I

ArticlesA. A. Cournot as the Great
Founder of Oligopoly
Theory─An Introduction

Information Exchanges
among Firms and Their
Welfare Implications(Part I)The Dual Relations between
the Cournot and Bertrand Models

Yasuhiro Sakai

Shiga University / Professor Emeritus

There is a historical episode by Joseph A.
Schumpeter (1883-1950), one of the greatest
economists in the 20th century. When he wasa distinguished professor at Harvard University in the 1930s, he surprised all the attending
students, one of whom was the young Paul A.
Samuelson (1915-2005), by saying the following
remark:1)
Listen, everybody. In the light of the history of
economic science, it seems to me that there exist
four great economists. Believe or not, three out of
those four are Frenchmen! Can you guess whothey really are?”According to Schumpeter’s opinion, F.
Quesnay (1694-1774), A. A. Cournot (180177) and M. E. L. Walras (1834-1910) werestrong candidates for such an exclusive economics club. Surely, Quesnay was very famousof newly inventing an economic table as theflow chart of all the economic activities in a nationa l e c onomy. Cournot wa s a b orn
mathematician, later applying a mathematical
approach to oligopoly theory. Walrus was reg arded as a pioneer of modern g eneralequilibrium theory, thus being admired somuch by Lionel W. McKenzie (1923-2008)
when I myself was a graduate student at the
University of Rochester in the 1960s.
There remains one question unanswered:
who should be the last of the four great economists according to Schumpeter’s preference?
One thing is for certain. The last person
1) See Yasui (1979).

034

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should not be a Frenchman. He could havebeen Adam Smith (1723-90), the author of theepoch-making book The Wealth of Nations(1776). Or possibly, Carl Menger (1840-1921),
an important member of the influential Austrian School? Or J. G. K. Wicksell (1851-1926),
the founder of the outstanding Swedish
School? Or J. M. Keynes (1883-1946), the author of the revolutionary book The General
Theory (1936)? Or perhaps Schumpeter himself ? No one really knows. Fortunately or
unfortunately, Schumpeter kept his mouth
shut until his death, thus contributing to thecreation of a great mystery in the history of
economic thought.
There was another interesting episode whichconnected Schumpeter with Cournot, when
Schumpeter was teaching economics at the
University of Bonn long before the Second
World War. The following question was asked
by Schumpeter to the young Nakayama, who
was then a visiting foreign student at Bonn andlater became a leading professor of modern
economics in the Japanese academics. “Mr.
Nakayama, please let me know how you havemanaged to study economics before coming toGermany. Nakayama’s answer was simple, yetgave Schumpeter a really nice surprise, “Yes,Sir. I have carefully read the works of Cournot,
Gossen and Walras under the direction of myJapanese mentor.”
Honestly speaking, in spite of his monumental works, Cournot has been mostly
underestimated with a few exceptions. One
outstanding exception was Schumpeter, who as
mentioned before, very highly evaluated
Cournot. In such long survey of papers, wewould like to gladly share this Schumpeter spirits, thus critically evaluating and freely

extending Cournot’s work on oligopoly to theworld of risk and uncertainty . In this connection, it is also worthy of attention to record thefollowing sentence by J. R. Hicks (1904-94),
one of the greatest economists in the 20th century:

Information Exchanges among Firms and Their Welfare
Implications (Part I)

Yasuhiro Sakai

The generally increased interest in mathematical
economics during the last few years has naturally
turned attention back to the work of Cournot,
the great founder of the subject, and still one ofbest teachers. It was Cournot’s creation of elementary monopoly theory which was the first
great triumph of mathematical economics; yet
Cournot had left much undone. It is not surprising that the endeavor to complete his work havebeen an attractive occupation for his successors.
(Hicks, 1935, p. 1).
As J.R . Hicks noted in his survey paper(1935), Cournot was regarded as the great
founder of the theory of monopoly and oligopoly, and still one of the best teachers in the
1930s. It was quite fortunate rather than unfortunate that Cournot had left much undone,
thereby the endeavor to make his work complete has been continuously an attractive task
for his successors until today. It is my sincere
hope that I will be one of his good successors.
More exactly speaking, almost 180 years havepassed since the publication of Cournot’s epoch-making book Recherches sur les principes
mathématiques de la theorie des richesses (1938).
It is appropriate as well as important to see
how and to what extent Cournot’s pioneering
work has contributed to the economics profession. One of the main goals of this paper is toshow that Cournot is academically alive andindeed very much alive, and continue to be so.

035

Cournot used to be called an insolent founding f ath er. Even b e f ore th e Mar g ina l
Revolution in the 1870s, he invented marginal
concepts such as marginal revenue, marginal
cost, demand elasticity, and the like. Morethan 100 years before the appearance of Game
Theory, he made full use of a very importantconcept of equilibrium in non-cooperative
games – Nash equilibrium. It is also worth
mentioning that it took as many as 45 years for
Cournot’s great book to be reviewed by Bertrand (1883).
The present paper aims to overview and evaluate the problem of information exchanges inoligopoly models, one of the most fashionable
topics in contemporary economics. It is intended to discuss a synthesis of the two
important fields, the economics of imperfectcompetition and the economics of imperfect
information.2)
Needless to say, the issue of informationtransmissions and exchanges among producers
is important not only from a theoretical pointof view, but also from antitrust policy implications. In the real economy, there exist several
types of institutions in which producers exchange their private information with eachother. Trade associations are among those information-pooling mechanisms. In order todetermine under what conditions the information exchanges among producers should be
encouraged or discouraged in terms of the consumer welfare or the total social welfare, it isfirst necessary to fully understand the working
and performance of oligopoly markets under
the conditions of imperfect information.
Let us consider a homogenous or a differentiated product market. Then this paper address
to the following set of questions. First of all,are firms with different demand and/or costfunctions willing to reveal or share information
about demand or cost? Next, how and to what
extent such information transmission affectconsumers or the whole society? Finally, are
the welfare implications of information exchang es sensitive to the number ofparticipating firms?
There exist a growing number of papers that
discuss those questions. The line of research
was initiated by Basar and Ho (1974) and Ponssard (1979), and continued by the explosion ofworks in the 1980s including Clark (1983),
Vives (1984), Okada (1984), Sakai (1985, 86, 87,89) , Gal-Or (1985, 86, 87), Shapiro (1986), andmany others. Besides, there have appeared anumber of remarkable papers in the 1990s andeven in the 2000s.3)
At the first glance, there appear no definite
answers in the existing literature, so that the
antitrust implications of information sharing
in oligopoly might be far from clear. In some
papers, firms are assumed to behave as Cournot
competitors whereas in others, they are regarded as Bertrand competitors. There may exist acommon risk or private (firm-specific) risks.
Risk may be about the demand or cost side.
Products may be homogenous or differentiated. Even if differentiated, they may besubstitutes, independent or complements.
When there exist more than two sources ofrisks, they may be positively or negatively correlated. Besides, the number of participation
firms may be just two, three, ..., or any finite
number.
It is generally expected that different models
leads to different consequences. The problem
of information exchanges in oligopoly models
has no exception for such a universal rule.

2) The imperfect competition revolution took place inmary promoters. The nature and significance of thisnew revolution was intensively discussed by Sakai (1982).
For the economic thought of risk and uncertainty, see
Sakai (2012).

the 1930s, with J. Robinson (1933), Chamberlin (1933)
and Stackelberg (1934) being its front runners. We
would add to say that another equally important revolution –the imperfect information revolution – happened in the 1970s in which Arrow (1970), A kerlof(1970), Stiglitz (1975a, 75b), and Spence (1974) were pri-

0363) See Sakai (1990), Vives (1990, 1992, 1999, 2001) andKühn & Vives (1994).

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Once a specific set of assumptions is made todescribe an oligopoly model to work with,however, a definite set of answers will be obtained. The following set of items must be
checked; (1) the type of competitors (Cournot-type or Bertrand-type), (2) the type of risks(demand or cost risks), (3) the nature of risks (acommon risk or private risks), (4) the degree ofphysical and stochastic correlation among firms(positively or negatively correlated) , and (5)the number of firms (two, three or any finite
number). In this paper, a wide variety of oligopoly models will successively be introduced,
and the problem how a change in one of thoseassumptions may result in a corresponding
change in some of welfare results will be the foc us o f inve sti g ati o n . Fo r th e s a ke o fpresentation and also subject to the space constraint, however, little or no attention will be
paid to some other related issues such as thoseof risk aversion, measurement errors, partial
sharing, garbling, first-mover versus secondmover advantages and the like.
While there may exist many possible models
regarding information exchanges in oligopoly
models as mentioned above, it is remarkable tosee that there is only one mathematical approach to such problems, namely the approachbased on game theory. As is well-known, gametheory has played a key role in integrating thetwo branches of economics into one, the Economics of Imperfect Competition and theEconomics of Imperfect Information. In fact,a recent body of work in oligopoly has been associated with the application of many concepts
borrowed from game theory, with the concept
of Nash equilibrium continuing to be a dominant one.4)

Even if each of various models aforementioned is set up on the stage, it is no easy task
to systematically analyze all the welfare effects
of information exchanges among firms, and toprovide clear-cut and intuitive interpretations
for the results. When the problem at issue is
too complicated to seize the essence of the matter, it is a well-established wisdom to break it
into several pats, and to examine the welfare
results piecewise before knitting them together.
As will be seen, the consequences of information exchanges among firms can be classified
under the four headings; namely, own andcross variation effects, and own and cross efficiency effects.
Interestingly enough, the first effect or theown variation effect represents how information flows affect the variability of each firm’s
strategic variable (each output for Cournot
models, or each price for Bertrand models),
whereas the second effect or the cross variation
effect shows how it influences the degree ofstrategic interdependence among firms. The
third and fourth effects demonstrate in which
direction information exchanges contribute to
the efficiency of resources on an industry-widebasis. In particular, whereas the own efficiency
effect is related to a better or worse correspondence between each stochastic parameter (thedemand intercept or the unit cost) and its associate strategic variable, the cross efficiency
effect is connected with a changed correspondence between the stochastic parameter of one
firm and the strategic variable of the other firm.
It will be seen that those four effects provide
quite useful tools by which to trace the welfare
implications of information exchanges amongfirms.

4) The theory of games was first invented as the joint

product of a born mathematician, von Neumann, and agreat economist, Morgenstern (1944), and later developed by the man with "beautiful mind," Nash (1951). For
its applications to oligopoly problems, see Shubik (1980),
J.M. Friedman (1977, 86), and more recent works bymany others.

Information Exchanges among Firms and Their Welfare
Implications (Part I)

Yasuhiro Sakai

037

As is seen in the contents mentioned above,the long series of papers consist of three parts:
namely, Part I, Part II, and Part III. The present paper corresponds to Part I, and aims to set
up a basic framework of differentiated duopoly
in the absence of any risks. The dual relations
between the Cournot and Bertrand models
will carefully be investigated. More specifically,
it is noted that the Cournot model with substitutable (or complementary) goods is the dualof the Bertrand model with complements (orsubstitutes). Part I also prepares for later discussions on extensive comparisons of moregeneral oligopoly models on the basis of thetype of competitors, the type of risks, and thenumber and nature of risks.
Part II will be discussed in the next paper. It
begins to deal with the world of risk and uncerta i nt y, w i th a d i s c u ss i o n o f th e m o s t
fundamental model: the Cournot duopoly
model with a common demand risk. It will befollowed by the same Cournot model with acommon cost, and by the corresponding Bertrand models with a common demand or costrisk. Part III will be the target of the last paperof the series. In this last part, many possible extensions of the duopoly results to very generaloligopoly situations will be carried out.

II

Alternative Duopoly
Models with and without
Risk Factors

2-1. The Dual Relations between
the Cournot and Bertrand Duopoly
Models─ The World of Perfect Information

As was noted above, there exist two types ofcompetition (Cournot or Bertrand), two more038

types of risk (demand or cost), and still two
more types of information structures (a common value or private values). Therefore, when
all the possible combinations are considered,
eight different types of oligopoly models will
have to be discussed. This could probably constitute a very repetitious and even tiresometask. Fortunately, there would be a great help
from the duality argument! Indeed, as will beseen below, there exist the nice dual relations
between the Cournot and Bertrand models in
the world of perfect information.
Let us pick up any two models. Then if theyshare the same formal structure and differ onlyin the interpretation placed on variables andparameters, we say that they are dual. One natural consequence of such duality argument isthat a proposition derived for one model canbecome a proposition for the other if variables
and parameters are duly interchanged between
the two models. In the light of the history of
economic thought, it is Cournot himself who
was close enough yet fell short of adopting
what we may now call a dual approach to oligopoly theory.
Cournot has established the important proposition that the output supplied under duopoly
is greater than the output supplied under puremonopoly. Since the market demand curve isusually downward sloping, this implies that theprice charged under duopoly is lower than theprice charged under pure monopoly.
While there is the wide range of physical interdependence between two outputs, Cournot
discussed only the two extreme cases, namely,
the case of perfect substitutes and the one ofperfect complements. This clearly indicates thelimitations of the original analysis of Cournot,
thus showing the necessity to extend it to a

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wider range of intermediate cases between
those of perfect substitutes and perfect complements.5)
The model we are going to analyze here is thefollowing non-stochastic duopoly model withdifferentiated products and/or cost differences.
On the production side, we have a duopolistic
sector with firms 1 and 2, each one producing a
differentiated product, and a competitive numéraire sector. Let x0 be the output of thenuméraire good, xi be the output of the i thfirm, and pi be its unit price (i = 1, 2). The unit
price of x0 is of course unity, namely p0 = 1.
On the consumption side, we have a continuum of consumers of the same type with utility
functions which are linear and separable in the
numéraire good. For tractability, it is assumed
that the utility function U of the representative
consumer is quadratic:
U = x0 + α1 x1 + α2 x2 –(1/2)(βx12 + 2βθ x1 x2 + βx22 ),p1 α1
1 θ x1=–βp2 α2
θ 1 x2 .
Now, assuming that α1 – α2θ > 0 and α2 –α1θ > 0, let us put
a1 = (α1 – α2θ) / β(1 – θ2) ,a2 = (α2 – α1θ) / β(1 – θ2) ,b = 1 / β(1 – θ2) .
It is then easy to see that these newly introduced parameters, a1, a2 and b, are all positive.
In the light of (2) and (3), it is not hard to obtain the following set of direct demand
equations:x1 = a1 – bp1 + bθp2 ,
x2 = a2 – bp2 + bθp1 .

(4)
(5)

Alternatively, in matrix notation, we have(1)

where αi and β are all positive, and the valueof θ lies between –1 and 1.
Let us assume that the consumer is supposed
to maximize U subject to the budget constraint, x 0 + p 1 x 1 +p 2 x 2 = m, where is mdenotes his given income. Then it can easily beseen that inverse demand functions are givenby the following set of linear equations:p1 = α1 –βx1 – βθx2,(2)p2 = α2 –βx2 – βθx1.(3)If we use matrix notation, we can summarize(2) and (3) as follows:x1 α1
1 – θ p1=–bx2 α2
–θ 1 p2 .
It is noted that the value of θ stands for agood measure of the substitutability of the two
products. In fact, x1 and x2 can be regarded as
substitutes, independent, or complements according to whether θ is positive, zero, or
negative.
We assume that the technology exhibits constant returns to scale, so that firm i has constantunit cost κi . Profits of firm i are provided by
Πi = ( pi – κi) xi. It is noted that Πi is not symmetric in p i and x i unless κ i vanishes. In
general, the Πi functions treat ( pi – κi) and xi
symmetrically. In order to make symmetrictreatment clearer, it is instructive to reformulate (2)–(5) in the following way:5) There is now a growing body of literature dealing

with the working and performance of oligopoly markets
under product differentiation, centering around the duality and efficiency comparison between Cournot andBertrand equilibriums. For its earlier works, see Singh& Vives (1984), Vives (1984), Okuguchi (1987), and Sakai(1986).

Information Exchanges among Firms and Their Welfare
Implications (Part I)

Yasuhiro Sakai039

p1 – κ1 = (α1 – κ1) – βx1 – βθx2 ,
p2 – κ2 = (α2 – κ2) – βx2 – βθx1 ,
x1 =(a1 – bκ1 + bθκ2)– b( p1 – κ1)+ bθ( p2 – κ2) , 
x2 =(a2 – bκ2 + bθκ1)– b ( p2 – κ2)+ bθ ( p1 – κ1). 

(2*)
(3*)
(4*)
(5*)

In matrix notation, these four equations may
be rewritten as follows:
p1 – κ1 α1 – κ1
1 θ x1=–βp2 – κ2 α2 – κ2
θ 1 x2 ,x1 α1 – bκ1 + bθκ2
1 –θ p1 – κ1=–bx2 α2 – bκ2 + bθκ1
–θ 1 p2 – κ2 .
As is well-known, the Cournot equilibriumis the Nash equilibrium in outputs, whereas the
Bertrand equilibrium is the Nash equilibriumin prices. In view of equations (2*)–(5*) or
their matrix notations aforementioned, the dual relations between the Cournot and Bertrand
models are given in Table 1.
There is an outstanding duality between the
Cournot and Bertrand equilibriums: Cournot
equilibrium with substitute (or complementary) outputs is the dual of Bertrand equilibriumwith complements (or substitutes). Once theCournot equilibrium strategies are determined,
the Bertrand equilibrium strategies are also
given by the duality argument. All we have to
do is to replace xi with (pi – κi), ( pi – κi) withxi,
(αi – κi ) with (ai – bκi – bθκj), β with b, andθ with (–θ) (i, j = 1,2; i ≠j).
More specifically, the equilibrium concept
we are going to use in this paper is the application of Nash equilibrium to many oligopoly
models of Cournot and Bertrand types. In the040

absence of any risks, we say that the pair (x1C,
x2C) of output strategies is an equilibrium if thefollowing conditions are met:
x1C = Arg Max x1 Π1 (x1 , x2C) ,x2C = Arg Max x2 Π2 (x1C, x2).
When the Cournot equilibrium is reached,
no firm has an incentive to deviate from it. It isnoted here that each firm’s profit is given by thefollowing equations:Π1(x1, x2) = (α1 – κ1 – βx1 – βθx2)x1 ,Π2(x1, x2) = (α2 – κ2 – βx2 – βθx1)x2 .
Then the reaction functions of firms 1 and 2
are provided by

Table 1 The Dual Relations between
the Cournot and Bertrand Models

Variables andParameters

Cournot
Model

Bertrand
Model

Strategic

x1

p1 – κ1

Variablesx2

p2 – κ2

Dependent

p1 – κ1

x1

Variables

p2 – κ2x2

α1 – κ1

a1 – bκ1 + bθκ2α2 – κ2

a2 – bκ2 + bθκ1

β

b

θ

–θ

Parameters

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2016 autumn / No.409

R1C : x1 = (1/2β) (α1 – κ1 – βθx2) ,p2B= Arg Max p2 Π2( p1B , p2).

R2C : x2 = (1/2β) (α2 – κ2 – βθx1) .

Since we have

The Cournot duopoly equilibrium under norisks can easily be depicted in Fig. 1. There are
the two charts (A) and (B) in the figure. The
left chart (A) indicates the case of substitutes(namely, θ > 0) in which the reaction curves
are negatively sloping.
In contrast, the right chart (B) shows thecase of complements (i.e., θ < 0) where the reaction curves are positively sloping.
As mentioned above, the Bertrand equilibrium with price strategies constitutes the dual of
the Cournot equilibrium with output strategies. Therefore, we say that the (p1B, p2B) ofprice strategies is an equilibrium if the following conditions are satisfied:
p1B = Arg Max p1 Π1( p1 , p2B),(A) θ> 0:
the Case of Substitutes

Π1( p1, p2) =( p1 – κ1) (a1 – bp1 + bθp2),
Π2( p1, p2) = ( p2 – κ2) (a2 – bp2 + bθp1) ,the reaction functions of firms 1 and 2 are given by
R1B: p1 = (1/2b) (a1 + bκ1 + bθp2) ,R2B: p2 = (1/2b) (a2 + bκ2 + bθp1) .
We can depict the Bertrand duopoly equilibrium under no risks in Fig. 2. The left chart (A)stands for the case of substitutes (θ > 0), in
which the reaction curves are positively sloping. The right chart (B) corresponds to thecase of complements (θ < 0), where the reaction curves are negatively sloping.

(B) θ< 0:
the Case of Complements

Figure 1 Cournot Duopoly Equilibriums under No Risks

Information Exchanges among Firms and Their Welfare
Implications (Part I)

Yasuhiro Sakai

041

(A) θ> 0:
the Case of Substitutes(B) θ< 0:
the Case of Complements

Figure 2. Bertrand Duopoly Equilibriums under No Risks

Comparison of Fig. 1 and Fig. 2 shows theexistence of dual relations between Cournot
and Bertrand. It is quite interesting to see that
Charts (A) and (B) of Fig. 1 respectively correspond to Charts (B) and (A) of Fig. 2 if priceand output variables are interchanged.
The duality argument is very convenient andreally powerful. However, it should not be almighty. It may sometimes break down. In fact,
as will be seen below, when we discuss consumer surplus, it surely breaks down!
It is a rather common practice in economics
that consumer surplus is measured by CS = U– x0 – ∑i pi xi. Therefore, if we make use of (1)– (3), we find the following CS formula:
CS = (1/2) ∑i(αi – pi) xi= (1/2)∑i {αi – ( pi – κi)} xi– (1/2) ∑i κi xi.It is easy to see that the formula does nottreat xi and (pi – κi) symmetrically. Consequently, the duality argument applies only to
profits and producer surplus, but not to consumer surplus and total surplus at all.
2-2. Introducing Risk Factors into
Alternative Duopoly Models
We are now ready to introduce risk factors
and investigate how the presence of demand or
cost risk affect the working and performance ofan oligopoly market. The problem here is that
there are so many ways of introducing stochastic factors into our model, depending on thetype of risk (demand or cost, a common value
or private values) faced by firms.6)
First of all, let us assume that risk is about thedemand side. For simplicity, suppose that α1(6)6) In the light of the history of economic thought, there

have been so many ways of introducing risk and uncertainty into economic models. For details, see Sakai(2010).

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and α2 are now random variables, and may bedescribed in the following way:
α1 = α+ε1 , α2 = α+ε2 .

(7)Here α denotes a stochastic demand common to all the firms, where as ε i shows astochastic demand specific to the i th firm (i =1, 2). It is noted that ε1 and ε2 may be positively or negatively correlated. For instance,
suppose that x1 and x2 respectively represent“one week trip in New York and Boston” andone week trip in California” as two attractive
goods in the tourist industry. Then α maymean the fluctuations of the yen-dollar exchange, whereas ε1 and ε2 respectively show theweather in the Eastern Coast and the one in
the Western Coast.
It is recalled that the Cournot and the Bertrand models are dual. If α 1 and α 2 arestochastic parameters in the former model, so
are a1 and a2 in the latter model. As was statedabove, the relations between these two set ofstochastic parameters must be indicated by thefollowing formulas:
a1 = (α1 – α2θ)/β(1 – θ2),
a2 = (α2 – α1θ )/β(1 – θ2).(8)Now, let us turn our attention to the case inwhich risk is about the cost side. Assume that
κ1 and κ2 are stochastic parameters, being written as follows:κ1 = κ + τ1, κ2 = κ + τ2.(9)

Here κ stands for a stochastic cost common
to all the firms, whereas κi shows a stochastic
cost specific to the i th firm (i = 1, 2). It is no-

Information Exchanges among Firms and Their Welfare
Implications (Part I)

ticed that τ 1 and τ 2 may be positively or
negatively correlated. For instance, let us consider the fluctuations of oil prices in the world.
The common parameterκmay represent the
dollar/yen exchange rate which fluctuates frequently but influences every firm’s cost at thesame ratio. Assume that τ1 and τ2 respectively
stand for the imported price of Iraq oil and theone of Venezuela oil. The Iraq oil and the Venezuela oil may rise or decline in the same
direction or in opposite directions, dependingon the domestic conditions of each country.
The question which would naturally arise iswhether or not the nice relationship between
the Cournot and Bertrand models remain intact in the presence of risk. On the one hand,
if risk is about the demand side, the parameters
α1 and α2 are random in the Cournot model,
the parameters a1 and a2 are random in the
Bertrand model (see Table 1). As a result, theintroduction of risk, whether it is common or
firm-specific, does not change the dual relation
between these two models.
On the other hand, if risk is about the costside, a completely new situation will emerge
since the simple duality argument is no longer
applicable. As can be seen in Table 1, when κ1and κ2 are random variables, they affect not
only parameters but also dependent variables in
the Cournot model, whereas they influence parameters as well as strategic variables in the
Bertrand Model. Therefore, the way how costrisk changes the relations between strategic anddependent variables in the Cournot model
must be different from the way how it changes
these relations in the Bertrand model. So whencost risk is introduced into an oligopoly model,
the Cournot equilibriums with substitute (orcomplementary) outputs are no longer the dual

Yasuhiro Sakai043

of the Bertrand equilibriums with complements (or substitutes).7)
In short, the duality argument is powerful,
but not almighty. As common sense tells us, it
may be helpful in some situations, it may not
be so in other situations. This shows the necessity for differentiating the case of demand risk
from the one of cost risk.

III

Concluding Remarks

In the above, we have intensively discussed
the information sharing in oligopoly and its
welfare implications. It is hoped that such discussions will lead to a synthesis of theeconomics of imperfect competition and theeconomics of imperfect information.
At the memorial Third Congress of the European Economic Association at the Universityof Bologna in 1988, Gray-Bobo as an invited
speaker impressed so many people by saying
the following:8)
A 150 years old book, written 15 years after Ricardo’s death by an almost entirely isolated man,can be so brilliantly argued that some of its parts
are still discussed today.  (Gray-Bobo, 1988, p. 2)
Almost 30 more years have passed since
Gary-Bobo’s interesting remark. It is true that
Augustin A. Cournot spent his isolated life as afirst-rate mathematician, later applied differential and integral calculus to the problem ofoligopoly. His courageous attempt to synthesize powerful mathematics and practical
economics , however, may now be regarded as a
towering landmark in the history of economic
thought. It would be fair to say that Cournot

7) For this point, see Sakai and Yamato (1989, 1990).

The usefulness and limitations of the duality argument
must always be kept in mind. Everything has a sunnyside as well as a shady side.

044

is so great because his doctrine is still alive after
180 yeas of its first publication.
In the absence of no risks, there exist the remarkable dual relations between Cournot andBertrand oligopoly models. In fact, the Courno t e qu i l i bri um wi th su b sti tuta b l e (orcomplementary) outputs is the dual of the Bertrand equilibrium with complements (orsubstitutes). Once the Cournot equilibrium
strategies are determined, so are the Bertrand
equilibrium strategies by the duality argument.
It is really one of main purposes of this paper
to discuss whether and to what extent introduction of risk factors into the Cournot or
Bertrand models would influence such duality
analogy.
On the one hand, if risk is about the demandside, the introduction of risk, whether is ofcommon type or of firm-specific type, does notchange the dual relation between the two models. On the other side, if risk is about the costside, a completely new situation has to emerge
since the simple duality argument is no longer
applicable.
In conclusion, we can say that the duality argument is powerful, but not almighty. It may
be useful in some situations, but it may not be
so in other situations. We must differentiate
the case of demand risk from the one of costrisk.
In this paper, we have worked with admittedly simple oligopoly models with or without
conditions of risk. We do believe, however,that the results obtained in this paper are fundamentally robust. Much work remains to beleft for future research.

8) See Gray-Bobo (1988), page 20. I still remember how

much I was excited when I read my paper on a new topic
on oligopoly theory before the huge audience at the University of Bologna, Italy. I really left my heart in the presumably oldest university in the world. We can surely
learn new lessons from old teachings!

THE HIKONE RONSO

2016 autumn / No.409

[Acknowledgement]This work was partly supported by theGrand-in-Aid for Scientific Research (C) No.
16K03837 from the Japanese Ministry of Education, S cience, Sp or t, Cu lture andTechnology. Editorial assistance by Dr. Masashi Tajima and the staff members of theCenter for Risk Research, Shiga University,
was gratefully acknowledged.
References
⦿Akerlof, G.A. (1970) “The Market for Lemons: Qualita-

tive Uncertainty and the Market Mechanism,” Quarterly Journal of Economics, Vol.84, pp.488-500.

⦿Harsanyi, J.C. (1967-68) “Games with Incomplete Infor-

mation Played by ‘Bayesian Players’,” Management Science , Vol. 14. Parts I, pp.159-182; Part II, pp. 320-334;
Part III, pp. 486-502.

⦿ Kühn, K.U. and Vives, X. (1994) “Information Ex-changes among Firms and Their Impact on Competition.” Discussion Paper, Institute of Analytical Economics, Univ. of Barcelona, No. 94-1, pp.1-126.

⦿ Li, L . (1985) “Cournot Oligopoly with Information

Sharing,” Rand Journal of Econnomics, Vol. 16, pp. 521536.

⦿ Nash, J.F. (1951) “Non-cooperative Games,” Annals of

Mathematics, Vol. 54, pp. 286-295.

⦿ Okada, A. (1982) “Information Exchange between Du-

opolistic Firms,” Journal of Operational Research Societyof Japan, Vol. 25, pp. 58-76.

⦿ Okuguchi, K. (1987) “Equilibrium Prices in the Ber-

⦿Arrow, K.J. (1970) Essays in the Theory of Risk-Bearing,

trand and Cournot Oligopolies,” Journal of Economic
Theory, Vol.42, pp.128-139.

⦿Basar, T. and Ho, Y. (1974) “Informational Properties of⦿ Ponssard, J.P. (1979) “Strategic Role of Information in⦿ Bertrand, J. (1883) “Book Review of Theorie mathema-⦿Robinson, J. (1933) The Economics of Imperfect Competi-

North Holland.

the Nash Solution of the Two Stochastic Non-zero-sum
Games,” Journal of Economic Theory, Vol. 7, pp. 370-384.

Demand Function in an Oligopolistic Market,” Management Science, Vol. 25, pp. 240-250.
tion, Macmillan.

tique de la richess social and of Recherches sur les principles mathematique de la theorie des richesses.” Journaldes Savants, pp. 499-508.

⦿Sakai, Y. (1982) The Economics of Uncertainty, Yukikaku

⦿ Clarke, R.N. (1983)“ Collusion and the Incentives for⦿ Sakai, Y. (1985) “The Value of Information in a Simple

Information Sharing,” Bell Journal of Economics, Vol.
14, pp. 383-394.

⦿Cournot, A.A. (1838) Recherches sur les principles mathe-matique de la theorie des richesses. Paris, Hachette.

⦿Friedman, J.M. (1986) Game Theory with Applications to

Economics, Oxford University Press.

⦿ Gray-Bobo, R. (1988) “Cournot: A Great Precursor of

Mathematical Economics,” Paper presented at the Third
Congress of the European Economic Association, Bologna, Italy.

⦿ Gal-Or, E. (1985) “Information Sharing in Oligopoly,”

Econometrica, Vol.53, pp. 329-343.

⦿Gal-Or. E. (1986) “Information Transmission–Courn-ot and Bertrand Equilibria,” Review of Economic Studies, Vol.53, pp. 279-292.

⦿ Gal-Or, E. (1987) “First Mover Disadvantages with Pri-

vate Information,” Review of Economic Studies, Vol. 54,
pp.259-270.

Publishing Company.

Duopoly Model,” Journal of Economic Theory, Vol. 36,
pp. 56-74.

⦿ Sakai, Y. (1986) “Cournot and Bertrand Equilibriums

under Imperfect Information,” Journal of Economics,
Vol. 46, pp. 213-232.

⦿Sakai, Y. (1987) “Cournot and Stackelberg Equilibriums

under Product Differentiation: First-mover and Secondmover Advantages,” Tsukuba Economic Review, Vol.18,
pp. 1-33.

⦿ Sakai, Y. (1989) “Information Sharing in Oligopoly:

Overview and Evaluation,” Discussion Paper, No. 89-1,
Institute of Social Sciences, University of Tsukuba, pp.
1-87.

⦿ Sakai, Y. (1990) Theory of Oligopoly and Information,

Tokyo, Toyo Keizai Publishing Company.

⦿ Sakai, Y. (2010) Economic Thought of Risk and Uncer-

tainty. Kyoto: Minerva Publishing Co.

⦿Sakai, Y. and Yamato, T. (1988) “Information Sharing in⦿ Hicks, J.R. (1935) “Annual Survey of Economic Theory:

Bertrand Oligopoly,” Discussion Paper, No. 88-1, Institute of Social Sciences, University of Tsukuba, pp. 1-28.

Information Exchanges among Firms and Their Welfare
Implications (Part I)

Yasuhiro Sakai

the Theory of Monopoly,” Econometrica, Vol. 3-1, pp.
1-20.

045

⦿ Sakai, Y. and Yamato, T. (1990) “On the Exchange of

Cost Information in a Bertrand-type Duopoly Model,”Economic Studies Quarterly, Vol.41, No.1, pp.48-64.

⦿ Shapiro, C. (1986) “Exchange of Cost Information in

Oligopoly,” Review of Economic Studies, Vol. 53, pp. 433466.

⦿ Shubik, M. (1980) Market Structure and Behavior, Har-vard University Press.

⦿ Singh, N. and Vives, X. (1984) “Price and Q uantity

Competition in a Differentiated Duopoly,” Rand Journal of Economics, Vol.15, pp.546-554.

⦿ Spence, M. (1974) Market Sig naling: Information

Transfer in Hiring and Related Screening Processes,
Harvard University Press.

⦿ Stackelberg, H. von (1934) Marktform und Gleichge-

wicht, Berlin, Julius Springer.

⦿Stiglitz, J. E. (1975a) “Information and Economic Anal-ysis,” in Parkin, M. and Nobay, A. R. (eds.), Current
Economic Problems. Cambridge University Press.

⦿Stiglitz, J. E. (1975b) “The Theory of ‘Screening’, Educa-

tion, and the Distribution of Income,” American Economic Review, Vol.65, pp.283-300.

⦿ Vives, X. (1984) “Duopoly Information Equilibrium:

Cournot and Bertrand,” Journal of Economic Theory,
Vol. 34-1, pp.71-94.

⦿Vives, X. (1990) “Information and Comparative Advan-

tage,” International Journal of Industrial Organization,
Vol. 8, pp.33-52.

⦿ Vives, X. (1992) “Trade Association Disclosure Rules,

Incentives to Share Information, and Welfare,” R AND
Journal of Economics, Vol. 21(3), pp.409-430.

⦿ Vives, X. (1999) Oligopoly Pricing: Old Ideas and New

Tools, MIT Press, 1999.

⦿Vives, X. (2008) Information and Learning in Markets:

The Impact of Microstructure, Princeton University
Press.

⦿Von Neumann, J. and Morgenstern, O. (1944) Theory ofGames and Economic Behavior, Princeton, Princeton
University Press.–⦿Yasui, T. (1979) Economics and My Academic Life, To-kyo, Bokutaku Publishing Company.

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THE HIKONE RONSO

2016 autumn / No.409

Information Exchanges among Firms andTheir Welfare Implications (Part I)
The Dual Relations between the Cournot and Bertrand Models

Yasuhiro Sakai

This long series of papers consist of threeparts. Part I is concerned with the basic dual
relations between the Cournot and Bertrand
models. Part II begins to deal with the worldof risk and uncertainty, with a discussion of theCournot duopoly model with a common demand risk as a starting point. It then deals withother types of duopoly models with a common
risk. Part III discusses more complicated problems such as private risks and oligopoly models.
All these three parts taken together aim to carefully outline and critically evaluate the problem
of information exchanges in oligopoly models,
one of the most important topics in contemporary economics.
The present paper corresponds to Part I. We
intend to give a synthesis of the economics ofimperfect competition and the economics ofimperfect information. The problem at issue is
how and to what extent the information exchanges among firms influence the welfare of
producers, consumers and the whole society.
In the real world, trade associations may be regarded as typical information exchangemechanisms. It is hoped that the welfare implications obtained in the paper will shed a newlight to the effectiveness and limitations of theindustrial policies adopted by governments.
Key words: Information exchange, oligopoly
models, welfare implications, dual relations,trade associations.

Information Exchanges among Firms and Their Welfare
Implications (Part I)

Yasuhiro Sakai

047