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This is to certify that the

dissertation entitled

ESSAYS ON MIXED OLIGOPOLY AND AGRICULTURAL R&D

presented by

Anwar Naseem

has been accepted towards fulfillment
of the requirements for

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6/01 c:/CIRCIDatoDu9.p65-p.15

ESSAYS ON MD(ED OLIGOPOLY AND AGRICULTURAL R&D

By

Anwar N aseem

A DISSERTATION
Submitted to

Michigan State University
in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

2002

ABSTRACT
ESSAYS ON MIXED OLIGOPOLY AND AGRICILUTURAL R&D
By

Anwar Naseem

Three essays are presented which explore how micro level interactions between
public and private research and development (R&D) translate into observable macro
effects. The framework of analysis is a mixed oligopoly model where a welfare
maximizing public firm is in competition with a profit—maximizing firm. Careful attention
is paid to how asymmetric objectives, spillovers and appropriability of research exert
strong influences on the behavior of competing firms.

The motivations for the dissertation are set out in the introductory chapter. In the
first essay, a simple model of an R&D race between a public and private firm is
presented. The essay presents three specifications of the general model, and seeks to
define and differentiate between the effects of public research appropriability and
research spillovers. In the second essay, the nature of the observed market structure and
R&D competition in genomics research is used as the basis for a comparative analysis of
research under a mixed oligopoly, pure oligopoly and monopoly when the timing of the
innovation outcome is uncertain (as in an R&D race), the winner-take-all assumption is
relaxed and the profits in later stages are a function of the R&D expenditures of prior
stages. In the third and final essay, cooperative and noncooperative behavior in the
context of mixed oligopoly models is examined. The essay attempts to investigate the

reasons for the relatively small number of partnerships between public and private firms

in spite of institutional arrangements that encourage them; and explores whether
partnerships with private firms are inconsistent with the objective of a welfare
maximizing public-sector firm. In particular we consider cooperative and noncoopertive
R&D between a public sector and private firm in a two-stage model where research in the
first stage is followed by further research and production in the second stage.

The results of the models are used to understand the changing market structure of
the agriculture research and development and their implications. The role of a
competitive public sector firm in agricultural R&D is highlighted and its welfare effects

studied.

Copyright by
ANW AR NASEEM
2002

Dedicated to my parents
S.M. Naseem and Zarina Naseem
East Lansing, MI, June 2002

ACKNOWLEDGEMENTS

There are numerous individuals that deserve my Sincere gratitude for their support
throughout my graduate program. In the writing of this dissertation, I have had the great
pleasure of working with Professor James Oehmke, who, as the dissertation supervisor,
has been a constant source of good ideas, encouragement and patient guidance. Professor
Oehmke not only extended to me the freedom to research, but also had the faith and
confidence in me to let me complete the dissertation while I worked at Rutgers
University. I am grateful to the other committee members for their comments and
suggestions. I was privileged to have had Professors Jay Pil Choi and Carl Pray as my
external thesis examiners, who graciously agreed to shoulder the burden of this essential
task. I am particularly indebted to Professor Pray for providing me the resources and time
to work on the dissertation while I was at Rutgers. I thank Professor Christopher Wolf for
his detailed reading of the manuscript and constructive criticism. I also appreciate the
comments of Professor Dave Weatherspoon, which has led to a sharper intuition of the
results in many instances.

The research issues explored in this dissertation were initially proposed and
discussed at the weekly meetings of the Biotechnology Interest Group (B.I.G.) at the
Department of Agricultural Economics. Members of the group, which included
Professors Oehmke, Wolf and Weatherspoon, as well as John Francis, Amie Hightower,
Dr. Mywish Maredia and Professor Kellie Raper, generously lent their time to listen, and

politely offered criticism, to my initially grand, but often confused research objectives.

vi

I would also like to express my appreciation to Professor Richard Bemsten, my
major professor, who has provided invaluable guidance and mentoring ever since I came
to Michigan State. From the very beginning, Professor Bemsten has taken a deep interest
in my intellectual development, and has provided great ideas, contacts and suggestions. I
could not have wished for a more caring major professor.

Finally, I thank the Food Security Project H and the Michigan Agricultural
Experiment Station for supporting my graduate studies through various research
assistantships. The support provided by the MSU/Ford Foundation pre-dissertation
fellowship for a study tour of the Philippines is also appreciated.

My graduate studies would not have been the same without the social and
academic challenges and diversions provided by fellow students. Greatly missed will be
the conviviality of the many meals shared with N azmul Chaudhry and Brady Deaton.
Also missed will be the Hayford House parties and the company of hosts David Mather
and Matthew Schaeffer. With so many other friends at MSU, especially Bocar Diagana,
Andrea Jeffers, Chris Penders, Julie Stepanek, and George Young, I have Shared many
diversely rewarding experiences, and my connection to each has been a continuous
source of enjoyment and sustenance. To Monika Tothova, I owe so much of my
happiness of the past three years. She has been my greatest distraction and staunchest
supporter as I struggled with the challenges of graduate research work.

My final thanks must be reserved for my parents, who have instilled in all their
children the importance of education and perseverance; and for my siblings, Farouk and
Saba, who have given me all the pleasures that comes from being their “little” brother, as

well as uncle to their children. I know of no greater joy than coming home to family.

vii

TABLE OF CONTENTS

List of Tables

List of Figures
Chapter I: Introduction

Chapter H: The Effect of Public Research Appropriability and Spillovers on
Private Research
H. l . Introduction
11.2. Literature Review
11.3. The Model
11.3.1. Specification 1: Generalizing Appropriability in D&D and
P-T models
11.3.2. Specification 2: Appropriability With Private Research
Capacity
11.3.3. Specification 3: Public R&D Spillovers
11.4. Comparative Statics and Welfare Analysis
H.5. Conclusions

Chapter 111: Should the Public Sector Conduct Genomics Research?
111.]. Introduction
111.2. Literature Review
111.3. The Model
111.3. 1. Monopoly equilibrium condition
111.3.2. Pure duopoly equilibrium condition
111.3.3. Mixed duopoly equilibrium condition
111.4. Comparative Analysis
111.4.]. Monopoly vs. Pure Duopoly
111.4.2. Monopoly vs. Mixed Duopoly
111.4.3 Mixed Duopoly vs. Pure Duopoly
111.5. Discussion
111.6. Conclusions

Chapter IV: Cooperative and Noncooperative R&D in a Mixed Duopoly with
Spillovers
IV. 1. Introduction
IV.2. The Model
IV.2.]. Regime 1: Mixed Duopoly in Both Stages
IV .2. l . l. Non-Cooperative Game
IV.2. l .2. Cooperative Game
IV .2.2. Regime 2: Mixed Duopoly in R&D Stage Only
N22. 1. Non-Cooperative Game

viii

xi

l6
l7

17

26
34
41
43

46
46
52
54
57
57
58
61
65
70
73
77
81

85
85
89
92
93
100
101
102

IV.3.

IV.4.
1V.5.

III.2.2.2. Cooperative Game
Specializing the Model
IV.3. l. Regime 1: Mixed Duopoly in Both Stages
IV .311. Non-Cooperative Game
IV.3. 1 .2. Cooperative Game
IV.3.2. Regime 2: Mixed Duopoly in R&D Stage Only

IV.3.2. 1. Non-Cooperative Game
IV.3.2.2. Cooperative Game
Welfare
Conclusions

Chapter V: Conclusion

Appendices

Chapter 11
Chapter 1H
Chapter IV

Bibliography

ix

104
106
106
106
114
118
119
122
126
128

131
139
140
146
155

168

LIST OF TABLES

Table 11.1: Effect of Spillover on the Public and Private Research

Table IV.1: Overview of the Cooperative and Noncooperative Game
Table IV.2: Summary of Results for §IV.3.2

Table A1H.1. Sufficient Conditions for Ranking Monopoly and Pure Duopoly
Research Effort

Table A111.2. Sufficient Conditions for Ranking Monopoly and Mixed Duopoly
Research Effort

Table A111.3a. Sufficient Conditions for Ranking Pure Duopoly and Mixed
Duopoly Research Effort

Table A111.3b. Sufficient Conditions for Ranking Pure Duopoly and Mixed
Duopoly Research Effort

Table AIV .3: Nash Equilibrium Values for the Noncooperative Case in Regime 1
Table AIV .4. Nash Equilibrium Values for the Cooperative Case in Regime 1
Table AIV.5: Comparing Equilibrium Values Between Cooperative and
Noncooperative Cases Under Regime 1

Table AIV .6. Nash Equilibrium Values for the Noncooperative Case in Regime 2
Table A1V.7: Nash Equilibrium Values for the Cooperative Case in Regime 2
Table A1V.8: Comparing Equilibrium Values Between Cooperative and
Noncooperative Case Under Regime 2

Table AIV .9: Socially Efficient Values at Different ,6 Values.

Table AIV .10: Comparison of Nash-Equilibrium Output and Research Across
Models

42

92

106

151

152

153

154

157
159

161
162
164

166
166

167

LIST OF FIGURES

Figure 1.1: Mergers and Acquisition by Diversified Biotechnology Firms
Figure 11.1: R&D Expenditures by Public and Private Sectors in the United States

Figure A11.2: Relationship Between R&D Intensity, Appropriability and Spillovers
Figure AII.3: Public and Private Best Response Under Varying Levels of a for
Specification 1.

Figure AII.4: Public and Private Best Response Under Varying Levels of on for
Specification 2.

Figure AII.5a: Public and Private Best Response Under Varying Levels of [3 for
Specification 3 ([3 =0.5). _

Figure AII.5b: Public and Private Best Response Under Varying Levels of B for
Specification 3 (,6 =1)

Figure A1150: Public and Private Best Response Under Varying Levels of B for
Specification 3 (,6 =0)

Figure A.1: Second Stage Mixed Market if Private Firm is the Winner of the R&D
Race -

Figure A.2: Second Stage Mixed Market if Public Firm is the Winner of the R&D
Race ‘

xi

10

140

141

142

143

144

145

150

150

CHAPTER I

INTRODUCTION

Research and development (R&D) in agriculture, both by public and private
sectors, has been a significant source of productivity improvements over the last century
(Huffman and Evenson, 1992; Alston, Pardey and Smith, 1997). New technologies, such
as biotechnology and information technology, provide hope that improvement in
efficiency will continue well into the new century. What is less clear is what impact, if
any, the comprehensive restructuring that has taken place in the agricultural R&D
industry, especially over the past decade, will have on market performance, economic
growth and social welfare.

The industry has undergone three distinct structural changes in recent years. The
most important, perhaps, is the change in relative levels of investments by public and
private sectors. Historically, the government has supported agriculture research on the
premise that the knowledge that results from research activities is, in general, a public
good (i.e., non-rival and non-excludable). Even when this is not the case, and some
returns to research can be captured by innovators, the incentives for the private sector
R&D investment may be weak. For this reason, public sector1 involvement in agriculture
research has been considerable, and during the first half of the past century (1906 to
1950) has been larger than the private sector’s (Huffman and Evenson, 1992). For the
latter part of the twentieth century, private sector research investments have risen

noticeably, both in absolute terms and relative to public funding. The rate of growth in

 

' References to “public” sector research pertains to all research performed in state agricultural experimental
stations, land grant and other universities, and the US. Department of Agriculture. “Private” research
pertains to all research activities conducted by private sector firms.

public research expenditure has, however, slowed since the mid-1970s, and virtually all
the growth in US. agricultural research expenditures can be accounted by private sector
investments (Fuglie, 2000). By the mid-1990’s private sector expenditures exceeded
those of the public sector by as much as 55 to 65 percent (Huffman and Evenson, 1992;
Fuglie 2000).

A second Structural change occurring in the agricultural R&D market is that the
composition of research conducted by the private sector has changed. During the 1960’s,
94% of the private sector research focused on machinery, chemicals, and post-harvest and
processing technologies. The private-sector focus was determined in large part because
the appropriability of research benefits was much easier in machinery, chemicals and
post-harvest innovations than in on-farm and agronomic technologies. However, with the
strengthening of the intellectual property rights (IPR) system, technological opportunities
offered by advances in biotechnology, along with an explicit policy to encourage more
public and private sector collaboration, has resulted in more private investment in
research areas that once largely fell in the public domain (e. g., plant breeding and
livestock improvement) (Fuglie et al., 1996). And while public sector research priorities
by technology area have remained relatively unchanged over the years (i.e., a continued
emphasis on basic research in crop and animal research), it is increasingly in competition
with the private sector in research areas that would be considered “basic.”2 Genomics

research, which has many of the attributes of basic research, is an example where both

 

2 As defined by the National Science Foundation, “the objective of basic research is to gain more complete
understanding of the subject under study without specific applications in mind. In industry basic research is
defined as research that advances scientific knowledge but does not have specific objectives, although it
may in fields of present or potential commercial interests.” (NSF, 1996)

private and public sectors compete to sequence the genetic code of several organisms,
cooperatively and noncooperatively.

A third structural change, and one which has received much scrutiny among
agricultural economists, has been the increasing consolidation in agriculture and related
sectors3. Mergers and acquisitions (M&As) have been the key drivers for the increased
consolidation (Fulton and Gianankas, 2001). Figure 1.] illustrates that the trend in M&A
activities by diversified biotechnology firms follows a cyclical pattern, with peaks

occurring in the late 80’s and 90’s.

Figure 1.1: Mergers and Acquisition by Diversified Biotechnology Firms

12

 

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<29 <23" <21" 21° <6" OP 9'” 9“ 01° 03’
9 9 9 9 9 9 9 ,9 .9 ,9

Year

Source: Kalaitzandonakes and Hayenga (2000).

 

3 Among studies that have documented, explained and/or related consolidation activity to innovation
measures include Kalaitzandonakes and Hayenga (1999), Fulton and Giannakas (2001), Oehmke et al.
(2000), and Brennan, Pray and Courtmanche (1999).

There is evidence that the consolidation in the vertically integrated seed,
biotechnology and chemical markets has resulted in concentration in specific output
markets as well as innovation markets. Kalaitzandonakes and Hayenga (2000) report that
in 1998, Monsanto and Pioneer Hi-Bred had, respectively, a market Share of 15% and
39% of the US. seed corn market, and 24% and 17% of the soybean seed market. The
cottonseed market is essentially controlled by Delta and Pine Land (with a 71% market
share) and Stoneville (with a 16% market share) (Kalaitzandonakes and Hayenga, 2000).
Brennan, Pray and Courtmanche (2000) provide preliminary evidence of concentration in
the plant biotechnology R&D market. Using firm level data on field trials of genetically
modified organisms (GMOS) in the US, they construct a four firm concentration ratio
for innovation in plant biotechnology. In 1998 the top four firms conducted 87% of all
field trials, which declined to 63% in 1995 and then rose to reach a high of 79% in 1998
(Brennan, Pray and Courtmanche, 2000). Whereas the evidence of concentration in the
plant biotechnology industry is compelling, its impact on output and innovation market
performance is ambiguous.4

This dissertation furthers an understanding of the implications of structural
changes in agricultural research and development from a theoretical standpoint. The
approach adopted is to model the micro-level interaction between innovating firms. In
particular, and in light of the changing R&D market, the dissertation seeks to address the

following research questions:

 

‘ Brennan, Pray and Courtmanche did not find a strong causal relationship between concentration and
innovation in the plant biotechnology industry.

How can we explain the differences in public-private research activities across
research areas? In other words, under what market and institutional conditions
does one sector conduct more R&D than the other?

What are the implications for a decline in public research activities on private
sector output? More generally, what, if any, is the nature of the causality
between private and public sector research?

Will the use of IPRS by both private and public sectors limit innovation? Our
underlying hypothesis is that stronger IPR will limit the amount of knowledge
sharing between research entities, and therefore may have implications for
market performance and welfare.

What iS the relationship between market concentration (in both product and
research markets) and research effort of the public and private sectors?

Under what conditions can the public sector affect market structure that
ensures competition and innovation? Consider, for example, that following the
enactment of the 1970 Plant Variety Protection Act, Butler and Marion (1985)
found little evidence of decline in competition in the seed industry which they
attribute to public plant breeding institutions that developed and released non-
protected varieties, thereby keeping concentration in check. Can one expect
that public sector research will be a source of competitiveness and innovation
in an increasingly concentrated market, and if so, under what conditions?
With the public sector coming under increasing pressure to form collaborative
research programs with private firms, what affect can be expected of such

cooperative research?

This dissertation addresses the above research questions in three related essays
(although no single essay Speaks to all the issues). The analytical basis for these essays is
a game-theoretic mixed oligopoly framework, wherein profit-maximizing private firms
and welfare-maximizing public sector firms compete, non-cooperatively, in product
and/or research markets. Due to the diverging objectives of public sector and private
sector firms in a mixed oligopoly setting, the outcome predicted by such a modeling
framework is different from the pure oligopoly case where only private-sector firms are
engaged in strategic behavior. The mixed oligopoly framework offers an approximate
representation of the agriculture research market, where both public and private firms
complement and compete with each other’s activities.5 The use of the mixed oligopoly
modeling as a tool to better understand the market structure and innovation as it relates to
agricultural research is the main analytical contribution of this dissertation.

Research questions 1 to 3 are addressed in the first essay. Differentiating between
appropriability affects and Spillovers we Show that these two affects can explain, in a
simple one-period innovation race model, the relative level of private and public sector
firm research. We first review the existing theoretical literature, which provide
contradictory results as to whether public research encourages or discourages private
research and later construct a general Coumot type game that includes the existing
models as special cases. We show that the contradictory results are due to different

interpretations of spillovers and appropriability. Moreover, we Show that the mechanisms

 

5 There does not exist, to our knowledge, a unified theory of mixed oligopoly. De Fraja and Delbono (1990)
and Nett (1991) review the existing models, all of which are a variation of the standard oligopoly models
where firms compete in output or prices, except that in a typical mixed oligopoly model, at least one firm is
welfare maximizing. In the agricultural sector, mixed oligopoly models have been used to study the
behavior of agricultural cooperatives in relationship to private firms (Tennback, 1995; Azzam and
Andersson, 2001).

through which public R&D impacts private R&D—whether through greater knowledge
flows or the ease with which the private firm can appropriate public sector firm’s
innovations—matter in determining the outcome of the R&D game. We interpret our
results in the context of strengthening of the IPR regime and changing resource allocation
of public and private sectors.

The second essay, which addresses research questions 4 and 5, is motivated by the
observed market structure and the nature of R&D competition in genomics research.
Genomics research (for example that on the human genome and to a lesser extent, rice
genome) has been characterized by intense competition between public and private
enterprises to sequence DNA. The successful sequencing of genes has profound
implications for downstream research, which depends on the accuracy and methodology
of sequencing6. We therefore model genomics research as a two-stage process, where the
research race in the first stage impacts the research of the second stage (the second stage
can also be interpreted as an output market). We compare the market performance of the
observed mixed oligopoly in genomics research vis a vis a monopoly and a pure duopoly.
We derive conditions that allow us to rank the amount of research that is performed by
each of the three markets and make Statements on how competition in genomics research
is affected by the presence of a public sector.

Our last essay addresses the issue of cooperative and non-cooperative behavior in
the context of mixed oligopoly models. Our interest here is to understand the reasons for

the relatively small number of partnerships between public and private firms (relative to

 

6 For instance, rice has related proteins for 85 percent of the proteins identified in cereals. The high
percentage of related genes could ease the identification of agronomically important genes in cereals.
Another prospect resulting from the research on rice genome is an increased ability to reveal the DNA
sequences of traditional n'ce varieties and wild species (Ronald and Leung, 2002)

the a much more frequent occurrence of cooperation among private firms). Further we
evaluate the merits of an oft-repeated criticism of partnerships between public and private
firms is that it compromises the public mission of the public organization and may also
give the partnering firm a competitive advantage over the other firms. We explore this
issue under two different types of market settings (one where the public sector firm
competes in the product market and another where it does not), and show that the results
are sensitive to the level of spillovers between firms. In particular, we find that for high
spillovers, profits of the private firm when it collaborates with the public sector firm are
actually lower than the noncooperative case or lower than the firm that does not
cooperate. Further it is found that cooperation is always a Pareto improvement for the
public sector firm, suggesting that incentives for the private firm to cooperate with the
private firm would be a welfare improving policy.

The concluding chapter of the dissertation summarizes the key findings of the
study and relates them to the structural changes that were described earlier. The
contribution of the research is highlighted and fixture work on empirically testing the

hypotheses generated by the models is suggested.

CHAPTER 11
THE EFFECT OF PUBLIC RESEARCH APPROPRIABILITY AND SPILLOVERS

ON PRIVATE RESEARCH

11.]. Introduction

In this chapter, we present a model to study the impacts of two significant
developments in agricultural R&D. First, as mentioned in the introductory chapter, is the
observation that there has been a shift over the last two decades in R&D effort from the
public sector to the private sector. Figure H.1, which plots the trends in R&D
expenditures (in real 1993 dollars) from 1970 to 1998, shows that since the mid 80’s
private sector R&D has surpassed that of the public sector. In 1992, the private sector
spent $3.5 billion (1993 dollars) on agricultural research, nearly one billion dollars more
than the public sector (Alston, Pardey and Smith, 1998). In addition, the private sector
moved into plant-breeding and veterinary research, areas traditionally dominated by the
public sector, spending 12% and 9%, respectively, of its funds in these areas. Since 1992,
the real value of public agricultural research expenditures has fallen (Alston, Pardey and
Smith, 1998). Today, agricultural R&D activities are dominated by firms such as
Monsanto, whose research budget in 1997 of approximately $1.2 billion (Monsanto,
1998) was approximately equal to the total amount spent by the US. public sector spent

on crop related activities.

Figure 11.]: R&D Expenditures by Public and Private Sectors in the United States

 

5,000
4,500 —
4,000 -
3,500 1
3,000 ~
2,500 4

  

 

 

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Source: Economic Research Service (2002)

A second observed phenomenon is the increasing use of intellectual property (IP)
protection, especially patents, by the public sector. IP protection, gives innovators, public
and private alike, a mechanism to appropriate the returns due to their R&D and hence
giving them an incentive to carry out further research. According to the Association of
University Technology Managers (AUTM) surveys, patenting activity in US. research
universities has been increasing since the Bayh-Dole Act of 1980, which sought to
encourage the transfer of publicly funded innovations to the private sector with an
emphasis on patenting. Prior to the Act, universities were patenting at a rate of 250
patents annually but have dramatically increased from 1600 patents issued to them in
1991 to about 2000 in 1998. For the period 1994-97, new patent applications and
licensing of 65 universities that responded to the survey, grew by 20.4% and 9.6%

respectively. This growth in the “commercialization” of academic research is seen as by

10

some as evidence of the benefit of university research as well as raising concerns about
the conduct and priorities of university research (Thursby and Thursby, 2000).

Standard economic theory suggests that if profit-maximizing firms are better able
to appropriate their rents due to R&D (and to the extent that stronger IPRS provide better
appropriability conditions), they will increase their research activity. This may not be
applicable to public sector organizations (as the casual observation of the data suggests),
since public sector organizations (henceforth, public sector firms), such as universities
and govemment-controlled entities, do not, generally, operate under the profit
maximizing principle. Instead, public sector firms are considered to be welfare-
maximizers and it is not clear, a priori, whether the ability to appropriate R&D returns
(through patenting, for example) should imply greater R&D intensity on the part of the
public sector firm. For example, if the public sector firm is able to appropriate returns of
its R&D, it would suggest that the knowledge/innovation that is being patented has “lost”
its public goods characteristics (i.e., it is no longer non-excludable) and has attained the
attributes of a private good. The more a research output becomes “privatized”, the less
compelling it becomes for the public sector firm to conduct such research.

A more holistic approach to understanding the incentives for research, for both
private and public sector firms, requires that we go beyond just the notion of
appropriability but situate it within the context of knowledge flows and spillovers. Before
so doing, it is important to be clear on the definition and distinction between the related
concepts of research appropriability and research spillovers that will be used in this and
subsequent chapters. Appropriability relates to the amount of rents captured by an

innovating firm due to its innovation. As such, it is a notion that applies after the

11

discovery process that leads up to the innovation. R&D Spillovers, on the other hand, are
most often defined as extemalities that arise from research effort of individuals or firms.
Geroski (1995) identifies three modes through which spillovers occur:

“they routinely arise when different agents discuss subjects of mutual interest, or when
research results are disseminated through publications and seminar presentations.
Spillovers can also be created when one agent observes the actions of another and makes
inferences about the thinking that lies behind those actions. Last but not least, spillovers
occur when a researcher paid by one firm to generate new knowledge transfers to another
firm (or creates a spin-off firm) without compensating his/her former employer for the
full inventory of ideas that travels with him/her” (Geroski, 1995)

This suggests that spillovers occur during the R&D process and not so much after
(Hinloopen, 2000). This is especially true for public R&D, where evidence suggest that
public research, especially research on biotechnology, plays an important role in the
innovation process of US industry (McMillan, N arin and Deeds, 2000).

R&D spillovers have been recognized to having an important effect on innovative
performance and productivity growth (Griliches, 1992). While Spillovers may increase
the productivity of recipients of R&D, and hence of R&D at the industry level, they may
also diminish appropriability (Cohen and Walsh, 2000). That is, the leakage of
knowledge flows from a firm may give other firms an edge in the research process and
use it to produce better or similar innovations; thereby reducing the market share and
profits for the firm from which the Spillovers initially arose. However low
appropriability does not necessarily imply the presence of large research spillovers.
Consider two firms conducting research independently and noncooperatively of each
other and without any research Spillovers between the two. If one firm innovates first, but

does not (or cannot) adequately protect its innovation such that the other firm is able to

effectively imitate and reproduce the innovation, the appropriability of rents for the

12

innovating firm are diminished and essentially shared among the two firms as the non-
innovating firm free rides off the innovation.

Figure AH.2, adapted from Cohen and Walsh (2000), helps conceptualize the
relationship between spillovers, appropriability and R&D intensity. In the schematic
representation of Figure AII.2 R&D iS undertaken by public and private sector firms.
Appropriability is characterized as the level to which different mechanisms, such as IP
protection through patents and plant variety protection, trade secrecy and first mover
advantage, are effective in protecting returns due to R&D. For the private firm
appropriability has a positive effect on the level of research conducted by it, as theory
would suggest. For the public sector firm, as suggested earlier, the ability to appropriate-
may or may not have a positive effect on the public R&D as the public sector firm may
leave such research to the private firm and concentrate on research through which its
welfare-objectives are better met (i.e., research that may be difficult to appropriate but
still is welfare increasing). The relationship between appropriability and spillovers may
also be different across public and private firms. If spillovers are large and significant
within an industry, then private firms will find it more difficult to appropriate rents to
R&D, hence the negative relationship. For the public sector firm, the lack of
appropriability due to greater spillovers, while decreasing any pure monetary gains to
itself, may in fact be welfare increasing if it results in significant productivity gains in the

economy. '

 

' Since the public sector need not be financially profitable, it can invest in research beyond the point which
the private sector can afford, with the express intent of creating spillovers to improve social welfare. Thus,
the appropriability problem that arises in private sector research is not an issue for the public sector. The
public sector’s objective of increasing social welfare may, in fact, be advanced through more public
research spillovers.

13

For both firms, however, the use of different appropriability mechanisms may
weaken the extent and value to competitors of spillovers. Some appropriability
mechanisms, such as trade secrecy, result in Spillovers information flows being less than
others, such as patents where one has to disclose the nature of innovation while offering
protection. We conjecture, however, that since public sector firm’s use of IP protection is
a fairly recent phenomenon it is more appropriate to consider Spillovers with and without
IP protection for the public sector firm. That is, in the case of the public sector firm, the
relevant issue is whether IP protection results in more or less spillovers occurring? We
surmise that since public sector firms serve the public good, they are more open to the
concept of sharing knowledge and explicitly allow spillovers to occur (as opposed to their
counterparts in the private sector, who tend to restrict such information flows). When a
public sector firm patents, while it still may have to disclose information in that patent, it
can restrict the use of the innovation to only a few agents by licensing the technology
exclusively.2 It is in this sense the IP protection by the public sector firm may lead to
lower spillovers.

Spillovers may also condition the incentives to conduct R&D by either

complementing firm R&D (Levin et al., 1987) or by substituting for it (Spence, 1984)3.

 

2 Maredia et. al. (2000), provide examples where exclusive licensing of a public sector firm’s innovation
may indeed increase welfare. However, since only a few firms were licensed the technology, the spillover
effects of such patenting and licensing may have been limited (relative to the case where the technology
was not patented and anyone could have availed them to it).

3 Empirical evidence does not always support the view that industries with high spillover rates will have
low R&D expenditures or that the latter are inversely related with the degree of spillovers. Bernstein and
Nadiri (1989) find that for some industries with high spillovers, firms perform quite well in terms of
dynamic efficiency and thus appear not to be discouraged by the implicit knowledge leakage. A similar
conclusion is reached in a survey conducted by Levin and colleagues (Levin et al., (1987)), where it was
found that the firms with the highest level of spillovers (computers, communications equipment, electronic
components, an aircraft) also ranked high in R&D intensity.

14

Lastly, industry R&D effort itself increases R&D spillovers across firms, as greater
amount of R&D implies a greater amount of Spillovers.

Although the literature has noted the importance of the magnitude of the
knowledge flows for the level of innovative activities that are obtained with competitive
or c00perative research among private firms (see DeBondt (1997) for a review),
substantially less attention has been given to spillovers from public sector firms within
the context of a mixed R&D market. The study of public research spillovers is important
especially at a time when publicly funded research institutions are protecting their
innovations for a variety of reasons such as increasing revenues through licensing,
‘defensive patenting’, and protecting and licensing commercially unattractive innovations
(Maredia et al., 2000). The gains from protection by a public sector firm however must
be weighed against the response that they generate among the private firms. If the
protection of innovations implies a restriction in the sharing of innovation and the
knowledge embodied in it, then the question of how such restrictions on public research
spillover effects the behavioral response of the private firm is not merely an empirical
one, but important also for its theoretical stipulations. In this paper we focus our attention
on theoretical aspects of public research spillovers to clarify the impacts that are due to
greater appropriability for public research and those that are due to greater knowledge
flows (spillovers). We present variations of the standard patent race models within the
context of a mixed market and under the assumption of asymmetric spillovers.4

The plan of this chapter is as follows. The next section briefly reviews the

literature on R&D in the context of mixed markets, highlighting key results that have

15

bearing on our modeling. The modeling section follows in which three Specifications,
variations of the standard R&D race models, are presented. The first Specification
generalizes two earlier models by parameterizing the degree to which public sector finn’s
R&D is appropriable. The second Specification further develops the first specification, by
requiring that the private firm have a research capacity to reap benefits of public sector
firm innovations. The last specification models public research spillovers as cost
reducing. In section four we provide a comparative analysis and their welfare

implications of the results of the three specifications. Section five concludes the paper.

11.2. Literature Review

Before presenting the theoretical framework of this paper, which builds on and
owes to the work of earlier researchers, a brief review of recent papers by Nett ( 1994),
Delbono and Denicolo (1993) and Poyago-Theotoky (1998), who have considered the
performance of mixed R&D oligopolies, is necessary. Nett’s analysis considers research
for a process innovation in a mixed duopoly with production. The analysis is cast under
certainty, without spillovers, where the objective of the public sector firm is output
maximizing (market-share maximizing). Under such a setup, Nett (1994) shows that a
private firm has an incentive to operate at lower variable cost but higher fixed costs than
a public sector firm. By choosing to shift a higher amount of its resources to the sunk cost
category, 3 private firm has more flexibility in its strategic behavior of the market game
than its public rival. Further, welfare in a private (pure) duopoly may exceed welfare in

the mixed duopoly. In contrast, Delbono and Denicolo (1993), in their modeling of R&D

 

" The asymmetry is due to spillovers arising only from public sector firms and not private, as our interest in

16

as a patent race, Show that a welfare—maximizing (not output-maximizing) public sector
firm actually contributes to an increase in social welfare relative to the private duopol y.
Having a public, welfare-maximizing firm compete with a private, profit maximizing
firm results in both firms reducing their R&D effort, which alleviates the over-investment
problem of the non-cooperative equilibrium (i.e., the case of the private duopoly).
Poyago-Theotoky (1998) extends the analysis of D&D to consider the under-investment
problem that arises from the ‘easy imitation’ of the reward. By modeling public sector
firrn’s research as not appropriable (i.e., the benefits of public research are distributed
among both private and public sector firms), P-T is able to reverse the results of D&D. In
particular she finds that the public sector firm will invest more in R&D than its rival
private firm, and when comparing welfare across different institutional settings, welfare
can be higher or lower in the mixed duopoly relative to the private duopoly, depending on
the size of the reward to innovation.

The D&D and P-T models serve as the point of departure for this chapter. In
particular we show that their models and results can be easily generalized by interpreting
the models in terms of appropriability as defined earlier. The effect of different levels of

public research appropriability on private research and welfare is discussed.

11.3. The Model
11.3.1. Specification 1: Generalizing Appropriability in D&D and P-T models
Consider a one-Shot non-cooperative game between a profit-maximizing private

firm (P) and welfare maximizing public sector firm (S), where the firms invest in R&D

 

this chapter is to primarily understand the impacts of appropriability and spillovers of public research.

17

with the aim of innovating. The firm that innovates first is awarded an exogenously

determined prize (W), which is the same for the two firms, i.e.,WS = WP = W .5 The

probability of success by firm i is a function of its R&D expenditure x,; where x,- is the
flow cost that firm i pays until one firm succeeds. The payoff function of the private firm

is specified as the present value of expected profits, net of R&D costs:6

00

VII) = Jexp{-(h[xp] + h[xS ]+ r)t} {—W (h[xP] +a’h[x5 ]) —xp]dt (11.13)
r
O

: (W/r)(h[xp]+ah[x5])—xp
h[xp]+h[xS]+r

 

(11.1b)

where r is the discount rate. Following the literature on innovation races, h[x,-], the hazard
function, is the instantaneous probability of innovating (Reinganum, 1989). The hazard
function is twice differentiable, strictly increasing and satisfies:

1. h[0] =0 = lim h'[x]

x9.»
2. h '[xi] > 0
3. hT4]<0
The parameter ais the appropriability parameter and lies on the closed unit interval [0,1].
The interpretation of the appropriability parameter within the present context is the

following. Most R&D race models have assumed that the returns to R&D for the winning

firm is value of the prize (W), with the losing firm getting nothing (winner take all

 

5 D&D justify this assumption on the grounds that “when the private firm is a perfectly discriminating
monopolist, whereas the public sector firm maximizes social welfare also in the product market.”

18

assumption). However if we assume that the public sector firm is unable (or unwilling) to
appropriate all of the returns to its research, than it is natural to assume that the private
firm benefits even if it were to lose the race. By assuming that public research is
imperfectly appropriable such that 0 S a $1 then the returns to the private research when
it loses are aW . The more appropriable public R&D becomes, due to increased patenting
by the public sector firm, the lower the opportunity for the private firm to copy public
sector firm’s innovation and reap benefits. In the D&D model it is implicitly assumed
that the public sector firm can appropriate all the returns to its research, thus a=0. P-T
considers the other extreme wherein returns to public research, should the public sector
firm win, is shared with the private firm (i.e., 0a=1.)7

Next, we specify the payoff to the public sector firm. The innovation entails a
social benefit, which is assumed to be equivalent to the prize obtained by the private firm.
Since only one innovation is in prospect, the public sector firm is indifferent as to who
wins the race; to the public sector firm the expected date of innovation is what matters.
Moreover, the public sector firm takes into account the R&D costs of both firms. It is in
these respects that the public sector firm is considered a welfare-maximizing firms. The

public sector firm’s payoff is specified as

 

6 A note on notation: to differentiate between the different payoff specification under consideration in this
paper, the superscript in Vii for j=1,2,3 represents the case being considered. The initial case that

generalizes the modeling of D&D and P-T is denoted case i=1, the other two cases (whose description
follows) are denoted 2 and 3.

7 To allow for comparison and consistency among the models, 1 use D&D’s specification as the basis. In P-
T’s original specification, the instantaneous resource cost of achieving a hazard rate x is given by fix]. The
function ){x} is a strictly increasing cost function satisfying the condition VxZO, y'[x]20, y ’[x]20. D&D’s
and P-T’s specification are equivalent in that they imply that there are decreasing returns to research
everywhere.

8 The underlying assumption here is that a faster pace of innovation in the economy is welfare increasing.

19

V51: J'exp{—(h[xP]+h[x5]+r)t}[-‘::(h[xp]+h[x5])—xp —XS:ldI (11.23)
0

: (W/r)(h[xp]+h[x5])—xp -XS
h[xp]+h[xS]+r

 

(II.2b)

It is important to note that appropriability parameter does not enter into the public
sector firm’s payoff. As a social-welfare maximizer, the public sector firm benefits
whether it wins or the private firm wins. Further, the fact that the private firm can easily
imitate public research firm when a =1, does not diminish the value of the prize. This is
an important point and requires emphasizing. We have already assumed that the value of
the innovation is the same across the two innovators and not related to who innovatesg.
Since the ‘prize’ to the public sector firm is social welfare whereas to the private firm it is
private profits when it is a perfectly discriminating monopolist, the value of the prize is
equal. The notion of appropriability would suggest that if the benefits of a firm’s
innovation are appropriated among several users (due to weak patent, for example) then
its profits should be lower relative to the case when it is able appropriate all the benefits
to itself. However, for the public sector firm profits, and more crucially the distribution of
profits among firms, are irrelevant, Since the public sector firm maximizes public welfare.
Since the concept of welfare constitutes individual firm profits, the public sector firm is
less concerned with whether it makes more profits (and hence someone else less) due to
increased appropriability as total social welfare remains unchanged given our

assumptions on equivalence of the prize. Appropriability would, however, matter to all

20

firms if all were profit-maximizing firms. In this case, an increasing ability to appropriate
returns by firm i (Should it win) would imply decreasing post-innovation profits for firm j
(the losing firm)”.

The two firms acting non-cooperativel y and simultaneously choose the R&D
expenditure in order to maximize their respective payoffs (i.e. equation 11.1 and 11.2).
From the first order condition for a maximum, one derives the following reaction curves

for the private and public sector firms, respectively, in the R&D space (xp,x5)
R}.[xp,x51= h '[xp](r+ (1 —a)h[x5])W - r2 — rh[xp]— rh[xs]+ rxph'ixpi = o (11.2)
R§[xp,xs] = h'[x5](W +xp)—r—h[xP]—h[x5]+xsh'[x5]= 0 (11.3)
The reaction functions defined by equations (11.2) and (11.3) are continuous and
will have a unique positive equilibrium x;[xs] and x;[xp] if R,-l[xp,xs] is decreasing
in x,, with R, positive at x,- = o and negative at x,- = co .” Choosing 1210] > 1/W and
h '[oo] = 0, for example, ensures a unique positive solution (flux; ) for every a. The
stability condition for the R&D race is equivalent to BR/ 8x, <0 (Lee and Wilde, 1980).
Nti (1999) Shows that the stability condition BR/ ax, < 0 is satisfied if the hazard rate
elasticity H [x] = xh '[x]/ h[x] is differentiable and non-increasing in x.

To characterize the Nash equilibrium in the context of public research

appropriability, we first describe the shape of the reaction curves. We limit our attention

 

9 This assumption fails when the use of the innovation by the public sector firm differs from that of the
private firm (D&D).
0 That is if W is the value of the prize then, under conditions of imperfect appropriability, the winning firm

may get W and the losing firm (1- a)W with 0.5 S a S l (the lower bound representing lowest
appropriability and the upper bound the highest appropriability).

21

to the private firm, as the public sector firm’s payoff structure remains unchanged and its
properties have been described by D&D”. Our interest here is to examine how
equilibrium research varies due to appropriability of public innovation. We formalize the

shape characteristics of the private reaction function in the following lemma:

Lemma 1].]: Define R};[x5,xp]=O as the private firm’s reaction function (11.3), and

R&[xs ,xp] 2 0 as the public sector firm’s reaction function (11.3). Then

: : X'=O .
a) R}>[O,XP]=Rg[Xs,O], or more compactly, x? 0 =x§P 0 where xi’ rs the

optimal R&D investment of firm i, given that the other firm does not invest.

b) Define MVRV as the marginal value product (MVP) of research from winning the
race, MVPL as the MVP of research from losing the race, and MC as the marginal

cost of research, then

1. Private reaction curve is positively sloped for M VFW - M VPL >MC
11. Private reaction curve is negatively sloped for MVRV -MVPL <MC

iii. Private reaction curve has a slope of zero for MVPW - MVPL =MC

c) Va,0$a$l 3a’,thai solves dx [x ]/dx =0
P S S

d) The private reaction function is bounded

Proof

 

“ This will be true for all the best response functions that we consider in the paper

'2 The public sector firm’s reaction curve is downward sloping if xs>xp and upward sloping for now. In
both D&D case and P-T case, there exists a unique Nash equilibrium of the innovation race, where D&D
shows that the public sector firm invests less in R&D than the private one at that equilibrium, whereas P-T
shows the opposite.

22

a)

b)

Setting x5 =0 into (11.2) and xp = 0 into (11.3) we get, in both instances,
h'[x,-](W +x,-)—h[x,-]—r =0 (11.4)

XS=0_ XP=0
Hence x,D —xS .

Implicitly differentiating equation (11.2) and exploiting the second order condition

we get

Sign [£11311] =sign[h'(xp)(1—a)W —r] (11.5)
de

Rearranging the term on the right hand side of equation (11.5), M E 0 iff

 

S
h [xp]W _ ah [xp]W 3] (1162!)
r r
01'
MVRV —MVPL a MC ("-7")

Recall that the cost of research is linear, x, hence the marginal cost of research is
1. What equation (H.7) shows is that when the difference between the marginal
value product of winning and losing is greater than the marginal cost of research,
then the private reaction curve is upward sloping and is downward sloping when
the difference is less than the marginal cost. The D&D result implies that for

a = 0 MVPW > MC , hence the reaction curve is upward sloping. Similarly, in the
P-T framework where ( 0! =1 ), MVPW = MVPL implies 0 < MC , hence the

reaction curve is downward sloping.

23

 

c) The solution to dxp[x5]/dx5 =0 requires 0' =1— . The zero slope of the

WhTXP]

r

I o . . = *

rrvate reaction curve rm lles that Vx , x =xx5 0. Thus, a =1—————

p P S P P x =0
Wh'[x,,,S ]

9

a critical value.

h'[xp]W(1-a’)

 

d) From part c of this lemma, we know that E l or after rearranging

r

the terms h '[xp] 3 —I———. Hence, x125 =0 < xp cannot exceed the finite value
W(l —a)

x‘;; Va,0 s a < a’ s1 implicitly defined by h 1x75] = ——’——. Similarly

W(1—a)

= . * . . .
xfis O < xp cannot be less then the finite value x% Va,0 _>. a > a 21 rmplrcrtly

defined by h '[x%] = mlr—j
—a

Figure AII.3 summarizes the main contents of this case, which graphs the public
reaction curve and the private reaction curves for five different values of 0:. These

graphs were generated in Mathematica using h[x] = x/(l + x), W = 100 and r = 0.1 '3.

The diagonal line from the origin is the 45-degree line, and the public reaction curve is
the curve originating from the xs axis. Five different private reaction curves are graphed,

representing, from left to right on the xp axis, the following values for 6! =1, 0.95, 0.75,

0.5 and 0. For this example, the critical value for alpha ( a* ), where dxp[xS]/dx3 = 0 is

calculated to be 0.89.

24

This case shows that the D&D and P-T models are special cases once we
parameterize the appropriability of public innovations. Without appropriability (as in
D&D) the private firm reacts to an increase in the public sector firm’s R&D investment
by increasing its effort for fear of being surpassed in the innovation race. When the
private firm can perfectly copy the public sector firm’s innovation (as in P-T), there is a
free-rider aspect to R&D such that the private firm leaves the R&D to its rival public
sector firm to undertake and then benefits through imitation. Such a situation leads to
under-investment in research and in equilibrium the public sector firm invests more in

research than the private one. By introducing appropriability as a parameter we Show that
. . * . , . .
these curvature properties hold; that 15 for a > a , the private firm 5 reaction curve rs

positively sloped and for a < a": for the private firm the curve is negatively sloped.

* . . . . . . . . *
Proposition 11.]: For a > a the equrlrbrrum public R&D rs mcreasrng 1n 0'. For a < at
equilibrium public R&D is decreasing in a for xp > x5 and increasing for xp < x5
Proof: The public reaction curve is increasing for xp > x3 and decreasing for xp < x5

(For a formal proof refer to D&D). Let 0 S a] < a2 51 then

Ri’,a1[xs ,xp] < RL,az[xS,xp]. From Lemma 11.1, the private firm’s reaction function is

. . * . * *
posrtrvely Sloped for a > a , decreasrng for a < a and has a slope of zero at a = 0.

=0

Also from Lemma 11.1 xfis = xg” =0. It follows then that for

 

'3 Several other functional forms can be chosen which meet Nti’s condition of a non-increasing hazard rate
elasticity. These include the power (x’a for 0 < fl < 1 ), logarithmic (10g[1+ x] ), and exponential

functions (1— e” ). We settle on the rational functional form as it is the easiest to solve.

25

* * :1: =1! * =1:
12a2>al>a >0,[X5,XP]aI <[X5,Xp]a2; for OSa2<al<a <1,

:1: =1: :1: :1: , =l= at: :1: :1: ,
[xS’xP1arl > [Isixplaz If xP > XS and {[x5]al < [xSlaZ’UPlal >[xPla2} If xP < XS

Taken together Lemma 11.1 and Proposition 11.1 have the following interpretation.
A decrease in the appropriability of public innovations ( a -> 1) decreases the incentive
for the private firm to do research since it can free ride off the innovations of the public
sector firm. The change in response by the private firm due to appropriability of public
innovations affects the Nash equilibrium, with lower appropriability resulting in

equilibrium conditions in which private R&D is lower and generally lower public R&D”.

11.3.2. Specification 2: Appropriability With Private Research Capacity

In this section, we consider a slightly different specification of that presented
earlier. We maintain the assumption that some public research may not be appropriable
such that the private firm is able to imitate the innovation of the public sector firm,
should the public sector firm win the R&D race. In this specification, however, we shall
assume that to be able to evaluate and employ the external public innovation, the private
firm must possesses a certain amount of internal research capacity that allows it to ‘copy’
public sector firm’s innovation. A weakness of the earlier specification is that it raises the
possibility that the private firm can benefit from the public sector firm’s innovation
without any research effort of its own”. As pointed out by Geroski (1995) even if the

appropriability is weak, it remains unclear as to the benefits recipient may accrue from

 

'4 Equilibrium R&D will increase with lower appropriability if X5>Xp

26

the use of its rival’s innovations. That is, it is not difficult to imagine scenarios where
firms must invest in R&D in order to learn, adapt and use a rival’s innovation”.
Therefore a more realistic condition would require that the firm undertake some research
to benefit from the imitation of the public sector firm innovation. Further, the more
research the private firm undertakes, the greater is the probability that it will be able to
benefit from the spillovers.

Mathematically, the appropriability of public sector firm’s R&D and that the
private firm maintain a research capacity to benefit from the loss of public research

appropriability is specified in the payoff of the private firm:
2 w W
VP = Iexp{—(h[xp] + h[x5 ] + r)t} [7(h[xp]+ ag[xp]h[xs ]) — xPJdt (11.7a)
0

: (W/T)(h[xp]+ag[xp]h[xs ])— xP
hlxp]+h[xs]+r

 

(11.8b)

Written this way, if the private firm wins the race, the flow of profits accruing to it is W.
On the other hand, if it loses the race to the public sector firm, the flow of profits is
reduced and becomes a function of the appropriability parameter (0t) and how much

research capacity the private firm accumulated during the race g[xp] , which has the

following properties
1. g[0] = 0

2. lim g[x] =1

x—>oo

3. g'[x]>0

 

'5 Substituting 01:1 and 10:0 in equation 1.1 we note that the payoff to the private firm is positive.

27

4. g "[x] < 0
The payoff for the public sector firm remains unchanged from that specified by both
D&D and P-T (see equation 11.2), as the appropriability is asymmetric.

As in the previous case, to characterize the Nash equilibrium in R&D space (xp,
xs), we derive, for each firm, the best response function and their shapes. Since the payoff
function for the public sector firm has already been considered, the properties of the
associated best response function for the public sector firm is not derived in this section.
What follows are the derivation for the properties of the private reaction function given
the payoff specified in equation (11.7a).

From the first order condition for a maximum, the reaction curve of the private

firm is

R12:[xp,xs]=(h'[xp](r+h[x5])W -r2 —rh[xS]—rh[xp]+rxph'[xp])+

Wahlxsl((r+ hlxsl+ hprl)g '[xPl-h'lxplglxpl) = 0

(11.8)

To facilitate interpretation, the reaction curve has been specified such that
spillover effect is separated from the no-spillover effect”. The term on the first line of
equation (11.8) is the private firm’s best response function without the appropriability
effect. The terms on the second line represent the change to the best response function
when the public sector firm’s innovation is not perfectly appropriable.

To help us identify the Nash equilibrium in the xp,x3 Space, we state the

following lemmas to establish the shape of the reaction curves

 

'6 Geroski (1995) mentions the study of learning in chemical processing industries by Lieberman (1984).
'7 The second line in equation (11.8) being the spillover effect.

28

Lemma [1.2: Define R%[x5,xp]=0 as the private firm’s reaction function (equation

(11.8)), and R§[x3,xp] = 0 as the public sector firm’s reaction function (equation (11.3)).

Then
a) R%[0,x,.] = R§lxs,0] or x7390 = x§P=0.
b) sz 20,ae l0,1| and xi} :0 < 52,; < x? a sufficient condition for
dxp[xS]/dx5 >0 (private firm’s reaction curve positively slope) is
Wh '[frp](1 —ag[ip]) > r.
c) Vx5,xp > 0,ae l0, 1| a necessary condition for dxp[xS]/dxs < 0 (private firm’s

reaction curve negatively sloped) is Wh'[xp](1-ag[xp]) > r.

Proof
a) As in Lemmas 11.1, setting x5 =0 into (11.8) and xp=0 into (11.3) we get

h “1' )(W + x,- ) " h(x,- ) - r = 0 , which implies x? =0 = ng =0

b) Implicitly differentiating equation 1.17, and assuming that the second order

condition is satisfied, we get

sign [£3] = sign [(Wh '[xP ](l - ag[xp ]) — r) + an '[xp](r + h[xp] + 2h[xs D] (11.9)
S

Equation (H.9) breaks down the curvature of the private reaction into two terms.

The second term is unambiguously positive Va,0 S or S1 and VxS 2 0. For the

private reaction curve to be unambiguously upward sloping requires that the first

29

term in equation (11.9) be positive. Hence, Vip , 0 < EP < x; , a suflicient
condition for (II.9)>0 is Wh'[52p](l -ag[ip]) > r
c) Va,0 < a S1 and VxS 2 0, equation (11.9)<0 iff Wh'[xp](1—ag[xp]) < r ,

VIP >0

The generality of our analysis does not allow for definitive characterization of the
private firm’s reaction curve. To facilitate interpretation of the private best response
function and to examine the forces at play that determine its shape, we can express
equation (11.9) in reduced form and in terms of the marginal value product as we did in

Lemma 11.1:

.31: g 0 iff F[g '[xP],h[xS ]]+(MVRv -MVPL[g[xpll) % r.
S

where MVRV = Wh '[xp] is the undiscounted marginal value product of winning and
MVPL[g[xp]] = aWh'[xp]g[xp]is the undiscounted marginal value of losing. Here the

marginal value of losing is a function of the ease by which the private firm can
appropriate the public sector firm’s research, which in turn is a function of an exogenous
appropriability parameter (a ), and the ability of the private firm to imitate the winner’s
prize. The a parameter is interpreted as the strength of the IPR regime. Whereas the
ability to imitate in specification 1 was only a function of the magnitude of the spillover
parameter, here it is also a function of how much research is undertaken by the private
firm itself during the course of the race that allows it to capture the shared knowledge.
The F function is interpreted as the marginal value change in the ability of the private

firm to imitate the public sector firm’s prize.

30

To illustrate the ambiguity in the response of the private firm to research by the
public sector firm, we provide two examples where the assumptions on functional form

and prize value will drive the results. We state these in the following propositions:

Proposition [1.2: VxS 2 0 ,6? =1, and O < r sufficient conditions for dxp[xS]/d.x5 < 0
for os x5 < x; and dxp[x5]/dxs >0for 0< x; < x5 are a) 0<< w and b) g '[0] —> oo

Proof: Consider 0 << W which from Lemma 11.2a implies 0 << 11:5 :0. That is the higher
the prize value, the greater the initial value of private research effort, given no research

by the other firm. If we assume g[x] , such that g“? :0] = 0 and g[ifis =0]=1, then

XS =0
de

dxs

 

<0 from equation (11.9). If we assume xp is bounded from below by zero (i.e.

x5 —-) oo , xp —> 0), then dxp / de >0 from equation 1.18. This implies that there exists

x; such that dxp/dx; =0

What Proposition 11.2 says is that it is possible to get a negatively sloped reaction
curve for the private sector in specification 3, if we assume that the prize value is “large”
and that the private firm can easily imitate the public sector firm’s winnings with only the
minimal imitative capacity of its own. This result is similar to the P—T case which also
showed a downward sloping private reaction curve, but differs in that the private reaction
curve is not negatively sloped everywhere. By decreasing its research effort the private
firm also decreases its ability to imitate and gain from the benefits of the public sector
firm’s prize. Thus at high levels of public research, the private will reverse the decline of

its own research.

31

Next, we show that under sufficiently strong assumption on the hazard rate, the g
function, and the best response function, the private best response function is increasing
in public research and that there exist a unique Nash equilibrium. Matters are simplified

further if we assume that g[x] = h[x]. Consider for example the rational function
fl[x] = x/(1+ x) or the exponential function f2[x] = 1 -— e-x , either of which satisfy the
concavity properties of the g and h functions. Further, if either f1 or f2 are specified as

the hazard rate then they also satisfy the stability condition (Rl- < 0) as both functions

have a non-increasing hazard rate elasticity.

Proposition [1.3: Assume that g[x] = h[x] , then VxS > 0 and 0 < a S 1 ,
pr[X81/d.XS > 0
Proof: With g[x] = h[x] , and from the first order condition for a maximum, the reaction

curve of the private firm is now

(h '[xP](r+ h[xs ])W - r2 - rhlel- ”11po “Pk M”) + (11.10)

Wah[x5 ]((r + h[xs ])h '[xp]) = 0

which can be re-written as

r(h[xp]- xph'[xp])
r + h[xS]

 

W(1+ah[xs])h'[xp]—r = > o (11.11b)

Implicitly differentiating equation. (11.10), we note that it follows from equation 11.1 lb

that

Sign[%:f] =sign[W(1+a(r + 2h[xs ])h '[xp]— r] > O

32

Proposition 11.3 essentially reverses the P-T result, which showed that the private
reaction function is unambiguously upward sloping for ,6 =1 . By requiring that the
private firm maintain a capacity to imitate the winnings of the public sector firm, and
under sufficiently strong assumptions about the capacity building function and hazard
rate, we Show that the result will converge to that of D&D. However our intuition differs
from that of D&D. Under Proposition 11.3, the private firm increases its research effort
for two reasons. Firstly it fears the competitive threat from the public sector firm and
secondly it realizes that a higher research effort results in greater ease by which it can
imitate the public sector firm’s winnings.

With the assumptions used to put forth Proposition 11.3, we can also Show that
with an increase in appropriability of public sector firm’s innovation, the equilibrium

R&D conducted by the public and private firms is lower.

Proposition [1.4: Assume g[xp] = h[xp] and let 0 S 0:1 < 0'2 S 1 , then

ixixinial < tx§.xi‘ala2

Proof: From Proposition 11.3 and Lemma 11.2, the public and the private reaction
functions are both positively sloped for xp > x5 (the private reaction curve is
unambiguously upward sloping). For 0 S a1 < a2 S 1 , Rlz)[x5 , xp]al < R33[xs ,xp]a2 , that
is the private reaction curve when the appropriability parameter is al lies below the one
when the appropriability parameter is 0'2. Since x? :0 = xg” :0 it follows than that

equilibrium R&D is lower or] than it is for a2.

33

The result of Proposition 11.4 is much more conclusive than its counterpart for
specification 1 (Proposition 11.1) It is also contrary to the earlier result where, recall, we
showed that if xp > x5 , then equilibrium public R&D is increasing with decreasing
appropriability. Under Proposition H.4, we have Shown the conditions under which R&D
may increase with decreasing appropriability.

Figure AII.4 graphs the public and private reactions curves in the R&D space.

11.3.3. Specification 3: Public R&D Spillovers

Our last specification considers the impact of public research spillovers on the
private firm and equilibrium properties of the R&D game.

A simple, yet realistic characterization of the R&D process is one where public
research reduces the effective cost of the private firm’s R&D. We define the effective

cost of the private R&D as the expenditure by the private firm (xp) less the R&D
spillovers that emanate from the public sector firm:

Cp=xp(1—flg[x5])20 (11.11)
where ,6 is interpreted as the Spillover parameter falling in the closed unit interval and the

function g[xi] signifies the amount of public R&D and has the following properties

5. g[O] = 0

6. lim g[x] =1
x—-)oo

7. g '[x] > O

8. g "[x] < 0

34

From our characterization of the private firm costs as that Specified in (11.11), the
marginal cost of research is decreasing in both the degree to which spillovers occur and
the level of public R&D. To account for cost saving research spillovers from the public

sector firm, we re-specify the payoff function of the private firm:
7

V; = Iexp{-(h[xp]+h[x5]+ r)t}[W (h[xp])—xp(1—flg[x5])]dt (11.12a)
0

= (W/r)(h[xp])—xp(l-flg[x5])
h[xp]+h[xS]+r

 

(11. 13b)

Cost reducing spillovers affect R&D effort of the public sector firm in an interesting
way. Recall that the public sector firm, as social welfare maximizer, takes into account
the cost of research by the private firm as well as its own. This implies that more research
by the public sector firm reduces the cost for the private firm and hence the public sector

firm’s total cost as well. The payoff for the public sector firm is specified as
2 w W
VS = Iexpl—(h[xp] + h[xS ]+ r)t} [TURF] + h[xs ]) - xp(1- ,Bg[xs ]) - x5 ]dt (H. 13a)
0

= (W/r)h[x5]+(W/r)h[xP]—xp(1_fig[xSD_xS
h[xS]+h[Xp]+r

 

(11. 14b)

Note that when fl = 0, this is model is equivalent to the one considered by D&D,
but it cannot be generalized to the P-T model for any value of fl . The two firms, acting
non-cooperatively and simultaneously, choose R&D expenditure (x,) to maximize their
payoffs. Using the first order condition for a maximum, one derives the following
reaction curves for firm P and S, respectively:

h'lxpl(r+htxsllw-r-hixsl-htxpl+xph'lel+]=0 (11.14)

3 =
RPl’P’xS] [flgixsl<r+hlxs1+hIxPI-xrh'leD

35

h'[xS](W +xp)—r—h[xp]-h[x5]+xsh'[x5]+

R3 , =
$le XS] 3x190?'[xsl(r+h[xP]+h[xSD’glxslthSD

]=0 (11.15)

Equation (11.14) and (11.15) simultaneously determine the equilibrium value of the

R&D expenditure x; and x; .18 The two terms that appear in (11.14) and (11.15) that did

not appear in the first order conditions of D&D specification (8:0 in equation (11.2), are

fights ](0) in (11.14) and ,Bxp(0) in (11.15). Both these terms reflect the presence of

spillovers in the technological competition. The private firm unambiguously benefits
from public Spillovers as it increases instantaneous profits. In this model then, the
presence of spillovers induces the private firm to speed up the expected date of
innovation. The effect of spillover on the public sector firm is less clear, as there are two
competing forces at play. Increased research expenditures by the public sector firm not
only increases the probability of success for the public sector firm but also decreases the
cost of private research, and hence increasing its instantaneous payoff. However greater
research expenditure implies a greater cost for the public sector firm as well as an
increase in the discount rate (the denominator in equation 11.14b, and hence a lower

payoff). Which of these two forces dominate will depend on the relative values of xp and

x5 , as well as the magnitude of the spillover parameter.

We formalize these statements and characterize the shape of the reaction curves

(equation (11.14) and (11.15) in the following lemmas:

 

'8 As in specification 1, the continuity of the reaction functions and existence of a unique solution follows
from quasiconcavity assumption of the objective function.

36

Lemma [1.3: For specification 3, define R;[x5,xp]=0 as the private firm’s reaction

function (equation (11.14)), and R§[X5,XP]=O as the public sector firm’s reaction

function (equation (11.15)). Then

a) R;)[O,Xp,fi]:R§[xS,O], 01' figs-:0 =X§P=O

b) Vfl 6 '0, 1| , the private reaction curve is positively Sloped
c) For ,6 =0, the private reaction curve is bounded.Vfle (0,1' , the private firm’s

reaction function is unbounded.
Proof:
a) Setting xs=0 into (11.14) and xp=0 into (11.15) we get

h'[x,-](W+x,-)-h[x,-]—r=0 (11.16)

XS=0__ XP=O
Hence xp —xS

b) Implicitly differentiating equation (11.14), and exploiting the second-order

condition we get:

[“9”]“2' n[h'[x5](h'[xp]W+rflg[xS]-—r)+ (11.17)

(138 '[xs 1(r+ (h[xp1+ h[xsl- xph'lxpl))
The second term on the right hand side of equation (11.17) is always positive due
to the concavity of the hazard function (h[xp] > xph '[xp] ). Rearranging the terms

in the FOC of the private firm (equation 1.7), shows that the first term is positive.

(h[xp I - xph '[XPD - flglxs 1011pr Xph '[XPD > 0
r + h[xS]

 

’1 11le + rflglxsl- r =

as 1681x31< l . Hence sign [92] >0.
dxs

37

c) To prove the last part of the lemma, consider 1.11, which requires that

 

h'IxP]W+r168[xs]>r or h'[xpl>r(]-rilg[xsl)

. If ,6 =0 then xp cannot

exceed the finite value x73 implicitly equals by h'[x7§]>firl-. For O<,6Sl the

reaction curve is unbounded as the value xi; is implicitly equal to

 

h.[x;]: ra—flgI-XSD

W . That is for h'[xp]=£, there exists x5 such that

 

r(1—.381Xsl) <8
W

A comparison of Lemmas H.1 and 11.3 clearly reveals the opposite effects that the
spillover parameter and the appropriability parameter has on the private firm when
spillover are modeled as either as cost reducing or increasing the ease in which the
spillovers can be imitated. 1n the former case, spillovers unambiguously increase the
amount of research by the private firm, whereas in the later case the private firm reduces
its effort and simply free-rides off the effort of the public sector firm.

Unlike in case 1, the public sector’s payoff function is also affected as a result of the

cost reducing spillovers. Consequently its properties need to be formalized as well.

Lemma [1.4: Vfl,0 < ,8 S1 , the public reaction function, defined by Rghcs ,xp] = 0, has

the following properties

a) lim fl‘S-< 0
XP—POCIXP

b) 1im 511»
XP—)°° dXP

38

V ,OS SI,R2 ‘ ’ 2R2 - , ff g'le]§ h'le]
c) .5 A1932 Slxs xplp, slxs rplg2 1 g[xs] r+h[xp]+h[xs]

Proof:

a) Implicitly differentiating the public sector firm’s best response function, we get

(h'lxsl— 81x5 I}? 1165 1)) +

. dx .
“3.43:1“... fl—g'leILL.hits].h[xPl]+hnpi[/i—g'“51-1] (“‘8’
8115] 1p Xp xp 81x5]

Consider the three terms in the right hand side of equation (11.18). The first two

terms are positive sz ,xp, ,6 > 0. From lemma 2 part a, we know that

lim0R§(xS,xP,fl) —> R&(x§”=0,0,fl) where x§P=0 >0. If we assume that
XP-)

8 Ix? =01

<1 then lim h'[xp][fl§;[—is—]-l
31x5

]—> -oo , implying that
X1340 g[xS]

lim éx—P< 0
XP—90 de

-—> O , implying that lim iii”— > 0

b) Here we note that lim h'[xp] fl-Lxsl-l
] XP—90dXS

xii—900 8 XS

 

C) R§[xs.xp]p, -R§[x5,xp]p2 =XP(151 "flfiglxs][:[[:S]]('+hIXP]+hIXS 11'”le I]
5

Given that 61 < ,62, for R§(xs,xp,fll) §R§(x5,xp,fl2) requires that

(M(r+h[xP]+h[xs ])—h'[x5]] E 0.
81x51

The message of Lemma 11.4 is that given the generality of the exposition, we are

unable to definitively infer the shape of the public reaction curve. However as in

39

specification 1, under sufficiently strong assumption on the hazard rate, the g function,
and the best response function we can guarantee the existence of a Nash equilibrium.

Assuming that g[x] = h[x] , we can show that for any )8 , the public reaction function is

positively sloped for xp > x5.

Proposition [1.5: Assume h[x] = g[x] , then Vfl e <0,1| and xP 2 x5 , the public reaction

curve in Specification 2 will be positively sloped.

Proof: With h[x] = g[x] , equation (11.18) is re-written as

sign [3;] = sign[h'[x5](1+ ,B(r + h[xS ] + h[xp])) -h'[xp]] (11.19)
P

whence the result follows from the concavity of h[x]

The importance of Proposition 11.5 is that, together with Lemma 11.3, the

existence of a unique Nash equilibrium for Specification 3 is now guaranteed. Notice that

in (11.19), the Sign remains ambiguous for x3 > xp, as %— § 0 if
P

h '[xS ](1+ fl(r+ h[xs]+h[xp]) é h'[xp]. What this suggests is that the spillover
parameter ( ,6 ) determines the range at which the public reaction curve is negatively
sloped and when it is positively sloped (for x5 > xp). The higher the spillover, the

smaller the range of xp values for which the public reaction curve is negatively Sloped.

Lemma [1.5: There exists a Nash equilibrium (x;,x; ) for the mixed duopoly R&D game

when there are cost reducing spillovers present.

40

Proof: Follows from the continuity of the reaction functions and Lemmas 11.2 and 11.3.
Figure A11.5.a, b and c illustrate the curvature properties of the public and private

reaction curves, for ,6 =0, 0.5 and 1, respectively. AS before these graphs were generated
on Mathematica where g[x] = h[x] = x/(1+ x), W=100 and r=0. 1. From the panels of
Figure AII.5, it is easily checked that for each specified ,6 there is a unique Nash

equilibrium.

11.4. Comparative Statics and Welfare Analysis

Table 11.] provides a comparison of how the spillover and appropriability
parameter affects private and public research effort. Under Specification 1, appropriability
of public sector firm’s innovations unambiguously lead to a decline in research by the
private firm. When the imitation of public innovation requires some level of private R&D
capacity, as in specification 2, a decrease in the appropriability of public innovation has
an unambiguous positive effect on private research only when we assume the equivalence

of g[x] and h[x]. When we model spillovers as cost reducing (specification 3) the

private firm increases its research effort to an increase in the spillover. The corresponding

result for the public sector firm is ambiguous until we make the simplifying assumption

that g[x] = h[x] , in which case 3x5 ldfl > 0.

41

Table 11.1: Effect of Spillover on the Public and Private Research

 

Model Specification Number

 

 

 

 

1 2 3
if]: <0 Ambiguous for g[x] at h[x] >0
312
>0 for g[x] = h[x]
git. 0 0 Ambiguous for g[x] at h[x]
312

>0 for g[x] = h[x]

 

 

 

 

where p = a for specifications 1 and 2 and p = ,6 for specification 3.

We next turn to the question of whether presence of a public sector firm in the
three specifications is a welfare improvement. We note that a finding of the D&D paper

was that social welfare is higher in a mixed duopoly than in a private duopoly (i.e. ,6 = 0
in specification 1). P-T, in her setup (fl =1) finds the result to be ambiguous and shows

that for low values of welfare prize the social welfare is higher in a mixed duopoly,
whereas for higher value of prize welfare in the private duopoly is higher. In specification
1, the parameterization of appropriability allows us to derive that level of appropriability

for which social welfare is maximized.

Proposition 4: In case 1, Va,0 S a S1, there exists 67 such that social welfare is

maximized .

Proof: A measure of social welfare is the public sector firm’s objective function given by

equation (II.2a). The social optimum is given by evaluating VSl at x5 = xp. This implies

42

 

in the mixed duopoly the social optimum is achieved if in the Nash equilibrium
:1: =1: :1: , ,
xp = x5 = x . From D&D’s analysrs we know that at x5 = xp, the publlc sector firm’s

reaction curve has a curvature of zero (dxs[xp]/ dxp = 0 ). It follows then that if the Nash

. . . . . . * * * * . ,
equrllbrrum rs to be at the socral optrmum xp = x5 = x , x must solve the prrvate firm 5

reaction best response equation (11.2), VW, r > 0, and

(r(W +x*)+Wh[x*])h'[x*]-r(r+2h[x*])

 

a~ : :1! *
Wh[x ]h'[x 1

What the above proposition suggests is that for any prize value, the amount that
the public sector firm is allowed to appropriate in the strategic game can be regulated to
achieve the social optimum. Consider a social planner that has prior knowledge of the
payoffs for the private and public sector firms. Given any Wand r, and the curvature
properties of the best response functions, it is relatively straightforward to calculate for
07 that results in the Nash being the social optimum.

For specification 2 and 3, the ambiguities that arise due to the nature of the best
response functions do not allow us to make a similar conclusion with regards to the effect
of the parameters on welfare. However, our results for these two cases confirm the D&D

findings that under the assumption when g[x] = h[x] , the mixed duopoly is welfare

increasing compared to the private duopol y.

11.5. Conclusions
We began this chapter with the observation that there has been a notable shift in

R&D effort of public and private firms (with the later taking on a more prominent role)

43

and that public sector firms have been patenting and licensing at a much faster pace than
they were in years prior to the Bayh-Dole Act. Standard economic theory would suggest
that the more a firm is able to appropriate its returns, the more likely it is that it will
conduct further research and increase its R&D expenditures. Why then, has the public
sector firm not responded to an increasing ability to appropriate the returns from its
innovation?

The contribution of this paper is to provide a theoretical model in which the
public sector may respond positively to an increase in its ability to appropriate the results
of its R&D by reducing its investments in R&D. This result hinges on the objective
function of the public-sector firm, namely that of welfare maximization. Recall, that the
public sector firm’s payoff does not change, either in specification 1 or 2, from the
parameterization of the appropriability parameter. It was pointed out that as a welfare
maximizer, the prize value and who wins the race was of a lesser concern then the pace of
innovation. While the firm itself may get higher returns from an increased ability to
appropriate, the welfare value remains unchanged (as it is distributed across all firms).
With “welfare” as the prize, the behavioral response of the public sector firm is
unaffected by the appropriability parameter. The effect of change in the appropriability
of public innovation however does impact private R&D. It was shown, in specification 1,
that the private firm’s response to an increase in the ability of the public sector firm to
appropriate its innovation is to increase its (private) R&D. In specification 2, while the
general result was ambiguous, specializing the model revealed that the private R&D

decreases with higher public R&D appropriability, and that the equilibrium public R&D

was lower with a greater ability to appropriate returns. This result possesses some
consistency with the reality of public R&D.

The parameterization of appropriability in specification 1 suggests that if the
public sector firm has leverage in how much of the returns it appropriates than it is
possible to attain a social optimum. It was found that if the public sector firm
appropriates all its returns and effectively barring the private firm from the use of its
innovation (through excessive protection of intellectual property), then it over-shoots the
social optimum. It also misses the social optimum mark when it allows the technology to
be essentially given away, giving rise to the free-rider problem and under-investment by
the private firm.

To clarify the distinction between the incentives that arise due to appropriability
issues and those that result from pure spillovers, specification 3 was presented to model
the latter. Whereas ‘appropriability’ concerns how much rent the public sector firm is
able to capture from its innovation, the notion of spillovers has to do with the degree of
knowledge flow during the R&D process. Stated this way, 0t can also be interpreted as
the spillover that occurs after the race and B is the spillovers during the race. Spillovers,
in specification 3, were modeled as cost reducing, and under reasonable assumptions
about the hazard function, we find that private reaction is unambiguously upward sloping
with greater spillovers resulting in higher private research. This result is consistent with
empirical evidence that public R&D positively impacts private sector R&D and

innovation (McMillan, Narin and Deeds, 2000).

45

CHAPTER III

SHOULD THE PUBLIC SECTOR CONDUCT GENOMICS RESEARCH?

111.]. Introduction

Over the past two and a half decades biotechnology has revolutionized major
portions of the human health and agriculture industries, transforming them into the life
sciences industry. Yet, potentially, the most revolutionary biotechnologies are still in the
early stages. Perhaps the most potent of these technologies iS genomics—the science of
sequencing all the genes of a given species to study the structure, function and evolution
of diverse organism. Genome research, however, is more than biology; it is also about
developing better drugs, foods, industrial products, and, in the case of agriculture,
improving plant and animal productivity and quality.

A striking feature of genomic research is the significant levels of investment by a
few dominant private firms in competition with an equally well-funded public sector that
seek to discover and subsequently patent important gene sequences. This observation is
made most pellucid by the private sector’s Celera Genomics Group challenge to the
longer-lived and more expensive, publicly funded Human Genome Project (HGP). The
Department of Energy and National Institutes of Health started HGP in 1990, at a cost of
approximately $2.2 billion over the course of the project. In 1992 Craig Venter, a
scientist with the HGP, left to form his own private company, Celera Genomics, and
claimed that the firm could sequence, using a different technique from the HGP, the
whole genome in less time and at a fraction of the cost (approx. $200 million). Celera’s
challenge to the publicly funded HGP signaled the start of the race for the sequencing of

the human genome, which was joined by numerous other Start-up companies looking to

46

capitalize on the potential of genomic research. The competition between Celera and
HGP so accelerated sequencing efforts that by late 2000 both projects were essentially
complete ahead of schedule.

Although less confrontational, the private sector is also at the forefront of
characterizing plant and animal genomes, sometimes subcontracting from the public
sector. In 1998, a consortium led by the International Rice Genome Research Program
(IRGSP) in Tsukuba, Japan, began efforts to sequence the rice genome. The participants,
which included primarily government and research foundation sponsored labs, took a
traditional approach to genome mapping known as the ‘stepwise sequence analysis’
(Bennetzen, 2002). This approach, while expensive and slow, provides the most precise
and complete sequence with a goal of 99.99% accuracy.l Soon after the IRGSP was
initiated, the Monsanto Co. began funding research to sequence the same variety of rice
as that by IRGSP. The Monsanto sequencing strategy was slightly different from that of
IRGSP, allowing it to sequence more of the genome with less time and cost. However,
Monsanto’s strategy does not provide enough information for highly accurate assembly
(Bennetzen, 2002).2 Much more recently, the Beijing Genomics Institute (BGI) and
Syngenta, a Switzerland based agricultural biotechnology company, independently
produced draft sequences of the rice genome by the quickest and least costly method, the

‘shotgun sequence analysis.’ Syngenta obtained 99.8% sequence accuracy, identifying

 

' By early 2002, the project had sequenced 15% of the genome.
2 In 2000, Monsanto abandoned its rice sequencing effort and donated its data to IRGSP.

47

more than 99% of the genes at 10% of the cost of the IRGSP strategy (Bennetzen, 2002)3.
On January 26, 2001 Syn genta and Myriad Genetics announced that they had sequenced
the rice genome and planned to provide their database to commercial customers, such as
seed companies or agricultural biotechnology companies. The competition from the
private sector has in turn spurred the IRGSP into advancing its calendar by almost four
years (to 2004) and increasing its budget. The Government of Japan pledged to increase
its annual rice genome research $60 million in 2000; a threefold increase over the
previous year.

The Human and Rice Genome Projects are classic examples of a research race
between firms where the objective is to be the first to discover, and subsequently obtain
patents, on important gene constructs. However, current innovation or R&D racing
models do not conform well to the type of behavior observed in genomics research for
two reasons. First, most models have assumed that the research race is between profit
maximizing private agents (Sabido, 1994; Reinganum, 1985). This is appropriate for a
variety of industrial research in which the public sector is not involved. However in the
agricultural sector, or in genomic research, there is a dominant public research sector
whose objective, it can be argued, is to maximize not profits but welfare.4 Thus, such
research is best characterized as a race between a welfare maximizing public

organization(s) and a profit maximizing private firm(s). In the context of genomics

 

3 A useful analogy in comparing the different strategies is to imagine the whole genome as being a large
puzzle. Without the knowledge of what the whole genome (puzzle) resembles, the genome is first broken
down into pieces (DNA strands) and the individual pieces subsequently sequenced. After the identification
(sequencing) of the smaller pieces has occurred the task of determining how the individual pieces relate to
each other and where they fall on the map follows (this is akin to putting puzzle together). While it is easier
to sequence smaller DNA strands, it is much more difficult to ‘rebuild’ the map with so many pieces.

‘ Following the literature on mixed oligopolies, we abstract here from moral hazard and internal
organization issues and define a public sector firm to be an entity whose objective is to maximize social
welfare, whereas a private firm would aim to maximize profit.

48

research, a private firm’s objective is to patent the genetic code for important proteins
and obtain royalties on the patent. Whereas the objective of a publicly funded entity is to
promote further innovations (and hence increase welfare) which it does by making the
genetic information more widely available, without regard to maximizing royalty
revenues. The asymmetric objectives of these two types of firms’ gives rise to a different
behaviors from the case wherein all the firms in the analysis are private.

A second reason why earlier patent racing models fail to capture the intricacies of
research such as genomics is in their assumption that the value of the prize is
exogenously determined (Sabido, 1994). Genomics is not Simply the identification of a
sequence of genes, but also involves understanding the properties and relationships of the
genetic code embodied in those genes. How accurately, and in what manner, the genetic
sequence has been identified has bearing on the ease of interpreting the functions of the
genes in the later stages. The amount of research done and the method employed in the
sequencing stage can also influence later stages of genomics research if, for example, one
assumes that more expenditure in the sequencing stage will, on average, lower the cost of
doing research in the later stages. For example, the scientific knowledge and tools used in
the sequencing the plant DNA have the potential of lowering the cost of breeding
varieties with agronomicaly desirably traits. These cost reductions primarily result in
more precision in transferring desirable genes to crops and reducing the time to breed
specific varieties. The implication for the winning firm is that it gains knowledge in the
process of the race that is useful and can be productively employed in further research
(by it or others). That is, the more research it expends on the race today, the greater is the

likelihood of winning the R&D race and the greater will be the cost savings on future

49

research or production. Moreover, if we do not assume a winner-take all situations, the
endogeneity of the prize value also has repercussions for the losing firms. That is, the
losing firm, which moves on to the next stage but uses the now inferior technology, will
face lower profits due to a decline in market share. The decline in market share for the
losing firm, and by extension profits, is a function then of the cost reduction implied by
winning the prize (for the winning firm) and how much research was expended to achieve
that prize (by the winning firm)6.

To capture more accurately the microfoundations of an R&D race observed in
genomics research, this chapter characterizes research as a two-stage process. The
research effort in the first stage reduces the cost of applied research or production in the
second stage. We use this framework to gain insights into a traditional theme in industrial
organization research that of the relationship between market structure and innovation.
Economists have been interested in this issue ever since Joseph Schumpeter’s seminal
work hypothesizing a positive correlation between market power and innovation.

Schumpeter (1934) argued that a few firms were more likely efficiently to develop and

 

5 The notion of ‘winning’ and ‘losing’ in the context of genome research may seem inappropriate as firms
cannot obtain patents for Simply sequencing a certain DNA strand nor are there any immediate commercial
benefits from the knowledge of such sequences. Nevertheless, it has been observed that private firms are
more reluctant to disclose their sequences (e.g., both Syngenta and Celera did not make their discoveries
public through a commonly used public database). By effectively using trade secrecy to protect their
sequences (especially large assembled sequences), the private firms hope to appropriate any return that may
arise at later stages. This is in contrast to public efforts in genomic research, who have made the sequences
available to all without strings attached.

6 For example, assume a strategic game between two firms in the output stage where the cost of production
for firm 1 before the innovation is 0.14197] where q, is the amount produced and 7 a technology
parameter. After the innovation race, the winning firm will maximize its profits

ltw = PIqwq, ]qw - C[qw, 7W,xw] choosing qw (where xw is the research expended in the racing stage
by the winner) and the losing firm maximizes II, = P[qw,q,]q, -c[q,, 7, = 7] choosing q, (assume
3c, dew
97; ex.
winning firm will be greater than that of the losing firm 71’; = fl';[}’;, 7;,x;]>7t; = 7t;[}’;, 7;,x;]

 

< O and 7w > 7, ). Solving for the equilibrium properties it can be shown that profits of the

50

employ more advanced technology than a competitive industry. Formal models of firms’
innovation-seeking behavior have evolved, that have either confirmed or refuted the so-
called ‘Schumpeterian tradeoff.’7 Similar to the mixed theoretical results, the findings of
the vast empirical literature on the Schumpeterian tradeoff are mixed and ambiguous as
no obvious relationship between industrial concentration and R&D performance emerges
from the data.

Specifically, we ask what kind of market structure, and circumstances, promote
R&D when the nature of R&D is as described for genomics. Does the public sector serve
a useful purpose by performing genomics R&D when all evidence suggests that there are
willing private firms that can do the same type of research more quickly and at lower
cost? Does public sector efforts have a role to play in genomics R&D? We show that this
question can be answered in the affirmative, if certain sufficient conditions regarding the
prize value and the nature of the innovation process are met. In the process of deriving
these conditions we also obtain conditions under which a monopoly research market
performs more R&D than a pure duopoly market and vice versa. These market structures
are chosen as they represent the observed (mixed duopoly) and alternative (duopoly and
monopoly) market structures in genomics researchg. It is not the intention of our analysis
to comment on the desirability of a mixed market over the other two (which requires
comparison to the first-best outcome), but rather to provide a set of sufficient conditions

whereby one would expect the public sector to conduct more R&D relative to firms in the

 

7 For a review of this literature see Karnien and Schwartz (1982) or van Cayseele (1998). Among the many
writers who subscribe to Schumpeterian are Scherer (1980) and Kamien and Schwartz (1982). The claim
has been challenged by Arrow (1962) and Dasgupta and Stiglitz (1980).

8 The high fixed costs associated with genomics R&D serves as an entry barrier, and hence a competitive
environment is not analyzed here.

51

other regimes. It is in this sense that we suggest that the public sector has a role to play in
conducting research of the kind observed in genomics R&D.

The paper is organized as follows. The next section briefly reviews earlier R&D
models on which our analysis is based. Section three introduces the modeling framework,
the assumptions and our approach in the comparing the different market structures.

Section four discusses the comparative analysis. Section five concludes the paper.

111.2. Literature Review

Before proceeding to describe our modeling framework, we briefly review the
relevant R&D models of Loury (1979), Lee and Wilde (1980) and Delbono and Denicolo
(1991). Loury (1979) modeled a one-shot non-cooperative game in which n identical
firms invest in R&D to innovate first. The first innovator is awarded an exogenously
determined prize. The probability of success by firm i is an exponential function of its

hazard rate (h[xi]) given no success to date. The loser of the race gets nothing and thus

suffers a loss equal to the cost of its R&D expenditure. The R&D cost for each firm is a
lump-sum x, expended at the beginning of the race. Lee and Wilde (1980) reformulated
Loury’s model assuming that the R&D expenditure is a flow cost that firm i pays until
any one firm is successful. The different specification of the cost of R&D results in a
different impact of an increase in the number of firms on the equilibrium individual R&D
effort. Whereas such an effect is negative in Loury’s model, it is positive in Lee and
Wilde’s model.

The contribution of the aforementioned models has been to illuminate the

relationship between the intensity of rivalry and R&D performance. However, since

52

neither Loury (1979) or Lee and Wilde (1980) explicitly modeled the product market, the
relationship between the structure of the product market and incentives to invest in R&D
was not examined. One rationalization for not allowing for an explicit Specification of a
product market has been that firms compete in prices in a homogenous product market so
that a Bertrand equilibrium results. In such a market, pre-innovation profits are zero (as
all firms share the same technology) and post-innovation, the winner, which has reduced
its own cost, will be the only active firm (Delbono and Denicolo, 1991).

Delbono and Denicolo (1991) show that when firms make positive profits in the
pre- and post-innovation Coumot-equilibrium markets (where the losers also make
positive profits, post innovation) the Loury result—that there is a positive relationship
between profits and equilibrium R&D effort—holds. This result is significant in that it
highlights the relationship between the number of firms and the equilibrium R&D effort
to be much more complex than in models where the prize value, or expected returns from
R&D, are exogenously given.

Now while the Delbono and Denicolo model allows for pre- and post-innovation
profits for all firms, the nature of the innovation and how much of an improvement it is
on a prior technology is still exogenously given. That is, at the end of the race, the winner
obtains the rights to technology that bestows on it an exogenously determined cost
advantage over the prior technology. This cost advantage is exogenously given and has
no relationship to the actual amount of research that the firm conducted in the R&D race.
As we have argued, there exists the possibility, particularly in genomics research, that the
R&D effort of the racing stage of the game results in further cost advantages for later

stages, such that a higher amount of research leads to a higher amount of product value

53

for the winning firm. It is this aspect of the R&D process, and the point of departure from

other models, that we highlight in our modeling framework and comparative analysis.

111.3. The Model

Consider a two—stage game. In the first stage, firms choose their R&D investment
and in the second stage they compete further by conducting more applied R&D or
competing in the product marketg. The first stage is modeled as an innovation race where
firms compete for the rights to an infinitely lived patent. The innovation embodied in the
patent allows firms to lower the cost of research in the second stage (or the cost of
production, if the second stage is modeled as a product market). Through backward
induction, the profits from the second stage determine the value of the first stage patent.
The firm that innovates first is awarded the patent and gets the exclusive right to use the
more productive technology forever. The losing firm, on the other hand, has to continue
using the pre-innovation race technology in the second stage and hence accrues a lower
profit than the winning firm, and possibly even lower than its own profits prior to
innovation.

The research effort employed in the R&D race not only determines the outcome
of the race, but also results in the generation of knowledge that is valuable to the winning
firm. This knowledge can be used in later stages to complement with the winning
technology and lower costs in those stages even further. In this respect the value of the
prize for the winner is endogenous and an increasing function of research expenditure in

the R&D race. R&D effort thus has a two pron ged direct effect on the winning firm;

 

9 The specific characteristics of the second stage are not of concern here. We note only that the payoff (or
prize) from undertaking R&D in the first stage is function of the market structure in the second stage and
the amount of research performed in the first stage.

54

allowing it to win the race and lowering its cost in later stages. The losing firm is also
affected by the amount of research effort employed by the winning firm in the first stage
(see footnote 6). Since we assume the strategic game in the second stage as well, an
increased market share for the winning firm from the lowering of its cost will imply
lower profits for the losing firm, cetris paribus.

To fix these ideas, assume that two firms play the following two-stage game. In
the first stage, firm i independently takes action, denoted x,-, regarding the current
research market. In a patent race set-up, x,- represents the flow cost of research where its

’thlx’] . The instantaneous

probability of being successful at or prior to date I is 1— e
conditional probability that firm i will be first to innovate at time t, given no success to

date, is therefore h[xi]. Firm i’s expected benefits after a discovery are determined by

both firms actions (bi,bj) and are denoted by WWi(bi[xi],bj[x,-]) if the firm emerges as the

winner and WLl-(bi[xj],bj[xj]) when it losses the race. In a Coumot set-up, b) would

represent output whereas in a Bertrand game it would be price. A strategy of firm i in this
entire game can then be written as sl- 2 (xi,b,-[o]) where bib] is a function specifying firm
i’s post innovation action conditional on first stage actions, in particular on the amount of
research done by the winning firm. Given (si, s j ), the payoff to the private firm i is:

ViIWWiiwuilxiilel =

°° _ . . (111.1a)
je (h[x'HhUJ]+r)t(h[x,-]Ww,-[x,-]/r+h[xj]WL,-[xj]/r+W,--x,-)dt

0

_ (1/r)(h[x,-]WW,-[x,-]+h[xjiwu1xj])+w,- -—x,~
_ h[xi]+h[xj]+r

 

(111.1b)

55

where

r is the discount rate;
x,- is firm i ’s R&D expenditure;
h[xi] is firm i’s instantaneous probability of innovating or the hazard rate. The

hazard rate, h[xi] , is twice differentiable, Strictly increasing and satisfies

1) h[0]=0= lim h'[x],2)h'[x,-]>0,3)h"[x,-]<O;

x-—)oo
WWi[x,-] is the value of innovation accruing to firm i if it wins the race;
WLi[xJ-] is the value of innovation accruing to firm i if it loses the race where j is

the winning firm;
,- is the pre-innovation benefits accruing to firm i.

As we shall see, the value of the innovation will be different for the private and
public sector firms. For the public sector firm the value of the an innovation is the total
welfare generated by it. For the private firm it is the value of the private benefits or
profits. Next, we proceed to characterize the equilibrium condition in R&D for the
market structures of interest, namely monopoly, pure duopoly, and mixed duopoly. We
make progress by first characterizing the best response function for each firm in the three

markets.

56

111.3.]. Monopoly equilibrium condition
The monopolist maximizes its payoff (equation (III.1)) where i=l=M), by

choosing xM.'0 This defines the first order condition for a maximum for the

av,"
xM

 

monopolist, =0, which is true if and only if RM [xM ] = 0 where

h' W —W — —h + h' +
RM IXM ] 5 My 1(' M [XM] M ) r [XMI XM [Jr/14] : 0 (111.2)
h[XM ]WWM [IM ](1 +h[xM ]/ r)

Equation (111.2) determines the equilibrium value of the R&D expenditure by the

- * 11
monopolrst xM ,

Following Delbono and Denicolo (1993), the difference between the profit from
winning and the firm’s current profit (WWM [xM ]—WM) measures the incentive to

innovate in the absence of rivalry and is referred to as the ‘profit incentive.’ If we assume

that WWM [xM ] > WM > 0, then presence of current profits in this model induces firms to

delay the expected date of innovation”:

111.3.2. Pure duopoly equilibrium condition
In the pure duopoly case (two profit maximizing firms), firm 1 chooses x) and

firm 2 chooses 1:2 to maximize the payoff function. Due to symmetry (n=2, therefore,

 

'0 The monopolist faces no rival, therefore x,=0.
11 In equation 111.2 and all subsequent equations, prime denotes first derivates and double prime second
derivative.

'2 Stated differently, if there were no current profits such that WM = 0 , then
WWM [xM ]- WM = WWM [xM ] , increasing the LHS of equation (111.2), relative to the case where
WM > 0

57

avD = 0
x0

xl=x2=xD), the best response function for each firm is defined by the condition

 

which is true, if and only if, RD[xD] = 0, where

h'[xD](WWD[xD]—WD)+
RDIXDJE (l/r)h'[xD]h[xD](WWD[xD]—WLD[xD])— =0 (111.3)

r— 2h[xD]+ th'[xD]+ h[xD 1WD [x010 + (2/ r)h[xD])
As in the monopolist case, the duopolist also faces a ‘profit incentive’ (WWD[xD]-WD)

which measures the incentive to invest in R&D in the absence of rivalry. However in a
duopoly there is rivalry as each firm anticipates research by the other. In the presence of
rivalry the incentive to invest in R&D is also reflected in the difference between the flow

of profits should it win the race and should it not, WWDIXDI‘WLDIXDI- Call this the

‘rivalry incentive.’ The presence of both pre-innovation profits and post-innovation
profits for the loser induce firms to delay the expected date of innovation, that is higher
pre-innovation and loser profits decrease the profit and rivalry incentives. The smaller the

profit and rivalry incentive, the more time it will take for innovations to arrive.

111.3.3. Mixed duopoly equilibrium condition

In the mixed duopoly the private and public sector firms choose x; but maximize
different payoffs. The private firm maximization problem remains unchanged from that
of a firm in the pure duopoly case. That is, it maximizes equation (111.1) (for i=2), where
we denote firm 1 as the private firm (P) and firm 2 as the public (S). Since symmetry no

longer holds (as the public sector firm’s payoff is different), the private firm’s best

58

response function is defined by the condition 51V; = 0 , which is true if and only if
xP

Rp[xp,x5]= 0 where

Ph'[Xp](Wwp[xP]—Wp)+
RP[XP’XS]E (1/I')h'[Xplhle](Wwp[Xp]—WLP[XSD-r— =0

_h[xp]—h[x5]+xph '[xp]+h[xP]Wv'1/P[xp](l+(l/r)(h[xp]+h[xS D-

 

 

(111.4)

The best response function of the private firm in the mixed duopoly is similar to
the one in pure duopol y. In both cases, there exists a profit incentive as well as a
competitive threat, faced by the private firm. The only difference between the two is that
the source of the competitive threat in the pure duopoly case is another, identically
specified private firm, whereas in the mixed duopoly case it is the welfare-maximizing
public sector firm.

The public sector firm in our model is a welfare maximizer to whom the value of
the prize embodies the social welfare. This means, first, that the “W’s” in the payoff
function are greater for the public sector firm than they will be for the private (more
discussion on the assumption and relative values of the prize follows). Second, the public
sector firm, as a social welfare maximizer, takes into account the flow cost of research
incurred by all firms in the economy. In the duopoly case where one firm is public and

the other private, the public sector firm’s payoff function is written as:

00

VS = Ie—(hle]+h[x5]+r)t(h[xS ]st[xS]/r+h[xp]WLS[xp]/r+W5 —xS —xp)dt (III.5a)
O

= (1")(hlxsstlxsl+hlelWlexPl)+Wp - xp -xs
h[xs]+h[x)o]+r

 

(III.5b)

59

The maximization of the equation 111.5 yields the public sector firm’s reaction curve

which is defined by the condition 2!; = 0, which is true, if and only if, RS[xS,xp]=0
XS

—

’h'ixs 1(st [xS]—WS)+
(1 / Oh '[Xplhlxs 1(Ww51xsl—W151xp1) "
r— h[xp] —h[x5]+ h '[xp](xp +xS ) +

_h[xs 1Ww51x51(1+(1/ r)(hlel+hIXSl)) _

RS[xS,xp] -=- = 0 (111.6)

 

 

The reaction curve of the public sector firm in the mixed oligopoly also reveals
that the public sector firm faces the ‘profit incentive’ and the ‘rivalry threat’ from the
opposing firm. For the public sector firm the ‘profit incentive’ is a misnomer (Since the
value of the prize to it is total welfare, and not private profits as the name implies),
although it is still the incentive to innovate in the absence of a rival. As with the earlier
case, the smaller the rivalry and profit incentives the later is the date of innovation

In summary, equations (111.2), (111.3), (111.4) and (111.6) are, respectively, the best
response functions for a monopolist, duopolist (in a pure duopoly), private firm in the
mixed duopoly and the public sector firm (in a mixed duopoly). In all these cases we see
that each firm faces a profit incentive and a competitive threat (except in the case of the
monopolist, where it does not face a rival). The profit incentive is a function of how large
the difference is between current profits and profits if the firm wins, and similarly the
competitive threat is a function of how big the difference in profits is between winning
and losing. Clearly, if the differences are small, then the firms will be conducting less
research, which will delay the expected date of innovation. For the public sector firm in
the mixed duopoly, the profit incentive and the competitive threat also matter (only that it

is not profits that the public-sector firm is after but welfare). But since it takes into

60

account the total R&D cost, the effect of R&D cost on the public sector firm (xp + x5 ) ,
relative to private firm (xp) , is to bring closer the expected date of innovation.

Lastly we note that in each of the four best response functions the presence of the

SW - . . . . .
term the term JL. Thls term, which reflects the marginal change 1n the prlze value
xi

due to a change in own research, is a direct consequence of our assumptions regarding the
endogeneity of the value of the prize. If we assume that profits are concave with respect
to own research then the endogenous nature of the prize value brings closer the expected
date of innovation. This implies all firms carry out a greater amount of research relative
to the case where the prize value is exogenous. To put it differently, when R&D
complements the innovation at a later stage, firms have an incentive to increase their
research effort. High amounts of research, or the so-called over-investment problem
(Dasgupta and Sti glitz, 1980), could therefore be explained by the presence of such a

complementary effect.

111.4. Comparative Analysis

Having established the nature of the game and market structure we now turn our
attention to the relative ranking of the equilibrium research effort by individual firms
(xM , x0, xp, and x5) as well the industry (xM , 2m, and xp+xs). First, however, we need
to make certain assumptions about the relative value of the prize in the different markets
and for the different firms. Since the relative values of W will determine the equilibrium
values of research effort for all the firms, we need to make our assumptions regarding

them explicit.

61

Assumption 1 .' The winning firm profits (for the private firms) or welfare (for the public

sector firm) are greater than current profits/welfare. That is le-[xi] > Wi- This

assumption ensures that the ‘profit incentive’ to innovate is always positive.
Assumption 2: In the pure duopoly the profits in the winning state are greater than or
equal to in the losing state. Moreover, profits are increasing in R&D for winning firm
and decreasing for the losing firm. Assume that firm 1 emerges as the winner and
acquires the rights to a superior cost reducing technology. In the second stage, the two
firms play a Coumot game. If we assume increasing costs of production in the second
stage, then it can be Shown that in equilibrium the winning firm will produce more than

the losing firm. Consider, for example, a second stage product market where P = a -bQ

is the inverse demand function (for Q = 4WD + q LD ). Assume that prior to the race, the

cost of production for both firms was C [ y,q] = yqz /2 where the parameter 7represents
the technological opportunity due to the successful research, such that 8C / 8y< 0. At the

completion of the race, the losing firm will continue to produce at the pre-innovation

cost, but the winner obtains a better technology such that it lowers its costs to

C[7,q] = 7q2 /2(1+ xD) where 7 < y. The (1+ x0) term reflects the fact that the

winning firm also gains from its research effort of the first stage. The post-innovation

maximization problems for the winning and losing firm are therefore, respectively

max WWD = PqWD — C[;7,qWD,x*D] (111.7)
4WD
max WLD = P‘ILD —C[}’,qLD] (111.8)
(ILD

62

The two firms acting simultaneously and non-cooperatively solve their maximization
problem. From the first order condition for a maximum it can be easily shown (see
appendix) that the standard Coumot equilibrium when costs are asymmetric will prevail,

that is13
l. ail/01x13] 2 QZDIXID]
2. W1;D[X*D] ZWZDIXBI
3. aWVDDpcBJ/ 8x3 >0

4. awZDlx’gi/ax; < o
Assumption 3 : In the mixed duopoly, no public production or further research takes
place in the second stage. Should the public sector firm win the race, it licenses its
technology to the private firm. Alternatively, if the private firm wins the race in the
mixed case, it does not face a rival in the second stage and assumes the role of a
monopolist. As we are assuming the same demand and technology characteristics across
the different market structures, it follows that the profits of the monopolist, should it
succeed in innovating (in the monopoly case), equal the profits of the winning firm in the
mixed duopoly, if and only if, the research effort by the two firms was equal in the first
stage”. On the other hand if the public sector firm wins the first stage race, and in the
absence of further research or production by the public sector firm in the second stage, it
licenses the technology to the private firm. While the private firm is still a monopolist
even with the license (by virtue of the fact that no rival exists), the terms of the license

are such that it is not allowed to produce at the profit maximizing level (where marginal

 

'3 The asterisks represent equilibrium values from the solution of the optimization problem
" The claim here is that if xM = x, = x then WW [x] = Wwp[x]

63

cost (MC) equal marginal revenue (MR )) but rather at a level where the welfare losses
associated with monopoly production are minimized, though not necessarily eliminated.
This is because the terms of the license also have to be incentive compatible in the sense
that the profits for the private firm from production using the new licensed technology are
greater than, or equal to, the profits associated with the older technology.

Therefore, it is reasonable to think that the welfare attained with a public sector
firm innovating is higher than the welfare attained when the innovator is a private firm,
simply because a public sector firm would license and contract its innovation to the

private firm which would have to price its innovation in order to maximize social welfare

taking into account consumer surplus. With Wv‘yshg] , as the total welfare generated by
. . . . . :1: :1:
the 1nnovatlon when the 1nnovator 15 the public sector firm, and WLSIXP] as the total
welfare generated when the innovator is the private firm. Finally, W&p[x;)] is the value
. . . :1: :1: ,
of the prrvate benefits when the prrvate firm 1nnovates, and WLp[xS] when 1t does not.
, , , :1: :1: :1: :1: :1: :1: :1: :1:
leen these definltrons, one has WWS[xS] > WLS[xp] 2 Wwp[xp] 2 WLp[xS] and

qZp1x§12 qivplxifl '5.

The generalized functional form of the hazard rate and the prize value does not
permit us to solve explicitly for equilibrium research effort in each case that could be
compared across the three different market structure. However, if we assume that the

. . . . . * * * * . .
equrlrbnum research condrtrons xM , xD , xp , and x5 solve therr respect1ve best response

function, we can derive a set of sufficient conditions to evaluate the relative magnitude of

 

'5 For a more formal treatment of this result, refer to the appendix.

64

research among the firms. To illustrate this approach, assume that, in a two firm strategic

* =1! , _
game, x,- and x j solve the best response functron Ri[x,-,xj] and R j[xi,x j] for firms:

and j, respectively (that is,R,-[x;-k,x xj-]= R j[x,- ,xJ-]= 0). Assume also that there rs an

asymmetric relationship between the two firms such that, a priori, we are unable to

. . . * * . . . .
deterrnrne the relative levels of research 1.e. x,- 5 x j. In such cases 1t 1s possrble to derlve
a set of sufficient conditions that will satisfy x: > xj, x: — -xj and x: < x; .We do so by

taking the difference of the best response function of one firm evaluated at the other

firm’s optimal research levels (i.e., R 11x; , x: ]) and the best response of the other firm

evaluated at its optimal level (i.e., Rip: ,x;] = 0 ). If we assume that the underlying value
. . . . . *

function for both firms 1s concave With a relatlve maxrmum at x,- and x:- , and that the

second order condition is satisfied (82Vi[x,-]/ 8x12 < O ), then we can claim that the
conditions under which R fix: ,x: ]—R,-[ ,xj *2] —0 implyx < x:-

We apply this strategy separately to compare and derive the sufficient conditions
to evaluate the equilibrium research effort between a) a monopolist and firms in a pure
duopoly, b) a monopolist and firms in the mixed duopoly, and c) firms in a duopoly and

firms in the mixed duopoly.
111.4. 1. Monopoly vs. Pure Duopoly

The first order condition of a monopolist evaluated at the equilibrium value for

the duopolist is expressed as:

65

h'lxgulexgi—WM)—r—h[x}')]+x’,3h'[xg]+

* , * * J (111.9)
h[xD]WWM[xD](1+ h[xD]/ r)

RM1x31=[

From this we subtract the best response function of a duopolist (equation (III.3))

evaluated at the equilibrium value of the R&D expenditure, x2):

R. U... 1_[h'lx31(wwplle—WD>+(1/rlh'llehiletwwoile-wwlxgll] _ O
D D _ :1: :1: :1: :1: 1 :1: :1: —
"'7' — 2h[xD]+ th '[XDl'i‘ h[xD ]WWDIXDKZ +(1 /r)h[xD])
(111.10)
Evaluating and collecting terms in the expression RM [xB]—- REUB] , we get
RM {x}; ] — 1:35)] = h[xgyt 1x310 / r)(Ww[x§,]— WWDix}; 1) +
h[XbKW/u [xbl-WwDIXBD +
(111.11)

h'le]((Wp-Wwplx;)1)+(WM[xEl-WM))+
1.1.310 +(1/r)h[r;3](w,'M [gr—awtptxgi)

We consider first the conditions under which the optimal research effort of the duopolist

in a (pure) duopoly is greater than that of the monopolist (i.e. x?) > x7” ), which is

implied by KL [xB]— RB[x*D] < 0. For this inequality to be satisfied it suffices that each
of the four terms (lines) in equation (111.11) be less than zero (the sufficient conditions).
Assumption 2 unambiguously implies that the first term is negative; the profits of the
winner are greater than that of the loser in a duopoly. The second term will be negative
(h[xguwj, [.3]..wa 1x31) <0) iff wj41x31< waixg] ; the marginal changes in
second period profits (at the duopolist’s optimal level) are greater for the winning

du0poly firm than it is for the monopolist. For the third term to be negative requires that

66

. . * ’1‘ . .
the condrtlon IWD ‘WDWIXDI |>| WM —WM [xD]| be met”. For the final term 1n equatlon

r
:1:

h[xD

 

(111.1 1) to be negative requires that < 2W5W[x;)]—W,;4 [x2] .'7

The implication and interpretation of these conditions are as follows. Prior to the
innovation the firms in a duopoly make equal but lower profits than the sole firm in a
monopoly. If the innovation in question results in significantly higher profits for the
winning firm (relative to the losing) in a duopoly, then these conditions are sufficient to

ensure that the optimal research effort by each firm in a duopoly will be greater than that
of a monopolist. It follows than that if x; > XX)! , than the industry wide research effort
in a duopoly will be greater than in a monopoly as well (i.e. 2x3 > x114 ).

Consider now the reverse case where a monopolist undertakes more research than

both firms in a duopoly (i.e., 2x?) < xxy )‘8. The conditions that would imply this result
would require that RM [2xg]—R;)[x;)] >0, which is expressed as:

RM [2x’l’yi— 1:51.31 = (2hrxgl- h[2xj'3]+ 2.31.1sz— x'fyh '[xBD +
(h'ixguwm _wWD[xg])+h12x;,ier [2x31—WM,)+
(1/r)h[x;]h'[xgnwwlxgi—WWDixgl)+ (111.12)
(1212):; m, [2x3 1 - lair}, ]WV'VDUZ) 1 +
(1/ r)(h[2x}3 ]zwj, [2x3 1 - 2h[x’,"J 12 We; 1)

 

'6 Consider the extreme case where the innovation is drastic allowing the winning firm in the duopoly to
serve the whole market then Wow [x;,] > Wotlxb] = 0. Having driven off the other firm, the duopoly

winner will act as a monopolist, such that Wowlxb1= WM [x3] .
c I l o I . ‘ ' .
'7 This being a consrstent, but stronger condrtlon to WWDDCD] > WM [xD]

'8 Which implies that x; < x;

67

A sufficient condition for (III.12)>0 is that each term in (111.12) be non-negative, which
we consider next. The first term is always positive due to the assumptions on the hazard
rate.‘9 For the second term to be greater than zero, we first note that it can be expressed as

the following inequality:

(1:115,qu —WDW[x}31)+h'[2x}3](WM[mm—WM,)
> (111.13)

h'[r"'D]((WDl -WDW[x;‘,])+(1/2)(WM [um—WM”)

If the right hand side of the inequality in (111.13) is non-negative then it follows that left

hand side is also non-negative. Thus, for
h ix}; ]((WD] — WDW 1x31) + (1 / 2)(WM [2x31- WM 1 )) >0 implies that
WD] “walxbll < l(1/ 2)(WM [2xg]—WM1) . That is, the change in profits for the

winning duopoly firm relative to pre-innovation profits is less than half the change for the
monopolist.

The third term will be less than or equal to zero since from assumption 2, we

know that WLD[x;)] S WWD[x;)] . Therefore, a sufficient condition for (111.12)>0 would

necessarily require that WLD[x;)]—WWD[xB] = O , implying that the gains from winning
and losing are the same in the duopoly. Such a situation would arise if there are perfect
spillovers allowing the losing duopolist to appropriate all the returns of the winning
firm’s prize. A weaker condition would require that the winning duopolist prize in the

second stage allow it only slightly higher profits than the losing firm’s profits

 

'9 Note that 2h[x;,] > h[2x,‘,] and Zth '[ZxL] > xgh '[x13]

68

WWlebl—WWDIXD] = 0. One implication of this condition would be that the innovation

is non—drastic and “small”.
As with the second term, the fourth term in equation (111.12) can be expressed as

an inequality such that
:1: 1 :1: :1: 1 :1: :1: ' :1: 1 :1:
’11szle [szl— h[xD]WleXD] > h[xD]((1/2)WM [ZxD]—WDW[xD]) 20.
Thus, for the fourth term to be non-negative requires that following condition be met:

aWM [2x3] 2 2 8WDW[x;)] .
8x3 8x2

This condition implies the slope of the profit function for the monopolist is relatively
constant (relative to the duopoly case). This sufficient condition is consistent with the
others in that innovation for the monopolist is significant but insignificant for the duopoly
winner.

The last sufficient condition for (111. l 2)>0 requires that the following inequality
be satisfied: (1 / r)(h[2x;)]2 Wj, [2.31- anagrwpbrzxgi) >0. Assume that the change

in profits for the monopolist due to first stage research is constant, such that

3W,- [ax*]/ Exit = ac for all a > 0. We have established that for the fourth term in

11‘ ’1:
W 2 2%. If we assume that

(111.12) to be non-negative requires
3x0 Bx D

W = 2.3WM [be = 2 aWowlxbl

* :1: than the last term in equation (111.12) can be
8x0 3x0 3x0

reduced to (1/ r)(h[2x;')]2 - h[xB]2)W)'W [x3]. This term will be non-negative if and only

if h[2x23]2 > h[xB]2 , which is always true. Table A111.1 provides a summary of the

69

sufficient conditions for ranking the research effort of a monopolist and firms in a

duopoly.

111.4.2. Monopoly vs. Mixed Duopoly

We next derive the sufficient conditions under which x; 2 xx! and x; 2 11:4
(which taken together would imply that x; + x; 2 xx, ). We first derive the sufficient
conditions for the equilibrium research of the private firm (in the mixed du0poly) to be
greater than that of the monopolist. This condition, x; > x7” , is implied by
Rp[x;4 ,x;]— RL [XL ]>O where Rp[x;4 ,x;] is the best response of the private firm in
the mixed duopoly (111.4) evaluated at the equilibrium research level of the monopolist

and the public sector firm, and R; [3‘14 ] is the monopolist best response (equation (111.9)
). Evaluating and collecting the terms in the expression Rp[x;4 ,x§]—R;4 [xXl ], we get

1111le axsl-RL [x14 1 = ’1'le ](WMl ”Wm”
12'th ](Wwp1xx4 i—WM 1x34 1) +
hut)"M 1+ (1 / r)h[x}'{, ]2)(W{Vp[x;4 1—W,;, {x}, ]+ (111.14)
(1/ r)h xx}, 1141:; ](Wwp[x;, ]—WLP[x;, ]) +
h[x§](h[x;, ]wv'vpux, ]— (1/ r))
A sufficient condition for Rp[x;4 ,x; ] — R114 [x714 ]>O is that each term in equation (111.14)
be non-negative. From our assumptions on the prize value, we note that WM] = WP] (pre—

innovation profits for the both the monopolist and the private firm in the mixed duopoly

are equal) and that wwpux, ] = WM [xj‘w] (post-innovation profits, when both monopolist

70

and the private firm are successful in research, are equal as well). These assumptions
imply that the first three terms in (111.14) all converge to zero. The fourth term is non—

negative and follows from assumption 1 (Wwp[x] 2 WLP[x]) . Thus the sufficient

condition for Rpm; ,x;]—R;4 [11:4 ]>0 is reduced to mu}; ]WV'VPuLp 1.
For equilibrium research of the monopolist to be greater than the private firm in

the mixed duopoly simply requires that we reverse the previous condition. That is, for

Rp[x;4 ,xg ] — R114 [xL ] <0 (which implies x; < xx, ) suffices that the sufficient condition

rh[x;4 ]Wt'yp[x;4 ] S] be met”. Further, since the profits from winning for the private

firm in the mixed duopoly will never be less than the profits from losing (i.e.,
Wwp[x] 2 WLP[x]), another sufficient condition that would guarantee x; < xx, is

Wwp[x] = WLP[x]. Recall that the when the private firm losses the race to the public

sector firm, it can either continue using the pre—innovation technology in the second stage
or license the technology from the public sector firm (which does not produce in the
second stage), and produce at level determined from the maximization of the public
sector firm’s objective (see appendix). The implication of the condition

pr [x] = WPL[x] is that the public sector firm essentially allows the private firm to earn

rents that the private firm would earn under the scenario where private firm actually had

won the race.

 

20 Note that the first three terms in (111.14) equal to zero

71

Next we derive the conditions for which the equilibrium research by the public

. . . 11= =1= . . . .
sector is greater than the monopolist (i.e. x5 > xM ) wh1ch 18 now implied by

=11 =1: =1 =1 . . . . .
RS[xp,xM ]—RM [xM ]>O. Evaluating and collecting the terms in this expressmn, we get

R5[x}';,x}'{4 ]-R,"{, {x}, 1: h'[x;4 1((WMl —WM[x}'{, ])+(WWS[x;, ]—W51))+
hixi'ii itwiisixfu 1—Wh 1x71, 1) +
(1/r>(hixli12<wt'vstxii i-Wli 1x71} 1) + htxia ihixl‘» iwi'vstxii i) +
x212 1x; 1- hix131+
(1 I r)h1x21h '[xL ](stle 1- Wis 1x791)
(111.15)

As before each line in equation (111.15) is a term, and it suffices that it be non-

negative to satisfy the inequality x; > x} . The first term will be non-negative if and
only if WM] —WM [x]; ]l S IWWS[x;4 ]—W51 , that is the absolute gains in profits for the
monopolist are less than the welfare gains to the public sector firm due to the innovation.

Thls suffic1ent condition is cons1stent w1th WWS[x;4 ] 2 WM [xM] which ensures that the

second and third terms are non-negative.

The fourth term will be non-negative if and only if x3311 '[xL ] 2 h[x;] . To

guarantee that the fourth term is non-negative it suffices that curvature of the hazard rate

be constant (i.e., a linear hazard rate h[x] = x). The last term in equation (111.15) is non-

negative for WW5 [x711 ] 2 WLS [x72]. While welfare will be greater when the public sector

wins as opposed to when it losses (assumption 3), this assumption requires that research

:1: 1k
effort, x, across the two states be equal. Clearly, when Jr," 2 Xp , then

72

=1: * , , =1: :1: ,
WWS[XM ] 2 WLS[xp] w1ll hold. However, if xM < X}; , then the comparlson between the

welfare in the two states remains ambiguous therefore one has to assume that

WW5“); ] 2 WLS[x;;] or that the public sector firm is always larger when the public

sector firm wins as opposed to when it losses.

Lastly, we state the sufficient conditions for the case where the equilibrium

. . . . * *
research by the monopollst 13 greater than the public sector firm (i.e., x5 < xM )2'. Thus,

relative to public’s welfare, the gains to the monopolist are now larger such that
Wv'vs[x;,1<wg,[x;41 and WMl —WM 111;, 1| >lWWS [x714 1—WS, (which would now
ensure that the first three terms in equation (111.15)). For the last two terms in (111.15) to

be negative simply suffices that x; < xL , and the sufficient conditions that are implied

by it. Table AIII.2 summarizes these results for the comparison between the monopolist

. . . :1: =1: :1: it: . .
and the firms in the duopolist. Note that whlle xp > xM and x5 > xM implles
at: * =1: , * at: :1: * , ,
xp + x5 > xM , we are unable to establish whether xp < xM and x5 < xM lmplles
:1: at: :1: . . . . * *
xp + x5 < xM . The best we can do here 15 to say that suffic1ent condltlons for xp < xM

at: :1: , , * at: *
and 213 < xM are the necessary condltlons for xp + x5 < xM .

111.4.3 Mixed Duopoly vs. Pure Duopoly
Our final comparative analysis is between the firms in the mixed duopoly and

firms in a pure duopoly. Here we seek to derive the sufficient conditions under which

 

2' As one would expect this implies reversing the sufficient conditions for x; > x; .

73

>1: :1: :1: >1: , , _ :i: :1: , ,
x0 2 xp and x0 z x5 . If the suffic1ent conditions for x0 > X}; are not inconSIStent for
. * * . .

those that would satisfy x0 > x5 , then we can claim to have found the suffiClent

. . * * * . . . . . . at: :1:
condition for 2x0 > x5 + xp. Similarly, ifthe suffiCient conditions for XD < xp and

* * . . . . .
xD < x5 are not inconSlstent, then those conditions would imply that the aggregate
research effort in the mixed market is greater than that in the duopoly market (i.e.,
* 31: *
2x0 < x5 + xp ).
. . . . . =1: =1: . .
We first examine the suffic1ent conditions that would satisfy xp > xD implied by

* * IF =1!

Rp[xD,xs]—RD[xD] >0, where

Rp[x*D,x;]—R;)[x;)] = h[xg1—h[x§1+

(1/r)h '[xg1h[x;1(ww[x;31—WWD1x}31)+

(1/r)h '[xg1h1xg1(WwpixL1-WLP1xE1)+
* * :1: (111.16)
h'[xD]((WD —WWD[xD])+ (Wwp[xD]-Wp))+
htxzi(wwplel—anlxzi)+

(l/r)hixBl(WwpixZi(hixBi+h[xEi>—Wwoix231(2hixiil))

As before the sufficient conditions for Rp[x;),x;]—R;)[xg]>0 require that each

term in (111.16) be non-negative. For the first term in (111.16) to be non-negative would

require that the equilibrium research effort by a firm in the duopoly be greater or equal

than that of public sector firm (x; 2 x; ). The second term will be non-negative if we

assume that profits for firms in the duopoly are equal, win or lose (i.e.

WLD[x;)] = WWD[x*D] ). The third term is always non-negative and follows from

74

. . . . . * * * *
assumption 2 and the earlier suffiClent condition x0 2 x5 (hence, Wwp[xD] 2 WLp[x5]).

For the fourth and fifth terms to be non-negative implies that the profits from winning in

the mixed duopoly case increases faster than in the pure duopoly case, such that

WV'VP1xg12 wwDixgi and WD —WWD[x}3 1| 5 [wwpixg1-WP . The last term will be non-

negative if and only if Wv'vpixguhixgi + h1x§ i) 2 W,'VP1x31(2h[x;31) .

A key sufficient condition in establishing that x*D < x; has been the assumption
that x?) 2 x2. Are the sufficient conditions that would satisfy x?) 2 1:; consistent with
those for x*D < x; ? We examine this next. For x?) 2 x; to hold implies that

II! it * *
R5[xp,xD]— RD[xD] S 0 also needs to hold, where

RS {x}, 1531- 1113th = (l / r)h '[xgihixg1(Ww1xB1-Ww01xg1)+
(l / r)h '[xg1h[x;1(wWS[xB1—WLS [x;1)+
h'1x231((wn -Wwpix31)+<wwslei—Ws>)+
h[xB](st[xBI-Wwplxi)])+
(l/r)h[x;')1(1+w,'VS[x;‘,1(h[x}‘,1+h[xj‘p1)—W,L,D[xg1(2h[x}31))
x212 111131-12le
(111.17)

For Rs[x;), x?) ] — 11315313 0, a sufficient condition is that each term in the

equation be less than or equal to zero. The first term will be less than or equal to zero by

. t :1: . . .
assumption 2, WLD[xD] S WWD[xD]. However, to be conSlstent With the suffiCient

. . . 11 . . . * 11
conditions that satisfy x?) < xp requires the stronger condition WLDlxD] =WWD[xD] .

75

which is maintained here as well. The sign on the second term is ambiguous, as

. . . . =1: *
assumption 2 no longer holds due to asymmetric research effort. That 18, Since x0 < xp ,

the term [st[x;)]—WLS[x;>]] cannot be signed without further assumptions on the

value of the welfare in the losing and winning states. Therefore, we require the stronger
condition than the public’s welfare in the winning state (evaluated at the research effort
of the duopolist) be equal to or less than the losing state (evaluated at the research effort
of the private firm in the mixed duopoly). The third and fourth term, taken together imply
that relative to the pre-innovation profits, the gains from innovation are greater to the

winning duopolist than they are for the public sector firm should it win. That is

wv'vsix31 s wthxgi and WD —WWD[xg 1| 2 les [xg1—WS . The fifth term will be less

than or equal to zero if and only if prS[x;)](h[x*D]+ h[x;]) < Wilt/D[x;)](2h[x;)]) +1 .
Note that this sufficient condition is consistent with our assumptions that x; > x; and
W115 [x3] > W,'VD[x;1 which implies st [xg1(h[x}31+ h[x*,,1) > wv'mixgkzhix; 1) . The
last term will be less than or equal to zero if and only if x;h '[x*D] < h[x;].

To derive the sufficient conditions that would guarantee the reverse relationship,

. * =1= =11 . . . . .
i.e. xp < xD < x5 , Simply implies that we reverse the above condition. For the sake of

brevity we present those results and a summary of the sufficient conditions in Table
AIII.3a. The conditions summarized in Table AIII.3a cannot, however, be used to make
statements on the aggregate research relationship between the mixed and pure duopoly.

To do so requires that we derive conditions for x0 2 x3 , xp , which implies

2x0 2 x5 + xp. Table AIII.3b summarize these sufficient conditions and follows from

76

equation (111.17). That is to derive the sufficient conditions that would simultaneously

satisfy x?) > x; and x?) > x; requires that both (111.16) and (111.17) be non-negative.

These sufficient conditions are summarized in the first column of Table III.3b. The

second column lists those sufficient conditions that would reverse this relationship, such

, =1: =1: :1: *
that they satlsfy xD < x5 and x0 < xp

111.5. Discussion

The sufficient conditions that have been derived, which allow us to rank the
research effon in the three markets, are primarily a function of three properties. First,
how the profits/welfare are distributed, post-innovation, in the pure and mixed duopoly.
Second, how the profits/welfare are increasing for the winning firm in research effort (x).
Lastly how hazard rate changes in x (or the curvature properties of the hazard rate). In all
the comparison we note that our assumptions about these three properties allow us to
derive the sufficient conditions and hence a particular ranking. What do the sufficient
conditions mean and what are the implications for genomics research?

Consider first our comparison of mixed and pure duopoly where our interest was
to explore the conditions under which a mixed duopoly would perform more research
than a pure duopoly (or vice versa; refer to Table AIII.3b). For the research intensity of
the mixed market to be greater than that of the pure duopoly it suffices that four
conditions be satisfied. The first condition, that profits be distributed equally across both
the winning and losing states in the pure duopoly, will be satisfied if one assumes that the
losing firm easily imitates the innovation. Since the current market structure resembles

mixed duopoly, this would suggest that one reason why the pure duopoly has not taken

77

hold is because intellectual property rights are ill defined. Indeed, the U.S. Patent and
Trademark Office has had difficulties in reliably defining the scope and scale of patents
in genomics and biotech in general (Padron and Uranga, 2001). Such uncertainty may
lead to greater likelihood of patent infringement and lowering of expected profits. Where
property rights protection is weak the winning firm will have difficulty appropriating the
returns of its innovation”. Another scenario where the first condition would also arise is
if the innovation is non-drastic such that the winning firm does not have a significant
advantage over the losing firm or the losing firm can invent around the innovation to
effectively achieve technological parity.

The second and third conditions relate to the profits/welfare function in the
winning state. The second condition is satisfied if we assume that the first-stage R&D
enters into the firm’s second stage profit function as a autonomous cost reduction. For

example, if we assume that the cost of production, post innovation, for the winning firm
is CIZ,qWD,x;)] =(Z)‘1WD —x;) , and for the losing firm C[7,qw] = (y)qLD then it can
be shown that WLi[xj]<WW,-[x;] for i: S ,P,D and ng ,x j > 0 (see appendix). Again,

given the presence of a mixed duopoly in genomics research, this implies that the
knowledge gained from the sequencing acts as “lump-sum” transfer from the first stage to
the second stage for the winning firm. This case would arise in genomics if the effect of

the sequencing stage has negligible or no effect on further research stages (such as

 

22 A simple example illustrates this point. Consider that post innovation, the winning duopolist is able to
lower its cost to C [ 7, qwm xL] . If the losing firm is able to imitate the technology than it too will have

the same technology and cost function, that would result in equilibrium conditions where both firms
produce the same amount and make equal profits (i.e. with all firms having the same cost function, the
symmetric result would hold).

78

breeding for better varieties) for firms in the pure duopoly but a positive effect for firms
in the mixed duopoly.

The third condition relates to the curvature properties of the profit (for the private
firm) and welfare (for the public) functions. That is if the profit function of the winning
private firm (in the mixed duopoly) and the welfare function of the public sector firm are
“more” concave in x than the profit function of the winning duopolist, then this condition
is satisfied. To the extent that private and public sector firms have taken a different
research strategies in genomics, the research production of firms in both stages in a pure
douopoly may indeed differ from that of the firms in the mixed. As stated in the
introduction to this chapter, the public sector firm has essentially pursued a strategy
where it expends more research resources in the initial stages so that it does not have to in
the latter. The private firms have proceeded to invest less now in the hopes to be the first
to sequence, even though it might entail more investments later to understand
significance of the sequenced genome. This implies that the public sector firm would
benefit more from the knowledge gained from the sequencing stage, and hence raising the
marginal productivity of successive stages.

The last sufficient condition for the mixed duopoly to perform more R&D than

that in the pure duopoly can be restated as such:

1+wv'vpixg1(h[xg1+h[x§1)-w,'VD[xf)1(2h[x",31)> h[xf ] >1 for it} > x; >0.

h[xD

 

A case where this condition is satisfied is when we assume that the hazard rate is

bounded such that h[x] —-> 1 as x —-) oo (e.g. a logarithmic function) and that equilibrium

R&D is “high” for both the firm in the private duopoly and the public sector firm (i.e.

79

h[x; 1

*

 

approaches 1 from above). Since Wpyp[xg](h[x;)]+h[x;]) > Wp'VD[x;)](2h[x;)]) ,

 

h[xD]
it follows that the left hand side will be greater than h[xf ] , iff, Wi'yp[x*D] >> Wt'VD[X*D].
h[xD]

An interpretation of this sufficient condition is that the innovation is drastic for the
winning private firm in the mixed duopoly and non—drastic for the winning duopolist (in
the pure duopoly).

We next turn to the interpretation of the conditions under which a mixed duopoly
would perform more R&D than a monopolist (refer to Table AIII.2). Three of the four
conditions are also related to the properties of the profit/welfare function and hazard rate.
If for example, the public sector firm’s welfare function is more concave in R&D than the
monopolist than the first condition is satisfied. The second sufficient condition is satisfied
if we assume that the hazard rate is relatively constant, for example a linear hazard rate.
The third condition requires that public welfare be strictly greater in the winning state

than in the losing state, an example of which was given earlier.

The fourth sufficient condition (rh[x;4 ]Wt'yp[x;, ] > 1) follows from the earlier

sufficient conditions and is met when those conditions suffice. Recall that when the

winning firm in the mixed duopoly wins it becomes a monopolist in the second stage.

Hence Wwp[x;, ] = WM [x714 ]. Since for the first sufficient condition to be met, it suffices

that the public sector firm’s welfare function be more concave than the monopolist, it
follows that it should be also more concave than the winning private firm’s profit

function in the mixed duopoly.

8O

In our comparison of the mixed duopoly with that of pure duopoly we also
derived a set of sufficient conditions, which, if met, ranked individual research effort at
the firm level (Table AIII.3a). The rankings, however, do not allow for an unambiguous
statement on whether at the industry level one market performs more R&D than the
other. Focusing on the sufficient conditions which ranks the amount of research
performed by the public sector firm above that of the private firms (either in the mixed
duopoly or the pure duopoly), we note that all the sufficient conditions relate to the losing
and winning profits of the private firms (which need to be equal) and the concavity
properties of the public sector firm’s welfare function in the winning state (the public
sector firm’s welfare function has to be more concave in R&D than the profit function of
the private firms).

Our analysis also presents a comparison of the pure duopoly market with that of a
monopoly market (Table AIII.1). Here, too, the sufficient conditions for ranking the
industry R&D in the two markets is function of curvature properties of the profit function
and the hazard rate as well as the distribution of profits in the duopoly. For example,
when the profits after the race are distributed evenly among winners and losers and the
profits for the monopolist are larger than it is for the winning duopolist, then it suffices

that the sole firm in the monopoly market performs more research than the firms in the

duopoly.

1116 Conclusions

The analysis of this chapter was motivated by the observation that in genomics
research, the amount of R&D expended to win an R&D race affects not only the

probability of success but also downstream profits. Further, we observed that the public

81

research sector is engaged in fierce competition with private firms in a variety of
genomics projects. Since it was not clear, a priori, the reasons for the public sector firm to
undertake genomics research (especially in light of the fact that activities are similar to
those of the private firms), we set out in this chapter to derive a set of sufficient
conditions under which R&D effort across different plausible and observed markets in
genomics research could be ranked. It was found that the sufficient conditions relate to

o the concavity properties of the profit/welfare function with respect to first stage

research,

0 the distribution of profits across firms in the duopoly markets,

0 the magnitude of the gains in the second (profit/welfare) from the innovation

relative to other firms as well as pre-innovation profits/welfare,

o the curvature properties of the hazard rate.

One interpretation of the concavity of profit/welfare is how useful first stage R&D (or
the knowledge gained in the racing stage) complements the innovation that is eventually
employed in applied research or production. For the firm for which the complementarity
effect is the greatest, the incentive to conduct more research will also be larger. Given the
disparate research strategies of the public and private firms in genomics, it was suggested
that the public sector firm by pursuing a different research strategy is able to exploit more
effectively the complementarities between the sequencing stage and research stages that
follow.

When comparing with the duopoly market, how the profits are distributed across the
two firms affects has implication for the sufficient conditions. Distribution of profits

across firms can be affected by the nature of the innovation (whether drastic or non-

82

drastic) and/or the ease to which the winning innovation can be imitated by the losing
firm. For example, if the innovation is minor such that the distribution of profits of the
two firms remains relatively unchanged then the incentives to innovate are less. Similarly
if the innovation is easily copied, due to weak property rights perhaps, then the incentives
remain weak. In such cases, we find that the mixed market undertakes more R&D then
the pure duopoly and a role for the public sector exists. Given the uncertain nature of
intellectual property rights in genomics, it can be argued that expected profits, and hence
incentives for research, for a private firm are low.

For firms in one market to perform more research relative to firms in another
market (e.g., firms in mixed relative to firms in pure duopoly), it also suffices that the
absolute profit/welfare gains (relative to the pre-innovation case) be greater than the other
market. For example, in the comparison between the mixed duopoly with that of the
monopoly market, a sufficient condition under which the public sector performs more
research than the monopolist is if the gains to public sector firm’s welfare are larger than
they are to the monopolist’s profits. It is not difficult to imagine cases where this would
be true. Consider a case where, prior to innovation, a firm holds a monopoly in the
breeding of a particular crop. The public sector firm, in competition with the monopolist,
is able to sequence the genome of the crop before the monopolist (i.e., the public sector
firm wins the race), and makes available to the monopolist the sequence for use in its
breeding program. Assuming that the public sector firm sets the terms for the use of the
genome such that the monopolist has to produce at the welfare maximizing level, it is
easy to envision in this case that the gains to public sector firm will be greater relative to

the gains to the monopolist.

83

In summary, this chapter has outlined the conditions under which a mixed
duopoly, and by extension a public sector firm, would conduct more research than other
plausible market structures. That a number of those conditions may resemble the
attributes of the genomics research may help explain why a mixed market has taken hold
as opposed other market structures. It is important to note that early research on genomics
was based solely on the efforts of the public sector, suggesting that the conditions derived
ranking the mixed market above others may have been stronger. As patent rights become
more defined and the research focus shifts from sequencing to more applied uses of the
genetic information, one can expect that the conditions that rank pure duopoly and
monopoly over mixed duopoly would prevail, and hence the role of the public sector

would diminish.

84

CHAPTER IV
COOPERATIVE AND N ONCOOPERATIVE R&D IN A MIXED DUOPOLY WITH

SPILLOVERS

IV. 1. Introduction

In the 70s and 803, there was well-known concern about the decline in the growth
of productivity in the U.S. economy as well as the loss of international competitiveness
amongst U.S. firms. It was felt that a lack of institutions to foster technology transfer and
research sharing among the various agents, both private and public, was a key factor for
the lack of vitality in the economy (Hall, Link and Scott, 2001). In response to these
concerns, government intervention, in the form of legislation, was required to encourage
the transfer of technology between public and private sector. As Day-Rubenstien and
Fuglie (2001) state, “these laws affected ownership rights to new technology developed
with federal funds and established new mechanisms for collaboration between public and
private researchers.” Research partnerships were an important element in the policy
response to global competition, with the explicit recognition that industry had to rely
heavily on U.S. universities and publicly funded institutions to ensure the success of the
research being undertaken.

To encourage research joint ventures among different research enterprises, the
1984 National Cooperative Research Act (NCRA) encouraged joint research among
private firms by providing for anti-trust exemption for private participants in such
research consortia. Further, to specifically encourage collaborative research amongst

private and public sectors, Cooperative Research and Development Agreements

85

(CRADAs) were implemented with the goal of developing and commercializing
technology. CRADAs are very specific arrangements defining how research
responsibilities are shared and results disseminated among the collaborating partners
(Day-Rubenstien and Fuglie, 2000). CRADAs have been credited for developing the anti-
cancer drug Taxol and commercializing useful agricultural technologies (Day-Rubenstien
and Fuglie, 2000).

By strengthening the linkages between publicly financed research with that of a
private firm, the primary policy aim was to increase the economic returns of public R&D
by disseminating those innovations and R&D outputs that have prospects for
commercialization. From the perspective of the private firm, the benefits of partnership
with a public ‘firm’ (such as a university), stems from the perception of the private firm
that the public institutions are engaged in “new” science. The public sector therefore
provides its private collaborators with research insights that anticipate future research
problems, especially with regard to the use of basic knowledge (Hall, Link and Scott,
2001).

While the benefits for both public and private firms may seem apparent, Hall,
Link and Scott (2001) report that of all research joint ventures registered under the
NCRA, only 15% involve universities. This observation leads them to ask whether
university research participation in such projects was not warranted because of the
characteristics of the research or whether certain institutional barriers prevented such
collaboration from taking place. And even though the number of joint ventures between
public and private institutions is not widespread (relative to partnerships amongst private

firms) concern has been expressed that such collaborations divert the public research

86

agenda from its central mission and may create an unfair competitive advantages for
specific firms (Cohen and Noll, 1995).

In this essay we develop a theoretical model of mixed duopoly to study the effects
of public-private partnerships. Specifically our objectives are to provide a better
characterization of the incentive for a private firm to collaborate with the public sector,
which could explain the low levels of partnerships between private firms and universities
reported by Hall, Link and Scott (2001). Second we examine whether public sector
objectives—to further social welfare through research—is compromised from partnering
with the private firm'. Third by introducing a welfare maximizing public sector firm in
the analysis of cooperative and non-cooperative R&D, we examine the robustness of
earlier results that examined c00perative research between only private firms. Lastly we
take issue with the standard assumption regarding the motivation for a public sector firm.
Most partial-equilibrium models of mixed oligopoly have assumed that the public sector
firm maximizes a social welfare function composed of the unweighted sum of consumer
surplus and producer surplus.2 Such an assumption is sufficient for a normative policy
analysis, not to mention that it is also practical and simplifies the analysis considerably.
However since it cannot be guaranteed that the public sector firm actually does or can
maximize the unweighted social welfare function, which would suggest that the analysis
merits an extentsion to allow for positivistic analysis. A weighted social welfare function
allows for a political economy perspective of the solution as well as simplifying our

analysis of cooperation of public and private firms.

 

1 Social welfare (of the public sector firm’s welfare) is defined as a the summation of producer profits and
consumer surplus.
2 Where producer surplus is simply the summation of individual profits.

87

Our approach and model are similar to the one found in the inspiring analysis of
d’Aspremont and Jacquemin (1988) (d’A&J) who studied the behavior of a R&D
performing firms in a (pure) duopoly. Their analysis consisted of comparing the
consequences of cooperative and noncooperative R&D in terms of a two-stage model of a
duopoly with R&D spillovers. They find that R&D agreements between otherwise
competing firms increases the R&D expenditure level at sufficiently large spillovers,
although the R&D is still lower then what would be achieved at the socially efficient
level. The significance of this result is that it is contrary to what one would expect, i.e. a
reduction of R&D due to less wasteful duplication.

In our model we introduce a welfare maximizing public sector firm that competes
with a profit maximizing private firm. As in d’A&.l we study the affects of cooperation
and non-cooperation between the two firms in the presence of spillovers and ask whether
the d’A&J results would also hold in the mixed duopoly case. We analyze two types of
markets, one where the public sector firm is a participant in both R&D and production
stages and another where it only engages in R&D. Our simulation results show that in all
instances, the public sector firm’s objective is advanced through partnerships although
that is not always the case for the partnering private firm. That, for certain ranges of
spillovers, private profits declines due to partnering with the public sector firm implies
that private firms may not have strong incentives to collaborate with the public sector
firm. When there is no participation by the public in the production stage we find that our
results are consistent with the observation of Hall, Link and Scott who suggest that
partnerships are more likely when private firm’s research is difficult to appropriate (i.e.

when spillovers are high).

88

The plan of this paper is as follows. We develop the general. model in section II,
introducing the structure of the two markets, the nature of the game and some general
solution concepts. We specialize the model in section H1 employing the example of
d’A&J for comparison. Simulation of the equilibrium values is provided in this section.
Section IV develops the social welfare criterion for the purposes of comparing the result

to an efficiency standard. Section V concludes the paper.

IV.2. The Model
Consider a duopoly consisting of a private firm (P) and a public sector firm (S)
producing a homogenous product. They face a negatively sloped demand P[Q] , where

Q is the aggregate output of this product (Q = qp + qS ). The two firms engage in

Coumot competition that proceeds in two stages. In the first stage the expenditure on
cost-reducing R&D is determined, which lowers the cost of production in the second
stage. In the second stage equilibrium levels of output are determined endogenously,
taking as given the R&D completed in the first stage. As in previous multi-stage models
of this nature, the solution is obtained recursively, where the equilibrium of the entire

game is a subgame perfect equilibrium. Each firm has a cost of production c,-[x,-,xj] ,

which is a function the amount of cost reducing research it undertakes (xi) , and the

amount of R&D conducted by the other firm xj which enters via ‘spillovers’ and where
the nature of the spillover is embodied in ci[x,-,xJ-]. The cost of undertaking R&D is

specified as R,-[x,-] , and it is assumed that there are diminishing returns to R&D.
(8Ri[x,-]/ax,- > 0 and dzie,-1x,-1/a(ir,-)2 > o ).

89

The spillovers signify that there are extemalities to R&D such that some benefits
of each firm’s R&D effort flows, without payment, to the other firm. What proportion of
those benefits are ultimately captured by the receiving firm is a function of how much
knowledge (embodied in an innovation) its competitor allows to flow to the other firm as
well as the usefulness and adaptability of that knowledge in the receiving firm’s own
production process3. If one assumes that the nature of the R&D discovery process and
innovation is similar across firms (i.e. the two firms have the same learning paths), then
the proportion of research that spills over is more a function of the former rather than the
latter. By this assumption then, the firm from which the spillovers originate has control
over how much knowledge is “shared” with its rivals via patents, trade secrets and other
intellectual property protection mechanisms“. However complete control of one’s
research may not always be possible especially if the research develops characteristics of
a public good or if IP protection is not a possibility. If the appropriability of research
returns is difficulty under existing intellectual property institutions the public sector will
usually dominate such research (as it occurs in basic research).

We describe two possible market regimes in the presence of a public sector firm.
The first regime is one where the public sector firm competes in both the R&D and
product markets. This market is akin to the one discussed in d’A&J, only that here we
replace one of the profit maximizing profit firms with a welfare maximizing public sector

firm. The second regime is more reflective of the true nature of the public-private

 

3 As in Chapter II, we differentiate here between ‘output’ spillovers and spillovers that occur during the
discovery process. Spillovers in this chapter refer to output spillovers as they reduce the cost of production
in the second stage without affecting the cost or productivity of the innovation process itself. They are not,
however, to be confused with appropriability, which relates to how much the returns of the innovation are
captured by the innovation.

4 That is, since its spillovers effect the production stage, the firm has control as to how much spillovers are
allowed.

90

interaction in R&D, in that the public sector firm is not engaged in any second stage
production, but the outputs of its first stage R&D effort spillover and reduces the
production costs of competing private firms in the second stage. For the second regime
we shall assume that there are two profit maximizing private firms that compete in the
second stage. The two private firms, join the public sector firm in first stage, and engage
in R&D competition. Whereas the incentive to conduct R&D for the private firms in this
market is due to their profit maximizing behavior, the public sector firm’s incentive are
due to its desire to increase welfare which occurs when the benefits of its research
spillover to other firms.

Following d’A&J and Suzumura (1992), for both these markers we examine two
equilibrium concepts. The first is one where firms act independently and
noncooperative] y throughout the game, so that the equilibrium of the second stage game
is Coumot-Nash equilibrium, and that of the entire game is a subgame perfect
equilibrium. The second equilibrium concept originates from collaborative R&D in the
first stage between the private and public sector firms; the public and private firms
coordinate their R&D in the first stage with the understanding that no further cooperation
will occur in the product market stage. We first provide some general equilibrium
concepts to the model. Later we specialize the model to correspond to the market
assumptions of d’A&J and compare the comparative statics and equilibrium concepts of
the different markets/games. Table IV.1 provides an overview of the two market regimes

that we study in this chapter and the sections in which they appear.

91

Table IV. 1: Overview of the Cooperative and Noncooperative Game

 

 

 

 

Regime l Regime 2
Non- o 2 firms (1 private, 1 public) 0 3 firms (2 private, 1 public)
Cooperative o 2 stages (R&D and output mkt.) o 2 stages (R&D and output mkt.)
Behavior 0 Both firms compete non— 0 Public sector firm not a
' cooperatively in the two stages participant in the output mkt
GM:§IV.2.].1 (second stage)
SM:§IV.3.] .l o All firms compete non-
cooperatively in the two stages
GM:§IV .2.2.1
SM:§IV.3.3.1
Cooperative 0 2 firms (1 private, 1 public) 0 3 firms (2 private, 1 public)
Behavior 0 2 stages (R&D and output mkt.) o 2 stages (R&D and output mkt.)

 

- Firms cooperate in the R&D stage
but compete non-cooperatively in
the output stage

GM:§IV.2. l .2

SM:§IV.3. l .l

 

0 Public sector firm not a
participant in the output mkt.
(second stage)

0 Public sector firm cooperates
with one of the private firms in
the R&D stage

GM:§IV.2.2.2

SM:§IV.3.3.2

 

GM refers to the generalized model and SM refers to the specialized model.

IV.2. l. Regime 1: Mixed Duopoly in Both Stages

In this market, private and public sector firms compete in both the R&D and the

production markets (or stages). Whereas the public institutions are known to actively

participate in the R&D stage, the notion that the public sector firm also competes at the

production level may seem less compelling. Our justification for explicitly involving the

public sector firm in the production stage is the observation that in many instances the

public sector firm contracts out its innovation (through a license for example) to a private

firm on terms that would be welfare maximizing and not necessarily profit maximizing.

The private firm’s incentive to participate in such a contractual arrangement would stem

92

 

from it gaining access to a superior technology or any transfers from the public sector
firm to fill the shortfall in profits from producing at the welfare maximizing level. A
second justification for having the public sector firm compete in both stages is to think of
the second stage as another R&D stage, where firms compete to produce a final
innovation that is homogenous across firms and sold non-exclusively to the consumers of
innovation in the second stage. For example in genomics research, the sequencing and
identification phase provides better tools for breeding and transformation of plants. The

final innovation, a transgenic plant, is then marketed for mass dissemination.

IV.2. l . l. Non-Cooperative Game
To characterize the second stage equilibrium, we begin with the classical model

of Coumot equilibrium. The objective of the private firm is to choose its output q p to

maximize its second-stage profits:

7: = P _ , -R ]
”(if P ([Ql CPlxp x5])qp PUP (IV.l)

The public sector firm maximizes social welfare where social welfare is
represented by the summation of consumer and producer surplus. The producer rent is

simply the sum of the profits of each firm in the output market (Itp + Its) . The consumer

surplus reflects the utility of the individuals and is calculated by taking the integral over

the demand function less the consumer payments. That is if P[Q] is the inverse demand

function and Q”: is the equilibrium output, the consumer rent is given by

$

Q
V[Q] = I P[Q*] dQ- P*Q*. The public sector firm’s maximization is thus

0

93

max w = va1Q1+ wS [(P1Q1—c51x5,xp1)q5 — Rstxs 1]+
qs (1v.2)

wP[(PlQl-cpixp,xsl)qp-Rplxp1]

where WC, W5 and wp are the weights assigned by the public sector firm to consumer

surplus, public sector firm profits and private firm profits, respectively. Assuming an
interior optimum for each firm and that the second order condition is satisfied, this

implies that the Nash—Coumot equilibrium must satisfy the following first order

 

conditions:
a”P[qP’qS] : P[Q*]+P'[Q*]q; ‘CPpr1xS]=0 (IV.3)
34p
aw , =1 . == . . 11
—[:q: ‘15] =va11Q ]+P [Q ](qup+WSqS)+WSP[Q l-WSCSIXS1XP]=O(IV°4)

It is convenient to rearrange the first order conditions so that they have an

. . . . . =1 =1 =1 ,
elastlc1ty interpretation. Letting Si = q1- / Q denote each firm 5 share of aggregate

output, we can write the first order conditions for the private and public sector firm’s as

*

P[Q*1[1+§f]=cpixp,xs1 (1v.5)

 

P1Q*1[ws + WPSP : W535 ]— wsc3[x5,xp] = —wCV '[Q*1 (IV.6)

where e is the elasticity of market demand. A common interpretation of the traditional
Coumot model is that it is in some sense the “in between” case of monopoly and that of

pure competition (Varian (1996), pp. 288). That interpretation however does not hold in

c ' ° 9, *
the present context. ConSider the “competitive case such that Si approaches zero so that
each firm has insignificant share of the market. Assuming equal weights

94

( WC = W5 2 wp =1 ), equal marginal costs across firms, and constant market elasticity,

the public sector firm will choose an output such that the marginal (increase) in consumer
surplus equals the marginal (decrease in) total domestic profits. de Fraja and Delbono
(1989) show that when the market is competitive, all firms, public and private, produce
an output such that their own marginal cost equals the price.

On the other hand if the product market is characterized as a mixed duopoly
where only a public sector firm competes with a private firm, then the private firm will

produce where the price is a constant markup on marginal cost in which case

P[Q*] 2 cp[xp,xs] . The production decision of the public sector firm is also a function

of the markup on marginal cost, where the markup by the public sector firm is always

:1:
greater than that of the private firm (i.e. [1+2]<(1+-1-)). In the absence of any
8 8

consumer concerns (WC = O) and assuming that the marginal cost of production is the

same across the firms, then the private firm will underbid the price of the public sector
firm, which can lead to comer solutions. If, however, the marginal change in consumer

surplus due to public sector firm production is large such that

*
P[Q*][] +2] > P[Q*](l +1) —V '[Q*] , then the public sector firm will underbid the
8 8

price of the private firm, which too can result in comer solutions.
For the maximization problem of the private and public sector firms, we also have
the second order conditions, which respectively take the form:

827:,»
amp)2

 

= 2P'iQ*i+ P"iQ*iq‘;, s 0 (N7)

95

32w
3(45)

 

2 = WCV "19*“ 2WSP'[Q*] + P"[Q*l(qu; + wsqg) s O (IV.8)

If we assume that the inverse demand and consumer surplus are twice

differentiable functions such that P '[Q] < O , P "[Q] .<_ O , V '[Q] > O , V "[Q] 2 O , then the

SOC for the private firm is always satisfied. However the SOC of the public sector firm
is satisfied only if appropriate restrictions are placed on the weights (i.e. cannot have

weights on consumer surplus, WC , too large relative to the profits of the two firms).

The existence and uniqueness of the Coumot equilibrium are well known and
have been studied by De Fraja and Delbono (1989, 1987), the issue of whether the
Coumot equilibrium is stable has not be adequately addresseds. It turns out that under the
assumption of equal weights for producer profits and consumer surplus in the public
sector firm’s objective, the model is indeed unstable. A sufficient condition for the
equilibrium to be stable is if one assumes that the weight of the public sector firm is
larger than that of the private firm in the public sector firm’s welfare (objective) function.

Lemma IV. I : A sufficient condition for the Coumot equlibrium in the output market of a

mixed duopoly to be stable is ”’5 > wp.

Proof: For the stability condition in the Coumot equilibrium to be met requires:

 

5 Stability in the Coumot equilibrium, or the Coumot adjustment process, is generally regarded as unsound
because the underlying assumption that rivals do not change their output during the course of the game is
often invalid. al-Nowaihi and Levine ( 1985) however point out that this only true if firms have full
knowledge of their own and rivals cost functions along with the demand curve. Under the more plausible
assumption of partial information, the Coumot adjustment process is an attempt at a solution and a
“reasonable approximation to a richer kind of consistent-expectiations formation.” (al-Nowaihi and Levine,
1985).

96

6% 3P1Q1+ PBZPIQ]
aqpaqs = 345 aqpaqs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 2 <1 (IV-9)
a mp 811191 a Pin
——5 2_+qP—7
(aqp) 34p (aqp)
2 2
a w (WS+WP)3P[Q1+(WPqP+WS 48) a P[Q] WC awe;
aqsaqp = P (8458”) (34") am 10)
32w 8an BZPIQl Moi '
2w +(w q +w q ) +WC
(aqs)2 5 34s P P S S (aqs)2 (3qu

 

If we assume symmetry, we have BPIQ] = BPIQ] ,
345 3411

BzPlQ] = azPlQ] :8211913210in and 3V1Q1= aVIQi
(34536111) (3411345) (3492 (34p)2

 

2 2 . It follows than that
(345) (MP)

the denominator in equations (IV.9) and (IV.lO) has to be larger than their respective
numerators for the stability condition to suffice. While the private firm’s reaction

function is always stable, the public sector firm’s reaction function is stable only for

“’5 > we but stability cannot be guaranteed for W5 5 wp.

Consider next the first stage of this game. The private firm, having chosen its

output that maximizes profits in the second stage (conditional on research level x,- ), now

will choose the research level that maximizes profits given its equilibrium output function

of xi. That is in the second stage it maximizes

max 7:; = (P[Q*[xp,x5 ]]-c[xp,xs ])q;[xp,x5 ]— Rp[xp] (IV.l l)
JrP

Similarly the public sector firm maximizes its welfare in the first stage given its output

choice in the second stage:

97

max W* : wCV[Q*[x5 , Xp]]+
xs

wS((P[Q*[x5,xp11~csixs,xp1)q§[x5,xP1—Rs[x51)+ (1v.12)
WP ((P[Q*[xp,x5 ll-CPIXP.X5 ])q;lxpixs ]— Rplxpl)

Applying the properties of the envelope theorem, the first order condition for the

maximization of the private and public objectives in the first stage simplify to:

 

P MJPP (-—PiQi clx2.xs*1)PPP[xP’PS]=Ri=txi>l (IV-13)
ax!) dxp

 

‘WS WC] "‘ WW]- CSIXS 35M)?
8X5 8x5

(IV.l4)

q*

1)]: WsRslxsl
3x5

 

+(PIQ l— cpixp.x§i)3:

dc [x , :1:
WP[ Pax:x5]4P

We also have the second order conditions for each firm which take the form:

 

 

Bop] BqPP =11 19ch dqu
__ __1 _ +P[Q*] 2"—R [16" 1<0 (1v.15)
P axp [asp ] 4,, am.)2 aunt.)2 P P

365 fig; 11 326's * 324.;
‘— —"1 — +P [Q 1_ '-
WS [P 3x5 ][axs ] qs 3(xs )2 8(x5 )2

* 2 2 *
WP —-aC—P fli—i —q; a PP +P[Q*1a 4P2 —R§[xs1<o
3x5 3x5 ( )2

 

(IV.l6)

 

 

The first order conditions (equations (IV.l3) and (IV .14)) simply state that the

two firms conduct research up to that point where the marginal cost of doing research

(R;[x,-]) is equal to the marginal benefits that accrue to the each firm. In the case of

private firm these benefits are its cost saving due to research and its increased second

stage profits brought about from a change in the quantity supplied as a result of the

98

R&D.6 The marginal benefits to the public sector firm, however is the aggregation of
marginal benefits due to its research in the second stage on both itself (i.e. the public
sector firm) and the private firm. This is a natural outcome of our assumption that the
public sector firm is a welfare maximizer that takes into account the quantity and research
output of all firms in maximizing its objectives.

Stability is also required in the second stage (Henriques, 1990) and takes the form

 

 

 

 

 

 

8271';
8111:3115 =| NP |<1 (IV.l7)
a ”P |socp
(dxp)2
82w"
axgaxf =|WSN5+WPNP|<1 (IV.18)
a W I SOCS |
(3x5)2

 

 

 

* 2 2 *
where N,- 2 —8i £9gL—l —q;-P a c, +P[Q*]—a—q'— for i=P,S i¢j and SOC,- is
x j 3x j 811in j 3x58): j

the second order conditions stated in equations (IV. 1 5) and (IV. 16). Without knowledge
of the nature of the Nash-equilibrium in the second (output) stage, we cannot claim that

the stability condition is satisfied. However we note that if the Nash-equilibrium is

. * * . . .
symmetric such that q p[xp,x5] = qS[xp,xS ] , then the reaction functions of the private

and public sector firm are stable in the R&D space as R;[x,-] > O for i = P, S .

 

6 Notice that if the second stage were a competitive market such that P[QP] = C5 [x5 , 117;] , the marginal
cost of research would simply equal the cost saving due to research.

99

IV.2. 1 .2. Cooperative Game
Next we consider the case where the two firms coordinate their research activities
in the first stage but remain competitors in the second stageP. When all firms have the
same profit-maximizing objective, the firms simply choose that level of research that
maximizes their joint profits. In the context of mixed duopoly, however, the notion of
joint profits is inappropriate as the two firms have divergent objectives; while the private
firm still seeks to maximize profits, the public sector firm maximizes welfare. However,
if one assumes that firms maximize their joint objectives, where joint objectives are
simply the summation of their individual objectives (i.e. profit for the private and welfare
for the public sector firm), then we can define joint objectives of the two firms, as a
function of xp and x5 , in the first stage as:
JP = W*[xp,xs]+ wjlr;[xp,xs]

= WCV[Q*[xp,xS ]]+ wSfl;[xP,x5 ]+ (WP + w] )Ir;[xp,x5] (IV. 1 9)

= wCV[QP[xp,x5= ]] + wslt;[xp,xs ] + wJplz’pr, x5]
where w, is the weight of private firm’s profits in joining the cooperative agreement.

We see now that having parameterizing weights allows us to analyze different forms of
cooperative arrangements. One could consider, for example, the case where the private
firms profits are weighted more than that of the consumer surplus or the profits of the
public sector firm (the weight will certainly be greater than if the private firm did not
collaborate). At the other extreme would be the case where the weights are skewed

towards the profits of the two firms, in which case the joint objectives would be in line

 

7 Since the second (output) stage of the market game is non-cooperative and therefore unchanged, we
restrict our analysis to the first stage which is now characterized by cooperation.

100

with the concept of joint profits considered by d’A&.l and Suzumura (for example,

WC = 0 and W”) 2 W5 in equation (IV.l9)).

The private firm chooses xp and the public sector firm chooses x5 in

maXimizmg J . Assuming an interior optimum for each firm, this means that a Nash-

Coumot equilibrium must satisfy the two first order conditions in the R&D space:

P :1: =11 *\ 1
_,,,S wq§+(P[Q*]—cs[x§,xp])aqs —
3x5 ()ij

=1)

8 * , * :1: =1: =1: :11 a
WJP CPIXP Xslqp+(P[Q I’CPle1xsl) qP
3x5 8x5)

 

.—.. wSR'SixS1 (1v.20)

 

 

 

 

 

 

P dc [x* x*] =1: =1: e a PM
—ws[—ia—Si-q§+(Pio l-cslxs,x1o]) ‘15 —
x O
P PP) =WJPRP1xP1 (iv.2i)

 

 

 

 

 

a *, * i. a s i. a *1
W”) CPle x5]qp+(P[Q ]—CP[xP1xS]) (111
Exp 83pr 1

The F.O.C. are similar to public sector firm’s F.O.C. in the non-cooperative game
and imply that research occurs at that level where marginal costs of research are equal to
its marginal benefits. The key difference resulting from the joint maximization of first
stage objectives is that both firms take into account the benefits of its research on that of
the other firm. Whereas the public sector firm was previously internalizing those benefits
in the non-cooperative case, the fact that R&D in the first stage is now combined, the
private is forced to internalize the benefits of its research on the public sector firm as

well.

101

IV.2.2. Regime 2: Mixed Duopoly in R&D Stage Only
In this section we consider a two-stage model of mixed oligopolistic competition
but in contrast to the previous section, we assume here that the public sector firm does

not engage in any production (i.e. ‘15 = O ). Instead we allow for quantity competition to

occur between two private firms in the product market (second) stage. In the first stage,
the two private firms, along with the public sector firm, decide on their cost-reducing
R&D either cooperatively or noncooperatively. The incentive for the public sector firm to
perform cost reducing R&D in the absence of any production, results from the spillovers
of its research to the private firms. To the extent that private firm’s benefit from public
sector firm’s research spillover, public welfare is increased and hence the public sector

firm’s welfare is advanced.

IV .2.2. 1. Non-Cooperative Game

In the non-cooperative case, no firm cooperates with any other in either of the
stages. Since only private firms are engaged in quantity competition in the second stage,
the standard Coumot result follows. That is, in the second stage, the two private firms

(denoted P1, P2) maximize their profits:

max 2:, =(P[Q]-c,-[x,~,xj,x5])qi -R,-[x,.1 (IV.22)
(It

for i,j=Pl,P2 and i¢j
where, as before, cost is a function of own research x,- , the research of the other private
firm x j , and the research of the public sector firm x5 . Assuming an interior optimum for

each firm, the Nash-Coumot equilibrium must satisfy the two first order conditions:

102

a”itch-11111

aq = Pio*l+ P'iQ’lqE’ —c.-1x.-,x 11xs 1 = 0 (N23)

i
Writing in terms of market elasticity of demand,

:1:
P[Q*l[1+iP—]=cilxg,xj.xsl (IV.24)
e
In the previous stage, the two private (and symmetric) firms join the public sector

firm in choosing R&D levels. The payoff to the two private firms and public sector firm

at this stage can be written as

It:=(P[Q*[xi,xj,x5]]—ci[x,-,xj,xs])q:[xi,xj,xs]—Ri[x,-] (IV.25)
* * Pl,P2 *
w =wCV[Q [xi,xj,x5]]+ Z wirri —R5[x5] (IV.26)

l

The first order conditions are

 

 

a ' 'Pl, *3 * :1: :11 It: at: at: a: P" *3 PP ' *
[_ cilx. x} xs]]qi +(P[Q ]—c[x,-,xj,x3) 41x. x] xs]=R,-[xi] (IV.27)
dxi axi

*

aqi

 

 

-Wi qt +(PIQPI-cile,xj.x§]) = R'sixs1 (IV.28)

PI’PZ aci [x:9x;,x;] :1-
8x5 8x5

As before, the first condition (IV .27) indicates that for each private firm the marginal
benefit from a unit of research should equal the marginal research cost, where marginal
benefit consists of the marginal cost saving due to research and the (commensurate)
increase in second stage profits. Equation (IV.28) indicates that the public sector firm
chooses a research level such than the weighted marginal (decrease in) total benefits from
research equals the marginal (increase in) the cost of doing research to itself. If one

compares these condition with the corresponding one under the non-cooperative case

103

(IV.l3) and (IV. 14), one observes that the quantity supplied by each firm is symmetric
when no production by the public sector firm takes place and that how much the research
the public sector firm undertakes is a function of how much public research is “used” by

the private firms. In the extreme case if R&D by the public sector firm has no effect on

the unit cost of the private firms such that do,- / 8x5 2 aq, / 8x; = 0 then x; = O.

IV.2.2.2. Cooperative Game

In Regime 2, cooperation implies that the public sector firm jointly undertakes
R&D with one of the private firms. Because of symmetry it does not matter as to which
private firm the public sector firm collaborates with, and henceforth we simply assume
that the collaboration is with P1 . The joint objective maximization is therefore much like
(IV.l9)with the profits of the collaborating private firm receiving a higher weight than

the case were it not to collaborate (for w] > wp] ). The joint objective maximization

problem of the public and firm P] in the first stage is

max W": = wCV[Q*[xi,xJ-,xs ]]+(WPI '1' WJ )fl;] '1' szfl;2 “RsLXSl
xS’xP‘ (IV.29)
* * *
= WCV[Q [xi1xj1xS 11+ WJPlflPl + WP2”P2 ‘Rslxsl

The payoff to the non-collaborating private firm (P2) remains unchanged from its payoff

in the non-cooperative case (i.e. equation (IV .25)) and is retained for i = P2. The first

order conditions from the maximization of (IV.29) and (IV.25) (for i = P2) are:

*

=1: 11: a - 1 a:
q,- +(PPQ 1‘61“) q, ]szPlRPllell (IV.30)
Pl Bxpl

dc,-
( WJPl Wm); ax

 

 

104

a . a a a i“ . i.
(_WJP1-WP2)Z[5::qi +(PlQ ]‘Ci)5:—P]=Rslxsl (IV-31)
i s

 

 

a :1: =1: 3 * ' *
[- 86112]qu + (P[Q l—sz) qPP = szlxpzl (IV-32)
xP2 3"?

Equations (IV.30) to (IV.32) determine, respectively, the choice of research level
for the cooperating private firm, public sector firm, and the non-cooperating private firm.
The private firm that collaborates with the public sector firm is now takes into account
the benefits of its research on all firm and chooses R&D where the total (weighted)
marginal benefits equal the (weighted) marginal costs. The private that does not
collaborate does not have to take into account the extemalities of its research on others
and chooses R&D where marginal benefits from doing R&D equal its marginal cost of
that R&D. While the marginal benefits of doing R&D for the cooperating firm are higher,
as it takes into account the marginal benefits of its rival), the cooperating firm may also
incur a higher marginal cost of R&D (due to the weight). Therefore, it remains
ambiguous whether the cooperating private firm performs more R&D than the other (or
vice versa), as the equilibrium R&D level is indeterminate. Table IV.2 provides a

summary of the key results of this section.

105

Table IV.2: Summary of Results for §IV.3.2.

 

Regime 2

 

Regime 1
Output Mkt. . W5 > WP
Non-
cooperative 0 Pvt. firm produces where MR = MC
solution 0 Public sector firm produces where

marginal consumer surplus equals total
marginal profits

0 No production by public
sector firm

0 Pvt. firms produce where
MR = MC

 

 

R&D Mkt. 1- Pvt. firm does R&D where marginal
Non- benefits (MB) equal MC
cooperative 1- Public sector firm does R&D where
SOlUtiOD (NC) weighted total MB equals MC

Cooperative 1- Public and private firms do R&D where
Solution (C) weighted total MB equals MC

 

 

Same as Regime 1 but with
two private firms

0 Public and C-private
firms do R&D where
weighted total MB
equals MC

0 NC-Pvt. firm does R&D
where MB = MC

 

IV.3. Specializing the Model

In this section we shall specialize the games of section 2 to provide us with a

sharper charecterization of the equilibrium conditions. In so doing, it will develop a

systematic methodology for analyzing the properties of a two-stage model of mixed

duopol y.

IV.3.]. Regime 1: Mixed Duopoly in Both Stages

The public and private firms face and inverse demand P[Q] = a —bQ where

Q = q P + qs is the total quantity produced and a,b > 0. Let the cost of production of

each firmbe cilqi,xi,xj]=(A—x,-—,6jxj)q,- for i=P,S i¢j.We assume that

0 < A < a and x,- + ,Bjxj S A. The flj parameter signifies that there are extemalities to

R&D, and by specifying a different spillover parameter for the two firms we are allowing

106

 

for asymmetries in spillovers to exist. The returns to research are assumed to be

diminishing, and hence we specify the cost of R&D as quadratic. As stated earlier, each

firm’s strategy is to choose a level of R&D (x,-) and a subsequent production choice

based on their R&D choice. We examine the cooperative and non-cooperative games

IICXI.

IV.3. 1 . l. Non-Cooperative Game

The profit of firm P and the welfare of firm S at the second stage, conditional on

R&D levels (xp,x5 ) are, respectively:

2
”P =PlQIqP 'chP — 7%) (IV-33)

The public sector firm maximizes social welfare where social welfare is represented by
the summation of consumer and producer surplus. For a linear demand function and

constant per unit costs ci , i = P, S , the public sector firm maximizes welfare choosing

qS
be2 1,,
W=WC -2— +WPflP+W57TS ( .34)

where ftp is the private firm’s profits as specified in equation (IV.33),

2
Its =P[Q]q‘(,= —c5qs —/i£25— is the public sector firm’s profits, and sz / 2 is the

consumer surplus. From the first-order conditions for a maximum for the private and
public sector firms, we derive the following reaction functions in output space, taking

R&D levels as given:

107

a“(15“CP

IV.35
2 b ( )

4101615] =

W5(a-C5)—(Wp +w5= —wC)bqp
(2W51 —Wc)b

 

qslqp1= (IV.36)

Whereas the second order condition for the private firm is always satisfied

(82717; / aq}; = —2b S 0 ), the second order condition for the public sector firm requires
that BZWI Bog = b(wC — 2wS) S O or WC S 2w5. Further, for the equilibrium to be stable
it suffices that |aq,- / Bq 1| < 1 (Henriques, 1990; Seade, 1980). The private firm’s reaction

curve in the output space will always be stable in this example, as ldqp / aqsl = l/ 2. The
stability condition for the public sector firm is different and requires that

aqs _WP+WS_WC|

a(1111

 

<l which is true for W5 > W,» and VwC or 2wC > 3W5 +wp.

 

 

 

WC -2WS

Taken together, the second order condition and the stability conditions imply that
3 + . . . . .
2W5 2 WC > —KS—2—Y£. Therefore for a stable eqmllbrlum to exrst, the public sector

firm’s profits must be given a higher weight than the private firm’s profits in the its

welfare maximization problem (W5 > we ).

From the equations (IV.35) and (IV.36), the Nash-Coumot equilibrium is

 

 

computed to be:
:1: w a—c -w a—2c +c
quxpixsl= C( P) S( P S) (IV.37)
(wC+wP—3ws)b
11: w —w a-c —w a—2c +c
qslxp,xs]=( P C)( P) S( S P) (IV.38)

(WC '1' WP - 3W5 )b

108

Notice that if the public sector firm maximized profits instead of welfare, such that
WC = wp = 0 , then the Nash Coumot equilibrium is completely symmetric.

At the preceding stage, define the first-stage payoff function of the private and

public sector firm’s as

2
air at: X
zrplxp.xsl = b(qP[xP,x5 l)2 — 77” (N39)
=1: x}; XE
W [XP,.Xsl=Fl[XP,Xsl-}’WP-§-—}’W57 (IVAO)

where Fl[xp,x5 ] , defined in the appendix, is the net benefits that the public sector firm
receives from the second stage, and q; and q; are the Coumot-Nash equilibrium
quantities. Then, the Nash equilibrium of the first-stage game, denoted by [x;,x;] is
characterized under the assumption of interior optimum and second—order conditions by
alt;[x;a,x;]/axp = O and 8W*[x;;,x§ ]/8x5 = 0. As was the case in the output space,
stability is also required in the R&D space such that lax, / 8le <1 (Henriques, 1990). To

explore these stability conditions we follow Henriques (1990) and employ reduced form
reaction functions in the R&D space. Matters are considerably simplified if we assume

that in the maximization of its objective function, the public sector firm places a higher

weight to the profits of its firm such that W5 = 2 , and wp = W5 =1 . Recall that these

weights will satisfy the stability conditions in the quantity space, and will be used, from

this point forward.

109

From the maximization of the first stage profits (for the private firm, eqn (IV.39))
and welfare (for the public sector firm, eqn (IV.40)), we have the following reduced form

reaction functionsgz

(2,6,) —3)((a—A)+x5 (3fls -2))/8b
32717;) /B(xp)2

 

xplxsl = (IV.41)

((G‘A)(18+flS)—XP(26-44flp —27fls +26flpfls))/16b
t‘12W"‘/r'l(x5)2

 

x5[xp] = — (IV.42)

where the denominator in the above two equations are the second order conditions that

must be less than 0 for an equilibrium. Differentiating (IV.41) and (IV .42) with respect to

xp and x5 yields, respectively:

911: = _ (3 - 25100.35 " 2)

(IV.43)
8x5 (3— 2,6,.)2 —8by

 

8x5 _ 26—27fls +(26fls —44)flp
dxp 44+(27fls —52)fls -32b}’

 

(IV.44)

The absolute value of these functions must be less than 1 for the stability condition to
suffice. Notice that for sufficiently large by the second order and the stability conditions
for both firms is satisfied for all values of the spillover parameters. For smaller values of
by, appropriate restrictions need to be placed on the spillovers that would satisfy the
equilibrium concepts. Henriques (1990), for example, shows for the pure duopoly case

that when by=1 , the non-cooperative model is unstable for low levels of spillovers,

 

8 With “’5 = 2 , and WP = W5 = l , we note that the second order condition for maximization of the each

firm’s objective requires that (3 - 2,6,, )2 / 8 < b}’ (for the private firm) and
44 '1' .65 (27,55 - 52) / 32 < b}’ (for the public sector firm).

110

which results in comer solutions.9 However if one were to scale up the research cost to,

say y: 5 , then the stability condition is satisfied Vfli 6 [0,1] for i = P,S . Since our

interest in this paper is to explore how the equilibrium values are affected by the presence
of a public sector firm as well as the consequences of cooperative and non—cooperative

R&D with such a firm, we shall assume that by are large enough to satisfy all the

conditions for an unique and stable equilibrium.'0

Table AIV.3 gives the Nash-equilibrium values for the R&D (x: ), output
:1: =1: :1: , :1: =1: :1: =1: :1: =1: :1: .
(qi [x5,xp] ), private firm’s profits Itp[qp[x5,xp],q5[xs,xp]] and public sector firm’s

welfare W*[q*p[x;,x;],q;[x;,x;>]] for different combinations of spillovers. For the
simulation we assumed a =10 , A = 7 , b =1, and y: 5 , but the comparative static

results that follow can be generalized for large range of values for a > A > 0 and
by>5”.
From the inspection of the simulation results we have the following comparative

static relationships:

1. VfliEIOJI, x; >x; and q; >q;;.For wC=wP=l and WS =2,itcan be

verified from equations (IV.37) and (IV.38), that q; > q; VB, 6 IO,1|.

 

9 . . ax, -2(2 - p)(2p- l)/9b
In the pure du0poly, non-cooperative case of d’A&.l and Henriques, — =

ax. (2(2—,B)2/9b)—y ’

J

 

which is unstable for fl < 0.17 and b = 7 = 1 .
'0 In the simulations that follow, we have assumed that 7 = 5 .

" For 0 < by < 5 , restrictions on the spillovers need to be placed so that an interior solution is obtained.

11]

2. Private firm R&D (.1173) , output (q?) , and profits (17;) are all decreasing in own

ax}; at); an;

3.511 PM!) P 315111

 

spillovers. That is < 0 , which implies that as appropriability of

the returns to private R&D declines, the private firm decreases its research. Public

*

 

sector firm output (qp) is increasmg in private firm spillovers 8‘15 > 0. With
P

higher levels of spillover, the public sector firm faces a lower unit cost of
production allowing it increase the quantity supplied.

3. For low levels of public sector firm spillover (,65 S 0.75 ), public research is
increasing in private firm spillovers. For high levels of public sector firm research
and high levels of private research spillovers, public research is decreasing in

private firm spillovers. For example when ,65 = 0.95 , then an increase in private
spillover from ,6]; = 0.75 to file = 0.95 results in a decline in public R&D (from

0.416 to 0.415).

4. Private firm R&D (x?) , output (q;) , and profits (12;) are all increasing in

:1: 11: :1:
Exp aqp aflp>0.

313$ ’afls P 3,55

public sector firm spillovers

5. The affect of public sector firm spillovers ( ,BS ) on public sector firm’s welfare is

ambiguous. We note however that with increasing public R&D spillover total
output (and by extension, consumer surplus) is increasing, whereas the
equilibrium price is declining.

Our interpretation of these results is as follows. Profit-maximizing firms react to a

loss of appropriability by decreasing their research effort. Our result is consistent with

112

this behavior; the easier it becomes for the public sector firm to employ the private firm’s

research (i.e. increasing flp ), the incentive to conduct R&D declines for the private firm
and hence a decline in equilibrium private firm R&D. The reaction of the public sector
firm to an increase in spillover is ambiguous; public R&D declines with increasing 165
but increases for very high .55 . This result is similar to the one observed in Chapter II

where, for specification 1, the decrease in appropriability of public research resulted,
initially, in a decline in public R&D but for very low appropriability of public research
( a >0.89), equilibrium public research increased public R&D increased with increasing

inability to appropriate the returns.

A decrease in the appropriability of public research (increasing 165 ), has a
positive affect on private firms R&D as well as total R&D. This implies than any decline
in public R&D is more than compensated for by increase in R&D by the private firm. In
the output market, we note that the quantity supplied by the public sector firm is
declining unambiguously in ,65 , while the quantity supplied by the private firm is
increasing in ,65 . Since total output is also increasing in ,65 , the decrease in the output
of the public sector firm is also more than compensated by the output of the private firm.

That the public sector firm’s output and profits (Its) unambiguously declines
with fls while its research does not may seem surprising. One would expect that an
increase in public sector firm’s R&D due to increasing .55 , would imply a proportionate
increase in output as the marginal cost of production is decreasing. For example, when

,35 increases from 0.75 to 0.95 in Table AIV.3, the marginal cost for the public sector

firm declines but its output (qS ) and profits (71’s ) are not increasing. The reason for this

113

is that with such high amounts of leakage of public research occurring, the private firm
benefits more and is able to reduce its marginal cost much more than the public sector
firm can. With a bigger decline in its marginal cost, the private firm is able to increase its
output, taking away some of the market share of the public sector firm. The equilibrium

results being that the private firm sees an increase in its output whereas the public sector

firm sees a decline. Equilibrium profits of the public sector firm (It; ) decline since it

incurs a higher research cost, the benefits of which it is unable to appropriate due to

higher spillovers.

IV.3.1.2. Cooperative Game
In the second game, under the regime where the public sector firm participates in
the second stage, the public and private firms cooperate in R&D, while remaining
competitors in the output market. The firms maximize their joint objectives, defined
earlier, as a function of R&D. In the context of the present market structure, joint
objectives are defined as:
1* = W" + at;
= wCV[q*P[x;>,x;],q;[x;>,x;a]]+ wsrtgtqillxio.1111.615471]]+
11111121411111.4111;[112.1111]+z11411x2.x}l,q§lx2.x7oil (1)145)
= wcviq;ix;,x;l.q§ [x;,x;11+ W5”;[qPPlx191xPPl1qglx1’1xPll+

* it II! 3|! * * *
WJ”P[qP[xP1xP]1qS[xP1xPll
where w, = wp +1 is the weight now given to the private firm in the joint objective

maximization. Since the second stage remains non-cooperative, we still require that the

stability condition be satisfied and hence retain the assumption that W5 > wp. However

114

in the first stage under cooperation a different set of stability and second order conditions

need to be satisfied for a Nash-Coumot equilibrium. The stability and second order
condition in this case are, respectively:

321*
dxidx j
821*
(axi)2

 

 

<1 for i,j=P,S andi¢j

 

 

 

2 *
a '12 S0 for i,j=P,S
(axi)

 

Both these conditions are satisfied for WC =1 and w] = “’5 = 2 and for sufficiently large

value of by (see appendix). As before, the following reduced form reaction functions are
obtained from the maximization of the first stage profits in this game'2

prxSl= (xS(38-52,6p —45,65 +238ppfl5)-27(a—A)(l+2fip))/16b (IV.46)
a J /d(xp)

Islxpl = (xP(38-5216P "15155+238fpfls);7(a-A)(2+fls))/16b (“’47)
a J /3(x5)

Table AIV .4 provides the simulation results for the Nash-Coumot equlibrium
values in the cooperative case for Regime 1. From the table, we observe that
Vfli el 0,] | q; > q; . Firm output is increasing in public sector firm spillovers.

* *
PPI—P,§P—P-S— > 0. Generally speaking public sector firm output is increasing in
353 3153

 

'2 Where we have assumed WC = WP =1 and W, = W5 = 2.

115

private spillovers, except when public sector firm spillovers are low (in which

*

 

 

 

                                                       

case it is decreasing). Industry output is increasing in spillovers 3% > 0
i
3x
2. Vfl-e axP P >.0)
' 31311 3.55

3. Private firm profits are increasing in public sector firm spillovers and generally
decreasing in own spillovers.l3 As such, the highest private profits are obtained
when there are low private spillovers and high public spillovers.‘4 Notice that at
high levels of private spillovers and low public spillovers private equilibrium
profits are actually negative. This occurs because the firm is not generating
enough revenue (due to lower output) to compensate the higher cost of R&D and
in spite of the lower unit cost of production. That is to say that the cost of R&D
has increased at a faster rate than the cost saving from production along with a
loss in market share

4. The affect of private and public research spillovers is ambiguous for public
welfare, although it is the highest at high levels of both private and public
spillovers.

The fact that public welfare is maximized when spillovers are high and the private
firm’s profits are maximized when private spillovers are low reflects the divergent
objectives of the two firms. When spillovers are high, both firms are not only able to
share each others research outputs, they also conduct more R&D. This lowers their

marginal cost of production allowing them to increase their output. The industry supply

 

'3 Except when public spillovers are low in which case private research is increasing in private spillovers.

'4 For example, flp = 0, 163 = 0.99 we get 71'; = 0.92706
1 l6

curve shifts to the inelastic portion of the demand curve where the consumer surplus is
greater and the profits to the private firm lower. Since the gains to the consumers are
higher than the losses to the private firm, public sector firm welfare is higher.

How do the cooperative and non-cooperative cases compare? The key difference

is in the way xp changes with respect to ,Bp; whereas it is decreasing in the non-

cooperative case it is increasing in the cooperative case. The joint objectives that arise
from cooperation imply that the private firm is now forced to internalize the consumer
benefits that arise from research. In some sense, the R&D stage is characterized by one
public sector firm, but with the greater weight being given to the profits of the producers
then the welfare of consumers. Whether it is advantageous for the private and public
sector firm to cooperate is function of how easy it is for the firms to imitate each other’s

R&D, which is captured by the spillover parameter. Table AIV.5 provides a matrix for
comparing key equilibrium values for total R&D ( X * ), private firm profits ( It; ),
consumer surplus VP , and public sector firm’s welfare between the two games and at

different level of spillovers. From Table AIV.5 we see that cooperation always increases

public sector firm welfare, regardless of the spillovers between firms. This is not the case
for the profits of the private firm, as profits (7!; ) are higher only if the spillovers are low.

That is when the research of the public and private firm is easy to appropriate (low 51')

then the private firm will obtain higher profits from cooperating than from not
cooperating. For most other levels of appropriability, however the private does not gain
much from collaborating. The intuition behind this result is that by cooperating with a

public sector firm, the private firm acts less like a profit maximizing firm and more like a

117

welfare maximizing ‘public’ firm. Since we have established that a public sector firm
(generally speaking) increases its R&D with higher rates of spillovers, the private firm

also will increase its R&D as ,3}: increases. The increased R&D it conducts in advancing

the joint objectives, however come at a cost of decreased private profits. That is the
private firm, by collaborating with the public sector firm, over-invests and beyond the
point where a profit maximizing firm normally would when spillovers are high. If

spillovers are low, the private and public sector firm gains from cooperation because they

are able to lower their R&D expenditures (xg < x}! C for low spillover) mitigating the

over-investment problem when research is appropriable.

IV.3.2. Regime 2: Mixed Duopoly in R&D Stage Only

In this section we specialize the model for the case where the public sector firm
does not compete in the second (output) stage and is only a producer of research in the
first (R&D) stage. Two symmetric private firms that compete in quantities in the second
stage undertake production. The two private firms also conduct (production) cost
reducing R&D alongside the public sector firm. Spillovers from research exist between
all three firms (two private and one public), such that all firms benefit from each other’s
research. As in the earlier case we analyze two types of games, first where all firms
conduct research non-cooperatively, and in the second we allow for cooperative behavior
between one of the private firms and the public sector firm. To allow for comparisons
between this market structure and the one discussed earlier, we assume the same cost

function and inverse demand of the earlier market.

118

IV.3.2. 1. Non-Cooperative Game

The second stage of the non-cooperative game is exactly as the one that appears in
d’A&J, except that in the present case the private firms also benefit from the spillovers of
a public sector firm. The profit of firm i at the second stage, conditional on first stage
R&D (x,- for i = Pl,P2,S ) is expressed as

2

711 =Pthq. -c.-q.- - 7%— (N48)

for i=Pl,P2 and i¢j.

where P[Q] = a —b(q,- +qj) is the inverse demand, ct = A —(x,- + ,Bpxj + ,Bsxs) is the

unit cost of production and (y/ 2)x,-2 is the cost of R&D. We assume that spillovers

between the two private firms are symmetric but allow for difference in spillovers
between the public sector firm and the private firm. The reaction functions in the output

space, taking R&D levels as given is

a—bqj-c,

V.4
2b (1 9)

qilqjl=

for i=Pl,P2 and i¢j.
As indicated by Henriques (1990), these output reaction functions meet the stability
conditions because of the assumptions on the cost, market demand and that R&D levels
are given. The Nash-Coumot equilibrium is computed to be

qf=a-2ci+cj (N50)
1 3b .

for i=Pl,P2 and i¢j.

119

In the first stage, we introduce a public sector firm that along with the two private firms
chooses R&D levels. The private firms choose that level of R&D that maximizes profits,
whereas the public sector firm maximizes welfare. Profits and public sector firm welfare
in the first stage are written as:

2

:1: at: x-
7t,-[xl-,xj,xs]=b(qi[xi,xj,x5])2—77’ (IV.51)

* * *
W21XP1Xsl= V[QPllel1xP21xS 1141111le1 =xP2=Xs 11+

P1112 . . .. x;
Z ”i [4P11Xp1ixpzixsliqP1lxp1ixp21Xsll‘77 (IV-52)
i
2 2 2
x x x5
=F21xPl1xP21x51_7%L‘7‘:_2_7’-2_

where F2[XP1,XP2,X51 is defined in the appendix. In the R&D space, the following

reduced form reaction functions are derived from the first order conditions for a

maximum of equations (IV.51) and (IV.52):

2(2—,6p)(a—A+xj(2,ap +l)+,BsxS)/9b

 

x-[x -,x5] = — * (IV.53)
I J 3272}- /(8x,-)2
for i=Pl,P2 and i¢j.

xsleltxP21= _4fls (2(0— A)+(XP1+XP2)(1+flP))/9b (W54)

 

32W; «at,»2

120

Assuming that the second order conditions are satisfied and that model is stable”,

there exists a unique solution satisfying an: lax,- =0 for i: Pl,P2 and i¢ j and

aw*/axS :0 for which:

11 (a— NO" .BP)

 

xi_ (IV.55)
4.—5by (2-flp)(1+flp) 4(fls)2
for i = Pl,P2
x;__ _—3fls(a A) (IV.56)

 

From the equilibrium values of R&D, we note that private research is increasing
in public spillovers and decreasing in private spillovers (Bxi ldflp < Oand ax,- /8,BS > 0 ).
Public research also is increasing in public spillovers 8x5 lafls > 0; public R&D is
however decreasing in private spillovers greater that half and increasing for spillovers
less than half(dx5 /a/3P < 0 for ,6]; > 0.5 and 3x5 /a,6,, > 0 for ,6}: < 0.5 )‘6. Our
simulation results (Table AIV .6) confirm these comparative statics, and we also observe
that firm output profits and public welfare are also increasing in public spillovers. Note

that firm output declines for flp > 0.5 , but firm profits are unambiguously increasing in

 

827:7 _ 2(2- 13,.)2

 

—}’S0 for i=Pl,P2 and

'5 The second order conditions are

 

 

 

 

 

 

13(11)‘ — 9b
2 1 2
8 W22 = 8(fl5) -}/SO. The stability conditions require that 85x —I 4165 (1+flp) )<I
8(xs) 9b 3x, WI—8(fls) +9byI<
Pi _ _ 2,6.(2—13.) |<1x._—|2(2—-fl,.)(l-2fl.)l
- 2 and:
3x5 2(2-,6,,) -9by| | 2(2- [3,.) —9by|<

'6 These comparative statics are given in the xappendix.

121

the private spillovers. The increase in producer surplus (industry profits) are however not
large enough to offset the losses in consumer surplus (due to declining output) which

occurs at higher levels of spillovers and is the reason as to why public sector firm welfare
(W§) declines at high spillovers.

When ,65 = O , the non-cooperative game under Regime 2 corresponds to the non-

cooperative game of the d’A&J. It is clear that the presence of a spillovers from a public

55 =0 )5

. . . >0
sector firm increases private firm research and output (i.e. xi < xi 5 and

qu=0 < (1,135 >0 for i = P1, P2 ).

IV.3.2.2. Cooperative Game

Assume now that the public sector firm and one of the private firms join to
maximize their joint objectives by choosing R&D levels in the first.'7 The second stage
remains unchanged, and the two private firms continue to compete in quantities, given
their first stage R&D. As was the case with Regime 1, ‘joint objectives’ imply that the
profits for the collaborating private firm are given a higher weight than the case where it
did not collaborate. Since we have implicitly defined the public’s welfare function to be
equally weighted (and equal to one) across consumers and producers (see equation
(IV.52)), the implication of cooperation is that the weight on the profits of the
cooperating firm (P1 in this case) is larger. The joint objectives of the cooperating firms

as function of R&D is than

 

'7 Because of symmetry, it does not matter here as to which private firm the public sector firm cooperates
with. We shall assume that the cooperating private firm is P1.

122

* * *
J [xP1x51=VIQPIIXPI1xP21xSI1qP2IXP11xP21xS 11+

* * *
WP175P1 [4P1 [xPl 1xP21xS l1qP21xPi 1xP2=xS 11+

* * *
7511214101le1 1XP21Xs 114112111111 1111121315 1] (W57)
_,:§_
2
2 2
x 2 x
= F3IxP11xP21xSl‘7%‘7xP2 —7’-2i

where WP] = 2 and F3 is defined in the appendix. The first stage profits of the firm that

does not cooperate with the public sector firm remains the same and are given by
equation (IV .51), only here it is just for P2 . As in the earlier cases, we derive the

following reduced form reaction functions in the R&D space:

((4+flp)(a -A)+xp2(ll—flp(32-1lflp)—2flsxs(4+pp))/9b

2 1 2 (IV.58)
8 J2 /d(xpl)

 

* *
J"11111111211151:-

2(2-flp)(a‘A+xPl(2flp +1)+flsxS)/9b
327!” /(dxp2)2

 

xp2[xpl,x3]=— (IV.59)

2,65 (5(a-A)+xp2(l+4flp)+xp](4+flp))/9b

2 , 2 (IV.60)
a .12 /(axs)

 

ISIXP11XP2]=‘

Assuming that the second order conditions are satisfied and that the reaction are
stable in the R&D space, we have the following unique solution for the equilibrium with

cooperation in this case:

 

 

le = 2(0 - A)(6-flp(1 1— flPUDI- flp(6-flp)+b}’)-4b}’ (IV.61)
$2 = 2(0’A)(2-16P)(3;)flP(4"16P)-2b7) (IV.62)
x; = 2(a-A)fls(flp(26-1 Vim-151110111) (N63)

 

D

123

where D is given in the appendix. Table AIV.7 presents the simulation results for the
equilibrium values for different levels of spillovers. Comparing across equilibrium
research levels for the three firms (the above equations and Table AIV .7), it is not clear
as to which firms conduct the most R&D or whether there are unambiguous benefits for
the private (in terms of profits) for cooperating with the public sector firm. From our

simulation we observe that:

1. VB; 6 [0,1] (for i = P,S ), qPPl > q;2. Further firm output is increasing in public

spillovers. aq: labs for i = P1, P2,S . By cooperating with the public sector

firm, the private firm produces above the profit maximizing level as it intemalizes
the gains to consumers. As such the quantity produced by the private firm if it

acted non-cooperative] y is lower than that produced when it acts cooperatively
(i.e. q}? > 4331’")

2. For all three firms, firm R&D is increasing in public spillovers (Bx: / ops > 0).
Whereas the R&D of the cooperating private firm and the public sector firm are
increasing with increasing private spillovers, it is decreasing for the non-
cooperative firm ( 8x: laflp > O for i = P1,S and 8x122 ldflp < 0). The industry

unit costs of production are lowered when more R&D is done and shared by as
many firms as possible. The cooperating firms would prefer this state as it
increases output and their joint objectives (even though it comes at a cost of lower
profits for the firm that cooperates). The non-cooperating firm however will
reduce its research level for higher private spillovers not only because it is able to

gain from the R&D of the other private firm (which is increasing its R&D) but

124

also because it gains (free-ride) of its rivals research. Interestingly the profits of
the non-cooperating firm are higher for higher private spillovers implying that

there are gains to cutting back its own in-house research and free riding off others.

3. Vfli e I0,l| (for i = P,S ), public welfare and the profits for the non-cooperating

firm are increasing in spillovers. For P1 , profits are increasing in public

spillovers ‘v’fls e I0,1I . Pl profits are however increasing for ,6}: < 0.5 and
increasing for flp > 0.5. Hence the reduction in profits for the cooperating firm

(for the BF > 0.5 cases) are offset by the increase in consumer surplus and the

profits of the non-cooperating, resulting in higher overall public sector firm’s
welfare.

In comparing the equilibrium outcome of the non-cooperative with that of the
cooperative case, we note that industry research and output are higher at all levels of
spillovers in the cooperative case (see comparative summary in Table AIV .8). The affect
of cooperation on profits is less clear. If both public and private spillovers are low, the
cooperating firm gains, as its profits are higher relative to the non-cooperative case.
When private spillovers are high, than the profits from cooperating are lower than they

are from non-cooperating. Notice however that a firm would make higher profits if it did

not cooperate than when it did at high levels of spillovers (i.e. for ,6}; > 0.5 ”$2 > 7:191) .

Generally speaking public welfare is higher in the cooperative case than it is in the non-
cooperative case. An exception appears to be for “low” private spillover and “medium”
public spillover whence the welfare in the non-cooperative case is higher. This may seem

somewhat counterintuitive considering that both consumer surplus and profits are higher

125

in the cooperative case than they are in the non-cooperative case. However since public

research is increasing in fig , so is the cost doing research. If the increase in the cost of

undertaking R&D is higher than the gains to producers and consumers, than public

welfare will be lower.

IV .4. Welfare
To put our results in perspective, we define in this section an efficiency criterion
to which the results are compared. We employ the d’A&J definition of social welfare

W[Q] as the sum of consumer’s surplus and the producer’s surplus”.

SW1Q1= Vin— CQ — 7X2 (N64)
where C = A — (1 + ,6)X is the unit cost of production. Given R&D, the socially optimal

output is given by

Q = (IV.65)

 

In the preceding stage, social welfare is now expressed as
:1! It 2 2
SW = b(Q 1X1) — yX (IV.66)

The social planner maximizes the social welfare function choosing R&D in the first.

From the first order conditions for a maximum"), the efficient level of R&D is

X11: = (a - A)(1+ fl) (IV.67)
2b7-(fl+1)2

 

'8 The social welfare here corresponds to the public sector firm’s welfare when no profit maximizing firms
are present. In this case the public sector firm’s maximization is equivalent to the social planner’s problem.

'9 The second order condition require that (1+ fl)2 / 4 < by

126

Table AIV.9 provides the simulation results for the socially efficient values of
R&D and output. It is easily verified that higher levels of spillovers result in more

production but also higher R&Dzo. Note as would be expected in the first best solution,

our simulation show that price equals marginal cost (P[Q*] = C ) when the social planner

allocates resources.

How do the cooperative and non-cooperative games of the two regimes compare
with the first-best solution? Table AIVJO provides a comparative summary of the
equilibrium output and welfare across the regimes, where, for the comparison to be
meaningful, we have assumed that private and public spillovers are equal
(flP = fls = ,5 )-

Our results confirm the earlier findings of d’A&J but also provide a new
perspective on how those results are affected by the presence and cooperation between
public and private firms”. Confirmation of the d’A&J result comes in the observation
that for “low” (symmetric) spillovers, noncooperation increases output quantities but at
high spillovers cooperation increases output quantities. Distinguishing between the two
regimes, the results reveal that the second best for output is due to Regime 1 in most
cases, except when spillovers are very high.

The comparison of industry research across the two regimes shows that for low
spillovers, the industry in the two regimes will over-invest relative to the first-best. Note

however that in all cases, the second best appears to be either the non-cooperative or

 

 

(a - A)((1 + W + 2hr)
2 2 > O
((1 +fl) - 2127)

2' Recall that the d’A&J analysis focused on the cooperative and noncooperative outcome between two
profits maximizing firms (i.e. a pure duopoly).

20 aQ‘/afl=x/b>0 and axgw/a:6=

127

cooperative equilibrium under the first regime. This result is in accordance with that of
the Chapter II, where we saw that the presence of public sector firm in an R&D race

tends to move the industry towards the socially optimal level of research.

IV.5. Conclusions

This chapter adapted the d’A&J framework to study the cooperative and
noncooperative equilibrium in the presence of a public sector firm that seeks to maximize
welfare. Our analysis was motivated by two issues, one positivistic and the other more
normative, concerning public and private sector partnerships The first issue is why more
partnerships among public and private sector R&D firms are not observed? A second
issue is whether the partnering of a public sector firm with that of a private one, will work
against the public sector firm’s ultimate objective of raising welfare?

We conducted our analysis first in a general framework where we found that a
sufficient condition for a stable equilibrium in mixed oligopoly is obtained only if the
public sector firm’s profits are weighted higher than that of the private firm in the
objective function of the public sector firm. The general model provides us with decision
criterion on how much output and R&D the public and private firms perform. A key
result of this section is that the private firm by jointly maximizing its objectives with a
public sector firm, conducts R&D at the point where weighted total marginal benefits
equals its marginal cost. As such, by internalizing the gains to its research that accrue to
itself and its collaborator, the private firm will conduct more R&D than it’s profit
maximizing level, assuming that there is perfect sharing of the innovation in the second

stage of the game which remains noncooperative.

l28

To provide more clearer and definitive results, we specialize the general model
using the framework of d’A&J. Specifically, we address the issue that while research
partnerships were encouraged as important element in the policy response to global
competition, the number of public-private partnerships has remained low, relative to
partnerships among private firms. The results of the model simulation provides why this
may be the case. If we assume that the public sector firm conducts both R&D and
production, as was the case in Regime 1, we find that profits from cooperating for the
private firm is not always unambiguously a Pareto improvement. Specifically we find
that for high spillovers, profits of the private when it collaborates with the public sector
firm are actually lower than the non-cooperative case (in Regime 1) or lower than the
firm that does not cooperate (in Regime 2). Hence the incentive to participate in a joint
collaborative effort will not be all that compelling for the private firm, when spillovers
are high. On the other hand when spillovers are low, then the profits for the private firm
from cooperating are higher than from not cooperating, suggesting that at very low levels
of R&D cooperation will take place.

Our finding that low levels of cooperation is due to high spillovers suggest that
the level of spillovers between private and public sector firms is an important ingredient
in the formation of research partnerships. That high level of spillovers may discourage
public-private partnerships, especially in agricultural research, is consistent with evidence
that agricultural research is characterized by the presence of large spillovers (Evenson,
2000; Fuglie et al., 1996).

Lastly we observed that under Regime 1, cooperation for the public sector firm’s

welfare was always a Pareto improvement. This result is also true under Regime 2, but

129

for a higher level of private spillover. This implies that collaboration between a private
and public sector firm does not compromise the objective of the public sector firm, and if
anything tends to increase it. Given that cooperation is a Pareto improvement for the
public sector firm and may not be for the private firm, the results suggest that private firm
be given incentives to undertake cooperative research. Such incentives could take the
form of subsidization of private research or that the profits of the private firm that
collaborates be weighted more heavily relative to consumer welfare. As long as the
decline in consumer welfare is offset by the increase in producer welfare, the public
sector firm’s welfare will be higher and the incentive for the private firm to cooperate

will exist.

130

CHAPTER V

CONCLUSIONS

The structural changes that have occurred in the U.S. agricultural R&D market
over the last two decades have served as the basic research motivation for this thesis. In
particular, we focused attention on the following observed facts: that a) there has been a
shift in the balance between public and private research activities, b) there has been an
increasing resort to intellectual property rights by both sectors and c) the phenomenon of
public and private research partnerships is on the rise, although not as common as one
would expect. The objective of this research was not to present a comprehensive
explanatory model of the agricultural R&D industry, but to provide a limited framework
for analyzing the effects and implications of the changing market structure through the
modeling of strategic micro-level interactions between firms in the two sectors.
Employing a mixed oligopoly framework, which aptly characterizes the nature of the
market structure in agriculture R&D, we presented theoretical models to investigate the
relationship of public and private research in the presence of appropriability and
spillovers (chapter II); a comparative analysis of non-competitive markets and where the
research has the attributes of genomics R&D (chapter III); and finally, an analysis of
cooperative and noncooperative R&D in the presence of the public sector firm (chapter
IV). Collectively, the results of this dissertation address the research questions that were
posed in the context of the recent structural changes in the agricultural R&D sector noted
in the introductory chapter. Next we summarize the key results and show how they relate

to the issues that we sought to explore.

l3]

In chapter 11, we show that the differential amounts of research being performed
by the public and private sectors can be understood through the modeling of
appropriability and spillover of public research. Our results demonstrate that if the
appropriability of public research is low, such that the private firm can easily imitate the
innovation of the public sector firm, the private sector loses the incentive to conduct its
own research. However, since the public sector firm is a social welfare maximizer, the
appropriability condition does not affect its behavior. As a welfare maximizer, the public
sector firm is only concerned with total welfare and not its distribution among the
producers. With total welfare remaining unchanged, the public sector firm’s response
function is therefore unaffected. This is in contrast to a private firm which will cutback its
R&D effort if the returns to its research are not appropriable (Poyago Theotoky, 1998).

The shift in the reaction curve of the private firm due to appropriability of the
public sector firm’s research will, of course, change the equilibrium level of R&D. While
the equilibrium research of the private firm is always lower as a result of lower
appropriability of public research, the effect on the public sector firm’s equilibrium R&D
is ambiguous. The ambiguity, however, provides us with another interpretation and
extension of the D&D and P-T results. A key finding of D&D was that a mixed duopoly
is welfare superior to the pure duopoly in spite of the distortion that prevails due to over-
investment in R&D. The comparable result in the P-T model is more ambiguous as social
welfare can be higher or lower than the private duopoly. In our analysis, the
parameterization of the appropriability condition allows for calculation of the first best
solution, whereby the mixed duopoly is not only welfare superior to the pure duopol y but

is equal to the welfare obtained by a social planner. This implies that regardless of the

132

value of the innovation, and as long as the public sector firm can choose the level of
appropriability of its innovation, the public sector firm can restore the correct incentives
for R&D.

We show that if we extend our analysis to account for private firm research
capacity, the D&D result can be confirmed and made more robust under a set of
sufficiently strong assumptions. Under these assumptions the private firm’s reaction is
unambiguously upward sloping and rises with increasing appropriability of the public
sector firm’s research. The D&D result is also confirmed if we assume that R&D
spillovers occur during the course of the race (as opposed to the appropriability condition
which are essentially spillovers after the conclusion of the race).

The implication of this result is that, in the presence of spillovers (both ex-post
innovation and during the course of the R&D race) an increase in public research can
result in an increase in private research. The effect of private research on public R&D is a
bit more ambiguous, but the results suggest that for ‘low’ (‘high’) levels of private R&D
an increase in private research reduces (increases) public research. An empirical test of
this hypothesis by Fuglie and Walker (2001) found that an increase in private breeding on
a commodity led to a small but statistically significant reduction in public research on
that commodity. Note that from a social welfare perspective, if higher spillovers lead to
an equilibrium condition in which both private and public R&D is higher, it may lead to a
reduction in welfare as one is moving away from the first best solution. That is, spillovers

may exacerbate over-investment, in which case the pure duopoly may perform better‘.

 

' The conditions under which this would occur is if the pure duopoly not spillovers occur, but they do in the
mixed as modeled for specification II and III in chapter 11.

133

Relating the approbiablity and spillover condition in our models to the use of
intellectual property protection by the public sector, our results suggest that the effect of
IP protection of public sector innovation needs to be evaluated within the context of the
response that the private firm elicits. The extreme cases of perfect protection and no
protection of public sector firm’s innovation are not welfare maximizing. Another
example showed that spillovers during the course of the R&D race may result in higher
private and public R&D in equilibrium, although the welfare may be lower because of
over-investment due to increased research activity.

While the primary focus of the analysis in chapter H was to investigate how
public research appropriability and spillover impacted the research activity of the private
firm, it also highlighted the regulatory role of a public sector firm in a research market.
Considering that a standard theme in the approach to market structure and innovation is
whether there is over- or under-investment in research, the results show the usefulness of
public R&D as part of technology policy to correct such distortionsz.

In chapter 111, we extended the analysis to consider, not the desirability of a
particular market structure per se, but the conditions under which a particular market
structure would imply higher levels of R&D in genomics. The modeling of chapter III
took into consideration some of the key elements of research in genomics, which is
characterized as a race and where the resources expended on the race impact the value of
the innovation or downstream profits/welfare. The results of the analysis suggest that the
distribution of profits among competing firms, the complementarity of the first stage

research with second stage output, and the curvature properties of the research function

 

2 Subsidies, taxes, patent laws and other legal regulation being some of the other technology policy tools
available to the government.

134

can, in large part, explain which market structure among monopoly, pure duopoly, and
mixed duopoly undertakes more research. In particular, we found that the R&D in the
mixed duopoly will be greater than the other markets if a) property rights are weak such
that firms can easily copy each other’s innovation, b) first stage research is not does not
complement well the downstream stages for firms in the pure duopoly, and c) the gains
from R&D activity are larger for firms in the mixed market relative to firms in the other
market. These sufficient conditions would generally be met for R&D on commodities for
which there is weak IP protection (such as non-hybrid crops), research effort of the R&D
race is limited to the race itself and limited in scope, and the market is not large enough
for the innovator to get a significant fraction of the total welfare. It was suggested that
these conditions resemble the current state of genomics R&D, as patent rights in
genomics are ill-defined leading to lower expected profits for firms, and private firms
pursue a research strategy in which the current sequencing effort does not significantly
impact research downstream. In contrast, the public sector firm, by pursuing a different
research strategy and expending more effort on the sequencing stage, may increase the
marginal productivity of its research effort.

The last essay explored the issue of c00perative and noncoopertive behavior in the
presence of a public sector firm. We showed in the essay that under certain conditions a
private firm may not have incentives to collaborate with a public sector firm as such
collaboration subverts its profit maximizing objectives as (under cooperation) it produces
where total benefits are maximized and not simply its private benefits. From the
perspective of the public sector firm, the cooperation with the private firm is welfare

improving so long as cooperation implies the maximization of joint objectives, which

I35

includes producer and consumer welfare. This result suggests that private firms be given
incentives to undertake cooperative research. Such incentives could take the form of
subsidization of private research or that the profits of the collaborating private firm be
weighted more heavily relative to consumer welfare.

The recent evolution of agriculture R&D industry structure is consistent with
theoretical models of R&D races in mixed oligopoly. The driving forces in these models
are spillovers, appropriability, cooperation, and competition. Although these models
show that the role of the public sector has shifted away from that of provider of a public
good (R&D) to that of a regulator of non-competitive effects, there remains an important
role for the public sector as provider of R&D.

The models explored in this study, while largely serving its purpose of examining
the issues raised in the introduction, have limitations leaving room for future work. All
the models of this study have been cast under the assumption of an oligopolistic market,
as our interest was to examine the nature of the behavioral response of the private and
public sectors. This assumption does not allow for an examination of how the intensity of
competition amongst several private and public sector firms affects innovation. The issue
of market performance was address in chapter 111, but only in the context of monopolistic
and oligopolistic markets. We argued that high fixed costs would not allow for a
competitive market. However, a more thorough treatment of this topic would allow for
competitive markets, one where both private and public sector firms are generalized to n
firms. Generalization to several firms also allows one to consider heterogeneity of firms
with respect to size and on how the welfare of such firms are weighted (a concept

introduced in chapter IV) in the public sector firm’s objective function.

136

A second limitation of our analysis relates to our implicit assumption that in the
mixed oligopoly the private and public sector firms produce a homogenous research
output and hence are in direct competition with each other. This is certainly true for the
genomics example, but may not be appropriate for research where public and private
sectors take on distinct and separate roles. That is, the public sector is generally
considered to be engaged in pre-technology and basic research whereas the private sector
conducts more applied research.

The purpose of the dissertation was to provide within the chosen game theoretic
analytical framework the range of choices of available market structures wherein public
and private firms can collaborate to produce the optimum level of R&D agricultural
research. These choices need to be narrowed down by further case-specific
parameterization of the model structure. Even within the broad framework chosen, there
is admittedly considerable scope for further extension and choice of more realistic
assumptions.

Lastly, the question of empirically implementing the model for a number of real
life situations also needs to be explored at some length. The main testable hypothesis that
have been generated by the models of this dissertation are

1) Greater ability by the public sector to appropriate its innovations will lead to a

decline in the private research. Greater research spillovers from public sector
research will lead to an increase in private sector research.

2) The rate of innovation in genomics for private and public sector firms differs.

The private firm employs a strategy in which it sequences the genome quickly

but at the expense of long-term profits. The public sector firm maximizes

137

3)

4)

long-term welfare gains by sequencing the genome more accurately and
thoroughly.

The presence of public research in agriculture, and especially biotechnology
research, reduces welfare losses associated with an imperfectly competitive
research market.

Private firms collaborate with public sector firms when research spillovers are

low.

138

APPENDICES

l39

Chapter 11

Figure AII.2: Relationship Between R&D Intensity, Appropriability and Spillovers

 

Industry R&D Intensity

 

 

 

 

 

 

 

 

 

 

Private R&D
+1-
Public R&D
4.‘ +
+ 5.4-1. «H-
Effectiveness of Approryability f R&D Spillover
Mechanism on: .
:' ”-
Private Firm Profits; ‘ e
"If: ........ ,

 

Public Firm Welfare '

 

 

Note: Dashed lines from the appropriability box relate to public R&D and solid lines

relate to private R&D.
Source: Adapted from Cohen and Walsh (2000).

140

Figure AII.3: Public and Private Best Response Under Varying Levels of or for
Specification 1.

m

 

 

 

 

 

 

 

Note: Simulation using h[x] = x/(1+x) , W = 100 and r =O.1

l4l

Figure AII.4: Public and Private Best Response Under Varying Levels of a for
Specification 2.

Pvt. best response

 

 

V
XS 45° (1:0 or=o.5 (1:1
10
Public
8’ 1:
response
6
4.
2
m X
10 20 30 40 p

Note: Simulation using h[x] = x/(1+x) , W = 100 and r = 0.1

142

Figure AII.5a: Public and Private Best Response Under Varying Levels of B for
Specification 3 (B=O.5).

Pvt best response

 

 

 

xs
10
K///////////’P;bT;c
8 best
1 response
6
O
4» 45
2t
* - x
20 4O 6O 80 p

Note: Simulation using h[x] = x/(l +x) , W = 100 and r = 0.1

143

Figure AII.5b: Public and Private Best Response Under Varying Levels of B for
Specification 3 (,6 =1)

 
    
   
 
   

Pvt. Best
response

  

Public best
response

 

- . ‘Xp
20 4O 6O 80 100

Note: Simulation using h[x] = x/(l +x) , W = 100 and r = 0.]

144

Figure AH.5c: Public and Private Best Response Under Varying Levels of [3 for
Specification 3 (B=0)

 

 

 

xs Pvt best
10 response

8 //#’/”//fi//,,,//rr”’

, Public

6 best

, response

4L 450

2

20 40 60 BOXP

Note: Simulation using h[x] = x/(1+x) , W = 100 and r = 0.1

145

Chapter 111

More on Assumption 2:

The second stage post innovation profits of the winning and losing doupolist is given by

equations 111.7 and 111.8:
max WWD = P[quWD — C17. mm (A. 1)
4WD
max Wu) = P[quu) - C17. qwl (A2)
4L0

where P[Q] = a — (4WD +qLD) is the inverse demand, C[_)_/,qWD,x;)] is the cost of

production for the winning firm and C [7, q L0] is the cost of production for the losing

firm. Consider three cost regimes, one where marginal cost is increasing (decreasing
returns to scale), and two others where marginal cost is constant but differ in how first
stage R&D affects production cost (a lump sump reduction in one case and a unit

reduction in another). Mathematically the three cost regimes are specified as:

2 2
1. Increasing costs: C[Z,qWD,x;)]= Z (qWD) , C[7,qLD]=7(—q-L§—),where

(1+xD) 2

 

z< 7, x2) 2 O

2. Constant costs with per unit cost decreasing with R&D:
C[_}_’,qWD,x;)] = (z-XBMWD , C[7,qLD] = (7)qLD. Assume, for the constant cost
cases, that a > y>z>0 and z> x0

3. Constant costs with lump sump reduction in total cost:

CIZflWD’xD] = (DQWD ‘xb, (517,qu = (1’)qu

146

 

Substituting the different cost specifications into equations (111.7) and (111.8), and from

the first order condition for a maximum of the profit expression, we can solve for the

equrlrbrrum quantrtres produced by the two firms in the second stage grven x0 and that a

winner to the R&D race has been determined.

1. If increasing costs then

 

 

(1:701ng a(1+x;))(1+ 7)
(1+ xB)(3+ 27) +_}_/(2+ 7)
. . a(1+xi) +7)
(11.01101: ,, -
(1 +xD)(3+ 27) +7(2 + 7)

2. If constant costs than

a — 27+ 7+ 2x?)
3

 

III III
QWDIXD] =

a: a; 0+7-27‘XB
4L01x0l= " 3

 

3. If constant costs with lump sump reduction due to R&D

III 31‘

It is easily verified that under the three regimes, gig/DUB] .>_ qZDDcB] ,

ijrxgizwzprxg], awviprxgyaxg >0 and awzprxgyaxg <0.

147

(A3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

More on Assumption 3:
When the mixed duopoly private firm wins, it faces no rival and its maximization
problem is simply the maximization of a monopolist. That is it maximizes the following

profit expression

max WWP = 1’qu — C17, (Max:121 ('49)
m

From the first order condition, the private firm chooses that output level for which

MR = MC . At that output level, profits and the public welfare can be calculated'. On the
other hand if the public sector firm emerges as the winner, since it does not participate in
the second stage it necessarily must introduce the technology through a licensing
arrangement with the private firm. The licensing arrangement is such that it maximizes

. the public welfare subject to the incentive compatibility condition of the private firm. The
output level that comes from solving this maximization is then used to calculate the
equilibrium profits for the private firm and welfare for the public sector firm. The

maximization problem for the public sector firm then is written as

max WW5: f P(Q)dQ—P*Q*+WLP[xS]+F s.t. WLprx512wp (A.lO)
qu

where

f P[QJdQ — P*Q* is the consumer surplus

F is the licensing fee charged by the public sector firm

WLPIxD] = Pqu - C[7,qu,x;4 ]— F is the private firm’s second stage profits when it

licenses the technology from the public sector firm (the winner)

 

' Public sector firm’s welfare in the second stage is defined as the summation of consumer and producer
surplus

148

WP = Pqp — C [7, q P] = WM is the private firms pre-innovation profits, where we assume
that pre-innovation profits of a private firm is equivalent to that of the monopolist, pre-
innovation.

From the first order conditions for a maximum of the public sector firm’s
objective function eqn. (A.10), the equilibrium levels of quantity produced and profits
can be calculated. It can be shown (see appendix) that in equilibrium
Wish; 1 > Wistxiir 2 Wiprxinr 2 WZptxi'ér and qZpthJ 2 qi‘vptxZJ.

To show that WW5”; ] > WZSIXP] 2 Wv;p[x;] 2 WZp[x;] , consider Figures Al and A2
below which assumes a linear inverse demand curve and increasing marginal cost. If the
winner of the R&D race is the private firm than it takes on a monopoly position (in the
absence of another firm). Monopoly profits are maximized where MR = MC (figure
A.1), in which case monopoly profits (ngplxh) are given by area C and consumer

surplus is given by area A. The public sector firm’s welfare is simply the aggregation of

monopolist’s profits and consumer surplus, i.e. Wishz] = A+ C .

Next consider the case where the winner of the R&D race is the public sector firm
that must license the technology to the private firm. However to prevent monOpoly
pricing, and the associated welfare loss, the public sector firm dictates the terms of the

licenses such that welfare is maximized (the first best welfare outcome) which occurs

where P = MC. In this case, the licensees’ profits (WZp[x;>]) are given by area E and the

consumer surplus is given by area D in Figure A.2. Comparing the two figures, it is clear

that wésrxfi] = D+ E >W,:g[x;] = A+C. Moreover wzprxh = E<W&P[x};] = E.

149

Figure A. 1: Second stage mixed market if private firm is the winner of the R&D race

MC

D=H®

 

 

 

 

Figure A.2: Second stage mixed market if public firm is the winner of the R&D race

MC

 

D = P(q)

 

 

 

150

 

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154

Chapter IV

1. Under Regime II, comparative statics in the noncooperative case are:

3x; _ 2(a—A)(2((flp —2)2 +4fl§)—9b7)

 

< O for “large” by

 

2
MP (“Mina-£00040)
it = _ 32(a - A)(,3p - 2)fls > 0
355

2
(4+2flp(1-flp)+8fl§-9br)

8x3 = 8(‘1 ‘A)(2flp(flp *1) +8fl§ +9by—4)

2
3'33 (4+2flp<1—/3p>+8fl§—9br)

> 0
for “large” by

 

.ai9.=- 16,63(a—A)(2,Bp—1) <0
afls (4+215P(1-flp)+8fl§ —9b7)2 for flp $0.5 (and >0 for ,6}; >05)

 

2. Definition of F] , F2 , F3and D

l
Flle’xS]E X

2b(WC + WP — 3W5 )2

 

f 2
(a —cp) WCWPQWC + WP)-
2ws(a-cp)(wg(a—c5)+waP(3a—4cp +c5)+ w%(c5 -cp))+

2

 

\2wg (a + op - 26‘s )2

155

"’5 (WC (4a2 -c}2> +8cPc5 —3c§ — 2a(3cp +c5))—2wp(5cz —4cP -Cs )(cP “CS )) +

 

)

1
F x ,x ,x E—X
2[P1 P2 5] 18b

(80- A17- +8<a — A>((1+flp><xp1 +xp2>+2xsfls )-
2xp1xm (7 + ,6P(7flp — 22)) + (x1291 + x%2)(11+ flpa 1,6,, -14)) +

\st (1 +flp)(XP1 + xP2)+8(XsflS)2 }

\

 

 

1
F x ,x ,x E—X
3[P1 P2 5] 18b

(10(0 — (4)2 +4((1 —A)((xpl(1+4flp)+xpz(4+fl)+5x3fl5 )-\
2xP1xP,(1 1+,6P(11,Bp —32)) +x,%2(19+ ,6P(13fip — 22)) +
x},, (13 + ,6P(19flp - 22)) +4xP2xS (4 + flpms +

\4xP1xs(1+4.3P)fls +10x§fls2 )

 

 

D a 2(flp —l)(6+15,6§ +,6P(1+,Bp(flp -4)-11,352))+
b7(35 485,, +17/3§ + 20x33) -18(by)2

156

3. Tables‘:

Table AIV.3: Nash Equilibrium Values for the Noncooperative Case in Regime 1.

 

 

 

 

 

 

 

 

 

 

 

 

fls —> 0.05 0.25 0.50 0.75 0.95
15p l
x} 0.204412 0.229652 0.257525 0.284426 0.306962
x; 0.418009 0.398702 0.389502 0.393119 0.404437
X’ 0.622421 0.628354 0.647027 0.677545 0.711399
4; 0.70487 0.791903 0.888018 0.980779 1.05849
(1.; 1. 81557 1. 74552 1. 67624 1.61771 1.5742
Q* 2.52044 2.53742 2.56426 2.59849 2.63269
0.05 0.; 6.57177 6.58982 6.59762 6.59266 6.58021
c; 6.77469 6.67067 6.54772 6.42073 6.30882
It; 0.39238 0.495261 0.622778 0.759682 0.884838
V[Q'] 3.17632 3.21926 3.28771 3.37607 3.46552
7:; 1.21132 1.12601 1.02561 0.922133 0.830124
W' 5.99135 5.96655 5.96171 5.98001 6.01061
P[Q*] 7.47956 7.46258 7.43574 7.40151 7.36731
x; 0.16224 0.183723 0.207294 0.229684 0.248098
x;- 0.437709 0.415794 o.403071 0.402941 0.410971
X* 0.599949 0.599517 0.610365 0.632625 0.659069
4; 0.64896 0.734891 0.829175 0.918736 0.992393
(1; 1.88621 1.81789 1.75048 1.69442 1.65374
Q,“ 2.53517 2.55278 2.57965 2.61315 2.64613
0-25 C; 6.52173 6.53828 6.54511 6.53964 6 .527
c; 6.81587 6.71233 6.59117 6.46811 6.36148
7!; 0.355344 0.45568 0.580104 0.71219 0.830961
V[Q’] 3.21353 3.25834 3.32731 3.41428 3 .501
It; 1.29991 1.22015 1.12592 1.02962 0.945177
W” 6.16871 6.15432 6.15926 6.18572 6.22231
P[Q*l 7.46483 7.44722 7.42035 7.38685 7.35387
0.50 x; 0 120058 0.13684 0.155172 0 172366 0.186282

 

 

' Tables IV .3 to IV. 10 are all based on calculating the equilibrium using the following parameter values
0:10, A=7, b=1,and 7=5.Notethat X =20, Q=Zqi andallomervaluesaredefined

in the text.

157

 

 

 

 

 

 

 

 

 

 

 

x5 0.453494 0.429507 0.413793 0.410338 0.415382
X: 0.573552 0.566347 0.568966 0.582704 0.601664
qP 0.600288 0.684199 0.775862 0.861829 0.93141
61; 1.94216 1.87582 1.81034 1.75646 1.71808
Q* 2.54244 2.56002 2.58621 2.61829 2.64949
C; 6.48648 6.50207 6.50862 6.50348 6.49148
Ci; 6.85727 6.75578 6.63793 6.51988 6.4191
7!; 0.324311 0.421315 0.541766 0.668474 0.780772

VlQ*] 3.23201 3.27684 3.34423 3.42772 3.50989
It; 1.37184 1.29816 1.21061 1.12163 1.04454
“1* 6. 30001 6.29447 6.30722 6.33947 6.37973

”9 1 7.45756 7.43998 7.41379 7.38171 7.35051
.1} 0. 0852334 0.0975079 0.110878 0.123313 0.133271
x; 0.46214 0.436841 0.419164 0.41346 0.416446
X * 0.547373 0.534349 0.530042 0.536772 0.549717
4; 0.568223 0.650052 0.739184 0.822083 0.888471
(1.; 1.97189 1.90661 1.84209 1.78924 1.75195
Q* 2.54012 2.55667 2.58128 2.61132 2.64042

0-75 C; 6.47394 6.49003 6.49768 6.49406 6 .4836
c; 6.89166 6.79328 6.67954 6.56659 6.47111
1:; 0.304715 0.398799 0.515658 0.637806 0.744979

V[Q*] 3.2261 3.26827 3.33149 3.40951 3.48592
71’; 1.41025 1.34051 1.25741 1.17332 1.1011
W" 6.35132 6.34809 6.36196 6.39395 6.43309

P[Q*] 7.45988 7.44333 7.41872 7.38868 7.35958
x; 0.0606795 0.0695127 0.0791146 0.0880039 0.0950821
x; 0.464984 0.438985 0.420266 0.413356 0.415224
X" 0.525663 0.508498 0.499381 0.501359 0.510306
4; 0.551632 0.631933 0.719223 0.800036 0.864383

0.95 (1% 1.98067 1.91539 1.8508 1.79795 1.76078
Q 2.5323 2.54733 2.57002 2.59798 2.62516
Cs 6.47737 6.49498 6.50458 6.50304 6.49445
c; 6.91607 6.82074 6.71075 6.60198 6.51046
It; 0.295093 0.38726 0.501634 0.620696 0.724556

V[Q*] 3.20626 3.24443 3.30251 3.37476 3.44574

 

158

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7:; 1.42099 1.35259 1.27117 1.18915 1.11915
W* 6.34334 6.33688 6.34649 6.37376 6.40858
P[Q*] 7 .4677 7.45267 7.42998 7.40202 7.37484
Table AIV.4. Nash Equilibrium Values for the Cooperative Case in Regime l.
'65 —3 0.05 0.25 0.50 0.75 0.95
310 i
x}; 0.09456 0.120671 0.150997 0.18405 0.215795
x; 0.357304 0.356545 0.37409 0.409371 0.45238
X * 0.451864 0.477217 0.525088 0.593421 0.668175
4;- 0.653303 0.726066 0.812712 0.909022 1.00258
4; 1.80582 1.75768 1.71262 1.67303 1.64039
Q”: 2.45912 2.48374 2.52533 2. 58206 2.64297
0.05 C; 6.63797 6.63742 6.61836 6.58143 6.53683
c; 6.88757 6.79019 6. 66196 6.50892 6.35444
72'; 0.404451 0.490768 0.6035 0.741635 0.888752
V[Q*] 3.02364 3.08449 3.18865 3.33351 3.49266
2:; 1.31133 1.2269 1.11667 0. 98056 0.833823
W“ 6.05074 6.02905 6.02549 6.03626 6.04905
P[Q*] 7.54088 7.51626 7.47467 7.41794 7.35703
x; 0.179065 0.197152 0.21997 0.246092 0.271666
x; 0.348215 0.354874 0.377775 0.416286 0.460788
X" 0.527279 0.552026 0.597746 0.662377 0.732455
4; 0.700866 0.762322 0. 84026 0. 929825 1. 01771
4; 1.79474 1.76123 1.72834 1.69866 1.674
9* 2.49561 2.52355 2.5686 2. 62848 2.69171
0-25 0; 6.60702 6.59584 6.56723 6.52219 6 .4713
c; 6.80352 6.71413 6.59114 6.44169 6.29058
71'; 0.411053 0.483962 0.585069 0.713172 0.851225
V[Q’] 3.11403 3.18415 3.29885 3.45446 3.62264
7:; 1.30742 1.23612 1.13679 1.00948 0.870319
W' 6.13992 6.14035 6.1575 6.18659 6.2145
P[Q*] 7.50439 7.47645 7 .4314 7.37152 7.30829
0.50 x2: 0.272536 0.28555 0.302799 0.323077 0.343091
x5 0.359341 0.367827 0.391858 0.430769 0.475023

 

 

 

 

 

 

 

159

 

 

 

 

 

 

X* 0.631877 0.653376 0.694656 0.753846 0.818114
‘15 0.720073 0.777829 0.852417 0. 938462 1.02249
45 1.85036 1.82185 1.79389 1.76923 1.74939
Q: 2.57043 2.59968 2.64631 2.70769 2.77188
Cs 6.50439 6.4894 6.45674 6.40769 6.35343
0; 6 .7095 6.62249 6.50127 6.35385 6.20564
7!; 0.332815 0.401171 0.497397 0.619763 0.751203
V[Q*] 3.30356 3.37916 3.50148 3 .6658 3.84165
7!; 1.3891 1. 32132 1.22515 1.10118 0.966061
“1* 6.41456 6.42298 6.44917 6.48793 6.52497
”9 1 7.42957 7.40032 7.35369 7.29231 7.22812
x}; 0.372782 0.382002 0.394778 0.410192 0.425557
x; 0.391542 0.396238 0.41671 0.452698 0.494644
X" 0.764323 0.778239 0.811488 0.86289 0.920201
(1; 0.708705 0.769426 0.845953 0.932115 1.0147
61; 1.97495 1.94221 1.91123 1.88548 1.86608
Q* 2.68365 2.71163 2.75718 2.8176 2.88077
0-75 C; 6.32887 6.31726 6.28721 6.23966 6.18619
6} 6.60764 6.51894 6.39687 6.25028 6.10453
It; 0.154848 0.227204 0.326012 0.448197 0.576861
V[Q*] 3.601 3.67648 3.80102 3.96943 4.14943
72’; 1.56695 1.49358 1.39228 1.26519 1.12944
W,“ 6.88974 6.89084 6.91159 6.948 6.98517
P[Q*] 7.31635 7.28837 7. 24282 7.1824 7.11923
x; 0.470929 0.476194 0.484629 0.495701 0.507195
x; 0.435395 0.432718 0.446615 0.477565 0.515951
X' 0.906324 0.908912 0.931243 0.973266 1.02315
(1; 0.678135 0.745729 0.827446 0.916166 0.999118
4; 2.13643 2.09292 2.05304 2.02154 1.99911
095 Q: 2.81456 2.83864 2.88049 2.93771 2.99823
Cs 6.11722 6.1149 6.09299 6.05152 6.00221
6} 6.5073 6.41563 6.29206 6.14613 6.00265
It; —0.0945683 -0.0107907 0.0975048 0.225061 0.355121
V[Q*] 3.96088 4.02895 4.14861 4.31507 4.49469
7:3 1.80824 1.72204 1.60883 1.47315 1.33271
W

 

 

 

 

 

 

 

I60

 

 

 

 

P[Q“)

7.4828
7.18544

 

 

7.46223
7.16136

 

7.46378
7.11951

 

7.48643
7.06229

 

7.51523
7.00177

 

Table AIV .5: Comparing Equilibrium Values Between Cooperative and Noncooperative
Cases Under Regime 1

 

 

 

 

 

 

 

.BS '9 Low Medium High
[3,, 1 (0.05) (0.5) (0.95)
XC<XNC XC<XNC XC<XNC
Low VC <VNC VC <VNC vC >VNC
(0.05) ”S >708“: 2:5 Org/C ”g >flgc
WC >WNC WC >WNC WC >WNC
Medium VC >VNC VC >VNC VC >VNC
(0.5) #5 > zgc ”S < nfic ”S <7t,IDVC
WC >WNC WC >WNC WC >WNC
.X(:>}XAK: X4:>jXAK: .XCE>}{NC
High VC >VNC VC >VNC VC >VNC
(0.95) ”S «QC ”,9 «2’0 ”S «ti-VG
chr>1VAK3 ‘VC7>‘VAK? ‘VCT>‘VAK?

 

 

 

 

161

 

Table AIV.6. Nash Equilibrium Values for the Noncooperative Case in Regime 2.

 

 

 

 

 

 

 

fls 7 0.05 0.25 0.50 0.75 0.95
fiP l
x}. 0.286169 0.289568 0.300733 0.321384 0.347336
x; 0. 0293506 0.148496 0.308444 0.494438 0.676859
X * 0.601688 0. 727633 0. 909909 1.13721 1.37153
4;; 1.10065 1.11372 1.15666 1.23609 1.33591
005 Q: 2 .2013 2.22745 2.31333 2.47219 2.67181
Cm 6.69806 6.65883 6.53001 6.29172 5.99228
”79: 1.0067 1.03076 1.11177 1.26971 1.48304
V[Q*] 2.42285 2.48076 2.67574 3.05586 3.56929
W' 4.43409 4.48714 4.66144 4 .9841 5.39003
P[Q*] 7.7987 7. 77255 7.68667 7. 52781 7.32819
x},- 0.258589 0.261682 0.271845 0.290657 0.314324
2:} 0. 029553 0.149533 0.31068 0.49827 0.682533
X* 0.546731 0.672897 0.854369 1.07958 1.31118
4;: 1.10824 1.1215 1.16505 1.24567 1.3471
025 Q: 2.21648 2.24299 2.3301 2.49135 2.69421
Cpi 6.67529 6.63551 6.50485 6.26298 5.95869
It}; 1. 06102 1.08656 1.17259 1.3405 1. 56769
V[Q*l 2.45638 2 .5155 2.71468 3.10341 3.62938
W" 4.57624 4.63272 4.81855 5.16373 5. 60013
P[Q*] 7.78352 7. 75701 7 .6699 7.50865 7. 30579
x},- 0.222332 0 .225 0.233766 0.25 0.270433
x; 0.0296443 0.15 0.311688 0 .5 0.685096
X" 0.474308 0.6 0.779221 1. 1.22596
4% 1.11166 1.125 1.16883 1.25 1.35216
050 Q: 2.22332 2.25 2.33766 2.5 2.70433
CPi 6.66502 6 .625 6.49351 6.25 5.94351
”in 1.11221 1.13906 1.22955 1.40625 1. 64551
V[Q*] 2.47158 2.53125 2.73233 3 .125 3.65669
W" 4 .6938 4.75313 4. 94856 5 .3125 5. 77432
P[Q‘] 7.77668 7.75 7.66234 7.5 7.29567
x},- 0.184706 0.186916 0.194175 0.207612 0.224517
0-75 1:;- 0.029553 0.149533 0 . 31068 0.49827 0. 682533
X * 0.398966 0.523364 0.699029 0. 913495 1.13157

 

 

 

 

 

 

 

I62

 

 

 

 

 

 

 

 

 

 

(1]); 1.10824 1.1215 1.16505 1.24567 1.3471
Q* 2.21648 2.24299 2.3301 2.49135 2.69421
6;),- 6.67529 6.63551 6.50485 6.26298 5.95869
7:},- 1.1429 1.17041 1.26308 1.44395 1.68867
V[Q*] 2.45638 2.5155 2.71468 3.10341 3.62938
W" 4.74 4.80042 4.99953 5.37063 5.84209
p[Q*] 7.78352 7.75701 7.6699 7.50865 7.30579
x},- 0.154091 0.155921 0.161933 0.173053 0.187027
x; 0.0293506 0.148496 0.308444 0.494438 0.676859
X" 0.337532 0.460339 0.632309 0.840544 1.05091
479i 1.10065 1.11372 1.15666 1.23609 1.33591
095 Q: 2.2013 2.22745 2.31333 2.47219 2.67181
CPi 6.69806 6.65883 6.53001 6.29172 5.99228
7’33.- 1.15207 1.1796 1.27232 1.45306 1.6972
V[Q*] 2.42285 2.48076 2.67574 3.05586 3.56929
W* 4.72483 4.78484 4.98253 5.3508 5.81834
P[Q*] 7.7987 7.77255 7.68667 7.52781 7.32819

 

163

 

Table AIV.7:

Nash Equilibrium Values for the Cooperative Case in Regime 2.

 

 

 

 

 

 

£5 7 0.05 0.25 0.50 0.75 0.95
flP l
x}, 0.300122 0.304608 0.319532 0.347944 0.38521
x332 0.284898 0.289156 0.303323 0.330294 0.36567
x:- 0. 0368146 0.186824 0.391955 0.64021 0.897788
X* 0.621834 0.780588 1.01481 1.31845 1.64867
4;] 1.11022 1.12682 1.18202 1.28713 1.42499
4:22 1.09576 1.11214 1.16663 1.27036 1.40642
005 Q: 2.20598 2.23895 2.34865 2.55749 2.83141
cm 6.68379 6.63423 6.46932 6.15538 5.74361
ch 6.69826 6.64891 6.48472 6.17215 5.76217
”:21 1.00741 1. 03775 1.14193 1.35404 1.65962
2&2 0.997774 1. 02782 1.131 1.34108 1.64374
V[Q‘] 2.43318 2.50646 2.75808 3.27037 4.00843
W" 4.43498 4.48477 4.64694 4.94082 5.29673
p[Q*] 7. 79402 7.76105 7.65135 7.44251 7.16859
x}, 0.324667 0.329619 0.346116 0.377613 0.41911
1552 0.25565 0.259549 0.272539 0.297341 0.330016
x; 0. 0375567 0.190648 0.400378 0.65522 0. 92115
X" 0.617874 0.779816 1.01903 1.33017 1.67028
4;] 1.14741 1.16491 1.22321 1.33452 1.48118
4332 1.09564 1.11235 1.16802 1.27432 1.41436
0.25 (2* 2.24305 2.27726 2.39123 2.60884 2.89553
cm 6.60954 6.55783 6.38556 6.05664 5.62329
c}; 6.66131 6.61038 6.44074 6.11684 5.69011
2:}, 1. 05302 1. 08539 1.19675 1.42447 1.75475
2:32 1.03704 1. 06892 1.17859 1.40286 1.72812
V[Q*] 2.51564 2.59296 2 .859 3.40303 4.19205
W" 4.60218 4 .6564 4.83357 5.15707 5.55363
plQ*] 7.75695 7.72274 7.60877 7.39116 7.10447
x}, 0.349429 0.354871 0.373022 0.407785 0.453797
0.50 x7? 0.222364 0.225827 0.237378 0 .2595 0.28878
x3 0.0383314 0.194641 0.409194 0.670992 0. 945824
X' 0.610125 0.775338 1.01959 1.33828 1.6884

 

 

 

 

 

 

 

164

 

 

 

 

 

 

 

 

 

 

 

q;1 1.17535 1.19366 1.25471 1.37164 1.52641
(1:32 1.11182 1.12913 1.18689 1.2975 1.4439
Q* 2.28717 2.32279 2 .4416 2.66914 2.97031
c2, 6.53747 6.48356 6.30369 5. 95922 5. 50328
c762 6 .601 6.54808 6.37151 6.03336 5. 58579
”7.1 1.0762 1.10998 1.22643 1.46568 1.8151
75,2 1.11253 1.14745 1.26783 1.51515 1. 87637
V[Q*] 2.61558 2.69767 2 .9807 3.56215 4.41137
W“ 4.80065 4.86039 5. 05637 5 .4174 5.86638
P[Q*] 7. 71283 7. 67721 7 .5584 7.33086 7. 02969
x761 0.371817 0.377705 0.397369 0.435125 0.485292
x382 0.19032 0.193334 0.203399 0.222725 0.248404
x;- 0.0389714 0.197943 0.416496 0.684104 0. 966439
X" 0.601108 0.768981 1. 01726 1.34195 1. 70013
4791 1.18729 1.20609 1.26889 1.38945 1.54964
4i>2 1.14192 1.16 1.22039 1.33635 1.49042
075 Q: 2.32921 2.3661 2.48928 2.7258 3.04007
cm 6.48349 6.42781 6.24183 5. 88475 5.41029
ch 6.52887 6 .4739 6.29033 5. 93785 5.46951
”:31 1. 06405 1. 09801 1.21532 1.45723 1.81263
75,2 1.21342 1.25216 1.38593 1.66181 2 .0671
V[Q*] 2.71261 2.79921 3.09826 3.71499 4 .621
w* 4.98629 5. 05142 5.26583 5.66404 6.16572
p[Q*] 7.67079 7.6339 7.51072 7.2742 6.95993
x;1 0.39016 0.396407 0.417289 0.457451 0.510956
x752 0.164541 0.167176 0.175982 0.19292 0.215484
x; 0. 0394021 0.200165 0.421419 0.692968 0. 980424
X* 0.594103 0.763749 1.01469 1.34334 1.70686
4;: 1.18657 1.20558 1.26908 1.39123 1.55395
0-95 4792 1.17529 1.19411 1.25702 1.378 1.53917
Q* 2.36187 2.39969 2.5261 2.76922 3.09312
Ch 6.45156 6.39473 6.20482 5.83955 5.35293
c}; 6.46284 6.4062 6.21688 5.85278 5.3677
”:51 1.0274 1.06057 1.17524 1.41236 1.76206
2;,2 1.31363 1.35604 1.50267 1.80584 2.25297

 

165

 

 

 

 

V[Q*] 2.78921
W" 5.12636
p[Q*] 7.63813

 

2.87926
5.1957
7.60031

 

3.19059
5.42452
7.4739

 

3.8343
5.85199
7.23078

 

4.7837
6.39566
6.90688

 

 

 

Table AIV.8: Comparing Equilibrium Values Between Cooperative and Noncooperative

 

 

 

 

 

 

 

 

 

 

 

 

 

Case Under Regime 2
fls -> Low Medium High
5,, 1 (0.05) (0.5) (0.95)
XC>XNC XC>XNC XC>XNC
Low VC >VNC VC >VNC VC >VNC
C NC NC C C C NC _ NC C C NC _ NC
(0-05) ”P1 >”P1 =”P2 >”P2 ”P1 >”P2 >”P1 -”P2 ”P1 >”P2 >”P1 -”P2
WC >WNC WC (WNC WC >WNC
XC>XNC XC>XNC XC>XNC
Medium VC >VNC VC >VNC VC >VNC
C NC_ NC C C NC_ NC C C C NC_ NC
(05) ”P1 <”P1 -”P2 <”P2 ”P1 <”P1 -”P2 <”P2 ”P2 >”P1>”P1 -”P2
WC >WNC WC >WNC WC >WNC
XC>XNC XC>XNC XC>XNC
High VC >VNC VC >VNC VC >VNC
C NC_ NC C C NC_ NC C C C NC_ NC
(095) ”P1 <”P1 -”P2 <”P2 ”P1 <”P1 -”P2 <”P2 ”P2 >”P1>”P1 -”P2
WC >WNC WC >WNC WC >WNC
Table AIV.9: Social] Efficient Values at Different Values.
3’
0.05 0.25 0.5 0.75 0.95
ng 0.354032 0.444444 0.580645 0.756757 0.943929
Q’“ 3.37173 3.55556 3.87097 4.32432 4.84066
SW 5.0576 5.33333 5.80645 6.48649 7.26099
C 6.62827 6.44444 6.12903 5.67568 5.15934
P[Q‘] 6.62827 6.44444 6.12903 5.67568 5.15934

 

 

 

 

 

 

 

166

 

Table AIV. 10: Comparison of Nash-Equilibrium Output and Research Across Models

 

0.05

0.25

 

 

XINC>X§>X§JC>X1C>XSW

SW NC C NC C
Q >Q1 >Q1>Q2 >Q2

 

X2C>X§C>X1C>XINC>XSW

SW NC C NC C
Q >Q1 >Q1>Q2 >Q2

 

Table AIV. 10 cont’d

 

0.5

0.75

 

 

X2C>X§VC>X1C>XSW>XINC

SW C NC C NC
Q >Q1>Q1 >Q2 >Q2

 

X2C>X$VC>X1C>XSW>X1NC

SW C C NC NC
Q >Q1>Q2>Q1 >Q2

 

Table AIV. 10 cont’d

 

0.95

 

 

Xf>XéVC>XIC>XSW >XlNC

SW C C NC NC
Q >Q2>Q1>Q2 >Q1

 

167

 

 

 

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