optimal control and differential games - …978-1-4615-1047-5/1.pdf · model revisited r.f. hartl,...

OPTIMAL CONTROL ANDDIFFERENTIAL GAMES

Essays in Honor ofSteffen Jorgensen

Advances in Computational Management Science

VOLUMES

SERIES EDITORS

Hans Amman, Eindhoven University of Technology, The Netherlands

Berc Rustem, Imperial College, London, United Kingdom

EDITORIAL BOARD

Christopher Deissenberg, University of Nantes, France

Arthur Farley, University of Oregon, USA

Manfred Gilli, University of Geneva, Switzerland

David Kendrick, University of Texas at Austin, USA

David Luenberger, Stanford University, USA

Rik Maes, University ofAmsterdam, The Netherlands

Istvan Maros, Imperial College, UK

John Mulvey, Princeton University, USA

Anna Nagurney, University of Massachusetts at Amherst, USA

Soren Nielsen, University of Texas at Austin, USA

Louis Pau, Ericsson, Alvsjo, Sweden

Edison Tse, Stanford University, USA

Andrew Whinston, University of Texas at Austin, USA

The titles published in this series are listed at the end of this volume.

OPTIMAL CONTROL AND DIFFERENTIAL GAMES

Essays in HonoT of Steffen JeTgensen

edited by

Georges Zaccour GERAD and HEC-Montreal

SPRINGER SCIENCE+BUSINESS MEDIA, LLC

Optimal control and differential games: essays in honor ofSteffen J0rgensen /edited by Georges Zaccour ISBN 978-1-4613-5368-3 ISBN 978-1-4615-1047-5 (eBook) DOI 10.1007/978-1-4615-1047-5

Copyright © 2002 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2002 Softcover reprint ofthe hardcover Ist edition 2002

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Contents

Preface

Contributing Authors

Part I Optimal Control

1Skiba Thresholds in Optimal Control of Illicit Drug UseG. Feichtinger and G. Tragler

2A Capital Accumulation Model with Debt Financing: The Steigum

Model RevisitedR.F. Hartl, P.M. Kort and A. Novak

vii

xi

3

23

3Turnpikes in multi-discount rate environments and GCC policy 39

evaluationA. Haurie

4An Ethical Behavior Interpretation of Optimal Control 53P.-D. Pineau

5From Love Dynamics to Relationship Marketing: The Case of Lin- 71

ear PartnersS.P. Sigue and F. Elloumi

6Existence and characterization of time-consistent monetary policy 87

rulesG. Sorger

Part II Differential Games

7Dynamic Oligopolistic Competition and Quasi-Competitive Behav- 107

iorE.J. Dockner and A. Gaunersdorfer

vi OPTIMAL CONTROL AND DIFFERENTIAL GAMES

8Recent Studies on Incentive Design Problems in Game Theory and 121

Management ScienceH. Ehtamo, M. Kitti and R.P. Hiimiiliiinen

9Computation of Markov Perfect Nash Equilibria without Hamilton- 135

Jacobi-Bellman EquationsG. Martin-Herran and J.P. Rinc6n-Zapatero

10Foreign Direct Investment and Localized Technological Spillovers 153M.-L. Petit and F. Sanna-Randaccio

11Impact of Retailer's Myopia on Channel's Strategies 179S. Taboubi and G. Zaccour

12Common Property Resource and Private Capital Accumulation 193N. Van Long and S. Katayama

13Pricing of Natural Resource under a Randomly Furcating Environ- 211

mentD. w.-K. Yeung

Publications of Professor Steffen J0rgensen 233

Preface

On behalf. of the contributors and participants at the Workshop onOptimal Control and Differential Games held in September 2002 in hishonor, it is my privilege to dedicate this volume to Professor SteffenJ0rgensen. With this book, we wish to honor his scientific achievements.

Professor J0rgensen's publications (in English) comprise two booksand more than 50 articles in refereed journals and proceedings volumes,in optimal control and differential games!. His theoretical contributionsdeal with certain structures of differential games that allow for analyticalsolutions (e.g., Dockner, Feichtinger, and J0rgensen (1985), J0rgensen,(1985, 1986) and J0rgensen and Zaccour (2001)). His most importantcontributions, however, are applications of optimal control theory anddifferential games to problems arising in economics and managementscience.

In marketing, he contributed to advertising dynamics and competition (e.g., J0rgensen (1982) and Dockner and J0rgensen (1988, 1992)),and to the design of dynamic optimal pricing strategies (J0rgensen (1983,2000), Dockner and J0rgensen (1984, 1988), Dockner, Gaunersdorfer andJ0rgensen (1996) and J0rgensen and Zaccour (1999)). Some of these papers have been devoted to the characterization of the impact of optimalpricing and advertising strategies on the diffusion of new products andtechnologies in monopolistic and oligopolistic markets. Still in marketing, Professor J0rgensen contributed recently to the literature dealingwith the coordination and conflicts in distribution channels and the design of mechanisms to reduce channel's inefficiencies (e.g., J0rgensen andZaccour (1996, 1999, 2002), J0rgensen, Sigue and Zaccour (2000, 2001),J0rgensen, Taboubi and Zaccour (2001) and J0rgensen and Kort (2002)).

Another area of interest has been the dynamics of the firm with special emphasis on investment and financing decisions and on productionand inventory strategies (e.g., J0rgensen (1982, 1984, 1986), Hartl andJ0rgensen (1985), van Schijndel, J0rgensen and Kort (1987), J0rgensen,Kort and van Schijndel (1989), J0rgensen and Kort (1993a,b,c, 2000,2001), J0rgensen, Kort and Zaccour (1999) and J0rgensen and Zaccour(2000)). Some of these contributions addressed the complex and still

1Professor Jorgensen's list of publications appear at the end of the volume.

viii OPTIMAL CONTROL AND DIFFERENTIAL GAMES

widely open problem of interaction between different functional areas ofthe firm. One more common denominator in this area is the use of pathsynthesizing procedure to obtain closed-form characterization.

Another applications area is resource and environmental economics(e.g., J0rgensen and Dockner (1985), J0rgensen and Sorger (1990),J0rgensen and Yeung (1996, 1999a,b), J0rgensen and Kort (1997),J0rgensen and Zaccour (2001a,b)). Here the focus has been the determination of equilibrium strategies for harvesting renewable resourcesand the design of incentives to sustain environmental cooperation.

Professor J0rgensen has also been interested in some unconventionalproblems for a management scientist such as optimal slidesmanship inconferences, amorous cycles, addiction and extramarital affairs! (e.g.,J0rgensen (1992), Hartl and J0rgensen (1990), Feichtinger, J0rgensenand Novak (1999), Dawid, Feichtinger and J0rgensen (2000)).

Finally, what also characterizes Professor J0rgensen's research effortis his interest in relating different models of the same topic to eachother. This is most evident from his differential game book (Dockner,J0rgensen, Van Long, and Sorger (2000)) and from his survey articles(e.g., J0rgensen, (1982, 1984, 1986, 1996) and Feichtinger and J0rgensen(1983)). These articles have been widely cited and have proved to beextremely valuable for researchers who want to get an overview of themost important contributions to the various fields.

Content of the volume

The volume is divided into two parts. Part I, Chapters 1-6, is dedicated to Optimal Control Methods and Applications and Part II (Chapters 7-13) to Differential Games in Economics and Management Science.

In Chapter I, G. Feichtinger and G. Tragler consider three models of control of illicit drug consumption and methadone treatment inpreventing blood-borne disease which have been validated with empirical data. They show that in all three cases, a positive feedback effect, i.e., state-dependent initiation, is responsible for the occurrence of"Skiba points" separating the basins of attraction of the multiple steadystates. In Chapter 2, R.F. Hartl, P.M. Kort and A. Novak study adynamic model of the firm where the later has debt financing as an additional means to provide funds for investment. The authors providea generalization of Steigum's model and show that, despite the fullyconcave setting, the optimal solution has history dependent equilibria.In Chapter 3, A. Haurie deals with the asymptotic behavior of optimal trajectories in infinite horizon control problems when the objectivefunction is a compromise between two averaged discounted sums of thesame stream of rewards. The paper derives some results regarding the

PREFACE ix

turnpike and zero-discounting and discusses the implications in the context of environmental management. In Chapter 4, P.-O. Pineau providesan ethical interpretation to optimal control theory. He shows that unless some religious beliefs or social considerations are taken into account,the decision-maker's actions become less ethical as time goes by. Themodel considered elicits some features of ethical behavior such as selfesteem, reputation and value of future. In Chapter 5, S.-P. Sigue andF. Elloumi introduce a formal model of relationship marketing using differential equations. The purpose of the model is to study how relationalcommitment is established, developed and maintained in a context ofsymmetric and asymmetric partners. One of their results is that neither a very low level of opportunism nor a very high level of trust arehealthy for the relational commitment. In Chapter 6, G. Sorger studiesexistence and properties of time-consistent monetary policy rules. Thelater is a function mapping any given output level of the economy to acorresponding rate of inflation. It is time-consistent if the central bankhas no incentive to deviate no matter the actual level of output. Theauthor characterizes existence of such time-consistent rules and showsthat they are generically non-unique.

In Chapter 7, E.J. Dockner and A. Gaunersdorfer consider an oligopolistic industry in the context where a regulator, or any other entity,imposes a provision of free access to reduce market power. The authorsshow that, irrespective of the information structure adopted by the players, the equilibrium price converges to the competitive level when thenumber of firms increase. The result holds for finite and infinite horizon games. In Chapter 8, H. Ehtamo, M. Kitti and R.P. Hama.liiinenstudy a principal-agent game and show how the linear wage contractcan be obtained by a three-phase adjustment process. The first two processes result in an incentive compatible Pareto optimal outcome and thethird process takes care of the agent's individual rationality. This approach is new to incentive problems and gives insights into the solutionof other related game problems of incomplete information. In Chapter 9,G. Martin-Hemin and J.-P. Rinc6n-Zapatero provide some new insightson the method they recently introduced to compute Markov perfect Nashequilibria without resorting to the use of Hamilton-Jacobi-Bellman equations. The method basically characterizes such equilibria by means of asystem of quasilinear partial differential equations. The main featuresare illustrated in the analysis of some resource games. In Chapter 10,M.-L. Petit and F. Sanna-Randaccio recall that although firms invest inforeign countries to gain some technological knowledge, recent literaturehas not fully taken into account this empirical fact. The aim of thispaper is to study how differences in degrees of technological spillovers

x OPTIMAL CONTROL AND DIFFERENTIAL GAMES

affect firms' incentive to innovate and their behavior, choice betweenexport or foreign investment, in an international market. In Chapter 11,S. Taboubi and G. Zaccour consider a differential game of a marketingchannel formed of one manufacturer and one retailer where the formercontrols her advertising expenditures and the later his marketing effortand retail price. They assess the impact of a myopic retailer, that is ashort term optimizer, on retail price, marketing efforts, manufacturer'sperformance and consumer welfare. In Chapter 12, N. Van Long andS. Katayama present a differential game model of exploitation of a common property resource where agents can also invest in private and productive capital and characterize the patterns of capital accumulationand consumption. They show that when agents are heterogeneous, themore productive accumulate more capital but all agents follow the sameconsumption and extraction rules. In Chapter 13, D.W.K. Yeung introduces a pricing dynamic game of common property resource wherethe future payoffs and the evolution of the resource stock dynamics areuncertain in the sense that the stochastic processes display a randomfurcating property. Nash equilibria are determined and implications forthe analysis of resource pricing are stated.

Acknowledgements

I would like to thank the authors for their contributions and P. Kortand G. Sorger for their help in drafting the Preface. I would like toexpress my gratitude to the Direction de la recherche, HEC-Montreal,for supporting the publication of this volume and to Carole Dufour andNicole Paradis of GERAD for helping in the organization of the Workshop and for having produced this Festschrift.

GEORGES ZACCOUR

Contributing Authors

ENGELBERT J. DOCKNERUniversity of Vienna, [email protected]

HARRI EHTAMO

Helsinki University of Technology, [email protected]

FATHI ELLOUMI

Athabasca University, [email protected]

GUSTAV FEICHTINGER

Vienna University of Technology, [email protected]

ANDREA GAUNERSDORFER

University of Vienna, [email protected]

RAIMO P. HAMALAINEN


RICHARD F. HARTL


ALAIN HAURIE

Logilab-HEC, University of Geneva, [email protected]

SEIICHI KATAYAMA

Kobe University, [email protected]

MITRI KITTI


PETER M. KORT

Tilburg University, the [email protected]

GUIOMAR MARTfN-HERRAN

Universidad de Valladolid, [email protected]

xii OPTIMAL CONTROL AND DIFFERENTIAL GAMES

ANDREAS NOVAK


MARIA-LUISA PETIT

Universita' di Roma "La Sapienza", [email protected]

PIERRE-OLIVIER PINEAU

University of Victoria, [email protected]

JUAN PABLO RINCON-ZAPATERO

Universidad de Valladolid, [email protected]

FRANCESCA SANNA-RANDACCIO

Universita' di Roma "La Sapienza", [email protected]

SIMON PIERRE SIGUE

Athabasca University, [email protected]

GERHARD SORGER

Queen Mary, University of London, [email protected]

SIHEM TABOUBI

HEC-Montreal, [email protected]

GERNOT TRAGLER

Vienna University of Technology, [email protected]

NGO VAN LONG

McGill University, [email protected]

DAVID WING-KAY YEUNG

Hong Kong Baptist University, [email protected]

G. ZACCOUR

GERAD and REC-Montreal, [email protected]

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