Learning and Collusion in New Markets with Uncertain Entry Costs ∗
Francis Bloch
Ecole Polytechnique
Simona Fabrizi
Massey University
and
Steffen Lippert
University of Otago
August 17, 2011
Abstract
This paper analyzes an entry timing game with uncertain entry costs. Two firms receive costless signals about the cost of a new project and must decide when to invest. We characterize the equilibrium of the investment timing game with private values, private and public signals. We show that competition leads the two firms to invest too early and analyze collusion schemes whereby one firm prevents the other firm from entering the market. We show that, in the efficient collusion scheme, the active firm must transfer a large part of the surplus to the inactive firm in order to limit preemption.
JEL Classification Codes: C63, C71, C72, D82, D83, O32 Keywords: Learning; Preemption; Innovation ; New Markets ; Entry Costs ; Collusion ; Private Information
∗
We are grateful to seminar participants at Erasmus University, Queen Mary, the Paris
School of Economics for helpful comments. We want to thank particularly Rossella Argen-
ziano, Nisvan Erkal, Emeric Henry, Hodaka Morita and Francisco Ruiz Aliseda for their
help and support.
1 Introduction
The development of new products and processes, the access to new markets are characterized by a high degree of uncertainty. Firms engage in long research process and accumulate information about the cost and benefits of the project before actually investing in new products or entering new markets. Project selection is often very fierce, and very few projects end up being implemented. For example, in the pharmaceutical industry, only a very small fraction of the molecules which are tested end up being patented.
A realistic description of investment in new products and markets must allow for learning about the cost and benefit of the investment.
When different firms or research teams compete to enter a new market, the dynamics of learning and competition may become very complex. Signals received by competitors may transmit information about the profitability of the market, affecting the incentives to acquire information and the speed of investment in the project. A further distinction exists between situations where information received by competitors is made public or kept private.
In the first case, information becomes a public good, resulting possibly in free-riding and low levels of information acquisition. In the second case, participants in the race must control the diffusion of information that their actions convey to their opponents, resulting in complex investment dynamics.
In particular, private information may lead to preemption and accelerate the race between the competitors.
In this paper, our objective is to better understand the interplay between learning and preemption in entry timing games, and to study simple mecha- nisms of collusion between firms. We contrast the outcome of an entry game among firms which cooperate in project selection, and between competitors with public and private information. We consider a simple compensating pay- ment cooperation mechanism, whereby one of the two firms pays the other team to stay out of the market, and show when this compensating payment enables firms to reach the collusive outcome.
Our analysis is motivated by situations where firms are engaged in inno- vation races to launch a new product, and upon success, one of the two teams is acquired by the other so that a single firm extracts the monopoly profit of the innovation. For example in the late 1990s and early 2000s, the computer industry saw an acquisition frenzy in which firms were bought out even before they had an initial success. After acquiring a Switch maker in 1993 (switches were the main competition to their core business, routers), Cisco acquired 69 companies between 1995 and 2000 (more than eleven companies per year).
Many of these acquisitions were early stage acquisitions that turned out not
to be viable. Cisco often did not learn about their targets: They bought out potential competitors early, before knowing whether or not their own R&D strategy is better. Subsequently, Cisco switched to an acquisition strategy, in which it acquired fewer, but more mature companies, after learning about their potential. 2
Similarly, threatened by looming patent expiry for blockbuster drugs as well as by competition from generics, and facing a declining product pipeline due to low in-house R&D productivity, big pharmaceutical com- panies have recently been acquiring biotechnology companies. For example, in 2009 Hoffmann-La Roche acquired the biotech company Genentech for nearly $47bn. Merck & Co. purchased a follow-on biologics platform from Insmed in 2010. However, contrary to Cisco in the 1990s and early 2000s, these pharmaceutical companies did these acquisitions in order to access their targets’ existing early-stage successes. This seems to suggest that competing teams sometimes wait until there is a success signal in early phases of the R&D process and then purchase their potential competitor.
On the other hand, in the pharmaceutical industry, companies at times engage in what to outside observers seems wasteful investment: Several phar- maceutical companies target the same biological mechanism with different molecules, leading to several drugs with the same therapeutic indications. 3 The phenomenon is called “me-too” drugs, and the presumption is that, after they see a competitor succeed with a particular mechanism, pharmaceutical companies engage in imitation and invest in finding a different molecule that target this mechanism. However, contrary to this commonly held belief, DiMasi and Faden (2011) show that many “me-too” drugs are not the re- sult of one company imitating another, but of parallel research targeting the same mechanism, with no one knowing which drugs will work until they have cleared their regulatory trials, often in rapid succession. 4 In these cases, it seems firms fail to cooperate and instead invest in competition with each other.
We construct a model of entry where firms initially ignore the fixed cost of entry. They gradually acquire signals about the entry cost through research
2
Between 2001 and 2003, they only acquired 10 companies.
3
For example, in the U.S., more than six anti-cholesterol drugs on the market, from Crestor to Zocor, are frequently advertised on television. The market for male sexual enhancements started with Pfizer Inc.s Viagra and now includes two other drugs, Cialis marketed by Eli Lilly and Co. and Levitra, which is sold by a partnership of GlaxoSmithK- line and Bayer Pharmaceuticals Corp.
4
See DiMasi and Faden (2011); and Economist article entitled “Me too! Me too!” (Apr
17, 2007; http://www.economist.com/blogs/freeexchange/2007/04/me_too_me_too).
and experimentation. We consider here the case of private values, where the profitability of entry differs across firms (as would be the case if uncertainty primarily involves the private cost of investment) and not the case of common values, where the profitability of innovation is the same for all teams (as would be the case if uncertainty primarily involved the value of the new market).
Our model results in project selection. In the cooperative outcome, only the most profitable firm should be allowed to enter. We show that com- petition results in preemption, as the two firms may, for some parameter values, choose to invest too early, before they learn the entry cost whereas it would have been optimal for them to wait until they learn their cost. If signals on entry costs are public, compensating payments can be used to en- force monopolization of the market. If the signals are private, firms do not observe whether their opponent has dropped from the race or not. Hence, over time, if they don’t observe investment by their opponent, teams become more optimistic as they believe that with high probability their opponent has decided not to invest (”no news is good news”). This dynamics of beliefs will eventually lead a firm to invest before it has learned the profitability of the innovation, aggravating preemption and resulting in an inefficient accel- eration of the entry race. Compensating payments can be designed in order to mitigate this preemption effect and lead to market monopolization. We characterize a lower and upper bound on the share of the surplus which is transferred to the firm which drops out of the race, and show that the optimal compensating payment will typically involve a large share of the surplus to be transferred to the trailing firm. However, the compensating payment cannot restore efficiency in all circumstances, and in fact, there exist states where no efficient, budget balanced, individually rational and incentive compatible mechanism can be designed.
Our analysis sheds light on situations of project selection, where two
independent firms run parallel research programs and a third party can en-
force a cooperative scheme to prevent inefficiencies. The third party can for
instance be a venture capitalist or a granting agency running competing re-
search projects, the editor of an academic journal or organizer of a scientific
conference who discovers that two teams of scientists are working on the same
problem. Our analysis suggests that selection should neither occur too early
(before the profitabilities of the projects are known), nor too late (when the
firms have become very optimistic about their prospects given that the other
firm has not entered). It also shows that the share of the surplus transferred
to the firm which is not selected should neither be too large (in which case
the selected firm may have an incentive to delay the research project) nor too
small (the higher the payoff transferred to the firm which is not selected, the smaller the gap between the payoffs of the leading and trailing firms, which reduces inefficiencies due to excess momentum.)
Our analysis is rooted in the literature on patent races in continuous time
pioneered by Reinganum (1982) and Harris and Vickers (1985). The first ex-
tensions of patent races allowing for symmetric uncertainty are due to Spatt
and Sterbenz (1985), Harris and Vickers (1987) and Choi (1991). Models
of learning in continuous time with public information have been studied by
Keller and Rady (1999) and Keller, Cripps and Rady (2005). Rosenberg,
Solan and Vieille (2007) and Murto and Valimaki (2010) extend the model
to allow for public signals. The model of preemption we consider is for-
mally closely related to Fudenberg and Tirole (1985)’s models of technology
adoption with preemption. Innovation timing games which can result either
in preemption or in waiting games have been studied by Katz and Shapiro
(1987). Hoppe and Lehmann-Grube (2005) propose a general method for
analyzing innovation timing games. Fudenberg and Tirole (1985)’s model
has been extended by Weeds (2002) and Mason and Weeds (2010) to allow
for stochastic values of the technology. However, none of these models allows
for private information. The closest papers to ours are the recent papers
by Hopenhayn and Squintani (2010) on preemption games with private in-
formation and Moscarini and Squintani (2010) on patent races with private
information. Moscarini and Squintani (2010) analyze a common values prob-
lem, where agents learn about the common arrival rate of the innovation. Our
model with common values shares the same characteristics as theirs, albeit
in a much simpler setting, and the results are closely related. Our model
with private values and preemption displays very different results. Hopen-
hayn and Squintani (2011)’s model is much more general than ours but only
covers situations where agents receive positive information over time. In our
model, research teams may either receive positive or negative signals about
the profitability of the research project, so that the results of Hopenhayn and
Squintani (2011) do not directly apply. However, the spirit of the analysis is
very similar, and we outline in the body of the paper the similarities and dif-
ferences between their results and ours. Cooperation among research teams
with private information has been studied in a mechanism design contest
by Gandal and Scotchmer (1993). In ongoing work, Akcigit and Liu (2011)
study cooperation in a patent race with learning and private signals, where
cooperation involves the disclosure of private signals.
2 The Model
2.1 Firms, new markets and entry costs
We consider two firms which may invest in order to enter a new market, launch a new product or exploit a new process. The monopoly and duopoly profits obtained after investment are fixed and given by π m and π d respec- tively. We suppose that π m > 2π d . The entry cost to the new market is uncertain, and can either take a high or low value, θ i = θ or θ i = θ. We consider the case of private values where costs are independently distributed among the two firms. Assuming that the two values of the cost are equiprob- able, the expected value of the entry cost is θ e = θ+θ 2 for both firms.
During the experimentation phase, each firm receives a costless perfectly informative signal about its cost according to a Poisson process with intensity µ. Hence, the probability that a firm receives a signal during the interval [0, t] is 1 − e −µt . With probability 1 2 the firm learns that it is of high type, and with probability 1 2 , it learns that it is of low type. We assume that the Poisson processes generating signals to the two teams are independent.
In the private values case, independence furthermore means that the signals received by the two teams are independent. In the common values case, the signals which are perfectly informative, are not independent.
We assume that high cost firms never have an incentive to invest, even if they receive monopoly profit. Low cost firms always have an incentive to invest, even if they receive duopoly profit. When entry cost remains unknown, research teams have an incentive to invest as monopolists but not as duopolists. Formally,
Assumption 1
θ ≤ π d ≤ e θ ≤ π m ≤ θ. (1)
2.2 Entry timing and strategies
At any date t = 0, ∆, 2∆, ..., both firms can choose whether to enter the market. (We will analyze situations where the time grid becomes infinitely fine, and ∆ converges to zero.) If team i enters the market, it pays the fixed cost θ i and starts collecting monopoly (or duopoly) profits immediately.
Investments to enter the market are immediately observed by the other firm.
Given Assumption 1, it is a dominant strategy for a high cost firm not
to invest. Hence, the only relevant choices are choices made by a firm which
learns that its cost is low, or by a firm which still ignores its entry cost. A
strategy specifies, after every possible history, a pair of probabilities with which the firm invests when it learns that its cost is low and when it ignores its cost. We consider perfect equilibrium strategies which maximize the firm’s expected discounted payoff after every possible history.
2.3 Cooperative benchmark
If the two firms cooperate, they either choose to enter immediately, and earn an expected profit of π m − θ, or wait until they identify whether one firm has e a low entry cost, and obtain an expected profit of
V C = µ
2µ + r (π m − θ)(1 + µ 2(µ + r) ).
The optimal cooperative choice depends on the values of the parameters.
Clearly, the incentive to experiment rather than enter immediately is increas- ing in the difference between the costs θ e and θ, and in the intensity of the Poisson process generating signals, µ. It is decreasing in the discount rate r and in the value of the monopoly profit π m . Finally, notice that if a single firm is present on the market, it will only be able to experiment with one of the two research processes, and obtain an expected payoff by experimenting given by
V O = µ
2(µ + r) (π m − θ).
This expected payoff is lower than the expected payoff obtained by two cooperating firms for two reasons. First, by experimenting in parallel, the two firms accelerate the rate at which signals arrive, as the Poisson process generating signals now has intensity 2µ instead of µ. Furthermore, in the private values case we consider, by experimenting on two projects in parallel, the teams draw two independent signals about the costs. In other words, even if the first team receives a signal that it has a high cost, there is a positive probability as the second team receives a signal that it has a low cost and implements the project.
2.4 Leader and follower payoffs
Suppose that one firm (the leader) invests first. In the private values model,
the second firm (the follower) will only follow suit if it learns that its cost is
low. Hence the expected value of the follower is given by
V F = µ
2(µ + r) (π d − θ).
The leader thus extracts monopoly profit as long as the other firm has not entered, and we compute the value of the leader after investment as:
V L = π m − µ
2(µ + r) (π m − π d )
3 Entry timing
3.1 Entry timing with public signals
We suppose that the signals received by the two firms during the experimen- tation phase are public. We first establish that a firm which learns that its cost is low invests immediately. Depending on the parameters, a firm which has not yet learned its cost will either choose to preempt at zero or to wait until it learns that its cost is low. In the preemption case, both firms invest with positive probability at zero, resulting in coordination failures. In the waiting game, we show that, because the leader’s and follower’s payoffs are independent of time, the equilibrium strategy is for both firms not to en- ter before they learn the value of their costs. Summarizing, we obtain the following characterization of equilibrium.
Proposition 1 In the entry timing game with public information, a firms which learns that its cost is low invests immediately. If V L − e θ > V F , pre- emption occurs and (i) a firm which ignores its cost invests with positive probability at any date t = 0, ∆, ... whenever the other firm has not invested and (ii) a firm invests immediately after it learns that the other firm has a high cost. If V L − θ < V e F , firms do not enter unless they learn that their cost is low.
Proposition 1 shows that the entry timing game is either a preemption
game (when V L − θ > V e F ), or a waiting game (when V L − e θ < V F ). Notice
that V F − V L = V O − π m < V C − π m . Hence, as compared to the cooperative
benchmark, competition results in excess momentum. In the competitive
entry timing game, firms have an incentive to invest too early, before they
learn the true value of their cost.
3.2 Entry timing with private signals
When signals are private, teams do not learn whether the other team has drawn a bad or good signal about its cost. Each firm holds beliefs γ t (θ) about the cost of the other firm. These beliefs evolve over time given the strategies and the observation of investments. In order to compute the beliefs, we let G(t, τ ) denote the probability that a firm which learns that its cost is low at date τ invests at t ≥ τ, and g (t, τ ) the instantaneous probability that the firm invests at date t. We also let h(t) denote the instantaneous probability that a firm which ignores its cost invests exactly at date t. Using Bayes’ rule, the beliefs at period t are then given by:
γ t (θ) = R t
0 [1 − G(t, τ )]µe −µτ dτ
A(t) ,
γ t (θ) = 1 − e −µt A(t) , γ t (e θ) = 2[e −µt − R t
0 e −µτ h(τ )]
A(t) where
A(t) = Z t
0
[1 − G(t, τ)]µe −µτ dτ + 1 − e −µt + 2[e −µt − Z t
0
e −µτ h(τ )].
We first establish that a firm which learns that its cost is low has an incentive to invest immediately:
Lemma 1 In the entry timing game with private signals, a firms which learns that its cost is low has an incentive to enter immediately.
Lemma 1 enables us to focus attention on situations where beliefs only involve situations where a firm has learned that its cost is high, or ignores its cost. In other words, we now consider beliefs:
γ t (θ) = 0,
γ t (θ) = 1 − e −µt 1 + e −µt − 2 R t
0 e −µτ h(τ) , γ t (e θ) = 2[e −µt − R t
0 e −µτ h(τ )]
1 + e −µt − 2 R t
0 e −µτ h(τ) .
It is easy to check that γ(t)(θ), increases over time. As t increases, the probability that the competing firm has a high entry cost increases. No news is good news: as time passes, each firm becomes more pessimistic about the cost of the other firm and thus more optimistic about its own prospects. This dynamics of beliefs is the driving force behind the dynamics of the model, as it ensures that firms which ignore their costs will eventually find it profitable to invest because they believe that the other firm has drawn a high cost and will never invest with high probability.
In order to characterize equilibrium strategies, we compute the expected discounted payoff of the leading team which is the first to invest at time t:
V L (t) = γ(t)(θ) + pi m γ(t)(e θ)V L = π m − γ(t)(e θ) µ
2(µ + r) (π m − π d ).
Because γ(t)(e θ) is decreasing over time, the value of the leader is increas- ing over time. As time passes, firms become more optimistic, and the value of the leader increases from V L = V L (0) to π m = lim t∞ V L (t). Notice that the value of the follower, V F , remains independent of time. In order to analyze the equilibrium entry times, we distinguish between three cases: (i) Case 1 when V L − θ > V e F ; (ii) Case 2 when π m − θ e ≥ V F ≥ V L − θ; and (iii) Case 3 e when V F > π m − θ. The three cases are illustrated below: e
Cases 1 and 3 correspond to the preemption and waiting cases in the timing game with public signals. Case 2 exploits the fact that beliefs evolve over time, and describes a new situation which is similar to the preemption game studied by Fudenberg and Tirole (1985). The expected payoff of the leading firm is initially lower than the expected payoff of the following firm, but is increasing over time and eventually becomes higher than the payoff of the following firm. As in Fudenberg and Tirole (1985), the unique subgame perfect equilibrium results in rent equalization. One firm invests at the first time at which the payoff of the leading firm is higher than the payoff of the following team, V L (˜ t) − θ e = V F . Formally,
Theorem 1 In the entry timing game with private signals, a firm which
learns its cost invests immediately. If V L − e θ > V F , preemption occurs at the
beginning of the game and both teams invest with positive probability at time
0. If π m − θ e ≥ V F ≥ V L − θ, in a symmetric equilibrium, rents between the e
leader and the follower are equalized and each firm invests with probability
Figure 1: Case 1: V L − e θ > V F
1
2 at time t ˜ such that: V L (˜ t) − θ e = V F . If V F > π m − θ, firms do not enter e unless they learn that their cost is low.
Theorem 1 shows that when signals are private, excess momentum due to preemption is higher than when signals are public. When π m − θ e ≥ V F ≥ V L − θ, firms do not invest before learning their cost when signals are public, e but rush to invest at time ˜ t when information about costs are private.This incentive to preempt earlier with private signals than with public signals is different from the result of Hopenhayn and Squintani (2011) who show that preemption is stronger with public signals than with private signals, when new information can only lead to improvements. In their model, publicity of information strengthens competition between agents, and results in higher preemption. In our model, publicity of information reduces competition as firms learn that the other firm is out of the market, and reduces preemption.
We now focus on Case 2 and analyze how the preemption time ˜ t depends on the parameters of the model. The preemption time is implicitly defined as the unique solution to the equation:
π m − µ
2(µ + r) π d − µ
µ + r (π m − π d ) e −µt 1 + e −µt − θ
2 − θr
2(µ + r) = 0. (2)
Implicit differentiation of equation (2) immediately results in the following
comparative statics:
Figure 2: Case 2: π m − e θ ≥ V F ≥ V L − θ e π m -
π d +
θ +
θ +
r -
µ ?
Table 1: Preemption time ˜ t – Comparative statics
An increase in the monopoly profit π m increases incentives to preempt and
results in a lower preemption time ; conversely, an increase in the duopoly
profit π d reduces incentives to preempt and lengthens preemption time. An
increase in entry costs (both θ and θ) makes entry more costly and results
in longer delays before preemption. When firms become less patient (r in-
creases), preemption time decreases. Changes in the Poisson arrival rate µ
have ambiguous effects, as an increase in µ simultaneously increases the pay-
off of the leader V L (t) and of the follower V F . For small values of µ, (µ close
to zero), the magnitude of the marginal effect of an increase in µ is higher
on the follower payoff than on the leader payoff, so that the preemption time
increases with µ. For large values of µ, (µ close to infinity), the comparison
is reversed, so that the preemption time decreases with µ.
Figure 3: Case 3: V F > π m − θ e
3.3 Efficiency comparison
We now compare the industry profits in the three r´ egimes of cooperation, competitive entry with public signals and competitive entry with private signals. We distinguish between four parameter regions, depending on the magnitude of the expected entry cost e θ:
1. −e θ > V C − π m : immediate entry in the cooperative r´ egime, and pre- emption at zero in both competitive r´ egimes
2. V C − π m > −e θ > V F − V L : delayed entry in the cooperative r´ egime, and preemption at zero in both competitive r´ egimes
3. V F − V L > −e θ > V F − π m : delayed entry in the cooperative r´ egime and in the competitive r´ egime with public signals, preemption at finite time ˜ t in the competitive r´ egime with private signals
4. V F − π m > −e θ: delayed entry in all r´ egimes.
We define the industry profits when both firms delay their entry until they learn that their cost is low as:
V S = µ
2µ + r [(π m − θ) + µ
2(µ + r) (2π d − 2θ)]
and the industry profits with preemption at finite time ˜ t as:
V P = (1 − e −(2µ+r)˜ t )[ µ
2µ + r [(π m − θ) + µ
2(µ + r) (2π d − 2θ)]]
− e −(µ+r)˜ t (1 − e −µ ˜ t ) µ
2(µ + r) (π m − θ) + e −(2µ+r)˜ t 2 µ
2(µ + r) (π d − θ)
It is easy to check that V C > V S > V P > 2V F . The following table lists the industry profits under the three r´ egimes in the four parametric configurations:
parameter region cooperative public private
−e θ > V C − π m π m − θ e 2V F 2V F
V C − π m > −e θ > V F − V L V C 2V F 2V F V F − V L > −e θ > V F − π m V C V S V P
V F − π m > −e θ V C V S V S
Table 2: Efficiency comparisons
Table 2 illustrates the three sources of inefficiencies due to competition in our model of entry. First, by competing on the market, the firms forgo the benefits of market monopolization, the difference between monopoly profits, π m and the sum of duopoly profits, 2π d . Second, by competing on the market, the firms pay twice the entry cost θ, whereas in the cooperative benchmark, only one firm enters. Finally – and this is the new element of the model we wish to emphasize– competition results in excess momentum, making firms enter the market before they learn their cost, whereas in the cooperative benchmark, they should wait until they learn their cost before entering.
4 Market foreclosure and cooperation
In this section, we analyze cooperation mechanisms which would allow the firms to reach the cooperative outcome. We focus attention on compensating payment schemes which are paid by one firm in order to compensate the other firm for not entering the market. Compensating payment schemes could be implemented at three different points in time:
• ex ante: payments are made before the firms learn their entry cost
• at the interim stage: payments are made by one firm after it learns its cost, when it ignores the cost of the other firm
• ex post: payments are made after the costs of both firms are common knowledge.
The best timing of compensating payments depends on the specific para- metric configuration. If −e θ > V C − π m , compensating payments should be made ex ante in order to prevent inefficiencies due to market competition and duplication of entry costs. If V C − π m > −e θ, the cooperative benchmark is reached when one of the two firms learns that its cost is low and we argue that the best timing for compensating payments is the interim stage. At the ex ante stage, if one of the firm pays the other firm to leave the market, it still needs to run the other firm’s project in order to select the project with the lowest cost. This requires ongoing cooperation on the part of the managers of the firm which has been bought off, and agency problems will arise, which may limit the gains from cooperation. When signals are public, cooperation may happen at the ex post stage, with compensating payments being paid to the follower firm only when it learns that its cost is low. When signals are private and firms cannot credibly convey information about their entry cost, ex post compensating payments can only be made after the follower firm has entered the market. In that case, the entry cost of the follower firm has been sunk, so that it becomes impossible to alleviate the duplication of entry costs with payments at the ex post stage.
We now concentrate our attention to compensating payments made at the interim stage when V C − π m > −e θ and signals are private. First note that compensating payments should satisfy individual rationality and incentive compatibility. Hence the utility obtained by the leader after the compensat- ing payment is paid at date t, U L (t) and the utility obtained by the follower U F (t) should satisfy the following two conditions:
U L (t) ≥ V L (t), (3)
U F (t) ≥ V F (4)
The first inequality stems from the individual rationality constraint of the
leader, who may choose to forgo the compensating payment. The second
inequality results from the follower’s incentive compatibility and individual
rationality constraint. As the leader cannot verify information about the
follower’s type, he ignores whether the follower has received a high cost signal
or not, and must pay the follower the fixed payment V F corresponding to a
firm ignoring its cost. As compensating payments are budget balanced, the sum of utilities received by the leader and follower must exactly equal the monopoly profit:
U L (t) + U F (t) = π m (5)
Given inequalities (3), (4) and equation (5), a necessary condition for the existence of a budget balanced, individually rational and incentive compatible transfer scheme is
V L (t) + V F ≥ π m .
As V L (t) is increasing, V L (0) < π m − V F and V L (∞) > π m − V F , there exists a unique date t ∗ , such that no budget balanced individually rational compen- sating payments exist if the first firm enters at date t ≥ t ∗ . This remark captures the following simple intuition. As time passes, firms become more optimistic about their prospects. If a firm enters at a late date, it will expect the other firm to have left the race and will not be willing to compensate the other firm at the level V F , which is the minimal level at which a firm which ignores its cost is willing to leave the race. Furthermore, this remark shows that there is no efficient, budget balanced and individually rational co- operation mechanism. To see this, consider a realization of the signals where no firm has learned its cost before t ∗ . Either the mechanism prescribes that one of the team invests before t ∗ , and the mechanism is inefficient because it will result in a high cost team investing with positive probability, or the mechanism prescribes to wait until one of the firm has learned it has a low cost, and the mechanism is inefficient because there is no budget balanced, individually rational compensating payment which prevents the other firm from entering the race.
We now consider the following problem: How should compensating pay- ments be designed in order to guarantee that, whenever one firm learns that it has a low cost before t ∗ , it is chosen to be the only firm operating on the market?
Proposition 2 A differentiable compensating payment scheme U F (t) imple- ments the cooperative benchmark when a firm learns that it has a low cost before t ∗ if and only if for all t < t ∗ ,
π m − θ < e 2U F (t) < 2r + µ
r + µ π m + U F 0 (t)
r + µ
Proposition 2 shows that efficient compensating payment schemes must be balanced to satisfy two requirements. First, the payment to the follower must be large enough to prevent early entry by firms which ignore their costs.
Second, the payment to the follower should not be too large, in order to give incentives to a firm which learns that its cost is low to enter immediately.
These two requirements provide an upper and a lower bound on the expected payoffs of the follower and leader firm and show that the cooperative surplus must be shared in a balanced way between the two firms.
In order to provide additional intuition, we specialize the model by as- suming that the compensating payment scheme assigns a fixed bargaining power to the leader and the follower firm, so that
U L (t) = V L (t) + α(π m V L (t) − V F ), U F (t) = V F + (1 − α)(π m − V L (t) − V F ),
We observe that U L (t) is increasing and U F (t) decreasing over time. The following graph illustrates these mappings for α = 0 and α = 1. It displays the maximal time t ∗ at which compensating payments can be implemented, shows that payoffs are independent of time if all of the bargaining power is given to the leader and that the gap between the payoff of the leader and follower is increasing in α
The cooperative benchmark can be implemented when a firm learns that it has a low cost before t ∗ if and only if
2V F ≥ π m − θ, e (6)
2V F + 2(1 − α)(π m − V L − V F ) ≤ 2r + µ
r + µ π m − V L 0 (0)
r + µ . (7) These conditions put a lower bound (but no upper bound) on the share of the bargaining surplus accruing to the leader. Notice that if the first condition (6) fails, it is impossible to guarantee that no early preemption occurs before t ∗ . In that situation, the value of α can be manipulated in order to increase preemption time. By reducing α, and giving a larger share of the surplus to the follower, the mechanism designer reduces incentives to preempt and increases the value of ˜ t. 5 The optimal compensating payment
5
