Runner-up Patents:
Is monopoly inevitable?
∗
Job Market Paper
Emeric Henry
†
November 11, 2005
Abstract
Exclusive patents sacrifice product competition to provide firms incentives to innovate. We characterize an alternative mechanism whereby later inventors are allowed
to share the patent if they discover within a certain time period of the first innovator.
These runner-up patents may reduce the research incentives but we show that under
very general conditions the benefits from increased competition on the product market
will outweigh these potential costs and render the system socially beneficial. We thus
prove that an efficient tradeoff between research incentives and deadweight loss can
be achieved without granting monopolies. Furthermore, we demonstrate that the time
window where later inventors can share the patent should become a new policy tool
at the disposal of the designer. This instrument will be used in a socially optimal mix
with the breadth and the length of the patent and could allow sorting between more
or less efficient firms in a differentiated patent policy.
∗
I wish to thank David Baron, Nick Bloom, Michael Boskin, Tim Bresnahan, Larry Goulder, Jonathan
Levin, Petra Moser, Patrick Rey, Suzanne Scotchmer, Jean Tirole, Nageeb Ali, Ben Ho, Prakash Kannan,
Ryan Lampe, Enrique Seira, for their comments and especially Douglas Bernheim for his generous support
and advice. I also thank participants at the public economics and IO seminars at Stanford University. This
research was supported by the Leonard W. Ely and Shirley R. Ely Graduate Student Fund through a grant
to the Stanford Institute for Economic Policy Research.
†
Correspondence Information: Department of Economics, Stanford University, 579 Serra Mall, Stanford, CA 94305-6072, email: ehenry@stanford.edu, tel: 650 796 2154
1
1
Introduction
In 1976 Eugene Goldwasser, after 20 years of work, successfully identified and isolated
the erythropoiesis protein (EPO), whose function is to produce red blood cells in the body.
His work attracted the attention of 3 companies, Biogen, Amgen and Genetics Institute,
that started research programs to isolate the human gene responsible for the production of
EPO. In 1983, Amgen succeeded and filed for a patent at the USPTO. Less than a year later,
Genetics Institute published similar results in the journal NATURE but realized its success
was economically pointless as a patent had already been submitted. The drug Epogen, subsequently sold by Amgen, became one of the most successful in history, generating revenues
of 2 billion dollars per year, an annual treatment costing 5000 dollars per patient. The cost
of production represents approximately 5 percent of the revenues and the initial investment
in research did not exceed 170 million dollars. Amgen appears to be fully exercising its
monopoly power at the expense of patients and government medical assistance programs.
This series of events raises the following question: would it be socially beneficial to let later
inventors, such as Genetics Institute, share the patent if they discover soon enough after the
first innovator?
The current patent system grants socially costly exclusive rights to first inventors to encourage risky investments in research. We want to determine if an efficient tradeoff between
incentives to innovate and deadweight loss on the product market can be achieved without
necessarily awarding monopolies. We therefore study an alternative mechanism whereby if a
first inventor files for a patent at time t, any other inventor who discovers before time t + T
will be allowed to share the patent and compete on the product market. The goal of this
paper will be to characterize these runner-up patents and determine under what conditions
they will increase social welfare. We will consider the time window T where other inventors
can share the patent, that we call intertemporal breadth, as a new tool at the disposal of the
policy maker. Will such a tool be used in a socially optimal mix with existing instruments
such as the length and the breadth of the patent? Can it be used to sort between more or
less efficient firms in a differentiated patent policy? All these questions will be answered in
a model where firms first invest in research and then potentially compete on the product
market.
Recent empirical evidence on patent races suggests that the case of Epogen, that we
previously described, is not just anecdotal and that runner-up patents could therefore have
a significant impact. Although patent races are typically unobservable, Cohen and Ishii
(2005) use patent interference cases in the US to overcome this lack of data. A patent examiner has to declare an interference if two inventors file for the same invention within a
short time period (3 months for lesser inventions and 6 months for more important ones).
The patent office then investigates the case more thoroughly to determine the identity of
the first inventor. They show that from 1988 to 1994, 0.6 percent of granted patents were
2
sector
chemicals
drugs
electrical
mechanical
biotech
overall
Table 1: Evidence on patent races
percentage of awarded percentage of valuable
patents in interference patents in interference
(lower bound)
0.92
5.11
2.15
11.94
0.34
1.89
0.43
2.39
4.71
26.16
0.63
3.5
declared in interference. However, these numbers are percentages of the total numbers of
patents declared and only a fraction of these will prove valuable. In Table 1, we present a
lower bound on the percentage of valuable patents that are declared in interference, where
we define a valuable patent as one that is at least renewed to term. These estimates suggest
that at least 3.5 percent of runner-up patents would be shared. This lower bound could
be as high as 12 percent for drugs and 26 for biotechnology. Considering the fact that an
interference is declared only if the filing dates are within 6 months of each other and that
we used a conservative estimate to measure the value of a patent, we realize that runner-up
patents could have quite a significant impact.
Encouraged by these empirical findings, we study this alternative mechanism in a theoretical model where firms first race for an innovation and then potentially compete in the
product market. We show that under very general conditions runner-up patents will increase
social welfare. We can separate their overall impact into two channels: they will affect both
incentives to innovate and deadweight loss on the product market. In most cases the first
effect will be negative as incentives to innovate will decrease. However, the second effect
is always positive, as this mechanism induces socially beneficial competition in the product
market. We show that under a very general condition, this second effect will more than
compensate the first and runner-up patents will increase social welfare. In particular, the
condition will be satisfied if social surplus under competition is high enough or if the research is expected to succeed quickly. We show that in some cases, surprisingly, the proposal
can even increase the incentives to conduct research. Both effects will then raise social surplus and the benefits of the reform will therefore be guaranteed. We find that runner-up
patents increase research incentives when the cost of research is a flow that can be stopped
at any point rather than a sunk investment or if the firms are differentiating their innovations. Indeed, runner-up patents will increase the probability that different inventions will
be available on the market and if the gains from product diversity are sufficient, the overall
returns to innovating could increase.
3
In this article, we introduce a new patent policy tool, the length of time when later
inventors can share the patent with the first innovator, that we call intertemporal breadth.
In our previous results, the existing patent policy tools such as the length and the breadth
of the patent were supposed fixed. In the second part of the paper, we ask a different
question: is the intertemporal breadth a redundant tool or can it achieve a more efficient
tradeoff between incentives to innovate and deadweight loss than the existing instruments?
We therefore examine the optimal mix with the length and the breadth of the patent. We
show that under very general conditions the intertemporal breadth will prove more efficient.
In particular, in terms of research incentives, it is less costly than the length of the patent
because it encourages competition only among firms that are engaged in the patent race.
Finally we demonstrate that the intertemporal breadth could become a very useful tool
in a differentiated patent policy. A major concern in the literature on patent races is that the
designer is unable to sort between more or less efficient firms. We show that she can achieve
this by offering a menu of patents where the inventor could tradeoff a higher intertemporal
breadth against a longer length. An efficient firm would be willing to take the risk of allowing other firms to patent during a certain time period to obtain a longer protection. A well
designed menu could therefore increase the incentives to innovate for the more efficient firm.
Furthermore, the intertemporal breadth is the only tool that can allow such a sorting.
Two articles by Maurer and Scotchmer (2002) and La Manna et al (1989), build on
similar concerns about the current patent system. Maurer and Scotchmer show that, if later
inventors or imitators were also allowed to patent, under the condition that research costs
of an imitator were more than half as high as those of the initial inventor, the first inventor
would be able to sign licensing contracts that would deter entry by competitors. As acknowledged by the authors, in practice, this condition could be rarely satisfied: it should be much
less costly for a firm to “reinvent” an existing product than to discover initially. In a model
where firms do not endogenously choose the amount to invest in research, La Manna et. al.
examine the following setup: during the entire life of the patent other inventors are allowed to
patent as well. They indicate how much the life of the patent would need to be increased to
create the same research incentives as under the current system. As opposed to Maurer and
Scotchmer, they ignore practical constraints such as reverse engineering and the disclosure
rule (the fact that the patent is currently disclosed 18 months after filing). We characterize,
in a model with endogenous choice of research, a different mechanism and this for a number
of reasons. First, we believe that the time period T during which later inventors are allowed
to share the patent with the first inventor, should become a new policy tool distinct from
the length. In particular, we show that this time period T will most often be set strictly
smaller than the length in a socially optimal mix of instruments. Second we argue that it
can be essential to distinguish between these tools in a differentiated patent policy. We show
that offering menus where firms tradeoff length against intertemporal breadth allows to sort
between different types. Finally, we analyze this mechanism taking into account practical
4
constraints such as the disclosure rule.
Our paper is organized as follows. In section 2 we present the environment that we
will consider throughout the paper. In section 3 we consider cases where runner-up patents
decrease incentives to innovate. In section 4 we study situations where the converse can
happen. In these initial sections, the breadth and the length of the patent were taken as
given. We modify this in section 5 and study the optimal mix of the intertemporal breadth
with these existing instruments. Finally, in section 6 we examine differentiated patent policy
and the potential use of menus that tradeoff length against intertemporal breadth.
2
The general environment
We present in this section the general model. Throughout this article we will discuss how
varying the environment will affect our main results.
2.1
Players
There are three agents involved in this model:
- Two risk neutral firms.
- A policy maker.
We restrict our attention to two risk neutral firms for a number of reasons. First, we
point out that if firms were sufficiently risk averse, runner-up patents would always increase
incentives to innovate. Indeed, one of the main effects of this system is to provide positive
profits in a wider range of cases; profits can now be obtained by a firm that is not the first to
invent. Therefore, for a sufficient degree of risk aversion, such patents would always appear
more attractive than the standard patents. To abstract from the impact of risk aversion and
concentrate on other effects, we suppose that firms are risk neutral.
2.2
Timing
The sequence of events is the following:
1) Research decision: at date 0 firms decide amount (x1 , x2 ) to invest in research. They do
not observe each other’s choice.
2) Patent race: firms race to obtain the invention and the research is modeled as a Poisson
discovery process of parameter λi = h(xi ).
3) At a random date t, first inventor succeeds. He decides when to commercialize the invention.
4) If the second firm invents between t and t + T , they share the patent and compete in the
market. If not, the first inventor obtains the exclusive patent
5
5) At t + L, where L is the length of the patent, the patent expires. We suppose that after
that date, there is perfect competition and the inventors get zero profits.
Note that we will vary a number of parameters throughout the article and determine
how the results are affected. Specifically we will alternatively define the amount spent on
research x as a sunk investment or as a flow cost that can be interrupted at any point.
This will significantly change the results. We will also suppose in some cases that it is
easy to reverse engineer a commercialized product: this will impact the strategic timing of
commercialization.
2.3
Preferences
Firms maximize their expected profits. They receive positive profits if they obtain the
exclusive patent or are allowed to share it. Specifically, in phase 4, if the first inventor gets
the exclusive patent, he obtains monopoly profits πm per period, until the patent expires at
t + L. If firms share the patent, they compete and obtain duopoly profits πd . We do not
specify here the nature of competition and therefore do not fix the values of πm or πd but
we will discuss in the following sections how it affects the results.
More specifically, we present here the expression of expected profits of firm 1 if firms
invest amounts (x1 , x2 ) in research. Let t be the date at which firm 1 invents and t2 the date
at which the competitor succeeds. The present value of the benefits of firm 1 at date 0 are
then:
πm −rt
e
if t2 > t + T
r
πm −rt
[e − e−rt2 ] + πrd [e−rt2 ] if t < t2 < t + T
r
πd −rt
e
if t2 < t < t2 + T
r
Therefore the expected benefits are:
Z
Z +∞
πm +∞ −rt
−h(x1 )t
E[π] =
e h(x1 )e
h(x2 )e−h(x2 )t2 dt2 dt
r 0
t+T
Z +∞
Z t+T
πm
πd
[ [e−rt − e−rt2 ] + [e−rt2 ]] h(x2 )e−h(x2 )t2 dt2 dt
h(x1 )e−h(x1 )t
+
r
r
0
t
Z
Z t
+∞
πd
e−rt h(x1 )e−h(x1 )t
h(x2 )e−h(x2 )t2 dt2 dt
+
r 0
max(t−T,0)
We will simplify and solve for the exact value of profits in the next section. Finally, we
point out that the policy maker is not an active player in the game. However she sets the
value of the patent policy tools beforehand in order to maximize social welfare. Therefore,
before making decisions, firms know the value of the length and intertemporal breadth of
the patent.
6
3
Runner-up patents and social welfare
We study in this section cases where runner-up patents decrease the incentives to innovate. We derive a very general condition that guarantees that their positive effect on deadweight loss dominates this negative impact on incentives and the system is overall socially
beneficial. This condition is first derived in a benchmark model where the only decisions
firms make is whether to enter the innovation race. We then show in a number of extensions
that these results are robust. In particular, a very similar condition is obtained in the case
where firms strategically determine the amount to invest in research.
3.1
Benchmark model
We present our initial results in a benchmark model that provides most of the intuition
for the more general case.
3.1.1
Model and assumptions
This model adds one major restriction to the general environment: firms do not decide
strategically the amount to invest in research. Specifically we add the following assumptions
to the previously described environment:
1) Firms only decide whether or not to enter the patent race; they do not decide on the
amount to invest in research x. Therefore the function h introduced in the previous section,
that maps investments into speed of the research process is degenerate. If firms enter the
race, they spend a fixed amount c on research and the speed of the Poisson process is fixed
at λ.
2) We denote the cost of the research c. Different inventions have different costs and the
decision maker cannot discriminate between inventions: he has to design a “one size fits all”
policy. For the entire set of potential inventions, we suppose that c is uniformly distributed
on [0, M ].
3) The length of the patent is infinite
4) It is not possible to reverse engineer the invention from a commercialized product.
We will relax all these assumptions in sections 3.2 to 3.4.
3.1.2
The two effects
As described in the introduction, runner up patents will impact both incentives to innovate and deadweight loss on the product market. We first show that, in this benchmark
model (which does not include product differentiation or strategic investment in research),
the first effect will be negative: runner-up patents decrease incentives to innovate. To prove
7
this result, we first describe in the following lemma the expected benefits of a firm at date
0 (at the start of the research process). The calculations follow from the expression given in
section 2.3. The benefits depend on whether the competitor enters the innovation race.
Lemma 1: Expected benefits are weakly decreasing in T and are given by:
λ πm
E[πa ] = λ+r
if only one firm enters
r
πm
λ2
λ
d
E[πb ] = 2λ+r [ r ] − [1 − e−(r+λ)T ) ][ πm −2π
] (λ+r)(2λ+r)
if both firms enter
r
Proof: See appendix
If firms produce exactly the same product, it is known that industry wide profits under
monopoly are bigger than under duopoly (πm > 2πd ), independent of the type of competition.
Therefore, lemma 1 shows that if both firms enter, increasing the intertemporal breadth T
will decrease expected profits E[πb ]. To understand the intuition of the result let us ignore
the waiting time and place ourselves in the context of a simple one period game where each
firm succeeds with probability p. Under the current patent system, the expected profits of a
firm engaged in the race is πcurrent = p(1 − p)πm + p2 0.5πm (if they both succeed, one will
be randomly picked to obtain the patent). Under the proposed reform, the profits become
πrunner−up = p(1 − p)πm + p2 πd . This explains why the comparison between 2πd and πm is
the relevant one; with a certain probability, monopoly profits are replaced by duopoly profits.
Firms, when making their decision whether to enter the race will compare expected
benefits to the cost of the project. We can therefore characterize their behavior according
to the value of c:
1) if c ∈ [0, E[πb ]], both firms enter the innovation race.
2) if c ∈ [E[πb ], E[πa ]], only one firm enters.
3) if c ≥ E[πa ], no firm enters.
Innovations are characterized by their costs. The consequence of Lemma 1 is therefore that
the range of innovations (or in other words costs) such that two firms enter will be decreased.
Therefore, runner-up patents will impose a social cost: over the entire set of possible innovations, in expectation, the product will be obtained later.
However we have to compare this cost to the benefits from the second effect. We have
already argued intuitively that this second effect on deadweight loss will always be positive
as runner-up patents encourage competition. This is shown formally in the following lemma,
where Sm (resp Sd ) refers to overall social surplus, sum of producer and consumer surplus,
under monopoly (resp duopoly).
Lemma 2: Expected social surplus is weakly increasing in T and is given by:
λ Sm
E[Wa ] = λ+r
[ r ] if only one firm enters and
2λ Sm
2λ2
m
E[Wb ] = 2λ+r [ r ] + [1 − e−(r+λ)T ) ][ Sd −S
] (λ+r)(2λ+r)
if both firms enter.
r
8
Proof: See appendix
The overall effect is therefore ambiguous and we provide in the next section conditions
under which the runner-up patents increase social welfare.
3.1.3
Do runner-up patents increase social welfare?
We now examine the problem faced by the policy maker who has to design an optimal
patent policy. We denote W (T ) the expected social welfare given the distribution of costs.
Specifically, W (T ) = Ec [1c∈[0,E(πb )] (E(Wb ) − 2c) + 1c∈[E(πb ),E(πa )] (E(Wa ) − c)]. As described
previously, we consider a “one size fits all” policy: the decision maker will choose T to maximize expected social welfare W (T ), given that this unique value will apply to a wide variety
of innovations with different costs. Furthermore, we will determine marginal social benefits
of the proposal at T = 0. As described in the introduction, under the current legislation the
application is made public 18 months after filing of the patent. If the proposal was evaluated
for large values of T, this disclosure rule would need to be modified or later inventors would
take advantage of the new policy by using the disclosed information to copy the invention.
We will examine this question in section 5 when we study the optimal mix of policy tools.
In the following proposition, the results are stated for small values of T.
Lemma 3: Runner up patents are socially beneficial (W 0 (0) > 0) if and only if
[2πd − πm ][Sm
r
− πm ] + 2πm [Sd − Sm ] > 0
λ+r
(1)
Proof: See appendix
Condition (1) guarantees that runner-up patents will increase social welfare. It involves
two types of parameters: demand parameters (profits and social surplus under monopoly or
duopoly) and a research parameter (i.e. λ, the speed of convergence of the research process).
However, this general condition is hard to interpret and we therefore provide in Proposition
1 a series of sufficient conditions. We can state the general conclusion of Proposition 1 in
the following way: it is hard to find counter examples where runner-up patents do not seem
socially beneficial.
Proposition 1: In the basic model, runner-up patents will be socially beneficial in a wide
variety of cases and in particular under the following sufficient conditions:
(a) If the social surplus under competition is high enough: Sd > 32 Sm
r
(b) If private and social benefits under monopoly are close: Sm λ+r
< πm
(c) For linear demand under Cournot or Bertrand competition.
(d) For constant elasticity demand (P (q) = q −1/η ) if the elasticity is not in the
9
range [1, 1.14] for Cournot competition or [1, 1.17] for Bertrand.
(e) If the research process is fast enough: λ ≥ λ∗ where λ∗ ≥ 0
(f ) If the interest rate is low enough: r ≤ r∗
Proof: See appendix
According to result (a), if the surplus under competition is high enough compared to
that under monopoly, runner-up patents will be socially beneficial regardless of the nature of
demand, competition or of the research process. The intuition for this result is the following:
if surplus is much higher under duopoly, the gains from increased competition (second term
in equation (1)) more than compensates the loss of incentives (first term), even in the worst
case scenario where profits under duopoly are zero. Note that this condition might not be
very hard to satisfy, in particular under Bertrand competition.
Result (b) states that if private and social benefits under monopoly are not too different, runner-up patents will always increase social welfare. In the exposition of the results,
we have chosen to separate the impact of these patents into two effects (incentives and
deadweight loss) and to present an increase in research incentives as always being socially
desirable. This simplification was aimed at providing a clear presentation of the different
impacts, although all the results were obtained by maximizing directly social welfare. Case
(b) illustrates the fact that decreasing incentives can actually be beneficial if the research
conducted under the current system was already excessive. If for example private and social
benefits under monopoly are equal (Sm = πm ), a unique firm chooses the socially optimal
amount of research. If a competitor enters the race, research in excess of this optimum
will therefore be conducted. Runner-up patents are then beneficial on two accounts: they
decrease incentives to innovate and decrease deadweight loss.
We also provide conditions for specific functional forms. Case (c) states that departing
from the current winner take all system is always beneficial if demand is linear, irrespective
of the type of competition. Case (d) provides the only counterexample we found where
runner-up patents do not necessarily increase social welfare. However, this result could be
due to the strange properties of constant elasticity demand when η is close to 1. In particular, if the elasticity is smaller than 1, the monopoly problem does not have a solution as
the monopolist cannot price on the inelastic portion of the curve. It is generally believed
that there cannot be such a case as a constant inelastic demand. Even if the demand is
elastic, but the elasticity is close to 1, the properties still remain peculiar. We believe that
this counter example is probably a consequence of these unrealistic properties.
Results (a) to (d) present characterizations that depend only on the specification of
demand and competition. Result (e) on the other hand depends only on the research parameter. It states that if the research process is not too slow, runner-up patents will be socially
10
beneficial. The intuition is the following: runner-up patents are costly in the sense that consumers have to wait longer for the invention. However, when λ is big, the discovery process is
fast and therefore this type of waiting cost will become negligible whereas the benefits from
increased competition are almost assured. Note that λ∗ can take the value 0, for instance in
the case of linear demand and runner-up patents are then socially beneficial for any value of λ.
In this benchmark case we obtained a general condition guaranteeing that runner-up
patents are socially beneficial. It appears this condition will be met in a wide range of
circumstances; the only counter-example we found seems to depend on the peculiar characteristics of the demand function. In the following sections we want to conduct a number of
robustness checks and therefore relax the different assumptions previously made.
3.2
The possibility of reverse engineering
In the benchmark model, we supposed that reverse engineering was impossible: the first
inventor was not exposed to the risk that other firms might be able to copy the product and
therefore, as soon as he obtained the invention, started to commercialize it. In practice, this
can be a good representation of reality for certain type of goods and not for others. In this
section we present results for the other extreme case where reverse engineering is costless
and immediate.
In such a case, the first inventor at the discovery date t has to decide on his stratλ
egy to commercialize: If πrd > λ+r
[1 − e−(r+λ)T ] πrd + e−(r+λ)T πrm his optimal strategy is to
commercialize immediately and share the market with the competitor. If the condition is
not satisfied, his optimal choice is to wait until t + T (or until the competitor invents) to
potentially obtain monopoly profits. Note however that we are interested in results for small
value of T. When T converges to 0, waiting is always the preferred option. As a result,
the first inventor if he succeeds at t, will wait until t + T before commercializing or until
the other firm succeeds if this occurs before t + T . Under these conditions, the following
proposition describes the condition such that runner-up patents increase social welfare.
Proposition 2: If reverse engineering is costless and immediate, runner-up patents are
socially beneficial (W 0 (0) > 0) if and only if
·
¸
r
[2πd λ − πm (λ + r)] Sm
− πm + 2πm [Sd λ − Sm (λ + r)] > 0
(2)
λ+r
In particular, there exists a λ∗ > 0 such that if the research process is faster than λ∗ (λ ≥ λ∗ ),
runner-up patents are socially beneficial (W 0 (0) > 0) and if it is slower, they are socially
harmful.
Proof: See appendix
11
Condition (2) is very similar to condition (1) given in Lemma 3, but is more restrictive,
as the first inventor has to wait before commercializing. We therefore cannot draw general
conclusions about the social benefits of runner-up patents independently of the speed of the
research process. What we can conclude is that, if the research process is fast, they will always be beneficial, for the same reasons as in the benchmark case. However, because of the
additional restriction, the benchmark speed λ∗ is now always strictly positive, irrespective of
the demand function (even for linear demand). Therefore, if the research is slow, runner-up
patents turn out to be socially damaging.
Runner-up patents therefore seem less attractive in this situation, although they are
still socially beneficial in a wide range of cases. However, we want to point out that the case
we consider here is an extreme one, and this for two reasons. First, in reality, the cost and
speed of backward engineering will take intermediate values. Second, it is doubtful that,
even if it was impossible to copy the product, the firm would have the industrial capacity
to commercialize immediately. Therefore, for small value of T, this cost of delayed commercialization would probably not be a real issue.
3.3
Finite length
The second assumption we relax is that the length of the patent is infinite. We find that
if the length L is finite, runner-up patents will be socially beneficial, W 0 (0) > 0 if and only
if:
·
¸
r
S0
[2πd − πm ]
(Sm + rL
) − πm + 2πm [Sd − Sm ] > 0
(3)
λ+r
e −1
We see that, as the length L becomes smaller, the condition becomes harder to satisfy.
The intuition is the following: if L is small, soon after invention, the market will be opened to
competition. Given such a length, the policy maker is then more concerned about providing
extra incentives to conduct research rather than finding ways to decrease the deadweight
loss. Runner-up patents thus become less attractive. However, this is only a static analysis.
We are answering the following question: given a certain length L, are runner-up patents
socially beneficial? We should determine how the length L and the time period T can be
jointly optimally set. Thus, it seems more interesting to ask the following questions: what
values of T and L achieve an efficient incentive-social welfare combination? Is T more efficient than L in achieving a tradeoff between incentives to innovate and deadweight loss or
is it just a redundant tool? We address these questions in section 5 where we determine the
socially optimal mix of different patent policy tools.
12
3.4
Patent races with sunk costs
In the benchmark model, we supposed that the only strategic decision the firms were taking was whether or not to enter the race. The costs, benefits and speed of the project were
fixed. We now suppose that the firms can strategically determine the quantity of research
they perform and thus the speed of the research process. If they spend at date 0 an amount
x, the instantaneous probability of discovery is given by h(x)e−h(x) . We suppose that this
investment x is sunk and we will see in the next section that changing this assumption can
have a big impact on the results. Overall, the results in this case prove to be very similar to
those of the benchmark model.
When firms invest strategically in research, we have to determine the Nash equilibrium
of the research phase. Before characterizing the symmetric equilibrium, we describe the expected benefits of firm 1 given that firms 1 and 2 spend respectively x1 and x2 on research.
Lemma 4: The expected benefit of firm 1 is decreasing in T
·
¸
πm
h(x1 )
πd
h(x1 )h(x2 )
−(r+h(x1 ))T
E[π1 ] =
+
(1 − e
)
r h(x1 ) + h(x2 ) + r
r
(h(x1 ) + r)(h(x1 ) + h(x2 ) + r)
·
¸
πd − πm
h(x1 )h(x2 )
−(r+h(x2 ))T
+
(1 − e
)
r
(h(x2 ) + r)(h(x1 ) + h(x2 ) + r)
Proof: See appendix
The first term corresponds to the expected profits under the current system (i.e. T = 0).
The second and third terms measure the expected effect of potential competition on the
product market. We see that, for the same reasons as in the previous section, taking x1
and x2 as given, increasing T will decrease expected profits. However, to determine how
runner-up patents affect the overall incentives to conduct research, we need to characterize
the symmetric Nash equilibrium of the game and to compute comparative statics. The results are presented in the following proposition.
Proposition 3: If firms are engaged in a patent race with sunk research costs, runner∗
up patents decrease the equilibrium amount of research: dx
(0) < 0
dT
where x∗ , the symmetric NE evaluated at T = 0 is given by:
h0 (x∗ )(h(x∗ ) + r) πm
=1
(2h(x∗ ) + r)2 r
Proof: See appendix
Note that first of all that, as expected, the expression characterizing the Nash equilibrium at T = 0 is identical to that obtained by Loury (1979) who studied patent races under
13
the current system. Proposition 3 also states that, as in the benchmark model, runner-up
patents will decrease research incentives. The intuition follows the same logic. For fixed investments, the expected profit is smaller than under the current system (Lemma 5) because
the industry wide profits under duopoly are smaller than under monopoly. Furthermore,
the cost is incurred initially independently of the other firm’s strategy. The overall effect is
therefore to decrease the expenditure on research in the Nash equilibrium.
As pointed out in the previous sections, this alternative mechanism can still be socially
beneficial. We describe in Proposition 4 a first characterization of cases where this is true.
The condition obtained depends both on demand parameters and on the equilibrium amount
of research. However, this amount x∗ is defined implicitly, and the result is therefore hard
to interpret. We then present in Proposition 5 a more interpretable characterization.
Proposition 4: If firms are engaged in a patent race with sunk research costs, runnerup patents are socially beneficial under the following condition:
·
¸
dx∗
Sm
r
Sd − Sm h2 (x∗ )
(0)
−
1
+
>0
(4)
dT
πm h(x∗ ) + r
r
2h(x∗ ) + r
In particular, it is the case if:
(a) Consumer surplus at monopoly prices is small enough Sm h(x∗r)+r < πm
(b) The research process is productive enough: if µ ≥ µ∗ when we assume
the following functional form h(x) = µx.
Proof: See appendix.
Result (a) follows closely result (b) in Proposition 1. It states that, if the consumer
surplus at monopoly price is small enough, runner-up patents will always be beneficial. This
result is also linked to one obtained by Loury in section IV of his paper. The author conducts a welfare analysis of patent races and concludes that firms conduct a socially excessive
amount of research. However, his result depends on the disputable assumption that the
social flow of benefits equals the private flow. In our context, this assumption would impose
the condition Sm = πm and in such cases runner-up patents would be socially beneficial.
As pointed out previously, if the amount of research is socially excessive under the current
system, then runner-up patents will be beneficial on two accounts: they reduce the amount
of wasteful research and encourage competition in the product market.
However, in most cases, the condition of case (a) will not be satisfied. Result (b) studies
a special functional form for h. This result is similar to one obtained in Proposition 1(e)
but the speed of the process is now endogenous and depends on the strategic choice of research effort x∗ . The result can be interpreted in the following way: the more productive
the research process, the more socially attractive runner-up patents become. The intuition
14
is the following: even when the research process becomes very productive (µ high), the equilibrium amount of research, because of strategic considerations, remains strictly positive. If
one of the firms was choosing an infinitesimal amount of research, the best response of the
other firm would be to invest slightly more to increase its chances of obtaining the exclusive
patent. Therefore, in equilibrium, a strictly positive amount is invested in research. We
can conclude that, as µ increases, the research process in equilibrium becomes faster. Thus,
the main cost of the mechanism which is a delay in the obtention of the product becomes
negligible whereas the benefits from expected competition become almost assured. Overall,
runner-up patents will therefore be socially beneficial in such a case.
The condition described in Proposition 4 depends both on demand parameters (Sm ,
πm ) and on the equilibrium amount of research x∗ which is implicitly defined. It is therefore
hard to interpret. The following result will prove useful to clarify our understanding.
Proposition 5: A sufficient condition guaranteeing that runner-up patents are socially beneficial (W 0 (0) > 0) is:
−[Sm − πm ][πm − 2πd ] + πm [Sd − Sm ] > 0
(5)
In particular, the condition is satisfied:
(a) If demand is linear, both for Cournot and Bertrand competition.
(b) If social surplus under competition is high enough: Sd > 2Sm .
Proof: See appendix.
The conditions obtained here are very similar to those derived in the base case (Proposition 1), only harder to satisfy. Indeed, we wanted a condition that does not depend on x∗ and
involves only demand parameters. We therefore had to impose stricter sufficient conditions.
The robustness checks we conducted were therefore conclusive: in a wide range of cases,
runner-up patents will increase social welfare, even when we consider the risk of reverse engineering or the strategic choice of investment in research. In the next section we show that,
in some cases, this new mechanism can even have a positive impact on incentives to innovate.
4
Runner-up patents can increase research incentives
In the situations studied in the previous section, we showed that even though runnerup patents were most of the time socially beneficial, they decreased research incentives.
Surprisingly, we show in this section that under certain assumptions, they can actually
increase incentives to innovate. This can occur if research costs are incurred as a flow or if
firms differentiate their inventions and there are sufficient gains from product diversity.
15
4.1
Patent races with flow costs
We showed that if research costs are sunk investments, runner-up patents decrease research incentives. In practice, a lot of projects also involve flow costs, for instance the cost
of financing the day to day work of research teams. These costs can be stopped at any point
in time. Under the current patent system, the loser would therefore halt his research effort
as soon as he observes the competing firm’s success. Lee and Wilde (1980) pointed out that
taking into consideration these flow costs changes several standard results on patent races.
We will discuss their work as we obtain further results.
We assume in this section that although the flow cost x can be stopped at any time,
its value is fixed at the start of the race. This corresponds to a situation where firms decide
on the experimental setup that will determine the flow of resources spent on research. We
point out in the appendix that this could also be a good approximation of the non stationary
game where the flow cost x can be adapted through time. The following result shows that
under a certain condition, runner-up patents can increase incentives to innovate.
Proposition 6: If firms are engaged in a patent race with flow costs, runner-up patents
will increase the incentives to conduct research under the following condition:
h(x∗ ) − x∗ h0 (x∗ ) > −
h00 (x∗ )
(2h(x∗ ) + r)(h(x∗ ) + r)
h0 (x∗ )2
(6)
where x∗ is the symmetric Nash equilibrium of the research phase for T = 0 and is given by:
(h(x∗ ) + r)πm =
r(2h(x∗ ) + r)
− rx∗
h0 (x∗ )
(7)
Proof: See appendix
The impact of runner-up patents on research incentives is therefore ambiguous. At the
Nash equilibrium, the left hand side is positive h(x∗ ) − x∗ h0 (x∗ ) > 0 (condition guaranteeing
that the equilibrium profits are strictly positive). The right hand side is also positive as h is
a concave function. The concavity of h will therefore determine if the condition is satisfied.
In particular, if h is linear h(x) = a + bx, the condition in Proposition 6 is satisfied. This
conclusion raises the question of what makes this case with flow costs different from the cases
where the firm made sunk investments in research. We give some intuition of the essential
differences.
In the case of flow costs the amount spent on research by one firm will depend on the
strategic decision of the other firm. For example in the current patent legislation, if firm j
discovers at time t, then firm i will stop searching at time t. The expected cost of firm i
is therefore decreasing in firm’s j research effort. This is also true with runner-up patents.
16
However, as T increases, the expected cost of one firm becomes less and less dependent on
the competitor’s effort. Indeed even if firm j discovers at t, firm i will pursue its research
until t + T . Note that in the extreme case where T = +∞, the cost becomes independent of
the other firm’s effort. Therefore if firm i is considering increasing its research effort, because
the cost of the other firm will decrease less with runner-up patents than under the current
legislation, firm j might not react as strongly. The overall effect might then be to bring
the equilibrium amount of research to a higher level. However we have only discussed the
effects on costs. The profits will also be less reactive to the effort of the other firm, creating
a countervailing effect. Overall, the condition of Proposition 6 will guarantee that the first
effect dominates the second.
The differences between flow and fixed costs had already been highlighted by Lee and
Wilde (1980). They found, contrary to Loury’s conclusion, that as the number of firms in
the industry increased, the equilibrium level of research increased as well. In the case of
fixed cost, competition did not spur innovation in that way. Our results seem to follow the
same logic. In the next section we present another situation where runner-up patents might
increase incentives to innovate.
4.2
Differentiated innovations
In the previous sections we followed most of the innovation literature and made the strong
assumption that firms engaged in the patent race develop exactly the same product. However, competing research teams might start with the same research idea but employ very
different methods and therefore develop significantly different products.
In this section we suppose that firms differentiate their inventions. We then find that
under specific conditions, runner-up patents will increase the incentives to innovate. The
intuition is that, when two firms share the market, the aggregate industry profits under
duopoly can be higher than under monopoly due to gains from product diversification. Indeed if consumers value product diversity, competing firms that sell different products can
jointly reap more profits as they tailor better to the desires of consumers. When the products
were exactly identical, this could never be the case and monopoly profits were always higher
than industry wide duopoly profits.
We revisit the results under the assumption of horizontal competition:
1) The product space is a circular city.
2) Firms observe each other’s initial positioning.
3) Agents incur a transport cost t.
Assumptions 1 and 2 guarantee that firms will indeed differentiate their innovations;
17
in a circular city, it is optimal for them to do so. Note that we made assumption 2 to
simplify the problem but could also have considered equilibria where firms randomize their
location. Under these assumptions, we obtain the following proposition characterizing how
the equilibrium amount of investment in research varies with the intertemporal breadth.
Proposition 7: If the transport cost is high enough (t > t∗ ), runner-up patents will in∗
crease the incentives to conduct research: dx
(0) > 0
dT
Proof: See appendix
The intuition of this result is simple. In equilibrium, it is optimal for the two firms
to differentiate their inventions. If the transport cost is high enough, the aggregate industry
profits under duopoly will be greater than under monopoly: 2πd > πm . For the same reason
as in section 3, this comparison between duopoly and monopoly profits is the relevant one
and guarantees in this case that runner-up patents will increase incentives to innovate.
We want to point out that the results would have been different if we had considered a
linear city. In a linear city, the optimal location for a monopolist is the center whereas it
is optimal for two duopolists to position themselves at the extremes. Therefore, the choice
of location will involve a subtle tradeoff between these two effects. In general, we can state
that increasing the intertemporal breadth T will increase product differentiation as it raises
the probability of being in competition. Furthermore, there are probably conditions that
guarantee a result similar to that of Proposition 7.
One objection to the ideas developed in this section could be that, if two products are
“far” enough, they should probably be covered by a different patent. This comment underlines the link between the intertemporal breadth and the classical breadth of the patent.
More generally, we explore the interaction with the existing patent policy tools, both breadth
and length, in the next section.
5
Optimal mix of patent policy tools
In this article, we introduced a new tool of patent policy: the length of time during
which later inventors are allowed to share the patent with the first inventor. We call this
instrument the intertemporal breadth. In the previous sections we supposed that the existing
patent policy tools (length and breadth) were fixed and determined the optimal value of the
intertemporal breadth. However, all these tools can tradeoff incentives to innovate against
deadweight loss in the product market and this in different ways. In this section, we ask a
different question: is intertemporal breadth a redundant tool or can it achieve a more efficient
18
tradeoff between these two effects? To answer this question we study the optimal mix of
these instruments. In particular we determine conditions under which a socially optimal mix
will include a strictly positive intertemporal breadth.
5.1
General method
We first describe the general method to determine the socially optimal mix between the
intertemporal breadth T and another tool I (that will be either the length or the breadth).
This method will employ the results of previous sections as building blocks. A socially optimal mix of the two instruments will provide the cheapest method in terms of social welfare
to achieve a specific amount of research. In mathematical terms, to determine the socially
optimal mix of T and I, we need to solve the following problem: maximize expected social
surplus S(T, I), given that the equilibrium amount of research x∗ (T, L) is greater than a
certain value x.
The method follows these steps:
1) Determine the Nash equilibrium of the research phase x∗ .
2) Determine the expected social surplus S(T, I) given the equilibrium amount x∗ .
3) Solve for (T ∗ , I ∗ ) solution to:
maxT,I S(T, I)
subject to T ≥ 0 and T ≤ L and x∗ ≥ x
4) Use the Kuhn Tucker conditions to obtain a condition such that T ∗ > 0.
5.2
5.2.1
Optimal mix with the length of the patent
Intertemporal breath: an efficient instrument
We first determine the optimal mix with the length of the patent. We return to the assumptions made in section 3 to abstract from the cases where runner-up patents can increase
incentives to conduct research. Indeed if that was the case, a higher intertemporal breadth
would assuredly be more efficient than a longer length: it would decrease deadweight loss and
also increase incentives to innovate. We abstract from these cases and suppose that there
is no product differentiation and the investment in research is a sunk cost incurred at date
0 (as in section 3.4). The following proposition provides a necessary condition guaranteeing
that a socially optimal mix will involve a strictly positive intertemporal breadth.
Proposition 8: The socially optimal mix (T ∗ , L∗ ) will require the intertemporal breadth
T ∗ to be strictly positive if the following condition is satisfied:
Sd − Sm
S0 − Sm
≥
0.5 πm − πd
0.5 πm
19
In particular:
(a) It is satisfied if demand is linear under Cournot competition
(b) For perfect Bertrand competition, the marginal effects of both tools are equal.
Moreover, at the limit, for small departures from perfect Bertrand competition,
the condition is strictly satisfied.
Proof: See appendix
A socially optimal mix is a combination of tools that guarantees a certain amount of
research at the lowest social cost. It is therefore natural that the condition obtained in
Proposition 8 compares the ratios of social benefits over loss of incentives from the two
instruments. A marginal increase of the intertemporal breadth T causes monopoly to be
replaced by duopoly at the margin. The social gain is therefore proportional to Sd − Sm
whereas the expected profit is reduced from monopoly profits with probability 0.5 to duopoly
profits. The loss in terms of incentives is therefore proportional to 0.5 πm − πd . A marginal
reduction in the length L of the patent means that perfect competition replaces monopoly
pricing at the margin and the social gain is S0 − Sm . At the same time, the profits will
fall to 0 due to perfect competition, and the marginal change in incentives will be πm . The
final condition obtained guarantees that the intertemporal breadth is more efficient than the
length at T = 0.
The condition obtained in Proposition 8 depends exclusively on the nature of demand
in the product market and the type of competition. In particular, result (a) states that for
linear demand and Cournot competition the intertemporal breadth will always be used in a
socially optimal mix. Result (b) is a general property independent of the type of demand.
It underlines an essential difference between the length and the intertemporal breadth. At
first glance, both tools seem very similar. Indeed, for perfect Bertrand competition, they
both lead to perfect competition and the condition is satisfied with equality: the marginal
effects of the two instruments are the same. However, if there is a small rigidity in prices,
the condition will always be strictly satisfied. Indeed, social surplus is maximal at Bertrand
prices and therefore the variations in social surplus are of second order whereas the change
in profits are first order variations.
This reflects an essential difference between these tools. They both encourage competition, but with the length this competition comes from outside imitators (for example
for pharmaceutical companies, firms producing generics) whereas with the intertemporal
breadth it involves only innovators inside the race. This difference has an impact on costs:
with intertemporal breadth this gain in social surplus can come at a lower cost as the innovators get a partial compensation (compensation if they are second in the race). Therefore,
it will generally prove to be less costly to achieve a certain increase in social surplus with
the intertemporal breadth.
20
5.2.2
Is it socially optimal to set T = L?
In the previous section we showed that the intertemporal breadth should be used in a
socially optimal mix as it proves to be a more efficient tool than the length to tradeoff incentives to innovate against deadweight loss. It is therefore natural to ask the question: under
these conditions is it socially optimal to set T ∗ = L∗ ? We provide in this section a simple
condition guaranteeing that the intertemporal breadth should be set strictly smaller than
the length. We show that if the initial innovation can lead to later inventions that provide
positive surplus, in a socially optimal mix, we will have T ∗ < L∗ .
The question of second generation products underlines once again the link between
runner-up patents and the disclosure rule. The time between filing and disclosure needs
to be bigger than the intertemporal breadth. If this wasn’t the case, a firm after disclosure
could reproduce the invention, file for a patent and thus be able to share the market with
the first inventor. Therefore, if T is big, runner-up patents impose an added cost: they can
delay the invention of subsequent innovations. If we take into account this cost, we can show
that in a socially optimal mix the policy maker will set T ∗ < L∗ .
Specifically, we make the following assumptions:
- The patent is publicly disclosed at t + T . We just mentioned that it cannot be disclosed
before that date. Furthermore, there is nothing to gain from disclosing it at a later date.
- From the date of disclosure, t + T , it takes a time period R to develop later innovations
building on the initial idea.
- The surplus from these later inventions is denoted D.
We chose to model the invention of second generation product as deterministic. This is
of course a very simplified model, but it allows us to exhibit the principal tradeoff as expressed in the following proposition.
Proposition 9: If later inventions generate a positive surplus (D > 0), in a socially optimal
mix, the intertemporal breadth will be smaller than the length T ∗ < L∗ .
Proof: See appendix
The result obtained in Proposition 9 is strong: even if subsequent inventions lead to
little added social surplus, in a socially optimal mix, the policy maker will not set the intertemporal breadth equal to the length. The intuition is the following: at T = L a marginal
increase in T has no effect on incentives or social surplus for the initial innovation. However,
increasing T has a first order negative effect on the social surplus from the second generation
product as it delays its invention. Therefore it will be optimal to reduce the intertemporal
breadth to avoid this delay cost and in the socially optimal mix, we will find that T ∗ < L∗ .
21
5.3
Optimal mix with the breadth of the patent
Whereas the length is a straightforward concept, the law specifies at what date patents
expire and competitors can enter the market, the breadth of a patent does not have a clear
legal specification. In general terms, it describes how different another innovation needs to
be not to infringe on the existing patent. From this intuitive definition, we understand that
a lot of discretion is left to the patent office and the courts in making this decision. This
concept of breadth has therefore been modeled in a variety of ways in the literature and
before presenting our own model, we describe some of these alternative approaches.
In the case of a process innovation (invention that is aimed at developing a technology to reduce the cost of production), Nordhaus (1972) defines breadth as the fraction of
cost reduction that does not become freely available to competing firms. Gallini (1992),
develops a related concept and models a wider patent as increasing the cost of imitation
for competing firms. Klemperer (1990) does not consider costs but studies directly a case
with differentiated products. He defines the breadth of the patent as the distance in the
characteristic space between the patented innovation and the non-infringing products of the
competitors.
Denicolo (1996) reconciles all these ideas in a model closer to ours where firms are engaged
in a patent race. In his model, the social surplus, the profits of the winner and of the loser of
the race, all depend on the breadth (social surplus and profits of the loser are decreasing in
breadth and winner’s profits are increasing). Whereas the previously mentioned authors gave
conflicting answers regarding the socially optimal mix, some advocating maximum length
and others maximum breadth, Denicolo concludes that the less efficient competition is in the
product market, the more likely it becomes that broad and short patents are socially optimal.
Denicolo’s model provides very clean predictions. However we believe it does not represent fully the incentives of the loser in the race. This was not of considerable importance
in the situation he was studying, but is essential when we want to examine the effect of
runner-up patents. In his model, the loser, as soon as the first inventor discovers, obtains a
certain profit, function of the breadth. The idea is that during the research process, the loser
accumulates knowledge that will allow him to compete more efficiently with the patented
product (for example it makes designing a non infringing version of the product less costly to
develop). We believe that the incentives are more profound. The loser is developing his own
product that, as we pointed out in the section on invention differentiation, could be quite
different from the patented innovation. He can potentially gain considerable profits from his
own invention in two situations: if his product is different enough from the patented innovation so that it is non-infringing or if he invents within T of the first innovator. We therefore
choose a different model to capture the idea of breadth, described in the next subsection.
22
5.3.1
The assumptions
We suppose there are only two possible locations for the final product. We define the
breadth as the probability α that these two positions will be covered by the same patent.
This should provide a good representation of reality. Indeed, the regulator cannot set an
absolute standard in terms of breadth. What she can do however is provide guidelines to
the patent office and judges to help them determine how strict they should be. The notion
of breadth as the probability of the strict regime therefore seems quite justified. A similar
vision is advocated by Lemley and Shapiro (2005) where they present patents as probabilistic
property rights.
More specifically in a model of horizontal differentiation, we suppose that:
1) There are two fixed positions l1 and l2 .
2) It is optimal for the two firms to differentiate their products and position themselves
respectively at l1 and l2 (verified for example for a circular city).
3) The breadth α is the probability that the two locations will be covered by the same patent.
5.3.2
Results
We use the general method previously described and determine in the following proposition a condition such that the intertemporal breadth will be set strictly positive (T ∗ > 0)
in a socially optimal mix.
Proposition 10: The socially optimal mix (α∗ , T ∗ ) will require the intertemporal breadth
T ∗ to be strictly positive if:
r(3h(x∗ ) + 2r)
πm ≥ πd
(h(x∗ ) + r)2
(8)
Furthermore, unless the locations are very far apart, so that decreasing the breadth increases the incentives for research, this condition will be satisfied.
Proof: See appendix
Decreasing the breadth increases the incentives for research only when the two positions
are far apart. The mechanism is similar to the one presented in the section on invention
differentiation: it occurs if the aggregate expected industry profits are much higher under
duopoly than under monopoly. In that case, the optimal choice of breadth is α∗ = 0. In
other words, the products are so different that they should be each covered by a different
patent.
In all other cases, the intertemporal breadth should be set strictly positive in equilibrium. The social benefits from increasing the breadth α or the intertemporal breadth T are
23
very similar: at the margin they replace monopoly profits by duopoly. However, using the
intertemporal breadth has an additional positive effect on research incentives. Indeed, each
firm knows it will only have a period of time T to innovate after the other firm succeeds.
Such competition is not generated by changing the classical breath. Therefore, intertemporal breadth provides more incentives to innovate and turns out to be a more efficient tool
that should be used in equilibrium. The conclusion of this section is once again that the
intertemporal breadth seems to be a very attractive tool. However, the model chosen here
ignores another potential benefit of breadth: a larger breadth is a protection against outside
innovators.
5.3.3
Breadth as protection
In this section we address the issue previously raised, by using a different model closer
to the approach chosen by Denicolo (1996). We therefore suppose that as soon as the first
inventor succeeds, the competitor obtains profits πL (α) but continues searching until t+T.
We also suppose that both profits and social surplus depend on the abstract parameter α
such that 1 − α measures breadth. πm (α) is decreasing in α, πL (α) and Sm (α) are increasing.
Proposition 11: The socially optimal mix (α∗ , T ∗ ) will require the intertemporal breadth
T ∗ to be strictly positive if:
0
Sd − Sm
Sm
(α)
>− 0
πm + πL − 2πd
πm (α)
Proof: See appendix
This condition, as in the case of length, compares the ratios of social surplus to research incentives that result from marginal variations of the intertemporal breadth and the
classical breadth. The social optimal mix provides the combination that allows for a certain
amount of research at the minimum social cost. However, because breadth is here defined
as an abstract concept, it is hard to provide a good interpretation.
We therefore describe a specific case where we want to capture the idea that other firms
can try and invent around the patent. We choose a model with horizontal differentiation a
la Hotelling where consumers are distributed on the line (0,1), and incur a transport cost t.
The two firms differentiate their products and are located at 0 and 1. They produce initially
2 goods of quality θ at zero costs. They are engaged in a patent race to obtain an innovation
b As soon as one invents, the other can produce a good of
that raises the quality to θ + θ.
b We find that in this specific application, the condition will always be satisfied
quality θ + αθ.
and therefore the intertemporal breadth will be used in a socially optimal mix.
24
6
Intertemporal breadth as a sorting tool
We showed in the previous sections that the intertemporal breadth could be an essential tool to increase social welfare in a uniform patent policy. We illustrate in this section
another potential use of this new instrument. The intertemporal breadth can be a very
efficient sorting tool in a differentiated patent policy. In particular we show it can resolve
a major concern expressed in the literature on patent races: by offering menus where firms
can tradeoff length against intertemporal breadth it will be possible to sort between more
or less efficient firms and provide extra research incentives to the most efficient.
In the previous sections, we tried to solve the problem of designing the optimal uniform
patent policy, as in most of the innovation literature (Klemperer (1990), Denicolo (1996)).
We supposed that the policy maker could not or did not want to discriminate between firms.
This is a good representation of reality if either the firms are homogenous or the policy
maker cannot discriminate between heterogenous firms. However, some articles have started
studying heterogeneous firms and in that context addressed the problem of designing a differentiated patent protection. Cornelli and Schankerman (1999) and Scotchmer (1999), in
models with asymmetric information, justify the use of patent renewal fees as an optimal
incentive scheme. Cornelli and Schankerman study a model where firms vary in their productivity: for an identical investment in research, a more productive firm will produce a more
socially valuable product. Therefore, a uniform patent life provides too much incentives for
the low productivity firm and too little for the high productivity one. They show that the
optimally differentiated patent policy can be implemented through a menu of patent lives
and associated renewal fees.
In the context of a differentiated patent policy, we introduce menus (T,L) such that
firms can tradeoff higher intertemporal breadth T against a longer length L of protection.
For example the patent office could offer to the first inventor the choice between the following
menus (T = 0 months, L = 18 years) and (T = 18 months, L = 22 years). We show in the
following sections that such menus will be extremely useful to increase social welfare.
6.1
Selecting cost efficient firms in patent races
An important concern expressed in the literature on patent races is that the current
uniform patent system provides the same research incentives to firms with low costs and
high costs of innovation. The incentives are therefore excessive for the less efficient firm and
insufficient for the more efficient one. Currently, it is not possible to sort between these two
types. We suggest in this section that using menus (T,L) can help achieve that goal.
We suppose that there are two firms engaged in a patent race that differ in their produc25
tivity. Suppose also that there are two contracts offered (T1 , L1 ) and (T2 , L2 ), with T1 > T2
and L1 > L2 . The high productivity firm, if it is the first to succeed, will not place a
high probability on the event that the less efficient competitor will succeed soon afterwards.
Therefore, if these menus are optimally designed, such a firm will choose the first menu,
involving a higher intertemporal breadth compensated by a longer length. Similarly, the less
efficient firm if it happens to be the first to succeed, will choose the second menu, preferring
the smaller value of T, because of the fear that the competitor will invent soon afterwards.
If the menus are optimally designed, they can therefore increase the research incentives of
the most efficient firm. Furthermore, the intertemporal breadth seems to be the only tool
that can allow such sorting in patent races. In the next section we examine in more detail
under which conditions it is possible to sort between firms of different types using only the
length and the intertemporal breadth.
6.2
Sorting and renewal fees
Cornelli and Schankerman (1999) and Scotchmer (1999) showed that an optimal differentiated patent policy can be implemented through a menu of patent lives and associated
renewal fees. More recently, Hopenhayn and Mitchell (2001) showed that using a menu of
breadth and length is more efficient as a sorting tool than using renewal fees and provide a
series of sorting conditions such that this is the case. We modify their argument to obtain
a similar result that states under which conditions length and intertemporal breadth are
sufficient to sort between different types. We point out that menus involving intertemporal
breath present a number of advantages. First, they are easily implementable, whereas menus
with breadth are quasi impossible to setup: the concept is already hard to define precisely,
it is probably impossible to contract on. Second, as we showed in the previous section, a
menu with intertemporal breadth is the only possible way of sorting in patent races. Finally
it avoids using monetary transfers such as renewal fees.
We introduce the following notations. Let R be the renewal fees and θ the type of
the inventor (θ could measure the productivity of the firm or the quality of its invention).
The profits of the inventor are given by Π(T, L, θ). We denote Πi the derivatives of profits
with respect to the ith variable.
Lemma 5: Under the following sorting conditions, an optimal differentiated patent policy will not use fees R (these will be set to zero):
1) Π1 (T, L, θ) is strictly decreasing in θ.
2) Π2 (T, L, θ) is strictly decreasing in θ.
3) Π(T, L, θ) is monotonic in θ.
Proof: See appendix
26
Renewal fees can be considered as a fixed cost that the inventor takes into account when
investing in research. The idea of patents is to reimburse such costs by granting monopoly
rights to provide the correct research incentives. Therefore a renewal fee will have to be reimbursed with socially costly market power and in that context, using intertemporal breadth
can be superior.
We need to present situations where these conditions are satisfied. We suppose θ measures
a specific notion of value, the “fertility” of the invention: a higher value of θ will correspond
to a lower expected arrival time of a non-infringing second generation product. Therefore
an invention of a higher type will generate more subsequent research. More specifically we
note P (t, θ) the probability that this subsequent product will be invented before t.
The timing is the following:
1) Two firms start at date 0 a race for the initial invention.
2) The first firm is successful at t and chooses a contract (T,L) among a menu of contracts.
3) If the second firm invents before t + T , they share the patent. If not the first inventor
obtains the monopoly.
4) After t + T , there is a probability that a second generation product will be invented.
5) If a second generation product is invented before t + L, the first product is excluded from
the market. However, inventors get licensing benefits. We suppose that, irrespective of the
value of θ, the first innovator (respectively innovators) will obtain a fixed licensing fee F2
(respectively 0.5 F2 ) from the second generation product inventors.
Proposition 12: If the licensing fee paid by the second generation product inventor is
high enough, F2 > F2∗ , the conditions of Lemma 5 will be satisfied and sorting between different types can be achieved using only menus that tradeoff length against intertemporal breadth.
Proof: See appendix
The intuition of this result is the following. If θ takes a high value, there is a high
probability that a non-infringing second generation product will be invented before the expiration of the patent. Therefore the marginal value of the length of the patent is relatively
low. On the other hand, if the fee paid by the second generation inventor is high, the expected profits of the original inventors, when they consider entry in the race are increasing
in θ. Therefore, for high values of θ, inventors will initially invest more in research, making
the endogenous speed of research faster. When one of them succeeds, he knows that there is
a high probability that the competitor will invent soon after. The marginal value of a lower
intertemporal breadth is therefore higher. A higher type is ready to tradeoff a smaller length
against a smaller value of intertemporal breadth, to decrease the probability of having to
share the patent.
27
The intuition of this result seems quite general. Inventions that are more valuable will
tend to attract more competitors and also, later on, more second generation innovators. The
marginal value of L will therefore be lower whereas the marginal value of a smaller T will
be higher. In such situations it will be possible to sort between different types using only
menus that tradeoff length against intertemporal breadth.
7
Conclusion
We have shown in this article that it is not necessary to grant monopolies to first inventors to achieve an efficient tradeoff between incentives to innovate and deadweight loss.
Indeed, runner-up patents, whereby later inventors can share the patent if they discover soon
enough after the first inventor, increase social welfare in a wide variety of cases. We also
showed that the time period where later inventors are allowed to share the patent with the
first innovator, that we call intertemporal breadth, could become an essential tool of patent
policy. We prove that it is often more efficient than the existing instruments, such as the
length and the breadth of the patent, and should thus be used in a socially optimal mix.
Finally we demonstrate that the intertemporal breadth could also be essential in another
dimension. In a differentiated patent policy, it could allow sorting between more or less
efficient firms and increase the research incentives of the most efficient.
This work could lead to a number of interesting extensions. Runner-up patents exacerbate the inherent conflict between patent and antitrust laws. In particular we will need to
ask the following questions: What licensing contracts should regulators allow between the
first inventor and his followers? Should royalty rates be explicitly forbidden? In general, as
runner-up patents are aimed at encouraging some degree of competition, royalty rates could
be socially harmful as they can allow at least partial collusion.
A second extension of this work could be aimed at understanding how runner-up patents
would influence the patenting of intermediate products. It will be difficult to answer such a
question using the current framework. A different type of model will have to be employed
that could allow for learning during the research process. On the one hand, with runner-up
patents, firms have less incentives to patent intermediate inventions because they know they
will not lose everything if the competitor patents before them. On the other hand, they
could increase incentives to patent intermediate results as the cost of disclosing their discovery is decreased: even if their competitors learn a lot from this disclosure and are quicker to
develop an improvement, runner-up patents might still allow them to share the patent. It
would be very interesting to study the balance between these two opposing forces.
28
We also believe that the idea of menus involving intertemporal breadth can lead to
further benefits. For example, Moser (2005) shows that patent laws have a larger influence
on the direction of innovation than on the absolute amount. In particular, in sectors where
secrecy is a good defence, patents will not be an attractive alternative. This secrecy is socially costly as knowledge is not shared. It will therefore be important to design ways of
bringing these firms back to the patent system. Menus that tradeoff intertemporal breadth
against breadth might be a way to achieve this. A firm for which secrecy is a good alternative might find the current uniform breadth not large enough. However, such a firm could
be willing to tradeoff a higher intertemporal breadth (similar to taking the risk of secrecy)
against a larger breadth. Such a menu could be attractive enough to make the patent system
a serious alternative. We plan to analyze the benefits of such menus in a model with second
generation products.
Finally, we want to point out that the ideas developed in this article could be applied
to other related fields. In particular, this work is related to the literature on contests and
tournaments (Moldavanu and Sela (2001) for instance), for which patents are one of many
applications. Some authors look at the optimal allocation of prizes and study the effects
of having more than one winner. However, they concentrate on how the number of prizes
affects the level of effort exerted during the contest and ignore the tradeoff with the change
in social surplus afterwards. It is possible that such a tradeoff could be important in other
applications such as labor markets or political competition. For instance, although having
more than one winner in a labour market tournament could decrease the incentives to exert
effort, it could later foster healthy competition or helpful collaborations.
29
8
Appendix
Lemma 1
We first present the expected benefits if only one firm enters:
Z
πm +∞ −rt −λt
E[πa ] =
e λe dt
r 0
If both firms enter, the calculations follow the expression given in section 2.3, taking
however the speed of research fixed at value λ:
Z
Z
Z
Z
πm +∞ −rt −λt +∞ −λt2
πd T −rt −λt t −λt2
E[πb ] =
e λe
λe
e λe
λe
dt2 dt +
dt2 dt
r 0
r 0
t+T
0
Z
Z
πd +∞ −rt −λt t
e λe
λe−λt2 dt2 dt
+
r T
t−T
Z +∞
Z t+T
πd
πm
+
λe−λt
[ [e−rt − e−rt2 ] + [e−rt2 ]] λe−λt2 dt2 dt
r
r
0
t
Integrating and factorizing, we obtain the result:
·
¸
2πd − πm
λ2
λ h πm i
−(r+λ)T )
+ [1 − e
]
E[πb ] =
2λ + r r
r
(λ + r)(2λ + r)
¥
Lemma 2
We calculate E[Wb ], the expected social welfare at date 0 if both firms enter the race.
The instantaneous social benefit if the two firms don’t innovate within T of each other is
the social surplus from monopoly Sm . If they invent within T of each other, Sm until the
loser innovates as well and then Sd . The 2 firms are symmetric, so we can restrict ourselves
to the case where t < t2 and then multiply the result by 2. We therefore obtain:
Z
Z
2Sm +∞ −rt −λt +∞ −λt2
e λe
λe
E[Wb ] =
dt2 dt
r 0
t+T
¸
Z +∞
Z t+T ·
Sm −rt
Sd −rt2
−rt2
−λt
[e − e
+2
λe
] + [e
] λe−λt2 dt2 dt
r
r
0
t
Hence the result given in Lemma 2. ¥
30
Lemma 3
We calculate the expected social welfare at date 0 for a certain value of T, given that c
follows a uniform distribution:
£
¤
W (T ) = Ec 1c∈[0,E(πb )] (E(Wb ) − 2c) + 1c∈[E(πb ),E(πa )] (E(Wa ) − c)
1
= E(πb )E(Wb ) + (E(πa ) − E(πb ))E(Wa ) − (E(πb )2 + E(πa )2 )
2
We then calculate the marginal value at T=0 and find:
·
¸
r
λ
λ2
0
(2πd − πm )(Sm
W (0) =
− πm ) + 2πm (Sd − Sm )
r(2λ + r) (2λ + r)
λ+r
Hence the expression given in Lemma 3. ¥
Proposition 1
(a) A sufficient condition for (1) is −πm [Sm − πm ] + 2πm [Sd − Sm ] > 0 which can be further
simplified to find the sufficient condition: Sd > 32 Sm .
(c) Under Cournot competition and linear demand, πd = 49 πm , Sm = 23 πm and Sd = 16
π .
9 m
r
We can therefore rewrite condition (1): λ+r < 4. This condition is satisfied for any λ and
any r.
Under Bertrand competition, the duopoly profits are 0, therefore the condition can be writr
ten: 2Sd − 2Sm − Sm λ+r
+ πm > 0. Using the results Sm = 32 πm and Sd = S0 = 2πm , we see
that this condition is always satisfied.
(d) We now suppose demand is characterized by constant elasticity. Specifically P (q) = q −1/η
and C(q) = cq.
Under monopoly pricing
1
πm = η1 ( η−1
)η−1 and Sm = 1−1/η
( η−1
)η−1 − c( η−1
)η .
ηc
ηc
ηc
Furthermore, under Cournot competition:
1 2η−1 η−1
1
πd = 4η
( 2ηc )
and Sd = 1−1/η
( 2η−1
)η−1 − c( 2η−1
)η .
2ηc
2ηc
Therefore, if we express condition (1) in this special case, we obtain the sufficient condition: (2η − 1)η−1 [7η − 2] + (2η − 2)η−1 [4 − 10η] > 0. This condition is always satisfied if
η > 1.14 and η < 1015 . As the second constraint is unreasonably high, we can ignore it.
Furthermore, for η < 1, the monopolist maximization problem does not have a solution. We
can therefore state that if η ∈ [1, 1.14], the condition will not be satisfied.
31
1
In the case of Bertrand competition, πd = 0 and Sd = c1−η [ 1−1/η
− 1]. Condition (1)
η
η η
2η−1
2
can be rewritten: − η−1 + 2[(η − η + 1)( η−1 ) − η−1 ] > 0. This condition is not satisfied for
η ∈ [1, 1.17].
(e) The condition is increasing in λ and is a continuous function. Furthermore, as λ → +∞,
r
− πm ] + 2πm [Sd − Sm ] → [2πd − πm ][−πm ] + 2πm [Sd − Sm ] > 0. Therefore
[2πd − πm ][Sm λ+r
∗
there exists a λ such that statement (e) is satisfied.
(f ) Taking the limit as r → 0, we find that the condition is satisfied at the limit. As
for result (e), because the condition is continuous in r, the corollary follows. ¥
Proposition 2
The first inventor in the case of costless reverse engineering and small values of T, waits
before commercializing the product. The expected profits become:
Z
Z
Z
Z
πm +∞ −r(t+T ) −λt +∞ −λt2
πd T −rt −λt t −λt2
E[πb ] =
e
λe
λe
dt2 dt +
e λe
λe
dt2 dt
r 0
r 0
t+T
0
Z
Z
Z
Z
πd +∞ −λt t+T −rt2 −λt2
πd +∞ −rt −λt t
−λt2
e λe
λe
dt2 dt +
λe
e
λe
dt2 dt
+
r T
r T
t−T
t
Integrating and factorizing, we obtain the result:
·
¸
h i
£
¤ 2πd
λ2
λ
−(r+λ)T πm
−(r+λ)T
E[πb ] =
e
+ 1−e
2λ + r
r
r (λ + r)(2λ + r)
(9)
Therefore, the terms that change from the previous section are the following:
πm λ(λ + r) πd 2λ2
+
r 2λ + r
r 2λ + r
Sm 2λ(λ + r) 2Sd λ2
=−
+
r 2λ + r
r 2λ + r
E(πb0 )|T =0 = −
E(Wb0 )|T =0
So the marginal social gains at T=0 become
·
¸
r
λ
λ2
([2πd λ − πm (λ + r)] Sm
− πm + 2πm [Sd λ − Sm (λ + r)])
W (0) =
r(2λ + r) 2λ + r
λ+r
0
Hence the expression given in Proposition 2. ¥.
Lemma 4
This calculation follows exactly the calculation of E(πb ) in Lemma 1, but with the speed of
the process taking different values for the two firms.
32
Proposition 3
We start by proving the second part that characterizes the Nash equilibrium of this game.
Firm 1 will choose x1 so as to maximize the expected benefits given by Lemma 4 minus the
cost x1 . The FOC are the following:
·
¸
πm h0 (x1 )(h(x2 ) + r)
h(x1 )
πd 0
h(x1 )
+
h (x1 )T
−
e−(r+h(x1 ))T
r (h(x1 ) + h(x2 ) + r)2
r
h(x1 ) + r h(x1 ) + h(x2 ) + r
¸
·
h0 (x1 )(h(x2 ) + r)
πd
h0 (x1 )r
−(r+h(x1 ))T
−
+ (1 − e
)
r
(h(x1 ) + r)2 (h(x1 ) + h(x2 ) + r)2
¸
·
πd − πm
h(x2 )
h0 (x1 )(h(x2 ) + r)
−(r+h(x2 ))T
+
=1
(1 − e
)
r
(h(x2 ) + r) (h(x1 ) + h(x2 ) + r)2
We are studying a symmetric Nash equilibrium. Therefore this equilibrium x∗ is solution to:
¸
·
2 2h(x) + r
πm (h(x) + r) + πd T (h(x))
e−(r+h(x))T
h(x) + r
·
¸
(2h(x) + r)2
−(r+h(x))T
+ πd (1 − e
) r
− (h(x) + r)
(h(x) + r)2
r(2h(x) + r)2
+ (πd − πm )(1 − e−(r+h(x))T )h(x) =
h0 (x)
Taking this result for T = 0 we obtain the FOC characterizing the equilibrium amount of
research.
To prove the first part of Proposition 3, we take the total derivative of the first order
conditions previously obtained.
·
¸
4h0 (x)2 (2h(x) + r) − h00 (x)(2h(x) + r)2
0
dx πm h (x) − r
+
h0 (x)2
2h(x) + r
(2h(x) + r)2
dT [πd h(x)2
+ πd (r + h(x))(r
− (h(x) + r))
h(x) + r
(h(x) + r)2
+ (πd − πm )(r + h(x))h(x)]= 0
Simplifying this expression, we obtain:
(πm − 2πd )(r + h(x))h(x)
dx
=
0
2
00 (x)(2h(x)+r)2
dT
πm h0 (x) − r 4h (x) (2h(x)+r)−h
h0 (x)2
To determine the sign of this expression, we use the FOC characterizing the equilibrium
∗
x:
h0 (x∗ )(h(x∗ ) + r) πm
=1
(2h(x∗ ) + r)2 r
33
So, we can express the denominator as:
r
(2h(x) + r)2
h00 (x)
− r4(2h(x) + r) + 0 2 (2h(x) + r)2 < 0
h(x) + r
h (x)
So overall, because πm > 2πd , we find that
dx∗
(0)
dT
< 0. ¥
Proposition 4
The social welfare function is given by:
·
¸·
¸
Sm 2h(x∗ )
Sd − Sm
2h(x∗ )2
−(h(x∗ )+r)T
W (T ) =
+ [1−e
]
−2x∗ (10)
∗
∗
∗
r 2h(x ) + r
r
(h(x ) + r)(2h(x ) + r)
Taking the derivative at T=0, we find the result given in Proposition 4.
Consequence (a) is straightforward.
We now prove result (b) for h(x) = µx. The equilibrium amount of research x∗ is solution to
the characteristic equation obtained in Proposition 3. The positive solution to this quadratic
equation is:
q
2
8 πµm + πrm2
π
r
m
x∗ =
−
+
8r
2µ
8
So a positive solution does exist for h(x) = µx.
At the limit, as µ → +∞, we find that W 0 (0) → +∞. Therefore, as the function is continuous in λ, there exists such a µ∗ .
Proposition 5
From the result of Proposition 4, we see that a sufficient condition is:
dx∗
Sm
Sc − Sm h2 (x∗ )
(0)[
− 1] +
>0
dT
πm
r
2h(x∗ ) + r
Furthermore, we can rewrite
(11)
dx
:
dT
dx
(πm − 2πd )(r + h(x))h(x)
= (2h(x)+r)(−2h(x)−3r) h00 (x)
dT
r
+ h0 (x)2 (2h(x) + r)2
(h(x)+r)
dx
| <
and because h00 < 0, we have | dT
Therefore a sufficient condition is:
−[
(πm −2πd )(r+h(x))h(x)
(2h(x)+r)(2h(x)+3r)
r
(h(x)+r)
Sm
(h(x∗ ) + r)2
− 1](πm − 2πd )
+ Sc − Sm > 0
πm
h(x∗ )(2h(x∗ ) + 3r)
34
(12)
∗
2
)+r)
Furthermore, if r < 1, h(x(h(x
∗ )(2h(x∗ )+3r) < 1 and we therefore obtain the sufficient condition
described in proposition 5. ¥
Proposition 6
Part I
We supposed in the main text that the value of the flow x is decided at the start of
the race, although it can be stopped at any time. We are in fact going to do the derivations
for the case where the flow can be adapted through time and the success of the competitor is
unobservable. We show that if we use a stationary approximation of that second game, the
results are the same as if the value of the flow was fixed initially. The reader can however
jump to Part II directly to see the derivations of Proposition 6.
In the case where the flow can be adapted through time, the game is non-stationary.
Indeed the posterior probability that the other firm has been successful between t − T and t
increases with t. However, we argue here that it is well approximated by a stationary game.
If it was stationary for t ∈ [0, T ], the game would look identical at any later date and it
would therefore be stationary thereon: for t > T , because the process is memoryless, the
probability that the other firm has succeeded between [t − T, T ] would be constant. However
it is not stationary in that initial time frame: the probability that the other firm has already
succeeded, is initially 0 for t = 0 and then increases between [0, T ].
We will approximate this game by a stationary game such that, even at t = 0, the
probability that the competing firm has already invented takes the steady state value. We
argue this is a good approximation for two reasons. Firstly the game is non stationary because of the strategic decision just after the starting date of the race (period [0, T ]). However,
the starting date of the race is not a very well defined concept. Secondly, we are interested
in results for small values of T, therefore the effect of this initial period will be negligible
and the steady state will quickly be reached.
We consider here the stationary game which is an approximation of the case with unobservable effort. There are two possible events:
(a) Either the competitor has invented between [t − T, t]
(b) Or he has not yet invented.
We start by calculating the expected benefits in case (a), when the competitor has already
invented. We denote here t3 = t2 + T , i.e. the date when it will no longer be possible to
patent, because the competitor will own exclusive rights.
Z t3
Z T
πc −rt
1 − e−rt
−h(x2 )t3
( e − x1
E(πa ) =
h(x2 )e
) h(x1 )e−h(x1 )t dtdt3
r
r
0
0
35
·
¸
¸
£
¤
£
¤
x1
πc h(x1 )
x1
r
h(x2 )
−h(x2 )T
E(πa ) =
−
1−e
+
1 − e−h(x1 +h(x2 ))T
r h(x1 ) + r
r h(x1 ) + r
r h(x2 ) + h(x1 )
·
¸
£
¤
πc + x1 h(x1 )
h(x2 )
−
1 − e−h(x1 +h(x2 )+r)T
r
h(x1 ) + r h(x1 ) + h(x2 ) + r
·
We now turn to case (b), where the competitor has not yet invented. The probability of
that case happening is e−h(x2 )T . In that case, the benefits are the same as in section 3.4:
E[πb ] = e−h(x2 )T [
πm
h(x1 )
πd
h(x1 )
h(x1 )
+
(1 − e−(r+h(x1 ))T )(
−
)
r h(x1 ) + h(x2 ) + r
r
h(x1 ) + r h(x1 ) + h(x2 ) + r
πd − πm
h(x1 )h(x2 )
+
(1 − e−(r+h(x2 ))T )(
)]
r
(h(x2 ) + r)(h(x1 ) + h(x2 ) + r)
Finally, we need to recalculate the expected cost that are different from those in section
3.4 as the flow can be interrupted at any point. The cost is characterized by the fact that if
t1 < t2 + T , firm 1 stops spending on research at t1 and if t1 > t2 + T , firm 1 stops at t2 + T .
So:
Z T
1
x1 [1 − e−rt ]h(x1 )e−h(x1 )t dt +
E(C) =
r
0
Z +∞
Z +∞
1
−rt
−h(x1 )
x1 [1 − e ]h(x1 )e
h(x2 )e−h(x2 )t dt2 dt1
r
T
t−T
Z +∞ Z t−T
1
+
x1 [1 − e−r(t2 +T ) ]h(x1 )e−h(x1 )t h(x2 )e−h(x2 )t dt1 dt2
r
T
0
Simplifying this expression, we find:
·
¸
x1
h(x2 )
−(r+h(x1 ))T
E(C) =
1−e
r + h(x1 )
r + h(x1 ) + h(x2 )
Therefore the expected profit can be written E[π] = E[πa ] + E[πb ] − e−h(x2 )T E(C). We
derive the first order conditions corresponding to this problem. To make the derivations
more readable we present the derivatives of the different terms separately.
The first term (derivatives for case (a)):
·
¸
¤
r(2h(x) + r)2
r(2h(x) + r)2 £
r(2h(x) + r)2
−h(x)T
πc
−
+
x
1
−
e
1
(h(x) + r)2
h0 (x)(h(x) + r)
(h(x) + r)2
·
¸
·
¸
¤
(2h(x) + r)2 −2h(x)T
h(x)(2h(x) + r)2 £
(2h(x) + r)2
−2h(x)T
1−e
+ x1 T
− x1
+
e
2h0 (x)
(2h(x))2
2
¸
·
¤
£
(2h(x) + r)h(x)
h(x)2
h(x)2 (2h(x) + r)
+ (πc + x1 )r
− (πc + x1 )
1 − e−(2h(x)+r)T
−
0
2
h (x)(h(x) + r)
(h(x) + r)
h(x) + r
h(x)2 (2h(x) + r) −(2h(x)+r)T
−(πc + x1 )T
e
h(x) + r
36
Plus the second term (derivatives for case (b)):
(2h(x) + r)2
− h(x)(2h(x) + r))e−(r+h(x))T
h(x) + r
(2h(x) + r)2
+ πd (1 − e−(r+h(x))T )(r
− (h(x) + r))
(h(x) + r)2
+ (πd − πm )(1 − e−(r+h(x))T )h(x) ]
+e−h(x)T [πm (h(x) + r) + πd T (h(x)
Is equal to the marginal costs
r + h(x) − xh0 (x) r(r + 2h(x))2
h(x)
(1 − e−(r+h(x))T
)
2
0
(r + h(x))
h (x)
r + 2h(x)
xh(x)
x
r(r + 2h(x))e−(r+h(x))T +
rh(x)e−(r+h(x))T ]
+ T
r + h(x)
r + h(x)
= e−h(x)T [
Taking these first order conditions at T = 0 we find the result in the second part of Proposition 6 characterizing the equilibrium.
Part II
To obtain the first result of Proposition 6 we totally differentiate the expression characterizing the equilibrium and take the value at T = 0. We note that the terms corresponding
to case (a) disappear and therefore the result in the case where the flow is fixed initially are
the same as for the stationary approximation of the other game.
·
¸0
r + h(x) (r + h(x) − xh0 (x))r(r + 2h(x))2
dx πm h (x) −
[
r + 2h(x)
(r + h(x))2 h0 (x)
(r + h(x) − xh0 (x)) r(r + 2h(x))2 (h0 (x)r)
+
(r + h(x))2
h0 (x)
(r + 2h(x))2
(h(x) + xh0 (x))(r + h(x)) − h0 (x)xh(x)
]
−r
(r + h(x))2
r + h(x) − xh0 (x) rh(x)(r + 2h(x))
dT [(2πd − πm )h(x)(r + h(x)) −
r + h(x)
h0 (x)
xh(x)
−
r(2h(x) + r) + xh(x)r] = 0
r + h(x)
0
Let’s first simplify the term corresponding to the derivative with respective to T. We can
simplify it as:
¸
·
rh(x)(r + 2h(x))
+ xh(x)r
dT (2πd − πm )h(x)(r + h(x)) −
h0 (x)
37
Using the result of Lemma 7 ((h(x∗ )+r)πm =
r(2h(x∗ )+r)
−rx∗ ),
h0 (x∗ )
we can simplify the expression.
dT [2(πd − πm )h(x)(r + h(x))] < 0
0
2
(x))(r(r+2h(x)))
For the first expression we have to calculate the derivative of: [ (r+h(x)−xh
].
(r+h(x))2 h0 (x)
Taking the derivative and simplifying we obtain:
dx [πm h0 (x) +
h00 (x)
r(2h(x) + r) − r]
h0 (x)2
Furthermore, we know that in equilibrium, (h(x∗ )+r)πm =
simplify further the expression and we find:
r(2h(x∗ )+r)
−rx∗ .
h0 (x∗ )
We can therefore
h00 (x∗ )
dx [h(x ) − x h (x ) + 0 ∗ 2 (2h(x∗ ) + r)(h(x∗ ) + r)]
h (x )
∗
The sign of
equivalent to
dx
dT
∗ 0
∗
is therefore going to be the sign of this second expression. So
h(x∗ ) − x∗ h0 (x∗ ) +
dx
dT
> 0 is
h00 (x∗ )
(2h(x∗ ) + r)(h(x∗ ) + r) > 0
h0 (x∗ )2
This is the condition given in Proposition 6. ¥
Proposition 7
Once the positions are fixed, the game is identical to the one in section 3.4. Therefore
whether incentives to innovate will decrease or increase with T will depend on the comparison between πm and 2πd . The value of profits is what changes compared to section 3.4.
Furthermore we know that if the transport cost is high enough, we can have 2πd > πm .
In particular it might be optimal for them to cover only part of the market and not sell to
customers in between.
Proposition 8
To determine the socially optimal mix of T and L we need to calculate the social surplus
and the conditions characterizing the equilibrium amount of research x∗ when the length L
of the patent is finite. It is a small modification of the results obtained in section 3.
The expected social surplus is given by:
¸
·
2h(x∗ )
Sd −rL
S0 −rL
Sm
−rL−h(x∗ )T
−h(x∗ )T
E[S] =
(1 − e
) − e (1 − e
)+ e
2h(x∗ ) + r r
r
r
∗ 2
Sd − Sm
2h(x )
∗
+ [1 − e−(r+h(x ))T ]
− 2x∗
∗
∗
r
(2h(x ) + r)(h(x ) + r)
38
Note that at the limit, when L → +∞, we find the social surplus of section 3.4. The introduction of a finite patent length decreases the time period where monopoly or duopoly
occurs in favour of a period of perfect competition, leading to the social surplus S0 .
We then determine x∗ .
The expected profit of firm 1 given firm 2 is investing an amount x2 is given by:
πm
h(x1 )
[1 − e−h(x2 )T e−rL ]
r
h(x1 ) + h(x2 ) + r
πd
h(x1 )h(x2 )
+ [1 − e−(r+h(x1 ))T ]
r
(h(x1 ) + r)(h(x1 ) + h(x2 ) + r)
πd − πm
h(x1 )h(x2 )
+
[1 − e−(r+h(x2 ))T ]
r
(h(x2 ) + r)(h(x1 ) + h(x2 ) + r)
πd
h(x2 )
πd −rL
h(x1 )
−
e (1 − eh(x2 )T )
− e−rL (1 − eh(x1 )T )
r
h(x1 ) + h(x2 ) + r
r
h(x1 ) + h(x2 ) + r
E[π1 ] =
Using the notation h∗ = h(x∗ ), the FOC in equilibrium are therefore given by:
∗
∗
πm (h∗ + r)(1 − e−rL e−h T ) − πm h∗ (1 − e−(r+h )T )
∗
∗
(h∗ )2 (2h∗ + r)
∗
−(r+h∗ )T rh (3h + 2r)
+ πd T e−(r+h )T
+
π
(1
−
e
)
d
h∗ + r
(r + h∗ )2
r(2h∗ + r)2
∗
∗
− πd T e−rL e−h T h∗ (2h∗ + r) − πd e−rL (1 − e−h T )r =
(h∗ )0
We rewrite this equation as G(x∗ ) = K.
The socially optimal mix is obtained by maximizing the following problem (where E(S)
and G(x∗ ) were determined above).
maxT,I E(S(T, I))
subject to T ≥ 0 and T ≤ L and G(x∗ ) ≥ K.
We can now write the Lagrangian of this problem:
L = E(S) + λ(K − G(x∗ )) + µ(−T ) + ν(T − L)
with λ ≤ 0, µ ≤ 0 and ν ≤ 0.
We want to examine under what conditions T ∗ = 0 cannot be a solution. If it is a so∂L
lution, the Kuhn Tucker conditions will impose: ∂T
(T ∗ = 0) = 0 and ∂L
(T ∗ = 0) = 0.
∂L
39
Furthermore, the complementary slackness conditions impose ν = 0. We first determine the
derivative with respect to intertemporal breath:
∂L
2(h∗ )2
Sd − Sm
(T = 0) =
(1 − e−rL )(
)
∗
∂T
2h + r
r
− λ[−πm (h∗ + r)h∗ (1 − e−rL ) + 2πd (h∗ + r)h∗ (1 − e−rL )] − µ
∂L
(T ∗ = 0) > 0, then T ∗ = 0 cannot be a solution. Using the fact that µ ≤ 0 we find
If ∂T
∂L
a condition such that ∂T
(T ∗ = 0) > 0. We call it condition (A):
£
¤
2(h∗ )2
−rL Sd − Sm
∗
∗
−rL
∗
∗
−rL
(1
−
e
)(
)
−
λ
−π
(h
+
r)h
(1
−
e
)
+
2π
(h
+
r)h
(1
−
e
)
≥0
m
d
2h∗ + r
r
To obtain the value of lambda we use the fact that
∂L
(T
∂L
= 0) = 0
∂L ∗
2h∗ −rL Sm − S0
(T = 0, L∗ ) =
e (
) − λπm (h∗ + r)re−rL = 0
∗
∂L
2h + r
r
We simplify the expression and find that condition (A) is equivalent to:
Sd − Sm
πm − 2πd
≥
S0 − Sm
πm
Consequence (a) is obtained by using the values of surplus and monopoly with linear demand
under Cournot competition.
For consequence (b) we consider Bertrand competition. Under perfect Bertrand competition, Sd = S0 and πd = π0 , so the condition is weakly satisfied. We examine what happens
if for a small departure from perfect Cournot competition. We suppose there is for example
a small rigidity in prices δ.
We use Taylor expansions of the surplus and profits:
Sd (δ) = Sd (0) + Sd0 (0)δ + Sd00 (0)δ 2 + ◦(δ 2 ).
Because surplus is maximal at Bertrand prices Sd0 (0) = 0. We do a similar expansion for
profits, using the fact πd = 0 to rewrite the condition:
S0 + Sd00 (0)δ 2 + ◦(δ 2 ) − Sm
πm − 2πd0 (0)δ − 2 ◦ (δ)
≥
S0 − Sm
πm
This can be rewritten:
Sd00 (0)δ 2 + ◦(δ 2 )
−2πd0 (0)δ − 2 ◦ (δ)
≥
S0 − Sm
πm
40
Because πd0 (0) > 0, irrespective of the nature of the imperfection, the condition will be satisfied at the limit. ¥
Proposition 9
We supposed that D is the social surplus from the set of subsequent innovations and that
it will be obtained after R (research) time from the disclosure date of the initial invention
(supposed to be T ). The expected social surplus obtained in Proposition 8 is modified to
include the new term corresponding to second generation products:
·
¸
Sm
2h(x∗ )
Sd −rL
S0 −rL D −r(T +R)
−rL−h(x∗ )T
−h(x∗ )T
E[S] =
(1 − e
) − e (1 − e
)+ e
+ e
2h(x∗ ) + r r
r
r
r
Sd − Sm
2h(x∗ )2
∗
− 2x∗
+ [1 − e−(r+h(x ))T ]
∗
∗
r
(2h(x ) + r)(h(x ) + r)
To simplify the problem we suppose that these subsequent innovations will not affect the
expected profit of the initial innovators. Therefore the initial conditions are not modified.
We examine the conditions such that T ∗ = L∗ can be a solution to the maximization problem.
∂L
The first condition is: ∂T
(T ∗ = L∗ ) = 0. This can be rewritten:
2h(x∗ )
−
De−r(T +R) − µ + ν = 0
∗
2h(x ) + r
The constraint T ≥ 0 will not bind and therefore by complementary slackness, µ = 0.
∂L
Furthermore, ν ≤ 0, therefore, if D > 0, the condition ∂T
(T ∗ = L∗ ) = 0 has no solution and in a socially optimal mix, we will have T ∗ < L∗ . ¥
Proposition 10
The breadth α is the probability that the courts will rule that the two inventions should
be covered by the same patent. Therefore with probability α, the profits take the same form
as in section 3 and with probability 1 − α, both firms can patent at any point. We therefore
note the profits:
E[π] = α E[πsame
patent ]
+ (1 − α) E[πdif f erent
patent ]
where
E[πsame
patent ]
=
πd
h1 h2
πm
h1
+
(1 − e−(r+h1 )T )
r h1 + h2 + r
r
(h1 + r)(h1 + h2 + r)
πd − πm
h1 h2
+
(1 − e−(r+h2 )T )
r
(h2 + r)(h1 + h2 + r)
41
E[πdif f erent
patent ]
=
πm
h1
πd
h1 h2
+
r h1 + h2 + r
r (h1 + r)(h1 + h2 + r)
πd − πm
h1 h2
+
r
(h2 + r)(h1 + h2 + r)
We solve the maximization problem and find the following first order conditions
∗ +r)T
πm (h∗ + r) + απd T e−(h
∗ +r)T
+απd (1 − e−(h
) [r
(h∗ )2 (2h∗ + r)
(h∗ + r)
(2h∗ + r)2
∗
−(h∗ +r)T
−
(h
+
r)]
+
α(π
−
π
)
(1
−
e
)h∗
d
m
(h∗ + r)2
(2h∗ + r)2
+(1 − α)πd [r ∗
− (h∗ + r)]
2
(h + r)
r(2h∗ + r)2
+(1 − α) (πd − πm )h∗ =
(h∗ )0
We now move to the second step and calculate the expected social surplus. Once again
it is a weighted sum of the usual expression (when it is the same patent) and the expression
when the two locations are not covered by the same patent.
W =
Sm 2h∗
2(h∗ )2
−(h∗ +r)T Sd − Sm
+
α
(1
−
e
)
r 2h∗ + r
r
(h∗ + r)(2h∗ + r)
2(h∗ )2
Sd − Sm
− 2x∗
+ (1 − α)
r
(h∗ + r)(2h∗ + r)
We finally move to the third step and examine the solutions of the constrained maximization
problem. We can write the Lagrangian of this problem:
L = W + λ(K − G(x∗ )) + µ(−T ) + ν(−α)
with λ ≤ 0, µ ≤ 0 and ν ≤ 0.
We want to examine under what conditions T ∗ = 0 cannot be a solution. If it is a so∂L
lution, the Kuhn Tucker conditions will impose: ∂T
(T ∗ = 0) = 0 and ∂L
(T ∗ = 0) = 0.
∂α
Furthermore, the complementary slackness conditions impose ν = 0. We first determine the
derivative with respect to intertemporal breath:
Sd − Sm 2(h∗ )2
∂L
(T = 0) = α
∗+r
∂T
r
2h
·
¸
2h∗ + r
(2h∗ + r)2
+λ α πd h2 ∗
+ α πd [r
− (h∗ + r)2 ] + α (πd − πm ) (r + h∗ )h∗ − µ
h +r
h∗ + r
∂L
If ∂T
(T ∗ = 0) > 0, then T ∗ = 0 cannot be a solution. Using the fact that µ ≤ 0 we find
∂L
a condition such that ∂T
(T ∗ = 0) > 0. We call it condition (A):
42
Sd − Sm 2(h∗ )2
r
2h¸∗ + r
·
2h∗ + r
(2h∗ + r)2
+λ α πd h2 ∗
+ α πd [r
− (h∗ + r)2 ] + α (πd − πm ) (r + h∗ )h∗ ≥ 0
∗
h +r
h +r
α
To obtain the value of lambda we use the fact that
∂L
(T
∂L
= 0) = 0
∂L
Sd − Sm
2(h∗ )2
(T = 0) = −
∂α
r
(h∗ + r)(2h∗ + r)
·
¸
(2h∗ + r)2
∗
2
∗
+λ −πd [r
− (h + r) ] − (πd − πm ) h
h∗ + r
We use this expression to find the value of λ. Therefore, the intertemporal breadth will be
used if:
£ ∗ +r ¤
πd h2 2h
+ D
h∗ +r
1−
>0
D
Where:
D = πd [r
(2h∗ + r)2
− (h∗ + r)2 ] + (πd − πm ) (r + h∗ )h∗
∗
h +r
This is equivalent to:
(2h∗ + r)2
πd [r
− (h∗ + r)2 ] + (πd − πm ) h∗ ≤ 0
∗
h +r
Simplifying this expression, we find the condition stated in proposition 10:
πm ≥ πd r
3h∗ + 2r
(h∗ + r)2
Proposition 11
The calculation of the equilibrium conditions and the social surplus are the same as in
Proposition 9. We can write the lagrangian of the problem:
L = W + λ(K − G(x∗ )) + µ(−T ) + ν(−α)
with λ ≤ 0, µ ≤ 0 and ν ≤ 0.
We can find the partial derivatives of the Lagrangian.
2(h∗ )2
Sd − Sm
∂L
(T = 0) =
(1 − e−rL )(
)
∗
∂T
2h + r
r
− λ[−(πm + πL )(h∗ + r)h∗ (1 − e−rL ) + 2πd (h∗ + r)h∗ (1 − e−rL )] − µ
43
and
∂L
S 0 (α) 2(h∗ )
0
(T = 0) = m
− λ[πm
(α)(h∗ + r)h∗ − πL0 (α)(h∗ )2 ]
∂α
2h∗ + r
Using the same methods as in Propositions 9 and 10, we find the following condition, guaranteeing that T ∗ = 0 cannot be a solution of the constrained maximization problem.
0
Sd − Sm
Sm
(α)
>−
0 (α) − π 0 (α)
πm + πL − 2πd
πm
L
h(x∗ )
h(x∗ )+r
We can then simplify this expression to find the one presented in Proposition 11.
Lemma 5
The proof is very similar to the proof given in Hopenhayn and Mitchell (2001) in Proposition
1 page 155. We make the same argument replacing breadth by intertemporal breadth, which
we argue is a much more practical solution. This changes the first sorting condition. We find
that Π1 (T, L, θ) is strictly decreasing in θ. Indeed increasing intertemporal breath decreases
the innovator’s expected profits whereas increasing the classical breadth increases them. The
condition is therefore reversed compared to the one found by Hopenhayn and Mitchell (2001).
Proposition 12
We make a simplifying assumption by imposing a restriction on P (θ, t): the second generation product cannot be invented before t + T (where t is the invention date of the initial
product). In the main text this is translated into the fact that stage 4 starts after t + T .
The expected profits of the first innovator when he is successful is given by:
Z
Z
L
−λT
π=e
e
−rt
L
[πm (1 − P (θ, t)) + P (θ, t)F2 ] dt +
0
Z
+(1 − e−λT )
0
L
0
Z
[
T
0
e−rt
πm − πc
(1 − e−rt2 )dt2 ]λe−λt2 dt
r
F2
[πc (1 − P (θ, t)) + P (θ, t) ] dt
2
Note that λ = λ(θ), it is endogenously set by the firms participating in the patent race. If
the licensing fee paid by the second generation product inventor is high enough, the expected
profits will be increasing in θ, and therefore λ(θ) will be an increasing function.
We can now determine if the sorting conditions are satisfied.
·
¸
·
¸
F
F −rL −λT
∂π
0
−rL −λT
−rL
−λT
0
= P (θ, t) (F − πm )e e
+ ( − πc )e (1 − e
) − λ (θ) πm − πc +
e e
∂L∂θ
2
2
We have λ0 (θ) > 0 and P 0 (θ, t), therefore,
∂π
∂L∂θ
44
< 0. The first condition is satisfied.
Z L
∂π
F2
0
−λT
= λ (θ) (1 − λ(θ)T ) e
[
e−rt [(πc − πm )(1 − P (θ, t)) + P (θ, t) ] dt
∂T ∂θ
2
0
Z L
πm − πc 0
F2
+ λ(θ)e−λT [
e−rt P 0 (θ, t)[πm − πc − ]] +
λ (θ)[e−λT − e−(λ+r)T ][1 − λT ]
2
r
0
So if λT > 1 and F2 > πm − πc , the second sorting condition will be satisfied. For F2
high enough, both these conditions will be satisfied. The benchmark value F2∗ is given as the
smallest value of the licensing fee such that these two conditions are satisfied and λ0 (θ) > 0.
45
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