Table Of ContentAkio Matsumoto
Ferenc Szidarovszky
Dynamic
Oligopolies
with Time
Delays
Dynamic Oligopolies with Time Delays
Akio Matsumoto Ferenc Szidarovszky
(cid:129)
Dynamic Oligopolies
with Time Delays
123
AkioMatsumoto Ferenc Szidarovszky
Department ofEconomics Department ofMathematics
ChuoUniversity CorvinusUniversity
Hachioji, Tokyo,Japan Budapest, Hungary
ISBN978-981-13-1785-9 ISBN978-981-13-1786-6 (eBook)
https://doi.org/10.1007/978-981-13-1786-6
LibraryofCongressControlNumber:2018958494
©SpringerNatureSingaporePteLtd.2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd.
Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721,
Singapore
Preface
The authors of this book spent several decades of performing research in different
areasofmathematicaleconomicsandespeciallyingametheoryuntilthemid-1990s
whentheymetinaconferenceinOdense,Denmark.Sincethen,theyworktogether
inexaminingdifferentdynamiceconomicmodelswithspecialattentiontodifferent
variantsofoligopolies.Thiscooperationresultedinmanyconferencepresentations,
over 75 research papers, an introductory book on game theory and a jointly edited
volume of selected articles in economic dynamics. They meet at least once every
year in Tokyo and either in Tucson (Arizona) or in Budapest.
Theobjectiveofthis bookistogive anoverview ofrecentdevelopments inthe
asymptotic analysis of different versions of dynamic oligopolies with special
attentionto thepresence ofdelayed information. Thisbook can beconsidered asa
continuation of some earlier works on oligopolies. The monograph of Okuguchi
(1976) summarizes the earliest developments up to 1975. Their multiproduct
generalizations arepresentedinOkuguchi andSzidarovszky (1999).The existence
anduniquenessoftheequilibriumwerethemainfocusinadditiontolineardynamic
extensions. In the case of linear models, local asymptotical stability implies global
stability making the asymptotical analysis relatively simple. Nonlinear oligopolies
are introduced and analyzed in Bischi et al. (2010); however, no time delays are
assumed in the models. The presence of delayed information is a realistic
assumption in real economies, since data collection, determination of acceptable
and optimal decisions and their implementations need time. This book is a sig-
nificant addition to the earlier works by investigating the effects of delayed infor-
mation on the asymptotical behavior of the state trajectories of the associated
dynamic systems.
The methodology developed and used in this book can be useful in other fields
of economic dynamics, especially when instantaneous information is not available
andthereforethedecisionmakershaveonlydelayeddata.Themethodologyandthe
actual research results of this book might have practical values when decision
makers base their actual decisions on sophisticated mathematical modeling and
methodology.
v
vi Preface
The authors strongly recommend this book to undergraduate and graduate stu-
dents, researchers, since the mathematical methodology and therefore the results
reported in this book can be generalized and modified accordingly to other similar
and even in more complex cases. This book also can help members of managerial
teams responsible to reach reasonable and profitable decisions under realistic
conditions.Theauthorsarethankfultothelargenumberofreviewersoftheearlier
papersoftheauthorsaswellastotheparticipantsofmanyconferencesonnonlinear
dynamics for their useful comments which helped to make the quality of the
published papers and therefore that of the corresponding sections of this book
better. The authors are also thankful to the editorial staff of the Tokyo office of
Springerfortheirassistanceinpreparingthefinalversionofthisbook.Further,the
authors appreciate financial supports: Graduate School of Economics of Chuo
University with the MEXT-supported Program for the Strategic Research
FoundationatPrivateUniversities2013–2017,theJapanSocietyforthePromotion
of Science (Grant-in-Aid for Scientific Research (C), 24530202, 25380238,
26380316 and 16K03556) and Joint Research Grant of Chuo University.
Hachioji, Japan Akio Matsumoto
Budapest, Hungary Ferenc Szidarovszky
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Dynamic Monopolies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Discrete Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Continuous Time Models with Fixed Delays . . . . . . . . . . . . . . . . 27
2.3 Continuous Time Models with Distributed Delays . . . . . . . . . . . . 37
2.4 Supplementary Notes and Discussions . . . . . . . . . . . . . . . . . . . . 55
3 Dynamic Duopolies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Discrete Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Continuous Time Models with Fixed Delays . . . . . . . . . . . . . . . . 74
3.3 Continuous Time Models with Distributed Delays . . . . . . . . . . . . 89
3.4 Supplementary Notes and Discussions . . . . . . . . . . . . . . . . . . . . 102
4 Dynamic Oligopolies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1 Discrete Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 Continuous Dynamics with Fixed Delays . . . . . . . . . . . . . . . . . . 139
4.3 Continuous Dynamics with Distributed Delays . . . . . . . . . . . . . . 152
4.4 Supplementary Notes and Discussions . . . . . . . . . . . . . . . . . . . . 166
5 Learning in Monopolies and Oligopolies . . . . . . . . . . . . . . . . . . . . . 167
5.1 Learning in Monopolies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.2 Learning in Oligopolies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.2.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.2.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.2.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.3 Supplementary Notes and Discussions . . . . . . . . . . . . . . . . . . . . 208
6 Oligopolies with Partial Cooperation . . . . . . . . . . . . . . . . . . . . . . . . 211
6.1 Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.2 Cartelizing Groups and Antitrust Threshold . . . . . . . . . . . . . . . . 225
6.3 Supplementary Notes and Discussions . . . . . . . . . . . . . . . . . . . . 234
vii
viii Contents
Appendix A: Stability Switching Curves with Fixed Delays.. ..... .... 237
Appendix B: Stability Analysis with Continuously Distributed
Delays .... .... ..... .... .... .... .... .... ..... .... 247
Appendix C: Comparison of Discrete and Continuous Dynamics.. .... 255
References.... .... .... .... ..... .... .... .... .... .... ..... .... 259
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 263
Chapter 1
Introduction
Gametheoryisoneofthemostimportantfieldsofmathematicaleconomics.Itisthe
basisof modeling situationswhen more than one decision makers (called players)
deal with an issue and the decision of each of them affects the outcome of all of
them.Iftheyselecttheirdecisionssimultaneouslyandindependentlyofeachother,
thenthegameisconsiderednoncooperativeandtheNashequilibriumprovidesthe
usualsolution.Theplayerscanexchangeinformationandtaketheinterestofothers
intoconsideration,whenthegameispartiallyorcompletelycooperative.Inthecase
ofleader–followergames,theplayersmaketheirdecisionssequentiallyandateach
timethedecisionsaremadeknowntoallplayers.Therefore,eachplayer’sdecision
dependsonthemostrecentchoicesoftheothers.Therearealsoseveralvariantsand
extensionsofthedifferentsolutionconceptsincludingStackelbergequilibria,sub-
gameperfectNashequilibria,BayesianandperfectBayesianequilibria;however,in
thisbookweconcentrateonlyontheclassicalNashequilibriumconceptandcases
ofpartialcooperation.
In the literature on game theory, the oligopoly models play a very special role.
Aclassicaloligopolymodeldescribesasituationwhenarelativelysmallnumberof
firmsproducethesameorrelatedproductswhicharesoldinthesamemarket,oroffer
thesameorrelatedservices.Themarketisassumedtobehomogeneouswithadeter-
ministicrelationbetweenthesuppliedquantitiesandmarketprices.Themostsimple
modelassumesoneidenticalproductwhichiscalledoligopolywithoutproductdif-
ferentiation,andifthefirmsproducedifferentproductsandthepriceofeachproduct
is a function of the quantities of all products, then the model is called oligopoly
withproductdifferentiation.Therearemanyvariantsofthismodelincludinglabor
managed,marketsharingandrent-seekinggames,multiproductoligopolies,group
equilibriumproblemsandoligopsonies.Theycanbeanalyzedsimilarlytothecases
shown in this book. Another distinction between the model types is the choice of
thedecisionvariablesofthefirms.InthecaseofCournotoligopolies,theproduced
and sold quantities are the choices of the firms and the unit prices are then deter-
minedbytheinversedemandfunctions.InthecaseofBertrandoligopolies,thefirms
©SpringerNatureSingaporePteLtd.2018 1
A.MatsumotoandF.Szidarovszky,DynamicOligopolieswithTimeDelays,
https://doi.org/10.1007/978-981-13-1786-6_1
2 1 Introduction
selectthepricesfortheirproductsandthenthemarketdemandfunctionsdetermine
their corresponding production quantities. Another distinction is the choice of the
timescale.Inthediscretecase,onlydiscretetimeperiodsareconsidered,whichare
assumedtobethenonnegativeintegersforthesakeofmathematicalsimplicity.In
thecontinuouscase,thetimeperiodscantakeanynonnegativerealvalues.
In the huge literature on oligopoly theory, first the existence and uniqueness of
theequilibriumwasthemainfocus,andthendynamicextensionsbecamethemain
researcharea.Informulatingdynamicextensions,usuallytwotypesofdynamisms
are introduced: simultaneous and sequential move models. In this book, we will
concentrateonthefirsttype.Ateachtimeperiod,thefirmsadjusttheiroutputlevels
orpricesbyoneofthefollowingconcepts.Inthecaseofbestresponsedynamics,
each firm determines its best response based on its belief about the choices of the
competitors and then adjust its output level or price into the direction toward its
best response. In the case of gradient dynamics, the firms compute their marginal
profitsandadjusttheirdecisionvariablesproportionallytothemarginalprofits.In
this book, examples willbe shown for both types of dynamic models. In the early
stages, only linear models were considered, when the asymptotic properties could
be easily examined and local asymptotical stability implied global stability. The
next stage was the introduction of nonlinearities into oligopolies by introducing
nonlinear price and cost functions as well as requiring bounded feasible decision
spacessuchasdefiningcapacitylimitsinCournotoligopolies.Thelocalasymptotic
behavior of the equilibria could be examined by selecting a Lyapunov function or
by local linearization. Global analysis can be performed by using an appropriate
Lyapunovfunction,computersimulationorinthediscretetimescalesbythetheory
of critical curves. All these models assumed that at each time period, each firm
had instantaneous information about the output volumes and prices of all firms;
however,inrealeconomiesthissimplifyingassumptionhadtobemodified.Before
afirmdecidesaboutitsoptimaloutputlevelorprice,ithastocollectallnecessary
information, determine its optimal choice and implement it. All these need time
leadingtoinformationlags,whicharetakenintoaccountinthemathematicalmodels
asdelayeddynamics.Ifthedelayisduetoinstitutionalorcontractualcircumstances,
thenfixeddelaysareassumed.Thelengthsofthedelaysareconsideredfixedasany
positive value in the continuous case and a positive integer in the discrete case. In
thecontinuouscase,thecharacteristicequationisamixedpolynomial–exponential
equation. In the discrete case, the introduction of additional shifting variables can
reducetheproblemtoahigherdimensionalmodelwithoutdelay.Ifthedelaysare
uncertain due to the different delays of a large number of firms, or the firms want
to react to average past information rather than to sudden market changes, then
continuously distributed delays are the appropriated approach. In the continuous
case,Volterra-typeintegro-differentialequationsareusedasmathematicalmodels.
In the discrete case, models with geometric delays describe the distributed delays
withappropriateweightingsequences.
The rest of this book consists of five chapters and three appendices. Chapter 2
discusses the most simple case of only one firm, called monopoly. Chapter 3 is
devotedtothecasesoftwofirms,whicharecalledduopolies,andChap.4discusses
