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Prices versus Quantities in a
Mixed Oligopoly
BY:
A mixed oligopoly is an oligopoly where firms face
different objective functions. Current literature
on mixed oligopolies, originating from Singh
and Vives (1984), suggests that quantities as
strategic variable dominates prices as strategic
variable when goods are substitutes (and vice
versa for complements). However, the dynamics
of pure quantity-setting competition (Cournot
competition) can become unstable as the number
of firms passes a certain threshold. This research
investigates a mixed oligopoly with one part of the
firms using prices as strategic variables, and the
other part using quantities.
The current literature on oligopolies revolves mainly
around two names: Cournot (1838) and Bertrand
(1883). These two forms of competition are both
commonly acknowledged, but lead to substantially
different predictions. Cournot competition assumes
firms use quantities as strategic variable, whereas
Bertrand competition assumes firms use prices as
strategic variable. Even when considering the exact
same market structure, these types of competition
lead to different outcomes. An important question
therefore is which type of competition gives a better
description of market outcomes.
Singh and Vives (1984) investigate a differentiated
duopoly where firms can choose whether to use price or
quantity as strategic variable. They find that competing
in quantities (prices) is a dominant strategy when
goods are substitutes (complements). These findings
are backed by other research as well, for example
Häckner (2000). Other research finds that dominance
is strongly dependent on the degree of substitutability
between goods (Sakai et al. (1995)), as well as qualitative
differences (Matsumoto and Szidarovsky (2010)). In all
this research a two-stage model is considered, where
in the first stage firms choose their strategy, and in the
second stage they determine the value of their strategic
variable. At first we assume that in the second stage
the Nash equilibrium of the corresponding subgame is
played, but later on we relax this assumption.
ROB GOEDHART
An important finding by Theocharis (1960) is that
Cournot-Nash equilibrium becomes unstable under
best-response dynamics as the number of firms gets
larger. This instability will typically decrease profits
and may make the quantity-setting strategy less
profitable than the price-setting strategy in the same
circumstances. It is unclear whether this instability
is indeed large enough to prevent the price-setting
strategy from being completely driven out by the
quantity-setting strategy. To study this we consider a
mixed oligopoly with an arbitrary number of firms. At
first we consider two model setups, which differ in the
information given to the firm about the other firms.
The first model assumes that firms enter the same
market each period, so that each firm is aware of the
number of price-setters and quantity-setters in their
market in every period. However, since we are looking
to introduce an evolutionary model that allows firms
to switch between strategies over time, we consider
a second model. In the second model we assume that
each period new markets are drawn from a very large
population. A certain fraction of this population uses
prices as strategic variable, and the other part uses
quantities. This structure is similar to Hommes et al.
(2013). The first model is used as a comparison for
the second model. Since the results of the models
are similar, this first model will not be discussed much
further.
32
Lemma 1. The model in terms of prices of the pricesetters, and quantities of the quantity-setters can be
written as
many markets of n firms are randomly drawn from this
population. Each period new random draws will be
made to match firms in markets of size n, and thus the
firms are not aware of the exact composition of their
market when it comes to number of quantity- or pricesetters. However, they are aware of the fraction of
the population of which the draws are made, and thus
they do know the probabilities of each possible sample
of n firms.
We assume that firms make a decision based on
naive best-responses, which means that each firm bestresponds to the quantities and prices set by the other
firms. Since the decision problem for each quantitysetter (price-setter) is the same by symmetry, we
assume that every quantity-setter (price-setter) sets
the same quantity (price). Such a system is called a
quasi-symmetric system. However, since in this model
the firms are not aware of the exact distribution of
their market, they use the binomial probabilities to
calculate the expected profits. This is because the
profits that come with a specific choice of quantity
or price may vary between different markets. We
therefore give an indication of the average equilibrium
profits by calculating the expected equilibrium profits.
The equilibrium price (quantity) of the price-setters
(quantity-setters) and their expected profits can be
written as a function of n, and .
Lemma 2. The Nash equilibrium prices and quantities
of price-setters and quantity-setters respectively, as
function of n, and equal
where
with
Here
is a k×n-k matrix with each element equal to
, and a k×k matrix with all diagonal elements equal
to one, and all the off-diagonal elements equal to .
Firms Know the Population Parameter ρ
This model uses a structure similar to Hommes et al.
(2013).We assume that there is a very large population
of firms where a fraction of the firms uses prices as
strategic variables, and a fraction 1- uses quantities.
We consider a two-shot game where in the first period
33
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Analogously to earlier research (e.g. Singh and Vives
(1984)) we consider the market game as a two-stage
game, where in the first stage each firm chooses its
strategy (price-setting or quantity-setting) and in the
second stage they choose the actual value of their
strategic variable. We assume that in the second stage
the Nash-equilibrium is played. In order to determine
this equilibrium we start with defining our market
demand structure.
We use a demand structure based on the one used
by Häckner (2000) and Matsumoto and Szidarovszky
(2010). We consider an oligopoly that consists of
n firms. In our model
represents the degree of
substitutability between the goods. We consider the
case where goods are imperfect substitutes, 0< <1.
Suppose that there are k price-setters in the market,
we then denote the vectors of prices and quantities
of price-setters by
and
respectively,
and the vectors of prices and quantities of quantitysetters by
and
respectively.
Without loss of generality, assume from now on that
the first k firms are price-setters and the last n-k firms
are quantity-setters. This means that the first k firms
will determine their prices ( ) while the last n-k firms
will determine their quantities (
). We have derived
that the model in terms of the strategic variables can
be written as
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and the resulting expected equilibrium profits equal
Assuming a sufficiently large population means that
one firm switching from one strategy to another
does not significantly influence the population
parameter . Therefore we compare
(
) and
(
). Numerical simulations show that the
equilibrium profits of the quantity-setters are
higher than these of the price-setters. Thus our
simulations suggest that for this model we have
Conjecture 1. The quantity-setting strategy yields
higher profits than the price-setting strategy
for all n, and . This represents the conjecture that in
the two-stage game as described in this section, it is always
more profitable to choose the quantity-setting strategy in
the first stage.
This result has important implications for the twostage game that we consider. It implies that in the
first stage each firm will choose the quantity-setting
strategy. In that case, in the second stage the Cournot
equilibrium will be the outcome. Note that this result is
not dependent on the actual degree of substitutability
between the goods.
The conjecture is in agreement with earlier research
(e.g. Singh and Vives (1984), Tanaka (2001)) that suggest
that the quantity-setting strategy is more profitable than
the price-setting strategy. Note that this result holds
only under the rational assumption that in the second
stage always the Nash equilibrium is played. However,
the assumption of perfectly rational agents has been
put into question by recent literature (originating from
Simon (1957)). In earlier research on oligopolies we
have also seen that Cournot competition (competition
where all firms are quantity-setters) under naive
expectations can become unstable as the number
of firms n in the market becomes larger ((Theocaris
(1960)). This makes it interesting to investigate the
stability of the system.
Stability: Firms Know the Population Parameter ρ
The first step to investigate the stability of the system
is to write the system as a quasi-symmetric dynamic
system, which is necessary to reduce it to a 2D system.
Remember that quasi-symmetric means that each
quantity-setter sets the same quantity, and each pricesetter sets the same price. In each period the firms best
respond (based on naive expectations) to the quantities
and prices in the previous period. This leads us to the
following dynamics of the price set by price-setters (1)
and the quantity set by the quantity-setters (2)
(1)
34
(7)
(2)
Since the system is linear, we can use the Jacobian of the system to determine the stability. A straightforward
computation shows that this Jacobian equals
(3)
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where
Since eigenvalues of this Jacobian are complicated functions of n, and , we will discuss stability of the
extreme cases of Cournot competition ( = 0) and Bertrand competition ( = 1). In the case of Cournot
competition there are no price-setters, so that the first equation is not relevant. We then only have to consider the bottom right element of (3) (the derivative of
with respect to ), which for = 0 simplifies to
(4)
35
Indications of the stability of the system for =(blue)
0.25,(green) 0.5 and (red) 0.75, with n on the horizontal
axis and on the vertical axis.The area above the graph
is the stable area, whereas the area below the graph is
the unstable area.
so that we find that Cournot competition is unstable
for
(5)
Evolutionary Population Parameter
In the case of perfect substitutes ( =1) this means that
the system is unstable for n > 3. This corresponds to
the special case investigated by Theocharis (1960). As
can be seen from (5), for every value of with 0<
<1 there exists a value n* such that the best-response
dynamics in Cournot competition are unstable if n>n*.
Similarly for Bertrand competition we consider only
the top left element of (3) (the derivative of
with
respect to ), which is for = 1 simplifies to
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(6)
which is always smaller than 1/2 for 0< <1.This means
that Bertrand competition is always stable for these
values of , independent of n.
The eigenvalues of the Jacobian (3) under mixed
competition are complicated functions of n, and . In order
to investigate stability we therefore use a numerical
approach. Figure 1 illustrates for each n the highest
value of for which the dynamics are unstable. When
dynamics are always stable for a particular value of
n, the critical value of is shown as zero. It can be
seen that as n grows larger, the value of for which
the dynamics become stable tends towards a value
slightly below 0.25, with only a small difference
based on . Differences between the three different
degrees of substitutability are more noteworthy in the
movement for low values of n.The higher the degree of
substitutability, the higher needs to be to have a stable
system for low values of n.This result is intuitively clear,
since a higher degree of substitutability means that
firms are more dependent on each other, and thus their
best-responses will be more narrowly connected. In the
extreme case of =0, changes in other firms’ strategies
would have no impact on the decision of the firm.
Previously we have assumed that the population
parameter remained unchanged over time. However,
as performance of the two different strategies is known
to the firms, this assumption does not seem very realistic.
The next step is thus to let this parameter depend on
the past performance of the strategies, so that better
performing strategies are more likely to be used. To
implement an evolutionary population parameter we
consider two dynamics. The first dynamic is based on
Anufriev and Hommes (2012), and the second dynamic
builds on replicator dynamics (Droste et al. (2002)).
We start by defining a performance measure for time t,
denoted by . This performance measure represents
the performance of the price-setting strategy compared
to the quantity-setting strategy, and it is based on
past profits of both strategies. We use a performance
measure of the following form
(7)
where represents the memory of the performance
rule. Next, we calculate the population parameter on
time t, , which for the first model is done according
to the following formula
(8)
where represents the idea that not all firms directly
update their strategy. The parameter represents the
intensity of choice, as a higher value of this parameter
causes firms to switch to the best performing strategy
more rapidly.
However, under the evolution described in (8)
will always converge to
= 0.5 when each of the
strategies is equally profitable. Since we would like to
capture the idea that
will remain constant in that
case (people will not change their strategy if both
strategies are equally profitable), instead of converging
to 0.5, we adapted the evolution of
to follow the
exponential replicator dynamics, similar to Droste et
al. (2002). Using the same performance measure
as
before, we now determine by
(9)
This function has the property that as long as the
price-setting strategy is more (less) profitable than
the quantity-setting strategy, the value of will slowly
move up (down). Note however that the extreme cases
of =0 and =1 are steady states. This can lead to
hghghbhhg
Figure 1. Stability of the System with Population Parameter
36
problems in the initial state of the process if is high. In order to deal with this we consider two possible solutions.
The first solution is to lower the value of , and increase the amount of time periods. This can be done without
loss of generality, since the time is not an absolute measure here. The general behavior of the dynamics thus
remains representative. The second solution introduces a small noise, such that steady
states are not available. Evolution according to the dynamics with noise is described by
(10)
Findings using the exponential replicator dynamics for
the population parameter are in agreement with the
findings using (8) for both of the mentioned solutions.
Starting in a stable state, with
sufficiently large, the
quantity-setting strategy will be most profitable and
thus lead to a decrease of . However, as the value
of
becomes too low the instability of the system
causes the price-setting strategy to be most profitable.
As a result the value of drastically increases again to
re-stabilize the system, and the cycle repeats. Although
we do notice that the price-setting strategy will not be
completely driven out, instability alone is not sufficient
to make this strategy most profitable. Good examples
of these dynamics are shown in Figure 2. We added
the critical value for instability (given the parameter
values) in the figure and observe that it is higher than
the turning point for the values considered. This is the
case for other combinations of parameter values as
well. As it seems, even when the system is unstable it is
still possible that the quantity-setting strategy is most
profitable. However we always observe a turning point
where this is no longer the case, and the system moves
back to a stable state to restart the cycle.
In the model with noise, as described in (10), a low
value of will lead to a fraction of price-setters slightly
below 0.5. This is similar to the findings when using
the model described in (8), and it happens because
the noise
is in that case too large compared to the
change caused by the difference in profits, so that the
fraction of price-setter will remain somewhere slightly
below 0.5. This can again be limited by reducing the
value of the shock further (e.g. to a uniform(0, 0.01)
distributed variable), but the general idea remains the
same. An example is shown in Figure 3.
Figure 2b. Exponential Replicator Dynamics
Typical dynamics using the exponential replicator
function (a) (9) or (b) (10) for. The vertical axis shows
, and the horizontal axes shows the period t. The
red line indicates the critical value of for the given
parameter values, as calculated in section 4. Parameter
values are n = 10, =0.75, =0.5, =0.4 and =(a) 50
or (b) 43000. Initial values are
=0.4,
= =0.01,
and
= =0.
Figure 3 - Exponential Replicator Dynamics with noise
Figure 2a. Exponential Replicator Dynamics
The evolution of the fraction of price-setters under
dynamics described in (10), when the value of is ‘too
low’.The vertical axis shows , and the horizontal axes
shows the period t. The red line indicates critical value
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to be uniformly distributed between 0 and 0.05 to represent a small but sufficient shock.
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where we choose
of for the given parameter values, as calculated in section 4.
Parameter values are n=10, =0.75, = =0.5 and =4000.
Initial values are =0.4, = =0.01, and
= =0.
Sakai, Y., S. Aguchi, H. Ishigaki (1995). “Price and
Quantity Competition: Do Mixed Oligopolies
Constitute an Equilibrium?” Keio economic
studies 32, 15-25.
Conclusion
Simon, H. A. (1957). “A Behavior Model of
Rational Choice,” Models of Man: Social and Rational, 196-279.
In this research we establish that instability caused by
competition in quantities is sufficiently large to allow the pricesetting strategy to remain in a mixed oligopoly. We observe a
cyclical switching between both strategies in most cases, but
the quantity-setting strategy is never able to drive out the
price-setting strategy. Thus, despite the fact that the quantitysetting strategy is always most profitable inside equilibrium,
the price-setting strategy can still be a more profitable option
when perfect rational play is no longer assumed.
Literature
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MSC-LEVEL | ECONOMETRICS
| SPECIALTY | NL
Anufriev, M. and C. Hommes (2012). Evolution of Market
Heuristics. Knowledge Engineering Review, 27, 255 - 271.
Singh, N. and X.Vives (1984). “Price and Quantity
Competition in a Differentiated Duopoly,”
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Tanaka, Y. (2001). “Profitability of Price and
Quantity Strategies in an Oligopoly,” Journal of
Mathematical Economics 35, 409-418.
Theocharis, R. D. (1960). “On the Stability of the
Cournot Solution on the Oligopoly Problem,”
Review of Economic Studies, 27, 133-134.
Anufriev, M., D. Kopányi, and J.Tuinstra (2013).“Learning Cycles
in Bertrand Competition with Differentiated Commodities
and Competing Learning Rules,” Journal of Economic
Dynamics and Control 37, 2562-2581.
Bertrand, J (1883). “Revue de la Théorie Mathématique
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Mathématiques de la Théorie des Richesses.” Journal des
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Cournot, A. (1838). “Researches into the Mathematical
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Entry Barriers,” Bell Journal of Economics 10, 20-32.
Droste, E., C.H. Hommes, and J. Tuinstra (2002). “Endogenous
Fluctuations under Evolutionary Pressure in Cournot
Competition,” Games and Economic Behaviour 40, 232-269.
Hackner, J. (2000).“A Note on Price and Quantity Competition
in Differentiated Oligopolies,” Journal of Economic Theory 93,
233-239.
Hommes, C., J.Tuinstra and M. Ochea (2013). “On the Stability
of the Cournot Equilibrium under Competing Learning
Rules,” Working Paper.
Matsumoto, A. and F. Szidarovszky. “Mixed Cournot-Bertrand
Competition in N-Firm Differentiated Oligopolies.” Working
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38
Rob Goedhart
Rob will start his PhD at IBIS UvA.
He obtained his Masters degree
Mathematical Economics at the
University of Amsterdam. This article
summarizes part of his master thesis,
which he wrote under supervision of
Prof. dr. J. Tuinstra.