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\\
Optimal Sharing Strategies in Dynamic Games of\\
Research and Development\\
by\\
Nisvan Erkal* and Deborah Minehart**\\
EAG 07-7 \ \ \ \ April 2007\\
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\bigskip
\bigskip
EAG Discussion Papers are the primary vehicle used to disseminate research
from economists in the Economic Analysis Group (EAG) of the Antitrust
Division. These papers are intended to inform interested individuals and
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and conclusions expressed herein are solely those of the authors and do not
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Information on the EAG research program and discussion paper series may be
obtained from Russell Pittman, Director of Economic Research, Economic
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Recent EAG Discussion Paper titles are listed at the end of this paper. To
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\bigskip
* Department of Economics, University of Melbourne, Victoria 3010, Australia.
** Economist, U.S. Department of Justice, Washington, D.C. 20530, USA. \
We have benefited from conversations with Eser Kandogan, Suzanne Scotchmer,
Ben Shneiderman, and members of the Research Division of IBM. Nisvan Erkal
thanks the Faculty of Economics and Commerce, University of Melbourne for its
financial support.
\begin{abstract}
This paper analyses the dynamic aspects of knowledge sharing in R\&D rivalry.
In a model where research projects consist of $N$ sequential stages, our goal
is to explore how the innovators' incentives to share intermediate research
outcomes change with progress and with their relative positions in an R\&D
race. We consider an uncertain research process, where progress implies a
decrease in the level of uncertainty that a firm faces. We assume that firms
are informed about the progress of their rivals and make joint sharing
decisions either before or after each success. Changes in the firms' absolute
and relative positions affect their incentives to stay in the race and the
expected duration of monopoly profits if they finish the race first. We show
that firms always prefer to have sharing between their independent research
units if they are allowed to collude in the product market. However, competing
firms may have either decreasing or increasing incentives to share
intermediate research outcomes throughout the race. If the lagging firm never
drops out, the incentives to share always decrease over time as the research
project nears completion. The incentives to share are higher earlier on
because sharing has a smaller impact on each firm's chance of being a
monopolist at the end of the race. If the lagging firm is expected to drop
out, the incentives to share may increase over time. We also use our framework
to analyze the impact of patent policy on the sharing incentives of firms and
show that as patent policy gets stronger, sharing incentives may decrease or
increase depending on whether or not the lagging firm has increased incentives
to drop out.
\end{abstract}
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\section{Introduction\label{introduction}}
The ability to create knowledge-based assets plays an increasingly important
role in determining firms' competitiveness in the market place. The goal of
this paper is to analyze dynamic aspects of knowledge sharing in research and
development (R\&D) rivalry. Knowledge sharing is an important way in which
firms can acquire the technological knowledge they need during their
innovation process. Firms are likely to benefit from sharing knowledge with
competitors. However, such alliances pose especially difficult challenges.
This leaves us with the following question. When would we expect cooperation
to emerge between competitors?
In the economics literature, knowledge sharing between rival firms has been
the focus of many papers. These papers have mainly studied firms' incentives
to share research outcomes at one point in time, either before the start of
research, as in the case of research joint ventures, or after the development
of a technology, as in the case of licensing.\footnote{See, for example,
Kamien (1989) on licensing, and Katz (1986), D'Aspremont and Jacquemin (1988)
and Kamien, Muller and Zang (1992) on research joint ventures. Patenting and
informal sharing between employees of firms are two other methods through
which knowledge may be disseminated between firms. See, for example, Scotchmer
and Green (1990) on early innovators' incentives to patent and Severinov
(2001) on informal sharing between employees.} In reality, the decision to
share intermediate steps with rivals may be an integral part of a dynamic
research process. Hence, the aim of this paper is to ask not whether but when
firms prefer to share their research outcomes during a research process and
what the emerging patterns of sharing activities are.
While sharing may cause researchers to benefit from each other's expertise and
generate better ideas, it may also result in a reduction in the commercial
value of their ideas. From a social welfare perspective, sharing of research
outcomes is desirable because it results in less duplication. Hence, it is
important to determine how close profit-driven firms come to maximizing
welfare. In the economics literature, knowledge spillovers are stated as one
of the most important reasons for rival firms to agree to share knowledge.
However, from a dynamic perspective, another important aspect is uncertainty.
The process of research is generally characterized by a high level of
uncertainty in the beginning. Progress in research can be described as a
decrease in the level of uncertainty that researchers face. Hence, one of the
novel aspects of this project is to focus on the role uncertainty plays in the
decisions to share knowledge and to analyze how firms' incentives to share
research outcomes change during a research process as the level of uncertainty
they face decreases.
We assume that research projects consist of several sequential steps.
Researchers cannot proceed to the next step before successfully completing the
prior step. Moreover, they cannot earn any profits before completing all steps
of the project. In a dynamic R\&D process, firms' incentives to share change
as their positions in the race change for two reasons. First, the expected
duration of monopoly profits for the leading firm depends on the progress the
firms make during the research process. Second, the probability that any two
firms will be rivals in the product market changes with progress.
An important feature of the model is that we assume the different steps of
research are symmetric in all respects except in regards to how far away they
are from the end of the project. In other words, the options and technology
available to the firms are the same in all steps of the research process. We
deliberately assume that there are no spillovers during the research process.
It has been stressed in the literature that firms may have higher spillover
rates and bigger appropriability problems in earlier stages of research than
in later stages of research.\footnote{See, for example, Katz (1986), Katz and
Ordover (1990), and Vonortas (1994).} Although the rate of spillovers may
shape the incentives to share, we show that it is not the only relevant
factor. Assuming that there are no spillovers between the research efforts of
different firms allows us to focus on the role uncertainty plays in knowledge sharing.
We assume that firms are informed about the progress of their rivals and make
joint sharing decisions either before or after each success. While sharing may
cause researchers to benefit from each other's expertise and help them avoid
wasteful duplication of R\&D, it may also result in a reduction in the
commercial value of their ideas. Because sharing decreases the lead of one
firm, it reduces the expected profits that the leader derives from finishing
the race first and being a monopolist for some period of time. This cost is
even greater if, but for the sharing, the lagging firm would drop out of the race.
Hence, the decision to share and the pattern of sharing activities critically
depend on the lagging firm's incentives to stay in the race in case of no
sharing. The results reveal that firms always prefer to have sharing between
their independent research units if they are allowed to collude in the product
market. Under rivalry, the incentives to share intermediate research outcomes
decreases monotonically with progress if the lagging firm is expected never to
drop out. The incentives to share are higher earlier on because there is more
uncertainty earlier on. Sharing has a smaller impact on each firm's chance of
being a monopolist at the end of the race.
If the lagging firm is expected to drop out, the incentives to share may
increase with progress. This is because earlier in the research process the
lagging firm may have a higher incentive to drop out and, hence, the leading
firm may have a higher chance of eliminating rivalry by not sharing. We also
illustrate that the incentives to share increase as the gap between the firms decreases.
We next use our framework to analyze the impact of patent policy on firms'
sharing decisions. The strength of patent policy can have an important impact
on firms' sharing decisions because it determines the costs of inventing
around patented technologies. We show that if a strengthening in patent policy
causes a change in the investment decision of the lagging firm at any of the
asymmetric histories, then sharing incentives in general get weaker.
Otherwise, they generally get stronger.
In addition to contributing to the literature on knowledge sharing, this paper
is also related to the literature on the management of innovation (Aghion and
Tirole, 1994a and 1994b). The design of an optimal R\&D strategy is a
multi-faced problem. Two aspects of this problem, regarding the intensity of
the R\&D effort and the riskiness of the R\&D projects chosen by firms, have
been dealt with extensively in the literature.\footnote{See Reinganum (1989)
for a survey of the papers that focus on the intensity of firms' R\&D efforts.
Bhattacharya and Mookherjee (1986), Klette and de Meza (1986) and Dasgupta and
Maskin (1987) analyze the riskiness of the research projects chosen by firms.
Cardon and Sasaki (1998) analyze whether firms prefer to work on similar or
different R\&D paths.} In this paper we are interested in analyzing how the
optimal strategies of firms change with progress. Other papers that have
studied how firms' optimal strategies change over time in a dynamic model of
R\&D are Grossman and Shapiro (1986 and 1987), Cabral (2003) and Judd (2003).
Grossman and Shapiro (1986 and 1987) analyze how firms vary their efforts over
the course of a research project. In an infinite-period race, Cabral (2003)
allow firms to choose between two research paths with different levels of
riskiness. He shows that the leader chooses a safe technology and the laggard
chooses a risky one. Judd (2003) shows that there is excessive risk-taking by
innovators. Our paper differs from these papers because we analyze how firms'
incentives to share and diversify change over the course of a research project.
The paper proceeds as follows. In the next section, we describe our set-up. In
Section \ref{benchmark}, we explore what happens if the firms are allowed to
collude in the product market. In Section \ref{ex-post-sharing}, we analyze
the effect of competition on the dynamic sharing incentives of firms in a
model with ex-post sharing contracts and $2$ research steps. We consider the
case of $N$ research steps in Section \ref{N-step}. In Sections
\ref{ex-ante-sharing} and \ref{asymmetric-firms}, we discuss extensions of our
basic model with ex-ante sharing contracts and asymmetric firms respectively.
After discussing the impact of patent policy on sharing incentives in Section
\ref{IP-policy}, we conclude in Section \ref{conclusion}.
\section{Model\label{model}}
\subsection{Research Environment\label{research-environment}}
Since we are interested in the effect of competition on firms' incentives to
share, we consider an environment with two firms, $i=1,2$, that each invest in
a research project. On completion of the project, a firm can produce output in
a product market. We assume the firms produce goods that may be either
homogeneous or differentiated, and that they compete as duopolists in the
product market.\footnote{We assume that the firms conduct the research to
solve the same technical problem. However, unmodelled differences in
production technologies can still lead them to produce differentiated
products.}
To capture the idea of progress, we assume that a research project has $N$
distinct steps of equal difficulty. Hence, we assume that the firms divide the
research project into different steps and that each firm defines the steps in
the same way. A firm cannot start to work on the next step before completing
the prior step, and all steps of the project need to be completed successfully
before a firm can produce output. There is no difference between the steps in
terms of the technology or the options available to the firms.
In the literature on multi-stage research, the phases of research are often
thought of as qualitatively different. For example, there may be two steps
identified as ''research''\ and ''development.''\ We do not make this
distinction, but rather we seek to derive endogenous differences in the
research phases that result from dynamics in the decisions made by the firms.
A basic intuition is that as firms approach the end of the research process,
their decisions might increasingly reflect the impending rivalry.
We assume that each firm operates an independent research facility. We model
research activity using a Poisson discovery process. Time is continuous, and
the firms share a common discount rate $r$. To conduct research, a firm must
incur a flow cost $c$ per unit of time.\footnote{We do not allow the firms to
choose continuous levels of research effort in our basic model. This
assumption can be motivated by presuming a fixed amount of effort that each
firm can exert, which is determined by the capacity of its R\&D division. As
an example, Khanna and Iansiti (1997) explain that given the highly
specialized nature of the R\&D involved in designing state-of-the-art
mainframe computers, firms in this industry find it very expensive to increase
their number of researchers available to them. We relax this assumption later
on and consider the case of continuous effort levels.} Investment provides a
stochastic time of success that is exponentially distributed with hazard rate
$\alpha$. This implies that at each instant of time, the probability that the
firm completes a step is $\alpha$. After completing a step, a firm can
immediately begin research on the next step. The successes of the two firms
are statistically independent. To represent the progress made by the firms, we
use the notation of a research history $(k_{1},k_{2})$, where $k_{i}$ stands
for the number of successes of firm $i$.
At each point in time prior to completing the project, a firm decides whether
or not to invest. A decision not to invest is assumed to be irreversible and
equivalent to dropping out of the game.\footnote{Later, we may relax this
assumption so that the decision not to invest can be reversed. We do not think
that the qualitative nature of our results will change.} Each firm is risk
neutral and makes decisions to maximize its discounted expected continuation
payoff given the strategy of the other firm. The payoff structures are more
fully described below. A firm that drops out earns a continuation payoff of
zero. Given the memorylessness nature of the Poisson process, if a firm is
conducting research, it will not stop unless there is a change in its relative
position in the research process. If the rival completes one of the steps
successfully, the firm may decide to drop out of the game at this point. We
implicitly assume that when one firm develops a step successfully, it does not
result in any technological spillovers. The successful firm can either keep
the innovation a secret or patent it. Patenting does not prevent the rival
from developing a non-infringing technology that serves the same purpose.
We will also allow firms to share their research. If one firm has completed
one (or more) steps that the other firm has not, the leading firm can share
its research with the lagging firm. After sharing, both firms can proceed to
the next research step.\footnote{Sharing in our model has the same impact as
patenting in Scotchmer and Green (1990). In both models, the lagging firm can
proceed to the next step after disclosure.} The timing of sharing decisions
and the contracts that govern the sharing process are described below.
Regarding the information structure, we make the following assumptions. The
firms will be able to share their research successes, but one firm cannot
acquire the rival's innovation without such sharing. For example, a firm
cannot observe the technical content of the rival's research without explicit
sharing.\footnote{Alternatively, we could assume that successful firms win
immediate patents. A leading firm could then prevent a lagging firm from
copying its research by enforcing its patent. If both firms complete the same
step, they win non-infringing patents.} Everything else in the game is common
knowledge. In particular, firms observe whether their rival is conducting
research as well as whether the rival has a success. Third parties such as
courts also observe this information.
We next consider the product market competition and the sharing process before
explaining how the firms' payoffs are represented.
\subsection{Product Market Competition\label{product-mkt}}
After a firm completes all stages of the research process, it can participate
in the product market. The firms produce goods that may either be homogeneous
or differentiated to some degree by unmodelled differences in the production
technologies. We represent the product market competition in the following
reduced form way.
If both firms have completed the research project, they compete as duopolists
and each earns a flow profit of $\pi^{D}\geq0$ forever. If only one firm has
completed the research project, the firm earns a monopoly flow profit of
$\pi^{M}>0$ as long as the other firm does not produce output. Here, $\pi
^{M}>\pi^{D}$. As a benchmark, we will consider the case that the firms make
production decisions to maximize their joint profits in the product market.
This results in a joint flow profit of $\pi^{J}$ where $\pi^{J}\geq2\pi^{D}$
and $\pi^{J}\geq\pi^{M}$. We use the notation $\widetilde{\pi}^{D}=\frac
{\pi^{D}}{r}$, $\widetilde{\pi}^{M}=\frac{\pi^{M}}{r}$, and $\widetilde{\pi
}^{J}=\frac{\pi^{J}}{r}$.
These payoffs are sufficiently flexible to capture various models of product
competition. For example, if the firms produce homogeneous products and
compete as Bertrand or Cournot competitors, then $\pi^{J}=\pi^{M}>2\pi^{D} $.
If the firms produce differentiated products, then $\pi^{J}>\pi^{M} $ and the
relationship between $\pi^{M}$ and $2\pi^{D}$ will depend on the degree of
product differentiation that exists between the products. For low levels of
product differentiation, $\pi^{M}>2\pi^{D}$; for high levels of product
differentiation, $\pi^{M}\leq2\pi^{D}$.\footnote{The magnitudes of each of the
profits $\pi^{D}$, $\pi^{J}$ and $\pi^{M}$ do not depend on the decisions
taken during the research phase. In future research, we may relax this
assumption.}
As an example, consider a demand function of the type $q_{i}=\left( a\left(
1-\gamma\right) -p_{i}+\gamma p_{j}\right) /\left( 1-\gamma^{2}\right) $,
where $0<\gamma<1$ so that the products are substitutes.\footnote{Singh and
Vives (1984) show how these demand functions derive from particular consumer
preferences.} The goods are more differentiated the higher is $\gamma$. It is
possible to show that $\pi^{M}\leq2\pi^{D}$ if and only if $\gamma$ is
sufficiently large.\footnote{The Hotelling models provide other examples of
differentiated duopoly that can correspond to these profits.}
From now on, we consider the case that there are $N=2$ steps in the innovation
process. In Section \ref{N-step}, we consider how our results extend to the
case of an $N$-step innovation process.
\subsection{Sharing of Research Outcomes\label{sharing}}
There are potential efficiencies in our model for firms to cooperate in the
research stage. Suppose that one firm successfully completes a stage of
research before the other firm does. We assume that the successful firm can
costlessly share this knowledge with the other firm, thereby saving the
lagging firm from having to continue to invest to complete the stage. From the
point of view of social efficiency, such sharing will always be efficient
because it prevents resources being spent to duplicate research results.
Because of the efficiencies of sharing, regulators in many countries encourage
sharing arrangements, especially in the early stages of research. Firms may
use a variety of contractual arrangements to govern the sharing process. There
may be some legal restrictions, however, that prohibit sharing contracts that
would inhibit competition in the output market. We want to consider firms
incentives to share research using legal contracts. To this end, we want to
classify contracts as either legal or illegal and limit our attention to legal
contracts. However, even in our relatively simple dynamic framework, there are
many contracts that might be written and it is not always obvious which ones
we might want to classify as anti-competitive. Our first approach will be to
consider two types of sharing contracts that are commonly observed in practice
and have also been analyzed elsewhere in the literature. Later, we may
consider a wider family of contracts.
The main of sharing contract we consider is \textit{ex post sharing }or
licensing, where the leading firm shares its results with the lagging firm in
exchange for a fixed fee. Sharing will occur whenever the joint profits of the
two firms are higher with sharing than without sharing. We do not place any
restrictions on the fee, but we assume that the successful firm (the leader)
makes a take-it-or-leave-it offer to the other firm (the follower). The
leader, therefore, has all of the bargaining power in the negotiation and will
offer a fee that leaves the follower just indifferent between accepting and
rejecting.\footnote{This division of bargaining power is appealing because it
insures that each firm earns the full return to its research effort. Other
divisions of bargaining power might also be considered. An existing literature
considers how other divisions of bargaining power in licensing arrangements
may affect the research incentives. See, for example, Katz and Shapiro (1985
and 1986), Shapiro (1985), Green and Scotchmer (1990 and 1995), Aghion and
Tirole (1994), and D'Aspremont, Bhattacharya and Gerard-Varet (2000).}
Because the research project has $2$ steps, there are six histories at which
one firm has more knowledge than the other. These are the histories\ $(1,0)$,
$(0,1)$, $(2,0)$, $(0,2)$, $(2,1)$ and $(1,2)$. We assume that if sharing
occurs, it physically occurs instantly once one of these histories is reached.
Given the memoryless nature of the Poisson process, this assumption is not
very restrictive. We also assume that when a leading firm is more than one
step ahead of the lagging firm, all the additional steps are shared, so that
the lagging firm catches up to the leading firm. This is a simplifying
assumption.\textit{\ }
We also consider a second type of sharing contract, \textit{ex ante sharing.
}We assume that at the histories $(0,0)$ and $(1,1)$, the firms can make a
joint decision about investing in the next research step and agree that once
the step is completed, both firms will have access to the
knowledge.\footnote{In histories where the firms do not have the same number
of successes, they can make a sharing agreement which involves both ex post
and ex ante sharing. This is a common occurrence in RJVs, where the firms
share their existing knowledge in an area in order to be able to work on the
same research question together.} The sharing agreement allows for contingent
payments between the firms when the step is completed and the physical sharing
of knowledge occurs. Knowledge sharing arrangements of this nature are often
referred to as research joint ventures (RJV). Formally, we assume that the
research technologies are not affected by the agreement. This means there are
no synergies between the firms in the research process.\footnote{Analyzing the
collaborative R\&D agreements of alliances and consortia registered under the
National Cooperative Research Act in the US, Majewski (2004) shows that when
the participants are direct competitors, they are likely to avoid spillovers.}
Rather, the RJV is an agreement that allows both firms to have access to a
success achieved by either one. Hence, it creates the opportunity for the
firms to avoid wasteful duplication of R\&D results. Alternatively, it allows
the firms to agree to have one of the two facilities shut down
altogether.\footnote{Such an asymmetric shut-down decision will never be
optimal in this model, whether or not the shut-down decision is reversible.}
\subsection{Histories and Payoff Structures}
To describe the game at any point in time, we need to specify how many firms
are still active in the game, how many successes each active firm has, and
whether there has been sharing. We use the following notation. Let
$(k_{1},k_{2})$ denote a research history where $k_{i}$ is the number of steps
that firm $i$ has completed. The histories can be partially ordered so that
$(k_{1},k_{2})$ is \textit{earlier} than $(k_{1}^{\prime},k_{2}^{\prime})$ if
and only if $k_{i}\leq k_{i}^{\prime}$ for $i=1,2$, with strict inequality for
at least one firm. In the following analysis, we refer to histories where
$k_{1}=k_{2}$ as symmetric histories and to those where $k_{1}\neq k_{2}$ as
asymmetric histories.
If a firm has dropped out of the game, we use $X$ to denote this in the
history. For example, to represent the history where firm $1$ is working on
the second step and firm $2$ has dropped out of the game, we use
$(1,X)$.\footnote{We do not extend the partial ordering to histories where a
firm has dropped out. This is because we will only need to refer to the
ordering at histories where both firms are still active in the game.}
Finally, to complete the description of the histories, we need to incorporate
the sharing decisions into our notation for the history of the game. At
symmetric histories, where there is no possibility of ex post sharing, we
continue to use the notation $(k,k)$ to denote that each firm has $k$
successes. At asymmetric histories, we need to indicate whether the firms have
made a sharing decision. At the instant that a firm achieves a success, we
denote the history as $(k_{1},k_{2})$ with $k_{1}\neq k_{2}$. At this point,
the firms make a sharing decision.\footnote{We assume that the sharing
decision takes place in the same instant of time as the success, but we are
using separate notation to capture the history before and after the sharing
decision. The history $(k_{1},k_{2})$ precedes the history $(k_{1},k_{2},NS)$
in the partial ordering of histories. The two histories have the same ordering
relative to all other histories in the game.} If the firms share, the history
becomes $(k,k)$ where $k=\max\{k_{1},k_{2}\}$. If the firms do not share, the
history becomes $(k_{1},k_{2},NS)$. For example, consider the history $(2,1)$.
If the firms share, then the history becomes $(2,2)$. If the firms do not
share, then the history becomes $(2,1,NS)$. In a continuation game at
$(k_{1},k_{2},NS)$, the firms do not get another chance to share until the
next innovation is achieved.
At any point during the research process, we denote the discounted expected
continuation payoff of firm $i$ starting at the history $\left( k_{1}%
,k_{2}\right) $ by $V_{i}(k_{1},k_{2})$. This payoff is developed recursively
from future continuation payoffs. Consider, for example, the continuation
payoff of firm $1$ at the history $\left( 1,0,NS\right) $, $V_{1}\left(
1,0,NS\right) $. Suppose there will not be any sharing between the firms at
any future history and the lagging firm will always choose to invest. If firm
$1$ develops the second step and firm $2$ continues to invest after firm $1$
develops the second step, then firm $1$'s continuation payoff is
$V_{1}(2,0,NS)$. If firm $2$ develops the first step before firm $1 $ develops
the second step and both firms stay in the game, then firm $1$'s continuation
payoff is $V_{1}(1,1)$. Hence, we have%
\begin{align*}
V_{1}\left( 1,0,NS\right) & =\int_{0}^{\infty}e^{-\left( 2\alpha
+r\right) }\left( \alpha V_{1}(2,0,NS)+\alpha V_{1}(1,1)-c\right) dt\\
& =\frac{\alpha V_{1}(2,0,NS)+\alpha V_{1}(1,1)-c}{2\alpha+r}%
\end{align*}
where the payoffs $V_{1}(2,0,NS)$ and $V_{1}(1,1)$ are similarly developed
from future continuation payoffs.
After a firm has finished the research process, it earns continuation profits
in the output market. To see how the payoffs are constructed, suppose firm $1$
is the leading firm and firm $2$ continues to research the second step. If
there is no possibility of sharing, then we are at the history $(2,1,NS)$.
Firm $1$ can produce output as a monopolist. In each instant of time, firm $2$
has a probability $\alpha$ of success. As soon as firm $2$ is successful, firm
$1$ starts to earn duopoly profits forever after. Hence, we have%
\begin{align*}
V_{1}\left( 2,1,NS\right) & =\int_{0}^{\infty}e^{-\left( \alpha+r\right)
}\left( \pi^{M}+\alpha\frac{\pi^{D}}{r}\right) dt\\
& =\frac{\pi^{M}+\alpha\frac{\pi^{D}}{r}}{\alpha+r}.
\end{align*}
If the firms decide to share, the continuation payoff of firm $1$ is equal to
$V_{1}\left( 2,2\right) =\frac{\pi^{D}}{r}$. The payoffs at other histories
are developed similarly using recursion.
%See text for a description of Figure 1
\subsection{Example of Game Structure}
To clarify the timing of decisions in the research phases of the game,
consider the following illustration. As in Scotchmer and Green (1990), we use
a discrete game tree as a stylized representation of the underlying continuous
time model. We assume that firms share using the ex post licensing
arrangements discussed above.\footnote{Scotchmer and Green (1990) include a
rigorous justification for this representation based on a dominant strategy
argument. That argument needs to be modified for our model in part because
unlike them, we model the investment decision as irreversible. However, the
basic idea is the same.}
At the beginning of the game, both firms simultaneously decide whether to
invest or not. As shown in the subgame depicted in Figure 1, after one of the
firms has made a discovery, the firms can jointly decide whether to share the
winner's discovery. As mentioned before, sharing brings them to a symmetric
position in the R\&D\ phase. If they decide not to share, the laggard decides
whether to continue to invest in order to develop the first innovation and the
leader decides whether to invest to develop the second innovation. If they
decide to share, they both simultaneously decide whether to invest in order to
develop the second innovation.
If the laggard (re)develops the first innovation before the leader develops
the second innovation, both firms start to invest to develop the second
innovation. If the leader develops the second innovation before the laggard
develops the first innovation, the firms again decide whether to share, this
time both of the innovations. If they agree to share, they start to compete in
the product market. If they agree not to share, the leader starts to produce
while the laggard decides whether to continue to invest.
\section{Benchmark Analysis\label{benchmark}}
Before solving the game, we consider a benchmark case where the firms
cooperate to maximize their joint profits. We assume that the firms make all
investment, sharing, and product market decisions jointly.
At each history prior to the product market, the firms jointly decide whether
to share the results of their research (if one firm is ahead) and whether one
or both firms will invest.\footnote{Throughout the paper, we assume that if a
firm stops researching before completion of the project, it cannot reenter the
game at a later date. This means that investment decisions are also
participation decisions. We make the assumption for simplicity, but consider
the consequences of relaxing it later.} Once one firm completes the research
project, the firms make joint decisions about the product market. We do not
make any assumptions about how the firms divide the joint profits, but we
simply assume that each decision is made to maximize the sum of the
continuation profits of the two firms.\footnote{We do not consider the
possibility that the firms can commit to decisions prior to making them, but
there is no dynamic inconsistency in the joint profit maximization problem.}
We have the following result.\footnote{The proposition is proved in the
appendix below.} \
\begin{proposition}
\label{benchmark-full-cooperation}Suppose that the firms maximize their joint
continuation profits. Then, at any history such that one firm has more
research successes than the other, the optimal sharing decision is for the
leading firm to share its research with the lagging firm. Given this sharing
pattern, the firms make investment decisions only at the symmetric histories
$(0,0)$ and $(1,1)$. At the history $(2,2)$, the firms cooperate in the
product market and earn joint continuation profits $\widetilde{\pi}^{J}%
=\frac{\pi^{J}}{r}$.
(i) If $\pi^{J}\geq\frac{2cr}{\alpha}+\frac{cr^{2}}{2\alpha^{2}}$, the optimal
investment decision is for both firms to invest at the histories $(0,0)$ and
$(1,1)$. At $(0,0)$, the joint continuation profits are $\frac{4\alpha^{2}%
}{r(2\alpha+r)^{2}}\pi^{J}-\frac{2\left( 4\alpha+r\right) }{(2\alpha+r)^{2}%
}c$.
(ii) If $\pi^{J}<\frac{2cr}{\alpha}+\frac{cr^{2}}{2\alpha^{2}}$, neither firm
invests at $(0,0)$ regardless of whether the firms would subsequently invest
at $(1,1)$. At $(0,0)$, the joint continuation profits are $0$.
\end{proposition}
Proposition \ref{benchmark-full-cooperation} reveals several features of our
model. First, it is jointly optimal for the firms to share a success as soon
as it is developed by either one of them and to move on to the next stage of
the R\&D\ process. This reflects the traditional justification for sharing
arrangements as a way for firms to avoid wasteful duplication. Second, if it
is optimal for one firm to invest, it is optimal for both firms to invest.
This is a feature of the Poisson discovery process that we are using. Indeed,
with flow costs of investment, if there were $N$ identical research
facilities, then it would be optimal for all of them to conduct research
simultaneously until one of the facilities achieves a success. This speeds up
the time to innovation, and the benefits of the time savings outweigh the
costs of running simultaneous facilities. Later, we discuss how our results
might change if we used a model of the research process that does not have
this feature.
The proposition illustrates how the cost and benefit parameters affect
payoffs. In the region where firms invest, their joint continuation profits at
the beginning of the game is increasing in $\alpha,$ the hazard rate for the
Poisson discovery process. A higher $\alpha$ means that the research is likely
to be successful sooner. The joint profits are also decreasing in the discount
rate $r$ and the flow cost of research $c$. Similar comparative statics
results obtain for the continuation profits at other points in the game.
\section{Ex-post Sharing\label{ex-post-sharing}}
The first type of sharing contract we consider is \textit{ex post sharing.
}Suppose that one firm completes a research step ahead of the other firm. We
consider a sharing contract where the leading firm shares its results with the
lagging firm in exchange for a fixed fee. Such sharing takes place as long as
it results in higher industry profits. Although a range of fees would
typically be acceptable to both firms, as discussed in Section \ref{sharing},
we will assume that the leader has all the bargaining power and sets the
licensing fee by making a take-it-or-leave-it offer to the follower.
For each specification of the basic parameters of the game, we use backwards
induction to find the subgame perfect equilibria in pure strategies. We
consider generic values of parameters.\footnote{In our proof, we divide the
space of parameters into regions such that the equilibrium set is constant on
each region. We do not consider parameters on the boundaries of the regions.
For these parameters, there can be multiple equilibria that exist only on the
boundary and not for a generic set of parameters.} In this section, we first
present the general properties of the equilibrium. Then, we present a full
characterization for all possible values of $\pi^{M}$ and $\pi^{D}$ using a
diagram and discuss how the equilibrium outcome changes as the profit levels change.
The primary benefit of sharing is that it avoids the wasteful duplication of
R\&D. The primary cost is the effect on output market competition. Because
sharing erodes the lead of one firm, it reduces the expected profits that the
leader derives from finishing the race first and being a monopolist for some
period of time. This cost is even greater if, but for the sharing, the lagging
firm would drop out of the race.\footnote{Scotchmer and Green (1990)\ examine
the effect of secrecy on the drop-out decision of the firm.} The trade-off
between monopoly and duopoly profits, has not been addressed at great length
in the patent race literature because patent models usually assume a
winner-take-all payoff structure.\footnote{Katz (1986) considers a model with
one stage of research such that firms first engage in cooperative and
independent R\&D and then compete in an oligopolist output market. Also, see
Cardon and Sasaki (1998) and Severnov (2001) for two more recent models of
innovation where the firms compete as differentiated duopolists.}
A central question is how the incentives to share change over time. Because
each of the research steps is identical from a technology standpoint, a
conclusion that sharing incentives must change over time is not obvious.
Certainly, if one firm is ahead of another, that may impact each firm's
individual choices. However, if we consider the histories $(1,0)$ and $(2,1)$,
it is not obvious that the sharing incentives should be any different. In both
cases, the leader is one step ahead of the follower. Sharing is socially
efficient in both cases and generates the same savings in terms of the
elimination of wasteful R\&D. The history $(1,0)$ is, however, earlier than
the history $(2,1)$. At the earlier history, there is more uncertainty to be
resolved before the firms enter the product market. We will consider how this
uncertainty affects firms decisions.
The number of steps that the lagging firm is behind is also a factor in firms'
sharing decisions. We control for this, however, by comparing histories such
that the lagging firm is a fixed number of steps behind the leading firm. This
implies that in an $N$-step research process, we compare sharing incentives at
all histories $(k+g,k)$ where $g$ is a fixed gap between the leading firm and
the lagging firm. The size of the gap can be as small as $1$ or as large as
$N-1$. When $N=2,$ we compare the sharing decisions at $(2,1)$ and $(1,0)$
where the leading firm is one step ahead of the lagging firm.
From a dynamic perspective, what matters is whether one history precedes
another. To consider dynamics, we compare histories $(k+g,k)$ and $(k^{\prime
}+g,k^{\prime})$. If $k2,$ the monotonicity property
holds for every subgame perfect equilibrium in Region B.
\end{proposition}
The monotonicity result of Proposition \ref{monotonicity} extends to Region B,
provided that $\frac{r}{\alpha}>2$. A no sharing decision is never followed by
a sharing decision. However, the monotonicity result cannot be further extended.
As we demonstrate in the proof of the proposition,\footnote{The formal
derivation of all the equilibria is in a companion appendix that is available
on request. In the appendix below, we derive the equilibrium in one region to
illustrate the backward induction. This example demonstrates a
non-monotonicity on the equilibrium path.} when $\frac{r}{\alpha}$ $<2$, there
exist values of $\pi^{D}$ and $\pi^{M}$ such that in equilibrium the firms do
not share step 1 at $(1,0)$, but do share step $2$ at $(2,1)$. The lagging
firm drops out at the history $(1,0,NS)$, but stays in at the later history
$(2,1,NS)$. The leader does not share at $(1,0)$ because this would maintain a
rival that is otherwise eliminated. Since the rival drops out at $(1,0,NS)$,
the history $(2,1)$ is not reached in equilibrium.\footnote{By symmetry, the
follower also drops out at the history $(0,1,NS)$. Thus, there is no path to
$(2,1)$.} Hence, the non-monotonicity pattern is not observed along the
equilibrium path. The parameter restriction means that the firms must be
relatively patient and good at research for this type of equilibrium to exist.
Sharing at $(1,0)$ allows both firms to work on step $2 $ and hastens the end
of the research phase. The firms are willing to forego this benefit when
$\frac{r}{\alpha}$ is not too large.
We also demonstrate an equilibrium in which a non-monotonic sharing pattern
arises on the equilibrium path. In equilibrium, the lagging firm drops out at
the history $(2,0,NS)$, but stays in at $(2,1,NS)$ and $(1,0,NS)$. The firms
do not share step $1$ at $(1,0)$, because this way they can reach the history
$(2,0,NS)$, at which point the lagging firm drops out. A non-monotonic sharing
pattern arises because after $(1,0,NS)$, the firms sometimes reach the history
$(1,1)$. Both firms invest in step $2$. The game may then proceed to the
history $(2,1)$, at which point the leading firm shares step $2$ with the
lagging firm.
For this type of equilibrium to exist, we need to impose a stronger
restriction on $\frac{r}{\alpha}$. As the proof shows, for $\frac{r}{\alpha
}>\frac{1}{2}(\sqrt{5}-1)$, there are no non-monotonic sharing patterns along
the equilibrium path. For $\frac{r}{\alpha}<\frac{1}{2}(\sqrt{5}-1)$ , there
exist values of $\pi^{D}$ and $\pi^{M}$ such that the non-monotonic sharing
pattern arises on the equilibrium path.
%See text for description of Figure 2.
Figure 2 depicts the equilibrium outcomes in the case when we see
non-monotonic sharing patterns both on and off the equilibrium
path.\footnote{For these parameters, there are multiple equilibria at $(0,0)$
in some of the regions. In the regions, both firms can be in or both firms can
be out at $(0,0)$. In the diagram, we selected the equilibrium such that both
firms invest at $(0,0)$. Otherwise, the region such that neither firm invests
at $(0,0)$ would be larger. A full description of all the equilibria including
all multiplicities is available in the companion appendix to the paper.} This
is the case that $\frac{r}{\alpha}<\frac{1}{2}(\sqrt{5}-1)$. The diagram lists
the sharing pattern for each region. For example, at the top left of the
diagram, the sharing pattern S,NS,NS describes the following sequence of
decisions: i) at $(1,0)$, the leader shares (S) step 1; ii) at $(2,0)$, the
leader does not share (NS) step 1; iii) at $(2,1)$, the leader does not share
(NS) step 2. The diagram shows that there are two regions with the
non-monotonic sharing pattern NS,NS,S. This pattern is non-monotonic because
the leader does not share step 1 at $(1,0)$, but does share step 2 at $(2,1)$.
The two regions with this sharing pattern are separated by a vertical line. In
the region to the left of the line, the follower drops out at the history
$(1,0,NS)$. Because of this, the history $(2,1)$ is not reached in the
equilibrium.\footnote{By symmetry, the follower also drops out at the history
$(0,1,NS)$. Thus, there is no path to $(2,1)$.} Thus, an observer of the game
would not observe a non-monotonicity. In the region to the right of the line,
the follower stays in the game at the history $(1,0,NS)$. Because of this, the
history $(2,1)$ is reached along the equilibrium path.
The diagram also shows the sharing patterns in other regions. Consider region
A. Here, we have that $\pi^{D}\geq\frac{cr}{\alpha}(2+\frac{r}{\alpha}).$ In
Region A, the lagging firm never drops out of the game and the sharing
patterns are monotonic. Of course, by Proposition \ref{monotonicity}, we
already knew that this result must hold. Consider how the sharing pattern
changes as $\pi^{M}$ increases, but with the value of $\pi^{D}$ held fixed.
For small values of $\pi^{M}$, the sharing pattern is S,S,S. As monopoly
profits increase, sharing breaks down at the history $(2,1)$.\footnote{Sharing
also breaks down at the histories $(2,0)$ and $(0,2),$ but we do not have any
other histories to compare these to with the same gap of 2 steps between the
leader and the follower.} As monopoly profits are increased further, sharing
eventually breaks down at the earlier history $(1,0)$ as well. This is a
monotonicity result for the comparative static analysis. As monopoly profits
$\pi^{M}$ increases, sharing breaks down, but it breaks down at later
histories first.
Two regions in the diagram have a sharing pattern of S,NS,S. Here, the leading
firm does not share step 1 at $(2,0)$, even though it does share step 1 at the
earlier history $(1,0)$. We do not interpret this as a non-monotonicity
result, because we only compare sharing decisions at histories where the gap
between the leader and the follower is the same.
On the far left of the diagram, for parameters $\pi^{D}<\frac{cr}{\alpha}, $
the leading firm is indifferent between sharing or not sharing at $(2,0)$.
Either way, the lagging firm drops out of the race and the decision does not
affect payoffs on the equlibrium path.
\section{$N$-step Research Process\label{N-step}}
In this section, we discuss some results obtained in a model with $N$ research
steps of equal difficulty.
As a starting point, we consider our benchmark model where the firms cooperate
to maximize their joint profits. The firms make all investment, sharing, and
product market decisions jointly. Proposition \ref{benchmark-full-cooperation}
extends in a straightforward way. At asymmetric histories, the optimal sharing
decision is for the leading firm to share its research with the lagging firm.
At the final history $(N,N)$, the firms cooperate in the product market to
earn joint continuation profits $\widetilde{\pi}^{J}$. If the joint
continuation profits are above a critical threshold, then both firms invest at
all earlier symmetric histories. Otherwise, at the start of the game, neither
firm invests.
We next turn to our monotonicity result that sharing declines over time.
Proposition \ref{monotonicity} extends to a model with $3$ research steps. We have
\begin{proposition}
When $N=3,$ the monotonicity property holds for every subgame perfect
equilibrium in Region A. Region A consists of all parameters such that
$\pi^{D}\geq\frac{cr}{\alpha}(3+3\frac{r}{\alpha}+\frac{r^{2}}{\alpha^{2}})$.
\end{proposition}
To prove the proposition, we derive the equilibria as we did for the case of
$N=2.$\footnote{The proof is available from the authors on request. The
calculations are straightforward, but long. In the appendix below, we derive
the parameter condition that defines Region A.} Region A is the set of
parameters such that the lagging firm would stay in the game at the history
$(3,0,NS)$. At this history, the lagging firm is as far behind as possible and
has no hope of ever earning monopoly profits. Because the lagging firm does
not have any bargaining power, its payoff at $(3,0,NS)$ is the payoff it would
get by conducting all three steps of research on its own and then producing in
the output market as a duopolist. Region A is all parameters such that this
payoff is positive.
The monotonicity property implies that if the firms share at the histories
$(k+1,k)$, then they share at $(k,k-1)$ for $k=1,2.$ At these histories, the
leading firm is one step ahead of the lagging firm. The monotonicity property
also implies that at the history $(3,1)$ when the leading firm is two steps
ahead, they share at the earlier history $(2,0).$
The monotonicity property cannot be strengthened to comparisons between
histories such that the leading firm is ahead by a differing number of steps.
For example, we find an equilibrium such that the firms share at $(2,1)$, but
do not share at the earlier history $(2,0).$ The reason is that at $(2,0)$,
the leading firm is further ahead and has more to give up in terms of forgone
monopoly profits.\footnote{When $N=2,$ we also found equilibria in Region B
such that the firms share at $(2,1),$ but not at $(2,0)$. The sharing pattern
arises because the lagging firm exits after the decision not to share. When
$N=3,$ we find this sharing pattern in Region A, even though the lagging firm
does not ever exit the game.}
We expect that Proposition \ref{monotonicity} could be extended further to a
model with $N$ research steps. The intuition behind the proposition is general
and does not depend on the assumption that $N=2$. The benefit of sharing is
the savings on R\&D costs. These cost savings do not change over time.
However, the cost of sharing, measured in terms of foregone monopoly profits,
do change over time. The advantage to the leading firm to being a fixed number
of steps ahead of the lagging firm increases over time as uncertainty is
resolved. The net effect is that the firms have decreasing incentives to share
as the game progresses
We have not proved Proposition \ref{monotonicity} for the general case of $N$
research steps, \ because the equilibrium calculations become too cumbersome.
Instead, we analyzed a related problem that we interpret as a partial
generalization of our monotonicity result. Consider any starting history
$(k+1,k)$ in the $N$-step model such that the leading firm is one step ahead
of the lagging firm. If the firms share at this history, then the new history
becomes $(k+1,k+1).$ If they do not share, then the new history is
$(k+1,k,NS).$ Assume that at all histories after $(k+1,k,NS)$ and $(k+1,k+1)$
the firms do not share and they also do not exit the game. Under this
assumption, we can derive formulas representing the firms' joint continuation
payoffs. We can compare the continuation payoffs from sharing and not sharing
at $(k+1,k)$. The benefit of sharing (which is equal to the cost savings by
the lagging firm) is an increasing (linear) function of the flow cost of
research $c.$ Because of this, there is a threshold cost $c(k+1,k)$ such that
the firms decide to share if and only if $c\geq c(k+1,k). $ Using numerical
analysis, we can show that the threshold cost $c(k+1,k)$ is increasing in $k$
for $N\leq20$.\footnote{We compared the payoffs by evaluating them on a
discrete grid of parameter values. The formulas appear to be sufficiently
continuous that we do not expect we missed any singularities in our
simulations. The computations are available on request.}
The finding suggests that sharing is more likely to occur at earlier
histories. The cost parameter $c$ is more likely to be above the sharing
threshold cost $c(k+1,k)$ at earlier histories than at later ones. The result
is different from Proposition \ref{monotonicity} because the assumptions about
firms' behavior after $(k+1,k)$ may not be consistent with any equilibrium.
However, the result is consistent with the intuition that the incentives to
share decline over time when firms never exit.
Our other result, Proposition \ref{non-monotonicity}, is that there are
parameter values in Region B such that a subgame perfect equilibrium does not
satisfy the monotonicity property. \ These non-monotonic equilibria continue
to exist as subgames of the $N$-step model. This is because the subgame that
begins at the history $(N-2,N-2)$ is a two-stage game. Parameters that support
the equilibrium are in Region B of the 2 step game. The parameters are also in
Region B of the $N$-step game because Region B grows as $N$
increases.\footnote{Region A shrinks as $N$ increases, because a lagging firm
has a lower payoff from staying in the game at $(N,0)$ than at $(N-1,0)$.
Region B grows as Region A shrinks, since they are complementary sets.} Thus,
Proposition \ref{non-monotonicity} continues to hold.\footnote{If a firm drops
out of the $N-$ stage game prior to the history $(N-2,N-2),$ then the
continuation equilibrium would still exist but would represent
off-the-equilibrium path behavior.}
\section{Ex-ante Sharing\label{ex-ante-sharing}}
In this section, we consider a second type of sharing contract. We assume that
at any (symmetric) history, the firms can make a joint decision about
investing in the next research step and agree that once the step is completed,
both firms will have access to the knowledge as in a RJV.
Consider a sharing contract that is signed at the beginning of the game. The
history is\ $(0,0)$. The firms both agree to conduct research on the first
step. The firms also agree that when one firm has a success, it will share the
success with the lagging firm. In exchange, the lagging firm will pay a fee to
the leading firm. At the time the contract is signed, the fee is contingent -
it is paid by the lagging firm to the leading firm at the instant of
innovation. We assume that the fee is set so that the lagging firm is
indifferent between paying the fee to get the result, and not paying the fee
and not getting the result. This means that the leading firm extracts the full
value of its success. Therefore, as was the case under our ex post sharing
contracts, both firms have efficient incentives to invest ex
ante.\footnote{The contract need not provide any direct incentives for
investment and would induce efficient effort even if the effort levels were
non-contractible.}
We consider a similar sharing contract at the history $(1,1)$. We then analyze
the sharing pattern to find whether the firms are more likely to share at
$(0,0)$ than at $(1,1)$. Our analysis shows that there is no difference in
terms of sharing incentives between the ex ante sharing contracts and the ex
post sharing contracts. Because of this, the dynamics of sharing over time are
essentially the same as before.
\section{Asymmetric Firms\label{asymmetric-firms}}
So far we have assumed that the firms are symmetric in their
research\ capabilities. In this section we relax this assumption and assume
that the firms can differ in their research capabilities. That is, different
firms may have different areas of expertise which make them perform better in
different stages of the research process. This is often the case, for example,
in the biotechnology industry (Greis et al., 1995). Large pharmaceutical
corporations form alliances with small research firms which perform the basic
research towards the development of a new product. After small research firms
successfully complete the initial stages of research project, large
pharmaceutical corporations work to bring the new product into the market.
To represent this kind of a scenario, we consider the $N=2$ model and assume
that one of the firms in our model is better at first-stage research and the
other firm is better at second-stage research. Let $\alpha$ stand for the
hazard rate of the more capable firm and $\beta$ stand for the hazard rate of
the less capable firm in each period.\footnote{Derivations of the results in
this section are available on request.}
Recall from our benchmark analysis in Section \ref{benchmark} that if the
firms can collude in the output market, then the optimal sharing decision is
for the leading firm to share its research with the lagging firm. Moreover, if
it is optimal for one firm to invest, it is optimal for both firms to invest.
With asymmetric firms, it is still the case that the optimal sharing decision
is for the leading firm to share its research with the lagging firm. However,
as far as the investment incentives of the firms are concerned, we can have
one of the following outcomes:\ (i) Neither firm invests, (ii) both firms
invest in both stages, (iii) only the efficient firm invests in both stages,
and (iv) only the efficient firm invests in the first stage and both firms
invest in the second stage. There are no equilibrium outcomes where only the
inefficient firm invests because if expected profits are high enough for the
less efficient firm to invest, they are high enough for the more efficient
firm to invest.
Under rivalry, the monotonicity result stated in Proposition
\ref{monotonicity} for Region A still holds. The boundary of Region A is again
defined by the incentives for the lagging firm to stay in the race at the
histories $\left( 2,0,NS\right) $ or $\left( 0,2,NS\right) $. We get the
same condition whether it is firm $1$ or firm $2$ that is the lagging firm.
In Region B, the asymmetry between the firms causes an asymmetry in the firms'
drop-out decisions. That is, there may be cases where the more efficient firm,
if it were the lagging firm, would stay in the market, but the less efficient
would not.
As far as the sharing conditions are concerned, since the hazard rate does not
enter the second-stage sharing condition, asymmetry does not affect the
sharing decision at the histories $\left( 2,1\right) $, $\left( 1,2\right)
$, $\left( 2,0\right) $ and $\left( 0,2\right) $. After the first success,
there may be cases where there would be sharing if it is the more efficient
firm that has the first success, but no sharing if it is the less efficient
firm that has the first success. If it is the more efficient firm that has the
first success, this implies that the firm that is less efficient in the
second-stage research is ahead and the firm that is less efficient in
first-stage research is behind. Both strengthen the incentives to share the
first success.
With the consideration of asymmetric firms, one question that arises is
whether we may have cases where only the efficient firm invests in each stage.
To consider this issue we need to change the assumption we have made so far
that once a firm exits the race, it cannot re-enter. Allowing re-entry, we can
show that firms with asymmetric hazard rates may find it optimal to
specialize. To see this, consider the extreme case where $\beta=0 $. It is
straightforward to show that only the efficient firm invests at $\left(
0,0\right) $. As soon as the first success arrives, the firm shares it with
the non-investing firm and drops out. Hence, without the prospective of
sharing the first success at $\left( 1,0\right) $, the firm would not have
invested at $\left( 0,0\right) $.
\section{Impact of Patent Policy\label{IP-policy}}
The framework we have developed can be used to investigate several policy
questions. Two important questions in patent policy are how strong patent
protection should be and how strict the non-obviousness requirement should be
(i.e., whether intermediate research outcomes should be patentable). In this
section, we focus on the first question.
So far we have assumed that once a firm successfully develops a research step,
it can either keep the technology a secret or patent it. Patenting does not
prevent the rival from developing a non-infringing technology that serves the
same purpose. In reality, the impact of patenting on the rival's progress
would depend on the strength of patent policy. Stronger patent policy would
make it harder for rival firms to invent around (Gallini, 1992).
This implies that the strength of patent policy can have an important impact
on firms' sharing decisions. Hence, we next use our framework to analyze the
impact of patent policy on firms' sharing decisions. Consider a variation of
our basic model where firms can choose between a continuum of research paths.
Different research paths are associated with different hazard rates. Firms
still must incur a flow cost $c$ per unit of time if they decide to invest. If
firm $i$ decides to invest, it can choose a research path that yields a hazard
rate $\alpha_{i}\in\left[ 0,\overline{\alpha}\right] $. We assume that the
research paths can be ranked in terms of their quality (i.e., how promising
they are) and that both firms rank the steps in the same way. Hence,
$\overline{\alpha}$ represents the hazard rate that is associated with the
most promising research path.\footnote{Derivations of the results in this
section are available on request.}
Clearly, if all of the research paths are available, the firms would choose
the research path that yields the highest hazard rate, $\overline{\alpha}$, at
the beginning of the race. After one of the firms is successful, it patents
the new technology. Patenting implies that if the rival continues to invest,
it must switch to a different research path, where it faces a lower hazard
rate.\footnote{Note that due to the memoryless nature of the Poisson discovery
process, such switching can happen independent of the lagging firm's initial
choice of research path.} Let $\alpha^{L}$ denote the maximum hazard rate that
the lagging firm can achieve without infringing the patent of the leader. A
stronger patent policy can be interpreted as corresponding to a lower
$\alpha^{L}$. Hence, we are interested in investigating how the sharing
incentives change as $\alpha^{L}$ decreases.
We divide the analysis into two parts depending on whether a strengthening in
patent policy changes the investment decision of the lagging firm at any of
the asymmetric histories. If patent policy gets stronger without affecting the
dropping out decision of the lagging firm at any of the asymmetric histories,
it can have two kind of effects on the sharing incentives. First, since a
decrease in $\alpha^{L}$ makes the lagging firm less efficient, it increases
the benefits from sharing. Second, since a decrease in $\alpha^{L}$
strengthens the leader's position, it increases the costs of sharing. The
first effect dominates everywhere in Region B. It also dominates in Region A
for sufficiently small changes in patent policy. For larger changes in patent
policy (i.e., larger decreases in $\alpha^{L}$), the second effect dominates
because the rival becomes considerably weaker.
If a strengthening in patent policy increases the set of histories where the
lagging firm drops out, then sharing incentives in general get weaker. This is
because as patent policy gets stronger, the position of the lagging firm gets
weaker and it is more likely to drop out. This reduces the sharing incentives.
However, this reasoning does not apply in the case when a strengthening in
patent policy causes the lagging firm to drop out at $\left( 2,1\right) $
and $\left( 1,2\right) $. In this case, the incentives to share step $1$ at
the histories $\left( 1,0\right) $, $\left( 0,1\right) $, $\left(
2,0\right) $ or $\left( 0,2\right) $ may get stronger relative to the case
when the lagging firm would not drop out at $\left( 2,1\right) $ and
$\left( 1,2\right) $.\footnote{More specifically, this is the case for
sufficiently high values of $\pi^{M} $.} This is because if the lagging firm
will not drop out at $\left( 2,1\right) $ and $\left( 1,2\right) $, the
leader chooses not to share the first step in order to prolong the time period
during which it can potentially earn monopoly profits. If the lagging firm
will drop out at $\left( 2,1\right) $ and $\left( 1,2\right) $, the firm
that first successfully develops the first step does not have to protect
itself by not sharing.
\section{Conclusion\label{conclusion}}
The paper considers the optimal pattern of knowledge sharing in the context of
technological competition. Developing a theoretical foundation for optimal
sharing strategies has important implications for the design of optimal as
well as efficient research environments.
We have analyzed how the incentives to share change over time as a research
project reaches maturity. The decision to share and the pattern of sharing
activities critically depend on the lagging firm's incentives to stay in the
race in case of no sharing. The results reveal under rivalry, the incentives
to share intermediate research outcomes decreases monotonically with progress
if the lagging firm is expected never to drop out. The incentives to share are
higher earlier on because there is more uncertainty earlier on. Sharing has a
smaller impact on each firm's chance of being a monopolist at the end of the race.
In many models of R\&D, there is an assumption that firms share at an early
research stage but not at a later one. This result shows that this sharing
pattern can be derived from the optimizing behavior of firms in a dynamic game
where the research technology does not change over time.
If the lagging firm is expected to drop out, the incentives to share may
increase with progress. This is because earlier in the research process the
lagging firm may have a higher incentive to drop out and, hence, the leading
firm may have a higher chance of eliminating rivalry by not sharing.
An analysis of the impact of patent policy on firms' sharing decisions reveals
that sharing incentives in general get weaker if a strengthening in patent
policy causes a change in the investment decision of the lagging firm at any
of the asymmetric histories. Otherwise, sharing incentives generally get
stronger. The framework can also be used to analyze whether patentability of
intermediate research outcomes is desirable and under what circumstances a
more tolerant treatment of research collaborations may be desirable.
\pagebreak
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\begin{center}
\appendix{\LARGE Appendix}
\end{center}
\section{Proof of Proposition 1}
We solve the model under the assumption that the two firms maximize their
joint payoffs. We derive continuation profits at each history working
backwards through the decision nodes.
At $(2,2)$, the firms cooperate in the output market to earn the joint flow
profit $\pi^{J}$ forever. Recall that $\pi^{J}\geq\max\{2\pi^{D},\pi^{M}\}$ so
that profits in the output market are greatest when the firms produce
cooperatively. The joint continuation profits are:\
\[
V_{J}(2,2)=\frac{\pi^{J}}{r}=\widetilde{\pi}^{J}%
\]
At the histories\footnote{The histories $(1,2)$ and $(0,2)$ are analyzed in
the same way as $(2,1)$ and $(2,0).$ We do not repeat the analysis here.}
$(2,1)$ and $(2,0),$ the leading firm shares all available research with the
lagging firm. This prevents the wasteful duplication of R\&D. The firms then
cooperate in the product market to earn joint continuation profits of
$\widetilde{\pi}^{J}$. Thus, we have that
\[
V_{J}(2,1)=V_{J}(2,0)=V_{J}(2,2)=\widetilde{\pi}^{J}%
\]
At the history $(1,1)$, if neither firm invests, the joint continuation
profits are $0$. If one firm invests (either firm), then the firm invests a
flow cost of $c$ and in each instant the probability of success is $\alpha$.
When the success arrives, the firms share the research and cooperate in the
product market to earn flow profits of $\pi^{J}.$ At $(1,1),$ the joint
continuation profits are:\
\[
V_{_{J}}(1,1)=\int_{0}^{\infty}e^{-(\alpha+r)}(\alpha\widetilde{\pi}%
^{J}-c)dt=\frac{\alpha\widetilde{\pi}^{J}-c}{\alpha+r}%
\]
If both firms invest, then each firm incurs a flow cost of $c$ and the flow
probability that at least one firm succeeds is $2\alpha.$The joint
continuation profits are:%
\begin{equation}
V_{_{J}}(1,1)=\int_{0}^{\infty}e^{-(2\alpha+r)}(2\alpha\widetilde{\pi}%
^{J}-2c)dt=\frac{2\alpha\widetilde{\pi}^{J}-2c}{2\alpha+r}
\label{JointProfitsat1-1}%
\end{equation}
\ Given these payoffs, the firms will either both invest or both not invest.
The firms invest if and only if $V^{J}(1,1)\geq0.$ This occurs\footnote{For
simplicity (but with some abuse of notation), we ignore non-generic parameters
such that some firm is indifferent between two actions, as would be the case
here if $\widetilde{\pi}^{J}=\frac{c}{\alpha}.$} if and only if $\widetilde
{\pi}^{J}\geq\frac{c}{\alpha}.$
Working backwards, we reach the history $(1,0)$. As at $(2,0)$ and $(2,1),$
sharing eliminates wasteful duplication of R\&D. Because the firms make
decisions cooperatively, there is no cost to them to sharing. Sharing either
strictly increases their joint continuation profits or has no effect on the
profits because the firms are in any event exiting the race. Without loss of
generality, we will assume that the firms share at $(1,0)$.
Finally, we consider the history $(0,0).$ If neither firm invests, their joint
continuation profits are $0.$ If both firms invest, then their joint
continuation profits are:\
\[
V_{_{J}}(0,0)=\int_{0}^{\infty}e^{-(2\alpha+r)}(2\alpha V_{_{J}}%
(1,1)-2c)dt=\frac{2\alpha V_{_{J}}(1,1)-2c}{2\alpha+r}%
\]
The continuation profits depend on whether the firms invest at $(1,1).$ If the
firms do not invest at $(1,1),$ then they clearly will not invest at $(0,0)$.
If the firms invest at $(1,1),$ then they invest at $(0,0)$ if and only if
$V_{_{J}}(0,0)\geq0$. This is the case if and only if $V_{_{J}}(1,1)\geq
\frac{c}{\alpha}.$ Using the expression for $V_{_{J}}(1,1) $ above, we find
that the firms invest at $(0,0)$ if and only if $\pi^{J}\geq\frac{cr}{\alpha}$
and\
\[
\pi^{J}\geq\frac{2cr}{\alpha}+\frac{cr^{2}}{2\alpha^{2}}\text{.}%
\]
The last inequality above implies the inequality $\pi^{J}\geq\frac{cr}{\alpha
}$. Thus, this inequality is a necessary and sufficient condition for both
firms to invest at $(0,0).$ If the firms invest, then their joint continuation
profits are\
\[
V_{_{J}}(0,0)=\frac{2\alpha V_{_{J}}(1,1)-2c}{2\alpha+r}=\frac{4\alpha^{2}%
}{r(2\alpha+r)^{2}}\pi^{J}-\frac{2(4\alpha+r)}{(2\alpha+r)^{2}}c\text{.}%
\]
\section{Proof of Lemma 1}
In a companion appendix that is available on request, we analyze all the
equilibria of the game. That analysis also proves the lemma. Here we take a
different approach. We focus on the payoff that a firm would earn by
conducting two steps of research on its own and then producing in the output
market as a duopolist. This payoff is necessarily a lower bound on any firm's
payoff at any history and in any equilibrium. This is because a firm always
has an option to complete two steps of research on its own to earn duopoly
profits or greater in the output market. We claim that in Region A, the payoff
of the lagging firm at $(2,0,NS)$ equals this payoff. To see this, consider
the decisions of the lagging firm beginning at $(2,0,NS)$. By the definition
of Region A, the firm does not drop out of the game at $(2,0,NS)$. Instead, it
completes the first step of research \textit{on its own} to arrive at $(2,1)$.
The firms may or may not share step 2 at $(2,1).$ Either way, because the
lagging firm has no bargaining power its payoff is the same as its payoff at
$(2,1,NS).$ In Region A, the lagging firm does not drop out at $(2,1,NS)$.
Instead, it completes the second step of research \textit{on its own} to
arrive at $(2,2)$. At $(2,2),$the firm is a duopolist. This shows that the
payoff of the lagging at $(2,0,NS)$ equals the payoff to a firm of conducting
two steps of research on its own and then producing in the output market as a duopolist.
We finish the lemma by computing the payoff to a firm of conducting two steps
of research and then producing in the output market as a duopolist. We work
backwards through time to compute the payoff.
After completing the two steps of research, the firm produces output as a
duopolist to earn $\widetilde{\pi}^{D}=\frac{\pi^{D}}{r}$. Working backwards,
suppose the firm has completed one step of research. To complete the second
step of research, the firm invests a flow cost of $c$ and in each instant the
probability of success is $\alpha$. The firm's expected payoff is
\[
\int_{0}^{\infty}e^{-(\alpha+r)}(\alpha\widetilde{\pi}^{D}-c)dt=\frac
{\alpha\widetilde{\pi}^{D}-c}{\alpha+r}%
\]
Again, working backwards, consider the first step of research. The firm again
invests a flow cost of $c$ and in each instant the probability of success is
$\alpha$. The firm's expected payoff is%
\[
\int_{0}^{\infty}e^{-(\alpha+r)}(\alpha\lbrack\frac{\alpha\widetilde{\pi}%
^{D}-c}{\alpha+r}]-c)dt=\frac{\alpha\lbrack\frac{\alpha\widetilde{\pi}^{D}%
-c}{\alpha+r}]-c}{\alpha+r}.
\]
This payoff is strictly positive if and only if
\[
\pi^{D}>\frac{cr}{\alpha}(2+r/\alpha).
\]
This is the inequality that defines Region A.
\section{Proof of Proposition 2 and Proposition 3}
We solve for the equilibria of the game for all parameter values in the
companion appendix to this paper that is available on request. That analysis
proves propositions 2 and 3. Figure \ref{equilibrium-outcomes} illustrates the
equilibria for an example with non-monotonicities in some regions. For readers
who do not wish to read the companion appendix, we show how to derive one of
the equilibria below. The equilibria for other parameter values are solved similarly.
\section{Derivation of a Non-Monotonic Equilibrium}
We solve the game in the following region of parameters:\ $\frac{cr}{\alpha
}(\frac{3}{2}+\frac{r}{2\alpha})<\pi^{D}<\frac{cr}{\alpha}(2+\frac{r}{\alpha
})$ and $2\pi^{D}(\frac{2\alpha+2r}{2\alpha+r})+c(\frac{2r}{2\alpha+r}%
)<\pi^{M}<2\pi^{D}+c$. This is a subregion of region B. A straightforward
calculation\footnote{This deriviation is available on request.} shows that the
region is non-empty if and only if $\frac{r}{\alpha}<\frac{1}{2}(\sqrt{5}-1)$
where $\frac{1}{2}(\sqrt{5}-1)\simeq0.62$. The equilibrium is also derived in
the companion appendix to this paper, where the region is labeled Region 6.
To find an equilibrium, we work backwards from the end of the game. We derive
the continuation profits at each history and solve for the equilibrium
actions. At asymmetric histories such as $(2,1)$ and $(1,2)$, the analysis of
the game is the same so we analyze only one of the histories.
\textbf{The last history is the history }$(2,2)$. At this history, the firms
have two successes each and are done with the research. \textbf{They produce
output and each earns discounted duopoly profits of }$V_{i}(2,2)=\widetilde
{\pi}^{D}=\frac{\pi^{D}}{r}$\textbf{.}
Working backwards,\textbf{\ the next history is }$(2,1,NS).$ \textbf{Firm }$1
$\textbf{\ is finished with its research and produces output. }Firm 2 has $1 $
success, and the firms have declined to share. Firm $2$ decides whether or not
to invest in step $2$. If firm $2$ invests, then its continuation profit is\
\begin{equation}
V_{2}\left( 2,1,NS\right) =\int_{0}^{\infty}e^{-(\alpha+r)}(\alpha
V_{2}(2,2)-c)dt=\frac{\alpha\widetilde{\pi}^{D}-c}{\alpha+r}.
\label{profit-noshare-2-1}%
\end{equation}
This payoff is positive because by assumption
\[
\pi^{D}>\frac{cr}{\alpha}%
\]
Hence\textbf{\ firm }$2$\textbf{\ invests at }$(2,1,NS)$. Firm 1 earns
monopoly profits until firm 2 completes the second step. The continuation
profit of firm 1 is%
\[
V_{1}\left( 2,1,NS\right) =\int_{0}^{\infty}e^{-(\alpha+r)}(\pi^{M}+\alpha
V_{1}(2,2))dt=\frac{\pi^{M}+\alpha\widetilde{\pi}^{D}}{\alpha+r}>0.
\]
The firms share at $(2,1)$ iff this maximizes their joint profits. Their joint
profits under sharing are%
\[
V_{J}(2,2)=V_{1}(2,2)+V_{2}(2,2)=2\widetilde{\pi}^{D}=\frac{2\pi^{D}}{r}%
\]
since when the firms share, the game reaches the history $(2,2)$. Joint
profits under no sharing are\
\[
V_{1}\left( 2,1,NS\right) +V_{2}\left( 2,1,NS\right) =\frac{\pi
^{M}+2\alpha\widetilde{\pi}^{D}-c}{\alpha+r}.
\]
We get $S\succ NS\Longleftrightarrow$%
\begin{align*}
2\widetilde{\pi}^{D}(\alpha+r) & >\pi^{M}+2\alpha\widetilde{\pi}^{D}-c\text{
or}\\
2\pi^{D}+c & >\pi^{M}.
\end{align*}
This condition holds in the region, and \textbf{the firms share step }%
$2$\textbf{\ at }$(2,1)$.
At the history $(1,1)$, each firm has one success. There is no sharing
decision to be made. The firms must, however, decide whether to invest to
develop the second step. Assuming firm $1$ invests, firm 2 will also invest
if
\[
V_{2}\left( 1,1\right) =\frac{\alpha V_{2}\left( 2,1\right) +\alpha
V_{2}\left( 1,2\right) -c}{2\alpha+r}=\frac{\alpha V_{J}\left( 2,1\right)
-c}{2\alpha+r}>0
\]
Since the firms share at $(2,1),V_{J}\left( 2,1\right) =2\widetilde{\pi}%
^{D}.$ Substituting, we get
\begin{equation}
V_{2}\left( 1,1\right) =\frac{2\alpha\widetilde{\pi}^{D}-c}{2\alpha+r}>0
\label{profit-1-1}%
\end{equation}
This simplifies to
\[
\pi^{D}>\frac{cr}{2\alpha}.
\]
This condition holds in the region, so firm $2$ invests. Hence, each firm
invests at $(1,1)$ if the other does. If firm $1$ does not invest at $(1,1)$,
the new history is $(X,1)$. Firm 2 invests if
\begin{equation}
V_{2}\left( X,1\right) =\frac{\alpha V_{2}\left( X,2\right) -c}{\alpha
+r}=\frac{\alpha\widetilde{\pi}^{M}-c}{\alpha+r}>0 \label{profit-1-X}%
\end{equation}
where $V_{2}\left( X,2\right) =\widetilde{\pi}^{M}$ because \textbf{at
}$(X,2),$\textbf{\ firm }$2$\textbf{\ produces output }as a monopolist. The
condition simplifies to $\pi^{M}>\frac{cr}{\alpha}$. The condition holds
because $\pi^{M}>\pi^{D}$and in this region $\pi^{D}>\frac{cr}{\alpha}$.
Hence, \textbf{firm 2 invests at }$(X,1)$. It follows that \textbf{both firms
invest at }$(1,1)$.
\textbf{At the history }$(2,0,NS)$\textbf{, firm 1 produces output.} The firms
have decided not to share. Firm 2 invests iff
\[
V_{2}\left( 2,0,NS\right) =\frac{\alpha V_{2}\left( 2,1\right) -c}%
{\alpha+r}>0.
\]
Since the lagging firm has no bargaining power, its earnings under sharing are
the same as its earnings under no sharing at the history $(2,1).$ The earnings
under no sharing, $V_{2}\left( 2,1,NS\right) ,$are given in
(\ref{profit-noshare-2-1}). Substituting and rearranging gives us%
\begin{align*}
V_{2}\left( 2,0,NS\right) & =\frac{\alpha^{2}\widetilde{\pi}^{D}%
-c(2\alpha+r)}{(\alpha+r)^{2}}>0\text{ or}\\
\pi^{D} & >\frac{cr}{\alpha}(2+\frac{r}{\alpha}).
\end{align*}
This condition fails in the region, so \textbf{firm }$2$\textbf{\ drops out at
}$(2,0,NS).$ (This result also follows from Lemma 1.)\
To see whether the firms share step $1$ at $(2,0),$ we compare the joint
profits under sharing with joint profits under no sharing. Joint profits under
sharing are $V_{J}\left( 2,1\right) =2\widetilde{\pi}^{D}$since if the firms
share, the game reaches the history $(2,1)$ and the firms share step $2$.
Joint profits under no sharing are $V_{J}\left( 2,0,NS\right) =V_{1}\left(
2,X\right) =\widetilde{\pi}^{M}$ since firm $2$ drops out of the game if the
firms do not share. In this region, we have that $\pi^{M}>2\pi^{D}.$ Hence,
\textbf{the firms do not share at }$(2,0)$. The lagging firm then drops out of
the game.
Working backwards from either $(2,0)$ or $(1,1),$ we next consider the history
$(1,0,NS)$. At this history, firm $1$ has one success and firm $2$ has no
successes and the firms have decided not to share. Each firm must decide
whether to invest. If firm 1 invests, then firm 2 also invests if
\begin{equation}
V_{2}\left( 1,0,NS\right) =\frac{\alpha V_{2}\left( 1,1\right) +\alpha
V_{2}\left( 2,0\right) -c}{2\alpha+r}>0 \label{profit-1-0}%
\end{equation}
We can substitute for $V_{2}\left( 1,1\right) $ from (\ref{profit-1-1}).
Moreover, $V_{2}\left( 2,0\right) =0$ since the firms do not share at
$(2,0)$ and the lagging firm drops out. Substituting and simplifying,
(\ref{profit-1-0}) becomes
\[
\pi^{D}>\frac{cr}{\alpha}(\frac{3}{2}+\frac{r}{2\alpha}).
\]
This holds in the region, so the lagging firm 2 invests at $(1,0,NS)$ if firm
1 does. It is straightforward to show that the leading firm $1$ invests at
$(1,0,NS)$ if firm $2$ invests. If firm $2$ does not invest, then the history
becomes $(1,X)$ and the leading firm invests as showed above. It follows that
the leading firm invests at $(1,0,NS)$ whether or not the lagging firm
invests. Thus, \textbf{both firms invest at }$(1,0,NS)$\textbf{. }
To see whether the firms share step $1$ at $(1,0),$ we compare joint profits
under sharing with joint profits under no sharing. If the firms share, the
game reaches the history $(1,1).$ Hence, joint profits are $V_{J}(1,1).$Joint
profits under no sharing are
\begin{equation}
V_{J}\left( 1,0,NS\right) =\frac{\alpha V_{J}\left( 2,0\right) +\alpha
V_{J}\left( 1,1\right) -2c}{2\alpha+r}=\frac{\alpha\widetilde{\pi}%
^{M}+\alpha V_{J}\left( 1,1\right) -2c}{2\alpha+r}. \label{jointprofit-1-0}%
\end{equation}
We have $NS\succ S\Longleftrightarrow$%
\[
\alpha\widetilde{\pi}^{M}+\alpha V_{J}\left( 1,1\right) -2c>(2\alpha
+r)V_{J}\left( 1,1\right)
\]
Substituting for $V_{J}(1,1)=2V_{2}\left( 1,1\right) $ from
(\ref{profit-1-1}) and simplifying, we have
\[
\pi^{M}>2\pi^{D}(\frac{2\alpha+2r}{2\alpha+r})+c(\frac{2r}{2\alpha+r})
\]
This inequality holds in the region, so \textbf{the firms do not share at
}$(1,0)$.
At the history $(0,0)$, assuming firm $2$ invests, firm $1$ will also invest
if \
\[
V_{1}\left( 0,0\right) =\frac{\alpha V_{1}\left( 1,0,NS\right) +\alpha
V_{1}\left( 0,1,NS\right) -c}{2\alpha+r}=\frac{\alpha V_{J}\left(
1,0,NS\right) -c}{2\alpha+r}>0
\]
Substituting using (\ref{jointprofit-1-0}) and (\ref{profit-1-1}) and
simplifying, this is%
\[
4\alpha\pi^{D}+(2\alpha+r)\pi^{M}>\frac{cr}{\alpha^{2}}(4\alpha+r)(2\alpha
+r)+2cr
\]
Since $\pi^{M}>2\pi^{D}$ in this region, the condition holds if
\[
(8\alpha+2r)\pi^{D}>\frac{cr}{\alpha^{2}}(4\alpha+r)(2\alpha+r)+2cr
\]
Since $\pi^{D}>\frac{cr}{\alpha}(\frac{3}{2}+\frac{r}{2\alpha})$ in this
region, the condition holds if
\[
(8\alpha+2r)\frac{cr}{\alpha}(\frac{3}{2}+\frac{r}{2\alpha})>\frac{cr}%
{\alpha^{2}}(4\alpha+r)(2\alpha+r)+2cr.
\]
This simplifies to
\[
2\alpha(2\alpha+r)>0
\]
which always holds. Hence firm $1$ invests at $(0,0)$ if firm $2$ invests.
Assuming firm $2$ does not invest, the history becomes $(0,X)$. Firm $1$
invests if
\[
V_{2}(0,X)=\frac{\alpha V_{2}(1,X)-c}{\alpha+r}=\frac{\alpha(\frac
{\alpha\widetilde{\pi}^{M}-c}{\alpha+r})-c}{\alpha+r}>0
\]
where we substituted for $V_{2}(1,X)$ using (\ref{profit-1-X}). Simplifying,
we get
\begin{equation}
\pi^{M}>\frac{cr}{\alpha}(2+\frac{r}{\alpha}). \label{stayin-X-0}%
\end{equation}
In this region, we have that
\[
\pi^{M}>2\pi^{D}\text{ and }\pi^{D}>\frac{cr}{\alpha}(\frac{3}{2}+\frac
{r}{2\alpha}).
\]
These two conditions together imply that (\ref{stayin-X-0}) holds. Hence,
\textbf{firm }$1$\textbf{\ invests at }$(0,X)$\textbf{. } It follows
that\textbf{\ both firms invest at }$(0,0)$\textbf{.}
\textbf{This completes the derivation of the equilibrium. The equilibrium is
unique. The equilibrium is non-monotonic because the firms share at }%
$(2,1)$\textbf{\ but not at }$(1,0)$\textbf{. The histories }$(1,0)$
\textbf{and} $(2,1)$\textbf{\ are both reached on the equilibrium path, so the
non-monotonicity arises on the equilibrium path. }
\section{Proof of Proposition 4}
A deriviation of all the equilibria of the game for $N=3$ is available on
request. That analysis proves the proposition. Here we derive the condition
given in the Proposition that defines Region A. As in Lemma 1, we focus on the
payoff that a firm would earn by conducting three steps of research on its own
and then producing in the output market as a duopolist. By the same reasoning
as in Lemma 1, this payoff equals the payoff of the lagging firm at $(3,0,NS)$
and is a lower bound on any firm's payoff at any history and in any
equilibrium. Therefore, Region A is all of the parameters for which this
payoff is positive.
We now compute the payoff. Suppose that the firm has completed the first step
of research on its own. As derived in Lemma 1, the firm's expected payoff from
computing the two remaining steps of research on its own is%
\[
\int_{0}^{\infty}e^{-(\alpha+r)}(\alpha\lbrack\frac{\alpha\widetilde{\pi}%
^{D}-c}{\alpha+r}]-c)dt=\frac{\alpha\lbrack\frac{\alpha\widetilde{\pi}^{D}%
-c}{\alpha+r}]-c}{\alpha+r}.
\]
Working backwards, consider the first step of research. The firm invests a
flow cost of $c$ and in each instant the probability of success is $\alpha$.
The firm's expected payoff is%
\[
\int_{0}^{\infty}e^{-(\alpha+r)}(\alpha\lbrack\frac{\alpha\lbrack\frac
{\alpha\widetilde{\pi}^{D}-c}{\alpha+r}]-c}{\alpha+r}]-c)dt=\frac
{\alpha\lbrack\frac{\alpha\lbrack\frac{\alpha\widetilde{\pi}^{D}-c}{\alpha
+r}]-c}{\alpha+r}]-c}{\alpha+r}.
\]
This payoff is strictly positive if and only if
\[
\pi^{D}\geq\frac{cr}{\alpha}(3+3\frac{r}{\alpha}+\frac{r^{2}}{\alpha^{2}}).
\]
This is the inequality that defines Region A.
\end{document}