A Model of Oligopoly
23
Julio de 2021 A Model of Oligopoly Hernán Vallejo Documento CEDE # 38
Transcript of A Model of Oligopoly
Jul io de 2021
A Model of Oligopoly
Hernán Vallejo
Documento CEDE
# 38
Serie Documentos Cede, 2021-38 ISSN 1657-7191
Edición electrónica. Julio de 2021
© 2021, Universidad de los Andes, Facultad de Economía,
CEDE. Calle 19A No. 1 – 37 Este, Bloque W. Bogotá, D. C.,
Colombia Teléfonos: 3394949- 3394999, extensiones 2400,
2049, 2467
http://economia.uniandes.edu.co
Impreso en Colombia – Printed in Colombia
La serie de Documentos de Trabajo CEDE se circula con propó-
sitos de discusión y divulgación. Los artículos no han sido
evaluados por pares ni sujetos a ningún tipo de evaluación
formal por parte del equipo de trabajo del CEDE. El contenido
de la presente publicación se encuentra protegido por las
normas internacionales y nacionales vigentes sobre propiedad
intelectual, por tanto su utilización, reproducción, comunica-
ción pública, transformación, distribución, alquiler, préstamo
público e importación, total o parcial, en todo o en parte, en
formato impreso, digital o en cualquier formato conocido o por
conocer, se encuentran prohibidos, y sólo serán lícitos en la
medida en que se cuente con la autorización previa y expresa
por escrito del autor o titular. Las limitaciones y excepciones
al Derecho de Autor, sólo serán aplicables en la medida en que
se den dentro de los denominados Usos Honrados (Fair use),
estén previa y expresamente establecidas, no causen un grave
e injustificado perjuicio a los intereses legítimos del autor o
titular, y no atenten contra la normal explotación de la obra.
Universidad de los Andes | Vigilada Mineducación Reconoci-
miento como Universidad: Decreto 1297 del 30 de mayo de
1964. Reconocimiento personería jurídica: Resolución 28 del
23 de febrero de 1949 Minjusticia.
Documento CEDE
Los documentos CEDE son producto de
las investigaciones realizadas por al menos
un profesor de planta de la Facultad de
Economía o sus investigadores formalmente
asociados.
A Model of Oligopoly
Hernan Vallejo
Universidad de los Andes
Facultad de Economıa
Bogota D.C., Colombia
July 27, 2021
Abstract
This article builds a simple model of oligopoly and uses it to make a detailed charac-
terization of the equilibrium prices; quantities; mark-ups; price elasticities of market
demand; price elasticities of residual demand; and welfare, all in terms of the param-
eters of the model. This is done under five different conjectures -Collusion, Threat,
Cournot, Stackelberg and Bertrand-. The results of the model are used do compar-
ative statics.
JEL Classification: C70, C71, D43, L13.
Keywords: Oligopoly, Collusion, Threat, Cournot, Stackelberg, Bertrand, mark-up
Un modelo de oligopolio
Hernan Vallejo
Universidad de los Andes
Facultad de Economıa
Bogota D.C., Colombia
Este artıculo construye un modelo simple de oligopolio y lo utiliza para hacer una caracter-
izacion detallada del equilibrio en terminos de precios; cantidades; mark-ups; elasticidades
precio de la demanda del mercado; elasticidades precio de la demanda residual y el bienes-
tar, en funcion de los parametros del modelo. Esto se hace bajo cinco conjeturas diferentes
-Colusion, Amenaza, Cournot, Stackelberg y Bertrand-. Los resultados del modelo se uti-
lizan para hacer ejercicios de estatica comparativa.
Clasificaciones JEL: C70, C71, D43, L13.
Palabras claves: Oligopolio, Colusion, Amenaza, Cournot, Stackelberg, Bertrand y
mark-up.
1 Introduction
This article presents a simple model of oligopoly. To do so, the same market demand
and the same cost structure per firm are used, in order to characterize the market equilib-
rium in terms of the parameters of the model, under five different conjectures -Collusion,
Threat, Cournot, Stackelberg and Bertrand-. The results are used to do comparative
statics.
2 Previous literature
The term oligopoly has been traced back by Chamberlain (1957), to St. Thomas
More´s “Utopia” (1518), where he used it to refer to the “the sale by the few”. Schlesinger
(1914) used the German word “oligopolische”, without a theory and without influence.
Oligopoly as a term was later used and massified by Chamberlein (1933) in his book “The
Theory of Monopolistic Competition”.
Cournot (1838) presented the first formal model of oligopoly, which remains at the
core of this theory and is regarded by authors such as Shapiro (1989), as one of the most
important -if not the most important- contribution to the topic.
Bertrand (1883) criticized Cournot and proposed that if players could choose between
quantity competition and price competition, price competition would prevail. With con-
stant returns to scale, the Bertrand price is equal to the competitive price, which is equal
to the marginal cost. Edgeworth (1897) critisized Cournot as well, pointing out that its
results had been shown to be wrong, regardless of the cost structures.
Shapiro (1989) made a thorough review of oligopoly theory, for static and dynamic
games. It has long been acknowledged that oligopolies have an equilibrium that can change
depending on the assumptions made and on the way firms interact among themselves. The
way firms interact among themselves, may also change across time.
Levin (1988) analyzed Collusion, Cournot and Stackelberg, and concluded that at
equilibrium, Collusion has greater prices than Cournot, and Cournot greater prices than
Stackelberg.
This article builds a simple model of oligopoly, that reflects some of the basic results
1
obtained in the literature, expressing them in terms of the parameters of the model.
Vallejo (2021) makes a similar approach for oligopsony in a factor (labor) market, which
has mirror opposite features to this article, in order to highlight explicitly that mirror
symmetry between oligopoly and oligopsony.
3 A Model of the Oligopoly
The model constructed in this article aims to characterize in a detailed way the
oligopoly equilibrium in the markets of goods and services, under the conjectures of Col-
lusion, Threat, Cournot, Stackelbeg, and Bertrand.
3.1 General Assumptions
In order to achieve the objectives of this article, the following assumptions are made.
There are many consumers of a good or service, which are price takers. The market
demand curve is linear, or is linearized around the relevant equilibrium, as:
p = a− bQ
Where p is the unit price and Q are the total quantities produced and sold in the
market, and a and b are positive parameters (a > 0 and b > 0).
Consider two identical firms A and B, that produce a homogeneous product. Firms
have identical total cost functions (TC), with a non-negative and constant marginal cost
d ≥ 0, and no fixed costs. Fixed costs are zero, or are sunk, in order to focus on the role
of conjectural variations, and not on the role of economies of scale.
Thus, the total cost function of any firm, as a function of a firm’s output q, can be
written as:
TC = dq
The marginal cost of any firm is thus constant, and non-negative, as pointed out before:
MgCi = d
Just as in monopoly or monopolistic competition, there is no supply curve in oligopoly,
2
since firms maximize profits and to do so, they sell at the highest possible price, which is
given by the demand curve. This implies that all combinations of prices and quantities
are on the demand curve, and thus, it is impossible to find a combination of prices and
quantities sold, that is independent of such demand curve.
Given the market demand, the market price is a function of the quantities produced
and sold by firms A and B:
p = a− b(qA + qB)
The market existence condition implies that the demand intercept on the vertical axis
is greater than the marginal cost (a > d). If this condition does not hold, the market
collapses.
3.2 Perfect Competition
Perfect competition is not an oligopoly equilibrium in its own right, since there is
no scope for strategic interactions in this market structure. However, its equilibrium is
relevant as a benchmark for the welfare impacts of the different conjectures considered
here, under oligopoly.
In perfect competition, the profit maximization condition is that price equals marginal
cost, so the equilibrium prices, total quantities and the profits of any firm Πi, can be
written as:
p = d
a− bQ = d
Q =a− db
Π = Πi = 0
Since price equals marginal cost, there is no mark-up (MU) in perfect competition.
Measuring such mark-up with the Lerner Index:
MU =p−MgC
p=d− dd
= 0
3
At the equilibrium price, the price elasticity of the market demand (ε) is:
ε = −∂Q∂p
p
Q
ε =1
b
da−db
ε =d
a− d> 0
Since the residual demand is defined as the demand faced by the firm, which may
differ from the market demand, and since the price is given to the firm under Perfect
Competition, the price elasticity of the residual demand (εi), tends to infinity:
εi →∞
Firms have no surplus and no profits, so social welfare (W ) is equal to consumer surplus
(CS):
W = CS =1
2(a− d)
a− db
W = CS =(a− d)2
2b
3.3 Collusion
Collusion (or cartel) is the only cooperative conjecture considered in this article. It
occurs when firms cooperate to maximize total profits. As such, they replicate monopoly.
The total profits of the cartel (Π) can be written as:
Π = [a− bQ]Q− dQ
The first order condition for profit maximization is:
∂Π
∂Q= a− b2Q− d = 0
4
Q =a− d
2b> 0
Given that the marginal cost is constant, there is no apriori mechanism to allocate
output between firms. However, in a symmetric equilibrium:
qA = qB =a− d
4b> 0
Substituting Q in the demand equation:
p =a+ d
2> 0
The mark-up under Collusion, measured with the Lerner Index, can be expressed as:
MU =
[a+d2− d]
a+d2
MU =
[2(a+ d− 2d)
2(a+ d)
]MU =
a− da+ d
> 0
The price elasticity of the market demand at equilibrium is:
ε =1
b
a+d2
a−d2b
ε =a+ d
a− d> 0
The price elasticity of the residual demand at the symmetric equilibrium is:
εi =1
2b
a+d2
a−d4b
εi =a+ d
a− d> 0
The profits of each firm are:
5
ΠA = ΠB =
[[a+ d
2
]− d]a− d
4b
ΠA = ΠB =(a− d)2
8b> 0
The total profits Π are:
Π = ΠA + ΠB =(a− d)2
4b> 0
The social welfare loss (WL) with the Collusion outcome, with respect to the welfare
under the perfectly competitive equilibrium, is:
WL =1
2
[(a+ d)
2− d] [
(a− d)
b− (a− d)
2b
]
WL =(a− d)2
8b
3.4 Threat
Threat is a conjecture where there is an established firm A that acts as a monopolist,
and another firm B considers entering the market. A decreases the price or expands
production to discourage the entry of B, and once B desists from entering the market,
A moves back to being a monopolist. Thus, the Threat equilibrium is the Monopoly
equilibrium, which is the Collusion equilibrium, but all the production is done by firm A,
and all the profits are for firm A. Thus, the Threat equilibrium can be characterized in
terms of the parameters of the model, as follows:
Q = qA =a− d
2b> 0
p =a+ d
2> 0
MU =a− da+ d
> 0
6
ε = εi =a+ d
a− d> 0
Π = ΠA =(a− d)2
4b> 0
WL =(a− d)2
8b
3.5 Cournot
Under the Cournot conjecture, firms chose the level of production that maximizes their
profits, given the production of the other firm. Output decisions affect market prices.
Given the Cournot conjecture, the profits of firm A can be written as:
ΠA = [a− b(qA + qB)]qA − dqA
The first order condition for profit maximization is:
∂ΠA
∂qA= a− bqB − 2bqA − d = 0
The reaction function (optimal strategy) for firm A is:
qA =a− d− bqB
2b
By symmetry it can be verified that the reaction function for firm B is:
qB =a− d− bqA
2b
The profit maximizing output of firm A, in terms of the parameters of the model, can
be found replacing the reaction function of B into the reaction function of A:
qA =a− d− b
[a−d−bqA
2b
]2b
7
2bqA = a− d−[a− d− bqA
2
]
4bqA = 2a− 2d− a+ d+ bqA
3bqA = a− d
qA =a− d
3b> 0
By symmetry, it can be verified that:
qB =a− d
3b> 0
Total output in the market is:
Q =2(a− d)
3b> 0
The market price under Cournot is:
p = a− b2(a− d)
3b
p =a+ 2d
3> 0
The mark-up estimated as before, is:
MU =
[[a+2d3
]− d
a+2d3
]
MU =a− da+ 2d
> 0
The price elasticity of the market demand at equilibrium, is calculated as:
8
ε =1
b
a+2d3
2(a−d)3b
ε =a+ 2d
2(a− d)> 0
The price elasticity of residual demand for each firm at equilibrium, is calculated as:
εi =1
2b
a+2d3
(a−d)3b
εi =a+ 2d
2(a− d)> 0
The profits of each firm and the total profits, are:
ΠA = ΠB =
[(a+ 2d)
3− d]
(a− d)
3b
Πi =
[(a− d)
3
](a− d)
3b
Πi =(a− d)2
9b> 0
Π =2(a− d)2
9b> 0
The welfare loss with the Cournot outcome, with respect to the welfare under the
perfectly competitive equilibrium is:
WL =1
2
[(a+ 2d)
3− d] [
(a− d)
b− 2(a− d)
3b
]
WL =(a− d)2
18b
3.6 Stackelberg
Under the Stackelberg conjecture, firms do not cooperate but rather, compete, with
one firm acting as a leader in the output produced and sold in the market, and the other
firm acting as a follower. Assume firm B is the leader and firm A is the follower, and
9
thus, firm A takes the output of firm B as given. Production decisions of A and B affect
the output market prices.
The reaction function for firm A is the same as in Cournot:
qA =a− d− bqB
2b
Firm B knows it is the market leader, and it is aware that the output decisions of A,
depend on its own output decision. In fact, B knows the reaction function of A. Thus,
profit maximization for firm B can be expressed as:
ΠB = [a− b(qA + qB)]qB − dqB
ΠB =
[a− b
(a− d− bqB
2b+ qB
)]qB − dqB
ΠB =
[aqB + dqB − bq2B
2
]− dqB
The first order condition for profit maximization is:
∂ΠB
∂qB=
[a+ d− 2bqB
2
]− d = 0
qB =
[(a− d)
2b
]Replacing the optimal output of the leading firm B, in the reaction function of firm A:
qA =
[(a− d)
4b
]Total output in the market under Stackelberg is:
Q =
[3(a− d)
4b
]Replacing in the demand curve, the price is:
p =
[(a+ 3d)
4
]
10
And the mark-up is:
MU =
[a+3d4− d]
a+3d4
MU =a− da+ 3d
> 0
The price elasticity of the market demand is:
ε =1
b
a+3d4
3(a−d)4b
ε =a+ 3d
3(a− d)> 0
The price elasticity of the residual demand of the leading firm, that has two thirds of
the market output, is:
εi =2
3b
a+3d4
(a−d)2b
εi =a+ 3d
3(a− d)> 0
The price elasticity of the residual demand of the following firm, that has one third of
the market output, is:
εi =1
3b
a+3d4
(a−d)4b
εi =a+ 3d
3(a− d)> 0
The profits for A, B and the total profits, are:
ΠA =
[[a+ 3d
4
]− d]
(a− d)
4b
ΠA =(a− d)
4
(a− d)
4b
ΠA =(a− d)2
16b
ΠB =(a− d)
4
(a− d)
2b
11
ΠB =(a− d)2
8b
Π =(a− d)2
16b+
(a− d)2
8b
Π =3(a− d)2
16b> 0
The welfare loss with the Stackelberg outcome, with respect to the welfare under the
perfectly competitive equilibrium is:
WL =1
2
[(a+ 3d)
4− d] [
(a− d)
b− 3(a− d)
4b
]
WL =(a− d)2
32b
3.7 Bertrand
In Bertrand, firms compete in prices, meaning that they set their prices to maximize
their profits, taking the price of their competitor as given. Starting with any price, for
example the Cournot price, which as shown before has a mark-up over the marginal cost,
one firm will have the incentive to decrease the price and capture the whole market. But
the other firm will do the same, and capture all the market. At equilibrium, one or both
firms will set the price equal to the marginal cost of production, and sell all the quantities
demanded in the market.
Thus, under the setting of this article, the Bertrand equilibrium replicates the perfect
competition equilibrium, since the equilibrium price equals the marginal cost of produc-
tion.
Profit maximization in Bertrand can be expressed as:
p = d
a− bQ = d
Q =a− db
12
Since the marginal cost is assumed constant in this model, there is no a priori mech-
anism to allocate production between firms. However, in a symmetric equilibrium:
qA = qB =a− d
2b
ΠA = ΠB = 0
MU = 0
At the equilibrium price, the price elasticity of the market demand is:
ε =d
a− d> 0
The price elasticity of the residual demand is:
εi =→∞
In this case, εi tends to infinity because the only price possible for a Bertrand producer
is the marginal cost. Else, a competitor would charge its marginal cost and the firm would
have no sales. The residual demand of any firm in this model under Bertrand is horizontal.
And as in perfect competition, firms will have no market power and no profits, and there
will be no welfare loss with respect to the competitive equilibrium.
The consumers’ surplus is:
CS = (a− d)a− d
2b
CS =(a− d)2
2b
And the welfare loss with respect to perfect competition is:
WL = 0
13
3.8 Summary
Graphically the results obtained in the model can be summarized as shown in figure
1:
Figure 1: Summary of the Results of the Model of Oligopoly
Source: author
A more detailed summary of the results is presented in table 1:
14
Table 1: Summary of the Results of the Oligopoly Model1
Market Collusion Threat Cournot Stackelberg Bertrand
qAa−d4b
a−d2b
a−d3b
a−d4b
a−d2b
qBa−d4b
0 a−d3b
(a−d)2b
a−d2b
Q a−d2b
a−d2b
2(a−d)3b
3(a−d)4b
a−db
p a+d2
a+d2
a+2d3
a+3d4
d
ΠA(a−d)2
8b(a−d)2
4b(a−d)2
9b(a−d)2
16b0
ΠB(a−d)2
8b0 (a−d)2
9b(a−d)2
8b0
Π (a−d)2
4b(a−d)2
4b2(a−d)2
9b3(a−d)2
16b0
MU a−da+d
a−da+d
a−da+2d
a−da+3d
0
ε a+da−d
a+da−d
a+2d2(a−d)
a+3d3(a−d)
da−d
εia+da−d
a+da−d
a+2d2(a−d)
a+3d3(a−d)
→∞
WL (a−d)2
8b(a−d)2
8b(a−d)2
18b(a−d)2
32b0
Source: author’s calculations
Within the simple model structure presented in this article, the algebraic results allow
us to conclude unambiguously, that, under the assumptions used:
Collusion and Threat replicate Monopoly, have the highest price; the lowest quantities
produced; the highest mark-up; the highest price elasticity of market demand; the highest
profits and the lowest welfare. When compared to Collusion, Cournot has a lower price;
higher total quantities produced; lower price elasticity of market demand; lower price
elasticity of residual demand; lower mark-up; lower profits and higher welfare. When
compared to Cournot, Stackelberg has a lower price; higher total equilibrium quantities;
lower price elasticity of market demand; lower price elasticity of residual demand; lower
mark-up; lower profits; and higher welfare. Bertrand has the lowest price; the highest
total equilibrium quantities; the lowest price elasticity of market demand, the highest
price elasticity of residual demand; the lowest mark-up; the lowest profits; and the highest
1The equilibria of Collusion and Bertrand are presented assuming the symmetric equilibrium.Given that there are no fixed costs, the equilibrium of Bertrand is identical to the equilibrium of PerfectCompetition, although in Perfect Competition there would be many firms.Note that given the assumptions of the model and in particular since a > 0, b > 0, d ≥ 0 and a > d, allthe results obtained are well defined.
15
welfare, when compared to any of the other conjectural variations considered in this
model, and replicates in this framework, the equilibrium that would prevail under perfect
competition.
Thus, in general, the conjectures with the lower prices have the higher equilibrium
quantities and vice-versa. These results are in line -and extend- those derived by Levin
(1988). The conjectures with the lower prices also have the higher welfare; the lower price
elasticity of market demand; the lower price elasticity of residual demand (except for the
price elasticity of residual demand in Bertrand); the lower mark-ups, the lower profits,
and vice-versa.
Note the counter-intuitive results regarding equilibria with lower mark-ups and lower
price elasticities of demand. However, both of these variables are endogenous in this
model and as such, this counter-intuitive results will be discussed in the next section of
this article.
4 Comparative Statics
The results presented in the previous sections, allow to perform comparative statics
that yield the following results:
4.1 All conjectures, except for Bertrand
In all conjectures except Bertrand, an increase in demand parameter a (an increase in
demand), leads to increases in equilibrium prices; output; profits; and mark-up. Increases
in a also lead to decreases in the price elasticity of market demand; the price elasticity of
residual demand; and welfare as compared to the welfare that would prevail under perfect
competition. And viceversa.
In these same conjectures, decreases in demand parameter b (an increase in demand),
lead to increases in equilibrium output and profits. The unit price; the mark-up; the
price elasticity of market demand; and the price elasticity of residual demand, remain
unchanged. Welfare would fall, as compared to the welfare that would prevail under
perfect competition. And viceversa.
16
In this scenario, decreases in the marginal cost d, at equilibrium, would lead to increases
in output, profits, and mark-up. Decreases in d would also lead to decreases in prices; the
price elasticity of market demand; the price elasticity of residual demand; and welfare,
when compared to the welfare under prefect competition. And viceversa.
As noted before, the results show -with the exception of the price elasticity of resid-
ual demand in Bertrand- a counter-intuitive positive correlation between mark-ups, and
the price elasticities of market demand and residual demand. However, in terms of the
parameters of the model and in particular, in terms of changes in the parameters a and
d, there is a negative correlation between mark-ups and price elasticities of demand, as is
standard in the economic literature.
4.2 Bertrand
In Bertrand, increases in demand (increases in a and/or decreases in b) or decreases
in the marginal cost d, lead to increases in output, with no changes in profits; mark-up;
the price elasticity of residual demand; or welfare -when compared to the welfare that
would prevail under perfect competition-. Decreases in the marginal cost would decrease
the unit price and the price elasticity of market demand. And viceversa.
Conclusions
This article has shown that given the model’s assumptions, when firms interact as in
Bertrand, they charge lower prices and have higher equilibrium quantities than when they
interact as in Stackelberg. When firms interact as in Stackelberg, they charge lower prices
and have higher equilibrium quantities than when they interact as in Cournot. And when
firms interact as in Cournot, they have lower prices and have higher equilibrium quantities
than when they collude.
In general, the conjectures with the lower prices have the the higher equilibrium quan-
tities; the higher welfare; the lower price elasticity of market demand; the lower residual
demand; the lower mark-ups, and the lower profits. viceversa.
In all conjectures except in Bertrand, increases in demand (as increases in a) and
17
decreases in the marginal cost, increase quantities; prices and mark-ups; and reduce welfare
when compared to perfect competition. Increases in demand (as decreases in b) also
increase quantities, mark-up and reduce welfare when compared to perfect competition.
In Bertrand, increases in demand (as increases in a) and decreases in marginal costs,
increase equilibrium quantities and reduce the price elasticity of market demand. Increases
in demand (as decreases in b) increase output. Decreases in marginal cost, decrease prices
and decrease the price elasticity of market demand. In Bertrand, none of the changes in
the parameters considered here alter the price elasticity of residual demand, the mark-up,
the profits, and welfare -when compared to the welfare under perfect competition-.
Although the results suggest in general, a positive -and counter-intuitive- correlation
between the mark up and the price elasticity of demand, these variables are endogenous
in this model. When the relevant parameters of the model change, there is a negative
correlation between these variables.
18
References
Bertrand, J., (1883), “Theorie Mathematique de la Richesse Sociale”,Journal des
Savants48: 499–508
Chamberlin, E. H. (1957) “On the Origin of ‘Oligopoly’”, The Economic Journal ,
Jun., 1957, Vol. 67, No. 266 (Jun., 1957), pp. 211-218, Oxford University Press
on behalf of the Royal Economic Society
Chamberlin, E. H. (1933), The Theory of Monopolistic Competition, Harvard Uni-
versity Press. d’Aspremont, C., Gabszewicz, J. J.and Thisse, J. F., (1979), “On
Hotelling’s ‘stability in competition’,Econometrica47: 1145–1150
Cournot, A. (1838),Recherches sur les Principes Mathematiques de la Theorie des
Richesses, Hachette, Paris
Edgeworth, F. Y. (1897) “La Teoria Pura del Monopolio” , Giornale degli Economisti
15: 13–31
Levin, D. (1988) “Stackelberg, Cournot And Collusive Monopoly: Performance An”,
Economic Inquiry; Apr; 26, 2; pg. 317
More, T. (1516/1967) “Utopia”, translated by Dolan, J. P. in Greene, J. J. and Dolan,
J. P. editors, The Essential Thomas More, New York: New American Library.
Shapiro, C. (1989) “Theories of Oligopoly Behavior” in Handbook of Industrial Or-
ganization, Elsevier, ch. 6, vol. 1, pp 329-414.
Vallejo, H. (2021) “Monopsony and minimum wages”, Documento CEDE No. 12.
19
