The Theory of Oligopoly with Multi-Product Firms

176 Pages·1990·3.265 MB·English

by  Prof. Koji Okuguchi, Prof. Ferenc Szidarovszky (auth.)

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contlnuatlon on page t 69
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
342
Koji Okuguchi
Ferenc Szidarovszky
The Theory of Oligopoly
with Multi-Product Firms
Springer-Verlag
Berlin Heidelberg GmbH
Editorial Board
H.Albach M. Beckmann (Managing Editor)
P. Dhrymes G. Fandei G. Feichtinger J. Green W. HildenbrandW. Krelle (Managing
Editor) H. P. Künzi K. Ritter R. Sato U. Schittko P. Schönfeld R. Selten
Managing Editors
Prof. Or. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Or. W. Krelle
Institut für Gesellschafts- und Wirtschaftswissenschaften
der Universität Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Authors
Prof. Koji Okuguchi
Department of Economics
Tokyo Metropolitan University
1-1-1 Yakumo, Meguro-ku, Tokyo, Japan
Prof. Ferenc Szidarovszky
Institute of Mathematics and Computer Science
University of Economics, Budapest
Dimitrov ter 8 Budapest, IX, Hungary
ISBN 978-3-540-52567-7 ISBN 978-3-662-02622-9 (eBook)
DOI 10.1007/978-3-662-02622-9
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© Springer-Verlag Berlin Heidelberg 1990
Originally published by Springer-Verlag Berlin Heidelberg New York in 1990
2142/3140-543210 - Printed on acid-Iree paper
Preface
In the mid nineteen-sixties both authors undertook independent
works in oligopoly and game theory. However, it was not until 1983 that
they formally met. Since then they have continued meeting yearly, either
in Budapest or Tokyo. Their collaboration has resulted in numerous pub
lications as well as in this work.
Essentially, this book has two origins. First, it originated in
previous results, either published or circulated in mimeograph form.
Finely sifting their results the authors constructed a concise reinter
pretation of their achievement to date. However this unifying process
led to the second origin. Reconsideration, particularly in this compre
hensive approach, generated new results. This was especially true
in the analysis of the existence, uniqueness and global stability
of the Cournot-Nash equilibrium for oligopoly with multi-product firms.
This book should be ideal for graduate students in economics or
mathematics. However, as the authors have firmly grounded their ideas
in the formal language of mathematics, the student should possess some
background in calculus, linear algebra, and ordinary differential and
difference equations. Additionally,the book should be useful to re
searchers in oligopoly and game theory as weil as to mathematically
oriented economists. The methodology developed for analyzing the exist
ence and stability of oligopoly equilibrium should prove useful also
in theoretical analysis of other economic models.
We are both very grateful to Professor W. Krelle for his careful
review and helpful suggestions. In addition, Koji Okuguchiwishes to
thank Professors Krelle, Bös and Selten for arranging his stay at the
Institut für Gesellschafts-und Wirtschaftswissenschaften, Universität
Bonn. It was here that some important results were obtained. Ferenc
Szidarovszky is indebted to the Department of Mathematical Sciences at
the University of Texas at El Paso and to the Department of Systems and
Industrial Engineering at the University of Arizona. Both of these in
stitutions offered ideal working conditions during his visiting profes
sorships. Additionally, Szidarovszky thanks Professor J. Szep of
the University of Economics in BUdapest,for his help during various
stages of research. Finally both authors thank Eva Nemeth, at the same
university, for her efficient and accurate secretarial work.
Contents
Chapter 1. Introduction. ......•............................•......... 1
Chapter 2. Oligopoly Garnes and Their Extensions ..•..•.........•...•.. 2
2.1. The Cournot Hodel and Its Variants ..•....•......•......•.... 3
2.2. Models wi th Product Differentiation ...•.•...........•......• 8
2.3. Mul tiproduct Hodels •....•.•.••....•...................•...•. 9
2.4. Group Equilibrium Problems .........•.•.•.••......•........• 11
2.5. Supplementary Notes and Discussions .•....•.•......•....•... 13
Chapter 3. Existence and Uniqueness Results ...•.•••••...•.•.•....•.. 14
3.1. Existence Results for Hultiproduct Oligopoly •••......•..... 14
3.2. Relation of Equilibrium Problems to Fixed Point and
Nonlinear Complementarity Problems ...•.....•.•....•.•••.... 19
3.3. Uniqueness and Properties of Equilibria in the Classical
Game •.•....•••••.•..•.•..••....•.•..••.•..••....••......•.. 25
3.4. Linear Oligopoly Markets ••••..••.••..•.......•.....•..•.... 31
3.5. Numerical Methods for Finding Cournot-Nash Equilibria ...... 34
3.6. Supplementary Notes and Discussions ••.•••.•.•.........•...• 38
Chapter 4. Dynamic Oligopoly with Discrete Time Scale •..•..•....•.•• 41
4.1. Classical Results •••••.•..•...•••..•.....•..•.••.....•.•... 42
4.2. Adaptive Expectations ..•••......•........•....•••......••.. 50
4.3. Combined Expectations ..•.•...••....•..••.••....••.••.•.•... 62
4.4. Sequential Adjustment Processes ...•..•...•....••.....•..... 69
4.5. Extrapolative Expectations .....•..•.•••..•..•••••...••.•... 76
4.6. Supplementary Notes and Discussions .•••.••..••.....••.....• 81
Chapter 5. Dynamic Oligopoly with Continuous Time Scale ••.....•.•... 82
5.1. Classical Resul ts ...•...........•..•..........•.....•.•..•. 83
5.2. Adaptive Expectations ....•....•...•..•.....•...•..•......•. 88
5.3. Combined Expectations •••••...•........•.•...•.•.....••..•.• 99
5.4. Extrapolative Expectations .............•.................. 105
5.5. Supplementary Notes and Discussions .......•...........•••. 108
Chapter 6. Extensions and Generalizations ..••.•.....•.......••.... 109
6.1. Quadratic Games under Discrete Time Scale ••••••...••....•• 109
6.2. Quadratic Garnes under Continuous Time Scale .••...•...•.•.. 122
6.3. Time uependent Hodels .......•.•....•.••....•.......•....•• 125
6.4. Nonlinear Oligopolies under Discrete Time Scale ....... ····132
6.5. Nonlinear Oligopolies under Continuous Time Scale·.·······144
6.6. Supplementary Notes and Discussions .•...........•........• 162
References
Chapter
INTRODUCTION
Since the appearance of the classic book by Cournot in 1838, in
creasing attention has been given to oligopoly. Oligopoly is astate
of industry where a small number of firms produce homogeneous goods or
close substitutes competitively.
Many models consider this situation as a static noncooperative
game, which is not repeated in time. In these models the central prob
lem is to find sufficient conditions which guarantee the existence and
uniqueness of the so called Cournot or Cournot-Nash equilibrium. This
concept will be defined and examined in Chapters 2 and 3.
The static models do not describe the real economic situations
properly since the firms produce and seil goods on the market repeated
ly over time. This fact implies that dynamic models which are able to
describe and analyse the dynamic behavior of firms are more apropriate.
These models can be divided into two main groups. In the first type the
time scale is assumed discrete, and the second assumed continuous. In
both types of models no time lag is assumed between producing and sell
ing the goods. At any time period the profit of each firm depends not
only on its outputs but also on the outputs of all other firms which are
unknown to the firms when they make their production decisions. Hence
at each time t ~ 0, each firm must form expectations on other firms'
most likely outputs.
Cournot examined this situation under discrete time scale and
assumed that in each per iod each firm believed that all its rivals'
outputs would remain the same as in the preceding per iod. This simpli
fying assumption has been modified and generalized by several economists
for oligopoly with or without product differentiation and with single
product firms. In this book two types of generalizations will be con
sidered for oligopoly with multiproduct firms: adaptive and extrapola
tive expectations.
The development of this book is as foliows. After discussing static
models in Chapters 2 and 3, dynamic models with discrete time scale will
be discussed in Chapter 4. In Section 4.1 expectations ! la Cournot will
2
be analysed and adaptive expectations will be discussed in Section 4.2.
The combination of these two types of expectations will be investigated
in Section 4.3. A special sequential adjustment process under expecta
a
tions la Cour not will be introduced in Section 4.4, and extrapolative
expectations will be discussed in Section 4.5. Models under continuous
time scale will be examined in Chapter 5.Sections 5.1, 5.2, 5.3 and 5.4
are the continuous time-scale counterparts of the corresponding dis
crete time-scale models discussed in Sections 4.1, 4.2, 4.3 and 4.5,
respectively. In Chapters 4 and 5 it is assumed that the price (or in
verse demand) functions are linear and that all cost functions are
quadratic. Under these assumptions quite strong stability conditions
would be derived. Special problems and further generalizations are
presented in Chapter 6. The generalizations of the results of Chapters
4 and 5 to quadratic games are introduced in Section 6.1 and 6.2 under
both discrete and continuous time-scales. All models so far mentioned
are based on the assumption that price and cost functions do not change
over time. In long-term models this is an unrealistic assumption since
inflation, innovation and any other circumstances may affect these
functions. In Section 6.3 we therefore consider models which incorporate
time dependent price and cost functions. The last two sections of the
book will discuss nonlinear models with both discrete and continuous
time scales. We added a supplementary section to each chapter in order
to discuss the relations of our results to earlier works and also to
outline further research areas.
Finally,we mention that all of our models are formulated in partial
equilibrium model framework in the sense that oligopoly is analysed
without considering its impacts on other industries and/or those from
other industries. In our models, however, multi-product firms are ex
plicitly introduced. All theoretical results so far obtained by other
economists on oligopoly without product differentiation as weIl as on
oligopoly with product differentiation but with single product firms
will emerge as special cases of our results.
Chapter 2
OLIGOPOLY GAMES AND THEIR EXTENSIONS
In this chapter several versions of the oligopoly game are intro
duced. The most simple model, the classical oligopoly game, will be
first described and then its extensions will be analysed.
3
2.1. The Cour not Model and Its Variants
Consider a market in which N firms produce a homogeneous good to
sell for a unit price which depends on the total output of the industry.1
Assurne that each firm faces a cost of production which depends only on
the output of the firm itself. If Lk denotes the production capacity of
firm k, then it can decide abo°ut its own output xk' which therefore
should satisfy the inequality ~ xk ~ Lk• Thus, the set of strategies
of firm k is given by the closed bounded interval [O,Lkl. Let p and Ck
(k=1,2, ••• ,N) denote the unit price function and cost function of firm
k. This market situation can be modelled as an N-person game where the
set of strategies of player (firm) k is the interval Xk = [O,Lkl, and
its payoff function (profit), can be formulated as
N
<Pk(x1,x2, ••• ,xN) = xkP( L xQ,) - Ck(xk). (2.1.1)
Q,=1
The Cournot otigopoty is an N-person noncooperative game defined by sets
Xk of strategies and payoff functions <Pk (i=1,2, ••• ,N). By using the
strategi~ form notations (see e.g.,Friedman, 1986) of N-person games,
Cournot oligopoly may be denoted by r = {N; X1 , •.• ,~;<P1' ••• ,<PN}.
The solution of the Cournot oligopoly is Nash-Cournot equilibrium
point which can be defined as follows.
Definition 2.1.1. A vector ~l f = (xl1f ' .•• ,xlNf ) is called a Nash-Cournot
equitibrium point of game r, if for k=1,2, •.• ,N,
(a)
(b) For arbitrary xk e Xk,
(2.1. 2)
In other words, the Nash Cour not equilibrium point is an N-tuple of
strategies for which each player naximizes his own payoff with respect
to his own strategy selection, given the strategy choices of all other
players. That is, no player can increase his payoff by changing his
strR~egy unilaterally.
Some authors refer to the above equilibrium point as an equilibrium
in pure strategies. Since in this book no mixed (i.e. probabilistic)
The economic interpretation of this property will be presented in
Footnote 4.
4
strategies will be considered equilibria will mean only pure strategy
equilibria.
A Cournot oligopoly does not necessarily have an equilibrium point.
The following example shows a duopoly (2-person game) in which no equi
librium point exists.
Exarnple 2.1.1. Select N=2, and define X1=X2=[O,;J. Set
s = xl + x2'
P (s) l-s,
Ck (xk) = { IOxk" if 0 < xk $ 21 ' (k=l ,2)
, 10, if xk = o.
Let (xl1f ,xl2f ) be an equilibrium point, if there exists any. Assume first
that xllf > o. Then
CJll (xl'x~) = xl (1-xl-x~)-10xl-5
-x12 -(9x1+x1xl2f +5) ,
and
Consequently for any xl' such that xl1f -x1 is sufficiently small positive
number,
hence (xl1f ,xl2f ) is not an equilibrium point. Assume next that xl1f =O. Then
but
We note that a similar example is presented in Okuguchi (1976).
Even in cases when equilibrium point exists the uniqueness of the
equilibrium point is not generally true. This is illustrated next.