Free Riding and Duplication in R&D
(Job Market Paper)
Tsz-Ning Wong
Department of Economics
The Pennsylvania State University
November 7, 2015
Abstract
We study a model of R&D race in the exponential-bandit learning framework (Choi
1991; Keller, Rady, and Cripps 2005), in which two heterogeneous …rms, each endowed
with an independent R&D process, choose when to irreversibly exit the R&D race. Each
R&D process can be either good or bad. In the absence of a research breakthrough
(innovation), a …rm becomes more pessimistic about its R&D process over time. We
show that strict patent protection may lead to excessive duplication of research e¤orts,
while the lack of patent protection leads to free riding and under-experimentation
of research opportunities. The choice of optimal patent system involves a trade-o¤
between duplication in the early stage of R&D when both …rms are optimistic and
under-experimentation in the later stage when one …rm has already exited and the
remaining …rm is pessimistic. Unless the …rms are homogeneous, the optimal patent
system is ine¢ cient. Nevertheless, we propose an asymmetric distribution policy that
implements the social optimum.
I would like to thank my advisor Kalyan Chatterjee for his constant encouragement and support. This
paper has bene…ted greatly from my discussions with S. Nageeb Ali, Sourav Bhattacharya, Martin Cripps,
Kaustav Das, Drew Fudenberg, Edward Green, Johannes Hörner, Navin Kartik, Vijay Krishna, and So…a
Moroni, as well as from the comments of the participants of the 26th Jerusalem School in Economic Theory
and the 26th International Conference on Game Theory at Stony Brook University. All remaining errors
are my own.
1
1
Introduction
The recent advancements in theoretical studies on strategic experimentation (e.g. Keller
et al. 2005) have renewed interests of economists to study experimentation in R&D races.
Various authors have applied learning model to investigate R&D races and obtained fresh
insights into this classic problem. Acemoglu et al. (2011) study a model of experimentation
in which …rms can copy the innovation of others. In the absence of patent protection,
free riding occurs in the form of delayed experimentation. They further show that an
appropriately designed patent system can implement the optimal allocation by encouraging
experimentation while maintaining e¢ cient transfer of knowledge ex post. Their analysis
suggests that patent could be an e¢ cient means of reducing free riding. On the other hand,
some authors (e.g. Das 2014b) argue that winner-take-all R&D race leads to excessive
duplication of research e¤orts. Since only one success counts, resources put into research
by competitors could be wasted.
In this paper, we analyze an exit game that allows us to study both free riding and
duplication in R&D races. In particular, we are interested in how a patent system may
create or resolve these ine¢ ciencies. In our model, each …rm is endowed with an independent
R&D process (project) and each chooses privately when to exit the R&D race irreversibly.
Each R&D process can be either good or bad and each has a di¤erent prior probability
of being good. We refer to the …rm whose project is more likely to be good as the strong
…rm and the other …rm the weak …rm. Given that a …rm’s project is good, a breakthrough
(innovation) occurs to her in a Poisson process as long as she remains in the R&D race.
Each …rm incurs a ‡ow cost to stay in the race. Once a breakthrough occurs, the game ends.
A parameter
2
1
2; 1
describes how the post-innovation surplus v is split between the
two …rms. The innovating …rm receives v and the non-innovating …rm receives (1
) v.
1
2,
both the
innovating …rm and the non-innovating …rm receive half of the surplus v2 . Thus,
can be
When
= 1, the innovating …rm receives all the surplus v. When
=
considered as a measurement of the patent strength.
We …nd that the heterogeneity in the …rms’projects has important welfare and policy
consequence. Unless the projects are homogeneous, strict patent, i.e.
2
= 1, is suboptimal
and leads to excessive duplication of research e¤orts.1 Imposing no patent protection,
= 12 , is likewise suboptimal and leads to excessive free riding behavior and under-
i.e.
experimentation of research opportunities. We further show that under the optimal patent
system, the strong …rm never over-experiments and the weak …rm never under-experiments.
As a result, duplication occurs in the early stage of R&D when both …rms are optimistic
and under-experimentation occurs in the later stage when the weak …rm has already exited
and the remaining strong …rm is pessimistic. Unless the projects are homogeneous, the
optimal patent system is ine¢ cient. Nevertheless, we propose an asymmetric distribution
policy that implements the social optimum. Under the optimal distribution policy, strict
patent protection is granted if the discovery is made by the strong …rm and innovation by
the weak …rm is only partially protected. Indeed, the optimal distribution policy may even
grant protection to the strong …rm when the discovery is made by the weak …rm. By doing
so, it discourages the weak …rm from wasteful duplication of the strong …rm’s e¤ort.
1.1
Related Literature
This paper belongs to the literature of strategic experimentation. Many works in this
literature take the payo¤ structure of the game as given. These works can be roughly
divided to two classes: models of “collaboration” in which experimentation has a positive
externality and models of “competitive experimentation”in which the success of one agent
imposes a negative externality to the others. Papers in the …rst class includes Keller et al.
(2005), Keller and Rady (2010), Bonatti and Hörner (2011), Murto and Välimäki (2011),
Rosenberg et al. (2013) and many others. Papers in the second class focus mostly on the
case of winner-takes-all contest, these includes Choi (1991), Malueg and Tsutsui (1997),
Chatterjee and Evans (2004), Mason and Välimäki (2010), Moscarini and Squintani (2010),
and Das (2014b).
More closely related to the present paper are works on contest and contract designs
with learning. Acemoglu et al. (2011) study a model of innovation and patent protection
1
Our notion of duplication is di¤erent from that of Chatterjee and Evans (2004) and Das (2014b). These
authors consider models with two lines of research; duplication occurs when two …rms conduct the same
line of research. In contrast, we say that duplication occurs in our model when too many …rms conduct
R&D relative to the e¢ ciency benchmark, even though each …rm has her own line of research.
3
similar to ours in spirit. Their model di¤ers from ours in that experimentation is completed
instantaneously in their model. As a result, beliefs do not evolve over time and they do not
address the issue of over-experimentation and under-experimentation over time. Moreover,
they focus ex ante identical …rms and do not consider asymmetric policy. Halac et al
(2015) study the optimal contest in a learning model with homogenous agents where the
designer chooses how to allocate a prize and what information to disclose over time about
contestants’successes to maximize the contestants’e¤orts. Our models di¤ers from theirs
in three respects. First, the patent designer in our model designs a patent to maximize
the total welfare of the contestants (…rms) instead of the total e¤orts. Second, we consider
contests with heterogeneous agents. Third, in our model, information about contestants’
successes is always fully revealed. These di¤erences lead to di¤erent trade-o¤ in the design
of optimal contest (patent). Moroni (2015) and Wagner (2015) consider dynamic moral
hazard models with multiple agents and learning. In contrast to our model, they assume
that fully dynamic, history-contingent contract with transfer can be made.
There is a sizable literature, pioneered by Nordhaus (1969), on patent policy focusing
various aspects of innovation and patent races. Much of this literature studies models
without learning and focuses on di¤erent trade-o¤s from those considered in this paper.
The design of optimal patent policy is studied by, for example, Denicolò (1999) and Judd,
Schmedders and Yeltekin (2012); see also the references therein.
The rest of the paper is organized as follows. Section 2 sets up the model. Section 3
presents the equilibrium characterizations. Section 4 studies the planner’s problem where
the …rms cooperate to maximize joint expected payo¤s and analyzes the welfare consequences of increasing the projects’qualities. Section 5 presents the characterization of the
optimal patent. Section 6 extends the model to allow general distribution policy and shows
that the optimal policy implements the planner’s solution. Section 7 discusses the roles of
the modelling assumptions and some possible extensions. Section 8 concludes. Some of the
proofs are relegated to the Appendix.
4
2
The Model
2.1
The Setup
Time is continuous and the horizon is in…nite. Two research …rms, 1 & 2, each endowed
with an R&D process (project), decide when to stop experimenting and exit the R&D race
(abandon the project). Each …rm incurs a ‡ow cost c to stay in the race. Once a …rm exits,
prohibitive sunk costs make re-entry infeasible. This assumption captures the idea that
large amounts of intellectual and …nancial resources are often needed for R&D activities.
Moreover, the decision to exit is private and unobservable to the competing …rm.
Each …rm’s project may be either good or bad. With prior probability pi;0 , …rm i’s
project is good, independent of …rm j’s project. At time t, a breakthrough (innovation)
occurs to …rm i with instantaneous probability equal to
if …rm i’s project is good and
she has not exited the race. On the other hand, a breakthrough never occurs if the project
is bad. We refer to the …rm whose project has a strictly higher prior probability of being
good as the strong …rm and use the upper case letter S to indicate her. The other …rm is
referred to as the weak …rm and the lower case letter w is used to indicate her. When the
prior probabilities are equal, the strong (weak) …rm refers to …rm 1 (2).
The game ends either when a …rm has achieved a breakthrough or when both …rms
exit.2 Suppose …rm i is the …rst to achieve a breakthrough, then …rm i receives an award
v while …rm j receives (1
) v, where
2
1
3
2; 1 .
Costs and pro…ts are undiscounted.
The description of the game is common knowledge among the …rms.
The parameter
1
2,
is referred to in this paper as the patent strength. When
both …rms earns the same amount v2 . When
0. The case of
equals to 1, the non-innovating …rm earns
= 1 is also referred to as the strict patent case,
2
equals to
2
1
2; 1
the weak patent
Strictly speaking, one should also specify the outcome of the game if both …rms choose to experiment
forever. However, our assumption of nonextreme prior makes sure that such a strategy is never optimal.
We, therefore, omit this detail.
3
The post-innovation payo¤s can be interpreted as follows. The innovation allows the …rms to produce
a good at zero cost. Suppose innovation occurs at time , for each t 2 [ ; + 1], the instantaneous demand
for the good is 1 if the price is weakly less than v and 0 otherwise. After + 1, there is no demand for
the good. The term of patent is given by 2
1. Between and + (2
1), the innovating …rm acts
as a monopoly. After the patent expires at + (2
1), the …rms engage in Cournot competition. In a
symmetric static equilibrium, each …rm produces 12 unit of the good at each time t 2 ( + (2
1) ; + 1].
The total payo¤s of the innovating and non-innovating …rms are thus v and (1
) v, respectively.
5
case, and
=
1
2
the no patent case.
Notice that the total surplus in the product market in our model is always v, independent of the patent system. This assumption allows us to abstract away from the trade-o¤
between ex ante incentives to innovate and ex post allocative ine¢ ciency and concentrate
on the trade-o¤ between experimentation and duplication. Moreover, we also focus on
a situation in which the consumer surplus is zero. The implication of adding consumer
welfare and e¢ ciency considerations in the picture will be discussed in Section 7.
A strategy of a …rm in this game is an exit time t 2 R+ , with the interpretation that the
…rm will exit at time t if no breakthrough has occurred prior to t. The solution concept we
used is Nash equilibrium. We focus on the pure strategy equilibrium that is total welfare
maximizing. An equilibrium strategy of …rm i is denoted by ti .
We make the following assumption on the prior (p1;0 ; p2;0 ).
Assumption (Nonextreme Prior): 8i = 1; 2, pi;0 2
c
v;1
.
Under this assumption, degenerate priors are ruled out. Unless a breakthrough occurs,
…rms is not sure that if a project is good. Moreover, the assumption that p1;0 ; p2;0 >
c
v
makes sure that both …rms must have incentives to experiment for a small period of time
under some patent arrangement. Otherwise, at least one of the …rms will always exit at
time 0 and the problem becomes one with a single …rm. Since pi;0
1, our assumption
entails that v > c.
2.2
Preliminaries
Since the arrival of a breakthrough is publicly observed and a …rm’s exit time is known to her
opponent in any equilibrium in pure strategies, given the …rms’common prior (p1;0 ; p2;0 ),
the …rms share a common posterior at each t
0 denoted by (p1;t ; p2;t ). Suppose at time t,
no breakthrough has occurred and …rm i has not exited, then, by Bayes’rule, the posterior
belief about …rm i’s project at time t is given by
pi;t =
pi;0 e t
:
pi;0 e t + 1 pi;0
6
(1)
The law of motion of the posterior beliefs can be derived accordingly, suppose that over
the time interval [t; t + dt), …rm i engages in R&D and has not achieved a breakthrough
by the end of the period, the belief about her project is updated to
pi;t + dpi;t =
pi;t (1
dt)
:
pi;t (1
dt) + 1 pi;t
Simplifying, we have
dpi;t =
pi;t (1
pi;t ) dt:
(2)
Once there is a breakthrough, the posterior belief jumps to 1. Notice that the expression
in (2) is not constant and is maximized when pi;t = 12 . Thus, when the prior probabilities
are di¤erent, the posterior beliefs don’t move at the same rate even when both …rms are
experimenting. Denote the belief path taken by two …rms that experiment forever by .
That is,
(p1;0 ; p2;0 )
(p1 ; p2 ) 2 [0; 1]2 : 9t 2 R+ ,8i 2 f1; 2g , pi =
pi;0 e t
pi;0 e t + 1 pi;0
Figure 1 illustrates some of the possible belief paths.
Figure 1: Belief Paths. Parameter Value:
= 1.
If …rm i exits at some time t0 , the posterior belief will stay at pi;t0 forever.
7
:
(3)
2.3
The Single Firm Decision Problem
In this subsection, we introduce and solve the single …rm problem, which is not only a
useful benchmark for the noncooperative problem, but also an important intermediate step
in solving it.
Suppose at time 0, …rm j exits R&D. Knowing that, …rm i chooses a time ti
0 to
exit conditional on no breakthrough. This problem, which we call the single …rm decision
problem, is standard, and the optimal policy is to exit when the posterior belief for …rm i’s
project reaches the threshold value
c
v,
which we also refer to as the single …rm threshold
belief. We have,
Lemma 1 Given that …rm j exits at time 0, …rm i’s optimal policy is to exit when the
c
posterior belief falls below the single …rm threshold belief
v.
The optimal exit time tyi ( )
has the expressions
tyi (
)=
8
>
<
1
log
>
:
1 pi;0
pi;0
c
v c
0
if pi;0 >
c
v;
if pi;0
c
v:
(4)
Moreover, …rm i’s continuation value at time t is given by
^ (pi;t ; ) =
W
8
>
< pi;t v
c
+ (1
pi;t )
>
:
c
log
1 pi;t
pi;t
c
v c
0
if pi;t >
c
if pi;t
c
v;
(5)
v;
and …rm j’s continuation value at time t is given by
W (pi;t ; ) =
where pi;t is given by (1).
8
>
<
pi;t
>
:
v c
v c
(1
0
) v if pi;t >
if pi;t
c
v;
c
v;
(6)
To see the optimality of the policy, we …rst write down …rm i’s objective function.
Suppose …rm i experiments at time t
0. At time t, the posterior belief that …rm i’s
project is good is pi;t , the instantaneous probability of a breakthrough conditional on a
8
good project is
and …rm i receives
expected payo¤ at time t is simply pi;t
v from a breakthrough. Thus, the instantaneous
v
c. To calculate …rm i’s payo¤ at time 0, the
instantaneous expected payo¤ must be weighted by the probability that a breakthrough
Rt
has not occurred by time t, which is expf 0 pi;s dsg. (Otherwise, the game has ended
and the experimentation at time t becomes irrelevant.) Thus, in the single …rm problem,
…rm i chooses ti to maximize the expression
Z
ti
(pi;t
v
c) e
0
Rt
0
pi;s ds
dt:
(7)
By (1), pi;t is strictly decreasing in t and tends to 0 as t increases. As a result, the
instantaneous expected payo¤, pi;t
v
c, is strictly decreasing and eventually becomes
negative and (7) is maximized at the point ti at which pi;ti
not exist, pi;t
v < c for all t
v = c. If such a point does
0. In this case, (7) is maximized at ti = 0. These two
cases lead to the two expressions for tyi ( ) in (4). The continuation values (5) and (6) are
then computed from the optimal exit time (4) by evaluating the integrals
^ (pi;t ; )
W
Z
tyi ( )^t
(pi;s
v
c) e
t
W (pi;t ; )
Z
tyi ( )^t
pi;s (1
t
) ve
Rs
t
pi;z dz
Rs
t
ds;
pi;z dz
ds;
Notice that even though …rm j exited R&D at time 0, her continuation value could still
be strictly positive. This is because as long as …rm i undertakes an innovation at some
point in the future, …rm j receives (1
) v without having to pay the ‡ow cost c. As we
shall see in Section 3, this possibility of being a copycat creates incentives for …rm j to
exit earlier than her single …rm benchmark to “free ride” on …rm i’s R&D e¤orts in the
noncooperative problem. Such behavior, however, is not necessarily bad for social welfare
as it may help to reduce duplications of research e¤orts, as we will …nd in Section 4.
9
3
Equilibrium Characterizations
In this section, we give several characterizations of the equilibria of our game. We …rst
provide a general characterization for all equilibria in pure strategies (Proposition 1). Then,
we divide equilibria in pure strategies into two distinct classes and study them separately.
An equilibrium is regular if and only if the strong …rm’s equilibrium exit time is weakly later
than the weak …rm’s. In Lemma 2, we completely characterize nonregular equilibria. After
establishing the existence of regular equilibrium, we provide characterizations of regular
equilibrium over di¤erent ranges of patent strength.
Building on Lemma 1, Proposition 1 provides the basic characterization of the equilibria
of our game, on which much of the subsequent analysis is based.
Proposition 1 Consider an equilibrium (t1 ; t2 ). Suppose ti
t that satis…es pi;t
v
c, which has the expressions
ti = tyi ( ) =
Moreover, tj satis…es
tj , then, ti is the smallest
8
>
<
1
log
>
:
IN B pj;tj ; pi;tj ;
1 pi;0
pi;0
c
v c
0
pj;tj
v
if pi;0 >
c
if pi;0
c
W pi;tj ;
v:
(8)
v:
c
0;
(9)
where the function W ( ; ) is given by (6). Moreover, if tj > 0, (9) also holds with equality.
Proposition 1 has two parts. The …rst part of Proposition 1 states that the equilibrium
exit time of the …rm that exits later in equilibrium must be identical to the optimal exit
time of the single …rm decision problem. In other words, the …rm whose equilibrium exit
time is higher acts as if the opponent chooses to exit at time 0.
To understand the characterization, suppose ti
tj and tj = 0. In this case, the
i
h
assertion is trivially true. Suppose tj > 0, consider a time t 2 0; tj when both …rms
are still active. The probability that a breakthrough has not occurred by that time is
Rt
expf 0 (p1;s + p2;s ) dsg. The posterior probability that …rm i’s (…rm j’s) project is
10
good is pi;t (pj;t ) and the instantaneous probability of a breakthrough by …rm i (…rm j) is
thus pi;t (pj;t ). It follows that …rm i’s instantaneous expected payo¤ at time t is given by
Rt
(pi;t + pj;t (1
)) v c. Moreover, with probability expf 0 j (p1;s + p2;s ) dsg, …rm
j exits at time tj and …rm i faces the single …rm decision problem from tj onwards. It
follows that given an exit time ti
V i ti ; t j
=
Z
tj
f(pi;t + pj;t (1
0
R tj
+e
Since ti
0
)) v
Z
(p1;s +p2;s ) ds
cg e
Rt
0 (p1;s +p2;s )
ti
(pi;t
v
c) e
tj
Rt
t
j
ds
pi;s ds
dt
!
dt
tj > 0, the …rst-order condition has an interior solution
@Vi ti ; tj
@ti
By (1), pi;t
tj , …rm i’s expected payo¤ is given by
v
jti =ti = e
R ti
0
R tj
pi;s ds
0
pj;s ds
pi;ti
v
c = 0:
(10)
c is strictly decreasing in t. Thus, (10) is both necessary and su¢ cient.
Next, consider the characterization of tj in the second part of Proposition 1. Since ti
is characterized by Lemma 1, given an exit time tj
ti , we can write …rm j’s expected
payo¤ as
Vj (tj ; ti ) =
Z
tj
0
+e
f(pj;t + pi;t (1
R tj
0
(p1;s +p2;s ) ds
)) v
cg e
Rt
0 (p1;s +p2;s )
ds
W pi;tj ;
dt
(11)
where W ( ; ) is given by (6). Di¤erentiating (11) with respect to tj and simplifying the
expression, we obtain …rm j’s marginal net bene…t of experimentation, M N B, at time tj
@Vj (tj ; ti )
=e
@tj
R tj
0
(p1;s +p2;s ) ds
fpj;tj
v
W pi;tj ;
cg
(12)
The …rst order condition implies that …rm j exits only if the M N B is nonpositive. We refer
to the term in the bracket in (12) as …rm j’s instantaneous net bene…t of experimentation
11
and denote it by
IN B (pj;t ; pi;t ; )
pj;t
|
v
W (pi;t ; )
{z
}
Instantaneous bene…t
The marginal net bene…t of experimentation
@Vj (tj ;ti )
@tj
c
|{z}
:
(13)
Instantaneous cost
di¤ers from the instantaneous net
bene…t of experimentation (13) in that the M N B is weighted by the probability that a
breakthrough has not occurred by time t. The M N B can be thought of as the net bene…t
from an in…nitesimal amount of additional experimentation at time t calculated at time 0.
The net bene…t is therefore “discounted”, as the additional experimentation may not be
carried out if a breakthrough has occurred prior to time t. On the other hand, the IN B can
be interpreted as the net bene…t from an in…nitesimal amount of additional experimentation
at time t calculated at time t. As a result, no discounting is needed. It is easy to see that
the M N B and the IN B always has the same sign. Thus, the condition (9) is equivalent
to the …rst order condition of …rm j’s problem. The advantage of working with the IN B
over the M N B is that the IN B refers only to the posterior (pj;t ; pi;t ). Neither the prior
(p1;0 ; p2;0 ) nor the current time t is involved. This allows us to analyze …rm j’s problem
using a 2-dimensional representation of the belief space.
To understand the meaning of (13), suppose, at time t, …rm j experiments for a short
amount of time dt and exits at t + dt. Three events may occur during this time interval.
Firstly, …rm i may achieve a breakthrough. The probability of this event is approximately
pi;t dt. If it occurs, …rm j receives (1
) v. Secondly, …rm j may achieve a breakthrough.
The probability of this event is approximately pj;t dt. If it occurs, …rm j receives
v.
(The probability that both …rms achieve a breakthrough between t and t + dt is of the
order of (dt)2 , which can be ignored when dt is small enough.) Thirdly, it may happen
that no …rm achieves a breakthrough during the time interval. The probability of this
event is approximately 1
(p1;t + p2;t ) dt. In this case, …rm j exits and receives the
continuation value W (pi;t + dpi;t ; ) at t + dt. Thus, …rm j’s continuation payo¤ at time t
12
is approximately
cdt + (pi;t dt) (1
) v + (pj;t dt) v + (1
(p1;t + p2;t ) dt) W (pi;t + dpi;t ; )
(14)
Moreover, for dt small enough, the following approximation holds:
W (pi;t ; )
(pi;t dt) (1
) v + (1
pi;t dt) W (pi;t + dpi;t ; )
(15)
This is because a breakthrough occurs to …rm i with probability approximately equal to
pi;t dt in a time interval with duration dt. Using (15), we can rewrite (14) into
W (pi;t ; ) + pj;t
v
W (pi;t + dpi;t ; )
c dt:
(16)
Further using the approximation
W (pi;t + dpi;t ; )
W (pi;t ; )
pi;t (1
pi;t ) W 0 (pi;t ; ) dt
(17)
to replace W (pi;t + dpi;t ; ) in (16) and ignore the term of the order of (dt)2 , we …nally
arrive at the following expression for the continuation payo¤ at time t;
W (pi;t ; ) + pj;t
v
W (pi;t ; )
c dt:
(18)
By exiting at time t, …rm j receives the continuation value W (pi;t ; ). Thus, (18) implies
that …rm j should never exit when the instantaneous net bene…t of experimentation (13)
is strictly positive, or, equivalently, …rm j should exit only if the instantaneous net bene…t
of experimentation is nonpositive, i.e. (9) holds. Notice that, in general, the objective
function (11) is nonconcave and condition (9) is necessary but not su¢ cient for optimality.
In the general characterization in Proposition 1, we make no assertion about the relationship between the exit times and the …rms’probabilities. In particular, the weak …rm
may exit later than the strong …rm, and vice versa. Distinguishing these two classes of
equilibria allows us to give …ner characterizations for each of them.
13
De…nition 1 An equilibrium (t1 ; t2 ) is regular if and only if the strong …rm’s equilibrium
exit time is weakly later than the weak …rm’s, i.e. tS
tw . An equilibrium that is not
regular is nonregular.
Intuitively, being more optimistic, the strong …rm should exit at a later time than the
weak …rm. Indeed, Proposition 2 con…rms this intuition and shows that such an equilibrium
always exists.
Proposition 2 There always exists a regular equilibrium.
While nonregular equilibrium may also exist, Lemma 2 shows that these equilibria exist
only in the case of extreme free riding in which the strong …rm exits immediately as the
game starts and free rides on the weak …rm’s e¤orts.
Lemma 2 In any nonregular equilibrium, the strong …rm must exit at time 0, i.e. tw >
tS ) tS = 0.
Whenever a nonregular equilibrium exists, the exit times of the weak and strong …rms
are uniquely pinned down by (8) in Proposition 1 and Lemma 2, respectively. Therefore,
nonregular equilibria are completely characterized by them. Next, suppose that a nonregular equilibrium exists so that the strong …rm has enough incentives to exit at time 0 and
free rides on the weak …rm’s e¤ort, then, the weak …rm, being more pessimistic, must have
enough incentives to do the same and free rides on the strong …rm’s e¤ort. This intuition
suggests that whenever a nonregular equilibrium exists, there exists a regular equilibrium
in which the weak …rm exits at time 0. Our next lemma shows that this is indeed the case.
Moreover, the regular equilibrium must be superior in terms of total welfare, as the strong
…rm is the remaining …rm in this equilibrium. ((5) and (6) are both increasing in pi;t .)
Thus, we have
Lemma 3 Whenever a nonregular equilibrium exists, there exists a regular equilibrium that
dominates it in terms of total welfare.4
4
In fact, Propositions 7 and 9 imply that any nonregular equilibrium is dominated by any regular
equilibrium.
14
As our focus is on the total welfare, from this point on, we will focus on regular equilibria.
To start, we de…ne
E( )
[ n
(p1 ; p2 ) 2 [0; 1]2 : pi
i=1;2
o
pj ; IN B (pj ; pi ; ) > 0 ;
(19)
which is the set of beliefs under which the weak …rm’s instantaneous net bene…t of experimentation (13) is strictly positive given the patent strength . Proposition 1 implies that
in a regular equilibrium the weak …rm will never exit when the posterior belief (p1;t ; p2;t )
is inside the set E ( ). Further de…ne
@E ( )
[ n
(p1 ; p2 ) 2 [0; 1]2 : pi
i=1;2
o
pj ; IN B (pj ; pi ; ) = 0 :5
We have,
Lemma 4 The set E ( ) is convex.
Our next lemma establishes a lower bound of the patent strength, below which the weak
…rm will exit at time 0 regardless of the prior (p1;0 ; p2;0 ). De…ne
r
c
:
v
(20)
We have,
Lemma 5 E ( ) is empty (@E ( ) =
c
v;
c
v
) if and only if
.
If E ( ) is empty, the weak …rm’s instantaneous net bene…t of experimentation (13) is
nonpositive everywhere on the belief space and thus nonpositive for all time t
0 regardless
of the prior (p1;0 ; p2;0 ). Thus, there is no point for the weak …rm to experiment at all. We
have,
5
This is a slight abuse of notation. We use @E ( ) to denote not the boundary of the set E ( ) but a
set that contains the boundary of the set E ( ). However, it is easy to show that when E ( ) is nonempty,
c
@E ( ) is simply the boundary of E ( ); when E ( ) is empty, @E ( ) =
; c v , as indicated in
v
Lemma 5.
15
Proposition 3 Suppose
, then in the unique regular equilibrium, the weak …rm exits
at time 0, and the strong …rm exits at tS given by (8).
. Before
Therefore, Proposition 3 completely characterizes regular equilibria when
turning into the characterizations of regular equilibria for
>
, we …rst establish a
fundamental result on comparative statics of the exit times (Proposition 4) through the
help of Lemma 6.
Lemma 6 Suppose
00
>
0
, then, for all pi
pj > 0, IN B (pj ; pi ;
00 )
> IN B (pj ; pi ;
To derive Lemma 6, substitute (6) into (13) and di¤erentiate, we have, for pi <
c
v,
@IN B (pj ; pi ; )
= pj v > 0;
@
c
for pi >
v,
@IN B (pj ; pi ; )
= pj
@
v+
An increase in the patent strength
pS;t >
c
v.
pi
v c
v
v c
(1
pi )
v 2 (1
( v
)c
c)2
:
(21)
has three e¤ects on the weak …rm’s IN B when
The …rst term is positive and it represents the direct incentives to the weak
…rm: since the award is higher, the bene…t of experimentation is higher. The second is
also positive and it represents the indirect incentives. Under a stricter patent, the bene…t
from the opponent’s breakthrough is lower, so there is less incentive to free ride. The third
term is negative and it represents the strategic e¤ect. An increase in the patent strength
increases the strong …rm’s amount of experiment through (8). This reduces the weak
…rm’s incentives to experiment. In general, the third e¤ect is not dominated by the other
two e¤ects. But this is the case when
.
Proposition 4 Suppose that under the patent system , …rm i conducts a positive amount
of experimentation in a regular equilibrium, i.e. ti > 0, then, an increase in the patent
strength to
0
>
strictly increases the amount of the experimentation by …rm i in the new
regular equilibrium.
16
0) :
The proof of Proposition 4 is straightforward. By Proposition 1, the strong …rm’s exit
time tS is given by (8) and is strictly increasing in
only if
when tS > 0. By Proposition 3, tw > 0
> . By Lemma 6, for all time t, the weak …rm’s M N B increases with . As a
result, the weak …rm must have incentives to experiment beyond the original equilibrium
exit time after an increase in .
By Lemma 6, as
increases, IN B (pj ; pi ; ) increases for all pi
4 and (19) then imply that E ( ) expands as
pj and
increases. De…ne
. Lemma
as the patent strength
at which the set E ( ) touches the point (1; 1). Substituting (6) into (13) and solving
IN B (1; 1; ) =
we …nd that
( v
(1
) v)
c = 0;
has the expression
1
c
+
.
2 2 v
It is straightforward to check that
< . Moreover,
(22)
> 12 , so that E
1
2
does not contain
the point (1; 1).
Figure 2: The set E( ). Parameter Values: (v; ; c; ) = (8; 1; 1; 0:5625).
It turns out that the parameter
is very useful for characterizing regular equilibria. To
see that, consider the weak …rm’s M N B at some t
(1) and (6) into (12) and solving the integral in exp
6
Rt
0
pi;s ds =
Rt
s
pi;0 e
s +1 p
0 pi;0 e
i;0
ds =
Rt
d(pi;0 e
0 pi;0 e
17
s +1
s
)
pi;0
tS in a regular equilibrium. Plugging
Rt
6
0 (p1;s + p2;s ) ds , we obtain the
= log pi;0 e
t
+1
pi;0
expression
@Vw (t; tS )
@t
= pS;0 pw;0 ( v (2
+ pw;0 (1
c (1
for t
1)
c) e
2v
pS;0 )
pS;0 ) (1
t
2
+ (1 2 ) c v
v c
!
c
cpS;0 (1
!
t
pw;0 ) e
pw;0 )
(23)
t.
tS . Observe that the M N B is a quadratic function of e
the weak …rm’s M N B is strictly concave in e
t;
when
< ,
Moreover, when
, it is convex in e
t.
This
distinction leads to di¤erent equilibrium characterizations (Propositions 5 and 6). To state
the result, de…ne the belief path
(p1;0 ; p2;0 )
(p1 ; p2 ) : pS =
[ (p1 ; p2 ) : 9t
pS;0 e tS
pw;0 e t
,
9t
2
(t
;
1)
,
p
=
w
S
pw;0 e t + 1 pw;0
pS;0 e tS + 1 pS;0
tS ,8i 2 f1; 2g , pi =
pi;0 e t
pi;0 e t + 1 pi;0
;
(24)
which is the belief path taken by a strong …rm that exits at time tS and a weak …rm that
experiments forever. The belief path
(p1;0 ; p2;0 ) is useful for the characterization of the
weak …rm’s strategy because choosing a time t to exit is equivalent to choosing a point on
the set
(p1;0 ; p2;0 ) to exit. This alternative representation allows us to analyze the weak
…rm’s problem graphically. We have,
Proposition 5 Suppose
< , given a nonextreme prior (p1;0 ; p2;0 ), there are three pos-
sible scenarios in a regular equilibrium.
1.
(p1;0 ; p2;0 ) \ E ( ) = : The weak …rm exits at time 0.
2. (p1;0 ; p2;0 ) 2 E ( ): The weak …rm exits when the posterior reaches the boundary of
the set E ( ).
3. (p1;0 ; p2;0 ) 2
= E ( ) and
(p1;0 ; p2;0 ) \ E ( ) 6= : There are two possible cases.
18
(a) The weak …rm exits at time 0.
(b) The weak …rm exits when the posterior reaches the boundary of the set E ( ) for
the second time. In this case, the posterior enters E ( ) prior to the exit and
leaves E ( ) at the time of the exit.
The scenarios in Proposition 5 are illustrated in Figures 3-6. The set E ( ) is represented
by the yellow (shaded) areas. The single …rm threshold belief for …rm 1 (2) is indicated
by the blue vertical (horizontal) lines. The dotted lines in the left panels represent the set
(p1;0 ; p2;0 ). Those in the right panels represent the equilibrium belief paths. In all of the
examples, …rm 1 is the strong …rm.
Figures 3(a) and 3(b): Regular equilibrium when
<
(scenario 1).
Parameter Values: (v; ; c; ; p1;0 ; p2;0 )=(8;1;1;0.53;0.95;0.4).
Figures 4(a) and 4(b): Regular equilibrium when
<
(scenario 2).
Parameter Values: (v; ; c; ; p1;0 ; p2;0 )=(8;1;1;0.53;0.65;0.5).
19
Figures 5(a) and 5(b): Regular equilibrium when
<
(scenario 3(a)).
Parameter Values: (v; ; c; ; p1;0 ; p2;0 )=(8;1;1;0.53;0.94;0.92).
Figures 6(a) and 6(b): Regular equilibrium when
<
(scenario 3(b)).
Parameter Values: (v; ; c; ; p1;0 ; p2;0 )=(8;1;1;0.53;0.85;0.76).
As a quadratic function, the M N B has at most two roots. This means that
(p1;0 ; p2;0 )
can intersect with @E ( ) at most twice. This leads to the three di¤erent scenarios listed
in Proposition 5, which we will discuss in turns.
Scenario 1 (Figures 3): If the belief path
the weak …rm’s M N B is nonpositive for all t
(p1;0 ; p2;0 ) never reaches the set E ( ),
0. It is thus optimal for the weak …rm
to exit at time 0 and the posterior belief for the weak …rm’s project stays at the prior
probability thereafter. After the weak …rm exits, the strong …rm continues to experiment
until the posterior belief for her project reaches the single …rm threshold belief. This leads
to the horizontal belief path in Figure 3(b).
20
Scenario 2 (Figures 4): If the prior (p1;0 ; p2;0 ) is inside the set E ( ),
(p1;0 ; p2;0 )
only intersects with @E ( ) at one point. This is because the posterior (p1;t ; p2;t ) must
eventually leave the set E ( ), leaving E ( ) and entering it again implies the quadratic
function (23) must have three or more roots, which is impossible. Therefore, as illustrated
by Figure 4(a), once the posterior leaves the set E ( ), it never gets back to it if the weak
…rm does not exit. This means that the weak …rm’s M N B is positive until the posterior
reaches the boundary of the set E ( ) and is negative thereafter. Thus, exiting at the
boundary is optimal.
Scenario 3 (Figures 5 and 6): If the …rms start with a prior (p1;0 ; p2;0 ) outside the
set E ( ) but the belief path reaches the set E ( ) at some point, then (23) implies that
(p1;0 ; p2;0 ) intersects with @E ( ) twice. In this case, the weak …rm’s expected payo¤ has
two local maxima. If the weak …rm experiments until the posterior reaches @E ( ) for the
second time, the weak …rm su¤ers losses when the posterior is outside E ( ) but eventually
gain when the posterior gets inside E ( ). On the other hand, if the weak …rm exits at time
0, such losses are avoided. Depending on how far the prior (p1;0 ; p2;0 ) is from the set E ( ),
either or both of these strategies can be optimal. In the numerical example illustrated by
Figure 5, exiting at time 0 is optimal, while in the numerical example illustrated in Figure
6, experimenting is optimal.
When
, (23) is convex in e
t.
This means that (23) has at most one root on
[0; tS ]. To see that, note that the M N B must be negative at tS , if there are two roots on
[0; tS ], the slope of the M N B function (23) must be positive at the …rst root and negative
at the second root. This contradicts the convexity of (23). Moreover, when (23) has a root
t0 on [0; tS ], it must be a maximum since the M N B is positive for any t < t0 and negative
for any t > t0 . When (23) has no root on [0; tS ], M N B is negative for all t
0, exiting at
time 0 is optimal. Thus, we have,
Proposition 6 Suppose
, given a nonextreme prior (p1;0 ; p2;0 ), there are two possible
scenarios in a regular equilibrium.
1. (p1;0 ; p2;0 ) 2
= E ( ): In this case, …rm 2 exits at time 0.
21
2. (p1;0 ; p2;0 ) 2 E ( ): In this case, …rm 2 exits when the posterior reaches the boundary
of the set E ( ).
The scenarios in Proposition 6 are illustrated in Figures 7 and 8. As in Figures 3–6, the
set E ( ) is represented by the yellow (shaded) area. The single …rm threshold belief for
…rm 1 (2) is indicated by the blue vertical (horizontal) lines. The dotted lines in the left
panels represents the set
(p1;0 ; p2;0 ). Those in the right panels represent the equilibrium
belief paths. In all of the examples, …rm 1 is the strong …rm.
Figures 7(a) and 7(b): Regular equilibrium when
(scenario 1).
Parameter Values: (v; ; c; ; p1;0 ; p2;0 )=(8;1;1;0.6;0.95;0.4).
Figures 8(a) and 8(b): Regular equilibrium when
(scenario 2).
Parameter Values: (v; ; c; ; p1;0 ; p2;0 )=(8;1;1;0.6;0.9;0.6).
In contrast to the case of
path
< , if the prior (p1;0 ; p2;0 ) is outside the set E ( ), the belief
(p1;0 ; p2;0 ) could never intersect with E ( ) (Figure 7(a)), the weak …rm’s M N B is
negative for all t
0, it is thus optimal for the weak …rm to exit at time 0 (Figure 7(b)).
22
If the prior (p1;0 ; p2;0 ) is inside the set E ( ), knowing that once the posterior leaves, it
will never enter the set E ( ) again (Figure 8(a)), the weak …rm will exit at the boundary
(Figure 8(b)).
Proposition 7 is a simple consequence of the previous results.
Proposition 7 Consider the weak …rm’s regular equilibrium exit time correspondence Tw :
1
2; 1
0
an
0
R+ . Tw has more than one element in at most one
exists, for all
2
1
2;
0
, 0 2 Tw ( ).
Suppose that two regular equilibria coexist at some
0
and 6,
2 ( ; ). Moreover, if such
0
2
1
2; 1
. By Propositions 3, 5
2 ( ; ) and one of the equilibria must have the weak …rm exiting at time 0.
Propositions 2 and 4 then implies that for all
<
0,
there is a unique regular equilibria in
which the weak …rm exits at time 0. Similarly, for all
>
0,
the weak …rm’s equilibrium
exit time must be at least as high as the higher equilibrium exit time at
0.
Thus, the
correspondence Tw has more than one element in at most one point.
4
Welfare Analysis
In this section, we analyze the welfare implications of our model. To provide a benchmark
of comparison, we …rst de…ne and solve the planner’s problem. Then, we de…ne the concepts
of under-experimentation and over-experimentation relative to the e¢ ciency benchmark.
In the end of this section, we apply these results to consider some comparative statics of
the total welfare in our model.
4.1
The Planner’s Problem
The objective of the planner is to maximize the sum of the expected payo¤s of the two
…rms. As such, how the prize v is allocated among the …rms is irrelevant. Therefore, we
de…ne,
De…nition 2 (The planner’s problem) A pair of exit times (t1 ; t2 ) for the two …rms
solves the planner’s problem if and only if it maximizes the sum of the expected payo¤ s of
the two …rms.
23
Given a pair of exit times (t1 ; t2 ) with ti
VP (t1 ; t2 )
Z ti
((p1;t + p2;t ) v
=
0
+e
R ti
0
(p1;s +p2;s ) ds
tj , the planner’s expected payo¤ is given by
Rt
2c) e 0 (p1;s +p2;s )
Z tj
(pj;t v c) e
ti
ds
dt
Rt
pj;s ds
ti
dt :
(25)
The expression for the case when ti > tj can be obtained easily by interchanging the roles
of i and j. Di¤erentiating (25) and the corresponding expression for ti > tj , we obtain
@VP (t1 ; t2 )
=e
@ti
where
UPi t; t0
R ti
0
pi;s ds
R ti ^tj
0
8
R
>
< t0 (pj;s v
t
pj;s ds
c) e
>
:
v
UPi (ti ; tj )
pj;z dz
t0 ;
pi;ti
Rs
t
0
ds if t
c ;
(26)
(27)
if t > t0 :
The function UPi (t; t0 ) is the continuation payo¤ of the planner after the planner retires
…rm i at time t given that …rm j exits at time t0 . It depends on the identity of the …rm
retired (…rm i), the current time t and the exit time t0 for the remaining …rm (…rm j). The
continuation payo¤ is zero if …rm i is last …rm to exit (i.e. t > t0 ). If t < t0 , …rm j continues
to experiment until t0 , conditional on the absence of a breakthrough. With probability
Rs
exp
t pj;z dz no breakthrough has occurred between times t and s, the planner incurs
instantaneous cost c to experiment; with instantaneous probability pj;s , a breakthrough
occurs and the planner receives v from a breakthrough. The continuation payo¤ UPi (t; t0 )
is thus obtained by integrating from t to t0 .
The derivative
@VP (t1 ;t2 )
@ti
in (26) represents the marginal net bene…t of experimentation
of …rm i at time ti to the planner. From the viewpoint of the planner, the instantaneous
bene…t from …rm i’s experimentation can be calculated as follows. With instantaneous
probability pi;ti , a breakthrough occurs and the value is v. If …rm i exits, the planner’s
continuation value becomes UPi (ti ; tj ). The term pi;ti (v
UPi (ti ; tj )) is thus the instanta-
neous bene…t. The instantaneous net bene…t of experimentation, pi;ti (v
24
UPi (ti ; tj ))
c,
is then weighted by the probability that no breakthrough has occurred until time ti ,
n R
o
R ti ^tj
ti
exp
p
ds
p
ds
. Since …rm j stops experimentation at time tj , the
i;s
j;s
0
0
probability that …rm j has not achieved a breakthrough at time ti depends on whether
ti > tj .
Our next lemma establishes that the planner’s marginal net bene…t of experimentation
for a particular …rm (26) satis…es the single-crossing property. As a result, the socially
e¢ cient exit time tzi for …rm i given …rm j’s exit time t0j is unique and pinned down by the
…rst order condition. Moreover, it also establishes a range for exit time t0j , so that …rm i’s
R&D e¤ort is a substitute of …rm j’s. In this range, as …rm j’s amount of experimentation
decreases, the socially e¢ cient amount of experimentation for …rm i increases. Notice that
any equilibrium exit time tj must be inside this range.
Lemma 7 Given an exit time t0j
0 for …rm j, there exists an exit time tzi
0 such that for
1 ;t2 )
> 0 and for all t 2 (tzi ; 1), VP (t
jti =t;tj =t0j < 0. Moreover,
@ti
h
i
1 p
y
y
1
the exit time tzi is decreasing in t0j for t0j 2 0; tj , where tj
log pj;0j;0 vc c .
all t 2 [0; tzi ),
VP (t1 ;t2 )
jti =t;tj =t0j
@ti
Lemma 7 implies that for each i 2 f1; 2g, the function tzi : R+ ! R+ is well de…ned. A
solution to the planner problem (t1 ; t2 ) must satisfy
tzi tj
= ti ;
for i 2 f1; 2g. Next, it is easy to see that the weak …rm must exit weakly before the strong
…rm under the planner’s optimal policy.
Lemma 8 It is never optimal for the planner to retire the weak …rm strictly later than the
strong …rm.
With Lemmas 7 and 8 at hand, we can characterize the planner’s solution easily. By
Lemma 8, the strong …rm exits later in the planner’s solution, thus, UPS (tS ; tw ) = 0,
the …rst order condition implies that tS must solve pS;tS v
c = 0, which, by lemma
7, is unique. Unlike the noncooperative problem, the corner solution is ruled out by the
25
assumption of nonextreme prior, which asserts that pS;0 >
c
v.
Then, Lemma 1 and (27)
imply that
^ (pS;t ; 1) :
UPw (t; tS ) = W
In other words, after the weak …rm has exited, the continuation value to the planner at time
t is the same as the continuation value of the strong …rm at time t in the single …rm problem
when
= 1. Intuitively, after the weak …rm has exited, the planner’s optimal action and
value must be identical to the strong …rm’s when the payo¤ externality is internalized.
De…ne the instantaneous net bene…t of experimentation of the weak …rm to the planner
IN BPw by
IN BPw p; p0
p
v
^ p0 ; 1
W
c;
(28)
where p is the probability for the weak …rm’s project and p0 is the probability for the strong
…rm’s project. The weak …rm’s …rst order condition implies
IN BPw pw;tw ; pS;tw
0.
Thus, we have,
Proposition 8 The solution to the planner’s problem (t1 ; t2 ) is given by:
1. tS is the unique solution to the equation pS;tS v = c and has the expression
y
t S = tS
2. If IN BPw (pw;0 ; pS;0 )
1
log
1
pS;0 c
pS;0
v c
(29)
0, then tw = 0. Otherwise, tw is the unique solution to
IN BPw pw;tw ; pS;tw
= 0:
(30)
As in the case of the noncooperative problem, we can illustrate the characterization of
the planner’s solution and the resulting dynamics visually by 2-dimensional …gures in the
26
belief space. Let E
be the set of beliefs at which the weak …rm’s IN B to the planner is
strictly positive. That is
E
[ n
(p1 ; p2 ) 2 [0; 1]2 : pi
i=1;2
Let @E
o
pj ; IN BPw (pj ; pi ) > 0 ;
denote the boundary of E . Proposition 8 implies the set E
(31)
is also the set of
beliefs at which the planner …nds it optimal to keep both …rms experimenting. In Figures
9(a) and 9(b), the set E
is represented by the green (shaded) area. If the prior (p1;0 ; p2;0 )
is inside the set E , the weak …rm exits once the posterior reaches the boundary of E ,
while the strong …rm continues to experiment until the posterior belief for her project
reaches the single …rm threshold belief
c
v,
as illustrated by Figure 9(a). If the …rms start
with a prior (p1;0 ; p2;0 ) outside the set E , the weak …rm exits immediately at time 0 and
the strong …rm again experiments until her posterior belief reaches the single …rm threshold
belief, as illustrated by Figure 9(b).
Figures 9(a) and 9(b): The planner’s solution. Parameter Values: (v; ; c)=(8;1;1).
(a) (p1;0 ; p2;0 )=(0.9;0.6). (b) (p1;0 ; p2;0 )=(0.95;0.4).
The planner’s solution provides a useful benchmark for welfare comparisons. Comparing
the …rst parts of Propositions 1 and 8, we …nd that the noncooperative exit time tS is always
less than the socially optimal exit time tS and strictly so if
< 1. This is intuitive. Unless
= 1, the strong …rm does not internalize the weak …rm’s gain from a breakthrough and
under-invests in R&D. As a result, the strong …rm exits too early relative to the planner’s
27
solution. Thus, we introduce the following de…nition of under-experimentation for the
strong …rm.
De…nition 3 The strong …rm under-experiments in a regular equilibrium (t1 ; t2 ) if and
only if tS < tS .
Since tS
tS , the strong …rm never over-experiments in our model. This is, however,
not true for the weak …rm. On the one hand, the social value of a breakthrough from
the weak …rm v is larger than its private value
v. On the other hand, the “outside
^ (pS;t ; 1) to the planner and W (pS;t ; ) to
option”, i.e. the value of exiting, at time t is W
^ (pS;t ; 1)
the weak …rm. Since W
W (pS;t ; ), comparing (13) and (28), the weak …rm’s
incentives to experiment in the noncooperative problem may exceed or fall short of the
planner’s. Therefore, the weak …rm’s equilibrium exit time may be higher or lower than
the planner’s exit time. However, having a higher equilibrium exit time does not necessarily
mean that the weak …rm over-experiments. This is because the socially optimal amount of
experimentation by the weak …rm tzw also depends on the strong …rm’s equilibrium exit time
tS . Thus, if the strong …rm exits prematurely, Lemma 7 implies that it would be socially
optimal for the weak …rm to experiment beyond the planner’s exit time to compensate the
loss. To deal with this issue, we provide a formal de…nition of under-experimentation by
comparing the weak …rm’s equilibrium exit time tw and the socially optimal exit time tzw
given the strong …rm’s equilibrium exit time tS .
De…nition 4 Given the strong …rm’s equilibrium exit time tS 2 [0; tS ], the weak …rm
under(over)-experiments in a regular equilibrium (t1 ; t2 ) if and only if tw < (>) tzw (tS ).
Note that when tS = tS , tzw (tS ) = tw . In this case, the weak …rm under-experiments
if and only if tw < tw . However, if the strong …rm experiments less than the amount in
the planner’s solution, i.e. tS < tS , by Lemma 7, the benchmark for the weak …rm’s exit
time tzw is in general larger than the weak …rm’s exit time tw in the planner’s solution. We
have,
28
Lemma 9 If the weak …rm’s exit time in a regular equilibrium (t1 ; t2 ) is strictly smaller
than the planner’s exit time for the weak …rm, i.e. tw < tw , then the weak …rm underexperiments in equilibrium, i.e. tw < tzw (tS ).
Lemma 9 is a useful shortcut in showing that the weak …rm under-experiments in
equilibrium. Without …nding the socially e¢ cient amount of experimentation tzw , one can
simply check the su¢ cient condition by comparing the weak …rm’s equilibrium exit time
with the planner’s. Together with the next lemma, it implies that the weak …rm cannot
over-experiment when
Lemma 10 For all
2 [ 21 ; ) (Lemma 11).
2 [ 21 ; ), E ( ) [ @E ( )
E .
To see that the weak …rm cannot over-experiment when
2 [ 12 ; ), note that, by
Proposition 5, the weak …rm either exits at time 0 or when the posterior reaches @E ( )
E . Since tzw (t)
0 for all t
0, if the weak …rm exits at time 0, she cannot over-
experiment. If the weak …rm exits at @E ( ), then p1;tw ; p2;tw 2 E . Thus, the weak
…rm’s IN B to the planner is strictly positive. Lemma 7 implies that tw < tzw (tS ) = tw .
By Lemma 9, the weak …rm under-experiments. In either case, the weak …rm does not
over-experiment. Thus, we have,
Lemma 11 Suppose
2 [ 21 ; ), the weak …rm cannot over-experiment in a regular equi-
librium.
Lemma 11 allows us to select the welfare superior equilibrium when there are multiple
equilibria.
Proposition 9 Whenever two regular equilibria coexist, the equilibrium in which the weak
…rm exits at a later time dominates the equilibrium with early exit in terms of total welfare.
By Proposition 7, two regular equilibria coexist only when
2 ( ; ). By Proposition
1, the amount of experimentation by the strong …rm is the same in both regular equilibria.
By Lemma 11, the weak …rm cannot over-experiment in equilibrium. Thus, the equilibrium
in which the weak …rm exits at a later time must dominate the other equilibrium in terms
of total welfare.
29
4.2
Comparative Statics
In this subsection, we consider the comparative statics of increasing the prior probabilities.
As illustrated by Examples 1 and 2, it turns out that increasing the prior probabilities
can have an adverse e¤ect on the total welfare. In Example 1, an increase in the prior
probability about the strong …rm’s project leads to more severe free riding by the weak
…rm. In Example 2, an increase in the prior probability about the weak …rm’s project
results in more severe duplication. In both cases, the total welfare decreases.
Example 1 (Adverse e¤ect of increasing the strong …rm’s probability) For each
2 ( ; ), we can construct examples in which an in…nitesimal increase of the prior
probability about the strong …rm’s project leads to a downward jump in the total welfare.
Suppose
2 ( ; ). By Lemma 5, E ( ) is nonempty. By the de…nition of
, E( )[
@E ( ) does not contain the point (1; 1). As illustrated by Figures 10(a) and 10(b), this
implies that there is a segment of the 45 line laying between E ( ) and (1; 1). If the prior
(p1;0 ; p2;0 ) is on the segment of the 45 line that is inside the set E ( ), both …rms will
experiment until the beliefs reach
c
v
on the lower boundary of the set E ( ). However,
as the prior moves outside the set E ( ) and up the 45 line, one of the …rms must exit
at time 0. This is because the time taken to reach E ( ) becomes arbitrarily large as the
prior gets close to (1; 1).7 The continuity of the …rms’ payo¤ functions implies that there
must exist a point on the 45 line at which …rm 2 is indi¤ erent between exiting at time
0 and experimenting until p2;t =
c
v
given that …rm 1 experiments until p1;t =
c
v.
At
such a point, there are two regular equilibria as given by …rm 2’s two best responses. By
Lemma 10, the set E ( ) [ @E ( ) is contained in the set E . (@E
is illustrated by
the blue dotted line in Figures 10(a) and 10(b).) By Proposition 9, the equilibrium with
immediate exit is dominated by the experimentation equilibrium in terms of total welfare.
Now, if the prior probability for …rm 1’s project increases for an arbitrarily small amount,
…rm 2 strictly prefers to exit at time 0 and the welfare superior equilibrium disappears. It
is easy to see that the same construction also works by moving up the belief path from any
point (p1 ; p2 ) 2 E ( ) and then increasing the prior probability for the strong …rm at the
7
If the prior is (1; 1), the posterior never enters E ( ).
30
point of indi¤ erence. In Figures 10(a) and 10(b), we identify such a point of indi¤ erence
numerically. The two equilibrium belief paths are illustrated in the two …gures, respectively.
Figure 10(a) (Figure 10(b)): Welfare superior (inferior) equilibrium.
Parameter Values: (v; ; c; ; p1;0 ; p2;0 )=(8;1;1;0.53;0.9202;0.9202).
Example 2 (Adverse e¤ect of increasing the weak …rm’s probability) The adverse
e¤ ect of increasing the prior probability about the weak …rm’s project is best illustrated by
the extreme case when the strong …rm’s project is known to be good. As a result, we will
depart brie‡y from the assumption of nonextreme prior in this example and assume that
p1;0 = 1. The continuity of the total payo¤ function implies that the same conclusion holds
when pS;0 is close but not equal to 1. Notice that, without discounting, there is no point to
have …rm 2 engaging in R&D at all in this case. This is because, even if p2;0 = 1, using
…rm 2 only speeds up the arrival of a breakthrough and has no e¤ ect on the total welfare.
c
With only …rm 1 engaging in R&D, the total welfare is simply v
arrives with probability 1 and its expected arrival time is
1
. (If both …rms’ projects are
good and both …rms experiment, the expected arrival time is
resulting in the same expected cost of
W (1; ) = (1
by v
c
c
.) Suppose
>
, since a breakthrough
1
2
and the ‡ow cost is 2c,
c
and p2;0
(2
1)v .
By (6),
) v, Proposition 1 implies that t2 = 0 and the equilibrium welfare is given
. On the other hand, if
c
(2
1)v
< p2;0 < 1, Proposition 1 implies that t2 > 0, the
equilibrium welfare is given by
v
c
(1
p2;0 ) 1
1
p2;0
p2;0
31
(2
c
1) v
c
c
:
This is because, with probability 1
incurred is 1
e
t2
c
p2;0 , …rm 2’s R&D process is bad and the expected cost
, where t2 =
1
log
1 p2;0
p2;0
(2
c
1)v c
. Therefore, as illustrated
by Figure 11, the total welfare decreases as p2;0 increases from p2;0 <
to
c
(2
1)v
c
(2
1)v
(point B)
< p2;0 < 1 (point A). Notice that both points are outside the set E , whose
boundary is illustrated by the blue dotted line. This indicates that the planner …nds it
optimal to retire …rm 2 at time 0 in both cases.
Figure 11: Increasing pw;0 decreases welfare. A =(1;0.7), B =(1;0.6).
Parameter Values: (v; ; c; )=(8;1;1;0.6).
5
Optimal Patent
In this section, we characterize the optimal patent, which we de…ne below.
De…nition 5 (Optimal Patent) A patent system represented by the patent strength
1
2; 1
is optimal if and only if
2
maximizes the sum of the expected payo¤ s of the two …rms
in the welfare maximizing equilibrium.
In the de…nition of the optimal patent, the welfare maximizing equilibrium is selected
whenever there are multiple equilibria. This choice makes sure that the total welfare
function is upper-semicontinuous. Our …rst lemma in this section rules out patent systems
that are too weak so that none of the …rms experiments in equilibrium.
Lemma 12 Under the optimal patent
, the strong …rm performs a positive amount of
experimentation in equilibrium, i.e. tS > 0.
32
Suppose that, under the optimal patent
, both …rms exit at time 0. Consequently,
the total welfare is zero. Such a patent system is strictly dominated by the strict patent
system. When
= 1, (6) implies that for all t
0, W (pS;t ; ) = 0. The assumption of
nonextreme prior and the characterizations in Proposition 1 then imply that both …rms
exit when the posterior belief for her project reaches the single …rm threshold belief
c
v.
That is, for i = 1; 2, pi;ti v = c. Under the strict patent system, the total payo¤ VP (t1 ; t2 )
is strictly positive, since all the integrands in (25) is positive and strictly so on a set of
positive measure.
Lemma 13 provides the basic characterizations of the optimal patent that drive lots of
the results in this section.
Lemma 13 Under the optimal patent
, the welfare maximizing equilibrium (t1 ; t2 ) sat-
is…es
1.
@VP (t1 ;t2 )
jtw =tw ;tS =tS
@tw
2. IN B pw;tw ; pS;tw ;
0.
= 0.
The …rst characterization of the optimal patent
in Lemma 13 states that the M N B
of the weak …rm’s experimentation at tw to the planner must be nonpositive. Intuitively,
if this is violated at the optimum and
2 [ 12 ; 1), one can strengthen the patent system
for a small amount and increase the total welfare. By Proposition 4 and Lemma 12, this
will weakly increase the exit time of the weak …rm and strictly increase the exit time of the
strong …rm. Since the strong …rms never over-experiment in equilibriums, the total welfare
must increase when the increase in patent strength is small enough.8 If
= 1, one can
check that the characterization is satis…ed.
The second characterization of the optimal patent
in Lemma 13 states that the weak
…rm’s IN B must be zero at her time of exit. This is trivially true when tw > 0. Suppose
the weak …rm’s exit time in the welfare maximizing equilibrium is 0 under the optimal
8
Although Proposition 7 implies that the weak …rm’s exit time in the welfare maximizing equilibrium
is in general discontinuous in . It is discontinuous at only one point and Proposition 9 implies that it
is upper-semicontinuous. Thus, for any
2 21 ; 1 and " > 0, we can …nd
> 0 small enough so that
tw ( ) tw ( + ) < tw ( ) + ".
33
patent
and IN B pw;tw ; pS;tw ;
< 0. In this case, the regular equilibrium must be
unique. Otherwise, by Proposition 9, the weak …rm’s exit time in the welfare maximizing
equilibrium would not be 0. The continuity of the equilibrium exit time tw then implies
one can …nd
> 0 small enough so that the weak …rm’s equilibrium exit time remains at
0 under the patent
+ . By Proposition 4 and Lemma 12, this strictly increases the exit
time of the strong …rm. Since the strong …rm always under-experiments in equilibrium, the
total welfare must increase as a result.
With Lemma 13 at hand, we can derive Propositions 10 and 11 easily.
Proposition 10 Under the optimal patent
, the weak …rm never under-experiments.
To prove Proposition 10, suppose the weak …rm under-experiments under the optimal patent
. By Lemma 7, tw < tzw (tS ) implies that
@VP (t1 ;t2 )
jtw =tw ;tS =tS
@tw
> 0, which
contradicts Lemma 13.
Proposition 11 Any patent
2 [ 21 ; ) is suboptimal. In particular, it is never optimal to
6= 12 .
impose no patent protection. i.e.
To prove Proposition 11, suppose by way of contradiction that
13, the weak …rm exits when IN B pw;tw ; pS;tw ;
By Lemma 10, @E (
)
2 [ 12 ; ). By Lemma
= 0. Thus, p1;tw ; p2;tw 2 @E (
).
E , which, by Lemma 7, implies that tw < tw . Lemma 9 then
implies that the weak …rm under-experiments in equilibrium, which contradicts Proposition
10. Notice that the fact that
6=
1
2
does not follow from Lemma 12, as the strong …rm
may still perform a positive amount of experimentation in the absence of patent protection.
Proposition 11 shows that some patent protection is always desirable. Strict patent,
however, is also suboptimal in general.
Proposition 12 Unless p1;0 = p2;0 , the strict patent system, i.e.
= 1, is not optimal.
To see why the strict patent system is in general suboptimal, note that when
= 1,
the strong …rm’s equilibrium exit time is socially optimal, i.e. tS = tzS (tw ). Therefore, the
strong …rm’s M N B at time tS to the planner is 0, i.e.
34
@VP (t1 ;t2 )
jtw =tw ;tS =tS
@tS
= 0. Thus, a
small departure of the strong …rm’s exit time from the optimum has a vanishing e¤ect on
the total welfare. While lowering the patent strength distorts the strong …rm’s exit time,
it also discourages the weak …rm from over-experimenting, which does occur in equilibrium
when p1;0 6= p2;0 and
= 1. If the decrease in
is small enough, the gain from the
discouragement e¤ect outweighs the distortion e¤ect. Thus,
= 1 is not optimal.
Example 1 in Section 4.2 suggests that an increase in the prior probability about the
strong …rm’s project may aggravate the free riding behavior of the weak …rm and have
an adverse e¤ect on the total welfare. This, however, cannot occur when the optimal
patent is used. This is because the weak …rm never under-experiments under the optimal
patent. Thus, encouraging the weak …rm to free ride and reduce experimentation is actually
bene…cial at the optimum.
Proposition 13 Under the optimal patent, an increase in the prior probability about the
strong …rm’s project always increases total welfare.
To understand Proposition 13, consider pS;0 as a parameter and the optimal patent
a function of pS;0 , denote the equilibrium exit times by t1 (
; pS;0 ) and t2 (
; pS;0 ) and
the total payo¤ by VP (t1 ; t2 ; pS;0 ), we have
dVP (t1 (
; pS;0 ) ; t2 ( ; pS;0 ) ; pS;0 )
dpS;0
@VP (t1 ; t2 ; pS;0 ) @VP (t1 ; t2 ; pS;0 ) @tS
@VP (t1 ; t2 ; pS;0 ) @tw
+
+
:
@pS;0
@tS
@pS;0
@tw
@pS;0
=
The …rst term
@VP (t1 ;t2 ;pS;0 )
@pS;0
represents the change in the total payo¤ due to an increase in
pS;0 holding the …rms’exit times …xed. This term is obviously positive, as the probability
of achieving a breakthrough by the strong …rm at each time t increases. The second term
represents the change in the total payo¤ through the change in the strong …rm’s exit time.
@VP (t1 ;t2 ;pS;0 )
Since the strong …rm never over-experiments in equilibrium,
0. Moreover,
@t
S
by (8),
@tS
@pS;0
> 0. The second term is also positive. The third term represents the change
in the total payo¤ through the change in the weak …rm’s exit time. By the …rst part of
@VP (t1 ;t2 ;pS;0 )
Lemma 13,
0. By the second part of Lemma 13, (9) holds at equality. Direct
@t
w
35
di¤erentiation yields
@tw
@pS;0
< 0. The third term is also positive. The desired conclusion
then follows from an application of the Envelope Theorem (e.g. Corollary 4 of Milgrom
and Segal (2002)).
The optimal patent, however, is not immune from the adverse e¤ect of increasing the
prior probability about the weak …rm’s project. This is illustrated by Example 3. Intuitively, when the weak …rm becomes more con…dent, it becomes more di¢ cult to discourage
her from experimenting. But this is exactly what is called for when the strong …rm’s project
is “too good” relative to the weak …rm’s.
Example 3 (Increasing the weak …rm’s probability under optimal patent) In this
example, we show that an increase in the prior probability about the weak …rm’s project
can have a negative e¤ ect on the total welfare even under the optimal patent. (In the
Appendix, we demonstrate this possibility analytically by taking derivative of the total payo¤ function with respect to pw;0 .) In the situation illustrated by Figure 12(a), the strong
…rm’s prior probability is much larger than the weak …rm’s. Under the optimal patent,
the weak …rm exits at time 0. By Lemma 13, (p1;0 ; p2;0 ) 2 @E (
). In Figure 12(b),
the weak …rm’s probability increases, yet the weak …rm is still too ine¢ cient to experiment under the optimal patent. In this case, the optimal patent is weakened by
that (p1;0 ; p2;0 ) 2 @E (
increases from
c
tation frontier
c
v,
v
to
> 0 so
). As a consequence of this, the single …rm belief threshold
c
(
)v
and moves away from the single …rm e¢ cient experimen-
which is illustrated by vertical blue dotted line. Thus, the strong …rm’s
equilibrium exit time further moves away from the social e¢ cient level after the increase in
pw;0 . Since the weak …rm never experiments in both cases, the total welfare decreases.
36
Figures 12: Increasing pw;0 decreases welfare under optimal patent.
6
(a) (v; ; c;
; p1;0 ; p2;0 )=(8;1;1;0.8068;0.95;0.2),
(b) (v; ; c;
; p1;0 ; p2;0 )=(8;1;1;0.6453;0.95;0.4).
Optimal Distribution Policy
We have shown in the previous section that unless the prior probabilities are equal, no
patent system can achieve e¢ ciency. This calls for a search for more general policies
that may improve social welfare beyond the optimal patent. Indeed, we …nd that one can
implement the planner’s solution by a simple distribution policy. To begin our discussion of
such a policy, we must …rst modify our game and develop a notion of incentive compatibility
in the reporting of innovations. That is because, while the …rms always report an innovation
immediately after a breakthrough to preempt her opponent’s attempt to patent under a
patent system, this is not true under general distribution policy. Therefore, we need to
make sure that the optimal policy we propose is also incentive compatible in reporting.
We augment the game introduced in Section 2 in the following ways. After a breakthrough occurs to …rm i at time t, …rm i exits R&D privately and stops paying the ‡ow
cost c. Firm i then chooses a t0i
t to report the innovation. The game ends with the …rst
report of a breakthrough or with both …rms exiting the R&D race. We now de…ne the class
of policies that we consider, which we call distribution policies, and a notion of incentive
compatibility in reporting given a distribution policy.
37
De…nition 6 A distribution policy is a pair (
1;
2)
2 [0; 1]2 , under which, if …rm i is the
…rst to report a breakthrough in the augmented game, the rewards to …rm i and …rm j are
given by
iv
and (1
i ) v,
respectively.
De…nition 7 A distribution policy (
1;
2)
is incentive compatible if and only if there exists
an equilibrium in which the …rms report an innovation immediately after a breakthrough.
De…nition 8 A distribution policy (
1
;
2
) implements the planner’s solution if and only
if it is incentive compatible and the …rms exit at (t1 ; t2 ) in equilibrium.
Our main result in this section identi…es a distribution policy that implements the
planner’s solution. It should be emphasized that while di¤erential policies could be e¢ cient,
we realize the di¢ culties of implementing such a policy in practice. As many works in the
literature of theoretical mechanism design, the optimal policy that we propose depends
on the details of the model. In particular, the optimal policy is sensitive to the prior
probabilities, which may be di¢ cult to obtain by the government. Di¤erential policy may
also be prohibited by laws or simply impossible to be carried out politically. Nevertheless,
the study of such policies is interesting from a theoretic perspective as it demonstrates
what is lacking in a patent system.
Proposition 14 The distribution policy described by
S
w
= 1 and
^ pS;t ; 1
W
w
;
= 1
v
(32)
(33)
implements the planner’s solution.
Notice that if p1;0 = p2;0 , by Proposition 8, the two …rm exits at the same time tw = tS
^ pS;t ; 1 = 0 and thus
in the planner’s solution. (5) and (29) imply W
w
w
= 1. The
optimal distribution policy in this case is simply the strict patent system, as we found in
Section 5. Unless p1;0 = p2;0 , the optimal distribution policy is not anonymous.
To see why the distribution policy (
1
;
2
) is incentive compatible, notice that the
strong …rm receives the maximum award by reporting immediately after a breakthrough
38
and thus cannot gain by delaying the report. On the other hand, the nonextreme prior
^ pS;t ; 1 < v. Thus,
assumption implies that W
w
award
w
w
v > 0. The weak …rm receives the
v for sure by reporting a breakthrough immediately. If she delays the report, she
gets at most
w
v and may get 0 if she is preempted by the strong …rm. Thus, none of the
…rms has incentives to delay a report.
To see why the equilibrium exit times of the …rms coincide with those of the planner,
consider the weak …rm’s problem. Since
S
= 1, the weak …rm never bene…ts from the
strong …rm’s innovation. The best response of the weak …rm is thus to follow the optimal
policy in the single …rm problem and exit when the posterior belief falls below the single
…rm threshold belief
c
w
v.
Thus, the weak …rm exits at the smallest t that solves
pw;t
The choice of
w
v
^ pS;t ; 1
W
w
c
0:
in (33) implies that the weak …rm’s exit time must be identical to tw in
equilibrium. Next, we would like to show that tS is a best response for the strong …rm.
First, observe that any t 2 [tw ; tS ) [ (tS ; 1) cannot be optimal, since the strong …rm can
improve by exiting at tS instead. Suppose tw > 0, to show that exiting at any t 2 [0; tw )
is suboptimal, consider the strong …rm’s M N B at each time t 2 [0; tw ), we have
@VS (t; tw )
@t
= e
e
e
e
> e
= 0
Rt
0 (p1;s +p2;s )
Rt
0 (p1;s +p2;s )
Rt
0 (p1;s +p2;s )
Rt
0 (p1;s +p2;s )
Rt
0 (p1;s +p2;s )
ds
pS;t
n
ds
pS;t
n
ds
pS;t
n
ds
pw;t
n
ds
pw;tw
v
W (pw;t ;
w
)
c
^ (pw;t ;
W
o
^ (pw;t ; 1)
v W
c
o
^ (pS;t ; 1)
v W
c
o
^ pS;t ; 1
c
v W
w
v
W (pw;t ;
w)
^ (pw;t ;
The …rst inequality follows from the fact that W
w
)
w)
o
c
0, the second from the fact
the total welfare must be maximized when all the externality is internalized (i.e. when the
39
full award v is given to the remaining …rm upon a breakthrough), the third from the facts
that pS;t
^ ( ; 1) is increasing and the fourth from Lemma 7. The expression in
pw;t and W
the last inequality is simply the planner’s M N B from the weak …rm’s experimentation at
time tw , which must be 0 if tw > 0, by Proposition 8. Thus, any t 2 [0; tw ) is suboptimal
as
@VS (t;tw )
@t
7
> 0.
Discussion
In this section, we discuss several assumptions and some possible extensions.
Discounting. For tractability, we have assumed no discounting in this paper. However,
many of the results, including the basic equilibrium characterization (Proposition 1), the
characterization of the planner’s solution (Proposition 8), and the optimal distribution
policy (Proposition 14), do not rely on this assumption. These results remains unchanged
after the value functions (5) and (6) are replaced by their counterparts under discounting.
Moreover, from a robustness perspective, small discounting would not qualitatively alter
our results. The main di¢ culty in analyzing the high discounting case is that the weak
…rm’s M N B (12) can no longer be expressed as a quadratic function as in (23). This makes
a detailed characterization of the weak …rm’s exit time di¢ cult.
Consumer welfare. Our analysis in this paper has abstracted away from the consideration of consumer welfare. To incorporate this aspect in the model, suppose an innovation
leads to producer surplus v and consumer surplus C > 0 in the post-innovation market
regardless of the patent system used.9 Since consumer surplus is irrelevant to the …rms’
decision, the characterizations of equilibrium as presented in Section 3 remain unchanged.
However, the welfare consideration in Section 4 and the characterizations of the optimal
patent in Section 5, however, should be adjusted accordingly. Since the value of an innovation to the planner is larger than that in the benchmark scenario as presented in Section
4, the planner will prefer both …rms to experiment more and the optimal patent will be
stricter. In particular, the strict patent system, i.e.
9
= 1, may be optimal if the consumer
This could be the case if the …rms are able to collude to restrict the total output to the monopoly
quantity in the post-innovation market.
40
surplus C is large enough. Finally, the optimal distribution policy no longer implements
the planner’s solution. This is because even if
S
= 1, the strong …rm does not internalize
the consumer’s gain from an innovation. The same analysis, however, can be applied to
…nd the optimal distribution policy.
Post-innovation market e¢ ciency. Classical analyses of the optimal patent (e.g.
Nordhaus 1969; Denicolò 1999) focus the trade-o¤ between ex ante incentives to innovate
and ex post allocative ine¢ ciency. That is, a strict patent system results in monopoly in
the post-innovation market, which is socially ine¢ cient. However, such ex post ine¢ cient
arrangement may improve the ex ante incentives to innovate. In such an environment,
optimal patent should be set to balance the two forces. In our model, the post-innovation
market e¢ ciency is assumed to be constant across patent systems. The total producer surplus v remains unchanged. Incorporating post-innovation market e¢ ciency consideration
would make the trade-o¤ between duplication and under-experimentation that we highlighted less transparent. In analyzing such a model, the planner’s problem as presented in
Section 4.1 is no longer a useful benchmark, as allocative e¢ ciency in the …rms’e¤orts is
no longer the only concern.
Reversible exit. In our model, we assume that the exit decision of a …rm is irrevocable.
To relax this assumption, the standard approach in the literature is to convexify the action
space so that each …rm can choose to invest e 2 [0; 1] in R&D at each point in time. A
…rm’s strategy in this alternative model is then a time-dependent path of e¤ort, which is
considerably more complex than the exit time in our model. We suggest the alternative
model here as a direction for future research.
8
Conclusion
In this paper, we analyze how a patent system may a¤ect the amount of experimentation
in an R&D race. We …nd that the optimal patent is sensitive to the qualities of the R&D
projects. With heterogeneous …rms, a strict patent system is not optimal and the choice of
optimal patent system involves a trade-o¤ between duplication in the early stage of R&D
and under-experimentation in the latter stage. An asymmetric distribution policy that
41
implements the social optimum is proposed.
References
[1] Acemoglu, D., K. Bimpikis, and A. Ozdaglar (2011), “Experimentation, Patents and
Innovation,” American Economic Journal: Microeconomics, 3, 37–77.
[2] Akcigit, U. and Q. Liu (2014), “The Role of Information in Innovation and Competition,” Journal of the European Economic Association, forthcoming.
[3] Besanko, D. and J. Wu (2013), “The Impact of Market Structure and Learning on
the Tradeo¤ between R&D Competition and Cooperation,”Journal of Industrial Economics, 61, 166–201.
[4] Bolton, P. and C. Harris (1999), “Strategic Experimentation,”Econometrica, 67, 349–
374.
[5] Bonatti, A. and J. Hörner (2011), “Collaborating,”American Economic Review, 101,
632–663.
[6] Chatterjee, K., and R. Evans (2004), “Rivals’Search for Buried Treasure: Competition
and Duplication in R&D,” Rand Journal of Economics, 35, 160-183.
[7] Choi, J. (1991), “Dynamic R&D Competition under "Hazard Rate" Uncertainty,”
RAND Journal of Economics, 22, 596–610.
[8] Das, K. (2014a), “Strategic Experimentation with Competition and Private Arrival of
Information,” Unpublished.
[9] Das, K. (2014b), “Strategic Experimentation with Heterogeneous Agents and Payo¤
Externalities,” Unpublished.
[10] Dasgupta, P. and J. Stiglitz (1980), “Uncertainty, Market Structure and the Speed of
Research.” Bell Journal of Economics, 11, 1–28.
42
[11] Décamps, J.-P. and T. Mariotti (2004), “Investment Timing and Learning Externalities,” Journal of Economic Theory, 118, 80–102.
[12] Denicolò, V. (1999) “The Optimal Life of a Patent when Timing of Innovation is
Stochastic,” International Journal of Industrial Organization, 17, 827–846.
[13] Gerardi, D. and L. Maestri (2012), “A Principal–Agent Model of Sequential Testing,”
Theoretical Economics, 7, 425–463,
[14] Halac, M., N. Kartik, and Q. Liu (2015), “Contests for Experimentation,” Unpublished.
[15] Halac, M., N. Kartik, and Q. Liu (2015), “Optimal Contracts for Experimentation,”
Unpublished.
[16] Heidhues, P., S. Rady, and P. Strack (2014): “Strategic Experimentation with Private
Payo¤s,” Unpublished.
[17] Hörner, J. and L. Samuelson (2014), “Incentives for Experimenting Agents,” RAND
Journal of Economics, 44, 632–663.
[18] Judd, K., K. Schmedders, and Ş. Yeltekin (2012), “Optimal Rules for Patent Races,”
International Economic Review, 53, 23–52.
[19] Keller, G., S. Rady, and M. Cripps (2005), “Strategic Experimentation with Exponential Bandits,” Econometrica, 73, 39–68.
[20] Keller, G. and S. Rady (2010), “Strategic experimentation with Poisson bandits,”
Theoretical Economics, 5, 275–311.
[21] Loury, G. (1979) “Market Structure and Innovation,” The Quarterly Journal of Economics, 93, 395–410.
[22] Malueg, D. and S. Tsutsui (1997), “Dynamic R&D Competition with Learning,”
RAND Journal of Economics, 28, 751–772.
[23] Mason, R. and J. Välimäki (2010), “Learning in Contests,” Unpublished.
43
[24] Milgrom, P. and I. Segal (2002), “Envelope Theorems for Arbitrary Choice Sets,”
Econometrica, 70, 583–601.
[25] Moroni, S. (2015), “Experimentation in Organizations,” Unpublished.
[26] Moscarini, G. and F. Squintani (2010), “Competitive Experimentation with Private
Information: the Survivor’s Curse,” Journal of Economic Theory, 145, 639–660.
[27] Murto, P. and J. Välimäki (2011), “Learning and Information Aggregation in an Exit
Game,” Review of Economic Studies, 78, 1426–1461.
[28] Nordhaus, W. D. (1969), Invention, Growth and Welfare, Cambridge: M.I.T. Press.
[29] Perry, M. and D. Vincent (2002), “The Optimal Timing of Procurement Decisions and
Patent Allocations,” International Economic Review, 43, 1035–52.
[30] Reinganum, J. (1982), “A Dynamic Game of R&D Patent Protection and Competitive
Behavior,” Econometrica, 50, 671–688.
[31] Rosenberg, D., A. Salomon, and N. Vieille (2013), “On Games of Strategic Experimentation,” Games and Economic Behavior, 82, 31–51.
[32] Rosenberg, D., E. Solan, and N. Vieille (2007), “Social Learning in One-Arm Bandit
Problems,” Econometrica, 75, 1591–1611.
[33] Thomas, C. (2011), “Experimentation with Congestion,” Unpublished.
[34] Wagner, P. (2012), “Learning From Strangers: Public Experimentation and Signalling,” Unpublished.
[35] Wagner, P. (2015), “Incentives for Innovation,” Unpublished.
44
Appendix
9
Proof of Lemma 1. Given an exit time ti
0, …rm i’s expected payo¤ at time 0 is given
by
pi;0
Z
ti
( v
t
ct) e
dt
e
ti
(1
cti
pi;0 ) cti
0
= pi;0 1
ti
e
c
v
(1
pi;0 ) cti :
(34)
The …rst order condition for optimality is thus
pi;0 e
tyi
v
c
(1
pi;0 ) c
8
>
< = 0 if ty > 0;
i
>
:
0 if tyi = 0:
Di¤erentiate the objective function (34) twice with respect to ti shows that it is strictly
concave. Thus, …rm i’s expected payo¤ has a unique maximizer given by (4). Moreover,
the continuation values of …rms i and j at time t are given by
W (pi;t ; ) = pi;t 1
e
n
o
max tyi t;0
c
v
(1
n
pi;t ) c max tyi
o
t; 0 ;
(35)
and
W (pi;t ; ) = pi;t
Z
tyi ^t
(1
t
= pi;t 1
e
) ve
n
o
max tyi t;0
(s t)
ds
(1
) v;
(36)
respectively. Plugging (4) into (35) and (36) results in (5) and (6).
Fact 1 gives the expressions for …rm i’s expected payo¤ and the …rst derivative of the
expected payo¤ given that …rm i uses the exit time ti and …rm j uses the exit time tj ,
where ti and tj are not necessarily equilibrium choices. These expressions are useful in the
proof of Proposition 2 and are provided here for easy reference.
Fact 1. Let i; j 2 f1; 2g, given that …rm j uses the exit time tj , …rm i’s expected payo¤
45
from using an exit time ti is given by
Vi (ti ; tj ) =
8 R
tj
>
)) v
>
>
0 f(pi;t + pj;t (1
>
>
R tj
>
R
>
t
>
< +e 0 (p1;s +p2;s ) ds tji (pi;t
Rt
0 (p1;s +p2;s )
cg e
v
Rt
tj
c) e
ds
pi;s ds
dt
if ti
tj
dt
(37)
Rt
R ti
>
>
0 (p1;s +p2;s ) ds dt
f(p
+
p
(1
))
v
cg
e
>
i;t
j;t
0
>
>
Rt
>
Rt
R tj
>
>
ti pj;s ds
: +e 0 i (p1;s +p2;s ) ds
dt
p
(1
)
ve
ti j;t
if ti < tj
Its …rst derivative is given by
8
>
>
>
>
<
@Vi (ti ; tj )
=
>
@ti
>
>
>
:
e
R ti
0
pi;s ds
e
R tj
ti
R ti
0
R tj
pj;s ds (p
0
(p1;s +p2;s ) ds fp
pj;t (1
) ve
Rt
ti
v
i;ti
i;ti
c)
if ti
( v
pj;s ds
dt)
tj ;
(38)
if ti < tj :
cg
which is continuous at ti = tj .
Proof of Proposition 1. In text.
Proof of Proposition 2. Without loss of generality, suppose p1;0
p2;0 . We would like
to show that an equilibrium (t1 ; t2 ) with t1
t2 exists. Suppose p1;0
v
c, we claim
that t1 = t2 = 0 is an equilibrium. Since p1;0
p2;0 , Lemma 1 implies that both …rms …nd
it optimal to exit at time 0 given that the opponent exits at time 0. Suppose p1;0
we claim that there exists an equilibrium (t1 ; t2 ) with t1
solution to p1;t1
v > c,
t2 , where t1 > 0 is the unique
v = c and pinned down by (8). This is done in two steps. 1. Given that
…rm 1 exits at time t1 , we show that …rm 2’s best response is to exit at some t2 2 [0; t1 ]. 2.
Given t2 , we show that t1 is indeed …rm 1’s best response.
1. Since p1;0
p2;0 , we have for all t > t1 ,
p2;t
v
p1;t
v < p1;t1
v = c:
Thus, by (38), the …rst derivative of …rm 2’s payo¤ function
@V2 (t;t1 )
@t
is strictly neg-
ative for all t > t1 . Since the payo¤ function is continuous, a best response t2 exists
but t2 2
= (t1 ; 1).
46
2. Given t2 2 [0; t1 ], we need to show that t1 is indeed a best response for …rm 1.
Di¤erentiate (38), we have, for all t 2 [t2 ; 1),
@ 2 V1 (t; t2 )
=
@t2
as
v > p1;t1
p1;t e
Rt
0
p1;s ds
R t2
0
p2;s ds
(
v
c) < 0;
@V1 (t;t2 )
jt=t1
@t
@V1 (t;t2 )
for t
@t
v = c. Since V1 ( ; t2 ) is strictly concave on [t2 ; 1) and
0, t1 maximizes V1 ( ; t2 ) on [t2 ; 1). Next, consider the derivative
=
2
[0; t2 ], we have
@V1 (t; t2 )
@t
Rt
0 (p1;s +p2;s )
= e
Rt
0 (p1;s +p2;s )
e
Rt
0 (p1;s +p2;s )
e
=
ds
fp1;t ( v
p2;t 1
e
(t2 t) (1
) v)
cg
ds
fp2;t ( v
p1;t 1
e
(t2 t) (1
) v)
cg
ds
fp2;t ( v
p1;t 1
e
(t1 t) (1
) v)
cg
@V2 (t; t1 )
;
@t
where the …rst inequality follows from the fact that p1;0
the fact that t1
p2;0 and the second from
t2 . This implies that the exit time t2 is at least as good as any
t 2 [0; t2 ) for …rm 1. To see this, consider any t 2 [0; t2 ), then
V1 (t2 ; t2 ) V1 (t; t2 ) =
Z
t2
t
@V1 (s; t2 )
ds
@s
Z
t2
t
@V2 (s; t1 )
ds = V2 (t2 ; t1 ) V2 (t; t1 )
@s
0;
where the inequality follows from the fact that t2 maximizes V2 ( ; t1 ). Since V1 (t1 ; t2 )
V1 (t2 ; t2 ), t1 is indeed a best response for …rm 1.
De…nition 9 The function fi : R ! R is de…ned by
fi (x)
p1;0 p2;0 ( v (2
+ pi;0 (1
(1
1)
c) x2
pj;0 )
p1;0 ) (1
2v
+ (1 2 ) c v
v c
p2;0 ) c
!
c
cpj;0 (1
!
pi;0 ) x
(39)
47
i
;
1
. By Proposition 1, the composite funclog x
@t
h
i
represents the M N B of …rm i for t 2 0; tyj in an equilibrium in which …rm i
which coincides with
tion fi e
t
@Vi t;tyj
jt=
h
on e
1
tyj
exits earlier than …rm j. Notice, however, that for t 2 tyj ; 1 , fi e
the M N B of …rm i in equilibrium. In fact, for t 2 tyj ; 1 , fi e
Proof of Lemma 2.
t
t
Without loss of generality, suppose p1;0
does not represents
>
@Vi t;tyj
@t
.
p2;0 . Since t2 > t1
0,
it is impossible to have t2 = 0. Since t2 > 0, by Proposition 1,
e
Since p1;t2
t2
=
1
p2;0
p2;0
c
v
c
:
p2;t2 , we have
f1 e
t2
= (p1;0
= e
=
0
1
v
p2;0
p2;0
(p1;s +p2;s ) ds
@V1 (t; t2 )
jt=t2
@t
and
We consider the two cases,
1. Suppose
R t2
p2;0 )
p1;t2
t2 ; 1
c
c
0:
> , separately.
, by (39), f1 (x) is concave. f1 e
most one x0 2 e
v
c
v
t2
0 imply that there is at
such that f1 (x0 ) = 0. (Otherwise, the derivative
@f1 (x)
@x
must changes from negative to positive between two roots, violating the concavity of
f1 (x).) Moreover, for such x0 , we must have
@ 2 V1 (t; t2 )
jt=t0 =
@t2
where t0
1
e
t @f1 (x)
@x
jx=x0 > 0;
log x0 . Therefore, t0 , if exists, is a local minimum of …rm 1’s payo¤
function. Therefore, any t 2 (0; t2 ) fails to satisfy either the …rst order condition for
optimality or the second order condition. We must have t1 = 0.
48
2. Suppose
v+c
2 v .
>
t2
f1 e
@f1 (x)
0
@x jx=x
By (39), f1 (x) is strictly convex in x. Since
0>
> 0 for some x0 2 0; e
c (1
t2
p1;0 ) (1
p2;0 ) = f1 (0) ;
. Strict convexity means that the slope of
t2 ; 1
f1 (x) must increase with x. Thus, for all x 2 e
t2 ; 1],
Thus, for all x 2 (e
f1 (x) > f1 e
t2
, we must have
0. Since f1 e
t
=
@f1 (x)
> 0.
@x
@V1 (t;t2 )
for
@t
t 2 [0; t2 ]. Any t 2 [0; t2 ) fails to satis…es …rm 1’s …rst order condition (9), which
contradicts the assumption of the nonregular equilibrium.10
Proof of Lemma 3.
Without loss of generality, suppose p1;0
p2;0 and a nonregular
equilibrium (0; t2 ) exists, we would like to show that a regular equilibrium (t1 ; 0) exists.
Notice that both t1 and t2 are uniquely pinned down by (8). Compare the …rst derivative of
…rm 2’s payo¤ function in the regular equilibrium (i.e. the M N B) and the …rst derivative
of …rm 1’s payo¤ function in the nonregular equilibrium, by (38), we have, for all t 2 [0; t2 ],
@V2 (t; t1 )
@t
= e
e
=
Rt
0 (p1;s +p2;s )
Rt
0 (p1;s +p2;s )
ds
fp2;t
v
W (p1;t ; )
cg
ds
fp1;t
v
W (p2;t ; )
cg
@V1 (t; t2 )
@t
Since (0; t2 ) is an equilibrium, the exit time 0 maximizes …rm 1’s payo¤. This implies that
the exit time 0 maximizes …rm 2’s payo¤ in the regular equilibrium. To see this, consider
any exit time t 2 [0; t2 ], then
V2 (t; t1 )
10
V2 (0; t1 ) =
Z
0
t
@V2 (s; t1 )
ds
@s
Z
0
t
@V1 (s; t2 )
ds = V1 (t; t2 )
@s
Notice that, in this case, a nonregular equilibrium fails to exist.
49
V1 (0; t2 )
0:
Thus, the exit time 0 weakly dominates any exit time t 2 (0; t2 ]. Next, for all t > t2 ,
@V2 (t; t1 )
@t
= e
e
Rt
0
Rt
0
p2;s ds
p2;s ds
R t1 ^t
0
R t1 ^t
0
p1;s ds
fp2;t
p1;s ds
fp2;t
v
W (p1;t ; )
v
cg
cg
< 0;
which implies that the exit time 0 weakly dominates any exit time t 2 (0; 1). Thus, the
exit time 0 is a best response for …rm 2. By Proposition 1, t1 = ty1 ( ) is a best response
for …rm 1.
Proof of Lemma 4.
We prove the lemma by showing that E ( ) is the intersection of
two convex sets. First, we claim that
E( ) =
o
\ n
(p1 ; p2 ) 2 [0; 1]2 : IN B (pi ; pj ; ) > 0
i2f1;2g
=
\
i2f1;2g
(
(p1 ; p2 ) 2 [0; 1]2 : pi >
v
c
W (pj ; )
)
To see that, consider (p1 ; p2 ) 2 E ( ). Then, by de…nition of E ( ), if pi
pj , then
IN B (pj ; pi ; ) > 0. But this also implies
IN B (pi ; pj ; )
= pi
v
W (pj ; )
c
pj
v
W (pi ; )
c
= IN B (pj ; pi ; )
> 0:
Conversely, any element (p1 ; p2 ) in the intersection must have either p1
which means that (p1 ; p2 ) 2 E ( ). Let
G (p) =
v
50
c
:
W (p; )
p2 or p1 < p2 ,
The convexity of E ( ) then follows from
v
v c
2
00
G (p) =
p
v
2
)2 v 2 c
(1
v c
v c
(1
> 0:
3
)v
Proof of Lemma 5. Suppose the set E ( ) is nonempty, there exists (p01 ; p02 ) 2 E ( ). If
p01 > p02 , then (p01 ; p01 ) 2 E ( ). Similarly, if p01 < p02 , then (p02 ; p02 ) 2 E ( ). This analysis
shows that the 45 line must intersect with E ( ) if it is nonempty. Suppose
, then for all p
would like to show that for all p < 1, (p; p) 2
= E ( ). Suppose
, we
c
v,
W (p; ) = 0, so
IN B (p; p; ) = p
Suppose p 2 [
c
v ; 1),
then,
v
v
c
0.
v
v
c
c
(1
c > 0, we have
IN B (p; p; ) = p
p
v
which is quadratic function of p. We have, for all p 2
@ 2 IN B (p; p; )
=
@2p
2
2
(1
v
c
)v
v;1
) v2
c
c
,
< 0:
and
IN B
c
v
;
c
v
;
= 0;
IN B (1; 1; ) =
Thus, if
c
v;p
0
@IN B (
c
v
@p
;
c
v
;
)
(2
1) v
c < 0:
c
> 0, there is a unique point p0 2
v;1
such that for all p 2
, (p; p) 2 E ( ). Otherwise, E ( ) is empty.
@IN B (p; p; )
jp=
@p
c
v
=
v
51
c
v
c
(1
)v
which is strictly increasing in
and equals to 0 when
= .
Consider an regular equilibrium (t1 ; t2 ). The strong …rm’s
@Vw (t;tS )
equilibrium exit times is characterized by (8). Since E ( ) = , for all t 0,
0.
@t
@Vw (t;tS )
Since (23) is a quadratic function with nonzero coe¢ cients,
= 0 in at most one
@t
Proof of Proposition 3.
point. Thus, tw = 0.
Proof of Lemma 6. Without loss of generality, suppose p1
p2 > 0, plugging (6) into
(13), we have
IN B (p2 ; p1 ; ) =
Suppose p1 <
c
v,
then
@IN B (p2 ; p1 ; )
@
8
>
< p2
>
:
v c
v c
p2
@IN B(p2 ;p1 ; )
@
= p2
v
> p2
v
= p2
v
p2
v
= p2
p1
v
p1
(1
v
)v
c if p1 >
c
v;
if p1
c
v:
c
= p2 v > 0. Suppose p1 >
c
v,
we have
v 2 (1
)c
p1 v c
v
2 +
v c
( v c)
c
v 2 (1
)c
p1 v c
1
v
+
v ( v c)2
v c
vc 1
p1 v c
+
v
v c
v c
!
pc
1
vc
p
v
c
1
pc
pc v +
v
v c
v
c
v
v
(1
p1 )
v c
v
v c
0
The …rst inequality follows from the fact that p1 >
c
v
and the second inequality follows
from the fact that the second term in the bracket is decreasing in
pc
v.
and our premise that
Proof of Proposition 4. Consider an regular equilibrium (t1 ; t2 ), suppose tS > 0 under
the patent system . If tS > 0, (8) implies
@tS
=
@
v
v
52
c
> 0:
(40)
Thus, tS increases with . If tw > 0, then, by Proposition 3,
> . Consider an increase
and let tw ( ) and tw ( 0 ) denote the equilibrium exit
@Vw (t;tS )
times of the weak …rm. We claim that tw ( ) < tw ( 0 ). By Lemma 6,
increases
@t
of the patent strength from
everywhere under
0.
to
0
Thus, the …rst order condition cannot be satis…ed at tw ( ). Moreover,
for any t < tw ( ),
V w tw ( ) ; t S ; 0
Vw t; tS ;
Z tw ( )
@Vw (t; tS ; 0 )
dt
=
@t
t
Z tw ( )
@Vw (t; tS ; )
>
dt
@t
t
= Vw (tw ( ) ; tS ; )
0
Vw (t; tS ; )
0:
Thus, any t
0.
tw ( ) cannot be optimal under the new patent system
We must have
tw ( ) < tw ( 0 ).
Consider an regular equilibrium (t1 ; t2 ), since pw;tS
pS;tS ,
@Vw (t;tS )
@Vw (t;tS )
by (38), for all t > tS , we have
< 0. By (39), fw e tS =
jt=tS
0.
@t
@t
Proof of Proposition 5.
Suppose
< , fw is strictly concave. Consider the three cases in Proposition 5.
1. Suppose
(p1;0 ; p2;0 ) \ E ( ) = , so that for all t 2 [0; tS ], fw e
t
=
@Vw (t;tS )
@t
< 0.
The weak …rm must exit at time 0.
2. Suppose (p1;0 ; p2;0 ) 2 E ( ), so
fw e
tS
@Vw (t;tS )
jt=0
@t
= fw (1) > 0. Since fw is quadratic,
0 implies that it has a unique root on e
so that fw e
tr
= 0. We must have, for all x 2 (e
tS ; 1
. Let e
tr
be the root
tr ; 1],
fw (x) > 0 and for all
@Vw (t;tS )
x 2 [e tS ; e tr ), fw (x) < 0. This means that for all t < tr ,
> 0 and for
@t
@Vw (t;tS )
all t > tr ,
< 0. Thus, the weak …rm must exit at time tr when the posterior
@t
reaches the set @E ( ).
3. Suppose (p1;0 ; p2;0 ) 2
= E ( ) and
fw (1)
(p1;0 ; p2;0 ) \ E ( ) 6=
0 and but fw (x) > 0 for some x 2 e
53
tS ; 1
. Thus, fw e
tS
0,
. Since fw is quadratic, there
must be exactly two roots on e
tS ; 1
. Let e
tr1
tr2
and e
be the two roots and
tr1 < tr2 . We must have,
8
>
<0
>
>
>
>
>
>
>
=0
>
>
<
t < t r1 ;
t = tr 1 ;
@Vw (t; tS )
> 0 tr 1 < t < tr 2 ;
>
@t
>
>
>
>
>
=0
t = tr 2 ;
>
>
>
>
:
<0
t > t r2 :
Thus, Vw ( ; tS ) has two local maxima, 0 and tr2 . If 0 is the global maximum, the
weak …rm exits at time 0. If tr2 is the global maximum, the weak …rm exits when the
posterior reaches the set @E ( ) for the second time.
pS;tS ,
Consider an regular equilibrium (t1 ; t2 ), since pw;tS
@Vw (t;tS )
by (38), for all t > tS , we have
< fw e tS
, by (39), fw is
0. Suppose
@t
Proof of Proposition 6.
convex. Consider the two cases in Proposition 6.
1. Suppose (p1;0 ; p2;0 ) 2
= E ( ), so fw (1)
0. Suppose
>
, strict convexity im-
tS ; 1
, fw (x) < max fw e tS ; fw (1)
0. Thus, for
@Vw (t;tS )
all t 2 (0; tS ) [ (tS ; 1),
< 0. Suppose
= , fw is linear, fw (0) =
@t
@Vw (t;tS )
< 0. In both
(1 p1;0 ) (1 p2;0 ) c < 0 implies that for all t 2 (0; 1),
@t
plies that for all x 2 e
cases, we must have tw = 0.
2. Suppose (p1;0 ; p2;0 ) 2 E ( ), so fw (1) > 0. Since fw is quadratic, fw e
0 implies that it has a unique root on [e
fw e tr
@Vw (t;tS )
@t
tS ; 1).
= 0. This means that for all t < tr ,
Let e
@Vw (t;tS )
@t
tr
tS
be the root so that
> 0 and for all t > tr ,
< 0. Thus, the weak …rm must exit at time tr when the posterior reaches
the set @E ( ).
Proof of Proposition 7. In text.
54
Fact 2. Given a pair of exit times (t1 ; t2 ), where ti
tj , the planner’s payo¤ is given
by
VP (t1 ; t2 )
Z
= p1;0 p2;0
tj
2 (v
2ct) e
2 t
dt +
ti
(v
pj;0 )
Z
tj
(v
t
2ct) e
dt +
dt
(ti +tj )
e
pi;0 ) pj;0
Z
Z
ti
(v
t
c (t + tj )) e
dt
ti
e
tj
(v
2ct) e
t
dt
tj
e
(ti + tj ) c
(1
p1;0 ) (1
!
(ti + tj ) c
p2;0 ) (t1 + t2 ) c
0
= p1;0 p2;0 1
(1
c
v
pj;0 )
1
e
tj
v
pi;0 ) pj;0
1
e
tj
v
+pi;0 (1
+ (1
(t1 +t2 )
e
p1;0 ) (1
2c
tj
+ e
2c
e
tj
e
(ti
ti
v
c
tj ) c
p2;0 ) (t1 + t2 ) c
Its …rst derivative is given by
8
>
>
pi;0 e
>
>
<
@VP (t1 ; t2 )
=
>
@ti
>
>
>
:
Notice also that
ti
(1
pj;0 e
@VP (t1 ;t2 )
@ti
Proof of Lemma 7.
tj
pj;0 e
tj
+1
+1
pj;0
v
pi;0 ) pj;0 e
pj;0
pi;0 e
ti
ti
c
+ (1
pj;0 ) c (tj
+1
pj;0 c
( v
c)
(1
ti )
pi;0 ) c
if ti
tj
if ti > tj
(41)
is continuous at t1 = t2 .
Given t0j
0, we …rst show that there is at most one tzi
0 that
satis…es
@VP (t1 ; t2 )
jt =tz ;t =t0 = 0:
i
i j
j
@ti
(42)
Suppose such a tzi exists, if tzi 2 [0; t0j ), di¤erentiate (41), we have
=
=
@ 2 VP (t1 ; t2 )
jt =tz ;t =t0
i
i j
j
@t2i
@VP (t1 ; t2 )
jt =tz ;t =t0
i
i j
j
@ti
(1
p1;0 ) (1
p2;0 ) c
(1
pi;0 e
p1;0 ) (1
tzi
55
(1
p2;0 ) c
pj;0 ) c < 0:
pi;0 e
tzi
(1
!
(ti + tj ) c
tj
0
+ (1
(t+tj )
c (t + tj )) e
tj
0
+pi;0 (1
Z
pj;0 ) c
(43)
If tzi 2 t0j ; 1 , then
@ 2 VP (t1 ; t2 )
jt =tz ;t =t0 =
i
i j
j
@t2i
Thus, for all t 2 [0; tzi ),
VP (t1 ;t2 )
jti =t;tj =t0j
@ti
+1
tzi
pj;0 pi;0 e
( v
> 0 and for all t 2 (tzi ; 1),
0. If (42) has no solution, then, limti !1
VP (t1 ;t2 )
jti =t;tj =t0j
@ti
t0j
pj;0 e
VP (t1 ;t2 )
jtj =t0j
@ti
c) < 0:
VP (t1 ;t2 )
jti =t;tj =t0j
@ti
< 0 implies that for all t
<
0,
< 0. Thus, tzi = 0 satis…es the requirement in this case.
Therefore, tzi (tj ) is well-de…ned for all tj 0. Berge maximum theorem implies that it
i
h
@tzi (tj )
y
= 0. If tzi (tj ) 2 (0; tj ),
is continuous. Consider tj 2 0; tj . If tzi (tj ) 2 (tj ; 1), then @t
j
then the implicit function theorem implies
@tzi (tj )
@tj
@ 2 VP (t1 ;t2 )
@tj @ti jti =tzi (tj );tj =t0j
=
@ 2 VP (t1 ;t2 )
jt =tz (t );t =t0
@t2i
i
j
i j
j
0;
which follows from (43) and
@
@tj
@VP (t1 ; t2 )
@ti
y
as pj;0 e tj ( v
i
h
y
tj 2 0; tj .
=
c) = (1
pi;0 e
ti
pj;0 e
tj
( v
c)
(1
pj;0 ) c
0.
(44)
pj;0 ) c. Thus, since tzi is continuous, it is decreasing for all
Proof of Lemma 8. Without loss of generality, suppose p1;0
p2;0 and suppose by way of
contradiction that t2 > t1 , then consider an alternative strategy with (t01 ; t02 ) = (t2 ; t1 ),
then
VP (t2 ; t1 )
= (p1;0
p2;0 )
VP (t1 ; t2 )
e
t1
e
t2
v
c
+e
t1
(t2
t1 ) c
> 0
Thus, it is without loss to consider only exit times with which the strong …rm exits weakly
later.
56
Proof of Proposition 8. In text.
By Proposition 1 and Proposition 8, the strong …rm in a regular
Proof of Lemma 9.
y
equilibrium (t1 ; t2 ) never over-experiments, i.e. tS < tS = tS . By Lemma 7, tzw (tS )
tzw (tS ) = tw . Thus, tw < tw implies tw < tzw (tS ). The weak …rm under-experiments.
By Lemmas 4 and 6, for all
Proof of Lemma 10.
0
;
1
2;
2
,
E ( 0 ). Thus, we only need to show that E ( )
E ( ) [ @E ( )
0
>
implies that
E . Consider the
di¤erence
^ (p; 1)
W
D (p) = v
v
W (p; ) :
Comparing (13) and (28), it’s easy to see that we only need to show that for all p 2 [
c
v ; 1),
D (p) > 0. Using (5) and (6), direct di¤erentiation yields
c
D00 (p) =
p2 (1
c
Strict concavity of D then implies for all p 2 [
D (1) = v
c
=
if D
c
v
> 0, then for all p 2
( v
(1
c
v
; D (1) . Since
) v)
, D (p) > 0.
c
D
v
^
W
= v
c
v
;1
2c
v
v+c
= v
=
D (p) > min D
1) v = 0;
(2
2c
v+c ; 1
v ; 1),
c
v
< 0:
p)
1
2
c
v+c
v
c
v+
+ 1
c
c
W
v
;
2c
v+c
v c
v+c
c
log 2
> 0
57
1
log
2c
v+c
2c
v+c
1
2
c
v
c
!!
v+c
v
2 v
Proof of Lemma 11. In text.
Proof of Proposition 9. In text.
Proof of Lemma 12. In text.
Proof of Lemma 13. Consider each part of Lemma 13 separately.
1. To prove the …rst part of Lemma 13, suppose
@VP (t1 ; t2 )
@tw
= e
e
R tw
0
R tw
0
= 1, then
(p1;s +p2;s ) ds
pw;tw (v
(p1;s +p2;s ) ds
pw;tw v
UPw (tw ; tS ))
c
c
= 0:
where the inequality follows from (27) and the fact that the welfare-maximizing
equilibrium must be regular, i.e.
@VP (t1 ;t2 )
jtw =tw ;tS =tS
@tw
tw
tS
tS .
Next, suppose
< 1 and
> 0, consider the derivative
@VP (t1 ; t2 ) @tS
@VP (t1 ; t2 ) @tw 11
dVP (t1 ; t2 )
=
+
;
d
@tS
@
@tw
@
we claim that it must be strictly positive, so that a small increase in
strictly
increases the total welfare. Since
< 1, tS < tzS (tw ) = tS . By Lemma 7, this
@VP (t1 ;t2 )
means
> 0. Moreover, by Lemma 12, tS > 0. Proposition 4 then implies
@tS
dVP (t1 ;t2 )
@t
w
that @ S > 0 and @t
0. Thus,
> 0, contradicting the assumption that
@
d
is optimal.
2. To prove the second part of Lemma 13, suppose
= 1, then the assumption of
nonextreme prior and Proposition 1 implies that IN B pw;tw ; pS;tw ;
suppose
< 1 and IN B pw;tw ; pS;tw ;
have tw = 0. If there exists an
0
< 0. Proposition 1 implies that we must
2 ( ; ) such that under
11
= 0. Next,
0,
two regular equilibria
If the weak …rm’s exit time in the welfare maximizing equilibrium is discontinuous at
, we replace
by its right hand limit, which always exists. The same argument suggests that a small increase in
strictly increases the total welfare.
@tw
@
58
coexist, then, Propositions 7 and 9 imply that
such that tw = 0 for
2 [ 12 ;
+
). Thus,
tS < tzS (tw ) = tS . By Lemma 7, this means
12, tS > 0, Proposition 4 implies that
@tS
@
<
0
and that there exists a
>0
@tw
= 0. Moreover,
< 1 implies that
@
@VP (t1 ;t2 ;pS;0 )
> 0. Moreover, by Lemma
@tS
> 0. As a result,
dVP (t1 ; t2 )
@VP (t1 ; t2 ) @tS
=
> 0:
d
@tS
@
Thus,
2
cannot be optimal. Next, suppose the regular equilibrium is unique for all
1
2; 1
. Then, Berge maximum theorem implies that the equilibrium exit time is
continuous in
. The fact that tw = 0 and IN B pw;tw ; pS;tw ;
< 0 implies that
0 must remain an equilibrium exit time for the weak …rm after a small increase in
dVP (t1 ;t2 )
@VP (t1 ;t2 ) @tS
. Using the same argument, we conclude that
=
d
@t
@ > 0 and
S
cannot be optimal. Thus, IN B pw;tw ; pS;tw ;
patent
= 0 must hold under the optimal
.
Proof of Proposition 10. In text.
Proof of Proposition 11. In text.
Proof of Proposition 12. Suppose pS;0 > pw;0 , we claim that the derivative
dVP (t1 ; t2 )
@VP (t1 ; t2 ) @tS
@VP (t1 ; t2 ) @tw
=
+
d
@tS
@
@tw
@
exists at
6, for all
i 2 f1; 2g,
= 1 and is strictly negative. It follows that
(45)
= 1 is suboptimal. By Propositions
2 [ ; 1], the regular equilibrium is unique. By the implicit function theorem,
@ti
@
exists for all
> . Thus, the derivative (45) exists at
= 1. Moreover,
the assumption of nonextreme prior and Proposition 1 imply that i 2 f1; 2g, ti > 0 when
= 1. Proposition 4 then implies that
@ti
j
@
=1
59
> 0:
Thus, the derivative (45) exists at
= 1. Since tS = tS = tzS (tw ) > 0,
@VP (t1 ;t2 )
@tS
= 0.
Moreover,
@VP (t1 ; t2 )
@tw
R tw
= e
0
R tw
< e
0
(p1;s +p2;s ) ds
pw;tw (v
(p1;s +p2;s ) ds
pw;tw v
UPw (tw ; tS ))
c
c
= 0:
where the inequality follows from the fact that tS = tS > tw . Thus,
designer gains by reducing
dVP (t1 ;t2 )
d
< 0. The
for a small amount from 1.
Proof of Proposition 13. As pointed out in the main text, we only need to show that
the derivative
dVP (t1 (
=
; pS;0 ) ; t2 ( ; pS;0 ) ; pS;0 )
dpS;0
@VP (t1 ; t2 ; pS;0 ) @VP (t1 ; t2 ; pS;0 ) @tS
@VP (t1 ; t2 ; pS;0 ) @tw
+
+
@pS;0
@tS
@pS;0
@tw
@pS;0
is strictly positive for all pS;0 > pw;0 . Consider the …rst term, we have
@VP (t1 ; t2 ; pS;0 )
@pS;0
n
co
c
+ 1 e tw
+ pw;0 e tw (tS
= pw;0 e tw 1 e tS v
n
o
c
+ (1 pw;0 ) 1 e tS v
+ ctS + c e tw (1
tw )
tw ) c
> 0;
where we have used the fact that for all x 2 R, ex
the second term, since tS
1+x and the fact that tS > 0. Consider
@VP (t1 ;t2 ;pS;0 )
tzS (tw ) = tS , by Lemma 7,
0. Moreover, By
@t
S
Lemma 12, tS > 0, (8) implies
@tS
1
=
> 0:
@pS;0
pS;0 (1 pS;0 )
The second term is also positive. Consider the third term, by the …rst part of Lemma 13,
60
@VP (t1 ;t2 ;pS;0 )
@tw
0. By the second part of Lemma 13, (9) holds at equality. By Propositions
6 and 11, the regular equilibrium is unique under the optimal patent. The implicit function
theorem implies that
@tw
=
@pS;0
@IN B (pw;t ;pS;t ;
@pS;t
)
@IN B (pw;t ;pS;t ;
@t
The weak …rm’s second order condition implies that
@IN B (pw;t ; pS;t ;
@pS;t
)
jt=tw
jt=tw =
)
@pS;tw
@pS;0
:
jt=tw
@IN B (pw;t ;pS;t ;
@t
v
pw;tw
v
where we have used the fact that tS > 0 so that
c
)
(1
v > pS;tS
jt=tw < 0. Moreover,
) v < 0;
v = c. Moreover,
di¤erentiating (1) yields
@pS;tw
=
@pS;0
(pS;0 e
e
tw
tw
+1
2
pS;0 )
> 0:
The third term is also positive.
Proof for Example 3. As in the proof of Proposition 13, we can show that the derivative
dVP (t1 (
=
; pw;0 ) ; t2 ( ; pw;0 ) ; pw;0 )
dpw;0
@VP (t1 ; t2 ; pw;0 ) @VP (t1 ; t2 ; pw;0 ) @tS
@VP (t1 ; t2 ; pw;0 ) @tw
+
+
@pw;0
@tS
@pw;0
@tw
@pw;0
(46)
is strictly negative for some (p1;0 ; p2;0 ) and apply the Envelope Theorem to obtain the
desired conclusion. Consider a sequence of nonextreme priors
and pn2;0 !
c
v,
pn1;0 ; pn2;0
with pn1;0 ! 1
we claim that (46) is strictly negative for n large enough. To do so, we …rst
show that for n large enough, t2n = 0.
Claim 1 For n large enough, the weak …rm exits at time 0 in the regular equilibrium under
the optimal patent
n,
i.e. 9N 2 N such that for all n
N , t2n = 0.
Suppose by way of contradiction that for each n 2 N, t2n > 0. By Propositions 6 and
11, for each n 2 N, the optimal patent
n
is larger than
61
and the regular equilibrium is
unique. For each n 2 N, the optimal patent
n
satis…es the …rst order condition
dVP (t1n ; t2n )
@VP (t1n ; t2n ) @t1n @VP (t1n ; t2n ) @t2n
=
+
d
@t1n
@
@t2n
@
Since for all n 2 N, IN B pn2;t ; pn1;t ;
and t2n > 0 implies that
n
n
pn2;t v
c for all t
0:
(47)
0, the fact that pn2;0 !
c
v
! 1. Direct di¤erentiating (8) yields,
@t1n
j
n!1 @
lim
=
n
=
v
v
c
> 0:
(9) and the implicit function theorem imply
@t2n
lim
j
n!1 @
Since
n
=
n
=
n
@IN B (pn
2;t ;p1;t ;
@
lim
n
n!1 @IN B (pn
2;t ;p1;t ;
@t
)
jt=twn ; = n
2v
=
> 0:
)
v c
n
jt=twn ; = n
! 1, we have
@V1 (t; t2n )
@VP (t; t2n )
jt=t1n = lim
jt=t1n = 0;
n!1
n!1
@t
@t
lim
Moreover, pn2;0 !
c
v
implies that t2n ! 0. As pn1;0 ! 1, t1n ! 1, we have, by (26),
@VP (t1n ; t)
jt=t2n =
n!1
@t
lim
v
c
c
c < 0:
(48)
Thus, (47) cannot be satis…ed for n large enough. We have reached a contradiction.
Claim 2 For n large enough, the derivative
negative under the optimal patent
dVP (t1 ;t2 ;p2;0 )
j(p1;0 ;p2;0 )=(pn ;pn )
dp2;0
1;0 2;0
(46) is strictly
n.
By Claim 1, for n large enough, t2n = 0 under the optimal patent
. Take any such n
and drop the superscript/subscript from this point on. Consider the …rst term in (46), we
62
have
@VP (t1 ; t2 ; pw;0 )
@pw;0
= pS;0 1 e (t1 +t2 )
pS;0
1
tw
e
v
+ (1
pS;0 )
+ (1
pS;0 ) (t1 + t2 ) c
1
c
v
e
2c
tw
v
+ e
2c
tw
e
tS
e
tw
c
v
(tS
tw ) c
= 0
@t
1
Moreover, by (8), the strong …rm’s equilibrium exit time is independent of pw;0 , so @p2;0
= 0.
@VP (t1 ;t2 ;pS;0 )
< 0. By the
The second term is also zero. Consider the third term, by (48),
@t
2
second part of Lemma 13, (9) holds at equality. By Propositions 6 and 11, the equilibrium
is unique under the optimal patent. The implicit function theorem implies that
@p2;t
@IN B(p2;t ;p1;t ; )
jt=t2 @p2;02
@p2;t
@IN B(p2;t ;p1;t ; )
jt=t2
@t
@t2
=
@p2;0
The weak …rm’s second order condition implies that
@IN B (p2;t ; p1;t ;
@p2;t
as W p1;t2 ;
(1
)v
)
jt=t2 =
v
@IN B(p2;t ;p1;t ;
@t
W p1;t2 ;
v. Moreover, since t2 = 0,
@p2;t2
= 1 > 0:
@p2;0
The third term is strictly negative.
Proof of Proposition 14. In text.
63
:
)
jt=t2 < 0. Moreover,
> 0;