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Dynamic models of research and development

Smrkolj, G.

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This thesis develops a new theoretical framework for modelingprocess innovations. In contrast to the existing literature, the global analytical approach pursued in the thesis makes it possible to study not only how different economic factors and economic policies affect R&D investments on existing markets, but also how they influence the formation of new markets. The strength of the analysis is that various stylized facts of innovation dynamics emerge from the same unifying framework. The analysis reveals possible hidden opportunity costs of existing competition policies that prohibit cartel formation, and provides new insights into the R&D investment decisions of firms whose innovation efforts are subject to imitation.

Grega Smrkolj (1983) holds a BA degree in economics from the University of Ljubljana, Slovenia (2006) and an MSc degree ineconomics from the University of Amsterdam (2007). In 2009,he obtained his MPhil degree in economics from the Tinbergeninstitute, majoring in econometrics. In the same year, he startedto work on his PhD thesis at the University of Amsterdam.His main research interests are industrial organization and economic dynamics. Currently, he is a lecturer at the University of Amsterdam.

Dynamic Models of Research and Development

Grega Smrkolj

University of Amsterdam

557

Dynam

ic Models of Research and D

evelopment G

rega Smrkolj

Dynamic Models of Research and Development

ISBN 978 90 361 0353 4

Cover design: Crasborn Graphic Designers bno, Valkenburg a.d. Geul

This book is no. 557 of the Tinbergen Institute Research Series, established through coopera-tion between Thela Thesis and the Tinbergen Institute. A list of books which already appearedin the series can be found in the back.

Dynamic Models of Research and Development

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus

prof. dr. D.C. van den Boom

ten overstaan van een door het college voor promoties

ingestelde commissie,

in het openbaar te verdedigen in de Agnietenkapel

op vrijdag 28 juni 2013, te 12:00 uur

door

Grega Smrkolj

geboren te Trbovlje, Slovenië

Promotiecommissie

Promotor: Prof. dr. J. Hinloopen

Co-promotor: Dr. ir. F. O. O. Wagener

Overige leden: Prof. dr. C. Fershtman

Prof. dr. P. M. Kort

Prof. dr. L. Lambertini

Prof. dr. R. Sloof

Faculteit Economie en Bedrijfskunde

Acknowledgements

I must admit that finishing the doctoral thesis is one of the most unsurprising events in my life,

at least for me. I do not remember when exactly I decided to obtain a doctorate, but it must

have been way before I decided it would be in economics. According to my grandparents, I

was still a young kid when I announced this big plan. The motivation for it was changing over

the years, but the goal was always firmly there. It was not an easy task. It involved a lot of

perseverance and sacrifice, but it was fun, I must say, to a large extent thanks to all the great

people I encountered on the way and who kindly provided me with guidance and support in

times when things did not go that smoothly.

First of all, I would like to express my sincere gratitude to my supervisor Jeroen Hinloopen

for his guidance, enthusiasm, and continuous support during my PhD studies. He always gave

me a lot of highly appreciated freedom at my research and teaching, but was never hesitant

to provide closer guidance when needed. I am also infinitely grateful to him for making

it possible for me to prolong my stay at this marvelous university and so be able to enjoy

pleasant work conditions and company of to me very dear people for an additional year.

My co-mentor and, as nicely called by Prof. Hommes at an occasion, my “private math

teacher” Florian Wagener also deserves special thanks. He introduced me to the world of

bifurcations and “higher” mathematics and was at all times a guide I could not imagine

being any better. Our weekly meetings kept me highly motivated and on track. While in our

conversations I often felt like a child who is thrown in the water before being able to swim, I

learnt from his profound remarks, and my attempts to decode them, more than I could have

imagined. I am also grateful for his continuous enthusiasm, for somehow always being able

to convince me that my problems are in fact excellent opportunities.

v

Besides my advisors, I would like to thank the rest of my thesis committee: Chaim

Fershtman, Peter Kort, Luca Lambertini, and Randolph Sloof, for agreeing to serve as

members of the committee and for the time they invested in reading the manuscript. Special

thanks go to Chaim Fershtman for inspiring me for dynamic models during his lectures at the

Tinbergen Institute.

My gratitude extends to the great co-workers at the University of Amsterdam: Adri-

aan, Arnika, Aufa, Flora, Jo, Liting, and especially Sander, thank you for always being so

supportive and ready to share a word of wisdom with me.

I consider myself extremely lucky to have got to know so many great people in the course

of my life in Amsterdam. It would be a daunting task to even try to list all of you here, yet

all the precious moments we shared together. You know who you are, and you are all very

special. Your involuntary participation in my pranks is also gratefully acknowledged. There

are, however, some who I need to address separately.

Gijs, Jordan, Roel – buddies, I cannot even imagine how boring my days at the Tinbergen

Institute would be without you. My memories go back to all our jokes and fun around the

homework sets. How many words did we redefine? How many ridiculous stories did we bring

to life? You! Thanks for all the smiles, for all the support, and for simply being who you are!

Natalya, you are like my soul sister. We share so many pleasures – horse riding, classical

concerts, good wine, movies, and hundreds more yet to come! Thanks for always being so

supportive, cheering and someone I can always count on.

Leontine, thanks for all the support through the years and for reviving my passion for

dancing. Our dance evenings are something I always look forward to with great excitement

after my workdays!

Gosia and Ona, my ex-housemates, it was nice with you and I owe you more for my

survival than I can write here. Ona, I will miss our chatter in the kitchen and your sofa

moments. Gosia, I need to say that it was always calming to see someone had even worse

sleeping habits than me. Changing a bulb or repairing something in the house way past the

midnight was something I probably would never have experienced without you. A desperate

housewife at times, but surely a distraction to die for.

vi

I am also lucky to have met two promising young Slovenian economists in Amsterdam,

who have both become good friends of mine – Janko and Timotej. Janko deserves special

thanks for constantly challenging my physical endurance, be it on a bike, in a swimming pool,

or in a go-kart. Thanks to him, I might live some years longer than I would have otherwise.

Due to his continuous cheering on the marriage market, there might still be some hope for me.

Discovering the beauties of the Netherlands is always an exciting task for me. Here I need to

express my gratitude to Timotej for our gang’s car adventures, which always turn out fun. I

just hope we never again forget where we parked the car.

I am also heavily indebted to Matjaž for always being everything that being a good friend

entails. We started to share big ideas already back in the elementary school and he has

always been very supportive of all my plans, calling me ‘Doctor’ before I have even finished

this dissertation. He also deserves thanks for always reminding me how beautiful my home

country Slovenia is.

I would also like to express sincere thanks to Tina, Nataša, and Nina, for all the great

moments during our studies in Slovenia and for supporting my decision to go study abroad.

Nataša, I will always cherish the memory of our weekly discussions of Hayek’s ideas along a

cup of tea in the very early morning train coupe. They were great inspiration for me!

I owe sincere thanks to all the great teachers in the secondary school who contributed a

great deal to my decision to pursue further studies in economics. Special thanks to Stanko

Miklic, Branko Ðordevic, Gabrijela Blaznek, and Vanda Macerl, for always openly believing

in my success.

I reserved the final and most important lines for my family. I consider myself an extremely

lucky child. My parents, Ksenija and Jano, and grandparents, Pavla, Pepi, and Fani, have

always provided me with everything I could wish, both emotionally and materially, many

times sacrificing their own well-being for mine. I am grateful to them for raising me up to

be the man who I proudly am today, for always believing in me, supporting me, and, what is

most important, for making me feel loved. They are the ones who invented a hundred and

one way of stimulating me to do schoolwork well in years when running in mud, playing

computer games and ringing on the doors of annoyed neighbors were more inviting activities

vii

than any book ever written could be. Without them, I would never have achieved what I have.

A special gratitude goes to my sister Kaja for sharing with me the precious bond only a sister

can. We had our share of fights, we had tears, but also lots of joy and that crazy laughter only

we will ever understand. I love you, my sister. More than words could ever say and I am

thankful for every day that you are in my life.

I am also heavily indebted to our ‘extended family’ members. In particular, to aunt Ema,

uncle Ivo; aunty Suzana, uncle Otmar, cousins Eva and Nina; aunt Dori, uncle Stane, cousins

Ana and Nejc; aunt Slavi, uncle Dušan, cousins Tina and Andraž: aunt Jana and uncle Jože;

Sara, Karmi, Jukic family, Gošte family, Tamše family, and our neighbors, Bokal family, Pilih

family, and Pirkovic family, for always believing in me. I feel privileged to have been a part

of your life.

My last words of deep appreciation and gratitude go to my late grandgrandmother Fani

Ivic, who left me while I was still a young boy, but whose good-heartedness, kindness, grace,

and love continue to live in my heart. The words she wrote in my memory book, “Be good

and honest and you will be welcome everywhere”, are deeply etched out in my soul. She is

an angel I could never disappoint.

Amsterdam, 18 February 2013

viii

Contents

1 Introduction 1

1.1 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Monopoly and Innovation 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 The system with positive production . . . . . . . . . . . . . . . . . 13

2.2.2 The system with zero production . . . . . . . . . . . . . . . . . . . 14

2.3 Steady-state solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Comparative statics . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Global optimality and industry dynamics

(bifurcations of optimal vector fields) . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Region I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 Regions II and III . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.3 Regions IV and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Appendix 2.A Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . 35

Appendix 2.B Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . 36

Appendix 2.C Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . 37

Appendix 2.D Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . 38

Appendix 2.E Proof of Remark 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Appendix 2.F Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 41

ix

Appendix 2.G Proof of Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . . 46

Appendix 2.H Proof of Lemma 6 . . . . . . . . . . . . . . . . . . . . . . . . . 46

Appendix 2.I Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . 47

Appendix 2.J Proof of Lemma 8 . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Cartels and Innovation 53

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Competition versus Collusion . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.1 Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.2 Collusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Collusion and the incentives to innovate . . . . . . . . . . . . . . . . . . . 69

3.6 Competition policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Competition and Innovation 79

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.1 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.1 Numerical Method of Lines . . . . . . . . . . . . . . . . . . . . . 95

4.4 Equilibrium strategies and industry dynamics . . . . . . . . . . . . . . . . 98

4.4.1 Value function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4.2 Policy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.3 Vector field and dynamics . . . . . . . . . . . . . . . . . . . . . . . 101

4.4.4 Leader versus follower . . . . . . . . . . . . . . . . . . . . . . . . 105

4.4.5 Stochasticity and R&D . . . . . . . . . . . . . . . . . . . . . . . . 106

4.4.6 Deterministic game and indifference curves . . . . . . . . . . . . . 109

x

4.4.7 Spillover effects and R&D . . . . . . . . . . . . . . . . . . . . . . 112

4.4.8 Market size and industry dynamics . . . . . . . . . . . . . . . . . . 123

4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Appendix 4.A Proof of Lemma 10 . . . . . . . . . . . . . . . . . . . . . . . . 129

Appendix 4.B Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 131

5 Summary 135

Bibliography 139

Samenvatting (Summary in Dutch) 149

xi

Chapter 1

Introduction

Around 1900, Swiss chemist Jacques E. Brandenberger invented a transparent film which he

named “cellophane”. On December 26, 1923, American chemical company DuPont acquired

from La Cellophane Société Anonyme, an organization to which Brandenberger had assigned

his various patents, the exclusive rights to its United States cellophane patents. DuPont’s

hopes for creating a lucrative market were shattered, however, by discovering that cellophane

could not be used to wrap up products that require moisture proofing, such as cake and

candy. It took DuPont chemist William H. Charch three years and thousands of tests to

develop a lacquer that made cellophane moisture proof, an invention that revolutionized the

packaging and merchandizing industry. As the manufacturing costs of cellophane continued

to decline due to DuPont’s ongoing process innovations, so did the prices of cellophane. All

this contributed to cellophane being used as wrapping material for a variety of products (from

food to jewelry), and for its use in various products (including batteries, scotch tape, and

dialysis machines). Since the mid-1930’s, cellophane has been manufactured continuously. It

is still used today.

The development of cellophane, from mind to market, is typical for the life cycle of many

new technologies. Research starts long before a prototype sees the light; development begins

long before the launch of a new product. However, ideas abound, but only a small fraction

leads to successful innovations. For instance, in 1979, Apple was enthusiastic about a design

for a computer mouse, discovered at Xerox research center. But Apple did not develop this

prototype further as the projected production costs of a single mouse would be over $400.1

Also, existing technologies tend to leave markets slowly due to incumbents’ R&D efforts.

For example, Edison’s invention of an electric light bulb was bound to replace the gas lamp.

Producers of the latter however prolonged the lifetime of this inferior technology through

continuous product innovations, including the introduction of the Welsbach mantle that made

gas lamps five times more efficient (Utterback, 1994).

These examples illustrate four stylized facts of R&D: (i) initial technologies (“ideas” or

“prototypes”) need to be developed further before they are suitable for large-scale production;

(ii) there are many initial technologies, but only a very limited fraction is developed further,

and from this fraction only a subset will constitute a successful innovation, (iii) production and

the search for further improvements take place simultaneously, whereby production starts only

after an initial stage of successful product development, and (iv) there are process innovations

for technologies that are destined to leave the market in the foreseeable future.

Existing models of R&D are not easily reconciled with these observations. Static models

of R&D are silent about the process from prototype to first releases of new products and

production technologies.2 Moreover, these models are forced to assume that unit costs, the

proxy for production technologies, are below the choke price (the lowest price at which the

quantity sold is zero). This assumption is quite unlikely to hold for new technologies in their

early stages of development.

Dynamic models of R&D are in principle equipped to capture the path from prototype to

market penetration. To date, however, neither “innovation race” models nor dynamic versions

of conceived static models do so adequately. In essence, models of innovation races examine

the time of completion of a cost-saving innovation of known magnitude, whereby the expected

time of completion is one-to-one related to R&D expenditures.3 These models exclude the

coexistence of production and R&D efforts. Moreover, the R&D process cannot fail, thus

1Years later, Apple came up with a new design which would only cost $25 to produce. This prototype wassubsequently developed into Apple’s famous single-button mouse.

2The seminal papers in this literature are d’Aspremont and Jacquemin (1988) and Kamien et al. (1992); seeDe Bondt (1997) for an overview.

3Seminal contributions here include Loury (1979) and Lee and Wilde (1980). Reinganum (1989) surveysthis literature.

2

transforming the R&D investment decision into a static one. The recently conceived dynamic

versions of static R&D models maintain the assumption that unit costs are below the choke

price at all times.4 Initial technologies that are “expensive” are excluded, by assumption, from

the analysis. Hence, R&D efforts always lead to the stable equilibrium. Put differently, every

initial technology will be developed further, and successfully so. Indeed, without exception

this literature provides analyses that are locally optimal only.

A distinguished feature of the solution approach taken in this thesis, as compared to the

received literature, is that we provide a truly global analysis. Be it in discrete or continuous

time, models of process innovation usually solve for the investment path relating the state

variable (unit cost c) to the control variable (the R&D effort k). Figure 1.1 illustrates a

typical such case. A firm is assumed to start with some (low enough) initial unit cost, the

value of which then decreases over time, due to continuous process innovation, to some

long-run equilibrium level (indicated by a dot). Typically, the equilibrium paths of different

competition regimes are then compared with respect to surpluses they bring about, and based

on this, subsequently, certain policy recommendations are outlined. Notice, however, the

underlying assumptions! Both regimes in Figure 1.1 are assumed to follow their respective

stable path to the equilibrium. In fact, the problem of development has been more or less

solved. The market is already formed, the production starts immediately and coexists with

R&D investment. There is no decision whether to pursue a particular project at all or not. The

R&D process cannot fail - both regimes converge to their respective equilibria. Furthermore, if

sufficiency conditions in such a model are not satisfied, the equilibrium path is not guaranteed

to be optimal even for those unit costs over which it exists. If one does not consider other

trajectories which also satisfy necessary conditions for an optimum, nothing can be said about

parameters for which the paths indicated in Figure 1.1 are optimal.

Our global solution concept remedies these shortcomings of the local approach in two

ways. First, all possible values of unit costs are considered (the shaded region in Figure 1.1 is

lightened up); in particular, also those above the choke price. This allows research efforts to

4This literature is still scant; it includes Petit and Tolwinski (1999), Cellini and Lambertini (2009), Lamber-tini and Mantovani (2009), and Kovác et al. (2010).

3

A (choke price)

Regime I

Regime II

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4C0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

k

Figure 1.1: Local approach.

precede production. Second, we do not limit ourselves to equilibrium paths but consider all

trajectories which satisfy necessary conditions and are as such also candidates for an optimal

solution. Hence, we do not limit ourselves to solutions that are only locally optimal. This

enables us to determine the location of critical points - points at which the optimal investment

function qualitatively changes. That is, we do determine the value of unit costs for which

R&D investments come to a halt or are not initiated in the first place. As the position of these

critical points depends on the conduct of firms, different regimes can lead to qualitatively

different long-run solutions despite starting with the same level of technology.

A special feature of the models considered in this thesis is that they are all non-convex

dynamic optimization problems. In such problems, a small change in parameters can lead

not only to quantitative changes in the solution structure, but also to qualitative ones (e.g.,

the number of equilibria changes, indifference points appear and disappear). If a small

change in parameters causes a change in one of the qualitative properties of the dynamical

system, the latter is said to undergo a ‘bifurcation’ (see Grass et al. (2008) and Kiseleva and

Wagener, 2010, 2011). The bifurcation analysis, which is needed for obtaining a complete

global solution, results in a bifurcation diagram that indicates for every possible parameter

combination the qualitative features of the solution structure and also provides the ‘bifurcating

values’ of the parameters at which these features change.

4

1.1 Outline of the Thesis

The results of this thesis are presented in three relatively self-contained chapters.

Chapter 2 introduces the global framework and characterizes the investment decisions

of a monopolist that are globally optimal. It shows that there exist four distinct stable

types of dynamics. These correspond well to the stylized facts of R&D mentioned earlier.

Furthermore, the notion of an indifference point – a point at which a firm is indifferent

between developing a technology further and opting out – is introduced and evaluated in

relation to model parameters.

Chapter 3 extends the analysis of the previous chapter to R&D cooperatives. It re-

examines the trade-off between the benefits of allowing firms to cooperate in R&D and the

concomitant increased potential for product market collusion. It shows how misleading, or at

least incomplete, the conclusions based on local analyses can be. In particular, it shows that

prohibiting collusion on the product market per se is not univocally welfare enhancing.

Chapter 4 represents a further extension to the model of Chapter 2. It considers two

firms which compete both in R&D and on the product market. This introduces the problem

of finding a solution to a differential game with two state variables, non-convex dynamics,

and a possibility of discontinuities in the firms’ investment functions. A special numerical

approach is taken to obtain an approximating solution. The chapter introduces a possibility

of asymmetric firms and random noise in the evolution of unit costs. It analyzes the role the

spillovers play in shaping the investments of the follower and the leader and the role they play

in determining the structure of a (newly formed) market.

5

Chapter 2

Monopoly and Innovation

2.1 Introduction

In this chapter, we provide a generalized description of the economics of R&D. In particular,

for any initial technology level c0, we characterize for a monopolist the investment and

production path that is globally optimal. Hence, in contrast to the existing literature, we do not

exclude initial technologies that are above the choke price, and we do not restrict ourselves to

solutions that are only locally optimal.1 Our model is, therefore, more easily reconciled with

the four stylized facts of R&D mentioned in Chapter 1.2

From a technical point of view, the problem of the monopolist is formulated as an optimal

control problem with an infinite time horizon. Its distinct characteristic is the presence of

multiple equilibria while at the same time the Arrow-Mangasarian sufficiency conditions

are not met. In models of this type, the qualitative properties of optimal solutions may

change if parameters are varied. We therefore use bifurcation analysis3, which constitutes the

mathematics of a qualitative change, to assess industry dynamics for all initial technologies in

1As it turns out, a non-trivial part of the parameter space yields stable paths with initial marginal costs sohigh that production would yield negative mark-ups. At that stage all the monopolist does is to invest in R&D inorder to bring down the costs of production.

2As this is the first analysis ever to consider the entire range of initial technologies, we restrict ourselvesto the monopoly case. Obviously, competitive forces play an important role as well in what we observe aboutR&D. These extensions are considered in subsequent chapters.

3Variations in parameter values can lead to qualitative changes in the solution structure (e.g., some steadystates lose their stability, indifference points appear, and so on). Such qualitative changes in the solution structuredue to smooth variations in parameters are called bifurcations (see Grass et al., 2008).

conjunction with other parameter combinations (including time preferences and the efficiency

of the R&D process).

Our analysis yields four distinct possibilities, three of which remain hidden in local

analyses: (i) initial marginal costs are above the choke price and the R&D process is initiated;

after some time production starts and marginal costs continue to fall with subsequent R&D

investments; (ii) initial marginal costs are above the choke price and the R&D process is

not initiated, yielding no production at all; (iii) initial marginal costs are below the choke

price and the R&D process is initiated; production starts immediately and marginal costs

continue to fall over time, and (iv) initial marginal costs are below the choke price and the

initiated R&D process is progressively scaled down; production starts immediately but the

technology (and production) will die out over time; the firm leaves the market. The strength

of our analysis is that all these cases can emerge from the same unifying framework.

The analysis is not without policy implications. It shows that market characteristics that

affect future profitability have an impact on the monopolist’s decision to develop further an

initial technology. If market regulations are such that future mark-ups are reduced a priori, it

could be that the range of initial technologies that is developed further will shrink. The loss

of total surplus this reduction brings about constitutes a hidden cost of market regulations

if the initial technology consists of marginal costs above the choke price. In that case, there

is no immediate production lost if the initial technology is not developed further. Yet, these

indirect costs should be taken into account when assessing properly the trade-off between

static and dynamic efficiency.

2.2 The model

Time t is continuous: t ∈ [0,∞). A single supplier produces at marginal costs c(t). In every

instant, market demand equals production

q(t) = A− p(t), (2.1)

8

where p(t) is the market price, q(t) is the quantity produced, and A is the choke price. The

monopolist obtains an initial, exogenous technology c(0) = c0. He can reduce this marginal

costs by exerting R&D effort k(t). Marginal costs evolve as follows over time:4

dc(t)

dt≡ c(t) = c(t) (−k(t) + δ) , (2.2)

where δ > 0 is the constant rate of efficiency reduction due to the ageing of technology and

organizational forgetting.5 The cost of R&D efforts per unit of time is given by:

Γ(k(t)) = bk(t)2, (2.3)

where b > 0 is inversely related to the cost-efficiency of the R&D process. Hence, the R&D

process exhibits decreasing returns to scale.6 The monopolist discounts the future with the

4One interpretation of the model is that c0 constitutes the initial level of technology corresponding to aproduct innovation which is then followed by subsequent process innovations. This idea can be traced back tothe pioneering work by Abernathy and Utterback (1975, 1978). Lambertini and Mantovani (2009) consider amultiproduct monopolist that simultaneously pursues process and product innovations. They show that the pathto the saddle-point steady state involves some substitutability between the two innovation types.

5The assumption of the positive rate of technology (know-how) depreciation is well established in thetheoretical literature (see Besanko et al. (2010) and Lambertini and Mantovani, 2009), as well as in the empiricalliterature (see Benkard, 2004). In Besanko et al. (2010), a firm whose gains in know-how through learningdo not outstrip the loss in know-how from forgetting moves down its learning curve and its marginal costincreases. Benkard (2004) shows that organizational forgetting is crucial to explaining the dynamics in themarket for wide-bodied commercial aircraft. As he puts it: “an aircraft producer’s stock of production experienceis constantly being eroded by turnover, lay offs and simple losses of proficiency at seldom repeated tasks. Whenproducers cut back output, this erosion can even outpace learning, causing the stock of experience to decrease”(Benkard, 2004, p. 590). In our model, know-how gains come from investments in R&D and not production, butthe main idea is the same. Furthermore, production costs include complementary inputs that are typically bought.Due to their inherent evolution over time, it becomes ever more costly to incorporate them into the productionprocess of a firm that is sluggish in R&D (e.g., installing a newer version of software on obsolete hardware cansignificantly slow down performance). The reason for this is that R&D efforts not only generate new informationwithin the firm, but also determine the firm’s ‘absorptive capacity’ - “the firm’s ability to identify, assimilate,and exploit knowledge from the environment” (Cohen and Levinthal, 1989, p. 569). Note that the equilibriafor a non-positive depreciation rate follow trivially. For δ is zero, consider δ very close to zero. Such smalldepreciation rate pushes the level of the initial technology that will not be developed further far beyond thechoke price as only minuscule investments are needed to reduce marginal costs over time. Indeed, for negative δ,every initial technology will be developed further as there is an exogenous reduction in marginal costs over time,yielding a steady state with no production costs.

6Again, we follow the literature with this assumption (see, e.g., d’Aspremont and Jacquemin (1988), Kamienet al. (1992), or Qiu (1997)). The evidence is mixed however. Schwartzman (1976) for instance documentssignificant economies of scale in R&D. But Madsen (2007) is unable to reject the hypothesis of constant returnsto scale in the number of patents applied for. Moreover, for the pharmaceutical industry, Vernon and Gusen(1974) and Graves and Langowitz (1993) concludes that firms experience decreasing returns to scale in R&D.And Adams and Griliches (1996) conclude that there are strong diseconomies of scale in the production ofscientific articles.

9

constant rate ρ > 0.

Instantaneous profit is:

π(q, k, c) = (A− q − c)q − bk2, (2.4)

yielding total discounted profit:

Π(q, k, c) =

∫ ∞0

π(q, k, c)e−ρtdt. (2.5)

The optimal control problem for the monopolist is to find controls q∗ and k∗ that maximize

the profit functional Π subject to the state equation (2.2), the initial condition c(0) = c0, and

two boundary conditions which must hold at all times: q ≥ 0 and k ≥ 0. Note that according

to (2.2), if c0 > 0, then c(t) > 0 for all t. The set of all possible states at each time t is given

by c ∈ [0,∞).

The model has four parameters: A, b, δ, and ρ. The analysis can be simplified by

considering a rescaled version of the model which, as defined in Lemma 1, carries only two

parameters: φ and ρ. That is, rescaling the model translates the four-dimensional parameter

space into a two dimensional one.

Lemma 1. By choosing the units of t, q, c, and k appropriately, we can assume A = 1, b = 1,

and δ = 1. This yields the following, rescaled model:

π(q, k, c) = (1− q − c)q − k2, (2.6)

Π(q, k, c) =

∫ ∞0

π(q, k, c)e−ρtdt (2.7)

˙c = c(

1− φk), c(0) = c0, c ∈ [0,∞)∀ t ∈ [0,∞) (2.8)

q ≥ 0, k ≥ 0 (2.9)

ρ > 0, φ > 0 (2.10)

with conversion rules: q = Aq, c = Ac, k = A√bk, t = t

δ, π = A2π, φ = A

δ√b, ρ = ρδ.

10

Proof. See Appendix 2.A.

The rescaled model introduces a new parameter φ = A/(δ√b), which captures the profit

potential of a technology in hand: a higher (lower) A implies higher (lower) potential sales

revenue, a higher (lower) b implies that each unit of R&D effort costs the firm more (less),

whereas a higher (lower) δ implies that each unit of R&D effort reduces the marginal cost by

less (more). Therefore, a higher (lower) φ corresponds to a higher (lower) profit potential of a

technology. For notational convenience, we henceforth omit tildes.

To solve the dynamic optimization problem, we introduce the current-value Pontryagin

function7

P (c, q, k, λ) = (1− q − c) q − k2 + λc (1− φk) , (2.11)

where λ is the current-value co-state variable. It measures the marginal worth of the increment

in the state c at time t when moving along the optimal path. As the marginal cost is a “bad”,

we expect λ(t) ≤ 0 along optimal trajectories.

Pontryagin’s Maximum Principle states that if the triple (c∗, q∗, k∗) is an optimal solution,

then there exists a function λ(t) such that c∗, q∗, k∗, and λ satisfy the following conditions:

1.) q∗ and k∗ maximize the function P for each t:

P (c∗, q∗, k∗, λ) = max(q,k)∈R2

+

P (c∗, q, k, λ), (2.12)

2.) λ is a solution to the following co-state equation:

−∂P∂c

= λ− ρλ ⇔ λ = q + (ρ− 1 + φk)λ, (2.13)

which is evaluated along with the equation for the marginal cost

c = c(1− φk) (2.14)

and the initial condition c(0) = c0.

7Also called pre-Hamilton or un-maximized Hamilton function.

11

Let q = Q(c, λ) and k = K(c, λ) solve the problem max(q,k) P (c, q, k, λ) for every (c, λ).

We define the current-value Hamilton function

H(c, λ) = P (c,Q(c, λ), K(c, λ), λ). (2.15)

The above necessary conditions (2.12)-(2.13) for an optimal solution are complemented by

two transversality conditions:8

limt→∞

e−ρtH(c, λ) = 0 (2.16)

and

limt→∞

e−ρtλc = 0. (2.17)

If problem (2.12) has a solution, it necessarily satisfies the following Karush-Kuhn-Tucker

conditions:∂P

∂q= 1− 2q − c ≤ 0, q

∂P

∂q= 0, (2.18)

∂P

∂k= −2k − φλc ≤ 0, k

∂P

∂k= 0. (2.19)

Conditions (2.18) imply that either i) q = 0 and c ≥ 1 or ii) q > 0 and c < 1.9 In particular:

q∗ =

(1− c)/2 if c < 1,

0 if c ≥ 1.

(2.20)

Conditions (2.19) imply that either i) k = 0 and ∂P∂k≤ 0 (implying λ ≥ 0) or ii) k > 0 and

∂P∂k

= 0. In particular:

k∗ =

−φ

2λc if λ ≤ 0,

0 if λ > 0.

(2.21)

The above conditions for optimal production yields two regimes of the state-control

8The necessity of (2.16), which allows exclusion of non-optimal trajectories, was proven by Michel (1982).Kamihigashi (2001) proves the necessity of (2.17).

9Observe that in the non-rescaled model, the analogous condition for positive production is that c(t) < A.

12

system. The first is characterized by positive production, in the second there is no production.

2.2.1 The system with positive production

In the region where c < 1, using (2.13), (2.14), (2.20), and (2.21), we obtain the following

optimal state and co-state dynamics:

c = c

(1 + 1

2φ2λc

),

λ = 12(1− c) +

(ρ− 1− 1

2φ2λc

)λ,

(2.22)

where λ ≤ 0.10 Equation (2.21) depends on the co-state and state variable. Note that the

correspondence between k and λ is one-to-one if, and only if, λ ≤ 0, and that therefore the

system can be re-written in the state-control form.

Differentiate equation (2.21) with respect to time to obtain the dynamic equation

dk

dt≡ k = −φ

2

(λc+ λc

). (2.23)

Use (2.13) and (2.14) for λ and c respectively. Note that equation (2.21) also implies that

λ = −2k

φc. (2.24)

Substitute this expression into (2.23), together with (2.22) and the expression for the optimal

output level (q = (1− c)/2), to obtain:

k = ρk − φ

4c(1− c). (2.25)

Hence, the system with positive production (c < 1) in the state-control form consists of the

10The differential equations in the text are all valid for λ ≤ 0. For a complete specification of optimal stateand co-state dynamics (including the case when λ > 0), we refer the reader to Appendix 2.F.

13

following two differential equations:

k = ρk − φ

4c(1− c),

c = c (1− φk) .

(2.26)

2.2.2 The system with zero production

In the region c ≥ 1, substituting (2.13), (2.14), and (2.24) into (2.23) and imposing q = 0,

gives the following expression:

k = ρk. (2.27)

Hence, the state-control system with zero production consists of the following two differential

equations: k = ρk,

c = c (1− φk) .

(2.28)

The state-co-state analogue is:

c = c

(1 + 1

2φ2λc

),

λ =(ρ− 1− 1

2φ2λc

)λ.

(2.29)

The Hamilton function, obtained by substituting (2.21) and appropriate expressions for q

from (2.20) into the Pontryagin function (2.11), is given by

H(c, λ) =

14(1− c)2 + 1

4φ2λ2c2 + λc if c ∈ [0, 1),

14φ2λ2c2 + λc if c ∈ [1,∞),

(2.30)

where λ ≤ 0. The state-control analogue is then

H(c, k) =

14(1− c)2 + k

(k − 2

φ

)if c ∈ [0, 1),

k(k − 2

φ

)if c ∈ [1,∞),

(2.31)

where k ≥ 0.

14

2.3 Steady-state solutions

The steady-state solutions of (2.26) and (2.28) are obtained by imposing the stationarity

conditions k = 0 and c = 0.

Lemma 2. The “no-production” state-control system (2.28) has no steady state in the region

where it is defined.

Proof. See Appendix 2.B.

The result of the above lemma is intuitive as a steady state in the “no-production” region

would imply that a firm maintains positive R&D investments without ever selling anything.

Consider now system (2.26). Imposing the stationarity condition k = 0, we obtain

kM =φ

4ρc(1− c) ≥ 0, ∀c ∈ [0, 1], (2.32)

where superscript M stands for Monopoly.11 Steady-state marginal cost follows from inserting

(2.32) into the state dynamics (2.14) and imposing the stationarity condition c = 0:

c = c

(1− φ2

4ρc(1− c)

)= 0. (2.33)

This yields:

cM = 0, cM =1

2± V, (2.34)

where

V =1

2

√1− 16ρ

φ2. (2.35)

Observe that V is real if, and only if, φ ≥ 4√ρ, in which case V ∈ [0, 1

2) and cM ∈ [0, 1).

The solution to the system is summarized in Proposition 1 and is depicted in Figure 2.1.

Proposition 1. If φ > 4√ρ, the state-control system with positive production (2.26) has three

steady states:

11All steady-state values have superscript M .

15

i) (cM , kM) = (0, 0) is an unstable node,

ii) (cM , kM) =(

12

+ V, 1φ

)is either an unstable node or an unstable focus, and

iii) (cM , kM) =(

12− V, 1

φ

)is a saddle-point steady state.

At φ = 4√ρ, a so-called “saddle-node bifurcation” occurs as the last two steady states

coincide and form a so-called “semi-stable” steady state.

If φ < 4√ρ, the state-control system with positive production has one single steady state: the

origin (cM , kM) = (0, 0), which is unstable.

Proof. See Appendix 2.C.

3 steady states

(c, k) = (0, 0) is the only steady state

SN

0 1 2 3 4 50

2

4

6

8

Ρ

Φ

Figure 2.1: Steady states of the state-control system. The saddle-node bifurcation (SN) curveis defined by φ = 4

√ρ. Passing through it from above, the two steady states other than origin

coincide, form a “semi-stable” steady state, and then disappear, leaving one steady state toexist only.

The intuition for the condition φ > 4√ρ in the above proposition is that developing a

16

technology further becomes an option (the saddle-point steady state exists) only if for a given

discount rate, the profit potential of a considered technology is high enough.

In the saddle-point steady state, instantaneous marginal cost, output, and profits are, respec-

tively:

cM =1

2− V, (2.36)

qM =1

4+V

2, (2.37)

πM =1

16(1 + 2V )2 − 1

φ2. (2.38)

2.3.1 Comparative statics

The relevant comparative statics are as follows (for a graphical illustration of these relations,

see Figure 2.2):12

Lemma 3. Steady-state marginal cost is increasing in b, δ, and ρ, and decreasing in φ;

steady-state quantity, profit, consumers’ surplus, and total surplus are decreasing in b, δ, and

ρ, and increasing in φ.

Proof. See Appendix 2.D.

Recall that δ accounts for an exogenous decrease in production efficiency due to the

ageing of technology. The monopolist does not need to exert the desired level of the R&D

effort at once but can smooth it over infinitely many instants. With a convex R&D cost

function this enables him to reduce his investment costs. The higher the rate at which the

technology depreciates, the less advantageous this smoothing will be, as a larger part of what

the monopolist has invested is lost in every next instant. Further, in the steady state, the

monopolist invests just enough to keep his marginal cost constant at the desired level. Hence,

the higher is δ, the more costly this maintenance will be, as a larger part of the technology

must be replaced in every period.

12The graphs are indicative only. They are drawn for the admissible range of parameters in the sense that forthese values the saddle-point steady state exists. However, and as discussed further below, the steady-state willnot be reached for all these parameter values.

17

(a) marginal cost (b) quantity

(c) profit (d) total surplus

Figure 2.2: Steady-state instantaneous values of the marginal cost, quantity, profit, and totalsurplus. Graphs are drawn for A = 10 and ρ = 0.1. Variables and parameters refer to thenon-rescaled model.

The higher is b, the more it costs a firm to implement a particular R&D effort in every

instant. As an increase in either δ or b makes implementing every desired R&D effort level

more costly, an increase (decrease) in b and/or δ increases (decreases) the steady-state level of

marginal cost and decreases (increases) the steady-state output, profits and total surplus, as

stated in Lemma 3 above.13

13Steady-state marginal cost in the non-rescaled model equals cM =(A−

√A2 − 16bδρ

)/2. For the

steady-state values of other variables of the non-rescaled model, see Appendix 2.D.

18

2.4 Global optimality and industry dynamics

(bifurcations of optimal vector fields)

Thus far we have focused on the steady state. Obviously, the stable manifold of the saddle-

point steady state is one of the candidates for an optimal solution. However, as the following

remark shows, it may not (always) be the true optimum.

Remark 1. The Pontryagin function of the optimization problem as given in (2.11) is not

jointly concave in the state and control variables. Hence, necessary conditions for an optimum

are not necessarily sufficient.

Proof. See Appendix 2.E.

Therefore, we have to carry out a detailed bifurcation analysis to assess the dependence

of the solution structure on the model parameters.14 15 This yields a complete overview of

how optimal R&D investment and concomitant production respond to changes in the initial

level of marginal cost and to varying characteristics of the R&D process. To analyze global

optimality, we have to inspect the complete state-control space, which is the set of solutions

(c(t), k(t)) of the state-control system considered as parameterized curves (trajectories) in a

plane. Its general form is sketched in Figure 2.3.

The dotted vertical line c = 1 separates the region with zero production from the region

with a positive level of production. In the left region, the parabola which achieves its maximum

at c = 1/2 is the locus k = 0. The loci representing points where c = 0 are the horizontal line

k = 1/φ and the vertical line c = 0 which coincides with the k-axis. Loci c = 0 and k = 0

intersect in the saddle-point steady state (S2) and the two unstable steady states (S1 and S3).

The black arrows in squares indicate the direction of trajectories in each respective region.

The horizontal arrows indicate an increase or a decrease in c, whereas the vertical ones refer to14For a recent discussion of the use of bifurcation theory in analyzing non-convex optimal control problems

with multiple equilibria, see Kiseleva and Wagener (2010).15Note that Remark 1 applies to all related papers as well, including Lambertini and Mantovani (2009),

Cellini and Lambertini (2009), and Kovác et al. (2010). Without further analysis it remains unclear whetherthese papers discuss truly optimal behavior. A sufficiency proof that does not rely on joint concavity of stateand control variables can be given by adapting the Stalford sufficiency theorem (Stalford, 1971) to an infinitehorizon setting, as for instance in Polasky et al. (2011).

19

0 0.5 1 1.5

c

kq > 0 q = 0

k = 0

c = 01φ

L1

S3

S1

S2

L2

L3

Figure 2.3: Illustrative sketch of the state-control space.

changes in k. A number of trajectories is indicated by grey arrows. Together, the trajectories

cover the entire space. However, there are only two trajectories leading to the saddle-point

steady state; these are called “stable paths” or “stable manifolds”. They are indicated by

the connected black arrows pointing towards S2. The lighter arrows pointing away from S2

indicate the two unstable paths. Trajectories spiral out from S3 in a counter-clockwise motion.

The solution to the state-control system (2.28), which yields trajectories in the region with

zero production, is given by:

k(t) = C1eρt, (2.39)

c(t) = C2e

(t− e

ρtφC1ρ

), (2.40)

where C1 and C2 are positive real constants.

A solution to the state-control system (2.26), which yields trajectories in the region with

positive production, cannot be obtained analytically; we therefore make use of geometrical-

20

numerical techniques.16

A particular trajectory is a candidate for an optimal solution if it satisfies all necessary

conditions: those given by Pontryagin’s Maximum Principle together with transversality

conditions (2.16) and (2.17). As Figure 2.3 shows, there exists a whole range of solutions

(c(t), k(t)) to the state-control system. These trajectories all satisfy the conditions of Pontrya-

gin’s Maximum Principle. By ruling out those trajectories that do not satisfy the transversality

conditions, we are left with the candidates for an optimal solution.

Lemma 4. The points in Figure 2.3 in the set

{(c, k) : c ∈ (0, 1), k = 0}

cannot be a part of any optimal trajectory.

Proof. See Appendix 2.F.

Lemma 5. All trajectories along which k → ∞ and c → 0 as t → ∞ can be ruled out as

optimal solutions.

Proof. See Appendix 2.G.

For instance, the trajectory denoted by L1 in Figure 2.3 is in the set of Lemma 5.

Lemma 6. The trajectory through the point (c, k) = (1, 0) satisfies the transversality condi-

tions (2.16) and (2.17).

Proof. See Appendix 2.H.

In Figure 2.3, the trajectory of Lemma 6 is labeled L3. We call this the “exit trajectory” as

it implies that the firm (eventually) exits the market. Indeed, at some stage the exit trajectory

enters the region where c ≥ 1, at which point both R&D investment and production come to a

halt.17

16See also Judd (1998) and Wagener (2003). All numerical simulations were done in Matlab.17Strictly speaking, the firm’s marginal cost continue to increase along the exit trajectory due to the positive

depreciation rate δ. However, we interpret any situation in which the firm stays inactive as if it has left themarket.

21

Notice that the stable path (denoted by L2 in Figure 2.3) to the saddle-point steady state

(S2) also satisfies the transversality conditions. Notably, along this trajectory, both c and k

converge to finite limits and, therefore, so does the co-state variable (2.24).

As the following corollary to the above lemmas shows, we are left with two candidates

for an optimal solution.

Corollary 1. The set of candidates for an optimal solution consist of the stable path of the

saddle-point steady state and the trajectory through the point (c, k) = (1, 0).

Proof. See Appendix 2.I.

To find an optimal solution from the candidates, we make use of the following lemma.

Lemma 7. If the triple (c(·), q(·), k(·)) satisfies the necessary conditions (2.12)–(2.17), then

Π(c, q, k) =

∫ ∞0

π(q, k, c)e−ρtdt

=1

ρP (c(0), q(0), k(0), λ(0))

=1

ρH (c(0), k(0)) , (2.41)

where H is the Hamilton function in state-control form.

Proof. See, e.g., Grass et al. (2008), ch. 3, pp. 161–162.

Lemma 7 is useful whenever several paths satisfy all necessary conditions for an optimal

solution and we have to determine which one yields the highest discounted profit flow.

According to Lemma 7, these flows are equal to the rescaled values of the Hamiltonian (2.31)

at the corresponding initial points.

Figure 2.4 provides an example in which the optimal solution is the stable path leading to

the saddle-point steady state. Note that the initial marginal costs can be higher than c = 1. In

that case, the optimal decision leads the firm to first produce nothing but to invest increasingly

in the reduction of its marginal costs. When marginal costs are below one, it starts producing.

22

0 0.5 1 1.50.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

c

k

S2

(a)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

t

q

(b)

0 2 4 6 8 100

0.5

1

1.5

t

c

(c)

0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

t

k

(d)

0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

c

Π(e)

Figure 2.4: State-control space (a), time paths for quantity (b), marginal cost (c), and R&Deffort (d), and the correlation diagram between the total discounted profit and marginal cost(e), respectively. All plots show curves for parameter values (ρ, φ) = (1, 8).

The level of R&D investments then gradually decreases to its long-run steady-state level. This

is exactly the scenario behind the story of cellophane. 18

Note that having a phase of zero production as a part of the stable path is possible only if

the range of initial marginal costs is not bounded to be below the choke price. In this phase

instantaneous profits are negative, but total discounted profits are positive; the monopolist

invest in R&D because once marginal costs are below the choke price, production becomes

profitable.19

In the example of Figure 2.4, the monopolist optimally selects an appropriate k such

18Another example is the petrochemical industry. Stobaugh (1988) finds that R&D efforts aimed at processinnovations increase every year in the period following the initial product innovation, only to decrease againafter that. He also documents that the probability of the next process innovation being major is decreasing overtime.

19Related, the literature on optimal technology adoption predicts a substantial time lag between the discoveryof a new technology and its adoption (see, e.g., Hoppe (2002) for an overview). Delaying the adoption providesadditional information as to the profitability of a new technology; it is also rational if a better technologyis expected to become available in the near future (see also Doraszelski (2004)). In this literature, however,technological progress is exogenous.

23

that (c0, k0) is located on the stable path. However, as the Arrow-Mangasarian sufficiency

conditions are not satisfied (see Remark 1), an investment policy that follows the stable path

is not necessarily optimal. Indeed, it could not even be a possible solution as the saddle-point

steady state does not exist for all parameter values (recall Proposition 1). That is, the solution

structure depends on the system’s parameters.

Figure 2.5 presents the bifurcation diagram, which shows the optimal R&D investments

for varying parameter values.20 Every combination of the exogenous parameters yields a point

on this diagram. It consists of five distinct regions, each representing a particular structure of

the set of optimal solutions. In general, either there exists an upper bound of initial marginal

cost below which the monopolist selects the stable path towards the saddle-point steady state,

or the (new) technology is always phased out by a firm (eventually) exiting the market. Hence,

if it exists, convergence to the saddle-point steady state along the stable path is locally optimal

only; it is never optimal globally.

Solutions to the optimization problem for parameters outside the shaded region in Figure

2.5 are characterized by the presence of threshold values; the stable path of the saddle-point

steady state is optimal if the initial marginal cost is below some threshold level, otherwise

the exit trajectory is optimal. The threshold point can be either an indifference point21 or a

repeller. Indifference points are initial states c = c0 for which the firm is indifferent between

several possible investment policies (i.e., the stable path and the exit trajectory). A repeller

differs from an indifference point in that at a repelling point, it is optimal to stay at that point

forever after.22

20The bifurcation diagram in Figure 2.5 is drawn for the parameters of the non-rescaled model.21Also called Skiba points or DNSS points (see Grass et al. (2008)).22More formally, a threshold point separates two basins of attraction: the initial values of the marginal cost

below these points constitute the basin of attraction of the saddle-point steady-state; the initial values of themarginal cost above these points constitute the basin of attraction to exit the market. An indifference point liesin both basins of attractions, a repeller in neither.

24

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

ρ/δ

II

IV

SN

V SN’

I

Aδ√b

ISN

IR III

IA

Figure 2.5: Bifurcation diagram. The uppermost curve represents parameter values forwhich the indifference point between the stable path and the exit trajectory is exactly atc = 1. SN stands for the “saddle-node bifurcation curve”, SN’ for the “inessential saddle-node bifurcation curve”, IA for the “indifference-attractor bifurcation curve”, and IR for the“indifference-repeller bifurcation curve”. ISN indicates the “indifference-saddle-node point”,which is the point from which indifference-attractor, indifference-repeller, saddle-node andinessential saddle-node curves emanate. Its coordinates are ρ ≈ 2.14 and φ ≈ 5.85. Romannumerals (I-V) indicate the corresponding parameter regions discussed in the main text. Theshaded region (regions IV and V) indicates the parameter region for which the firm alwaysexits the market. Notice that the axes are excluded from the admissible parameter space. Thecurve representing indifference points at c = 1 obtains a value of φ ≈ 2.00 for ρ = 1× 10−5.

2.4.1 Region I

In Region I, the point of indifference is above one. Therefore, the system will converge to

the saddle-point steady state for all initial technologies c0 ∈ (0, 1].23 A typical example is

depicted in Figure 2.6. The arrows indicate the direction of motion along optimal trajectories.

The two regions (production, no production) are split by the dotted vertical line c = 1. For a

given ρ, Region I is characterized by a relatively high φ, that is, by relatively large demand

23Note that c0 = 0 corresponds to an unstable steady state; for this trivial value of the marginal cost, themonopolist stays at that point forever.

25

0 0.2 0.4 0.6 0.8 1 1.2 1.49 1.8 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

c

k

S2

1 1.1 1.2 1.3 1.4 1.49 1.6 1.7 1.8 1.9 2−0.02

−0.01

0

0.01

0.02

0.03

c

Π

Π2

Π1

Figure 2.6: State-control space and the total profit function when the indifference point is inthe region with zero production. The plots are drawn for parameter values (ρ, φ) = (1, 8).

and/or high R&D efficiency. In such a favorable environment, the monopolist can compensate

for his early stage losses if, initially, production is postponed as that would yield negative

mark-ups. There exists, however, a finite upper bound c > 1 for the initial marginal costs c0

beyond which future profits are not enough to compensate for short-run losses (c ≈ 1.49

in the case considered). In that case, the initial technology will not be developed further. In

Figure 2.6, the point of indifference (indicated by a dashed vertical line) occurs where the

total profit function for the stable path (Π1) in the right-hand picture obtains a value of zero

(which is the value of the total profit function for the exit trajectory Π2 in the region c ≥ 1).

As the total profit beyond this point is negative (the firm makes a net loss), the firm prefers not

to invest at all. Starting from any initial value of marginal cost below the indifference point,

the firm’s optimal decision is to follow the stable path. The exact position of the indifference

point is defined in Lemma 8.

Lemma 8. If an indifference point between the stable path and the exit trajectory does exist

in the region with zero production (c ≥ 1), it is given by the value of the marginal cost c ≥ 1

for which the point (c, k) =(c, 2

φ

)lies on the stable path. For c0 = c, total discounted profits

are zero for both trajectories.

Proof. See Appendix 2.J.

Region I is bounded from below by the curve for which the point of indifference is exactly

at c = 1. This particular situation is depicted in Figure 2.7: for all initial values of the marginal

26

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

c

k

S2

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.05

0

0.05

0.1

0.15

0.2

c

Π

Π1

Π2

Figure 2.7: State-control space and the total profit functions when the indifference point isexactly on the boundary between the two regions (c = 1). The plots are drawn for parametervalues (ρ, φ) = (1, 6.655).

cost c0 ∈ (0, 1), the optimal-solution trajectory is the stable path of the saddle-point steady

state, while for c0 ∈ (1,∞), it is optimal not to produce and invest anything; at c0 = c = 1,

either strategy is optimal. The right-hand picture shows that the total profit Π1 for the stable

path is zero at exactly c = 1. For comparison, the total profit Π2 on the exit trajectory, which

passes through the point (c, k) = (1, 0), is plotted up to its conjugate point (the last point on

the trajectory where marginal cost is still decreasing; see also footnote 26 below).

The stable path in the region with zero production (c ≥ 1) is characterized by a decreasing

k and an increasing c as t→ −∞ (see (2.39)-(2.40) and Figure 2.6). From Lemma 8 follows

that the indifference point between the stable path and the exit trajectory is exactly at c = 1

when the stable path reaches k = 2/φ exactly at c = 1 (see Figure 2.7). As investment

is increasing in φ (for a given ρ), we can find for every ρ a high enough φ such that the

indifference point is either at c = 1 or in the region with c ≥ 1. A higher φ for a given ρ

in Region I just moves the indifference point further beyond c = 1. Consequently, Region I

is not bounded from the right. Moreover, as k decreases along the stable path as t→ −∞,

at some instant the stable path, having a value of k at c = 1 above k = 2/φ, reaches the

point with k = 2/φ for some c > 1 as t→ −∞. Therefore, for parameters in Region I, we

always have an indifference point at some c = c ≥ 1 and, in particular, the convergence to

the saddle-point steady-state is for these parameters guaranteed if c0 ∈ (0, 1], but not for all

c0 ∈ (1,∞).

27

Figure 2.8 illustrates how the indifference point varies with the parameters.24 Clearly, for

a given discount rate ρ, the more efficient is the R&D process, the larger will be the range of

initial marginal costs that lead the monopolist to select the stable path. Likewise, a higher

choke price corresponds to a higher value of initial marginal cost at the indifference point.

This has an important implication: market characteristics that influence future production,

and hence profitability, affect whether or not a monopolist starts investing in developing a

new technology. That is, if market regulations are such that future mark-ups are reduced,

for instance because the monopolist is forced to price closer to its marginal costs, the range

of initial technology levels that repay development shrinks. The loss of total surplus this

reduction brings about is a hidden cost of competition policies if the indifference point is

above c = 1, as in that case there is initially no production. Not pursuing the development

of the technology does consequently not surface as an immediate reduction in total surplus.

Yet, these indirect costs should be taken into account when assessing properly the trade-off

between static and dynamic efficiency.

2.4.2 Regions II and III

In Region II, the indifference point c is below c = 1. This implies in particular that at the

indifference point the monopolist produces a positive quantity. Figure 2.9 illustrates this

case. The indifference point occurs where the two respective total profit functions intersect,

as indicated in the right-hand picture. For the particular parameter values considered in

Figure 2.9, the indifference point occurs at c ≈ 0.61. Although it is also profitable to invest in

R&D for some initial technology level c0 ∈ (c, 1), the technology is not promising enough

for the monopolist to select the stable path towards the saddle-point steady-state. Rather, he

invests in R&D at some smaller rate that retards the decay of the technology level optimally;

but eventually the technology will leave the market, and the monopolist with it.25

For an even higher initial level of marginal cost (c0 ≥ 1), the monopolist does not initiate

24Figure 2.8 is drawn for the parameters of the non-rescaled model.25Note that if c0 ∈ (c, 1), the alternative strategy of producing but not investing anything in R&D, which

corresponds to “following” the c-axis below the exit trajectory in Figure 2.9, yields lower total discounted profitsthan following the exit trajectory (recall Lemma 4).

28

0 1 2 3 4 50

5

10

15

20

25

30

35

40

ρ/δ

Aδ√b

c = 4

c = 3

c = 2

c = 1

c = 5

Figure 2.8: Dependence of the indifference point c on the model parameters. Curves aredrawn for five fixed values of c. Given ρ, a higher value of A or a lower value of b and/or δincreases the value of marginal cost at the indifference point, and vice versa. This effect islarger, the smaller the value of ρ. An increase in ρ reduces the critical value of marginal cost,ceteris paribus.

any activity; the new technology is not developed at all and the monopolist does not enter the

market.26

Utterback (1994) documents several examples of technologies where process innovations

continue to be realized although it is known that these technologies will leave the market

in the near future. For instance, steam-powered saws and conveyors were first used in the

ice harvesting industry after the introduction of machine-made ice. And self-setting shutters,

small plate cameras, and celluloid substitutes for glass were introduced in the dry plate

photography at the time that Kodak introduced roll film cameras. These examples correspond

26Conjugate points P1 and P2 indicate the last element on the respective trajectory (when starting from itsend) where the marginal cost is still increasing/decreasing. Considering a particular trajectory beyond this pointwould give us two possible k-points for each c, of which the one beyond the P point is suboptimal. Observe thatin the plot of the profits, the value of the total profit for the exit trajectory (starting from its end) increases allalong the way to P2 (where it obtains a cusp point), beyond which it starts decreasing.

29

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

c

k

P2 S

3P

1S2

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

c

Π

P2

P1

Π2

Π1

Figure 2.9: State-control space and the total profit functions when the indifference point iswithin the region with positive production. The plots are drawn for parameter values (ρ, φ) =(1, 5). The optimal-solution trajectories are marked by thick lines. The two candidates for theoptimal solution, which both spiral out of the unstable steady state S3, are the stable path andthe exit trajectory.

well to the scenario predicted by the “exit trajectory”.

Region III is similar to Region II in that for c0 ≥ 1 the firm does not initiate any activity

and only for sufficiently low initial marginal costs c0 ∈ (0, cR) there is convergence to the

saddle-point steady state. There is, however, a small qualitative difference between the two

regions: in Region III, the threshold point is not an indifference point but a repeller (see

Figure 2.10). Starting close to the repeller, the firm will “linger” near to it for a long time

before deciding on developing the technology (if c0 < cR), or on phasing it out eventually (if

c0 > cR).27

Region III is separated from Region II by a type-2 indifference-repeller bifurcation curve

(IR1(2)), at which a repeller changes into an indifference point.28 Fixing φ and varying ρ, it

can be shown that the exit trajectory of a repeller lies below that of an indifference point.29

This is also intuitive: a lower discount rate ρ (in the case of an indifference point) implies

larger investments in R&D as the monopolist values future profits relatively more. Hence, for

each initial value of marginal cost, the exit trajectory corresponding to a lower discount rate27Observe in Figure 2.10 that in the case of a repeller, the two trajectories never occur at the same time.

Hence, the corresponding total profit functions (the right-hand picture) do not intersect.28Before the bifurcation (Region II), the unstable steady state (S3) has two complex eigenvalues with positive

real parts. The stable path and the exit trajectory both spiral out from it, which yields an indifference point tothe left of the unstable steady state (focus). At the bifurcation, both eigenvalues become real and equal, suchthat all trajectories move away from the unstable steady state in the direction of a unique eigenvector. After thebifurcation, the steady state has two real positive eigenvalues and two corresponding (and distinct) eigenvectors.

29Fixing ρ and varying φ leads to the same conclusion.

30

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

c

k

cR

S2 S

3

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.005

0.01

0.015

0.02

c

Π

Π1

Π2

Figure 2.10: State-control space and the total profit functions when there is a repellerpoint (cR) in the region with positive production. The plots are drawn for parameter values(ρ, φ) = (4, 8.2). The stable path of the saddle-point steady state and the exit trajectory bothflow out of a repelling point, which is the unstable node S3 (corresponding to c = 0.6098).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

c

k

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.05

0.1

0.15

0.2

0.25

c

Π

Figure 2.11: State-control space and the total profit function for the case with no saddle-pointsteady state. Parameter values are (ρ, φ) = (1, 3). In this situation (and any other situationin which φ < 4

√ρ), the exit trajectory covers the whole space and it is the only solution.

Observe that the total discounted profit is positive for c < 1.

lies above the exit trajectory corresponding to a higher discount rate.

2.4.3 Regions IV and V

Region IV is separated from Region III by a saddle-node bifurcation curve, given by φ = 4√ρ.

This curve represents parameter combinations for which the saddle and the unstable node

S3 coincide and form a semi-stable steady state. After the bifurcation, both the saddle and

the unstable node disappear. In such a case (see Figure 2.11), the only solution left is the

exit trajectory: for c0 ≥ 1 the firm never initiates any activity, while for c0 ∈ (0, 1) the

31

firm does invest in R&D but it will leave the market at some future instant. This is again

intuitive as in Region IV we have that φ is relatively small (for a given ρ). That is, R&D is

relatively inefficient and/or market demand is relatively small. Note that in Figure 2.11, R&D

investments are initially increasing over time for very low values of marginal costs. These

investments reduce the pace at which the monopolist will leave the market; for low initial

marginal cost it is profitable to slow down this process in order to profit optimally from the

relatively favorable initial technology.

Finally, Region V resembles Region IV in that only the exit trajectory is optimal (see

Figure 2.12). This is in spite of the fact that points in Region V have higher φ values than

points in Region IV with the same value of ρ, and that the saddle-point steady state (S2)

together with its stable path (denoted by T1) exists. A distinctive feature of Region V is that

the exit trajectory (denoted by T2) intersects the vertical line through the saddle-point steady

state. Whenever this is the case, it is never optimal to be at the saddle, or to follow any stable

path leading towards it. Rather, the exit trajectory is optimal, yielding a higher discounted

profit flow: that is, the value of the Hamiltonian at any point of the exit trajectory is higher

than at points with the same value of marginal cost on the stable path.30

Observe that in the upper right graph of Figure 2.12 total discounted profit evaluated along

the stable path (Π1) is always below that evaluated along the exit trajectory (Π2). For the sake

of completeness, the second stable path (denoted by T4) and the unstable path (denoted by

T3) are plotted jointly with the corresponding total discounted profit functions (Π4 and Π3,

respectively). The latter are also both lower than Π2. Therefore, as said, for parameter values

in Region V, the exit trajectory is always optimal and the saddle-point steady state is never

approached, neither from above nor from below.31 Put differently, the saddle-point steady

state is not even locally optimal.

The IA1 curve that bounds Region V from above is an indifference-attractor bifurcation

curve. At the IA1 bifurcation (depicted in the second row of Figure 2.12), the exit trajectory

30This follows from the fact that the derivative of the Hamilton function with respect to k is negative fork < 1/φ (which is also the k-coordinate of the saddle-point steady state).

31The second stable path is not drawn in the previous figures so as not to blur the exposition. The policy thatit induces implies that the long-run optimal marginal cost is just higher than the initial value of the marginal cost.

32

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

c

k

T4

S2

T3

T2

T1

S3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

c

Π

Π4 Π

2

Π3

Π1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

c

k

T4

S2

T2

S3

T1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

c

ΠΠ

4

Π2

Π1

Figure 2.12: State-control space and the total profit functions before and at the indifference-attractor bifurcation. The plots in the upper two graphs (before the bifurcation) are drawnfor (ρ, φ) = (0.5, 3.1). The plots in the lower two graphs show that the bifurcation occurs at(ρ, φ) ≈ (0.5, 3.32).

(T2) coincides with the unstable path (T3) and yields a higher total discounted profit than

the stable path converging to the saddle-point steady state from the right (see also Kiseleva

and Wagener (2011) and Wagener (2003)). Hence, there is no convergence to the saddle

from above. However, the exit trajectory does not intersect any vertical line in the region

to the left of the saddle-point steady state. Consequently, the left-hand stable path remains

the only candidate for an optimal solution. That is, for initial marginal cost lower than the

saddle-point steady-state value, the optimal trajectory approaches the saddle-point steady

state from below.32

All in all, before the bifurcation, the exit trajectory lies below the unstable path of the

saddle; after the bifurcation, it lies above it. At the bifurcation, the two trajectories change their

32Notice that for the values of initial marginal cost lower than the saddle-point steady state, the saddle is alsoapproached from below for parameter values in Regions I, II, and III as the left-hand stable path of the saddlepoint remains the only candidate for the optimal solution in the respective region.

33

relative positions, the saddle-point steady state becomes locally optimal and an indifference

point is created. That is, we enter Region II. In other words, moving from Region V, where

the firm always exits the market, by increasing φ while keeping ρ fixed, we arrive at Region

II, where staying in the market might be an optimal solution, depending on the initial level c0

of the marginal cost. The interpretation is straightforward: staying in the market becomes an

option if R&D becomes more efficient and/or if market demand increases.

The curve bounding Region V from below is an inessential saddle-node bifurcation curve

(SN′1). It is similar to the SN1 curve in that two new steady states appear when crossing

it. However, it is inessential for the optimization problem: the bifurcation does not entail a

qualitative change of the optimal strategy, as the steady states involved in the SN′1 bifurcation

are not associated with the optimal strategy; as above, the exit trajectory below the unstable

path of the saddle remains the optimal trajectory. Consequently, while the state-control system

does bifurcate, the structure of the optimal solutions does not.

Together, Region IV and Region V represent the parameter region for which there is never

convergence to the saddle-point steady state because it is either not optimal to approach it

(Region V) or because there is no saddle-point steady state at all (Region IV).33

2.5 Concluding remarks

Existing models of R&D are difficult to reconcile with empirical aspects of R&D. These

include the observation that many initial technologies (“prototypes” or “ideas”) need to be

developed further before they can enter the market, that only a minority of initial technologies

is successfully brought to the market, that production starts only after an initial stage during

which the technology is developed further, and that process innovations are implemented

for technologies that are destined to leave the market. In this chapter, we have developed a

dynamic model of R&D that describes the R&D investment decision of a monopolist, and

33Cellini and Lambertini (2009) restrict the values of the marginal cost to c ∈ [0, 1] and assume that thesaddle-point steady state is always optimally approached. Clearly, their conclusions hold only for parametersin Region I as only there we have convergence to the steady state for initial marginal costs c ∈ (0, 1]. A lookat Figure 2.5, however, reveals that four other regions exist where the convergence is not guaranteed for allc ∈ (0, 1].

34

that is better aligned with these empirical observations.

A distinguishing feature of our analysis is that we provide a global analysis, as opposed to

all related papers. Accordingly, we do not limit initial technologies to carry marginal costs

that are below the choke price. The part of the parameter space for which such technologies

could eventually lead to the saddle-point steady state is not negligible indeed. More generally,

we have shown that there always exists a critical value of initial marginal cost above which

the firm does not initiate any (R&D) activity; the saddle-point steady state is never globally

optimal.

Our analysis suggests a careful look at competition policies that reduce mark-ups in

products markets. Reduced future profits diminish the profitability to develop further initial

technologies, in particular those that come with marginal costs above the choke price. Not

developing further these technologies does not surface as a cost of competition policy as there

is no production yet that will be taken from the market.

Appendix 2.A Proof of Lemma 1

A rescaled variable or parameter is distinguished by a tilde: for instance, if π denotes profit,

then π denotes profit in rescaled variables. The profit function in the original (non-rescaled)

model is:

π = (A− q − c) q − bk2 (2.42)

Using the conversion rules given in Lemma 1, we obtain:

π = (A− q − c) q − bk2

= (A− Aq − Ac)Aq − b(A√bk

)2

= A2(

(1− q − c) q − k2)

= A2π

35

The equation for the evolution of marginal cost over time is in original variables given by:

c(t) = c(t) (−k(t) + δ) . (2.43)

Write c(t) = c(1δt). Then

dc

dt=

dc

dt

dt

dt

=c

δ

=

(1− 1

δk

)c.

Setting k = A√bk and substituting it in the previous equation, we obtain:

dc

dt=

(1− A

δ√bk

)c.

It is now natural to introduce φ = Aδ√b.

Note that if c = c/A, then ˙c = c/A and

˙c =(

1− φk)c

Observe finally that if t = tδ, then e−ρt = e−ρt if and only if ρ = ρ

δ.

Appendix 2.B Proof of Lemma 2

As ρ > 0, the equilibrium condition k = ρk = 0 implies that k = 0. But then c =

c (1− φk) = c. This is equal to 0 if, and only if, c = 0. But this cannot be, as in the system

with zero production we have c ≥ 1.

36

Appendix 2.C Proof of Proposition 1

Assume φ > 4√ρ. The stability of the steady states can be analyzed by evaluating the trace

and determinant of the following Jacobian matrix:34

JM =

∂c∂c

∂c∂k

∂k∂c

∂k∂k

=

1− φk −φc

−φ(1−2c)4

ρ

. (2.44)

At c = k = 0, the trace τ of the matrix JM is given as

τdef= tr JM = 1 + ρ > 0,

its determinant ∆ is

∆def= det JM = ρ > 0,

and its discriminant D is

Ddef= τ 2 − 4∆ = (1− ρ)2 > 0.

Hence, this steady state is an unstable node.

Evaluating the Jacobian matrix at

k =φ

4ρc(1− c), c =

1

2+ V =

1

2+

1

2

√1− 16ρ

φ2, (2.45)

we obtain τ = ρ > 0 and

∆ =

√φ2 − 16ρ

[φ+

√φ2 − 16ρ

]8

= φ2V

(1

4+

1

2V

), (2.46)

34Note that the trace of the respective Jacobian matrix is equal to the sum of eigenvalues, while its determinantis equal to their product. If the real part of each eigenvalue is negative, then the steady state is asymptoticallystable. If the real part of at least one of the eigenvalues is positive, then the steady state is unstable. Moreparticular, if one eigenvalue is real and positive and the other one real and negative, the steady state is said tobe a saddle. In this last situation, there are four special trajectories, the separatrices, two of which are forwardasymptotic to the saddle, while the other two are backward asymptotic to it. The union of the former separatriceswith the saddle form the stable manifold of the saddle; analogously, the union of the latter with the saddle formthe unstable manifold.

37

which is clearly positive if φ > 4√ρ. Hence, this steady state is also unstable. The discrimi-

nant takes the value

D = ρ(8 + ρ)− 1

2φ(φ+

√φ2 − 16ρ

), (2.47)

which is zero for φ = φ0 =√

ρ(8+ρ)2

(4+ρ), negative for φ > φ0, and positive for 4

√ρ ≤ φ < φ0.

The steady state is an unstable node if D > 0 and an unstable focus if D < 0. In the latter

case, the eigenvalues of JM are complex conjugates with positive real parts.35

Finally, evaluating the Jacobian matrix at

k =φ

4ρc(1− c), c =

1

2− V =

1

2− 1

2

√1− 16ρ

φ2, (2.48)

we obtain τ = ρ > 0 and

∆ =φ2 − 16ρ− φ

√φ2 − 16ρ

8. (2.49)

If φ > 4√ρ, then ∆ < 0, and the eigenvalues are real and have opposite sign. Therefore,

(2.48) is a saddle-point steady state of the system. Observe that for φ = 4√ρ, the two steady

states (2.45) and (2.48) coincide at c = 1/2 and k = 1φ

. A saddle-node bifurcation occurs at

these parameter values, where the two equilibria collide and disappear.36

Substituting the expression for the steady-state marginal cost of the steady states other than

the origin into (2.32), the expression for the optimal investment in the steady state simplifies

to kM = 1φ

.

Appendix 2.D Proof of Lemma 3

As we deal with both versions of the model, we use tildes in the notation of rescaled variables

and parameters to avoid ambiguity. We first prove the stated relations between parameters and

steady-state values for the rescaled model. Take φ > 4√ρ, such that the saddle-point steady

state exists. The saddle-point steady state values of the marginal cost, output, and profit are

35Note that the eigenvalues of JM are r1,2 = 12

(τ ±√τ2 − 4∆

).

36In Figure 2.3, this corresponds to the locus c = 0, which is the horizontal line k = 1φ , becoming tangent to

the locus k = 0 at its peak.

38

given in (2.36)-(2.38). Consumer surplus and total surplus are, respectively:

CSM

=(1− p)q

2=

1

2(qM)2 =

1

32(1 + 2V )2 , (2.50)

T SM

= πM + CSM

=3

32(1 + 2V )2 − 1

φ2. (2.51)

Substituting the expression for V given in (2.35) in the steady state expressions (2.36)-(2.38)

and (2.50)-(2.51) and taking appropriate derivatives, we obtain:

∂cM

∂φ= − 8ρ

φ2√φ2 − 16ρ

< 0,

∂cM

∂ρ=

4

φ√φ2 − 16ρ

> 0,

∂qM

∂φ=

4ρ

φ2√φ2 − 16ρ

> 0,

∂qM

∂ρ= − 2

φ√φ2 − 16ρ

< 0,

∂πM

∂φ=

2

(1 + ρ+ ρφ√

φ2−16ρ

)φ3

> 0,

∂πM

∂ρ= −

1 + φ√φ2−16ρ

φ2< 0,

∂CSM

∂φ=ρ+ ρφ√

φ2−16ρ

φ3> 0,

∂CSM

∂ρ= −

1 + φ√φ2−16ρ

2φ2< 0,

∂TSM

∂φ=

2 + 3ρ+ 3ρφ√φ2−16ρ

φ3> 0,

∂TSM

∂ρ= −

3 + 3φ√φ2−16ρ

2φ2< 0.

All inequalities follow straightforwardly form the admissible values of the parameters.

We now prove the stated relations for the original model. Applying the conversion rules

39

defined in Lemma 1, we obtain:

cM =A−

√A2 − 16bδρ

2,

qM =A+

√A2 − 16bδρ

4,

πM =A2 − 8bδ(δ + ρ) + A

√A2 − 16bδρ

8,

CSM =1

2(qM)2 =

1

32

(A+

√A2 − 16bδρ

)2,

TSM =3A2 − 8bδ(2δ + 3ρ) + 3A

√A2 − 16bδρ

16.

Take A > 4√bδρ, such that the saddle-point steady state exists. Then:

∂cM

∂b=

4δρ√A2 − 16bδρ

> 0, (2.52)

∂cM

∂δ=

4bρ√A2 − 16bδρ

> 0, (2.53)

∂qM

∂b= − 2δρ√

A2 − 16bδρ< 0, (2.54)

∂qM

∂δ= − 2bρ√

A2 − 16bδρ< 0. (2.55)

It follows from the definition of consumer surplus (CSM = 12

(qM)2) that the effect of b and

δ on CSM is qualitatively the same as their effect on qM . Moreover, we have:

∂πM

∂b= −δ(δ + ρ)− Aδρ√

A2 − 16bδρ< 0,

∂πM

∂δ= b

(−2δ + ρ

(−1− A√

A2 − 16bδρ

))< 0,

∂TSM

∂b=

1

2δ

(−2δ + ρ

(−3− 3A√

A2 − 16bδρ

))< 0,

∂TSM

∂δ=

1

2b

(−4δ + ρ

(−3− 3A√

A2 − 16bδρ

))< 0.

Again, all inequalities follow straightforwardly from the relevant values of the parameters.

40

Appendix 2.E Proof of Remark 1

The Hessian matrix of the Pontryagin function with respect to control and state variables is

D2(q,k,c)P =

∂2P (·)∂q2

∂2P (·)∂q∂k

∂2P (·)∂q∂c

∂2P (·)∂k∂q

∂2P (·)∂k2

∂2P (·)∂k∂c

∂2P (·)∂c∂q

∂2P (·)∂c∂k

∂2P (·)∂c2

=

−2 0 −1

0 −2 −φλ

−1 −φλ 0

. (2.56)

The leading principal minor of the first order of this matrix is −2. The leading principal

minor of the third order, which is also the determinant, of the above matrix is equal to

2 + 2φ2λ2. As the leading principal minors of odd order do not have the same sign, the

matrix is indefinite. Consequently, the Pontryagin function is nowhere jointly concave in

state and control variables. Hence, the Arrow-Mangasarian sufficiency conditions are not

satisfied.37

Appendix 2.F Proof of Lemma 4

To prove the lemma, we consider the state–co-state form of the solution, given by:

c =

c(1 + 1

2φ2λc

), 0 ≤ c < 1, λ ≤ 0;

c, 0 ≤ c < 1, λ > 0;

c(1 + 1

2φ2λc

), c ≥ 1, λ ≤ 0;

c, c ≥ 1, λ > 0;

(2.57)

λ =

12(1− c) +

(ρ− 1− 1

2φ2λc

)λ, 0 ≤ c < 1, λ ≤ 0;

12(1− c) + (ρ− 1)λ, 0 ≤ c < 1, λ > 0;(ρ− 1− 1

2φ2λc

)λ, c ≥ 1, λ ≤ 0;

(ρ− 1)λ, c ≥ 1, λ > 0;

(2.58)

37For details of sufficiency conditions, see, for instance, Grass et al. (2008).

41

H(c, λ) =

14(1− c)2 + 1

4φ2λ2c2 + λc, 0 ≤ c < 1, λ ≤ 0;

14(1− c)2 + λc, 0 ≤ c < 1, λ > 0;

14φ2λ2c2 + λc, c ≥ 1, λ ≤ 0;

λc, c ≥ 1, λ > 0.

(2.59)

Introduce the characteristic function χS of a set S by

χS(x) =

1 if x ∈ S

0 if x /∈ S,(2.60)

differential equations (2.57) and (2.58) can be rewritten more compactly as:

c = c+ χ(−∞,0](λ)1

2φ2λc2

def= F (c, λ) (2.61)

and

λ = (ρ− 1)λ− χ(−∞,0](λ)1

2φ2cλ2 + χ[0,1)(c)

1

2(1− c) def

= G(c, λ). (2.62)

The state–co-state space is presented in Figure 2.13. We first consider the trajectories in the

region where c ≥ 1 and λ > 0, which are the solution to the following canonical system:

c = c,

λ = (ρ− 1)λ,

(2.63)

given by λ(t) = C1e(ρ−1)t and c(t) = C2e

t, where C1 and C2 are positive constants. As

mentioned in the main text, every candidate for an optimal solution must necessarily satisfy

the transversality condition limt→∞ e−ρtλ(t)c(t) = 0. As we now show, the trajectories

in the region considered violate this condition. Moreover, they also violate the following

transversality condition:38

limt→∞

e−ρtH(c, λ) = 0, (2.64)

38In words, the above condition means that the present value of the maximum of the Pontryagin function (thepresent value of the Hamiltonian) converges to zero when time goes to infinity.

42

0 0.5 1 1.5

−0.4

−0.2

0

0.2

0.4

0.6

c

λ

S1 λ = 0

c = 0

λ = 0S

3

q = 0, k = 0q > 0, k = 0

q > 0, k > 0 q = 0, k > 0

Figure 2.13: Illustrative sketch of the state–co-state space (for ρ < 1). The dotted verticalline c = 1 separates the region with zero production from the region with a positive level ofproduction, whereas the horizontal line λ = 0 separates the region with positive investmentfrom the region with zero investment. The loci c = 0 and λ = 0 intersect in the two unstablesteady states (S1 and S3) and the saddle-point steady state S2 (not indicated). A number oftrajectories is indicated by black curves: the arrows point in the direction of the flow. Thethick curve indicates the stable path leading to the saddle-point steady state.

which in this case coincides with the first transversality condition, as shown below. It follows

from Michel (1982) that condition (2.64) is also a necessary condition, and hence that it

allows exclusion of trajectories which verify the other necessary conditions in an infinite

horizon optimization problem. The value of the Hamiltonian function evaluated along the

considered trajectories is H(c, λ) = λ(t)c(t) = C1C2eρt; it follows that

limt→∞

e−ρtH(c, λ) = limt→∞

e−ρtλ(t)c(t) = C1C2 6= 0. (2.65)

Hence, no trajectory in the region given by the restrictions c ≥ 1 and λ > 0 can be optimal.

Consider now trajectories in the region with 0 < c < 1 and λ > 0. These trajectories are

43

the solution to c = c,

λ = 12(1− c) + (ρ− 1)λ.

(2.66)

We show that any trajectory in this region sooner or later enters the region with c ≥ 1 and

λ > 0.

Observe that the equation c = c has as solution c(t) = C3et, where C3 is a positive

constant. Hence, along any trajectory in this region, c is increasing. From (2.58) follows that

if λ = 0, then

λ =1

2(1− c) > 0

for all c ∈ (0, 1). Hence, trajectories in the region with 0 < c < 1 and λ > 0 cannot exit this

region through the line segment {(c, λ) : c ∈ (0, 1), λ = 0}.

We now show that they also cannot exit through the point (c, λ) = (1, 0).39 Let x = (c, λ)

and x0 = (c0, λ0) = (1, 0). Furthermore, let F : D → R2 be a vector function defined as

F(c, λ) =(F (c, λ), G(c, λ)

),

where F is defined in (2.61) and G in (2.62).40 Its domain is D = R+×R ⊂ R2. We are then

looking for a solution to a 2-dimensional nonlinear autonomous dynamical system of the form

x = F(x), x(0) = x0. (2.67)

Take 0 < r < 1. Then the set

D0 = Br(x0) = {x ∈ D : ‖x− x0‖ ≤ r}, (2.68)

39If they exited through this point, they would satisfy the transversality conditions as λ = 0 for λ = 0 andc ≥ 1.

40Though continuous, functions F and G are not differentiable: F is not differentiable with respect to λat λ = 0, whereas G is not differentiable with respect to c at c = 1. Using Peano’s existence theorem, thecontinuity of F implies that at least one solution to (2.67) exists. However, continuity is not enough to guaranteeuniqueness. An additional condition that needs to be fulfilled to guarantee uniqueness of solutions, at least insome neighborhood of x0, is that F is locally Lipschitz in x at x0.

44

is a neighborhood of x0. First, we show that the restriction of the functions F and G on D0 (a

compact subset of D) is Lipschitz.41 Consider first the function

F (c, λ) = c+ λχ(−∞,0](λ)φ2c2

2.

Write for brevity χ = χ(−∞,0] and set h(λ) = χ(λ)λ. Then,

|h(λ)− h(0)| = |χ(λ)λ− 0| = |χ(λ)λ| ≤ |λ| = 1 · |λ− 0|. (2.69)

Hence, h is Lipschitz on D0. As on compact sets continuously differentiable functions are

Lipschitz, as well as sums and products of Lipschitz functions, it follows that F is Lipschitz

on D0. Consider now on D0 the function

G = (ρ− 1)λ− λ2χ(−∞,0](λ)φ2c

2+

1− c2

χ[0,1)(c).

The first term of this expression is a linear function; the second term is the function h

introduced above times a differentiable function. That the final term is Lipschitz at c = 1 is

demonstrated in the same way as for the function h. It follows that G is Lipschitz on D0.

We have proved that the functions F and G are locally Lipschitz at x0. Consequently, so

is the vector function F. By the Picard-Lindelöf Theorem, the system (2.67) has a unique

solution in a neighborhood of x0. As this point is on the exit trajectory, no other trajectory

can pass through it, in particular no trajectory from the region with λ > 0. As c > 0 in the

region with 0 < c < 1 and λ > 0, as shown above, all trajectories in this region exit through

{c = 1, λ > 0} and enter the region with c ≥ 1 and λ > 0 as t→∞. We have already shown

that none of these can be optimal.

Moreover, trajectories through points in the set {(c, λ) : c ∈ (0, 1), λ = 0} satisfy λ > 0;

they move in the region for which λ > 0 and hence cannot be a part of any optimal trajectory.

Due to a one-to-one correspondence between λ and k as given in the first part of (2.21), this

41Simply put, a function f : D → Rn is said to be Lipschitz on B ⊆ D if there exists a constant K > 0 suchthat ‖f(x)− f(y)‖ ≤ K‖x− y‖, for all x, y ∈ B. See Kelley and Peterson (2010), ch. 8, and Sohrab (2003),ch. 4, for the introduction to concepts and precise definitions of terms used in this section.

45

set corresponds to the set {(c, k) : c ∈ (0, 1), k = 0} in the state-control representation of the

solution. As we have shown, all trajectories leading to these points violate the transversality

conditions which every optimal trajectory must necessarily satisfy.

Appendix 2.G Proof of Lemma 5

Assume that there is an optimal investment schedule for which c→ 0 and k →∞ as t→∞.

The instantaneous profit function of the firm reads as

π = (1− q − c)q − k2. (2.70)

As c→ 0 along the trajectories considered, they sooner or later enter the region with positive

production, where we have q = (1− c)/2. Substituting this expression into (2.70), we obtain:

π =(1− c)2

4− k2. (2.71)

As c → 0, (1 − c)2/4 approaches its upper bound of 1/4. The second term in the above

equation, −k2, decreases beyond all bounds as k →∞. As t→∞, there is therefore a time

t0 such that π(t) = 0 for t = t0, and π(t) < 0 for all t > t0.

Changing the investment schedule to k(t) = 0 for all t ≥ t0 would yield a higher value of

total discounted profits Π, contradicting the assumption.

Appendix 2.H Proof of Lemma 6

We know that the state-control system with zero production is:

k(t) = ρk(t),

c(t) = c(t) (1− φk(t)) .

46

At k = 0, this reduces to: k(t) = 0, c(t) = c. Hence, marginal costs increase to infinity

along the exit trajectory as t → ∞. However, as λ = 0 for c ≥ 1 along the exit trajectory,

the transversality condition (2.17) is satisfied. Observe that along the exit trajectory k > 0

for 0 < c < 1 and k = 0 for c ≥ 1. Consequently, it follows from (2.21) that λ = 0 at

c = 1. Then, from (2.58) we have that indeed λ = 0 for all c ≥ 1. Increasing marginal costs

along the exit trajectory sooner or later exceed the value of 1, for which the value of H(c, λ)

becomes 0 (see (2.59)). Hence, the transversality condition (2.16) is satisfied as well.

Appendix 2.I Proof of Corollary 1

In this appendix, we prove that the only candidates for an optimal solution curve are the

stable path of the saddle-point steady state and the exit trajectory. In particular, we show

that any solution curve of the state-control system, given in (2.26) and (2.28) and depicted in

Figure 2.3, starting at a point (c0, k0) with c0, k0 > 0 either (i) ends on the stable path, (ii) ends

on the exit trajectory, (iii) gives rise to a control k(t) that goes to infinity, and then satisfies

the condition of Lemma 5, or (iv) passes through the line segment {(c, k) : c ∈ (0, 1), k = 0},

and is then excluded as an optimal solution by Lemma 4. We note that the vector field defined

by the state-control system is always locally Lipschitz, and that therefore the theorem of

existence and uniqueness of trajectories through a given initial point holds.

There are two situations, determined by the location of the maximum (12, k∗) of the k = 0

isocline which is the quadratic function c 7→ φ4ρc(1− c). In the first situation k∗ ≥ 1/φ, in the

second k∗ < 1/φ.

If k∗ ≥ 1/φ, the state-control space S = {(c, k) : c > 0, k > 0} can be partitioned in

three regions S1, S2, and S3. The first set is defined as follows:

S1 = {(c, k) : 0 < c < 1, k > k∗},

where k∗ = φ/(16ρ) is the maximum of c 7→ φ4ρc(1 − c). To define the second set, we

note that the trajectory γ of the state-control space that passes through the point (c, k) =

47

(1, φ/(16ρ)), when continued backwards in time, necessarily has a second intersection with

the line c = 1. The first of these intersections, as time decreases, is denoted (c, k) = (1, k∗),

where 0 < k∗ < 1/φ. Let D be the region bounded by γ and the line c = 1. Then,

S2 = {(c, k) : c ≥ 1, k > 0}\D.

Finally,

S3 = S\(S1 ∪ S2).

From the state-control equations

c = c(1− φk), k = ρk − φ

4c(1− c),

and the fact that k∗ > 1/φ, it follows that everywhere in S1 we have c < 0 and k > 0. It

follows by Lemma 5 that no trajectory in this region can be optimal.

In region S2, the state-control equations read as

c = c(1− φk), k = ρk.

We claim that every trajectory in this region must leave it through the half-line `1 = {(c, k) :

c = 1, k ≥ k∗}. Note first that the boundary of S2 consists of `1, the curve γ, the line

segment `2 = {(c, k) : c = 1, 0 < k ≤ k∗}, and the half-line `3 = {(c, k) : c ≥ 1, k = 0}.

As γ and `3 are parts of trajectories of the state-control system, no trajectories can leave S2

through them. Moreover, we have c ≥ 0 on `2, so exit through this part of the boundary is

also impossible. The remaining possibilities are to leave through `1, as claimed, or to remain

in S2 indefinitely.

To show that the latter alternative is impossible, note that since k = ρk, any trajectory

in S2 will eventually satisfy k > 2/φ. But then c < −c, and this implies that eventually c

should satisfy c = 1, leaving the region S2 towards S1. As no trajectory in S1 can be optimal,

this now extends to all trajectories in S2.

48

It remains to analyze the trajectories in S3. They can leave that region through the line

segments `2, `4 = {(c, k) : 0 < c < 1, k = 0} or `5 = {(c, k) : 0 < c < 1, k = k∗}, as the

remaining parts of the boundary are trajectories of the state-control system, or through the

point (1, 0) on the exit trajectory.

Trajectories leaving through `2 enter S2 and therefore cannot be optimal. As noted above,

trajectories leaving through `4 are excluded by Lemma 4 from optimality. Trajectories leaving

through `5 enter S1 and again cannot be optimal. Of all the trajectories leaving S3, only those

on the exit trajectory are thus candidates for optimal solutions.

It remains to discuss the trajectories that remain in S3 for all time. By the Poincaré-

Bendixon theorem, since S3 is bounded, the limit set of such a trajectory is either a steady-state

point or a closed curve. The latter possibility can be ruled out as the area enclosed by the curve

would be invariant (cf. Wagener, 2003). The only steady state in S3 that can be approached

by a trajectory is the saddle, if k∗ > 1/φ, or the semi-stable steady state, if k∗ = 1/φ, and

this shows the result.

If k∗ < 1/φ, the situation is much simpler. Define in that case

S1 = {(c, k) : 0 < c < 1, k > 1/φ},

S2 = {(c, k) : c ≥ 1, k > 0},

S3 = S\(S1 ∪ S2).

As above, points in S1 cannot be optimal as a consequence of Lemma 5; points in S2 eventually

end up in S1 and are excluded by the same reasoning; and as there are no saddle points in S3

and the trajectories leaving through `4 cannot be optimal, the only remaining candidate is the

exit trajectory.

49

Appendix 2.J Proof of Lemma 8

From (2.31), we know that the Hamiltonian for c ≥ 1 is given by:

H(c, k) = k

(k − 2

φ

); (2.72)

and from Lemma 7, we know that the comparison of the total discounted profits of each two

candidate optimal paths amounts to comparing the values of the respective Hamiltonians in

the initial point of each respective path. As the Hamiltonian in the case of zero investment and

zero production is zero, the indifference point between the stable path and the exit trajectory

in the region with zero production must be the point at which the Hamiltonian evaluated along

the stable path obtains the value of zero.

H(c, k) = k(k − 2

φ

)= 0

⇒ k = 0 or k = 2φ

The solution structure (2.39)-(2.40) tells us that the stable path in the region with c ≥ 1

decreases as t→ −∞, assuming that the stable path covers the respective region. Observe

that the derivative of the Hamiltonian with respect to k is ∂H(c,k)∂k

= 2(k − 1

φ

), which is

positive for k > 1/φ and negative for k < 1/φ.

If the stable path enters the region with zero production at all, it enters this region at the

point where k > 1φ

; this follows directly from equation (2.28).

The same equation implies that k is decreasing along the stable path as t→ −∞, assuming

the stable path enters the zero-production region. The above conclusions lead us to distinguish

three cases. First, if the stable path crosses the boundary line c = 1 at k > 2φ

, then the value

of the Hamiltonian is decreasing along the stable path in the region with c ≥ 1 as t→ −∞,

passes zero at k = 2φ

and is negative afterwards. In this case, the indifference point is the value

of the marginal cost c > 1 that corresponds to the point(c, 2

φ

)on the stable path. Second, if

the stable path reaches c = 1 at exactly k = 2φ

, then the indifference point is c = 1. Third, for

lower values of k on the stable path at c = 1, the value of the Hamiltonian evaluated along the

50

stable path in the zero-production region is negative, such that the point of indifference (if at

all) must be in the region with positive production.

51

Chapter 3

Cartels and Innovation

3.1 Introduction

There are compelling reasons for rival firms to set up R&D cooperatives. These “organizations,

jointly controlled by at least two participating entities, whose primary purpose is to engage

in cooperative R&D” (Caloghirou et al., 2003) allow risks to be spread, secure better access

to financial markets, and pool resources such that economies of scale and scope in both

research and development are better realized. In the words of John Kenneth Galbraith (1952,

pp. 86 – 87, emphasis added): “Most of the cheap and simple innovations have, to put it

bluntly and unpersuasively, been made. Not only is development now sophisticated and

costly but it must be on a sufficient scale so that success and failures will in some measure

average out.” Moreover, R&D cooperatives internalize technological spillovers - the free flow

of knowledge from the knowledge creator to its competitors.1 Sustaining R&D cooperatives

is thus perceived to diminish the failure of the market for R&D.2

However, as Scherer (1980) observes: “the most egregious price fixing schemes in

American history were brought about by R&D cooperatives”, an observation that confirms a

widely-aired suspicion (see, e.g., Pfeffer and Nowak (1976), Grossman and Shapiro (1986),

1Bloom et al. (2007) estimate that a 10% increase in a competitor’s R&D is associated with up to a 2.4%increase in a firm’s own market value. Not surprisingly, internalizing technological spillovers is one of the primereasons for firms to join an R&D cooperative (Hernan et al., 2003; see also Roeller et al., 2007).

2This motivates in particular why independent firms are allowed to cooperate in R&D. See Martin (1997)for an overview of the policy treatment of R&D cooperatives in the E.U., the U.S., and Japan.

and Brodley, 1990).3 The channels through which cooperation in R&D can facilitate product

market collusion have been examined in a number of theoretical studies (see, e.g., Martin

(1995), Greenlee and Cassiman (1999), Cabral (2000), Lambertini et al. (2002) and Miyagiwa,

2009). As Martin (1995) puts it: “common assets create common interests, and common

interests make it more likely that firms will non-cooperatively refrain from rivalrous behavior.”

(Martin, 1995, p. 740).4 While price fixing may lead to a reduction of standard surplus

measures, in this chapter we challenge the view that extending cooperative behavior to the

product market necessarily diminishes consumer surplus and total surplus.

Geroski (1992) argues that it is the feedback from product markets that directs research

towards profitable tracks and that, therefore, for an innovation to be commercially successful

there must be strong ties between marketing and development of new products. Jacquemin

(1988) observes that R&D cooperatives are fragile and unstable. He reasons that when there

is no cooperation in the product market, there exists a continuous fear that one partner in

the R&D cooperative may be strengthened in such a way that it will become too strong a

competitor in the product market. Preventing firms from collaborating in the product market

may therefore destabilize R&D cooperatives, or prevent their creation in the first place. Our

focus is on the incentives to develop further an initial technology (‘ideas’). In general, we find

that product market collusion fosters R&D investment incentives because more of the ensuing

economic rents can be appropriated by the investing firms. As a result, if firms collude, they

will bring more initial technologies to full maturation. And this is unambiguously welfare

enhancing.

Static models of R&D predict total surplus to go down if members of an R&D cooperative

3Goeree and Helland (2008) find that in the U.S. the probability that firms join an R&D cooperative hasgone down due to a revision of antitrust leniency policy in 1993. This revision is perceived as making collusionless attractive. Goeree and Helland (2008) conclude that “Our results are consistent with RJVs [research jointventures] serving, at least in part, a collusive function.” Related evidence is reported by Duso et al. (2010).They find that the combined market share declines if partners in an RJV compete on the same product market(“horizontal RJVs”), while it increases if members of the RJV are not direct rivals (“vertical RJVs”). Thelaboratory experiments of Suetens (2008) show directly that members of an RJV are more likely to collude onprice.

4In a similar vein, Fisher (1990, p. 194) concludes that “...[firms] cooperating in R&D will tend to talkabout other forms of cooperation. Furthermore, in learning how other firms react and adjust in living with eachother, each cooperating firm will get better at coordination. Hence, competition in the product market is likely tobe harmed.”

54

collude in the product market.5 But a static view of the world necessarily ignores an important

aspect of R&D: time. It takes time for an initial idea to be developed towards a marketable

product; continuous process innovations gradually reduce production costs (Utterback, 1994).

In this chapter, therefore, we develop a dynamic model of R&D to examine the welfare

implications of product market collusion by firms of an R&D cooperative. This analysis

builds upon our discussion in Chapter 2, where we developed a global framework for an

innovating monopolist.

Static models of R&D also predict that the marginal benefit of any R&D investment

increases if firms collude in the product market. That is, firms are willing to spend more

resources on R&D if the intensity of product market competition is diminished through some

collusive agreement.6 This suggests that any initial idea (that is, any initial level of marginal

costs) is more likely to be developed further if firms collude in the product market. Therefore,

in a formal analysis, no level of initial marginal costs should be excluded from the analysis,

in particular marginal costs that exceed the choke price (that is, the lowest price at which

the quantity sold is zero). Moreover, requiring marginal costs to be below the choke price at

all times implicitly imposes R&D activity and production to coexist at all times. Surely this

assumption is quite unlikely to hold for new technologies at their early stages of development.

Research starts long before a prototype sees the light; development begins long before the

launch of a new product. To properly assess the welfare implications of product market

collusion induced by an R&D cooperative, this development phase should be included in the

analysis.

Therefore, a distinguishing feature of our approach is that we provide a global analysis.

That is, we consider all possible values of initial marginal costs, including those above the

5d’Aspremont and Jacquemin (1988) are the first to show that a scenario where firms cooperate in R&D andcollude in the ensuing product market yields a lower total surplus than the situation where firms cooperate inR&D only.

6Again, d’ Aspremont and Jacquemin (1988) are the first to show this formally. This touches upon thedebate between Schumpeter Mark I (“...new combinations are, as a rule, embodied, as it were, in new firmswhich generally do not arise out of the old ones but start producing beside them;...in general it is not the ownerof stage-coaches who builds railways”; Schumpeter, 1934, p. 66) and Schumpeter Mark II (“As soon as we gointo the details and inquire into the individual items in which progress was most conspicuous, the trail leads notto the doors of those firms that work under conditions of comparatively free competition but precisely to thedoors of the large concerns...and a shocking suspicion dawns upon us that big business may have had more to dowith creating that standard of living than with keeping it down”; Schumpeter, 1943, p. 82).

55

choke price. Hence, we allow research efforts to precede production.7 Also, we do not

limit ourselves to an analysis of equilibrium paths but we consider all trajectories that are

candidates for an optimal solution. This enables us to determine the location of critical points

- points at which the optimal investment function qualitatively changes. In particular, we

determine the value of marginal costs for which R&D investments are terminated, and for

which they are not initiated at all. The size of these critical cost levels is affected by firm

conduct. Extending the R&D cooperative agreement to product market collusion can lead to

qualitatively different long-run solutions, despite starting from an identical initial technology.

For a global analysis we have to use bifurcation theory. This gives us a bifurcation

diagram that indicates for every possible parameter combination the qualitative features of any

market equilibrium. Like in the monopoly case discussed in Chapter 2, it yields four distinct

possibilities: (i) initial marginal costs are above the choke price and the R&D process is

initiated; after some time production starts and marginal costs continue to fall with subsequent

R&D investments; (ii) initial marginal costs are above the choke price and the R&D process

is not initiated, yielding no activity at all; (iii) initial marginal costs are below the choke price

and the R&D process is initiated; production starts immediately, marginal costs continue

to fall over time, and the steady-state is reached that is characterized by continuous R&D

investments, and (iv) initial marginal costs are below the choke price and the initiated R&D

process is progressively scaled down; production starts immediately but the technology (and

production) will die out over time; the firms leave the market. To date, the literature has

considered possibility (iii) only, and only partially so.

We then compare two different scenarios across these possibilities. In the first scenario,

labeled ‘competition’, firms cooperate in R&D and compete on the concomitant product

market. In the second scenario, labeled ‘collusion’, cooperation in R&D is extended to

collusion in the product market. We then compare the qualitative properties of these two

scenarios in order to assess the potential set-back of R&D cooperatives in that they can serve

7Here we deviate from the related literature that, with no exception, restricts the analysis to initial levels ofmarginal costs that are below the choke price (cf. Petit and Tolwinski (1999), Cellini and Lambertini (2009),Lambertini and Mantovani (2009), and Kovac et al. (2010)). As will become clear below, this restrictionexcludes a crucial part of the parameter space.

56

as a platform to coordinate prices.

Our analysis yields three key findings: (i) if firms collude, the range of initial marginal

costs that leads to the creation of a new market is larger, (ii) collusion in the product market

accelerates the speed with which new technologies enter the product market, and (iii) the set

of initial marginal costs that induces firms to abandon the technology in time is larger if firms

do not collude in the product market. Related, we show that there are parameter configurations

whereby collusion in the product market yields higher total surplus. We thus qualify the

conclusion of Petit and Tolwinski (1999, p. 206) that “[collusion] is socially inferior to other

forms of industrial structures”, a conclusion that is based on a local analysis and which can be

seen as representative for the related literature.

Our results are not without policy implications. When designing antitrust policies, it is

important to understand that these policies not only affect current markets, but also markets

that are not yet visible. Preventing firms from colluding in the product market reduces the

number of potential R&D trajectories that successfully lead to new markets. In itself this

constitutes a welfare loss. However, because not developing further an initial technology

does not surface as a direct surplus loss, this welfare loss remains hidden. It is left for future

research to assess empirically the size of this hidden cost.

3.2 The model

Time t is continuous: t ∈ [0,∞). There are two a priori fully symmetric firms which both

produce a homogenous good at constant marginal costs. In every instant, market demand is:

p(t) = A−Q(t), (3.1)

where Q(t) = q1(t) + q2(t), with qi(t) the quantity produced by firm i at time t, and where

p(t) and A are respectively the market price at time t and the choke price.

Each firm can reduce its marginal cost by investing in R&D. In particular, firm i exerts

R&D effort ki(t) and as a consequence of these investments, its marginal cost evolves over

57

time as follows:dcidt

(t) ≡ ci(t) = ci(t) (−ki(t)− βkj(t) + δ) , (3.2)

where kj(t) is the R&D effort exerted by its rival and where β ∈ [0, 1] measures the degree

of spillover. The parameter δ > 0 is the constant rate of decrease in efficiency due to the

ageing of technology and organizational forgetting.8 Both firms have an identical initial

technology ci(0) = cj(0) = c0, which is drawn by Nature. The cost of R&D efforts per unit

of time Γi(ki(t)) takes the form

Γi(ki) = bk2i , (3.3)

where b > 0 is inversely related to the cost-efficiency of the R&D process. Hence, the R&D

process exhibits decreasing returns to scale (Schwartzman (1976); see also the discussion in

Chapter 2). Both firms discount the future with the same constant rate ρ > 0. Either firms’

instantaneous profit therefore equals:9

πi(qi, Q, ki, ci) = (A−Q− ci)qi − bk2i , (3.4)

yielding total discounted profit:

Πi(qi, Q, ki, ci) =

∫ ∞0

πi(qi, Q, ki, ci)e−ρtdt. (3.5)

The model has five parameters: A, β, b, δ, and ρ. The analysis can be simplified by

considering a rescaled version of the model which, as defined in Lemma 9, carries only three

parameters: β, φ, and ρ (see Appendix 2.A in Chapter 2 for the proof).

Lemma 9. By choosing the units of t, qi, qj , ci, cj , ki, and kj appropriately, we can assume

8Assuming an exogenous depreciation rate is common in the literature. See footnote 5 in Chapter 2 for adiscussion.

9Implicitly we assume here that firms know market demand in advance of production. Relaxing thisassumption would make the analysis more complex, while it would not alter any of our conclusions as to thecomparison of the competitive and collusive scenario, provided that under both scenarios firms have identicalexpectations about future demand.

58

A = 1, b = 1, and δ = 1. This yields the following rescaled version of the model:

πi(qi, Q, ki, ci) = (1− Q− ci)qi − k2i , (3.6)

Πi(qi, Q, ki, ci) =

∫ ∞0

πi(qi, Q, ki, ci)e−ρtdt (3.7)

˙ci = ci

(1−

(ki + βkj

)φ), ci(0) = c0, ci ∈ [0,∞)∀ t ∈ [0,∞) (3.8)

qi ≥ 0, ki ≥ 0 (3.9)

ρ > 0, φ > 0 (3.10)

with conversion rules: qi = Aqi, qj = Aqj , ki = A√bki, kj = A√

bkj , ci = Aci, cj = Acj ,

πi = A2πi, πj = A2πj , φ = Aδ√b, t = t

δ, ρ = ρ

δ.

The rescaled version of the model introduces a new parameter φ, which captures the profit

potential of a technology in hand: a higher (lower) A implies higher (lower) potential sales

revenue, a higher (lower) b implies that each unit of R&D effort costs the firm more (less),

whereas a higher (lower) δ implies that each unit of R&D effort reduces the marginal cost by

less (more). Therefore, a higher (lower) φ corresponds to a higher (lower) profit potential of a

technology. For notational convenience, we henceforth omit tildes.

3.3 Competition versus Collusion

This section derives the necessary conditions for optimal production and investment schedules

in case firms cooperate in R&D, but compete in the product market (a scenario labelled

‘competition’), and in case firms cooperate in R&D and collude in the product market (a

scenario labelled ‘collusion’).

3.3.1 Competition

Both firms operate their own R&D laboratory and production facility, and while they select

their output levels non-cooperatively, they adopt a strictly cooperative behavior in determining

59

their R&D efforts so as to maximize joint profits. These assumptions amount to imposing

a priori the symmetry condition ki(t) = kj(t) = k(t).10 As ci(0) = cj(0) = c0, this implies

that ci(t) = cj(t) = c(t). Equation (3.8) thus reads as:

c = c(1− (1 + β)φk). (3.11)

It may seem reasonable to assume that when firms cooperate in R&D, they also fully share

information, that is, to assume the level of spillover to be at its maximum (β = 1; see Kamien

et al., 1992). For the sake of generality, we do not a priori fix the value of β at its maximal

value. There are also intuitive arguments for not doing so as there might still be some ex post

duplication and/or substitutability in R&D outputs if firms operate separate laboratories (see

also the discussion in Hinloopen, 2003).

The instantaneous profit of firm i is

πi(qi, Q, k, c) = (1−Q− c)qi − k2, (3.12)

with Q = q1 + q2, yielding its total discounted profit over time:

Πi(qi, Q, k, c) =

∫ ∞0

πi(qi, Q, k, c)e−ρtdt. (3.13)

As firms cooperatively decide on their R&D efforts, the only independent decisions are those

of production levels. However, as quantity variables do not appear in the equation for the

state variable (3.11), production feedback strategies of a dynamic game are simply static

Cournot-Nash strategies of each corresponding instantaneous game.

Maximising πi over qi ≥ 0 gives us standard Cournot best-response functions for the

product market:

qi(qj) =

12(1− c− qj) if qj < 1− c,

0 if qj ≥ 1− c.(3.14)

10We consider symmetric equilibria only. See Salant and Shaffer (1998) for a specific example of a staticmodel of R&D in which it is optimal for firms in an R&D cooperative to make unequal investments.

60

This expresses the fact that the constraint qi ≥ 0 is binding when qj ≥ 1 − c. Solving for

Cournot-Nash production levels, we obtain

qN =

13(1− c) if c < 1,

0 if c ≥ 1.

(3.15)

Consequently, the instantaneous profit of each firm is

π(c, k) =

19(1− c)2 − k2 if c < 1,

− k2 if c ≥ 1.

(3.16)

The dynamic optimization problem of the R&D cooperative reduces to finding an R&D effort

schedule k∗ for each firm that maximizes the total discounted joint profit of the two firms,

taking into account the state equation (3.11), the initial condition c(0) = 0, and the boundary

condition k(t) ≥ 0 which must hold at all times. Note that according to (3.11), if c0 > 0,

then c(t) > 0 for all t. The state space of this problem is the interval [0,∞) of marginal cost

levels.

To solve this problem, we introduce the current-value Pontryagin function (also called

pre-Hamilton or un-maximized Hamilton function)11

P (c, k, λ) =

19(1− c)2 − k2 + λc(1− (1 + β)φk) if c < 1,

− k2 + λc(1− (1 + β)φk) if c ≥ 1,

(3.17)

where λ is the current-value co-state variable of a firm in the R&D cooperative. The co-state,

or shadow value, measures the marginal worth of the increment in the state c for each firm

in the cartel at time t when moving along the optimal path. As marginal cost is a “bad”, we

expect λ(t) ≤ 0 along optimal trajectories.

We use Pontryagin’s maximum principle to obtain the solution to our optimization problem.

11We omit a factor of 2 for combined profits to obtain the solution expressed in per-firm values. Due tosymmetry, maximizing the per-firm total profit corresponds to maximizing the two firms’ combined total profit.

61

Maximising over the control k ≥ 0 yields

k = −1

2λc(1 + β)φ, (3.18)

whenever the value on the right hand side is nonnegative, and k = 0 otherwise. The maximum

principle states further that the optimizing trajectory necessarily corresponds to the trajectory

of the state-costate system

c =∂P

∂λ, λ = ρλ− ∂P

∂c,

where k is replaced by its maximizing value. For λ ≤ 0, relation (3.18) gives a one-to-one

correspondence between the co-state λ and the control k. We use this relation to transform

the state-costate system into a state-control system which an optimizing trajectory has to

satisfy necessarily as well. This system consists of two regimes (following the two part

composition of the Pontryagin function). The first one corresponds to c < 1 and positive

production (q = (1 − c)/3). The second one corresponds to c ≥ 1 and zero production.12

The state-control system with positive production consists of the following two differential

equations:13

k = ρk − (1+β)φ

9c(1− c),

c = c (1− (1 + β)φk) .

(3.19)

12Recall from Lemma 9 that A = 1 in the rescaled model. In the non-rescaled model, the analogousconditions for positive and zero production are c(t) < A and c(t) ≥ A, respectively.

13Our closed-loop solution differs from that of Cellini and Lambertini (2009), who consider the case whenmarginal cost is always lower than the choke price. This is so because their proof that the open-loop andclosed-loop solutions coincide is flawed by the fact that in their derivation of the closed-loop solution, players’output choices are not properly treated as functions of the state variable. The derivations of the authors implicitlyassume that if marginal cost within the R&D cooperative changes, the opponent’s quantity does not change,which is the violation of the feedback principle underlying the closed-loop solution. It is also counterintuitiveas firms in the R&D cooperative are supposed to jointly decide on their R&D efforts taking into account thatmarginal cost in any period affects the ensuing Nash-equilibrium profits. Our calculations also show that thesolution of Cellini and Lambertini (2009) yields situations in which per-firm profits in the competitive scenarioexceed those of the collusive scenarios, which obviously contradicts the notion that in case of collusion firmsmaximize joint profits.

62

The state-control system with zero production is given by

k = ρk,

c = c (1− (1 + β)φk) .

(3.20)

3.3.2 Collusion

If firms collude, they adopt a strictly cooperative behavior in determining both their R&D

efforts and output levels. These assumptions amount to imposing a priori the symmetry

conditions ki(t) = kj(t) = k(t) and qi(t) = qj(t) = q(t). Equation (3.8) reads therefore as:

c = c(1− (1 + β)φk). (3.21)

The profit of each firm in every instant is:

π(q, k, c) = (1− 2q − c)q − k2, (3.22)

yielding its total discounted profit over time:

Π(q, k, c) =

∫ ∞0

π(q, k, c)e−ρtdt. (3.23)

The optimal control problem of the two firms is to find controls q∗ and k∗ that maximize the

profit functional Π subject to the state equation (3.21), the initial condition c(0) = c0, and

two boundary conditions which must hold at all times: q ≥ 0 and k ≥ 0.14 Notice again that

according to (3.21), if c0 > 0, then c(t) > 0 for all t.

The current-value Pontryagin function now reads as:

P (c, q, k, λ) = (1− 2q − c) q − k2 + λc (1− (1 + β)φk) , (3.24)

where λ is the current-value co-state variable. It now measures the marginal worth at time t

14Again, due to symmetry, maximizing per-firm total profit is the same as maximizing the two firms’combined total profit.

63

of an increment in the state c for a colluding firm when moving along the optimal path.

The necessary conditions for the solution to the dynamic optimization problem consist

again of a state-control system which has two regimes. As in the competitive case, the first

regime corresponds to c < 1 and positive production (q = (1 − c)/4), while the second

corresponds to c ≥ 1 and zero production.

The state-control system in the region with positive production reads as

k = ρk − (1+β)φ

8c(1− c),

c = c (1− (1 + β)φk) ,

(3.25)

whereas the state-control system with zero production is

k = ρk,

c = c (1− (1 + β)φk) .

(3.26)

3.4 Analysis

The systems (3.19) – (3.20) and (3.25) – (3.26) have the same structure as the state-control

system for the monopoly problem considered in Chapter 2. Indeed, consider the system

c = c (1− (1 + β)φk) , (3.27)

k = ρk − α(1 + β)φc(1− c)χ[0,1](c), (3.28)

where χA(c) = 1 if c ∈ A and χA(c) = 0 if c 6∈ A. The monopoly system in Chapter 2 as

well as systems (3.19) – (3.20) and (3.25) – (3.26) are instances of the system (3.27) – (3.28),

with α = 1/9 for the competitive scenario, α = 1/8 for the collusion scenario, and α = 1/4

for the monopoly studied in Chapter 2.

Therefore, the analysis of system (3.27) – (3.28) corresponds to the analysis of the system

in Chapter 2. Accordingly, we confine ourselves to stating the principal results of system (3.27)

– (3.28), including its bifurcation diagrams. All proofs are in the appendices of Chapter 2.

The first result gives the properties of the steady states of the state-control system.

64

Proposition 2. Let

D =1

4− ρ

α(1 + β)2φ2.

Depending on the value of D, there are three different situations.

1. If D > 0, the state-control system with positive production (3.25) has three steady

states:

i. (cC , kC) = (0, 0) is an unstable node,

ii. (cC , kC) =(

12

+√D, 1

(1+β)φ

)is either an unstable node or an unstable focus,

and

iii. (cC , kC) =(

12−√D, 1

(1+β)φ

)is a saddle-point steady state.

2. At D = 0, there are two steady states:

i. (cC , kC) = (0, 0), which is an unstable node, and

ii. (cC , kC) =(

12, 1(1+β)φ

), which is a semi-stable steady state.

3. If D < 0, the origin (cC , kC) = (0, 0) is the unique steady state of the state-control

system with positive production, which is unstable.

The system consequently exhibits a saddle-node bifurcation at D = 0.

The stable manifold of the saddle-point steady state is one of the candidates for an optimal

solution. As neither the Mangasarian nor the Arrow concavity conditions are satisfied, the

stable manifold is not necessarily optimal. Note that Proposition 2 already implies that there

should be other candidates for optimality as there is a parameter region for which there is no

saddle point, and hence no stable manifold to it. We have the following result.

Corollary 2. The set of candidates for an optimal solution consist of the stable path of the

saddle-point steady state and the trajectory through the point (c, k) = (1, 0).

The thick black lines L1 and L2 in Figure 3.1 indicate these two candidates. In this figure,

the dotted vertical line c = 1 separates the region with zero production from the region of

65

0 0.5 1 1.5

c

k

q > 0 q = 0

k = 0

c = 01φ

L1

S3

S1

S2

L2

L3

Figure 3.1: Candidate trajectories in the state-control space.

positive production. We label the trajectory L2 the “exit trajectory”, as both firms eventually

leave the region 0 < c < 1 of positive production when following this trajectory.

To assess the dependence of the solution structure on the model parameters, we carry out a

bifurcation analysis. This consists in identifying the parameter values at which the qualitative

structure of the optimal dynamics changes. These ‘bifurcating’ values bound open parameter

regions such that the optimal dynamics are qualitatively equal for all parameter values in the

region (see Wagener (2003), Kiseleva & Wagener, 2010, 2011). As for any point in the region,

a sufficiently small change in the parameter value does not lead to a qualitative change of the

dynamics, these regions are said to characterize stable types of dynamics.

In Chapter 2, we identified four distinct stable types. Figure 3.2 illustrates these types;

Figure 3.3 shows the corresponding bifurcation diagram.

The first type is one of a “Promising Technology”, where there is an indifference thresh-

old15 in the region of no production. In an optimal control problem, an indifference threshold

is a point in state space where the decision maker is indifferent between two optimal trajecto-

ries that have distinct long-term limit behavior. In case of a Promising Technology, there is a

point c > 1, such that for 0 < c0 ≤ c, it is optimal to start developing the initial technology,

15Also known as Skiba, Dechert-Nishimura-Skiba or DNSS point; see Grass et al. (2008).

66

0 0.5 1 1.3 1.50

0.05

0.1

0.15

0.2

0.25

c

k

S2

(a) Promising Technology

0.2 0.4 0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

c

k

e+

S2

(b) Strained Market

0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

c

k

cR

S2 S

3

(c) Uncertain future

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

c

k

(d) Obsolete technology

Figure 3.2: The four stable types of dynamics.

ending up in the saddle-point steady state in the region of positive production. In particular,

this happens for initial values of c that are in the no-production region. If c0 ≥ c, it is optimal

not to initiate R&D efforts as in this case potential future profits do not suffice to compensate

for losses that would be incurred in the initial periods during which firms would invest in

R&D but would not produce yet. Note that for c0 = c, there are two entirely different R&D

investment policies, which are, nevertheless, both optimal.

The second type corresponds to a “Strained Market”, where there is an indifference

threshold in the region of positive production: 0 < c < 1. The new feature here is that for

c ≤ c0 < 1, the firm eventually leaves the market. It is optimal however to invest in R&D to

slow down the technological decay.

We label the third type, which occurs only in a small part of the parameter space, the

“Uncertain Future”. Instead of an indifference point, here a repelling steady state divides the

67

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

ρ/δ

IRA(1+β)

δ√b SN

SN’

IAIV Obsolete technology

II Strained market

I Promising technology

III Uncert. fut.

ISN

Figure 3.3: Bifurcation diagram for β = 1.

initial states that optimally converge to the steady state with positive production and those

with no production region. If the system starts exactly at the repelling point, it stays there

indefinitely; when it starts close to it, it stays there for a long period of time, after which it

converges to one of the steady states.

The fourth type typifies the dynamics of an “Obsolete Technology”. Whatever the initial

state, the firms let the technology decay and eventually leave the market. In the region of

positive production, the decay is again slowed down by R&D investments.

In the bifurcation diagram, the uppermost curve represents parameter values for which

the indifference point is exactly at c = 1. At the saddle-node curve (SN), an optimal repeller

and an optimal attractor collide and disappear. The curve SN’ corresponds to saddle-node

bifurcations in the state-control system that do not correspond to optimal dynamics. At

the indifference-attractor bifurcations (IA),16 an indifference point collides with an optimal

attractor and both disappear. Finally, at an indifference-repeller bifurcation, an indifference

16For the terminology, see Kiseleva & Wagener (2010, 2011).

68

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

ρ/δ

A(1+β)

δ√b

IR

SN’

III

IAIV

I

SN

II

ISN

Figure 3.4: Bifurcation diagram. The plot depicts the bifurcation curves for the competitivescenario (grey) together with those of the collusive scenario (black). Throughout, the collusivecurves pass through lower values of A(1+β)/(δ

√b) (non-rescaled variables) when compared

at equal values of ρ/δ (non-rescaled).

point turns into an optimal repeller. The central indifference-saddle-node (ISN) bifurcation

point at (ρ, φ) ≈ (2.14, 8.78) organizes the bifurcation diagram. The curve representing

indifference points at c = 1 obtains a value of φ ≈ 2.998 for ρ = 1× 10−5.

3.5 Collusion and the incentives to innovate

Having characterized the global optimum of both the competitive and the collusive scenario,

we can compare their respective bifurcation diagrams. These are superimposed in Figure 3.4.

Qualitatively, there is no difference between the diagrams. There are, however, important

quantitative differences which the following observation summarizes:

Numerical observation 1. Over the entire parameter space we observe that if firms collude,

the bifurcation curves lie below the concomitant curves in case firms compete.

69

This observation has two corollaries. First, the “Promising Technology” region is larger

if firms collude. Put differently, if firms collude, the situation where firms first invest in

R&D, and only after some initial development period start producing, is more likely to occur.

Second, if firms collude, the “Obsolete Market” region is smaller. That is, due to collusion, it

is less likely that firms either do not develop further an initial technology, or that they invest

in R&D only to abandon the technology in time.

Numerical observation 2. Over the entire parameter space we observe that whenever a

threshold value of initial marginal costs exists in both scenarios (be it an indifference point or

a repeller), it is larger if firms collude.

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

c

k

c1 c2

(a)

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Π

cc1 c2

(b)

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

c

CS

c2c1

(c)

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

c

TS

c2c1

(d)

Figure 3.5: State-control space (a), total discounted profit (b), consumer surplus (c), and totalsurplus (d), when the indifference point is in the region with zero production. Parameters:(β, ρ, φ) = (1, 0.1, 2.25). Grey curves correspond to competition, whereas the black onescorrespond to colllusion. For all c0 ∈ (c1, c2), the collusive scenario brings about higherconsumer surplus and total surplus than the competitive scenario.

The implications of this observation are twofold. First, if firms collude, the set of initial

70

technologies that are developed further and that lead to the saddle-point steady state is larger.

Figure 3.5 illustrates this implication. If the initial technology draw c0 falls in the non-empty

interval (c1, c2), the firms will develop the technology and this will eventually give rise to

a new market, but only if firms collude. If they compete, neither firm will develop the

technology.

Note that a higher value of initial marginal cost implies larger early-stage losses because

there is no profitable production yet. Obviously, these losses are more quickly off-set by

future profits if firms collude, due to higher mark-ups. Therefore, under collusion, firms can

afford to invest more in R&D prior to production, and thereby to bring down over time a

higher initial level of marginal cost.

2 4 6 8 10 12 14 161

2

3

4

5

6

7

8

9

10

A(1+β)

δ√b

c

ρ/δ = 0.1

C1

C2

ρ/δ = 0.5

ρ/δ = 1

∆1c

∆2c

Figure 3.6: Dependence of the indifference point c on model parameters. Curves are drawnfor three fixed values of ρ. Curves for competition (dotted) lie below the curves for collusion(full).

For this situation, Figure 3.6 illustrates some comparative statics of the indifference

points. Obviously, these points are positively related to market size and R&D efficiency.

Note however that also the difference ∆c between ccompetitive and ccollusive increases if the R&D

71

process becomes more efficient and/or if the market size becomes larger, the more so the

lower the discount rate is. This inelasticity corresponds in Figure 3.6 to a larger slope of the

convex curves. Because future mark-ups are positively related to both market size and R&D

efficiency, an increase in either one of these has a larger (positive) effect on future profits if

firms collude. And these future benefits feature more prominently in total discounted profits

if the discount rate is lower. Put differently, indifference points occur at smaller values if the

discount rate goes up, all else equal (cf. the relative location of C1 and C2 in Figure 3.6).

A particular situation arises when the indifference point under collusion is above the

choke price, while it is below the choke price if firms compete. This is the case for all points

in Figure 3.4 in between the two bifurcation curves that separate the Promising Technology

region from the Strained Market region. In any such a situation, only colluding firms may

develop further a technology which requires investments in advance of production. Competing

firms never develop it further as for them the exit trajectory is optimal for all initial costs

above the choke price. Obviously, the latter scenario yields a lower total surplus.17

Second, if firms collude, the set of initial technologies that triggers no investment in R&D

at all or that induces firms to select the exit trajectory is smaller. Figure 3.7 illustrates this for

the Strained Market region. The strained investment circumstances, in the sense of a high

depreciation rate in comparison to the market size and R&D efficiency, induce competing

firms to exit the market in due time for all c0 > c1. In contrast to this, colluding firms exit

the market only for c0 > c2, which is again due to larger mark-ups in the product market.

Initial technologies c0 in the interval (c1, c2) are therefore brought to full maturation only by

colluding firms, which leads to a direct welfare gain of collusion.

So far we can conclude that due to collusion (i) it is more likely that we have a Promising

Technology, and if so, that it is more likely to be developed further, (ii) it is less likely that

we have an Obsolete Technology, and if so, it is more likely that firms invest in R&D, albeit

temporarily, and (iii) if the technology causes a Strained Market or if it induces an Uncertain

Future, it is less likely that it will be taken of the market in due time. In sum, due to collusion

it is more likely that firms invest in R&D, and that these investments eventually lead to a

17In the next section, we discuss what our analysis implies for competition policies.

72

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

c

k

c1 c2

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

c

Π

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

c

CS

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

c

TS

(d)

Figure 3.7: State-control space (a), total discounted profit (b), consumer surplus (c), andtotal surplus (d), when the indifference point is within the region with positive production.Parameters: (β, ρ, φ) = (1, 0.1, 2). Grey curves correspond to competition, whereas the blackones correspond to collusion. Full curves correspond to the stable path, whereas the dottedones to the exit trajectory. Dots indicate the saddle-point steady state. For all c0 ∈ (c1, c2), thecollusive scenario brings about higher consumer surplus and total surplus than the competitivescenario.

steady state with positive production.

The next observation is about the intensity of the R&D process as such.

Numerical observation 3. Over the entire parameter space we observe that whenever both

scenarios trigger either the exit trajectory or the stable path towards the saddle-point steady

state, the trajectory of the collusive scenario lies above that of the competitive scenario.

This observation implies the following. First, whenever both scenarios lead to the saddle-

point steady state, marginal costs in the collusive scenario are lower than in case of competition,

because colluding firms have invested more in cost-reducing R&D to arrive at the long-run

73

equilibrium. Put differently, collusion yields a higher production efficiency. Second, if the

initial technology leads to production after some initial development period only, colluding

firms will enter this production phase more quickly. That is, at every instant of the pre-

production phase, colluding firms invest more in R&D in order to bring some initial level of

marginal costs below the choke price. As a result, less favorable initial technologies will be

brought to the market if firms collude. Third, colluding firms abandon obsolete technologies

at a lower pace. This implication, that a monopolist holds on longer to a technology that

is bound to leave the market, has a similar vein as the argument of Arrow (1962), that a

monopolist has less incentive to invest in R&D than an otherwise identical but perfectly

competitive market, because by doing so the monopolist replaces current monopoly profits by

future (higher) monopoly profits. Here, of course, the alternative for the colluding firms is to

exit the market more quickly (rather than staying in the market as a monopolist, as in Arrow,

1962), an alternative that for them is not optimal (see Figure 3.8).

3.6 Competition policies

Summarizing the results of the previous section, we have found that the collusive scenario is

more R&D intensive: R&D investment levels are higher and the set of initial technologies

that is developed further is larger. The price to be paid for this increased innovation intensity

is the higher mark-up in the product market. It should therefore not come as a surprise that

the welfare comparison between the two scenarios yields a mixed picture.

First, as alluded to in the previous section, if the firms develop an initial technology in

such a way that this leads to a positive production steady state, then this always yields a higher

total surplus over the alternative of no R&D investment at all. Indeed, in Figure 3.5, for all c0

∈ (c1, c2), the collusive scenario is the better alternative.

Numerical observation 4. Over the entire parameter space we observe that whenever both

scenarios have an indifference point above the choke price, the collusive scenario yields higher

consumer surplus and total surplus than the competitive scenario for all initial technologies

in between the two indifference points.

74

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

c

k

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

c

Π

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

c

CS

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

c

TS

(d)

Figure 3.8: State-control space (a), total discounted profit (b), consumer surplus (c), and totalsurplus (d), when the exit trajectory is an optimal solution. Parameters: (β, ρ, φ) = (1, 1, 2).Grey curves correspond to competition, whereas the black ones correspond to collusion. Thecompetitive scenario brings about higher consumer and producer surplus than the collusivescenario for all c0.

This observation qualifies the argument that R&D cooperatives make it easier for firms

to collude in the concomitant product market and that this is necessarily welfare reducing.

Obviously, this fails to be the case for all c0 in the interval (c1, c2). It is also not necessarily

valid in situations where collusion induces firms to select the stable path while competition

induces them to exit the market (recall Figure 3.7).

For competition authorities, a particularly difficult situation arises when the initial draw c0

out of (c1, c2) is above the choke price (c0 > 1). Then the welfare costs of prohibiting firms

to collude in the product market do not surface because no production is affected by this

prohibition. There is no production yet, and because collusion is prohibited, there will be no

production in the future. Yet, in this case, prohibiting firms of an R&D cooperative to collude

75

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

c

TS

(a)

3.4 3.6 3.8 4 4.2 4.4 4.6 4.80

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

c

TS

c? c2c1

(b)

Figure 3.9: Total surplus when the indifference point is in the region with zero production.Parameters: (β, ρ, φ) = (1, 10, 50). Grey curves correspond to competition, whereas theblack ones correspond to collusion. c? ≈ 3.6, c1 ≈ 4.01, c2 ≈ 4.74. For all c0 ∈ (c?, c2), totalsurplus is higher if firms collude in the product market.

in the product market is welfare reducing. To the extent that competition policies are designed

to enhance total surplus, a general prohibition of product market collusion is not first-best

per se. At the same time, and more in line with traditional views, Figures 3.5 and 3.7 suggest

that if both scenarios induce firms to select the stable path towards the saddle-point steady

state, the competitive scenario yields a higher total surplus (Figure 3.8 contains a similar

suggestion in case both scenarios induce firms to select the exit trajectory).18 However, this is

not necessarily the case, as Figure 3.9 illustrates. Although both scenarios would induce firms

to select the trajectory towards the saddle-point steady state, for all c0 ∈ (c?, c2), total surplus

is higher if firms collude in the product market. In this example, the discount rate is high:

ρ = 10, which corresponds, for instance, to δ = 0.01 and ρ = 0.1. Also, the initial marginal

costs have to be ‘high’ for the collusive scenario to outperform the competitive scenario

in terms of consumer surplus and total surplus. In such an environment, the higher R&D

investments and the reduced importance that is attached to future surplus are favorable for the

collusive scenario: if firms collude, they reach the production stage more quickly, a benefit

that more than off-sets the concomitant welfare loss of increased mark-ups in the future.19 To

18As noted above, over the entire trajectory, collusion yields more R&D investments. Insofar higherinvestment levels as such are desirable, the case for prohibiting collusion in the product market is weakened.

19More precisely, a higher discount rate ρ = ρ/δ implies either a higher discount rate ρ or a lower δ. With alower δ, any cost reduction takes longer, such that whenever future benefits are discounted, the time differencein reaching the production stage between the scenarios becomes more pronounced.

76

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

p

LI

c

(a)

0 2 4 6 8 10 12 14 16 18 20−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

t

Π

π

(b)

Figure 3.10: Marginal cost, price, and Lerner Index (a); total discounted profit and instan-taneous profit (b), for collusion. Parameters: (β, ρ, φ) = (1, 0.1, 2.25) and starting pointc0 = 2.

illustrate further what difficulties competition authorities face, consider Figure 3.10. Among

others, it shows the development of the Lerner index over time towards its long-run level of

0.92 for the parameter configuration of Figure 3.5, where the initial draw c0 = 2 is from the

interval (c1, c2). This case illustrates what has been alluded to by Lindenberg and Ross (1981,

p. 28): “[The Lerner index] does not recognize that some deviation of P from MC comes

from ... the need to cover fixed costs and does not contribute to market value in excess of

replacement cost.”20 The high value of the Lerner index is due to collusion, which, in this

case, is welfare enhancing. Indeed, this example suggests that the court was right in its ruling

of US vs. Eastman Kodak (1995) when it concluded that “Kodak’s film business is subject

to enormous expenses that are not reflected in its short-run marginal costs.” More generally,

it illustrates the difficulty in designing optimal competition policies for high-tech industries.

This is illustrated further if one considers instantaneous profits and total discounted profits, as

in Panel (b) of Figure 3.10. Clearly, after a while, the former are much larger than the latter.

But the high instantaneous mark-ups should not be considered as a signal of potential welfare

losses, because if it had not been for these mark-ups, in the long run there would have been

no market at all.

20See Elzinga and Mills (2011) for a critical assessment of the use of the Lerner index; see also Armentano(1999).

77

3.7 Concluding remarks

We present an analysis of R&D cooperatives whereby the phase prior to production is taken

into account, because it is well known that collusion triggers the incentives to invest in

R&D. Our global analysis shows that if firms collude in the product market, the set of

initial technologies that is developed further increases, and that, in particular, more initial

technologies are brought to full maturation. This is a direct welfare gain of product market

collusion. Also, the probability that an initial technology induces firms to leave the market

altogether is reduced, which again is welfare enhancing.

Our analysis presents a problem for competition policy because it shows that prohibiting

collusion in the product market per se is not univocally welfare enhancing. It also shows

that the associated welfare costs might not surface because a prohibition of product market

collusion affects R&D investment decisions prior to the production phase. Any decision not

to develop further some initial technology does not materialize as a welfare cost because no

production is affected (yet).

78

Chapter 4

Competition and Innovation

4.1 Introduction

Contemporary markets are flooded with imitations – it is hard to find a business model, a

good, or service that is not a variation or an adaptation of some earlier version. Recently,

even Samsung’s lawyers could not tell the difference between Samsung’s Galaxy Tab and

Apple’s iPad in court.1 Imitators often even outperform innovators in business results (e.g.,

both Visa and Mastercard enjoy larger market shares than the first credit card issuer Diners

Club, and currently Samsung’s lead over Apple in smartphone market share has been widened

further2). Similar observations led Levitt to claim already back in 1970s that “Imitation is

not only more abundant than innovation, but actually a much more prevalent road to business

growth and profits” (Levitt, 1966, p. 63). A business strategy specialist Oded Shenkar in his

recent book, titled “Copycats: how smart companies use imitation to gain a strategic edge”,

even talks about an “imovation challenge” – companies that want to succeed need to fuse

innovation and imitation as in the future it will not be possible anymore “to rely on innovation

or imitation alone to drive competitive advantage” (Shenkar, 2010, p. 169).

Hardly any business idea is immune to imitation. In the words of Arrow (1962, p. 615):

“No amount of legal protection can make a thoroughly appropriable commodity of something

1http://gizmodo.com/5849803/even-samsung-cant-tell-the-difference-between-its-tablet-and-ipad (accessedAugust 2, 2012)

2http://www.totaltele.com/view.aspx?ID=475297 (accessed August 2, 2012)

so intangible as information. The very use of the information in any productive way is bound

to reveal it, at least in part. Mobility of personnel among firms provides a way of spreading

information. Legally imposed property rights can provide only a partial barrier, since there

are obviously enormous difficulties in defining in any sharp way an item of information and

differentiating it from similar sounding items.”

Indeed, Mansfield (1985) finds that rivals have information about new products or pro-

cesses in 12 months or less. Similarly, Cabellero and Jaffe (1993) in their analysis of patent

citations conclude that diffusion of information about innovations is so rapid that it can be

regarded as being instantaneous. Shenkar (2010) concludes that the pace of imitation is nowa-

days increasing with the increased codification of knowledge and the advance of globalization:

“In 1982 generics constituted a mere 2 percent of the U.S. prescription drug market, but by

2007 they made up 63 percent. In the early 1990s Cardizem lost 80 percent of the market

to generic substitutes within five years; a decade later, Cardura lost a similar share in nine

months; and Prozac, an Eli Lilly blockbuster drug, lost the same market share in only two

months” (p. 6). On the other hand, Vonortas (1994), for instance, claims that the features of

technological knowledge itself obstruct others from copying it. As he puts it, “technological

knowledge involves a combination of poorly-defined and often incomplete know-how and a

set of highly codified information which is hard to acquire and utilize effectively” (p. 415). In

practice, the extent to which R&D (Research and Development) information leaks or spills

over to competitors will most likely depend on specific characteristics of a particular industry

and product in question.

The spillovers are prevalent in business practice, yet our understanding of their role in

innovation activities of firms is still incomplete. On one hand, spillovers could improve market

performance as imitators make market more competitive. On the other hand, innovators facing

the danger of having their returns to research investments reduced by imitators could be less

willing to innovate in the first place. Furthermore, the effect of spillovers on the innovation of

asymmetric firms remains ambiguous as well. Empirical studies show that the follower often

strives to catch up with the leader (see, e.g., Lerner, 1997). If imitating the leader is easier,

the follower is able to progress faster and the industry is less likely to become monopolized.

80

However, the final result depends also on the way the leader responds to spillovers. The

leader might increase his innovation efforts in order to widen the lead and hopefully drive

the follower out of the market, or he might decrease his innovation efforts as the follower is

free-riding on them.

The analysis of spillovers in the existing literature is typically limited to their effects

on the innovation activities already in place. That is, the existence of the market and R&D

process is already assumed and the question left then is how spillovers affect the (size of)

R&D efforts. A distinguishing feature of our approach is that we do not limit ourselves to the

question of how much to invest on a given market but recognize that any such a question is

preceded by the question of whether to invest at all.3 Specifically, we pay special attention to

the determination of indifference points in a firm’s investment function. At these points, a firm

is indifferent between developing a given technology further and opting out. Consequently,

we are able to analyze not only how spillovers affect the investments on existing markets, but

also how they influence the likelihood that a new market will be formed, and if so, how does

its likely structure (monopoly or oligopoly) relate to the level of spillovers.

Like in previous chapters, our focus here is on process innovation. That is, the firms

increase their production efficiency by exerting R&D efforts. This higher production efficiency

in turn makes them stronger competitors on the Cournot product market. We allow for initial

unit production costs of firms (representing initial technology levels) being above the choke

price (the lowest price at which the quantity sold is zero) and we explicitly take firms’ product

market participation constraints into account. In consequence, firms’ R&D process and

production do not need to coexist at all times and firms can enter or exit the product market

and initiate or cease their R&D processes at different times. Here our work builds upon

the previous chapters, where we observed that the existing literature on strategic process

innovation holds on to the assumption of “low enough” initial production costs, and thereby

implicity imposes the coexistence of production and R&D at all times, without a proper

3Elmer Bolton, a scientist-manager at the DuPont company, one of the most innovative corporationsin American business history, was famous for saying to company’s chemists who in his opinion lacked theawareness that the success of the company depends on its products being commercially exploitable: “This is veryinteresting chemistry, but somehow I don’t hear the tinkle of the cash register” (Hounshell and Smith, 1988).

81

justification. Obviously, this assumption is in contradiction with the real life observation that

for great many new technologies, research starts long before the first prototype sees the light.

In our critical re-assessment of competition policies on R&D cooperatives in Chapter 3, we

showed how incomplete and misleading conclusions based on this restrictive assumption can

be. Furthermore, our model allows for asymmetric positions of firms, such that we are also

able to study investment decisions of the leader in relation to the follower at different levels

of spillovers and firms’ relative positions.

The rigor of our approach has its price - indifference points in firms’ policy functions

bring about a possibility that the latter exhibit discontinuities, which in turn implies a possible

existence of multiple regions of non-differentiability in the value functions. This poses

significant problems to numerical schemes. We progress by first exploiting the fact that

adding random noise to the R&D process smooths up the policy functions and value functions,

making numerical schemes easier to implement. We then consider a solution to the related

stochastic optimization problem, interesting in its own right, as an approximating solution to

the deterministic game when the noise level tends to zero. That is, we obtain a solution to the

deterministic game as a vanishing viscosity solution (see Basar and Olsder, 1995, Ch. 5.7).

We solve for a feedback Nash equilibrium of the differential game, characterized by a system

of highly nonlinear implicit partial differential equations, by a variant of the numerical method

of lines (Schiesser, 1991): we transform the system of partial differential equations into the

system of ordinary differential equations and consider the solution to the latter as an initial

boundary value problem. Solution methods for the latter require the values of the solution to

the game at the boundaries of the state space over which we seek a solution before they can

proceed. The problem is that true values at all boundaries are ex ante not known to us. We

solve this problem by exploiting the fact that the characteristics of the associated first-order

Hamilton-Jacobi-Bellman partial differential equations leave the state space at the boundaries.

This implies that the solution in the interior of the state space is unaffected by the precise

specification of the boundaries, possibly excepting a small strip along the boundaries. This

enables us to obtain an accurate approximating solution over an interior region of interest.

We show that in general duopoly results in the product market only if initial asymmetries

82

between the firms are not too large.4 The duopoly on the product market is characterized by

regression toward the mean phenomenon: asymmetries between the firms tend to vanish over

time.

Our results qualify the indication in the literature that larger spillovers might prevent the

monopolization of an industry (Petit and Tolwinski, 1999). We show that this is true only

when initial production costs of the leader are high and so also his incentives to exert R&D

efforts are high. At relatively lower unit costs of the leader, when additional R&D efforts

benefit the leader progressively less and the follower progressively more, the incentives of

the leader to exert R&D efforts can be rather low. This makes it harder for the follower to

catch up with the leader. Notably, the ability to copy is not worth much when there is little to

copy. Consequently, lower cost asymmetries can suffice to induce the monopolization of the

industry at larger spillovers.

We show that through increasing complementarities in R&D, larger spillovers always

increase the chance that an expensive technology that calls for investments in advance of

production will be brought to production. Though, the level to which such a technology is

developed can be lower due to lower R&D investments of firms that try to free-ride on each

other along the way. In this sense, spillovers increase production efficiency only up to a point.

We show that larger random shocks to firms’ production costs are favorable to the like-

lihood that a technology will be developed further as firms are stimulated by the chance of

a favorable shock to their production costs in the future more than they are destimulated by

the equal chance of an unfavorable shock. Stochasticity also increases the likelihood that the

product market will be competitive as the chance of a larger favorable shock in the future

increases the endurance of the follower.

We find comparably large investments of firms at low spillovers and high initial unit costs

of both firms. There, a small cost advantage of one firm leads to a behavior that can be

considered predatory: the leader exerts high R&D efforts which are profitable in that they

induce the follower to give up. When firms start from a symmetric situation, their behavior,

4This conclusion is similar to Doraszelski (2003) who finds action-reaction behavior in a patent race modelwith history-dependent R&D stocks: the follower catches up with the leader provided his initial stock ofknowledge is of sufficient size and gives up otherwise.

83

however, resembles a preemption race: each firm invests a lot trying to win the race in which

a small lead suffices for gaining a monopoly position.

4.2 Model

The dynamic game is defined in continuous time and over an infinite horizon: t ∈ [0,∞).

There are two firms which potentially both compete in a market for a homogenous good with

demand given by

p(t) = max {A− qi(t)− qj(t), 0} , (4.1)

where p(t) is the market price, qi(t) is the quantity produced by firm i = {1, 2}, qj(t) is the

quantity produced by its rival (i 6= j), and A is the choke price (the lowest price at which the

quantity sold is zero). At the outset of the game, each firm obtains an exogenous technology

ci(0). For simplicity, we assume that firms may differ in their production cost, but they are

identical in every other aspect. While both firms produce with constant returns to scale,

each firm can reduce its unit cost ci(t) > 0 by investing in R&D. This process is subject to

spillovers. Firm i exerts R&D effort ki(t) ≥ 0 and as a consequence of these investments, its

unit cost (state variable) evolves over time according to

dcidt≡ ci(t) = ci(t)(−ki(t)− βkj(t) + δ), (4.2)

where kj(t) is the R&D effort exerted by its rival and where β ∈ [0, 1] is a degree of spillovers.

Notice that equation (4.2) is not linear and that consequently the game is not linear-quadratic.

Low values of β correspond to strong intellectual property protection and the ability of firms

to prevent involuntary leaks of information. The reverse is true for high values of β. We treat

the value of β as given for firms.5 Observe in (4.2) that the smaller the ci, the smaller the

effect of particular ki on ci. Further innovations require increasingly more R&D efforts. The

5In general, β may be one of a firm’s strategic variables. See Katsoulacos and Ulph (1998) and Amir,Evstigneev and Wooders (2003) for an attempt to endogenize the degree of spillovers. Von Hippel (1988)provides empirical evidence for firms being consensually involved in information sharing. See also Shenkar(2010). Amir, Amir and Jin (2000) allow for spillovers to differ between firms.

84

parameter δ > 0 is the constant rate of efficiency reduction due to the ageing of technology

and organizational forgetting. Exerting R&D effort is costly. This cost is per unit of time

given by

Γi(ki(t)) = b(ki(t))2, (4.3)

where b > 0 is inversely related to the cost-efficiency of the R&D process. In assuming

decreasing returns to R&D, we follow the bulk of the literature (see Chapter 2 for a discussion

of model’s assumptions). Both firms discount the future with the same constant rate ρ > 0.

The instantaneous profit of firm i is:

πi(t) =

(A− qi(t)− qj(t)− ci(t)) qi(t)− bki(t)2 if p(t) > 0,

−ci(t)qi(t)− bki(t)2 if p(t) = 0,

(4.4)

yielding its total discounted profits over time:

Πi =

∫ ∞0

πi(t)e−ρtdt. (4.5)

4.2.1 Rescaling

Our model depends on five parameters: A, b, δ, β, and ρ. Some of these can be set to 1 by

choosing the measurement scale of units appropriately. The five-dimensional parameter space

then reduces to a three-dimensional one.

Lemma 10. By choosing the units of t, qi, qj , ci, cj , ki, and kj appropriately, we can assume

A = 1, b = 1, and δ = 1. The state equation changes to

ci(t) = ci(t)(1− (ki(t) + βkj(t))φ) (4.6)

with φ = A/δ√b.

Proof. See Appendix 4.A.

The new parameter φ = A/δ√b captures the profit potential of a technology: a higher A

85

implies higher potential sales revenue, a higher b implies more costly R&D efforts, whereas

a higher δ implies that each unit of R&D effort reduces the marginal cost by less. Hence, a

higher (lower) φ corresponds to a higher (lower) profit potential of a technology.

4.2.2 Equilibrium

In our two-firm differential game, each firm tries to maximize its total discounted profits by

selecting a strategy which specifies a quantity produced and an R&D effort exerted at each

point in time.

When selecting their strategies, firms have a lot of possibilities. In case firms use open-

loop strategies, they precommit themselves at the outset of the game to a fixed schedule of

actions over the entire planning horizon. That is, they specify an entire time path of quantities

produced and R&D efforts exerted at the beginning of the game and commit themselves to

stick to these preannounced plans over the entire horizon no matter what. Alternatively, firms

can use feedback strategies. In this case, strategies of firms are functions of time and the

current values of state variables (see Basar and Olsder, 1995). Firms are not required to

precommit. If the unit cost of a competing firm is reduced at some point in time, the opponent

reacts by choosing a quantity and an R&D effort level that take this change in unit costs

into account. As players condition their actions on the current state of the system and react

immediately to any changes in state variables, feedback strategies capture the essence of

strategic interactions. Consequently, while harder to derive, feedback Nash equilibrium is a

more satisfactory solution concept than open-loop equilibrium. We therefore seek a solution

to our dynamic game in the class of feedback Nash equilibria. The latter are derived by a

dynamic programming approach and are by construction strongly time consistent or subgame

perfect (players have no incentives to deviate unilaterally at any stage of the game).6

In our case, a feedback strategy for firm i specifies its quantity produced and R&D effort

exerted (control variables) for every possible combination of the two firms’ unit costs (state

6In the literature, strategies that depend on the current value of state variables (and on the time variable)only are also called Markov strategies. Equilibria in which all players use Markov strategies are called Markovequilibria. If such equilibria are also subgame perfect, they are called Markov-perfect equilibria (see Maskinand Tirole, 2001). Feedback Nash equilibria are by definition Markov perfect.

86

variables).

Product market and equilibrium output levels

We assume that firms compete in a product market by strategically setting their output levels.

The analysis of the product market is simplified by the fact that quantity variables, unlike R&D

efforts, do not appear in the equations for the state variables. Hence, production feedback

strategies of the dynamic game, associating output levels with unit costs, qi = ψi(c1, c2), are

static Cournot-Nash strategies of each corresponding instantaneous game.

Proposition 3. A strategy profile ψ∗(c1, c2) = (ψ∗1(c1, c2), ψ∗2(c1, c2)), where

i) ψ∗i (c1, c2) =1− 2ci + cj

3if 2c1 − c2 < 1, 2c2 − c1 < 1 (4.7)

ii) ψ∗i (c1, c2) = 0, ψ∗j (c1, c2) =1− cj

2if 2ci − cj ≥ 1, cj < 1 (4.8)

iii) ψ∗1(c1, c2) = 0, ψ∗2(c1, c2) = 0 if c1 ≥ 1, c2 ≥ 1 (4.9)

is a feedback Nash equilibrium of a quantity setting duopoly in the product market (i, j =

{1, 2}, i 6= j).

Equilibria on the product market are illustrated in Figure 4.1. Both firms produce positive

amounts only for combinations of unit costs corresponding to the shaded region. There, the

market price is higher than the unit cost of each firm (the first case in the above proposition).

In the duopoly on the product market, each firm earns a profit of

gi(c1, c2) =(1− 2ci + cj)

2

9. (4.10)

Outside the Duopoly region, there is either a monopoly on the product market or there is no

production at all. The curve E1 is the “entry/exit” curve for firm 1. Below it, the market price

87

is lower than firm 1’s unit cost, such that it is in the interest of firm 1 not to sell anything.

Analogously, E2 is the “entry/exit” curve for firm 2. Above it, the market price is lower

than firm 2’s unit cost, such that there firm 2 optimally does not sell anything. In the region

“Monopoly of Firm I”, firm 2 does not sell anything, whereas firm 1 can sell a positive amount

at a price above its unit cost. Consequently, firm 1 is a monopolist on the product market

there. Analogously, in the region “Monopoly of Firm II”, firm 2 is a monopolist. The two

regions – “Monopoly of Firm I” and “Monopoly of Firm II” – correspond to the second case

in Proposition 3. Passing from the Duopoly region through Ei in the direction of ci axis,

firm i stops producing (gi(c1, c2) = 0), whereas the other, more efficient firm switches to a

monopoly output and earns a monopoly profit

gj(c1, c2) =(1− cj)2

4. (4.11)

In the region “No production”, the unit costs of both firms are higher than the choke price

(A = 1). As firms could sell a positive amount only at negative mark-ups, neither firm

produces (the last case in Proposition 3).

In sum, the sales profit of firm i is given by

gi(c1, c2) =

(1− 2ci + cj)

2/9 if 2c1 − c2 < 1, 2c2 − c1 < 1,

(1− ci)2/4 if 2cj − ci ≥ 1, ci < 1,

0 otherwise,

(4.12)

where gi is a continuous function. The total instantaneous profit is the sales profit gi diminished

by the R&D expenditure k2i :

πi = gi(c1, c2)− k2i . (4.13)

Observe that the substitution of equilibrium output levels in firms’ profit functions has

resulted in the profit function of firm i being dependent only upon unit costs and its R&D

effort. Consequently, the problem of the firms is reduced to finding optimal R&D efforts.

88

c 2=

2c 1-

1

c1= 2 c2

- 1

E2

E1

E2

E1

Monopoly of Firm I

Duopoly

No

production

Monopoly of Firm II

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 c10.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

c2

(a)

Figure 4.1: Product-market activity.

Problem statement

To sum up formally, in our two-firm differential game, each firm maximizes its pay-offs

Πi =

∫ ∞0

[gi(c1, c2)− k2i

]e−ρtdt (4.14)

through its choice of the control ki = γi(t, c1, c2) ≥ 0, subject to state equations (ci > 0)

ci = ci (1− (ki + βkj)φ) , (4.15)

i = 1, 2.

Equilibrium R&D strategies

A feedback strategy of firm i, ki = γi(t, c1, c2) expresses firm i’s R&D efforts as a function

of the time variable and both firms’ unit costs. Subsequently, we define a feedback Nash

equilibrium as a pair of strategies γ∗ = (γ∗1 , γ∗2), such that strategy γ∗i , if it exists, maximizes

the present discounted value of firm i’s profits, given that the other firm pursues its strategy

89

γ∗j . That is, for firm i, γ∗i solves

γ∗i = arg maxγi≥0

∫ ∞0

πi(γi, γ∗j )e−ρtdt, (4.16)

subject to the state equations. Introduce the value function W i(t, c1, c2) as the value of the

above maximand. That is, let W i be the maximum present discounted value of profits that

can be earned by firm i, given that the other firm pursues its equilibrium strategy. Then, if

γ∗ is a feedback Nash equilibrium solution to our dynamic game, there exist functions W i,

satisfying the so-called Hamilton-Jacobi-Bellman equations (see Basar and Olsder, 1995, Ch.

6.5) in a suitable sense:

−W it = max

γi≥0

[(gi(c1, c2)− γ2i

)e−ρt +W i

ci(1− (γi + βγ∗j (c1, c2))φ)

+W icj

(1− (γ∗j (c1, c2) + βγi)φ)],

(4.17)

where i = {1, 2}, i 6= j. We adopt the convention that a subscript to the value function

indicates a partial derivative of that function with respect to each subscripted variable.

Introduce the current-time value function V i(t, c1, c2) by settingW i = V ie−ρt. That is, V i

equals the profits earned when firm i starts in the state (c1, c2) at time t and invests optimally,

while firm j pursues its equilibrium strategy. The equations in (4.17) then transform into the

following reduced Hamilton-Jacobi-Bellman equations:

ρV i − V it = max

γi≥0

[gi(c1, c2)− γ2i + V i

ci(1− (γi + βγ∗j (c1, c2))φ)

+ V icj

(1− (γ∗j (c1, c2) + βγi)φ)],

(4.18)

where the unknowns are the value functions V i.

The equations (4.18) above are formulated under the assumption of continuous differ-

entiability of V i. If this assumption is not valid, V i does not satisfy the equations (4.18) in

a classical sense. In such a case, these equations are understood in the sense of viscosity

90

solution (see Crandall and Lions, 1983). If V i is differentiable at some (c1, c2), then

γ∗i (c1, c2) = max

{−1

2φ(V ici

+ βV icj

), 0

}. (4.19)

It has been shown in Chapter 2 that in the case of a single firm, the value function is not

differentiable over the entire space. So-called indifference or Skiba points appear for certain

parameter values. These are points where a firm is indifferent between developing and not

developing a technology further. At these points, the optimal investment function has a jump,

whereas the value function obtains a kink. We expect the same phenomenon to occur in the

case of two competing firms. In Chapter 2, we were able to construct the value function

using Pontryagin’s Maximum Principle. In the present situation, where we are dealing with

a dynamic game, we are not able to obtain a solution along these lines. In the next section,

we therefore propose a method to obtain a numerical approximation to the value function.

In a nutshell, this method consists in considering a related stochastic optimization problem,

where the noise intensity σ depends on a parameter ε. As ε→ 0, the Nash equilibria of the

stochastic problem tend to those of the deterministic problem.7

4.3 Computation

First, we consider the state equation (4.2) with a noise term added. Specifically, let the state

of firm i evolve according to the following stochastic differential equation

dci = ci (1− (ki + βkj)φ) dt+ ci√

2εdBi, (4.20)

where Bi(t) is a standard Brownian motion or Wiener process, and where ε denotes the

noise level. Note that this equation is of the form dc = µ(c, k)dt + σ(c)dB, where µ and

σ are drift and diffusion, respectively, of a controllable Itô process c (see Kloeden and

7Kossioris et al. (2008) numerically compute a non-linear feedback Nash equilibrium for a differential gamewith a single state variable, limiting themselves to a class of continuous feedback rules. Dockner and Wagener(2006) study necessary conditions for feedback equilibria in games with a single state variable. Through anauxiliary system of differential equations they are also able to find non-continuous feedback strategy equilibria.

91

Platen, 1995). Hence, we make firms face some randomness in their unit costs.8 Firms

then maximize their expected current and future profits. That is, firm i solves the problem

maxγi≥0E∫∞0πi(γi, γ

∗j )e−ρtdt. The conditions that characterize a feedback Nash equilibrium

of the stochastic game are then given by the following coupled second-order parabolic partial

differential equations (see Basar and Olsder, 1995, Ch. 6.7)9

ρV i − V it = max

γi≥0

[gi(c1, c2)− γ2i + V i

ci(1− (γi + βγ∗j (c1, c2))φ)

+ V icj

(1− (γ∗j (c1, c2) + βγi)φ) + c2i εVici,ci

+ c2jεVicj ,cj

],

(4.21)

i = {1, 2}. Some motivation for the above formulation is in place. We conjecture on the

basis of the results obtained in Chapter 2 that the value functions of the deterministic game

are non-smooth. This makes numerical schemes to approximate the value functions hard to

implement. For this reason, we exploit the fact that adding stochasticity to the state equations

translates into two additional second-order terms being added to the Hamilton-Jacobi-Bellman

equations (c2i εVici,ci

and c2jεVicj ,cj

, respectively). We expect these two terms to have the effect

of artificially smoothing-up the value functions in (4.18) in the regions where the functions

change most rapidly. Then, hopefully, the solution to (4.21) will resemble a solution to

(4.18) when the viscosity coefficient ε ↓ 0. We call in this way obtained solution to (4.18)

a vanishing viscosity solution (see Basar and Olsder, 1995, Ch. 5.7). A benefit of this

formulation is that equation (4.21) is not only an equation for an approximating solution to

(4.18), but represents an equilibrium condition for the value function of the related stochastic

dynamic game which is interesting in its own right.

In the stochastic process (4.20), we let diffusion σ be a function of ci. In particular,

8Equation (4.20) can be rewritten as

cici

=

(1 +√

2εdBidt

)− (ki + βkj)φ,

where the change in the unit cost depends on two factors: i) the results of the R&D process (the second bracket)and ii) some randomness which is out of a firm’s control (the first bracket). In a sense, one can see this asperturbing the depreciation rate δ in the original model (4.2) with white noise.

9Notice that there is no stochasticity in the Hamilton-Jacobi-Bellman equations – the expectation operatorand randomness have been eliminated by using Itô’s lemma.

92

the variance of noise decreases as ci decreases. This prevents the system to jump over to

negative values of ci when the values of unit costs get close to zero: the stochastic process

(4.20) satisfies the so-called Feller condition (Feller, 1951), such that for any ci(0) > 0,

the value of ci(t) remains strictly positive with probability one for all times t.10 However,

when numerically solving (4.21), the dependence of second-order terms on the state variables

becomes inconvenient. In the next step, we therefore apply a diffeomorphic transformation of

the state variables

ci = e−xi , (4.22)

converting them to more convenient coordinates. Using Itô’s formula, ci(t) in the state equa-

tion (4.20) then transforms into a stochastic process xi(t) with constant diffusion strength11

dxi = ((ki + βkj)φ− 1 + ε) dt−√

2εdBi. (4.23)

The Hamilton-Jacobi-Bellman equations take the form:

ρV i − V it = max

γi≥0

[gi(x1, x2)− γ2i + V i

xi((γi + βγ∗j (x1, x2))φ− 1 + ε)

+ V ixj

((γ∗j (x1, x2) + βγi)φ− 1 + ε) + εV ixi,xi

+ εV ixj ,xj

].

(4.24)

To solve the infinite horizon problem, we consider a family of finite horizon problems over

[0, T ], T →∞, with a terminal value of zero. That is, V i(T ) = 0, as at time T nothing is left

for the firms. These finite horizon problems can all be solved simultaneously by reversing

the direction of time. Thus, we introduce time to completion, s = T − t, as a new time

10This formulation is also intuitive. It means that the more efficient a firm gets, the smaller is the probabilityof large unexpected changes in its unit costs. As an efficient firm has already done a great deal of R&D, it isquite realistic to expect that such an experienced firm is in a better position to avoid undesired, positive shocksto its unit costs. Moreover, as the firm has already reaped many fruits of its R&D endeavors, it is also realisticto expect that unexpected discoveries leading to large further reductions in unit costs are less likely. In fact,in his analysis of the petrochemical industry, Stobaugh (1988) documents that the probability of next processinnovation being major innovation decreases over time.

11Precisely put, a simple logarithmic transformation xi = ln(ci) would already achieve our aim. We take aninverse of ci under the logarithm as we find it convenient that the main region of interest (region where ci < 1)is on positive axes and so also the steady states of the state vector field have positive coordinates. On a relatednote, the applied logarithmic transformation is convenient also for the reason that we can obtain a solution overthe same range of the state variable (after the ex-post inverse transformation) by effectively solving over a muchsmaller range of it (e.g., solving for xi ∈ [a, b] enables us to obtain a solution over ci in the range of [e−b, e−a],a, b ∈ R. This allows for a significant saving on grid points.

93

variable. This has the effect of transforming the terminal condition V i(T, x1, x2) = 0 into an

initial condition V i(0, x1, x2) = 0, where V i(s, x1, x2) is a value function of the time-reversed

problem satisfying

ρV i + V is = maxγi≥0

[gi(x1, x2)− γ2i + V ixi((γi + βγ∗j (x1, x2))φ− 1 + ε)

+ V ixj((γ∗j (x1, x2) + βγi)φ− 1 + ε) + εV ixi,xi + εV ixj ,xj ].

(4.25)

Let V iT (t, x1, x2) denote the value function of a game with finite horizon [0, T ]. It then

holds that V i(T, x1, x2) = V iT (0, x1, x1). Hence, once we have obtained a solution for Vi

over s ∈ [0, S], we can recover the value function V iT for the entire family of finite horizon

problems [0, T ] with T ∈ [0, S].

Observe now that in our game, profit functions as well as state equations are autonomous.

That is, they do not explicitly depend on the time variable. Furthermore, the discount rate and

all other parameters are constant throughout the game, the stochastic shocks are independent,

the firms are fully informed at the outset of the game and do not learn anything new about

the game over the course of time. As a consequence, with infinite horizon, the continuation

game at some subsequent instant is identical to the game at the initial instant. In other words,

if a firm finds itself in some state (x1, x2), the rest of the game is the same whether this

situation occurs at some time t1 or t2. We therefore expect that the value function of the

infinite horizon game is time-invariant. That is, we expect that V iT (t, x1, x2) → V

i(x1, x2)

as T → ∞.12 But then also the time-reversed value function must become time-invariant

as the solution proceeds to infinity. That is, as s→∞, V i becomes a function of x1 and x2

alone: V i(t, x1, x2) → Vi(x1, x2) and thus V is = 0, where V i(x1, x2) solves the stationary

12Notice that W i(t, x1, x2) = Vi(x1, x2)e−ρt, where W i is the value of profits discounted to the initial time

of 0 (defined as in (4.17)). That is, while Vi

is time-independent, W i is not. This is intuitive. While, for a givenx1 and x2, V

iis the same for some t1 and t2 (t2 > t1), W i is not as V

i(t2) occurs later and is so worth less

than Vi(t1) when evaluated at the initial time of 0.

94

Hamilton-Jacobi-Bellman equations

ρV i = maxγi≥0

[gi(x1, x2)− γ2i + V ixi((γi + βγ∗j (x1, x2))φ− 1 + ε)

+ V ixj((γ∗j (x1, x2) + βγi)φ− 1 + ε) + εV ixi,xi + εV ixj ,xj ].

(4.26)

To obtain an approximating numerical solution to our infinite horizon game, we therefore

let the solution to (4.25) progress in s towards infinity and stop once V is is sufficiently close to

zero.13 Equilibrium strategies corresponding to the so computed value functions are then also

stationary and can be expressed as functions of x1 and x2 only: k∗i = γ∗i (x1, x2), where

γ∗i (x1, x2) = max

{1

2φ(V ixi + βV ixj

), 0

}. (4.27)

4.3.1 Numerical Method of Lines

The computation of a feedback Nash equilibrium amounts to solving the Hamilton-Jacobi-

Bellman equations given in (4.25). In effect, this leads to determining solutions to a system

of two coupled non-linear implicit two-dimensional partial differential equations. We solve

this system using the so-called numerical method of lines (see Schiesser, 1991), whose main

idea is the following. We discretize spatial derivatives of V i by evaluating their algebraic

approximations over the pre-specified grid points, but we leave the time variable continuous.

This leads to a system of ordinary differential equations to which usual numerical methods

for solving initial value problems can be applied.14

We use a finite difference discretization of partial derivatives. For this, we introduce a grid

in space, xi,1 < xi,2 < ... < xi,n, where n is the number of grid points in one dimension. We

assume constant grid spacing ∆xi = (xi,n− xi,1)/(n− 1), such that xi,` = xi,1 + (`− 1)∆xi,

13This approach resembles what is in the literature known as a method of false transients (see Schiesser,1991), where a time derivative which is not part of the original problem is added to a partial differential equationin order to transform it into a well-posed initial value (Cauchy) problem. It is then expected that this additionalterm will have an insignificant effect on the final solution. In our case, we however deal with true transientsas Vis (or V it ) is a true part of the Hamilton-Jacobi-Bellman equation (corresponding to a finite-horizon game)and approaches true zero only in limit (when the horizon of the game approaches infinity and the game itselfbecomes stationary).

14We wrote the code for computations in Fortran 95. The code uses double precision arithmetic. Thecriterion for the convergence of a solution is that the value of the L2-norm of Vs is below 1× 10−12. Auxiliarycalculations and plots were executed in MATLAB and Mathematica.

95

for ` = 1, ..., n. The obtained tensor grid is taken to be square (∆x1 = ∆x2 = h). At each

point of the grid, (x1,k, x2,m), we then replace first-order and second-order derivatives by

second-order central finite differences15, e.g.,

∂

∂x1V i(t, x1,k, x2,m) ≈

V i(k+1,m) − V i

(k−1,m)

2h, (4.28)

∂2

∂2x1V i(t, x1,k, x2,m) ≈

V i(k−1,m) − 2V i

(k,m) + V i(k+1,m)

h2. (4.29)

This leads to a system of n × n ordinary differential equations, which we solve using a

third-order Runge-Kutta method (Judd, 1998).16

Boundary conditions

We already motivated our choice of the initial condition V i(0, x1, x2) = 0. To solve the system

of differential equations (4.25), we also need to specify boundary conditions corresponding to

the four sides of the grid square.17 The problem is that the value of a solution at all boundaries

is ex ante not known to us. We address this delicate matter in Appendix 4.B where we argue

that the misspecification of the boundary conditions only results in a significant error in a

15We find the second-order scheme a good compromise between accuracy and the minimization of oscillation.That is, while higher-order schemes are in principle expected to increase accuracy, they also increase thepossibility of undesired oscillation in the solution (derivatives of increasing order have more roots betweenwhich the solution can oscillate).

16In presented plots, we set n = 200, which leads to a square state space grid of 40, 000 points over[−2.5, 4.5]× [−2.5, 4.5]. The accuracy of the numerical calculation can be increased by increasing the numberof grid points, which reduces ∆x. To prevent the solution from becoming unstable, we place an upper limit onthe time step ∆t in the Runge-Kutta method. In particular, we require the time step to satisfy two conditions.The first one is the Courant-Friedrichs-Lewy condition, specifying ∆t < ∆x

v , where v is a maximum driftvelocity. The second one concerns time required for diffusion to be captured ∆t < ∆x2

2ε . We observe that thelatter condition is non-binding for small ε.

17We already noted that the probability that unit costs reach the value of zero is zero. As the boundary isnot attainable, the boundary condition at ci = 0 is not needed from mathematical perspective. However, whensolving the system of partial differential equations with a finite differences method, the boundary condition atci → 0 is needed despite being mathematically redundant.

96

small region along the boundaries. In short, we select the following boundary conditions:

∂

∂x1V1(s, x1,1, x2,`) = 0,

∂

∂x1V2(s, x1,1, x2,`) = 0, (4.30)

∂

∂x2V1(s, x1,`, x2,1) = 0,

∂

∂x2V2(s, x1,`, x2,1) = 0, (4.31)

∂

∂x1V1(s, x1,n, x2,`) = 0,

∂

∂x1V2(s, x1,n, x2,`) = 0, (4.32)

∂

∂x2V1(s, x1,`, x2,n) = 0,

∂

∂x2V2(s, x1,`, x2,n) = 0, (4.33)

where ` = 1, ..., n.

Time paths

Once we have obtained the numerical approximation of the value functions in (4.25) and

from them the equilibrium feedback strategies, we can simulate the investment paths of firms.

For low values of ε, equation (4.23) with only the drift term generates a good approximation

of the evolution of the state variables over time. Hence, we solve the following system of

ordinary differential equations:

xi = (γ∗i + βγ∗j )φ− 1 + ε, xi(0) = x0i , (4.34)

i = {1, 2}, where γ∗i (x1, x2) and γ∗j (x1, x2) are obtained from (4.27) after replacing deriva-

tives of the value functions with their numerical approximations and x0i is firm i’s initial

value of unit cost. We solve the above system of ordinary differential equations over the time

interval [0, T ] by a third-order Runge-Kutta method (Judd, 1998), where we limit terminal

time T so that the values of xi remain within the state grid. We calculate any necessary value

of variables between the grid points by using cubic splines interpolation (Judd, 1998).

The state vector field assigns to each point in the (by interpolation refined) grid a vector

(x1, x2). Steady states are the grid points corresponding to a zero vector (x1, x2) = (0, 0). To

97

analyze the stability of steady states, we approximate the Jacobian matrix

J =

∂x1∂x1

∂x1∂x2

∂x2∂x1

∂x2∂x2

(4.35)

in each steady-state point of the vector field. We then compute the eigenvalues of the Jacobian

matrix and compare their signs.18 All derivatives are approximated by second-order central

differences.

4.4 Equilibrium strategies and industry dynamics

In this section, we present the results of the numerical analysis. We discuss strategic interac-

tions between firms as implied by their value and policy functions. Furthermore, to obtain

insight into possible evolutions of the game, we analyze state vector fields and time paths of

certain variables of interest.

We begin by examining the case of moderate spillover effects (β = 0.5). Afterwards, we

confront our conclusions with the case of low and high spillovers. The presented plots are

all drawn for φ = 8 and ρ = 1.19 The dynamics at this parameterization is representative of

all the cases in which firms have an incentive to develop further a technology which requires

R&D efforts prior to production. We discuss this at greater length later on, in Section 4.4.8.

4.4.1 Value function

Figure 4.2 shows the value functions and R&D efforts for ε1 = 0.125 and ε2 = 0.0156. As

the value functions are symmetric in the sense that V2(x2, x1) = V1(x1, x2), it is sufficient

to consider just V1. Note that ε1 > ε2 and that ε1 graphs are smoother. Note also that large

values of xi correspond to small values of ci.20 As Figure 4.2 shows, a firm’s value function

18If the real part of each eigenvalue is negative, the steady state is asymptotically stable. If the real part ofat least one eigenvalue is positive, the steady state is unstable. More particular, if one eigenvalue is real andpositive and the other one real and negative, the steady state is a saddle.

19ρ = ρ/δ is a rescaled discount factor (see Appendix 4.A).20Recall that ci = e−xi . Hence, negative (positive) values of xi correspond to unit costs above (below) the

choke price (A = 1). At the latter, x = 0. As xi →∞, ci → 0.

98

(a) (b)

(c) (d)

Figure 4.2: Value functions (top) and R&D efforts (bottom) for ε = 0.125 (left) and ε = 0.0156(right). In both cases β = 0.5.

is negatively related to a firm’s own unit cost and positively related to the unit cost of its

competitor. The smaller a firm’s unit costs for a given cost of its competitor, the better a firm’s

competitive position and so the larger the profits a firm is able to reap. The highest, left part

of the value function corresponds to unit costs for which firm 1 is a monopolist. Firm 1’s

relative cost advantage keeps its competitor out of the market. For lower values of firm 2’s

unit costs, both firms are (eventually) active in the market (recall Figure 4.1). This change

of the regimes is marked by a steep decline in the value function of the incumbent firm. For

relatively high values of own unit costs (the region of the southern valley), the value of the

game for firm 1 is zero as the firm finds it optimal to stay inactive.

99

(a) (b)

Figure 4.3: R&D efforts for firm 1 (left) and firm 2 (right). β = 0.5, ε = 0.125.

4.4.2 Policy function

The profits a firm is able to reap from the product market are determined by a firm’s cost

efficiency. The latter is costly in the sense that due to a positive rate of technology depreciation,

a firm needs to invest in R&D not only to increase its efficiency (relative to its competitor),

but also to maintain it. The equilibrium R&D efforts are shown in Figure 4.2. As equation

(4.27) indicates, we can decompose two effects underlying R&D efforts. We call the first

effect, which corresponds to the relation between the firm’s value of the game and its own unit

cost, V ixi , a pure cost effect, and the second one, which corresponds to the relation between

the firm’s value of the game and its competitor’s unit cost, βV ixj , a feedback cost effect.

Observe that, for a given unit cost of its competitor, the R&D effort of a firm increases with

decreasing own unit cost over the region of zero production and decreases shortly thereafter.

This is driven by the pure cost effect. This effect is always positive and is present whether the

competing firm is active or not. It is independent of strategic considerations in the sense that

it concerns the relation between a firm’s own production costs and its profits. When initial

unit costs are high (but still low enough for a firm to pursue further development), there are

huge benefits for a firm to exert R&D efforts as this reduces the amount of time needed to

reach the production phase. Consequently, R&D efforts are high. The lower the unit costs, the

100

more efforts it takes to reduce them further. This, together with lower tendency of technology

to depreciate for lower unit costs (see equation (4.20)), leads the firm to optimally invest the

less, the lower its unit costs.

On the contrary, the feedback cost effect corresponds to the fact that due to spillovers,

any R&D effort a firm exerts contributes also to the reduction of a competitor’s production

costs, which retroactively affects the firm’s profits through the product market competition.

The feedback cost effect is always negative and depends positively on the level of spillovers.

In the extreme case of zero spillovers, this effect is null. The feedback cost effect underlies

industry dynamics through strategic considerations discussed in what follows.

4.4.3 Vector field and dynamics

The R&D efforts that both firms exert influence the way in which unit costs evolve over time

through the drift term (see equation (4.23)). This evolution of costs as governed by the drift

term is summarized by the drift vector field in Figure 4.4. Let xεi (t) be a solution to (4.23).

Notice first that if ε = 0, then (4.23) is a deterministic ordinary differential equation with a

unique deterministic solution (given k1 and k2). If however ε > 0, then (4.23) is a stochastic

differential equation whose solution xεi (t) is a stochastic (random) process. Consequently, in

a deterministic game, as approximated by a stochastic game with a small ε, the drift vector

field shows how costs evolve for every possible initial position. However, in a stochastic

game, the drift vector field only shows the most likely evolution of costs for finite times. This

is further illustrated in plot (a) of Figure 4.5. The bold curve indicates a possible path of xi

over time, xi = xi(t), as implied by the drift vector field in Figure 4.4. At every finite time t,

the value xi(t) converges to the mean of the distribution of trajectories xεi (t) of the stochastic

differential equation (4.23) in the feedback strategy Nash equilibrium as ε→ 0. The variance

of the distribution increases over time due to the cumulation of random shocks (see Theorem

2.2.2 in Freidlin and Wentzell, 1998). Plot 4.5b shows how the actual costs fluctuate around

the drift path.21 In what follows, we therefore use a drift path as an approximation of the

evolution of the game over time.21The stochastic paths were calculated using the Euler-Maruyama scheme (see Kloeden and Platen, 1995).

101

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

x 1

W2S

I1

E1

I2

S1

I2

W2U

W2U

W1S

W2S

W1U

E1

W1U

S4

I2

I1

I1

I1*

I2*

E2

E2

S3

W1S

S2

Figure 4.4: Drift vector field. β = 0.5, ε = 0.125.

In Figure 4.4, the x1 = 0 loci (labeled I1) and x2 loci (labeled I2) intersect in four steady

states of the drift vector field: S1 and S2 are saddles, S3 is a nodal source, whereas S4 is

a nodal sink. Invariant manifolds of the two saddles are labeled by letter W . Stable and

unstable manifolds of S1 are labeled by W S1 and WU

1 , respectively. Similarly, W S2 and WU

2

are, respectively, a stable and an unstable manifold of S2. E1 and E2 are the product market

“entry/exit” curves of firm 1 and firm 2, respectively. They were introduced in Figure 4.1

and are here redrawn in the new coordinates. In the region above the indicated 45-degree

diagonal, firm 1 has a cost advantage over firm 2, whereas the reverse is true in the region

below the diagonal. For combinations of unit costs lying exactly on the diagonal, the firms

are equally efficient. Observe that the vector field below the indicated 45-degree diagonal is a

mirror image of the field above the diagonal. This follows from the symmetry of the feedback

equilibrium which is best visible in Figure 4.3.

102

0 1 2 3 4 5t-2

-1

0

1

2

3

4

x

(a)

0 5 10 15 20−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

t

x 1

(b)

Figure 4.5: (a) Drift as a change of the mean value of a stochastic process; (b) Drift path(bold line), corresponding to the drift vector field in Figure 4.4, and two different realizationsof a stochastic path for unit cost of firm 1 (x1(0) = x2(0) = −0.1,∆t = 0.01, T = 20).

Notice that W S1 and W S

2 are separatrices which divide the state space into two domains.

The first domain is a basin of attraction of the asymptotically stable steady state S4. Every

motion starting in this domain converges to S4 as t → ∞. In this domain, eventually both

firms are active on the product market. In the second domain, the unit cost of at least one firm

diverges to infinity; we are left either with a monopoly or no market at all.

We now analyze possible evolutions of the game by jointly looking at Figure 4.3 and

Figure 4.4. In the region south-west from S3, the unit costs of both firms are “very” high

and above the choke price, such that both firms decide to refrain from developing further the

initial technology. Future expected profits are not high enough to compensate for investments

needed to bring technology to the production phase. Technically, unit costs flow towards

infinity due to a positive depreciation rate.

Left to the stable manifold of S1, labeled in Figure 4.4 by W S1 , the cost advantage of

firm 1 over firm 2 is so large that the latter gives up on R&D (see figure 4.3). When cost

asymmetries are large, the profits the less efficient firm earns on the product market are low.

This reduces the ability of firm 2 to compensate for R&D investments needed to bring its

technology to the product market and catch up with firm 1. It turns out that left to W S1 the

cost asymmetries are just so large that firm 2 cannot even afford to battle depreciation of its

own technology, thereby succumbing to its more efficient competitor. Firm 2 does produce

103

only when its initial unit costs are already sufficiently low (the region between E2 and W S1

curve), such that it can profitably sell a positive quantity in a competitive product market.

However, its product market activity is only temporary as the large cost advantage enables

the more efficient firm 1 to squeeze firm 2 out of the market. Thus, firm 2 does eventually

neither produce nor invest in R&D. Its unit costs in this region always tend to flow towards

infinity, which is due to a positive depreciation rate. Intuitively, though, we always interpret

any situation in which a firm stays inactive as if this firm has left the market.

The investment of firm 1 depends on its initial unit cost. The firm decides to enter the

market for all initial costs that in Figure 4.4 correspond to x1 above the I∗1 curve which flows

through S3. For initial unit costs above the choke price (x1 below the horizontal part of E1

curve), the firm does at first produce nothing but invests increasingly in the reduction of

its unit costs. Once its unit costs have been reduced below the choke price, the firm starts

producing as it can now sell at positive mark-ups. The level of R&D efforts and unit costs

then gradually decrease to their long-run optimal levels (x1 approaches the unstable manifold

WU1 which asymptotically converges with I1 isocline). For unit costs above the choke price,

instantaneous profits of firm 1 are negative as there is no production yet. Firm 1 initiates R&D

as it expects future profits will more than compensate for initial investments. There exists

a finite upper bound on unit costs beyond which expected future profits are not enough to

compensate for short run losses (unit costs corresponding to x1 below the I∗1 curve). In this

case, the initial technology is not developed further. Observe how the direction of vectors in

Figure 4.4 changes its sign when passing through the I∗1 curve. While firm 2 benefits from

R&D efforts of firm 1 through spillovers, this effect is not strong enough to bring firm 2 onto

the market. It, however, slows down the rise in the discrepancy between the two firms’ unit

costs.

In the south-eastern part of the state space, below the W S2 curve, the situation is reversed.

It is now firm 2 whose cost advantage leads to its monopoly. For all initial unit costs on the

right side of the I∗2 curve passing through S3, firm 2 brings a technology on the market, while

firm 1 is sooner or later forced out of business.

In the north-east region of the state space, between the W S1 and W S

2 manifolds, the cost

104

asymmetries are moderate. Eventually, a product market duopoly emerges as for all initial

costs in this region, each firm sooner or later brings a technology on the product market. It

is interesting to observe that the asymptotically stable steady state S4 lies on the 45-degree

diagonal. This implies a kind of a regression toward the mean phenomenon, where any initial

difference in the unit costs between firms tends to vanish over time.22 We have noted that

above the 45-degree diagonal, firm 1 has a cost advantage, which is to the left of W S1 large

enough to squeeze firm 2 out of the market. However, to the right of W S1 , this is not the

case any more. Notice that W S1 curve travels along the edge of the precipice in the policy

function of firm 2 (see the right plot in Figure 4.3). While left to W S1 firm 2 gives up on

R&D, right to W S1 , it invests heavily to catch up with firm 1. Firm 1 exerts less R&D efforts

than firm 2, however, it prolongs its cost supremacy through positive spillover effects arising

from relatively high R&D efforts of firm 2. When its initial costs are very low, firm 1 for

some time even sits back on R&D (observe the basin in the northern region of the firm 1’s

policy function in Figure 4.3) and retards its technology decay optimally by relying mostly

on spillovers from the R&D efforts of its zealous counterpart.23 Namely, when unit costs of

firm 1 decrease relative to firm 2, an additional unit of firm 1’s R&D effort benefits firm 2

progressively more than firm 1 itself, which diminishes firm 1’s incentives for own R&D (this

follows directly from the formulation of unit costs in (4.2)). The story is analogous when we

are on the other side of the diagonal, where firm 2 has a relative cost advantage. In both cases,

a dominant firm gradually loses its lead.

4.4.4 Leader versus follower

We have noted that whenever cost asymmetries are large, the less efficient firm is squeezed

out of the market. Only when initial asymmetries are not too large, that is, when we are within

the basin of attraction of S4, both firms steer the evolution of their costs so that they remain

active in the product market. In this latter case, any initial asymmetry between the firms tends22We say “tends to” as it de facto vanishes only in light of a deterministic game interpretation. In a stochastic

interpretation, only the gap between the mean values of the two unit costs narrows and eventually closes.23A typical example of a large firm relying on inventions by smaller firms is Microsoft, whose competitors

“have long complained that the rest of the industry has served as Microsoft’s R&D lab” (New York Times, 4thAugust, 1991, p. 6).

105

to vanish over time.24 Figure 4.6 illustrates further how in the latter case, costs and R&D

efforts evolve over time along the drift path. In the first two plots, firms already start with

relatively low costs, (x1, x2) = (1.5, 1) or (c1, c2) = (0.22, 0.37). We see that the follower

exerts more R&D efforts than the leader, which gradually reduces the gap between the two

firms’ unit costs. In plots 4.6c and 4.6d, both firms start with unit costs above the choke price,

(x1, x2) = (−0.2,−0.3) or (c1, c2) = (1.22, 1.35). With own unit costs very large, the more

efficient firm 1 initially invests a lot in R&D, in particular, more than firm 2. Firm 1 exploits

its cost advantage to enter the product market first and earn temporary monopoly profits. The

situation changes in the course of time and eventually the follower invests more than the

leader. This causes the gap between the unit costs to shrink (the evolution of costs is similar

to that in 4.6a and is omitted for brevity). The quantity each firm produces increases over

time together with decreasing unit costs. As firm 1 is more efficient than firm 2, firm 1 at all

times produces more. However, with the gap between unit costs gradually narrowing, the gap

between quantities is narrowing as well.

4.4.5 Stochasticity and R&D

It is an interesting question how the R&D efforts of firms relate to uncertainty. A look at

Figure 4.2 reveals that both the value function and the policy function are smoother for higher

levels of noise in unit costs. To investigate this further, we plot the value function and policy

function of firm 1 for different fixed values of firm 2’s unit cost. In Figure 4.7, we fix c2 at

such a high value that firm 1 is a monopolist (c2 = 11.76; for reference, A = 1). Then, we can

directly compare our solution with the deterministic monopoly solution, obtained in Chapter 2.

We observe that while the deterministic value function has a kink, the stochastic value function

is smooth. The fact that the value function corresponding to a higher noise level lie above the

one corresponding to a lower noise level suggests that stochasticity increases expected profits.

We see that the stochastic value function converges to the deterministic monopoly value

function as ε ↓ 0. For ε = 0.0156, the stochastic solution is already almost indistinguishable24Once the drift path has reached the asymptotically stable steady state S4, the actual unit costs fluctuates

around S4. Of course, over a very long time, eventually a large shock may occur, driving one of the firms out ofthe market (its unit cost diverges to infinity).

106

0 1 2 3 4 5 6 7 8 9 10

0.1

0.15

0.2

0.25

0.3

0.35

t

c i(t)

firm 1firm 2

(a)

0 1 2 3 4 5 6 7 8 9 10

0.1

0.15

0.2

0.25

0.3

t

k i(t),

k1(t

)+k 2(t

)

firm 1firm 2total

(b)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

t

k i(t),

k1(t

)+k 2(t

)

firm 1firm 2total

(c)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

q i(t),

Q(t

)

firm 1firm 2total

(d)

Figure 4.6: Time paths: follower versus leader.

107

from the deterministic one, the absolute difference between the two solutions at the kink

being 0.0012. The deterministic policy function is discontinuous at the point of indifference,

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

c1

V1 (c

1,c2)

ε = 0.5

ε = 0.25

ε = 0.125

ε = 0.0156

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

c1

k 1

ε = 0.5

ε = .25

ε = 0.125

ε = 0.0156

(b)

Figure 4.7: Value functions (a) and policy functions (b) of firm 1 for varying levels of noise εwhen the unit cost of firm 2 is fixed at c2 = 11.76. The full line corresponds to the deterministicmonopoly solution.

where the firm is indifferent between developing a technology further or staying out. This

discountinuity is smoothed out by stochasticity. The policy function of the stochastic model

is smooth and everywhere differentiable. It is interesting to observe that stochasticity makes a

firm invest in R&D over the values of unit costs for which a firm in the deterministic setting

already gives up. The firm in the stochastic setting still invests a bit at larger costs in hope

of a favorable shock, for which it sacrifices some investments at lower unit costs – the R&D

efforts are smoothed out. While R&D efforts exerted at large costs might as such not be

sufficient to bring a technology to the production phase, they at least retard the decay of a

technology for some time during which hopefully a favorable shock arises. Higher uncertainty,

therefore, leads to more opportunistic behavior of firms, which increases the chance that the

development of expensive technologies will be pursued further. Computations shows that

this opportunistic behavior also increases the relative size of the region of the state space for

which eventually duopoly tends to appear on the product market (in the drift vector field, the

basin of attraction of S4 spreads out with increasing noise levels).

108

4.4.6 Deterministic game and indifference curves

In this section, we take a closer look at the deterministic game. We hope that its solution

resembles well the solution to the stochastic game with a small noise level. This hope was in

part verified in the previous section, where we saw that the two solutions are close at least at

the boundaries where only one of the firms is active.

As shown in Figure 4.7, the policy function in the deterministic monopoly solution is

discontinuous at the indifference point. Only for initial costs below the indifference point

a monopolist continues to invest in R&D and stays active in the product market. In what

follows, we analyze the existence of indifference points in the deterministic competitive game.

We define the deterministic indifference point of a firm as a value of its unit cost at which a

firm is indifferent between developing a technology further and exiting the market. In general,

indifference points do not coincide with points at which the R&D effort of a firm is zero. This

is the case only when an indifference point is in the region of zero production.25 When an

indifference point is at the value of a unit cost at which a firm produces, an exiting firm might

still invest a bit in order to slow down the speed at which it leaves the product market (cf. the

notion of the ‘exit trajectory’ in previous chapters).26

In our stochastic setting, firms steer the evolution of their unit costs through the drift term.

The direction of this steering is summarized in the drift vector field (see Figure 4.4). The

lower the noise level and so the lower the random shocks to the unit costs, the smaller the

deviations of actual costs form their drift path. For zero noise, the drift path is the actual path.

We saw in our discussion of the drift vector field above that the isoclines I∗1 and I∗2 around

S3 and the separatrices W S1 and W S

2 play an important role in the motion of the drift – they

are boundaries of different basins of attraction. With noise approaching zero, these curves

therefore converge to the boundaries of basins of attraction of actual unit costs. Consequently,

our conjecture, on which we further elaborate below, is that these curves converge to the union

of indifference points of the deterministic game.

25Clearly, when there is no uncertainty involved, a firm has no reason to initiate investment if it never plansto enter the product market.

26This is true only for the deterministic game. Whenever ε > 0, the R&D efforts are always positive on bothsides of an approximating indifference point due to smoothing-out.

109

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

c1

k 1

(a)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

−1

0

1

2

c1

c 1

(b)

Figure 4.8: Policy function of firm 1 (a) and drift of its unit cost (b). The full line correspondsto the deterministic monopoly solution, the dotted line to the stochastic game with ε = 0.0156.The unit cost of firm 2 is fixed at c2 = 11.76.

Consider first the state space left to S3 in Figure 4.4, where the cost of firm 2 is relatively

high. We have already observed that firm 1 steers its unit cost downwards for initial unit

costs corresponding to x1 above the I∗1 curve, and upwards otherwise. In Figure 4.8, we

consider the left most part of the state space as, again, there we can compare our solution

with the deterministic monopoly solution. We know that the deterministic policy function

is discontinuous at the point of indifference. As plot 4.8b shows, the deterministic drift is

also discontinuous at the indifference point, it is negative left to the indifference point (the

unit cost continues to decrease) and positive to the right of it (the unit cost flows towards

infinity as a technology decays due to zero research activity). On the contrary, in the stochastic

setting, the policy function is smooth, as is the drift. The point of zero drift in plot 4.8b

corresponds to the point on the I∗1 isocline. The difference between the latter and the

deterministic indifference point is for ε = 0.0156 already within the second decimal point.

This difference decreases further with lowering the noise level as the point on I∗1 converges to

the deterministic indifference point. We expect the same relation between the points on the

isocline and indifference points to hold also for other parts of the I∗1 isocline. Analogously,

for firm 2, indifference points correspond to points on the I∗2 isocline below S3.

We have already noted that the stable manifold W S2 act as a separatrix. For unit costs

above it, firm 1 invests relatively a lot and steers its unit cost towards S4. For unit costs below

110

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

c1

k 1

(a) ε = 0.125

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

c1

k 1

(b) ε = 0.0526

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

c1

k 1

(c) ε = 0.0156

Figure 4.9: Policy functions of firm 1 for a given value of firm 2’s unit cost, c2 = 0.4232.The vertical line corresponds to the point on the stable manifold W S

2 .

it, the firm still invests a bit in hope of a shock (random discovery) that would reduce its unit

cost, but progressively less so as its unit cost increases due to technology depreciation, thereby

steering its unit cost towards infinity. In the absence of a sufficiently large favorable shock, the

firm gradually gives up on R&D. Figure 4.9 shows plots of firm 1’s R&D efforts for a given

value of firm 2’s unit cost. The vertical line corresponds to the point on the W S2 curve. As

plots show, with decreasing ε, the policy function straightens up at the point corresponding to

W S2 as the latter converges to the deterministic indifference point at which the policy function

breaks off and the discontinuity arises. Right to the indifference point, the less efficient firm

gives up, whereas left to it, the firm invests heavily in R&D in order to catch up with the more

efficient competitor.

Analogously, points on W S1 converge to the indifference points of firm 2 as ε ↓ 0. We

also observe that steady states S1, S2, and S3 can lie in the regions of zero production

(see Figure 4.4). Clearly, in a deterministic game, there cannot be steady-state points in

such regions as that would imply a situation in which a firm invests at all times but never

produces. These steady states are the implication of the continuity of the stochastic drift.

They correspond to points at which the drift of both unit costs is zero. Like other points on

the two manifolds, when a stochastic game transforms into a deterministic one, these steady

states converts into the points of discontinuity. While at a stochastic steady state the drift is

zero, at the indifference point the drift is multi-valued and so there can be no steady state

corresponding to such a point. This is visible in plot (b) of Figure 4.8 above and fits nicely

into our picture of convergence.

111

We summarize our observations in the following conjecture.

Conjecture 1. In the deterministic game, the union of the stable manifold W S2 and the I∗1

isocline to the left of S3 approximates an indifference curve of firm 1 as ε→ 0. Likewise, the

union of the stable manifold W S1 and the I∗2 isocline below S3 approximates an indifference

curve of firm 2 as ε→ 0.

This conjecture is illustrated in Figure 4.10. Observe that indifference curves divide the

state space into four regions i) the region of eventual duopoly, ii) the region of eventual

monopoly of firm 1, iii) the region of eventual monopoly of firm 2, and iv) the no market

region, in which initial unit costs are too high for either firm to consider further development

of a technology. The indifference curve of a firm roughly outlines the “foothills” of that firm’s

policy function (see the bottom right plot in Figure 4.2).

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

No Market

x1

Eventual Monopolyof Firm 1

firm 1 indifferent

Eventual Competition

Eventual Monopolyof Firm 2

firm 1indifferent

firm 2indifferent

E1

E2

E2

E1

firm 2indifferent

Figure 4.10: Conjectured indifference curves of the deterministic game.

4.4.7 Spillover effects and R&D

In this section, we compare the case with moderate spillovers (β = 0.5) with that of low and

high spillovers. The larger the level of spillovers (the larger the β), the more the R&D efforts

112

(a) β = 0.1, ε = 0.125 (b) β = 0, ε = 0.25

(c) β = 0, ε = 0.125 (scale preserving) (d) β = 0, ε = 0.125 (in full)

Figure 4.11: Preemption & predation at low spillovers. The plots show the policy function offirm 1 for different levels of spillovers and uncertainty.

that a firm exerts benefit its competitor, and thus the larger the role of feedback cost effects in

shaping a firm’s policy function.

Low spillovers

For low levels of spillovers, the policy function exhibits a sharp and narrow bulge, as visible

in Figure 4.11.27 The lower the level of spillovers and/or noise, the more pronounced and

sharp the bulge. Figure 4.12 shows the drift vector field corresponding to the case with β = 0.

27The case with low spillovers is the hardest one to integrate numerically. For small values of ε, the bulgeincreases and sharpens dramatically, posing problems for the stability of the numerical scheme. Solutions forlower ε require increasing refinements of the grid, which in consequence rather considerably affects the speed ofcalculations. For this reason, the lowest noise level we currently present for this case is ε = 0.125.

113

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

x 1

E2

I2

I1

I2*

W1S

S1

S3

S2

S4

W2S

E1

W2U

E1

I1*

W1U

E2

Figure 4.12: Drift vector field for β = 0, ε = 0.125.

Observe that W S1 and W S

2 separatrices are in the vicinity of S3 practically indistinguishable

from the diagonal – a small difference in unit costs is enough to drive the less efficient

firm out of the market. This region of proximity corresponds to the location of the bulge.

With increasing levels of spillovers and/or noise, the bulge becomes thicker and lower; the

separatrices shift away from the diagonal, implying that a larger cost advantage is needed to

drive the opponent out of the market (compare with Figure 4.4).

As Figure 4.13 shows, on the diagonal within the canal, each firm invests a lot trying to

reduce its production costs as fast as possible and so increase its chances of survival (the

vertical line in the figure corresponds to symmetric costs). For a symmetric initial position,

firms are engaged in a preemption race where the one that falls sufficiently behind the other is

driven out of the market.

Observe that the bulge attains its top above the diagonal (left to the vertical line in

114

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.5

1

1.5

c1

k 1

ε = 0.125

ε = 0.177

c1 = 0.98

Figure 4.13: Policy function of firm 1 for a value of firm 2’s unit cost corresponding to theregion of the bulge. β = 0, c2 = 0.98.

Figure 4.13) and sweeps sharply down on the other side of the diagonal (the policy function

has a steep slope on the right side of the vertical line in Figure 4.13). The firm with a cost

advantage therefore invests heavily (but briefly), whereas the follower is induced to give up.

This additional R&D effort of the leader can be considered predatory in a sense that it is

profitable only for its effect on the exit decision of the follower, but unprofitable otherwise.28

The predatory nature of these investments is confirmed by the fact that such large investment

asymmetries never occur when the likelihood that a rival remains viable is negligible (e.g., at

very high levels of a rival’s unit cost) or the ability of a firm to influence this likelihood is

negligible (e.g., in the case of large spillovers where large investments would to a great extent

benefit the competitor).

The extent of predatory efforts is positively related to the easiness with which the leader

can induce the follower to give up. Recall that the bulge grows with the spillovers and noise

level approaching zero (see Figure 4.11). At low spillovers, it is easier for the leader to induce

the follower to exit as the latter cannot count on catching up with the leader by copying the

results of the leader’s R&D efforts. Thus, the lower the spillovers, the easier it is for the leader

to achieve his dominance by exerting R&D efforts and so the larger are his incentives for

extensive predation. Next, when the probability of large unexpected changes in costs is large,

28In declaring an action predatory, we follow Cabral and Riordan (1997) who define an action as predatory if“i) a different action would increase the likelihood that rivals remain viable, and ii) the different action wouldbe more profitable under the counterfactual hypothesis that the rival’s viability were unaffected” (p. 160).Our interpretation is similar to that of Borkovsky, Doraszelski and Kryukov (2012) who consider predatoryinvestment in a dynamic quality ladder model.

115

the follower does not give up that fast when falling behind as it is still possible for him to catch

up the leader if he has a run of luck. In this case, the leader needs to achieve a relatively large

cost advantage to induce the follower to give up. However, due to large randomness in costs,

the effect of the leader’s R&D efforts on the likelihood of achieving such an advantage is

low. Consequently, his incentives for predatory investments are low as well. On the contrary,

when there is low uncertainty in cost movements, a small cost advantage is sufficient to drive

the other firm out of the market and the effect of the leader’s R&D efforts on the likelihood

of achieving a needed cost advantage is large. As a consequence, the leader’s incentives to

engage in extensive predation are large as well.29

High spillovers

Figure 4.14 and Figure 4.15 show the policy function and the drift vector field, respectively,

for β = 0.9. Figure 4.16 jointly plots the indifference curves (separatrices) for β = 0 (L),

β = 0.5 (M), and β = 0.9 (H).

When the level of spillovers is high, the R&D efforts of one firm benefit the other firm to

a large extent. As each firm tries to free-ride on the other firm’s R&D efforts, the incentives

to exert much R&D efforts can be rather small. This standard conclusion in the literature

is in part confirmed by our calculations – R&D efforts decrease over the bulk of the state

space as the level of spillovers approach one. However, there is an important exception,

depicted in Figure 4.14. The policy function for large spillovers exhibits a pronounced bulge

spreading into the “southern” part of the state space (i.e., x1 low, x2 ≈ −0.1). The size of

this bulge increases with spillovers (first traces of it appear in the policy function for β = 0.5

in Figure 4.3). The intuition is the following. Notice that exerting R&D efforts is costly

29We have seen that the situation in which one firm invests heavily while the other negligibly small can bea feature of the equilibrium of the stochastic game as large investments are optimal in that they influence thelikelihood the rival is induced to give up on R&D and exit the market. Clearly, it is hard to imagine that such asituation could be a feature of the equilibrium of a deterministic game as there is no probability of the rival’smarket viability involved. We observe that the bulge narrows and sharpens as ε ↓ 0 (refer also to Figure 4.13).We therefore conjecture that in a deterministic case, the bulge corresponds to the point of discontinuity in thepolicy function, whereas his left and right foothills determine the R&D efforts of the leader and the follower,respectively. While still asymmetric, the difference in R&D efforts is comparably much smaller. In the region ofthe bulge, indifference curves are tangent to the diagonal (recall the tangent behavior ofWS

1 andWS2 separatrices

in Figure 4.12). For symmetric initial costs, both firms eventually produce, whereas a minuscule asymmetryalready leads to a monopoly.

116

(a) ε = 0.125 (b) ε = 0.0156

Figure 4.14: Policy function of firm 1 for β = 0.9.

(recall the quadratic cost function in (4.3)). When spillovers are large and so R&D efforts

of the firms complement each other well, the firms facing a convex cost function are able

to circumvent diseconomies of scale in R&D to a large extent. This reduction in the costs

of R&D enables the two firms to competitively bring on the market a technology which a

single firm cannot profitably develop itself. This explains why, provided the cost asymmetries

are not too large, a firm in a duopoly market sometimes does invest in further development

of a technology whereas a monopolist does not (the bulge). The implication of this is best

visible in the drift vector field in Figure 4.15, where we observe that the steady state S3 (the

indifference point of two firms) corresponds to a higher value of the firm 1’s unit cost than

the leftmost part of the I∗1 isocline (the indifference point of a monopolist). Furthermore,

Figure 4.16 shows that SH3 corresponds to a higher value of unit costs than either SL3 or SM3 .

The larger the spillovers, the larger the joint savings in R&D costs and so the more infant the

initial technology that the firms can afford to develop further. In contrast to the cases with

lower spillovers, in the case with high spillovers, W S1 and W S

2 separatrices also form a wide

arc around S3. Therefore, high spillovers not only increase the range of initial technologies

that are developed further, but also the likelihood that the ensuing product market will be

competitive.

Comparing the drift vector field of β = 0 with that of β = 0.5, we observe that the basin

of attraction of the steady state S4 is wider in the latter case (see Figure 4.16). The W S1 and

117

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

x 1

I1

E2

E1

I1*

E1

S1

S2

S3

S4

I2

I2

I1

W1S

W1U

W2S

W2U

E2

I2*

Figure 4.15: Drift vector field for β = 0.9, ε = 0.125.

W S2 separatrices spread out. This suggests that it takes a larger cost asymmetry for the less

efficient firm to leave the market when spillovers are higher. In particular, the exit of any firm

is much less likely when both firms already produce (for β = 0.5, larger parts of separatrices

lie outside the production region bounded by E1 and E2 curves). The larger the spillovers,

the more the follower can benefit from the R&D investments of the leader and so the more

disadvantaged it must be to give up. This point was already raised by Petit and Tolwinski

(1999) claiming that “[...] for a duopoly consisting of unequal competitors free diffusion of

knowledge may be a way to avoid market concentration” (p. 204).

In contrast to the aforementioned authors, our analytical framework allows us to draw

much more precise conclusions as it makes it possible for us to obtain indifference sets over

the entire state space, which in turn enables us to compare entry-exit investment decisions

of firms for all possible initial positions. In particular, we show that the pro-competitive

118

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

x 1

S4HS

4L S

4M

S3H

S3M

S3L

Figure 4.16: Comparison of indifference curves (separatrices) between β = 0 (black), β = 0.5(red), and β = 0.9 (blue). ε = 0.125.

benefit of larger spillovers does not hold for all levels of spillovers and costs. Observe how

in Figure 4.16 the separatrices corresponding to β = 0.9 intersect those corresponding to

β = 0.5. While for high initial unit costs of firms larger spillovers still make duopoly on

the ensuing product market more likely, this does not hold for lower values of initial unit

costs as there the less efficient firm is sooner squeezed out of the market when spillovers are

larger. Behind this result are two countervailing effects of spillovers. The first effect is a pure

spillover effect – the larger the spillovers, the more one firm is able to free ride on the other

firm’s R&D efforts and so the easier it is for the follower to overcome any initial asymmetries.

This effect is positively related to the level of spillovers and contributes to widening the region

of eventual product market duopoly. The second effect is the feedback cost effect, which is

also positively related to the level of spillovers, however, it contributes to narrowing the region

of eventual product market duopoly. When the unit cost of a firm is large, an additional unit

of R&D effort benefits this firm a lot (pure cost effect dominates). However, when the unit

119

cost of the firm is lower, so is the impact of an additional unit of R&D on its costs (the factor

ciki in (4.2) decreases with ci for a given ki). If the unit cost of the follower is sufficiently

larger, it can well happen that the additional R&D effort of the leader benefits the follower

more than the leader himself (ciki < cjβki). As lower costs of the follower then through the

product market competition negatively affects the leader’s profits, this reduces the leader’s

incentives to invest in R&D. This feedback effect, which negatively affects the leader’s R&D

efforts, is stronger, the larger the spillovers. Consequently, the larger the spillovers, the less

asymmetry in costs it takes for the leader to optimally stop his R&D efforts. Observe how

the region of zero R&D efforts above the diagonal spreads out in the policy function as the

spillovers increase (compare Figure 4.3, Figure 4.11, and Figure 4.14). This explain why

larger spillovers might in fact increase the likelihood that the market will be monopolistic.

After a certain level, further increases in spillovers decrease the leader’s incentives to invest

rather significantly, which makes it harder for the follower to catch up with the leader. The

follower’s possibilities to copy the leader’s R&D results do increase further with increasing

spillovers, however, the problem is there is now very little or nothing to copy. In the two

north-eastern regions between the intersecting separatrices in Figure 4.16, the leader in case

of β = 0.9 invests relatively less than in case of β = 0.5 and this effect of lower investments

by the leader dominates the pure spillover effect. Consequently, while for larger spillovers,

the follower is driven out of the market, for smaller spillovers, he continues to catch up with

the leader. In sum, increasing spillovers favors a socially desirable outcome only up to a

point.

Whenever one firm gives up, the unit cost of the remaining firm converges to the same

long-run optimal monopolist’s level irrespective of the level of spillovers. The separatrices

WU1 for the three different levels of β asymptotically converge as t→∞ (not shown). The

same holds for separatrices W S2 . However, the long-run steady-state level of unit costs in

case of duopoly (S4) does depend on the level of spillovers. As Figure 4.16 shows, in our

rescaled coordinates SH4 < SL4 < SM4 . That is, the long-run unit costs are the lowest in case

of β = 0.5, the second lowest in case of β = 0, and the highest in case of β = 0.9. This

suggests that the spillovers decrease steady-state costs only up to a certain level, beyond which

120

further increases in spillovers start to increase the long-run costs. We illustrate this further by

time plots in Figure 4.17. We select an asymmetric initial position which lies in the basin of

attraction of S4 for all the three levels of spillovers: (x1, x2) = (1.68, 0.81), corresponding to

(c1, c2) = (0.186, 0.440).

Plot 4.17a shows total R&D efforts of the two competing firms over time for different

levels of spillovers. We see a typical effect of increasing spillovers – the total industry

R&D efforts decrease as firms increasingly free-ride on each other. However, due to larger

complementarities between R&D efforts at larger spillovers, the effective efforts of the firm

i and the firm j (ki + βkj and kj + βki, respectively) might be larger at larger spillovers

despite the firms’ lower de facto R&D efforts (ki and kj , respectively). Plot 4.17b shows

that this is indeed the case when spillovers increase from β = 0 to β = 0.5. While for

β = 0.5, the industry R&D efforts are relatively lower at all times, the effective industry R&D

efforts are larger for most of the time. This explains why in the latter case, the unit costs

converge to a lower long-run level than in the case with β = 0. We see that among the three

regimes, the industry R&D efforts are comparably the lowest for β = 0.9. At the beginning,

the leader in the latter case invests very little as he free-rides on the efforts of the follower.

These smaller investments are not offset by larger spillovers, such that the effective efforts

are much lower than in the other two cases. This changes over time as the leader himself

starts to invest more when the follower gradually reduces his efforts over time. However, as

plot 4.17c reveals, lower effective investments at the beginning very much slow down the

speed at which unit costs decrease. In the case of β = 0.9, the unit costs of both the leader

(full line) and the follower (dotted line) decrease much slower than in the other two cases.

Moreover, the gap between the follower and the leader also closes more slowly. These slower

and lower reductions of costs as a consequence of smaller investments are the reason that

the total quantity offered in the market is for β = 0.9 at all times the lowest among the cases

considered (see plot 4.17d). The case with β = 0.5 offers the largest total quantity, whereas

the quantity for β = 0 is close to that for β = 0.5 but a bit lower.

Calculations show that total profits monotonically increase with spillovers. This is the

effect of higher complementarities in R&D outputs that allow for significant savings on R&D

121

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

t

ki(t)

+kj(t)

β = 0

β = 0.5

β = 0.9

(a)

0 1 2 3 4 5 6 7 8 9 100.2

0.25

0.3

0.35

0.4

0.45

0.5

t

(1+

β)(k

i(t)

+kj(t))

β = 0

β = 0.5

β = 0.9

(b)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

t

c i(t),

c j(t)

β = 0

β = 0.5

β = 0.9

(c)

0 1 2 3 4 5 6 7 8 9 100.45

0.5

0.55

0.6

0.65

t

Q(t)

β = 0

β = 0.5

β = 0.9

(d)

Figure 4.17: Time paths for varying levels of spillovers. ε = 0.125. In plot (c), the full linescorrespond to the leader, whereas the dotted lines correspond to the follower.

122

costs. However, the consumers are not necessarily any better for it. As our comparisons

indicate, there exists a threshold level of spillovers after which further increases in spillovers

do not benefit consumers. At large spillovers, the free riding effect induces firms to invest less

and the consequent lower production efficiency, to the detriment of consumers, also induces

them to produce less.

4.4.8 Market size and industry dynamics

The presentation so far has been focused on the dynamics that occurs at parameters (φ, ρ) =

(8, 1). Our calculations over a wide range of parameterizations show that dynamics at this

particular parameterization is representative of the subset of the parameter space for which

both coordinates of the unstable steady state S3 are negative, i.e., S3 corresponds to unit costs

above the choke price. This means that it can be profitable for firms to develop further a

technology which requires R&D efforts before the production can profitably start. We focused

on this subset of the parameter space, which we label “promising technology”, as we find

it most relevant – for great many new technologies, research starts long before a prototype

sees the light. We now briefly consider two other possibilities suggested by our solution to

the monopoly case (cf. the bifurcation diagram in Chapter 2). In the first one, which we call

“strained market”, a technology corresponding to initial unit costs above the choke price is

never developed further, whereas that already in the production phase is developed further

only if it is already sufficiently developed, such that it does not require “too much” additional

R&D efforts. If a technology is still relatively undeveloped such that firms need to exert

a lot of R&D efforts to maintain and develop it further, it is gradually discontinued by all

firms in the market. In the second case, which we call “obsolete technology”, it is always in

the interest of any firm to exit the market at some optimal speed as a low demand makes it

unprofitable to maintain a decaying technology.

Consider first the case of a strained market, represented by (φ, ρ) = (5, 1). Notice that a

lower φ for a given discount rate corresponds to a lower demand and/or higher costs of R&D

(recall the rescaling in Lemma 10). In general, lowering φ moves S3 and S4 closer together,

123

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

x 1

E1

E2

I2

I1 E

1

S2

S4

W1S

W2S

W2U

E2

W1U

S3

S1

Figure 4.18: Drift vector field for β = 0.5, (φ, ρ, ε) = (5, 1, 0.125).

contracting the region of the state space for which there is a duopoly on the product market

(the region between the W S1 and W S

2 separatrices). When demand decreases or R&D costs

rise, a much smaller lead is needed to induce the follower to give up. Figure 4.18 indicates

this for the case of β = 0.5. Observe that neither a monopolist nor any of the two competing

firms develop further a technology which would require investments prior to production –

the basin of attraction of S4 is compressed and fully contained within the production area

bounded by the E1 and E2 curves (compare with Figure 4.4).

Figure 4.19 shows the drift vector field for β = 0.1 and (φ, ρ) = (5, 1). There are now

only two steady states – a nodal source S3 and a saddle point steady state S∗1 . The two saddles

(S1 and S2) and the nodal sink (S4) have colluded and formed a new steady state S∗1 . This new

saddle has two manifolds – the unstable manifold WU1 and the stable manifold W S

1 which

lies on the diagonal of the state space. The implication of this is that the region of duopoly is

now compressed into a line segment which originates in S3, passes through S∗1 and continues

to infinity. Only for symmetric initial position lying on this line both firms keep producing

124

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

x 1

E2

I1

E1

E2

I2

E1S

3

S1*

W1S

W1U

W1U

Figure 4.19: Drift vector field for β = 0.1, (φ, ρ, ε) = (5, 1, 0.125).

and steer their unit costs towards the long-run equilibrium level of S∗1 .30 However, any initial

asymmetry makes the less efficient firm to gradually exit the market. At low spillovers, a small

lead is enough to induce the follower to give up. Observe how the diagonal acts like a repeller

– on each side of it, the motion is away from it. Clearly, in a stochastic game interpretation,

any symmetry is only temporary, such that for most of the time, one firm always diverges out

of the market.

In sum, it is noteworthy that for less favorable market and R&D conditions, the asymmetry

emerge for low spillovers – initial asymmetries lead to asymmetric outcomes (a firm with an

initial cost advantage becomes a monopolist, whereas the other firm exits the market).

As spillovers increase, S∗1 transforms into two saddles and a nodal sink. The region of

duopoly becomes a proper region, as in the case of β = 0.5. With increasing spillovers, the

two saddles (S1 and S2) move aside and the region of duopoly enlarges. After some point,

30If an initial position happens to lie on the diagonal above S∗1 , firms find it optimal to decrease their efficiency

towards a higher long-run level which is less costly to maintain.

125

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

x 1

E2

I1

E1

I2

E2

E1

S3

W1U

W1S

W2S

W2U

S4

S2

S1

Figure 4.20: Drift vector field for β = 0.9, (φ, ρ, ε) = (5, 1, 0.125).

however, the saddles start approaching each other and so the region of duopoly starts to

contract. This effect is visible when comparing the drift vector fields for β = 0.1 and β = 0.5

with the drift vector field for β = 0.9 (see Figure 4.20).

Figure 4.21, which shows a zoomed comparison between β = 0.5 and β = 0.9, indicates

that this contraction of the duopoly region at higher spillovers is again not universal as the

region of duopoly for β = 0.9 remains wider at larger levels of unit costs. All in all, our

conclusion is similar as before – after a certain level, larger spillovers start to reduce the

duopoly region (the region of regression toward the mean) as smaller investments of the

unmotivated leader makes it harder for the follower to catch up.

The drift vector field in Figure 4.22 shows the drift motion for the case of obsolete

technology represented by (φ, ρ) = (2, 1). We see that both unit costs diverge towards infinity.

At this parameters, the demand is so low and the R&D process so costly that both firms find

it optimal to eventually leave the market. They might still invests in R&D at some smaller

rate that retards the decay of the technology optimally, but eventually both the R&D and

126

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.2

0

0.2

0.4

0.6

0.8

x 1

x2

E2

E1

S3H

S3M

Figure 4.21: Comparison between β = 0.5 (red) and β = 0.9 (blue).

production will terminate and the firms will exit the market.

4.5 Concluding remarks

In this chapter, we study feedback Nash equilibria of a dynamic game in which firms enhance

their production efficiency through R&D endeavors. The model allows for the possibility

that initial unit costs of firms are above the choke price. Firms’ product market participation

constraints are also explicitly taken into account. As a result, R&D efforts and production do

not necessarily coexist at all times. Furthermore, by allowing for asymmetric initial positions,

the model provides insight into the investment relations between the market leader and the

follower and their entry-exit decisions in relation to the level of spillovers, market size, the

efficiency of R&D, and uncertainty in unit costs.

Our results qualify the indication in the previous literature that higher spillovers might be

socially beneficial as they might obstruct the monopolization of the industry by preventing

127

−2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

x2

x 1

Figure 4.22: Drift vector field when exit is always optimal. (β, φ, ρ, ε) = (0.5, 2, 1, 0.125).

the lagging firm from falling too much behind the leader. We show that this pro-competitive

effect of larger spillovers holds only up to a certain point, after which further increases in

spillovers start reducing the region of the state space for which there is duopoly on the product

market. While higher levels of spillovers indeed make it easier for the follower to copy

the R&D results of the leader, the latter might have very little incentives to invest knowing

that its R&D efforts will benefit its competitor to a large extent (possibly even more than

the leader himself). As the follower has then little to copy, smaller initial asymmetries can

induce the monopolization of the industry at larger spillovers. The pro-competitive effect of

larger spillovers however remains at high initial unit costs as there the leader himself has high

incentives to invest.

We show that larger spillovers always increase the likelihood that some initial technology

which requires investments in advance of production will be developed further. With a convex

R&D cost function, larger spillovers enable the firms to save more on R&D costs, which in

consequence enables the firms to bring on the market expensive technologies which the firms

with lower complementarities in R&D cannot afford to. In this sense, larger spillovers are

always conducive to R&D. The other thing, however, is to which level these technologies are

developed. The long-run steady-state unit costs decrease by increasing spillovers only up to a

certain point, after which they start increasing. At first, larger spillovers more than compensate

for lower actual R&D efforts of firms, such that effective R&D efforts increase with increasing

128

spillovers. In consequence, larger spillovers lead to duopoly with lower steady-state unit

costs and larger quantity produced. After a point, however, larger spillovers are not anymore

enough to compensate for lower actual efforts of firms that try to free-ride on each other,

leading to lower effective R&D efforts along the equilibrium path. Consequently, further

increases in spillovers lead to higher steady-state unit costs and lower quantity produced.

We show that the region for which there is duopoly on the product market is characterized

by regression toward the mean phenomenon, where asymmetries between the firms tend to

vanish over time.

Next, we find that at low spillovers, a preemption race occurs at relatively high initial unit

costs where initially symmetric firms invest a lot trying to win the race in which a small lead

suffices for monopolizing the industry. We also find that at low spillover levels, the leader

may engage in large investments that can be considered predatory in the sense that they are

profitable only in inducing the less efficient opponent to surrender.

We find that larger noise in unit costs induces the firms to develop further the technologies

which would be left intact at smaller noise. This investment of firms is stimulated by a higher

chance of a large favorable shock to unit costs at larger noise. Larger noise also widens the

duopoly region as the follower is less inclined to exit the market when there is a higher chance

of a large favorable shock in the future. As both favorable and unfavorable shocks are always

equally likely, the firms appear cautious in the sense that they rather invests more than be later

sorry for giving up on a technology too early.

Appendix 4.A Proof of Lemma 10

A rescaled variable or parameter is distinguished by a tilde: for instance, if π denotes profit,

then π denotes profit in rescaled variables. Set A = 1, b = 1, δ = 1, and define the following

conversion: qi = Aqi, qj = Aqj , ki = A√bki, kj = A√

bkj , ci = Aci, cj = Acj , πi = A2πi,

πj = A2πj , φ = Aδ√b, t = t

δ, ρ = ρ

δ. The state equation in the rescaled variables is then

˙ci(t) = ci(t)(

1−(ki(t) + βkj(t)

)φ). (4.36)

129

The new instantaneous profit function is:

πi(t) = (1− qi(t)− qj(t)− ci(t))qi(t)− ki(t)2, (4.37)

whereas the total discounted profit is

Πi =

∫ ∞0

πi(t)e−ρtdt. (4.38)

To prove this, consider first the profit function of firm i, which is in the original (non-rescaled)

model given by:

πi = (A− qi − qj − ci) qi − bk2i .

Using the conversion rules given above, we obtain:

πi = (A− qi − qj − ci) qi − bk2i

= (A− Aqi − Aqj − Aci)Aqi − b(A√bki

)2

= A2(

(1− qi − qj − ci) qi − k2i)

= A2πi

The equation for the evolution of the unit cost over time is in original variables given by:

ci(t) = ci(t) (−ki(t)− βkj(t) + δ) .

Write ci(t) = ci(1δt). Then,

dci

dt=

dcidt

dt

dt

=1

δci

= ci

(1− 1

δki − β

1

δkj

).

130

Setting ki = A√bki and kj = A√

bkj , and substituting them in the previous equation, we obtain:

dci

dt= ci

(1−

(ki + βkj

) A√bδ

).

It is now natural to introduce φ = Aδ√b. Notice that if ci = ci/A, then ˙ci = ci/A and

˙ci = ci

(1−

(ki + βkj

)φ). (4.39)

Observe finally that if t = tδ, then e−ρt = e−ρt if and only if ρ = ρ

δ. Despite dealing with the

rescaled model, we omit tildes in the main text so not to blur the exposition.

Appendix 4.B Boundary conditions

To solve our system of differential equations, we need to specify eight boundary conditions

(four for each player’s value function), corresponding to the four sides of the grid square.31 The

problem is that the true value of a solution at all boundaries is ex ante not known to us. In what

follows, we show that the solution in the interior of the state space is unaffected by the precise

specification of the boundary conditions, excepting a small strip along the boundaries. The

reason for this is that the characteristics of the associated first-order Hamilton-Jacobi-Bellman

partial differential equations leave the state space at the boundaries.

As explained in the main text, the grid square follows from discretizing each of the two

state variables. That is, we introduce xi,1 < xi,2 < ... < xi,n, where n is the number of

grid points in one dimension, i = 1, 2. At each boundary value of xi, we impose Neumann

conditions, which we motivate as follows.

From a deterministic solution to a monopolist’s problem (see Chapter 2), we know that a

firm pursues R&D only if its initial unit cost is small enough. We therefore conjecture that

31We observe ex post that the Nash equilibrium is symmetric; that is, V1(x1, x2) = V2(x2, x1) andγ∗1 (x1, x2) = γ∗2 (x2, x1). This information can be used to halve the dimension of the problem and therebyspeed up (reparameterized) calculations. Define V1(x1, x2) = V(x1, x2) and γ∗1 (x1, x2) = γ∗(x1, x2). ThenV2(x1, x2) = V(x2, x1) and γ∗2(x1, x2) = γ∗(x2, x1). In this reformulation, the unknown are then only V andγ∗. We do not pursue this here as our code is written so that it allows for asymmetric equilibria. While this waywe do sacrifice speed a bit, the solution itself is not affected.

131

Figure 4.23: Solution at the boundary.

in the competitive case, for sufficiently large own unit costs (small xi), firm i stays inactive,

whereas firm j is a potential monopolist. The value of the game is then zero for firm i and

small changes in its unit costs have no effect on either its own or its competitor’s value

function. The situation at sufficiently large unit costs of firm j is analogous. From this, the

following boundary conditions follow:

∂

∂x1V1(s, x1,1, x2,`) = 0,

∂

∂x1V2(s, x1,1, x2,`) = 0,

∂

∂x2V1(s, x1,`, x2,1) = 0,

∂

∂x2V2(s, x1,`, x2,1) = 0,

where ` = 1, ..., n.

The situation at low unit costs is more subtle. We expect, however, that whenever one firm

has very low unit cost, the effect of this unit cost on each firm’s value of the game is small at

the margin. We approximate this effect by zero, from which the following conditions follow:

∂

∂x1V1(s, x1,n, x2,`) = 0,

∂

∂x1V2(s, x1,n, x2,`) = 0,

∂

∂x2V1(s, x1,`, x2,n) = 0,

∂

∂x2V2(s, x1,`, x2,n) = 0,

where ` = 1, ..., n.

132

In calculations presented in the main text, we usually set xi,1 = −2.5 and xi,n = 4.5,

corresponding to ci = 12.1825 and ci = 0.0111, respectively. To see how boundary conditions

affect the solution, we vary the range of the grid and compare the so obtained solutions. It

turns out that the interior solution is very much unaffected by varying the grid. Figure 4.23

compares the solution to the policy function of player 1 at the upper boundary of x1 for

different values of xi,n: xi,n = 4.5, xi,n = 5, and xi,n = 5.5, respectively. We see that

consecutive solutions diverge only in the very close proximity of the boundary, where the

solution corresponding to a smaller grid range sweeps sharply down to zero. Hence, by

specifying a large enough range of the grid, we can always obtain a good approximation over

the interior region of interest. The correctness of our solution is further confirmed by the first

plot in Figure 4.7, which shows that for a given large value of firm j’s unit cost, the solution

for firm i converges to the deterministic monopoly solution as ε ↓ 0.

133

Chapter 5

Summary

This thesis studies process innovation in global, dynamic perspective. In contrast to the

existing literature, all possible values of firms’ unit costs, including those above the choke

price, are considered. This allows research efforts to precede production. Furthermore, the

analysis is not limited to equilibrium paths, but considers all trajectories that are candidates

for an optimal solution. This includes the determination of critical points - points at which the

optimal investment function qualitatively changes. In particular, it includes the determination

of the value of unit costs for which R&D investments are terminated and for which they are

not initiated at all. The size of these critical cost levels is affected by the conduct of firms. In

consequence, different market regimes can lead to qualitatively different long-run solutions,

despite starting from an identical initial technology. This global approach makes it possible to

see not only how different variables of interest (e.g., spillovers, R&D efficiency) affect the

investments on existing markets, but also how they influence the likelihood that a new market

will be formed, and if so, how its likely structure relates to them.

Chapter 2 introduces the global framework by characterizing investment and production de-

cisions of a monopolist that are globally optimal. A distinct characteristic of the optimization

problem is the presence of multiple equilibria while at the same time the Arrow-Mangasarian

sufficiency conditions are not met. In models of this type, the qualitative properties of optimal

solutions may change if parameters are varied. For a global analysis, we therefore have to

use bifurcation theory. This gives us a bifurcation diagram that indicates for every possible

parameter combination the qualitative features of any market equilibrium. The analysis yields

four distinct possibilities: (i) initial unit costs are above the choke price and the R&D process

is initiated; after some time production starts and unit costs continue to fall with subsequent

R&D investments; (ii) initial unit costs are above the choke price and the R&D process is not

initiated, yielding no production at all; (iii) initial unit costs are below the choke price and the

R&D process is initiated; production starts immediately and unit costs continue to fall over

time, and (iv) initial unit costs are below the choke price and the initiated R&D process is

progressively scaled down; production starts immediately but the technology (and production)

will die out over time; the firm leaves the market. The strength of the analysis is that all these

cases can emerge from the same unifying framework.

Chapter 3 extends the analysis of an innovating monopolist in Chapter 2 to the case with

two firms involved in different levels of cooperation. It compares two different scenarios

across the four types of industry dynamics outlined in Chapter 2. In the first scenario, firms

cooperate in R&D and compete on the concomitant product market. In the second scenario,

cooperation in R&D is extended to collusion in the product market. The chapter then compares

the qualitative properties of these two scenarios in order to assess the potential set-back of

R&D cooperatives in that they can serve as a platform to coordinate prices. It yields three

key findings: (i) if firms collude, the range of initial unit costs that leads to the creation

of a new market is larger, (ii) collusion in the product market accelerates the speed with

which new technologies enter the product market, and (iii) the set of initial unit costs that

induces firms to abandon the technology in time is larger if firms do not collude in the product

market. In particular, it is shown that there are parameter configurations for which collusion

in the product market yields higher consumer surplus as well as total surplus. This analysis

presents a problem for competition policy because it shows that prohibiting collusion in the

product market per se is not univocally welfare enhancing. It also shows that the associated

welfare costs might not surface because a prohibition of product market collusion affects

R&D investment decisions prior to the production phase. Any decision not to develop further

some initial technology does not materialize as a welfare cost because no production is visibly

affected.

136

Chapter 4 considers a differential game in which two firms compete both in R&D and

on the product market. To capture strategic interactions between firms in as realistic way as

possible, a solution to the game is sought in the class of feedback Nash equilibria, where the

possibility of discontinuous investment functions requires a special treatment. A solution

to the stochastic game is considered as an approximating solution to the deterministic game

when the random noise level tends to zero. A numerical approximation to the value function

is obtained using a variant of the numerical method of lines. The chapter analyzes how

spillovers affect the investments of firms, how they influence the likelihood that a given initial

technology will be developed further and, thus, a new market will be formed, and how the

likely structure of a new market relates to them. The obtained results qualify the indication

in the literature that higher spillovers might be socially beneficial as they might prevent the

monopolization of the industry. It is shown that this pro-competitive effect of larger spillovers

holds only up to a certain point and that, consequently, lower cost asymmetries can suffice

to induce the monopolization of the industry at larger spillovers. Moreover, it is shown that

through increasing complementarities in R&D, larger spillovers increase the chance that an

expensive technology that calls for investments in advance of production will be brought to

production. Though, the level to which such a technology is developed can be lower at larger

spillovers due to lower R&D investments of firms that try to free-ride on each other along the

way. In this sense, spillovers increase production efficiency only up to a point. Stochasticity

is found to increase the likelihood that a given initial technology will be developed further as

well as the likelihood that the ensuing product market will be competitive. The reason for this

is that a higher chance of a large favorable shock to unit costs at larger levels of random noise

stimulates firms to invest more. It also enhances the endurance of the follower. Furthermore,

both predation and preemption can emerge in the model as a consequence of firms’ optimal

decisions.

There are several possible directions for future research. All the models in this thesis allow

for only a single technological process being developed by a firm at a time. A natural extension

is to allow for several technologies, with different levels of substitutability on the product

market, being developed simultaneously. Next, this thesis considers process innovation only.

137

The global framework needs to be extended so to allow for a proper combination of process

and product innovation. The analysis in Chapter 4 opens up a possibility of studying the

incentives of asymmetric firms for cartelization and the stability of different R&D cooperatives

among asymmetric members in global perspective. In all models considered, the level of

spillovers was assumed to be exogenous. However, in practice, firms usually select, or at least

influence, the amount of information they share strategically. This calls for the endogenization

of spillovers within the global framework. Furthermore, in contrast to Chapter 2 and Chapter 3,

the study of the differential game in Chapter 4 was in many ways limited to some representative

configurations of parameters. A more complete analysis of solutions to differential games

awaits the development of bifurcation methods for non-convex optimization problems with

multidimensional state space, which are still in their infancy.

138

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147

Samenvatting (Summary in Dutch)

Dit proefschrift bestudeert procesinnovaties vanuit een globaal, dynamisch perspectief. In

tegenstelling tot de bestaande literatuur worden àlle mogelijke waarden van de kosten per

eenheid product bestudeerd, inclusief waarden boven de zogenaamde afkapprijs; de prijs

waarboven geen vraag bestaat. Dit maakt het mogelijk om investeringen in Onderzoek &

Ontwikkelings (O&O) voorafgaand aan het productieproces te onderzoeken. Bovendien

is de analyse niet beperkt tot evenwichtspaden maar onderzoekt het alle mogelijke paden

die kunnen leiden tot een optimale oplossing. Dit omvat de bepaling van omslagpunten -

punten waarop de optimale investeringsfunctie kwalitatief verandert. In het bijzonder wordt

vastgesteld bij welke waarde van de kosten per eenheid product de O&O investeringen worden

beëindigd en voor welke waarde het O&O proces in het geheel niet wordt opgestart. De

hoogte van deze kritieke kostenniveaus worden beïnvloed door het gedrag van bedrijven.

Derhalve kunnen verschillende marktregimes leiden tot kwalitatief verschillende uitkomsten

op de lange termijn, ook al is de initiële technologie identiek. Deze globale benadering maakt

het niet alleen mogelijk om te bezien hoe de relevante variabelen (zoals spilllovers en O&O

efficiëntie) de investeringen in bestaande markten beïnvloeden, maar ook hoe zij van invloed

zijn op de kans dat een nieuwe markt wordt gevormd, en hoe de structuur daarvan hieraan

gerelateerd is.

Hoofdstuk 2 introduceert het globale raamwerk waarin een monopolist de investerings-

en productiebeslissingen neemt die globaal optimaal zijn. Een onderscheidend kenmerk van

het optimalisatieprobleem is de aanwezigheid van meervoudige evenwichten terwijl tegeli-

jkertijd niet aan de voldoende voorwaarde à la Arrow-Mangasarian voldaan wordt. In dit

type model veranderen de kwalitatieve eigenschappen van optimale oplossingen wanneer de

parameterwaarden veranderen. Voor een globale analyse moeten we daarom de bifurcatiethe-

orie gebruiken. Deze theorie resulteert in een bifurcatiediagram dat voor iedere mogelijke

combinatie van parameterwaarden de kwalitatieve eigenschappen aangeeft van ieder mark-

tevenwicht. De analyse heeft vier mogelijke uitkomsten: (i) de initiële kosten per eenheid

product liggen boven de afkapprijs en het O&O proces wordt gestart; na verloop van tijd

begint de productie en dalen de kosten per eenheid product met de opvolgende investeringen

in O&O; (ii) de initiële kosten per eenheid product liggen boven de afkapprijs en het O&O

proces wordt niet gestart, zodat er in het geheel geen productie plaatsvindt; (iii) de initiële

kosten per eenheid product liggen onder de afkapprijs en het O&O proces wordt gestart;

productie start onmiddellijk en de kosten blijven dalen met verloop van tijd, en (iv) de initiële

kosten per eenheid product liggen onder de afkapprijs en het gestarte O&O proces wordt

afgebouwd; productie start onmiddellijk maar de technologie (en de productie) zal uitsterven

na verloop van tijd; de firma verlaat de markt. De kracht van de analyse is dat alle vier

uitkomsten kunnen voortkomen uit hetzelfde overkoepelende raamwerk.

Hoofdstuk 3 breidt de analyse van de innoverende monopolist uit hoofdstuk 2 uit naar de

situatie met twee firma’s met verschillende vormen van samenwerking. Het vergelijkt twee

verschillende scenario’s voor de vier verschillende marktuitkomsten zoals geïdentificeerd

in hoofdstuk 2. In het eerste scenario werken beide firma’s samen op het gebied van O&O,

maar tegelijkertijd concurreren zij in de productmarkt. In het tweede scenario wordt de O&O

samenwerking uitgebreid tot samenspanning in de productmarkt. Dit hoofdstuk vergelijkt

de kwalitatieve eigenschappen van deze twee scenario’s om de mogelijke nadelen van O&O

samenwerkingsverbanden – die als platform kunnen dienen voor prijsafspraken – te kunnen

evalueren. Dit levert drie belangrijke resultaten op: (i) als er sprake is van samenspanning dan

is de bandbreedte van de kosten per eenheid product waarin een nieuwe markt ontstaat groter;

(ii) samenspanning in de productmarkt vergroot de snelheid waarmee nieuwe technologieën op

de markt komen, en (iii) de bandbreedte van de kosten per eenheid product waarbij bedrijven

de technologie afstoten is groter wanneer de firma’s niet samenspannen op de productmarkt. In

het bijzonder wordt aangetoond dat er parametercombinaties bestaan waarbij samenspanning

in de productmarkt leidt tot een hoger consumentensurplus en een hoger totaal surplus. Deze

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analyse schildert een dilemma voor mededingingsbeleid omdat het laat zien dat het verbieden

van samenspanning in de productmarkt niet onomstotelijk leidt tot verhoging van de welvaart.

Het laat tevens zien dat de gerelateerde welvaartskosten zich niet gemakkelijk laten meten

omdat een verbod op samenspanning in de productmarkt de O&O beslissing beïnvloedt nog

voordat er überhaupt sprake is van productie. De beslissing om een initiële technologie niet

verder te ontwikkelen betekent dat een mogelijke welvaartswinst niet gerealiseerd wordt, maar

dit blijft onzichtbaar omdat er geen verandering van productie waar te nemen is als gevolg

van deze beslissing.

Hoofdstuk 4 bestudeert een differentiaalspel waarin twee firma’s concurreren in zowel

O&O als in de productmarkt. Om de strategische interactie tussen de firma’s zo realistisch

mogelijk weer te geven wordt de oplossing van dit spel gezocht in de categorie van de

feedback Nash evenwichten waar de mogelijkheid van discontinue investeringsfuncties een

aparte benadering vereist. De oplossing van het stochastische spel wordt beschouwd als een

benadering van het deterministische spel wanneer het niveau van witte ruis tot nul nadert. Een

numerieke benadering voor de waardefunctie wordt verkregen middels een variant van de

lijnenmethode. Dit hoofdstuk analyseert hoe spillovers van invloed zijn op de investeringen

van firma’s, op de kans dat een gegeven initiële technologie verder wordt ontwikkeld en er

zodoende een nieuwe markt gecrëerd wordt, en hoe de mogelijke structuur ervan gerelateerd

is aan de spillovers. De verkregen resultaten kwalificeren de aanwijzingen in de literatuur dat

hogere spillovers maatschappelijk wenselijk kunnen zijn aangezien ze de monopolisering van

de sector kunnen voorkomen. De analyse laat zien dat het mededingingsversterkende effect

van van grotere spillovers zich maar tot op een bepaalde hoogte voordoet en dat derhalve

een lagere kostenasymmetrie al voldoende kan zijn om tot monopolisering van de markt te

leiden bij grotere spillovers. Bovendien laat de analyse zien dat door middel van toenemende

complementariteit in O&O, grotere spillovers de kans vergroten dat een dure technologie –

die investeringen voorafgaande aan de productie noodzakelijk maakt – in productie wordt

genomen. Echter, het niveau waarop een dergelijke technologie wordt ontwikkeld kan lager

uitvallen bij grotere spillovers door firma’s die met lagere O&O investeringen proberen mee

te liften, zowel tijdens de ontwikkelfase als daarna. Zo bezien verhogen spillovers de efficiëtie

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van het productieproces maar tot op zekere hoogte. Ook verhoogt stochasticiteit de kans dat

een gegeven initiële technologie verder wordt ontwikkeld en vergroot het ook de kans dat de er

concurrentie zal zijn op de productmarkt. De reden is dat bij een hoger niveau van witte ruis er

een grotere kans is op een gunstige schok van de kosten per eenheid product en dit stimuleert

de firma’s om meer te investeren. Anders gezegd, het vergroot het uithoudingsvermogen van

de technologische volger. Tenslotte kunnen zowel predation als pre-emption in het model

ontstaan als gevolg van de optimale beslissingen van de firma’s.

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The Tinbergen Institute is the Institute for Economic Research, which was founded in1987 by the Faculties of Economics and Econometrics of the Erasmus University Rotterdam,University of Amsterdam and VU University Amsterdam. The Institute is named after the lateProfessor Jan Tinbergen, Dutch Nobel Prize laureate in economics in 1969. The TinbergenInstitute is located in Amsterdam and Rotterdam. The following books recently appeared inthe Tinbergen Institute Research Series:

506 T. KISELEVA, Structural Analysis of Complex Ecological Economic Optimal Manage-ment Problems

507 U. KILINC, Essays on Firm Dynamics, Competition and Productivity508 M.J.L. DE HEIDE, R&D, Innovation and the Policy Mix509 F. DE VOR, The Impact and Performance of Industrial Sites: Evidence from the

Netherlands510 J.A. NON, Do ut Des: Incentives, Reciprocity, and Organizational Performance511 S.J.J. KONIJN, Empirical Studies on Credit Risk512 H. VRIJBURG, Enhanced Cooperation in Corporate Taxation513 P.ZEPPINI, Behavioural Models of Technological Change514 P.H.STEFFENS, It’s Communication, Stupid! Essays on Communication, Reputation

and (Committee) Decision - Making515 K.C. YU, Essays on Executive Compensation - Managerial Incentives and Disincentives516 P. EXTERKATE, Of Needles and Haystacks: Novel Techniques for Data-Rich Eco-

nomic Forecasting517 M. TYSZLER, Political Economics in the Laboratory518 Z. WOLF, Aggregate Productivity Growth under the Microscope519 M.K. KIRCHNER, Fiscal Policy and the Business Cycle – The Impact of Government

Expenditures, Public Debt, and Sovereign Risk on Macroeconomic Fluctuations520 P.R. KOSTER, The cost of travel time variability for air and car travelers521 Y.ZU, Essays of nonparametric econometrics of stochastic volatility522 B.KAYNAR, Rare Event Simulation Techniques for Stochastic Design Problems in

Markovian Setting523 P. JANUS, Developments in Measuring and Modeling Financial Volatility524 F.P.W. SCHILDER, Essays on the Economics of Housing Subsidies525 S.M. MOGHAYER, Bifurcations of Indifference Points in Discrete Time Optimal

Control Problems526 C. ÇAKMAKLI, Exploiting Common Features in Macroeconomic and Financial Data527 J. LINDE, Experimenting with new combinations of old ideas528 D. MASSARO, Bounded rationality and heterogeneous expectations in macroeco-

nomics

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529 J. GILLET, Groups in Economics530 R. LEGERSTEE, Evaluating Econometric Models and Expert Intuition531 M.R.C. BERSEM, Essays on the Political Economy of Finance532 T. WILLEMS, Essays on Optimal Experimentation533 Z. GAO, Essays on Empirical Likelihood in Economics534 J. SWART, Natural Resources and the Environment: Implications for Economic Devel-

opment and International Relations535 A. KOTHIYAL, Subjective Probability and Ambiguity536 B. VOOGT, Essays on Consumer Search and Dynamic Committees537 T. DE HAAN, Strategic Communication: Theory and Experiment538 T. BUSER, Essays in Behavioural Economics539 J.A. ROSERO MONCAYO, On the importance of families and public policies for child

development outcomes540 E. ERDOGAN CIFTCI, Health Perceptions and Labor Force Participation of Older

Workers541 T.WANG, Essays on Empirical Market Microstructure542 T. BAO, Experiments on Heterogeneous Expectations and Switching Behavior543 S.D. LANSDORP, On Risks and Opportunities in Financial Markets544 N. MOES, Cooperative decision making in river water allocation problems545 P. STAKENAS, Fractional integration and cointegration in financial time series546 M. SCHARTH, Essays on Monte Carlo Methods for State Space Models547 J. ZENHORST, Macroeconomic Perspectives on the Equity Premium Puzzle548 B. PELLOUX, the Role of Emotions and Social Ties in Public On Good Games:

Behavioral and Neuroeconomic Studies549 N. YANG, Markov-Perfect Industry Dynamics: Theory, Computation, and Applications550 R.R. VAN VELDHUIZEN, Essays in Experimental Economics551 X. ZHANG, Modeling Time Variation in Systemic Risk552 H.R.A. KOSTER, The internal structure of cities: the economics of agglomeration,

amenities and accessibility553 S.P.T. GROOT, Agglomeration, globalization and regional labor markets: micro evi-

dence for the Netherlands.554 J.L. MÖHLMANN, Globalization and Productivity Micro-Evidence on Heterogeneous

Firms, Workers and Products555 S.M. HOOGENDOORN, Diversity and Team Performance: A Series of Field Experi-

ments556 C.L. BEHRENS, Product differentiation in aviation passenger markets: The impact of

demand heterogeneity on competition

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