, on the right of M2 further entries proceed as market demand increases. Finally, while the mass of new subscribers strongly depend on the initial user set m, the critical threshold does not. However, the degree of coordination among potential entrants increases as m decreases as there are more externalities to be exploited (see proposition 1). 6 Comments on the literature The previous section has shown that for m < m̄ the candidate policy θ∗ (m̄) is the unique threshold beyond which a mass (m̄ − m) of idle Þrms Þnds it optimal to move simultaneously. This was done by showing that θ∗ (m̄) satisÞes the necessary and sufficient conditions of optimality for a single Þrm that Þnds it optimal to enter with the others. It is also shown that once entry has exhausted the positive externalities, new Þrms will enter following the standard competitive rule (5) where in equilibrium the option value of waiting does drop to zero. In this respect, our model is an extension of the dynamic equilibrium in a competitive industry presented by Leahy (1993) and Dixit and Pindyck (1994, ch.5).27 Contrary to that model we allow the Þrms to experiment positive externalities before the industry reaches the size where negative externalities apply. We Þnd that Þrms invest simultaneously once the industry proÞtability has developed sufficiently to allow them to capture all the externalities and to recover the option value of waiting. In a duopoly model, Nielsen (2002) predicts a result similar to ours, namely that the Þrms invest simultaneously at the market proÞtability given by the duopoly proÞt.28 Thus Nielsen’s result (2002) holds more generally in a free entry competitive framework. 27 Baldursson (1998) extends Dixit and Pindyck’s model considering Cournot-Nash competition. His analysis indicates that although qualitatively the investment process is similar in oligopoly and competitive equilibrium, oligopoly quantitatively slows down investment. 28 Huisman (2001, ch. 8) extends the Nielsen (1999) model introducing asymmetry into the investment costs of the Þrms. Although cost asymmetry may reduce the positive externality effect, both Þrms invest simultaneously and early in anticipation that the other will invest early as well. 36 Obviously simultaneous investments may arise under circumstances very different from those considered here. For example in Bartolini (1993), simultaneous investment is driven by a constraint on the total size of the industry. He considers a competitive industry where the Þrms initially enter following the optimal policy as in Result 1, until a “critical” size is reached. At this “critical” size, rent competition generates a “competitive run” that immediately Þlls the rest of the quota. During this run the Þrms experience a reduction of current proÞts in the attempt to capture the rent that the constraint on the industry size is expected to generate. Unlike Bartolini, in our model a run is generated by the maximization of the rent associated with the positive externalities. These rents will be dissipated in the future by competitive entry of Þrms with negative externalities. Moreover, as entry is not constrained, the negative externalities do not lead to proÞt reduction during the run. To see this formally, let’s start by imposing the free entry zero-proÞts condition at m̄. That is: ∗ ∗ β1 V (m̄, θ (m̄)) − K ≡ A(m̄)θ (m̄) u(m̄)θ∗ (m̄) −K =0 + ρ−α (42) Unlike Bartolini (1993), at the end of the run equation (42) implies A(m̄) < 0, which gives (28) as optimal entry policy.29 Secondly, substituting (38) into the extended smooth pasting condition (22) and letting b → m̄, we obtain: ¸ · ¤ u(m) dθ∗ Φ1−β 1 £ ∗β β1 β1 1 =0 (43) u(m̄) − u(m) β 1 θ + β 1 (ρ − α) ρ − α dm The term inside square brackets is always positive (i.e. there is no value ∗ m ∈ (m, m̄) that makes it nil), and (43) holds with dθ = 0. That is, all dm Þrms in the range (m, m̄) must enter at θ = θ∗ (m̄). In Grenadier (1996) on the other hand, simultaneous investment occurs because two Þrms rush to enter a declining real estate market that will otherwise leave space only for one Þrm. As developers see the market falling they realise that if they continue to wait and none of them decide to invest, they will be shut out of the market. Grenadier refers to this occurrence as a “recession-induced construction boom”, however it occurs only if the initial 0 29 If M = m̄ is the constraint on the total size of the industry then A( m̄) = 0 and eq. (11) in Bartolini (1993) gives u( m̄)θ∗ ( m̄) = (ρ − α)K < π ∗ . However, as by assumption 6 M could be arbitrarily large this excludes “competitve run” in our model. 37 level of demand is greater than the level that induces to optimally invest as a follower. In Moretto (2000), simultaneity arises because of a bandwagon effect on entry costs. Two Þrms are engaged in an “attrition” game generated by the presence of incomplete information plus positive externalities (“network beneÞts”) on the investment costs: i.e. it is more expensive to go Þrst than to adopt the technology coordinately or later on when others have already done so. Although the Þrst-mover disadvantage leads to sequential investment, if the asymmetry between Þrms is not too high the investment occurs as a cascade: i.e. the beneÞts of going second after the Þrst Þrm has invested induces the second to follow suit.30 At the opposite end, Huisman and Kort (1999) show that simultaneous investments may arise also in the presence of negative externalities. The model considers a preemption game where two identical Þrms are active on a market and have the option to make an irreversible investment in a new technology which results in higher proÞt ßow. Although, in general, the presence of a Þrst-mover advantage leads to a preemption equilibrium where one Þrm plays the role of leader, the condition of both the Þrms being already active on the output market where they compete does not exclude the possibility of both Þrms investing at the same time. This happens in particular when the Þrst-mover advantage is so low that both the Þrms prefer to delay investment and invest at a later time jointly. Maison and Weeds (2001) show the same result in a similar duopoly model. Although they consider the simultaneous presence of negative and positive externalities, the only case in which both Þrms enter simultaneously is when they know that if the investment occurred sequentially, the leader would lose out considerably once the follower decided to enter. Finally, all these recent works are built upon the seminal paper of Farrell and Saloner (1986). These authors present a two-agent model of technology investment with uncertainty about the timing of the investment, positive externalities and irreversibility where each agent has to invest exogenously at random opportunities driven by a Poisson process. They count cases of preemption equilibrium as well as cases of joint adoption. However, if agents are allowed to invest at any time and not just at occasional chances, many of the features found by Farrell and Saloner would disappear leaving the basic 30 Dosi and Moretto (1996, 1998) also examine a war of attrition game induced by spillover beneÞts on the cost of adopting a “green” technology. They show that auctioning green investment grants is a better policy to stimulate simultaneous investment than standard subsidies that lower investment costs. 38 coordination problem due to the positive externalities. 7 Conclusion In this paper we have offered an initial investigation into the effect of competition on the irreversible investment decisions under uncertainty of the Þrms as generalization of the “real option” approach. We have considered a product market that allows simultaneous treatment of two different cases, namely those of positive externalities for low level of market size and negative externalities for high level of market size. The latter case corresponds to the traditional competitive industries in which the investment of one Þrm lowers the proÞtability of the others. In this case, Þrms invest sequentially as the market proÞtability develops. The former case corresponds to industries in which investments are mutually beneÞcial: the investment of one Þrm increases the proÞtability of other Þrms’ investments. In this case we Þnd that Þrms invest simultaneously after the proÞtability of the market has developed sufficiently. By sufficient we mean the proÞt level that triggers a Þrst investment under negative externalities; this trigger determines endogenously the optimal start-up size of the industry. Not excluding further improvements, putting together these theoretical results may help to explain both the recent rapid and sudden development that has occurred for internet investments, for example the setting up of dotcoms on the World Wide Web for e-commerce, and the many prolonged start-up problems while awaiting market development as, for example, the story of the digital fax machines shows (Rohlfs, 2001).31 Some extensions can be easily incorporated such as the inclusion of Þnitelylived capital projects, stage investments, growth options and operative options that lead to suspension or deÞnitive abandonment of the investments. The model also permits study of the efficiency of the investment-entry pattern. Is the equilibrium investment-entry time efficient? Does the efficient entry pattern occur in equilibrium? This study can be conducted considering the cooperative solution where the investment decisions are determined by maximizing the sum of the Þrms’ value functions or introducing a true 31 Both these are examples of interlinked network services competiti vely supplied. Each consumer enjoys network externalities not only with respect to the consumers of his or her own supplier. The history of the fax also illustrates the importance of interlinking in making the demand grow to solve the start-up problem. 39 social value function. Finally a more substantial modiÞcation concerns the comparison with the case in which there is a monopolist which possesses all the investment opportunities. Although intuitively the start-up problem in this case is much simpler, of particular interest is the analysis of the startup conditions and the optimal network size. In the speciÞc, where network externalities are present, it may be proÞtable for the monopolist to sacriÞce proÞts in the short-run in the hope of raising prices in the future after the demand has grown and consumers are enjoying network effects. 40 A Appendix A.1 Properties of F F (T 2), F (T 1; T 2), M(T 1) and L(T 1; T 2). Lemma 1 M(T 1 ) > F F (T 2 ) Proof. Recalling that T 2 = inf(t > 0 | θ = θ∗ (m)) where θ∗ (m) ≡ β 1 (ρ−α) , subsituting in F F (T 2 ) we obtain the following function: β −1 u(m) 1 µ u(m) f f(m) ≡ − u(m̄) ¶β 1 β1 + β1 − 1 µ u(m) u(m̄) ¶β 1 −1 , for m ∈ [0, m̄) (44) which is with M(T 1 ) ≡ ff (m̄). Now, making use of the variable x(m) = u(m) u(m̄) monotonically increasing in m, with x = 1 for m = m̄ and x = 0 for m = 0, we are able to simplify (44) as: f f(x) ≡ − (x)β 1 + with ff(1) = 1 , β 1 −1 β1 (x)β 1 −1 , β1 − 1 for x ∈ [0, 1) f f(0) = 0. Now, taking the derivative of ff (x) with respect to x gives ff 0 (x) = −β 1 (x)β 1 −1 + β 1 (x)β 1 −2 = β 1 (x)β 1 −1 (x−1 − 1), which is always positive for x < 1 (i.e. for m < m̄). Lemma 2 F F (T 2 ) < F (T 1 ; T 2 ) < M(T 1 ) Proof. As F F (T 2 ) ≡ f f(m̄−i ), this follows directly from application of the properties of f f(m). Lemma 3 L(T 1 ; T 2 ) < F F (T 2 ). Proof. Let’s deÞne the function l(x) ≡ ff (x) + g(x), where g(x) ≡ 1 − β 1−1 (x)β 1 + β β−1 x − 1 and x(m) = u(m) . As g(0) = −1, g(1) = 0 and u(m̄) 1 1 1 g0 (x) = β β−1 [1 − (x)β 1 −1 ] > 0 for all x, yields l(x) ≤ f f(x) for all x ∈ [0, 1). 1 Therefore, simple considerations show that L(T 1 ; T 2 ) ≡ l(m+i ) < ff (m+i ), where the last inequality follows from the inÞnitesimal dimension of the ith Þrm (see Þgure 4 below). 41 ff, l 1 0.5 0 0 0.25 0.5 0.75 1 x -0.5 -1 Figure 4: ff (x) and l(x) with β 1 = 2. A.2 Monotonicity property of M, F and L Let’s consider the case in which the Þrms coordinate at t > T 1 with θt > θ∗ (m̄). By the shape of θ∗ (m), it is always possible to Þnd m̃ < m̄ such that θt = θ∗ (m̃) > θ∗ (m̄). Then the payoff of m̄ − m Þrms coordinating at θt is equivalent to the payoff, starting with m̃ active Þrms, of m̄ − m̃ Þrms that do not enter at T 1 and all wait until T 2 before entering. ¶β ¸µ ∗ θ (m̄) 1 u(m̄)θ∗ (m̃) −1 M(t) = ρ−α θ∗ (m̃) ¶β µ ¶β −1 µ u(m̃) 1 β1 u(m̃) 1 , + = − u(m̄) β 1 − 1 u(m̄) · 42 (45) for m̃ ∈ [m, m̄) and: ¶β 1 ¸µ ∗ θ (m̃) u(m̄)θ∗ (m̃−i ) −1 F (t; T ) = ρ−α θ∗ (m̃−i ) µ ¶β µ ¶β −1 β1 u(m̃−i ) 1 u(m̃−i ) 1 = + , u(m̄) β1 − 1 u(m̄) 2 · (46) for m̃ ∈ [m, m̄) Noting that as now m̃ goes from m̄ to m as t goes from T 1 to T 2 , the ³ ´β 1 t term θ∗θ(m) < 1 represents the discount factor. Applying Lemma 1 and 2 the following Lemma can be directly proved: Lemma 4 1) M(t) ≡ ff (m̃), and ∂M(t) ≡ ∂f∂fm̃(m̃) < 0, for all m̃ ∈ [m, m̄); ∂t 2) m̃−i ) ≡ ∂f f∂(m̃ < 2) F (t; T 2 ) ≡ ff (m̃−i ) < f f(m̃) ≡ M(t), and ∂F (t;T ∂t 0, for all m̃ ∈ [m, m̄); 3) if we allow t to increase towards T 2 (or equivalently m̃ → m ) we obtain: lim2 M(t) = lim2 F (t; T 2 ) = F F (T 2 ) t→T t→T Although for the payoff of a player who coordinates when the other fails to do so, L(t; T 2 ), we cannot refer directly to the f f(m) function, we are able to show that: 2 2 ) ) ≡ ∂L(t;T < 0, for all m̃ ∈ [m+i , m̄); Lemma 5 1) ∂L(t;T ∂t ∂ m̃ 2 2) limt→T 2 −∆T L(t; T ) = F F (T 2 − ∆T ). Proof. First, the payoff L(t; T 2 ) is deÞned for all m̃ ∈ [m+i , m̄). By (10), this follows from the deÞnition of T 2 − ∆T = inf(s > t | θs = θ∗ (m+i )) as the Þrst time to which the rivals respond by entering. Second, evaluating directly the payoff for θt = θ∗ (m̃) we get: ¶β ¸µ ∗ θ (m̄) 1 u(m+i )θ∗ (m̃) −1 L(t; T ) = + ρ−α θ∗ (m̃) · ¸µ ∗ ¶β 1 (u(m̄) − u(m+i ))θ∗ (m+i ) θ (m̄) + ρ−α θ∗ (m+i ) 2 · or 43 · ¶β ¸µ β 1 u(m+i ) u(m̃) 1 L(t; T ) = −1 + β 1 − 1 u(m̃) u(m̄) · ¸µ ¶β β 1 u(m̄) − u(m+i ) u(m+i ) 1 β1 − 1 u(m+i ) u(m̄) 2 (47) with L(T 2 − ∆T ; T 2 ) ≡ F F (T 2 − ∆T ). As only the Þrst term on the r.h.s of (47) depends on m̃, taking the derivative we get: µ ¶β −1 · ¸ ∂L(t; T 2 ) u(m̃) 1 u0 (m̃) β 1 u(m+i ) ∂L(t; T 2 ) 1 u(m+i ) ≡ = β1 −1− ∂t ∂ m̃ u(m̄) u(m̄) β 1 − 1 u(m̃) β 1 − 1 u(m̃) µ ¶β 1 −1 0 · ¸ u(m̃) u (m̃) u(m+i ) = β1 −1 ≤0 for all m̃ ∈ [m+i , m̄) u(m̄) u(m̄) u(m̃) A.3 Proof of proposition 1 Proof. To prove the Þrst part of proposition 1 let’s Þrst consider the difference F F (T 2 − ∆T ) − L(t; T 2 ). This difference is always positive for all t ∈ (T 1 , T 2 − ∆T ), i.e. for m̃ ∈ (m+i , m̄), and null in T 2 − ∆T , i.e. at m̃ = m+i . F F (T 2 − ∆T ) − L(t; T 2 ) = · ¶β ¸µ u(m̄) β1 u(m+i ) 1 = − −1 β 1 − 1 u(m+i ) u(m̄) ¶β ¸µ ¶β ¸µ · · u(m+i ) 1 u(m̃) 1 β 1 u(m̄) − u(m+i ) β 1 u(m+i ) −1 − − β 1 − 1 u(m̃) u(m̄) β1 − 1 u(m+i ) u(m̄) µ ¶β 1 · ¶β 1 ¸µ u(m+i ) u(m̃) 1 β 1 u(m+i ) −1 = − β1 − 1 u(m̄) β 1 − 1 u(m̃) u(m̄) By Lemma 3, if t tends to T 1 , i.e. m̃ → m̄, it follows directly that: µ ¶β u(m+i ) 1 1 β 1 u(m+i ) 2 1 2 F F (T − ∆T ) − L(T ; T ) = +1 − β1 − 1 u(m̄) β 1 − 1 u(m̄) = −g(m+i ) > 0 44 However, if t tends to T 2 − ∆T, i.e. m̃ → m+i , by Lemma 5 we get that F F (T 2 − ∆T ) − L(t; T 2 ) tends to zero. Let’s now consider the difference M(t) − L(t; T 2 ). Also this difference is always positive for all t ∈ (T 1 , T 2 − ∆T ), i.e. for m̃ ∈ (m+i , m̄), and null in T 2 − ∆T , i.e. at m̃ = m+i . M(t) − L(t; T 2 ) = · ¸µ ¶β β 1 u(m̄) u(m̃) 1 = − −1 β 1 − 1 u(m̃) u(m̄) ¶β ¸µ ¶β ¸µ · · u(m+i ) 1 u(m̃) 1 β 1 u(m̄) − u(m+i ) β 1 u(m+i ) −1 − − β 1 − 1 u(m̃) u(m̄) β1 − 1 u(m+i ) u(m̄) · ¸µ ¶β 1 · ¸µ ¶β u(m̄) u(m+i ) u(m̃) u(m+i ) 1 β 1 u(m̄) − u(m+i ) β1 − − = β 1 − 1 u(m̃) u(m̃) u(m̄) β1 − 1 u(m+i ) u(m̄) " µ ¶β 1 µ ¶β 1 # β1 1 1 u(m̃) u(m+i ) = − [u(m̄) − u(m+i )] β1 − 1 u(m̃) u(m̄) u(m+i ) u(m̄) By Lemma 1 and 3, if t tends to T 1 , i.e. m̃ → m̄, it follows directly that: " # µ ¶β −1 β1 1 1 u(m+i ) 1 u(m+i ) [u(m̄) − u(m+i )] − = β1 − 1 u(m̄) u(m+i ) u(m̄) u(m̄) # " ¶β −1 µ u(m+i ) 1 β 1 u(m̄) − u(m+i ) >0 1− = β1 − 1 u(m̄) u(m̄) On the contrary, if t tends to T 2 − ∆T, i.e. m̃ → m+i , by Lemma 4 M(t) − L(t; T 2 ) tends to zero. Putting together these results we can, Þnally, take the derivative of s̃−i (t) with respect to t : ∂s̃−i (t) ∂s̃−i (t) ≡ = ∂ m̃ ∂t ∂L(t;T 2 ) [F F (T 2 − ∆T ) − M(t)] + ∂M(t) [L(t; T 2 ) − F F (T 2 − ∆T )] ∂t ∂t = >0 [M(t) − L(t; T 2 )]2 The proof of the second part follows directly from Lemmas 4 and 5 and the deÞnition of T 2 . In fact, T 2 → ∞ as m → 0 and recalling that m̃ goes 45 from m̄ to m as t goes from T 1 to T 2 it follows directly that: F F (T 2 − ∆T ) − L(t; T 2 ) = 1 lim 2 T →∞ and lim M(t) − L(t; T 2 ) = 2 T →∞ Finally, the third part derives from the fact that 46 ∂β 1 ∂σ 2 < 0. β1 β1 − 1 References [1] Bartolini L., (1993), “Competitive Runs, The Case of a Ceiling on Aggregate Investment”, European Economic Review, 37, 921-948. [2] Bernheim, B.D., Peleg, B. and M. Whinston, (1987), “Coalition-proof Nash Equilibrium I: Concepts”, Journal of Economic Theory, 42, 1-12. [3] Cox, J.C., and S.A. 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STREBULAEV (lvx): Multiple Unit Auctions and Short Squeezes Anders LUNANDER and Jan-Eric NILSSON (lvx): Taking the Lab to the Field: Experimental Tests of Alternative Mechanisms to Procure Multiple Contracts TangaMcDANIEL and Karsten NEUHOFF (lvx): Use of Long-term Auctions for Network Investment Emiel MAASLAND and Sander ONDERSTAL (lvx): Auctions with Financial Externalities Michael FINUS and Bianca RUNDSHAGEN: A Non-cooperative Foundation of Core-Stability in Positive Externality NTU-Coalition Games Michele MORETTO: Competition and Irreversible Investments under Uncertainty_ (l) This paper was presented at the Workshop “Growth, Environmental Policies and Sustainability” organised by the Fondazione Eni Enrico Mattei, Venice, June 1, 2001 (li) This paper was presented at the Fourth Toulouse Conference on Environment and Resource Economics on “Property Rights, Institutions and Management of Environmental and Natural Resources”, organised by Fondazione Eni Enrico Mattei, IDEI and INRA and sponsored by MATE, Toulouse, May 3-4, 2001 (lii) This paper was presented at the International Conference on “Economic Valuation of Environmental Goods”, organised by Fondazione Eni Enrico Mattei in cooperation with CORILA, Venice, May 11, 2001 (liii) This paper was circulated at the International Conference on “Climate Policy – Do We Need a New Approach?”, jointly organised by Fondazione Eni Enrico Mattei, Stanford University and Venice International University, Isola di San Servolo, Venice, September 6-8, 2001 (liv) This paper was presented at the Seventh Meeting of the Coalition Theory Network organised by the Fondazione Eni Enrico Mattei and the CORE, Université Catholique de Louvain, Venice, Italy, January 11-12, 2002 (lv) This paper was presented at the First Workshop of the Concerted Action on Tradable Emission Permits (CATEP) organised by the Fondazione Eni Enrico Mattei, Venice, Italy, December 3-4, 2001 (lvi) This paper was presented at the ESF EURESCO Conference on Environmental Policy in a Global Economy “The International Dimension of Environmental Policy”, organised with the collaboration of the Fondazione Eni Enrico Mattei , Acquafredda di Maratea, October 6-11, 2001 (lvii) This paper was presented at the First Workshop of “CFEWE – Carbon Flows between Eastern and Western Europe”, organised by the Fondazione Eni Enrico Mattei and Zentrum fur Europaische Integrationsforschung (ZEI), Milan, July 5-6, 2001 (lviii) This paper was presented at the Workshop on “Game Practice and the Environment”, jointly organised by Università del Piemonte Orientale and Fondazione Eni Enrico Mattei, Alessandria, April 12-13, 2002 (lvix) This paper was presented at the ENGIME Workshop on “Mapping Diversity”, Leuven, May 16-17, 2002 (lvx) This paper was presented at the EuroConference on “Auctions and Market Design: Theory, Evidence and Applications”, organised by the Fondazione Eni Enrico Mattei, Milan, September 2628, 2002 2002 SERIES CLIM Climate Change Modelling and Policy (Editor: Marzio Galeotti ) VOL Voluntary and International Agreements (Editor: Carlo Carraro) SUST Sustainability Indicators and Environmental Valuation (Editor: Carlo Carraro) NRM Natural Resources Management (Editor: Carlo Giupponi) KNOW Knowledge, Technology, Human Capital (Editor: Dino Pinelli) MGMT Corporate Sustainable Management (Editor: Andrea Marsanich) PRIV Privatisation, Regulation, Antitrust (Editor: Bernardo Bortolotti) ETA Economic Theory and Applications (Editor: Carlo Carraro) 2003 SERIES CLIM Climate Change Modelling and Policy (Editor: Marzio Galeotti ) GG Global Governance (Editor: Carlo Carraro) SIEV Sustainability Indicators and Environmental Valuation (Editor: Anna Alberini) NRM Natural Resources Management (Editor: Carlo Giupponi) KNOW Knowledge, Technology, Human Capital (Editor: Gianmarco Ottaviano) IEM International Energy Markets (Editor: Anil Markandya) CSRM Corporate Social Responsibility and Management (Editor: Sabina Ratti) PRIV Privatisation, Regulation, Antitrust (Editor: Bernardo Bortolotti) ETA Economic Theory and Applications (Editor: Carlo Carraro) CTN Coalition Theory Network