Evolutionary Dynamics in a Cournot duopoly
with Strategic Complements and Heterogeneous
Heuristics
Nikki van Ommeren
June 27th 2014
Abstract
In this paper the stability threshold of the equilibrium in a Cournot Duopoly under evolutionary pressure is investigated with its main focus on dierentiation level of goods, which is measured by the parameter γ. Every successive period rms are able to choose between two behavior rules, with more advanced rules requiring extra information costs. For the Cournot versus Nash rule, the rms on the market produce the Nash equilibrium quantity for all values of γ. For the Cournot versus Rational rule and information costs associated with the Rational rule, the results are less stable, with a two-cycle when the products are complements (γ < 0), chaos for γ ∈ [0.4, 0.7] and small oscillations for γ > 0.7. In the absence of information costs the quantity and fraction dynamics are similar, but more stable.
1 Introduction
After the development of the rst formal model of a duopoly by Augustin Cournot, many papers have been published on this subject. Most of them assume quantities in the Cournot oligopoly to be strategic substitutes. For this assumption it is shown by Hommes, Ochea & Tuinstra (2011) that under evolu-tionary competition the Nash equilibrium for the duopoly is stable for dierent sets of behavior rules. In this paper the assumption of strategic substitutes is relaxed and it focuses on the case where quantities are strategic complements instead of substitutes.
The original model described by Cournot (1897) is a market with two players selling water in each period. The water is a homogenous good and in this model the players compete on quantities. Each player chooses, independently and simultaneously the amount of water he wants to produce. To maximize his pro ts he rst makes an assumption of the quantity oered by the other player. If after a number of turns all players will oer the same quantity in every successive period, the corresponding quantity is the stable Nash solution.
Whether the solution of the game is stable, depends on various factors, for example the assumptions they make about the behavior of other players and the number of players on the market. This last property is studied in the paper of Theocharis (1960) for the n-player oligopoly game. He assumes each seller shows the naive best response behavior, which means all sellers assume the quantities of the competitors to remain unchanged in the coming period. It is shown that only for the duopoly the Cournot solution is stable, for three players there are nite oscillations around the equilibrium and for more than three players the equilibrium is never stable.
The stability of the equilibrium for the evolutionary version of this model is investigated by Hommes et al. (2011), relaxing the constraint all players show the best-response behavior. In their model it is possible to switch between dif-ferent types of behavior rules showing stability for two players on the market. Their research restricts quantities of the oligopoly model to be strategic substi-tutes, in this paper the stability conditions for the evolutionary duopoly model with strategic complements is investigated. It will be of interest whether the same stability threshold can be found for dierentiated products.
An example of a duopoly with strategic complements can be found in Cournot (1897). It describes the production of brass, where the producer needs both cop-per and zinc to produce an unit of brass. Each of the items is useless unless used with the other item, so copper and zinc are perfect complements. Another example of a market with complementary goods is the production of pieces for electronic devices. To produce an iPhone components from dierent companies are brought together, without one of those components the others are useless, so the components can be seen as complementary goods.
The Cournot duopoly with complements is mentioned in the paper of Sing & Vives (1984), where the pro ts for Cournot and Bertrand competition with strategic complements and substitutes are compared. They show that in Cournot quantity competition with complements rms enhance lower pro ts, lower values of consumer surplus and hence lower general welfare.
The base for the model used in this paper is the one for the dierentiated duopoly proposed by Dixit (1979) in which the demand function allows goods to be complements or substitutes. In this model rms have constant marginal costs, no xed costs and no capacity limits. All rms on the market will try to maximize their pro ts. The behavior of the competing rms in the studied Cournot duopoly is modeled in terms of dierent behavior rules. This paper analyzes three dierent types of behavior rules, namely the naive best-response rule (as in Theocharis, 1960) and the more sophisticated, but possibly more costly Nash and Rational rules. Firms playing the Nash rule are aware of the quantity associated with the Nash equilibrium and will choose to produce this quantity. Firms using the Rational rule are aware of the quantity the other rm will produce in the upcoming period and are therefore more capable of choosing the quantity that will optimize their pro ts. There are more costs associated with the Nash and Rational rule, because rms need more information about the market and therefore it may be harder to implement those rules (as in Droste, Hommes & Tuinstra, 2002).
In the evolutionary model of Cournot competition speci ed in this paper, every discrete-time period rms can switch between two types of heuristics, taking the past performance of the rule into account (as in Droste et al, 2002). The past performance is measured by the realized pro ts minus the associated costs, which are available for all rms on the market. This results in only the best performing rules to be used and the less unsuccessful rules to be dropped. They show that whenever the dynamical system is close to the equilibrium, all rms will use the naive rule because there are no costs associated with it and therefore generates higher pro ts. The opposite happens when the dynamical system is far from the equilibrium.
In this paper the global dynamic behavior of the evolutionary model is stud-ied by means of numerical simulations under evolutionary pressure. For higher values of evolutionary pressure rms are more likely to drop the unsuccessful heuristics from the previous period (as in Hommes et al, 2011). In the case of strategic substitutes they conclude that for higher values of evolutionary pressure, with rms switching more often between the dierent heuristics, the equilibrium becomes less stable for more than three players.
In the Duopoly model described in this paper, the results are stable for all sets of behavior rules. In the rst part of the simulations rms are able to switch between the Cournot and Nash rule nding stability for all values of γ, which denotes the extent to which the products are complements or substitutes. In the following part rms are able to choose between the Cournot and Rational rule, with stable results for substitutes in the absence of information costs and a two cycle when the products are complements. Less stable results are found when there are information costs associated with the Rational rule, with chaos for γ ∈ [0.4, 0.7], small oscillations for γ > 0.7 and a two cycle for γ < 0.
Section 2 presents the Cournot model proposed by Dixit (1979) with a linear demand structure and makes some adjustments to investigate the model with dierentiated goods. It also introduces three dierent types of behavior rules: the Cournot, Nash an Rational rule. Section 3 enables rms to switch between the dierent types of behavior rules and introduces the evolutionary version of the model. Section 4 shows the resulting global dynamic behavior of the composed model, rst of the Cournot versus Nash rms followed by the Cournot versus Rational rms. Finally, the concluding remarks are described in the discussion in Section 5.
2 The Cournot Duopoly Model
This section will describe the most important features of the basic Cournot model, the adjustments made in this paper to be able to study dierentiated products and it will introduce three dierent types of behavior rules used by the rms in this model.
2.1 The linear model for dierentiated goods
The basic Duopoly Cournot model considers a sector in an economy where two rms produce homogenous products and compete on quantities. Both rms on the market have market power, which means the output level of both rms has in uence on the prices on the market. In this paper the assumption of homogenous products is relaxed and it will investigate the case of dierentiated products. Therefore the parameter γ is introduced, which indicates the extent to which the goods are complements or substitutes (as in Dixit, 1979). The quantities of the products are denoted by qi and the prices by pi with i =
1, 2 for rm one and two respectively. The consumers maximize their utility function: U(q1, q2)−P
2
i=1piqi. As in Singh & Vives (1984) the utility function
is assumed to be quadratic and strictly concave and is given by: U(q1, q2) =
α1q1+ α2q2−
(β1q21+2γq1q2+β2q22)
2 .
The inverse demand function can be derived as follows: dU (q1,q2)
dq1 = α1−
2β1q1−2γq2
2 = α1− β1q1− γq2. So the inverse demand functions for rm 1 and
2 respectively are given by:
p1= α1− β1q1− γq2
p2= α2− β2q2− γq1
Note that the prices and quantities should always be positive, so pi > 0
and qi > 0. For illustration purposes the symmetric case is considered, where
α = α1= α2 and β = β1= β2. Therefore the inverse demand functions can be
simpli ed to:
p1= α − βq1− γq2
p2= α − γq1− βq2
As in the model of Dixit (1979) constant marginal costs are assumed, so c = c1= c2, therefore the pro t function of rm i is given by:
π1= p1q1− cq1= (a − βq1− γq2)q1− cq1
π2= p2q2− cq2= (a − γq1− βq2)q2− cq2
Each rm is assumed to produce the quantity that will maximize its pro ts, which gives the rst order condition: ∂π1
∂q1 = α − 2βq1− γq2− c = 0. Implicitly
this de nes the reaction curves for the rms 1 and 2: R(q2,t) = q1= α − c 2β − γ 2βE(q2,t) R(q1,t) = q2= α − c 2β − γ 2βE(q1,t)
The reaction curves give the marginal returns of the pro t equations. The eect of q2on the marginal return of the pro ts of rm 1 is equal to ∂
2π 1
∂q1∂q2≡π12, where
when π12< 0(Bulow, Geanakoplos & Klemperer, 1985). When we consider the
case of perfect complements, where β = −γ, the derivative of the reaction curve is ∂R(q2) dq2 = − γ 2β = β 2β = 1
2 and because the symmetric case is considered, the
same derivative will be found for the other rm. In the case of complementary goods it is assumed that γ < 0 and β > 0, therefore value of π12is indeed always
positive for complements and it implies that an increase of the output of the other rm by one unit leads to an increase of the output of rm i by −γ
2β and
to an increase of 1
2 in the case of perfect complements.
It is assumed there exists an unique Cournot-Nash equilibrium, because the alpha and beta values are equal, the solution of this equilibrium is symmetric. The associated quantity q∗ can be derived from the reaction functions of both
rms, by lling in the reaction of the other rm as the expectation of quantity the other rm will produce.
q1= α − c 2β − γ 2β( α − c 2β − γ 2βq1)
Solving this equation for q1 the solution of the symmetric Nash equilibrium
is equal to:
q∗= α − c 2β + γ
which is identical to the equilibrium found by Singh & Vives (1984).
2.2 Behavior Rules
Whether a rm learns to play the Nash equilibrium q∗depends on the behavior
of the rm and to be more speci c, the expectations on the quantity the other rm will produce. This paper considers three dierent ways of making expec-tations about the other rm's production level in the next period, the naive Cournot rule and the more sophisticated Nash and Rational rules.
If rms use the Cournot rule, the so called Cournot rms, they will base their expectations on the opponent's output in the previous period. Each rm assumes that the quantities of the other rm will remain unchanged in the coming period. This leads to the following dierence equations:
qC1,t= R(q2,t−1)
qC2,t= R(q1,t−1)
which is similar to the best reply rule described by Theocharis (1960).
Another approach is to assume rms have perfect information about the quantity produced by the other rm in the next period. In this case each rm is able to derive the Nash equilibrium and the production level is equal to the associated Nash equilibrium quantity. This type of behavior stabilizes the dy-namical system and the produced quantities will merge to the Nash equilibrium. So for rms 1 and 2 respectively:
qN1 = q∗ qN2 = q∗
The third type of behavior is the Rational rule. They are assumed to have complete information about the quantity produced by the other rm on the market and optimize their output using this information. Therefore the output level of the Rational rms is the solution to the following reaction curves:
q1R= R(q2)
q2R= R(q1)
In this paper it is investigated what happens if rms are able to switch between two behavior rules, therefore the next section introduces the switching model.
3 Switching between Behavior Rules
In this section rms are allowed to switch between two of the three behavior rules discussed in the previous section. In the rst part rms are able to choose between the Cournot and the Nash rule and in the second part between the Cournot and Rational rule. Because the assumption of identical products is relaxed, there exist two dierent markets for the goods produced by rm 1 and 2 when β 6= γ. Therefore two parameters are introduced ρ1,tand ρ2,tindicating
the fraction of rm 1 and 2 respectively using one heuristic and the fraction (1 − ρ1,t)and (1 − ρ2,t)using the other heuristic in period t, with ρt∈ [0, 1].
3.1 Cournot vs. Nash rule
In the following model ρ1,t represents the fraction of rm 1 playing the Nash
rule and therefore the Nash equilibrium quantity and (1 − ρ1,t) the fraction of
rm 1 playing the naive best reply rule in period t, the same applies for ρ2,tand
rm 2. The inverse demand function can be represented as follows: p1= α − β(ρ1qN1 + (1 − ρ1)q1C) − γ(ρ2q2N + (1 − ρ2)q2C)
p2= α − β(ρ2qN2 + (1 − ρ2)q2C) − γ(ρ1q1N + (1 − ρ1)q1C)
With the pro t function for the Nash and Cournot rm:
πN1 = (p1− c)q1N
= (α − β(ρ1qN1 + (1 − ρ1)q1C) − γ(ρ2qN2 + (1 − ρ2)q2C) − c)q N 1
πC1 = (p1− c)q1C
= (α − β(ρ1qN1 + (1 − ρ1)q1C) − γ(ρ2qN2 + (1 − ρ2)q2C) − c)q C 1
The derivation of the production level of the Nash rms is straightforward because it is equal to the quantity of the symmetric Nash equilibrium, so for the Nash rm 1 and 2 the quantity produced in every period equals:
q1,tN = qN2,t= q∗= α − c 2β + γ
To derive the Cournot quantity for rm 1 it is assumed that the full fraction of rm 1 plays the Cournot quantity so ρ1,t= 0. Therefore the pro t function
of rm 1 can be simpli ed as follows: πC1,t= (α − βq
C
1,t− γ(ρ2,t−1qN2,t−1+ (1 − ρ2,t−1)qC2,t−1) − c)q C 1,t
Firms attempt to maximize their pro ts and therefore the quantity produced by Cournot rm 1 in period t can be found by setting the derivative of the pro t function equal to zero: δπ1,tC
δqC 1,t
= α−2βqC
1,t−γ(ρ2,t−1qN2,t−1+(1−ρ2,t−1)q2,t−1C )−c =
0. Solving this equation for qC
1,t, the production level of Cournot rm 1 can be
expressed as follows: q1,tC = α − γ(ρ2,t−1q N 2,t−1+ (1 − ρ2,t−1)q2,t−1C ) − c 2β = q∗− (γ 2β)(1 − ρ2,t−1)(q C 2,t−1− q∗)
Because the symmetric case is considered a similar expression can be found for the quantity produced by Cournot rm 2:
q2,tC = α − γ(ρ1,t−1q N 1,t−1+ (1 − ρ1,t−1)q1,t−1C ) − c 2β = q∗− (γ 2β)(1 − ρ1,t−1)(q C 1,t−1− q∗)
The quantity produced by each rm depends on the fraction of rms using the Nash rule in the previous period. Therefore the dynamical system should also include how these fractions vary over time. Firms will base their choice for a certain behavior rule on the past performance of the rules. It is assumed that these past performances are available to all rms on the market and are measured by the realized pro ts minus the associated costs. These costs are based on the dierence in complexity of the behavior rules, where it requires more information or more eort to implement the more sophisticated Nash rule
than the naive Cournot rule. Therefore with the former rule higher costs are involved, so κ ≡ κN − κC with κ ≥ 0 , where κN and κC denotes the costs
associated with the Nash and Cournot rule respectively. To express the fraction dynamics it is necessary to compute the pro t dierential. The fractions of Nash rms ρtevolves over time according to:
ρ1,t= G(πN1,t−1− π C
1,t−1− κ)
as in Hommes et al. (2011). The pro t dierential, πN,t−1− πC,t−1, can be
computed from the pro ts of playing the Nash and Cournot rule in the previous period. πN1,t−1− πC1,t−1 = (α − c) (q∗− q C 1,t−1) − (q∗− q C 1,t−1)(βq C 1,t−1+ γq C 2,t−1 +βq∗ρ1,t−2+ γq∗ρ2,t−2− βρ1,tq1,t−2C − γρ2,t−2qC2,t−2)
Rewriting the rst part of the formula to be more capable to see the dierence from the Nash equilibrium.(α−c) q∗− qC
1,t = (α−c) (2β−d+γ)(2β − d + γ)(q ∗− qC 1,t) = (α−c) 2β−d+γ(2β−d+γ)(q ∗−qC
1,t) = (2β−d+γ)q∗(q∗−qC1,t). Now the pro t dierential
can be expressed in terms of the Nash equilibrium:
π1,t−1N − π1,t−1C = (2β − d + γ)q∗(q∗− qC1,t−1) − (q∗− qC1,t−1)(βq1,t−1C + γqC2,t−1
+βq∗ρ1,t−1+ γq∗ρ2,t−1− βρ1,t−1q1,t−1C − γρ2,t−1q2,t−1C )
To be able to compare the results with the results found by Hommes et al. (2011), the same logit dynamics for the evolutionary pressure are used. Therefore the fraction dynamics of rm 1 and 2 can be expressed as follows:
ρ1,t= 1 1 + exp(−z(πN 1,t−1− πC1,t−1− κ)) ρ2,t= 1 1 + exp(−z(πN 2,t−1− πC2,t−1− κ))
The parameter z ≥ 0 is introduced to measure the value of the evolutionary pressure, also called the selection pressure. This denotes the extent to which rms are likely to switch to the more successful behavior rules of the previous period. In Section 4.1 the quantity and fraction dynamics of rm 1 and 2 are used to explore the global evolutionary dynamics for the Cournot vs. Nash rms by means of simulations in E&F Chaos.
3.2 Cournot vs. Rational Rule
In the next model the usage of the parameter ρ1,t is similar to that of the
rms on the market instead of the fraction of Nash rms. Again (1−ρ1,t)equals
the fraction of rms using the more naive Cournot rule. The expression for the inverse demand function is:
p1= α − β(ρ1q1R+ (1 − ρ1)q1C) − γ(ρ2q2R+ (1 − ρ2)q2C)
p2= α − β(ρ2q2R+ (1 − ρ2)q2C) − γ(ρ1q1R+ (1 − ρ1)q1C)
with qR
i the quantity produced by the Rational rms 1 and 2.
Straightfor-ward the pro ts of the Rational and Cournot rms on this market equal:
πR1 = (p1− c)qR1 = (α − β(ρ1q1R+ (1 − ρ1)q1C) − γ(ρ2qR2 + (1 − ρ2)qC2) − c)q R 1 π1C = (p1− c)qC1 = (α − β(ρ1q1R+ (1 − ρ1)q1C) − γ(ρ2qR2 + (1 − ρ2)qC2) − c)q C 1
The Rational rms are assumed to have all the information about the pro-duced quantities and the composition of the market in the upcoming period and are therefore capable of optimizing their pro t function using the quantities and fraction of period t. The Rational rm i does not make the naive assumption that ρ1= 1when it sets its production level and therefore the quantity produced
by the Rational rms depends on the fraction of rm i choosing this behavior rule as can be seen in the following derivation of the qR
1,t.
dπ1,tR
dqR 1,t
= α − β(2ρ1,tq1,tR + (1 − ρ1,t)qC1,t) − γ(ρ2,tq2,tR + (1 − ρ2,t)qC2,t− c = 0
Solving this equation for qR
1,t, the amount produced by Rational rm 1 can
be written as:
q1,tR = α − c − β(1 − ρ1,t)q
C
1,t− γ(ρ2,tqR2,t+ (1 − ρ2,t)q2,tC)
2βρ1,t
and likewise for the Rational rm 2:
q2,tR = α − c − β(1 − ρ2,t)q
C
2,t− γ(ρ1,tqR1,t+ (1 − ρ1,t)q1,tC)
2βρ2,t
The quantity produced by Rational rm 1 still depends on the quantity produced by Rational rm 2, to lose this dependency the second expression can be substituted in the rst one:
q1,tR = α − c − β(1 − ρ1,t)qC1,t− γ(ρ2,t( α−c−β(1−ρ2,t)q2,tC −γ(ρ1,tqR1,t+(1−ρ1,t)q1,tC ) 2βρ2,t )) 2βρ1,t +γ(1 − ρ2,t)q C 2,t 2βρ1,t
Solving this equation for qR
1,t, the quantity produced by Rational rm 1 can
be expressed as1: qR1,t = 1 (4β2− γ2)ρ 1,t (cγ − 2cβ + 2αβ − αγ − 2β2q1,tC + γ2q1,tC + 2β2ρ1,tq1,tC −γ2ρ 1,tq1,tC − βγq C 2,t+ βγρ2,tqC2,t)
and similar for the Rational rm 2:
qR2,t = 1 (4β2− γ2)ρ 2,t (cγ − 2cβ + 2αβ − αγ − 2β2q2,tC + γ2q2,tC + 2β2ρ2,tq2,tC −γ2ρ2,tq2,tC − βγq C 1,t+ βγρ1,tqC1,t)
As can be seen from this formula the Rational rm 1 uses the fraction and quantity dynamics of the current period from the other rms on the market and from their own market. This is dierent from the Rational rule used by Hommes et al. (2011) because there it is assumed Rational rms are not aware of the fraction dynamics of the current period. The quantity dynamics for the Cournot rms can be computed as in the previous section, again it is assumed that the quantity produced of the Cournot rm 1 is computed assuming the full fraction of rm 1 plays the Cournot rule, so ρ1,t = 0. When the Cournot
rm optimizes his quantity using the opponent's quantity of the last period, the quantity dynamics for qC
1are equal to:
qC1,t=α − γ(ρ2,t−1q
R
2,t−1+ (1 − ρ2,t−1)q2,t−1C ) − c
2β Extracting qR
2,t−1 of the formula by substituting the quantity produced by
Rational rm 2 into the expression:
qC1,t=α − γ(ρ2,t−1q R 2,t−1+ (1 − ρ2,t−1)q2,t−1C ) − c 2β q1,tC = α − γ(ρ2,t−1((4β2−γ12)ρ 2,t−1(cγ − 2cβ + 2αβ − αγ − 2β 2qC 2,t−1+ γ 2qC 2,t−1 2β
−2β2ρ
2,t−1qC2,t−1− γ2ρ2,t−1q2,t−1C − βγqC1,t−1+ βγρ1,t−1q1,t−1C ))
2β +(1 − ρ2,t−1)qC2,t−1)) − c
2β This can be simpli ed to:
q1,tC = 1 2γ2− 8β2(−2cγ + 4cβ − 4αβ + 2αγ − γ 2qC 1,t−1+ 2βγq C 2,t−1+ γ 2qC 1,t−1ρ1,t−1 −2βγqC 2,t−1ρ2,t−1)
In order to run the simulations it is necessary to compute the fraction dynamics and therefore the pro t dierential of the Rational and Cournot rms in the previous period: π1,t−1R − π1,t−1C = (α − c)(q R 1,t−1− q C 1,t−1) − (q R 1,t−1− q C 1,t−1)(βq C 1,t−1+ γq C 2,t−1 −βρ1,t−1qC1,t−1− γρ2,t−1q2,t−1C + βρ1,t−1q1,t−1R + γρ2,t−1qR2,t−1)
Now it is possible to express the fraction dynamics for rm 1: ρ1,t= G(π1,t−1R − π C 1,t−1− κ) = 1 1 + exp(−z(πR 1,t−1− πC1,t−1− κ))
with κ ≡ κR− κC the cost dierential of implementing the Rational and
Cournot rule respectively,κ ≥ 0 and z the evolutionary dynamics as described in the previous section. When the conditions for the xed points of the Cournot quantities are computed, it is found that the Nash equilibrium of Singh & Vives (1984) is a xed point of this 4-dimensional dynamical system for ρ1= 0. It is
not possible to get a closed form-solution for the fractions ρ1and ρ2,therefore
the stability conditions of this system could not be computed. The resulting global dynamics of this evolutionary system can be found in subsection 4.2.
4 Results
In this section the global dynamics of the evolutionary system of the Cournot vs. Nash rms and the Cournot vs. Rational rms are shown by means of simulations in E&F Chaos. To perform these simulations the quantity and fraction dynamics from subsections 3.1 and 3.2 are used, with logit dynamics for the evolutionary pressure. In order to compare the results with the results of the paper of Droste et al. (2002) and Hommes et al. (2011) the same values are chosen for the parameters of the model with a = 17, β = 1 and c = 10. In the rst of these two papers it is argued that for dierent values of these parameters the same results are obtained. For the γ the values dier between −β and β, which in this case equals -1 and 1, with γ = −1 for perfect complements and γ = 1for perfect substitutes.
4.1 Cournot vs. Nash rms
This section considers the global dynamic behavior for the evolutionary model where rms can switch between the Cournot and Nash rule. In the absence of information costs, so whenκN = κC and therefore κ = 0, the Cournot rms will
never outperform the Nash rms. This is because when qC
1 > q∗it will be most
pro table to produce less than the Nash equilibrium quantity and therefore the q∗ is always closer to the optimal quantity and the Cournot rms can never achieve higher pro ts than the Nash rms. This result is independent of the value of γ and it is line with the results of Hommes et al. (2011). Note however that, when there are information costs, Cournot rms are always performing better at the Nash equilibrium. This is because qC
1 = q N 1 = q∗ and therefore in this period πC 1 = π N
1 , but with higher implementation costs for the Nash rms,
so the Cournot rms outperform the Nash rms. In Section 2 it is shown that q∗= α−c
2β+γ and therefore the amount of goods
produced at equilibrium will always be higher for complementary goods. Be-cause it is assumed that β ≥ 0 and γ < 0 for complementary goods, the value of the denominator will be smaller when the goods are complements and therefore the q∗will be higher.
The global dynamics of this system are illustrated in Figure 1 by means of simulations for the model with the values of the parameters α, β and c as mentioned before. In Figure 1a there are no information costs (κ = 0) and the result are independent of the value of the evolutionary pressure with z > 0. It shows the quantity dynamics for the Cournot rms for dierent values of γ, the extent to which the rms are producing complements or substitutes. The produced quantity of the Cournot rms is equal to the Nash equilibrium quantity and therefore this system is stable. It con rms that for negative values of γ, when the goods are complements, the production level is higher than for positive values of γ. The associated fraction dynamics are stable with half of the rm choosing the Nash and the other half choosing the Cournot rule. Because the symmetric case is considered, the same result is found for the fraction of rm 2.
In Figure 1b it is shown what happens to the fraction dynamics when there are information costs (κ = 0.5), this result is independent of the value of γ and corresponds to the results found by Hommes et al. (2011). It can be seen that for increasing values of z the fraction of rm 1 choosing the Nash rule decreases. The Figures 1c and 1d show time series of the quantity produced by the Cournot rms and of the fraction of Nash rms for dierent starting values and κ = 0, z = 2.8 and γ = −1. It illustrates that the quantities and fractions always converge to the equilibrium quantity, independently of the initial values.
(a) Bifurcation diagram (qC
1, γ) (b) Bifurcation diagram (ρ1, z)
(c) Time series (qC
1, t) (d) Time series (ρ1, t)
Figure 1: Bifurcation diagrams of Cournot vs. Nash rms with α = 17, β = 1 and c = 10. (a) Quantity dynamics for dierent values of γ of the Cournot rm 1 with k = 0. (b) Fraction dynamics for increasing evolutionary pressure z and κ = 0.5.(c)-(d) Time series for dierent initial values of qC1 and ρ1 with κ = 0,
z = 2.8and γ = −1.
4.2 Cournot vs. Rational rms
Instead of choosing between the Cournot and Nash rule in this section rms are able to choose between the Cournot and Rational rule, where ρi denotes the
fraction of rm i choosing the Rational rule with i = 1, 2. The global dynamics of the evolutionary model are shown in Figure 2 with the parameters for α, β and c as chosen as in subsection 3.1. In the absence of information costs κ = 0, the quantity dynamics for the Cournot rm 1 and for the total production level of rm 1 are shown in Figure 2a and 2b for dierent values of γ and high evolutionary pressure (z = 8). In Fig 2a it can be seen that for negative values of γ the Cournot rms oscillate between the Nash equilibrium and a higher production level. For γ = 0, the Cournot rms produce the q∗, because the
production level of the other rm has no in uence on the price of their own good. When the value of γ is positive the Cournot rms produce less than the equilibrium quantity. Fig 2b illustrates what happens to the total production level of rm 1, with the total production level equal to: qt= ρ1qR1 + (1 − ρ1)q1C.
a higher production level for all values of γ, the same results can be found for rm 2. In Fig 2c the associated fraction dynamics are shown for the same parameters of κ and z. For complementary goods (γ < 0), rm 1 switches every period between the Rational and Cournot rule and when the goods are substitutes a higher fraction of rm 1 chooses the Rational rule, with ρ1= 0.9
in the case of perfect substitutes.
Figure 2d, 2e and 2f show the quantity and fraction dynamics with informa-tion costs (κ = 0.5) and high evoluinforma-tionary pressure (z = 8). The dynamics of Fig 2d are similar to the one in Fig 2a where there are no information costs, with the same two cycle for γ < 0. For values of γ between 0.4 and 0.7 there is more instability in the quantity produced by the Cournot rms, this corre-spondents to the instability for the fraction dynamics in Fig 2e, which shows unstable results for the fraction of Nash rms between the same values of γ. The unstable quantity dynamics for γ = 0.5 and with information costs are also shown for increasing values of evolutionary pressure in Fig 2f. It can be seen that the quantity dynamics are unstable for β > 6.8 with its rst period doubling at β = 1.5.
(a) Bifurcation diagram (qC
1, γ) (b) Bifurcation diagram (q1, γ)
(c) Bifurcation diagram (ρ1, γ) (d) Bifurcation diagram (q1C, γ)
(e) Bifurcation diagram (ρ1, γ) (f) Bifurcation diagram (qC1, z)
Figure 2: Cournot vs. Rational rms with α = 17, β = 1 and c = 10, for (a)-(c) there are no information costs κ = 0 and in (a)-(e) there is high evolutionary pressure z = 8. (a) Quantity dynamics for the Cournot quantity. (b) Quantity dynamics for the total production level. (c) Fraction dynamics for dierent values of γ. In (d)-(f) there are information costs with κ = 0.5. (d) Quantity dynamics of the Cournot rms. (e) Fraction dynamics for dierent values of γ. (f) Quantity dynamics for the Cournot rms with γ = 0.5 and increasing values of z.
5 Discussion
In this paper the stability threshold of the equilibrium in a Cournot Duopoly under evolutionary pressure with dierentiated good is investigated. Where the basic Cournot model focuses on an economy with perfect substitutes, the research of this paper relaxes this assumption and looks at the case where the products are complements or weak substitutes. It is found that when rms are able to switch between the naive Cournot rule and the Nash rule, the rms on the market produce the quantity associated with the Nash equilibrium in every successive period and for all values of γ, which denotes the extent to which the products on the market are complements or substitutes. When rms are able to choose between the Cournot and Rational rule and there are information costs κassociated with the Rational rule, the results are less stable, with a two-cycle when the products are complements (γ < 0) , chaos for γ ∈ [0.4, 0.7] and small oscillations for γ > 0.7. In the absence of information costs κ, the quantity dynamics of the Cournot rms are equivalent for γ < 0, more stable for γ > 0, with small oscillations in the region 0.4 to 0.7 and stable results for γ > 0.7.
The results for the evolutionary model with rms able to switch between the Cournot and Nash rule correspond to the similar research done by Hommes et al. (2011). They show that for perfect substitutes in the Duopoly model the quantities produced by the rms on the market always converge to the stable Nash equilibrium. The research done is this paper extends this model and shows that for all values of γ the quantities will converge to the Nash equilibrium quantity with increasing production levels for decreasing values of γ. This can be explained from the fact that when the goods are complements, the output level of the other rm has a positive correlation with the price of its own good, therefore yielding into higher prices on both markets and a higher value of the Nash equilibrium quantity.
However there are some dierences in the results for the evolutionary model when rms are able to switch between the Cournot and Rational rule. For this model Hommes et al. (2011) found that when the goods are perfect substitutes, there is always a stable equilibrium for two players on the market, independent of the value of the information costs or the height of the evolutionary pressure. In this research it is shown that when there are information costs associated with the Rational rule and the evolutionary pressure is high, the rms oscillate between two quantities when the products are perfect substitutes. One of them is the Nash equilibrium quantity as found by Singh & Vives (1984) and one a slightly higher production level. The quantity produced by the Cournot rms at γ = 0 is always equal to the Nash equilibrium quantity, because the Cournot rms only respond to the quantity produced by the other player on the market and at γ = 0 the products are independent, so there is no dependency between the two goods on the market.
The dierence in the results found in this paper and by Hommes et al. (2011) is presumably caused by the distinct simulation of the behavior of the Rational rms. In his paper the Rational rms are assumed to know the quantity produced by the Cournot and Rational rms in the successive period, but are
not aware of the composition of rms in the next period and are therefore forming expectations over all possible mixtures of behavior rules. In this paper rms are assumed to not only have all the information about the quantities, but also about the mixture of rms on both markets in the upcoming period. However the biggest impact on the change in the behavior of the Rational rms is that they are able to respond to the fractions in their own market. In the model of Hommes et al. (2011) all the goods were homogenous so there was only one market, but with the introduction of the parameter γ a second market arises and the Rational rm 1 takes into account the fraction of rm 1 selecting the Rational rule. Therefore the quantity of the Rational players of rm i also depends on the value of ρi for i = 1, 2. This assumption is made because for a
Rational rm i, which is aware of the fraction ρi in the upcoming period, it is
not likely to ignore this information and make the naive assumption that ρi= 1
when it sets its production level.
In this model there are only two players on the market and it would be of interest what would happen with the stability of the equilibrium when there are more rms on the market. As described in the introduction Theocharis (1960) discovered that when there are more than 3 Cournot players on the market the equilibrium is never stable, further research has to point out if the same stability threshold can be found for dierent values of γ. It is of interest to see what happens to the stability when there are more rms on one of the two existing markets with the same values for β and when there are rms introduced with dierent products, thereby introducing an extra market and parameter.
Acknowledgment
I would especially like to thank Dr. M.I. (Marius) Ochea for his guidance on building the model and encouragements during the writing of this thesis.
References
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