research article effects of size, composition, and

Research ArticleEffects of Size Composition and EvolutionaryPressure in Heterogeneous Cournot Oligopolies withBest Response Decisional Mechanisms

Fausto Cavalli1 Ahmad Naimzada1 and Marina Pireddu2

1Department of Economics Management and Statistics University of Milano-Bicocca20126 Milano Italy2Department of Mathematics and Applications University of Milano-Bicocca 20125 Milano Italy

Correspondence should be addressed to Marina Pireddu marinapiredduunimibit

Received 12 November 2014 Revised 31 March 2015 Accepted 31 March 2015

Academic Editor Baodong Zheng

Copyright copy 2015 Fausto Cavalli et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study heterogeneous Cournot oligopolies of variable sizes and compositions in which the firms have different degrees ofrationality being either rational firmswith perfect foresight or naive best response firmswith static expectations Each oligopoly canbe described using its size and composition that is the fraction of firms that are rational We take into account two frameworksone in which the decisional rules are exogenously assigned and the other in which the firms may change their heuristics Weconsider a switching mechanism based on a logit rule where the switching propensity is regulated by a parameter which representsthe evolutionary pressure In the fixed fractions setting we prove that in general the composition has a stabilizing effect whileincreasing the oligopoly size leads to instability However we show that for particular parameters settings stability is not affectedby the composition or the firms number Similarly in the evolutionary fractions setting we analytically prove that when marginalcosts are identical increasing the evolutionary pressure has a destabilizing effect Nevertheless focusing on particular exampleswith different marginal costs we are able to show that evolutionary pressure may also have a stabilizing or a neutral role

1 Introduction

An oligopoly is a market structure which consists of onlya few firms that in order to decide their productions takeinto account at the same time both their own actions andthose of their competitors Since the first formal theoryof oligopoly developed by Cournot in 1838 (see [1]) theresearch focused on the decisional mechanisms that firmsmay adopt in choosing their strategies and on the differentbehaviors of the oligopoly models obtained combining suchrules in particular when the firms are not fully rational sothat they have to adjust their production level over timeIn this case the oligopolistic competition can be modeledusing a discrete dynamical system The investigation aboutoligopolistic dynamical models can be subdivided into tworesearch strands concerning homogeneous oligopolies inwhich all the firms adopt the same adjustment mechanismand heterogeneous oligopolies in which there are at least twofirms that adopt different decisional mechanisms

Starting from the seminal contributions by Palander[2] and Theocharis [3] several homogeneous frameworkshave been investigated (see eg the works by Lampart[4] Matsumoto and Szidarovszky [5] and Naimzada andTramontana [6]) involving different economic contexts orbehavioral rules See [7] for a survey on nonlinear oligopoliesand the references therein The common feature of suchmodels is that from the equilibrium stability point of viewthe situation can severely change when the oligopoly sizeincreases In particular the size has in general a destabilizingrole

The literature about heterogeneous oligopolies usuallydealing with a limited and fixed in advance number of firmshas instead developed in the direction of studying the possibleconsequences arising from considering different rationalitydegrees also on the stability of Nash equilibria In particularsuch works in view of giving a foundation to the conceptof Nash equilibrium investigate under which conditions thedynamics converge towards the Nash equilibrium or towards

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 273026 17 pageshttpdxdoiorg1011552015273026

2 Discrete Dynamics in Nature and Society

a different attractor To that research strand belong forinstance the contributions by Agiza and Elsadany [8 9] byAgiza et al [10 11] by Ahmed and Agiza [12] by Angelini etal [13] by Bischi and Naimzada [14] by Canovas et al [15]by Cavalli and Naimzada [16 17] by Cavalli et al [18 19]by Elabbasy et al [20] by Ji [21] by Li and Ma [22] byMatsumoto [23] by Naimzada and Sbragia [24] by Pu andMa [25] by Tramontana [26] and by Tuinstra [27]

Ourwork aims to investigate the role of oligopoly size andcomposition for heterogeneous competitions In particularwe consider an economy characterized by linear demandfunction and we assume linear total costs for the firmsThe heterogeneity we study is in terms of the degree ofrationality of the firms as we consider two different kindsof informational endowments Both the firms decide theirproduction level using a best reply mechanism in order tomaximize their profits with respect to the production levelsof all their competitors However rational firms have fullinformational and computational capabilities to solve suchoptimization problem and are endowed with perfect foresightso that they exactly know their competitors next periodstrategies On the other hand naive best response firms arenot able to predict such strategies and thus they assumethat their opponents next time production levels remainthe same as in the current period that is they have staticexpectations Finally we allow each group of firms to havedifferent marginal costs The approach of considering themarket split into two groups of firms identical within eachgroup is common in the oligopoly literature as shown forinstance by the works of Anufriev et al [28] Bischi et al[29] Cavalli et al [18] Droste et al [30] and Matsumotoand Szidarovszky [5] The theoretical rationale for such anapproach is that although it concerns one of the simplestpossible forms of heterogeneity among firms besides keepingthe study analytically tractable it allows obtaining interestingresults for example with respect to equilibrium stability andchaotic dynamics which can be straightforwardly interpretedand linked to the heterogeneity in rationality

In the adjustment mechanism of the less rational bestresponse firms because of capacity and financial constraintswe introduce a constraint on the strategy variation sothat firms can not immediately increase or decrease theirproduction level by arbitrarily large quantities but they adaptit towards best reply strategy with a certain reaction speed Tothis end we introduce a limiting mechanism similar to thatused in [31] in microeconomic frameworks and in [32 33] inmacroeconomic settings On the contrary due to the highdegree of rationality of the rational firms we assume thatthey are able to freely modify their production level in everyperiod

We study two different settings considering the case offixed fractions in which the behavioral rules are exogenouslyassigned and do not change and that of evolutionary frac-tions in which the firms can switch between the differentmechanisms on the basis of the realized performances of thefirms To exploit the competition between all the oligopolistswe adopt a switching mechanism based on the logit rule asproposed by Brock and Hommes in [34] The performancesof the behavioral rules are evaluated taking into account

the profits that the rule allowed achieving in the previousperiod In particular we assign additional fixed informationalcosts to the rational firms as a consequence of their highdegree of rationality The switching between mechanisms isregulated by the so-called evolutionary pressure parameterwhich describes the propensity of a firm to switch to themostprofitable rule

In the existing literature the study of heterogeneousoligopolies of generic size can be found in the works byAnufriev et al [28] Banerjee and Weibull [35] Droste etal [30] and Gale and Rosenthal [36] However only thesetting and the aims of the contribution by Droste et alare similar to those investigated in the present work as inboth papers the economy is characterized by the same lineardemand function and the firms may be both rational andbest response However in [30] it is considered an infinitepopulation of firms that are randomly matched in pairs ateach discrete-time period and that play a symmetric Cournotduopoly game so that the influence of oligopoly size onthe dynamics can not be studied and only the evolutionaryfractions framework is investigatedMoreover no productionlimiters are taken into account and a replicator switchingmechanism is considered which is different from that studiedin our contribution Another substantial difference is thatin [30] firms compute their best response with respect tothe average industry output while in our work all the firmscompete against each other Finally in [30] the firms haveidentical cost functions while in our paper theymay possiblybe different

About the fixed fractionsmodel ourmain results concernthe effects of the oligopoly size and composition and of theevolutionary pressure on the stability of the equilibriumwhile when evolutionary fractions are considered we studythe role of the evolutionary pressure focusing on whathappens with either identical or different marginal costsMoreover for the model with fixed fractions we investigatethe effects of the production limiters and of the reactionspeed In such setting we also show that marginal costs donot affect the local stability of the unique steady state whichcoincideswith theNash equilibrium and that its loss of stabil-ity occurs through a flip bifurcation Furthermore we provethat even if for most configurations the oligopoly size have adestabilizing effect there exist oligopoly compositions whichremain stable as the number of firms increases differentlyfrom the other results about homogeneous oligopolies in therelated literature

For the evolutionary fractions model we prove that thereis a unique equilibrium fraction which in general is notanalytically computable We then show that the effects ofevolutionary pressure can change depending on the marginalcosts differenceWhen the technology of the firms is identicalor when the best response firms are more efficient than therational ones increasing the evolutionary pressure alwaysleads to instability Conversely we provide simulative evi-dence of the fact that if the marginal cost of the rationalfirms is sufficiently smaller than the marginal cost of thebest response firms evolutionary pressure either can haveno effect or may stabilize the dynamics Such results whichare made possible by studying the effect of different marginal

Discrete Dynamics in Nature and Society 3

costs are in contrast to the existing literature in whichevolutionary pressure only has a destabilizing role

The remainder of the paper is organized as follows InSection 2 we describe the static Cournot competition andthe decisional mechanisms In Section 3 we introduce andstudy the fixed fractions model In Section 4 we deal withthe evolutionary fractions model Conclusions and futureperspectives are collected in Section 5

2 Oligopolistic Cournot Game

In the model we will deal with in the present paper agentsdiffer for their degree of rationality In what follows we firstintroduce the static game the decisional mechanisms arethen described in Section 21

We consider a homogeneous market characterized by 119873

firms that produce quantities 119902119894 119894 = 1 119873 of the same

good for which we assume a linear inverse demand function

119901 (119876) = 119886 minus 119887119876 (1)

where 119876 = sum119873

119894=1 119902119894 is aggregate demand and 119886 119887 are positiveconstants In particular in such formulation 119886 representsthe market size We assume that the firms have linear costfunctions

119862 (119902119894) = 119888119894119902119894 (2)

where 119888119894isin 1198881 1198882 119894 = 1 119873 are two possibly different

(constant) marginal costs In such setting the industry con-sists of two groups of 120596119873 and (1minus120596)119873 firms identical withineach group respectively where 120596 represents the fraction offirms with marginal cost 1198881

The previous framework sets up a game in which theplayers are the119873 oligopolists the admissible strategies are allthe nonnegative production choices 119902

119894ge 0 and the payoff

functions are the profit functions

120587119894= 119902119894 (119886 minus 119887119876) minus 119888

119894119902119894 119894 = 1 119873 (3)

Weonly consider the situation inwhich all firms have nonnullproduction levels and so we focus on the internal Nashequilibrium namely the equilibrium consisting of strictlypositive strategies We derive its expression in the next result

Proposition 1 Let

119886

gt max 1198881 +119873 (1198881 minus 1198882) (1minus120596) 1198882 minus119873120596 (1198881 minus 1198882) 0 (4)

Then there is a unique internal Nash equilibrium for our gamefor which

119902⋆

1 =119886 minus 1198881 minus 119873 (1198881 minus 1198882) (1 minus 120596)

119887 (119873 + 1) (5a)

119902⋆

2 =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1) (5b)

where 119902⋆

1 is the equilibrium strategy of the firms havingmarginal cost 1198881 and 119902

⋆

2 is the equilibrium strategy of thosehaving marginal cost 1198882

Proof Firstly we compute the best responses of the 119894th playerof the first group and of the 119896th player of the second groupwith respect to the remaining players strategies This can bedone by deriving the first order conditions on payoffs of thetwo groups obtaining

119886 minus 119887 (1198761minus119894 +1198762) minus 21198871199021119894 minus 1198881 = 0119886 minus 119887 (1198761 +1198762minus119896) minus 21198871199022119896 minus 1198882 = 0

(6)

where119876119895 119895 = 1 2 are the aggregate strategies of the players of

the 119895th group and1198761minus119894 (resp1198762minus119896) is the aggregate strategyof all the players belonging to the first (resp second) groupexcept the 119894th (resp 119896th) player From (6) we obtain the bestresponse functions

1199021119894 =119886 minus 1198881 minus 119887 (1198761minus119894 + 1198762)

2119887 (7)

provided that 119886 minus 1198881 minus 119887(1198761minus119894 + 1198762) gt 0 and

1199022119896 =119886 minus 1198882 minus 119887 (1198761 + 1198762minus119896)

2119887 (8)

provided that 119886 minus 1198882 minus 119887(1198761 + 1198762minus119896) gt 0 It is easy to see thatboth (7) and (8) satisfy second order conditions In order toobtain the equilibrium strategy by solving (7)-(8) we noticethat players belonging to the same group are identical so thatthey have the same equilibrium strategy

119902⋆

1119894 = 119902⋆

1

119902⋆

2119896 = 119902⋆

2 (9)

and consequently

119876⋆

1 = 120596119873119902⋆

1

119876⋆

2 = (1minus120596)119873119902⋆

2 (10)

We obtain the internal Nash equilibrium (5a) and (5b) byinserting the previous expressions in (7) and (8) and solvingthe resulting 2 times 2 system provided that (4) is satisfied

We observe that in theNash equilibrium it holds that 119902lowast1 gt

119902lowast

2 if and only if 1198882 gt 1198881

21 The Decisional Mechanisms In this work we aim tostudy the dynamics arising when the two groups of firmsadopt different decisional mechanisms In particular we willsuppose that not all the firms are fully rational so that adynamic adjustment process naturally arises The behavioralrules we focus on are best reply mechanisms with eitherperfect foresight (rational firms) or static expectations (bestresponse firms) Rational firms have complete informationaland computational capabilities so that they are able to opti-mally respond to the other players strategies and are endowedwith perfect foresight about the next period production levelof their competitors Best response firms have a reduceddegree of rationality with respect to rational firms as theyassume their opponents next period production to be thesame as in the past period


In the remainder of the present section we will obtainthe reaction functions of each group of players Without lossof generality we assume that rational firms have marginalcost 1198881 Moreover we denote by1198761119905 the aggregate productionof rational firms and by 1198762119905 the aggregate output of bestresponse players both at time 119905 while wewill denote by1198761minus119894119905the aggregate production at time 119905 of all rational firms exceptfirm 119894 We remark that in principle both the best reply ofrational and of best response players may be null Since weonly want to focus on positive dynamics for the productionlevels we will not deal with such situations

Rational Players If we consider a generic rational firm 119894 wehave that its profit is given by

1205871119894 = 1199021119894119905+1 (119886 minus 119887 (1198761minus119894119905+1 + 1199021119894119905+1 +1198762119905+1))

minus 11988811199021119894119905+1(11)

where we denoted by 1199021119894119905+1 the strategy of firm 119894 at time119905 + 1 by 1198761minus119894119905+1 the aggregate quantity at time 119905 + 1 of allthe rational players but the 119894th and by 1198762119905+1 the aggregatestrategy of the best response players at time 119905 + 1

As noticed in the proof of Proposition 1 since all rationalplayers are identical we can compute the reaction function ofa generic rational player with respect to the correctly foreseenstrategies of the best response players

Proposition2 If1198762119905+1 lt (119886minus1198881)119887 then the reaction functionof a generic rational player 1199021119894 at time 119905 + 1 one denotes by1199021119905+1 is given by

1199021119905+1 =119886 minus 1198881 minus 1198871198762119905+1

119887 (120596119873 + 1) (12)

Proof We can compute the best response of the 119894th rationalplayer with respect to the aggregate strategies of the remain-ing rational players and of the best response players acting asin the proof of Proposition 1 obtaining from the first orderconditions of (11)

1199021119894119905+1 =119886 minus 1198881 minus 119887 (1198761minus119894119905+1 + 1198762119905+1)

2119887 (13)

provided that 119886minus1198881minus119887(1198761minus119894119905+1+1198762119905+1) gt 0The second orderconditions are indeed satisfied Since all the rational players

are identical we can set1198761minus119894119905+1 = (120596119873minus1)1199021119894119905+1 in (13) andsolving the resulting equation with respect to 1199021119894119905+1 we find(12) as desired

Best Response Players Best response players are able tooptimally respond to the other players strategies but sincethey do not have perfect foresight they rather use staticexpectations The reaction function of the generic 119896th playerat time 119905 + 1 we denote by 1199022119896119905+1 is then the best response to119876minus119896119905+1 which is the aggregate strategy at time 119905 + 1 of all the

remaining players namely

1199022119896119905+1 =119886 minus 1198882 minus 119887119876

minus119896119905

2119887 (14)

provided that 119876minus119896119905

lt (119886 minus 1198882)119887

3 Fixed Fractions Oligopoly Model

Our purpose is to study a heterogeneous oligopoly that isa competition in which not all the firms adopt the samebehavioral rule To this end we suppose that there is alwaysat least one rational firm and at least one best response firmnamely 120596 isin [1119873 1 minus 1119873] Moreover we assume that theinitial production levels of the best response firms are thesame which in particular means that their production levelsare identical Finally since we only focus on economicallyinteresting positive production levels in what follows weconsider initial data that give rise to positive trajectories Suchassumptions allow us to identify the generic rational and bestresponse playerrsquos strategies respectively with

1199021119905 = 1199021119894119905

1199022119905 = 1199022119896119905(15)

which give 1198761minus119894119905 = (120596119873 minus 1)1199021119905 1198762119905 = (1 minus 120596)1198731199022119905 and119876minus119896119905

= 1205961198731199021119905 + ((1 minus 120596)119873 minus 1)1199022119905 and to rewrite (13) and(14) as

1199021119905+1 =119886 minus 1198881 minus 119887 (1 minus 120596)1198731199022119905+1

119887 (120596119873 + 1) (16)

1199022119905+1 =119886 minus 1198882 minus 119887 (1205961198731199021119905 + ((1 minus 120596)119873 minus 1) 1199022119905)

2119887 (17)

Combining (16) and (17) we have

1199022119905+1 =119886 minus 1198882 minus 119887 (120596119873 ((119886 minus 1198881 minus 119887 (1 minus 120596)1198731199022119905) (119887 (120596119873 + 1))) + ((1 minus 120596)119873 minus 1) 1199022119905)

2119887 (18)

Gathering in the previous expression the terms with 1199022119905 weobtain

1199022119905+1

= minus 1198871199022119905

2119887(minus119887120596 (1 minus 120596)119873

2

119887 (120596119873 + 1)+ (1minus120596)119873minus 1)

+119886 minus 1198882 minus 119887120596119873 ((119886 minus 1198881) (119887 (120596119873 + 1)))

2119887

= minus1199022119905

2(minus120596 (1 minus 120596)119873

2

(120596119873 + 1)+ (1minus120596)119873minus 1)

+119886 minus 1198882 minus 120596119873 ((119886 minus 1198881) (120596119873 + 1))

2119887


= minus1199022119905

2(minus120596 (1 minus 120596)119873

2+ ((1 minus 120596)119873 minus 1) (120596119873 + 1)(120596119873 + 1)

)

+(119886 minus 1198882) (120596119873 + 1) minus 120596119873 (119886 minus 1198881)

2119887 (120596119873 + 1)

= minus1199022119905

2((1 minus 120596)119873 minus (120596119873 + 1)

(120596119873 + 1))

+(119886 minus 1198882) + 120596119873 (1198881 minus 1198882)

2119887 (120596119873 + 1)

=1199022119905

2(2120596119873 + 2 minus 1 minus 119873

(120596119873 + 1)) minus

(1198882 minus 119886) minus 120596119873 (1198881 minus 1198882)

2119887 (120596119873 + 1)

(19)

which gives

1199022119905+1 = 119891 (1199022119905)

= (1minus(1 + 119873)

2 (1 + 120596119873)) 1199022119905

minus1198882 minus 119886 minus 120596119873 (1198881 minus 1198882)

2119887 (120596119873 + 1)

(20)

However we suppose that the best response player doesnot immediately choose the new production level 1199022119905+1 buthe cautiously moves towards it This also reflects the factthat in concrete situations firms can only handle limitedproduction variations for example because of capacity orfinancial constraints (see [31 page 507]) so that the maxi-mum admissible upper and lower output variations may bedescribed by means of two positive parameters 1198861 and 1198862as 1198861 and minus1198862 respectively The possibility of varying 1198861 and1198862 allows investigating the effects of the output limitationdegreeThe values assigned to such parameters depend on thekind of commodity under analysis and on its unit productiontime and are calibrated on the base of empirical research indifferent oligopolistic markets

The cautious behavior of the best response player may bemodeled in several ways introducing for instance an inertialparameter which anchors the next time production to thestrategy 1199022119905 as done by Matsumoto and Szidarovszky in [5]In our contribution we adopt a different cautious adjustmentsimilar to that used by Du et al in [31] which is obtained by(20) as follows

1199022119905+1 = 1199022119905 +119892 (120574 (119891 (1199022119905) minus 1199022119905)) (21)

where 119892 R rarr R is the nonlinear sigmoidal map defined as

119892 (119909) = 1198862 (1198861 + 1198862

1198861119890minus119909 + 1198862

minus 1) (22)

with 1198861 1198862 positive parameters playing the role of horizontalasymptotes and restricting the possible output variation asfrom period 119905 to 119905 + 1 1199022 can increase at most by 1198861 anddecrease at most by 1198862 Indeed 119892 is a strictly increasingfunction which takes values in (minus1198862 1198861) and as shown in

Proposition 3 its introduction does not alter the steady statesof (20) Parameter 120574 gt 0 in (21) represents the speedof reaction of the agents with respect to production levelvariations

Variation limitation mechanisms similar to (21) havebeen also applied to different variables in macroeconomicsettings for instance in [32 33 37ndash40] As regards the specificshape of the output variation limiters in our work insteadof describing the output variation through a piecewise linearmap we chose to deal with the sigmoidal function in (22)which is differentiable and allows an easier mathematicaltreatment of the model We stress that 119878-shaped functionsalike to the one in (22) have already been considered in amacroeconomic framework in the Kaldorian business cyclemodel in [41] and in a microeconomic setting in thetatonnement process in [42]

In view of the subsequent analysis it is expedient tointroduce the map 120593 (0 +infin) rarr R associated with thedynamic equation in (21) and defined as

120593 (1199022) = 1199022 +119892 (120574 (119891 (1199022) minus 1199022)) = 1199022

+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596119873)))1199022+(1198882minus119886minus119873120596(1198881minus1198882))(2119887(119873120596+1))) + 1198862

minus 1)

(23)

We remark that in the oligopolistic competition described by(21) the decisionalmechanisms are exogenously assigned anddo not change

31 Stability and Bifurcation Analysis We start studying thepossible steady states of (21) and their connections with theNash equilibrium We have the following straightforwardresult

Proposition 3 Equation (21) has a unique positive steadystate which coincides with the Nash equilibrium (5a) and (5b)

Proof The expression for 119902⋆

2 can be immediately found bysolving the fixed point equation 120593(119909) = 119909 with 120593 as in (23)noticing that no other solutions exist The expression for 119902

⋆

1directly follows by that for 119902⋆2 and by (16)

The next results are devoted to the study of the localstability of theNash equilibriumWe also present several sim-ulations to confirm the theoretical analysis and to investigatethe loss of stability As we are going to prove marginal costsdo not affect the stability so in all the simulations we set1198881 = 1198882 = 01 Moreover unless otherwise specified we set1198861 = 3 1198862 = 1 and 120574 = 1 in the sigmoid function in (22)

Lemma 4 For the dynamical equation in (21) the steady state119902⋆

2 = (119886minus 1198882 +119873120596(1198881 minus 1198882))(119887(119873+ 1)) is locally asymptoticallystable if

12057411988611198862 (1 + 119873)

4 (1198861 + 1198862) (1 + 120596119873)lt 1 (24)


Proof A direct computation shows that

1205931015840(119902⋆

2 ) = 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596119873) (25)

Since 1205931015840(119902⋆

2 ) lt 1 is always satisfied the local asymptoticstability conditions require just that 1205931015840(119902⋆2 ) gt minus1 which isfulfilled for 12057411988611198862(1 + 119873)(4(1198861 + 1198862)(1 + 120596119873)) lt 1 Thisconcludes the proof

We conclude the present section by showing that whenour dynamic equation loses stability at the steady state aperiod-doubling bifurcation occurs

Lemma 5 For the dynamical equation in (21) a period-doubling bifurcation occurs at the steady state 119902

⋆

2 = (119886 minus 1198882 +

119873120596(1198881 minus 1198882))(119887(119873 + 1)) when 12057411988611198862(1 + 119873)(4(1198861 + 1198862)(1 +

120596119873)) = 1

Proof According to the proof of Proposition 1 the steadystate 119902

lowast

2 is stable when 1198651015840(119902lowast

2 ) gt minus1 Then the map 119865 satisfiesthe canonical conditions required for a flip bifurcation (see[43]) and the desired conclusion follows Indeed when1198651015840(119884lowast) = minus1 that is for 12057411988611198862(1+119873)(4(1198861 + 1198862)(1+120596119873)) =

1 then 119884lowast is a nonhyperbolic fixed point when 12057411988611198862(1 +

119873)(4(1198861 + 1198862)(1 + 120596119873)) lt 1 it is attracting and finally when12057411988611198862(1 + 119873)(4(1198861 + 1198862)(1 + 120596119873)) gt 1 it is repelling

In the next corollaries we will derive the stability con-ditions for 120574 120596119873 1198861 and 1198862 respectively in view of under-standing whether they have a stabilizing or a destabilizingeffect and we will illustrate our conclusions through somepictures We stress that those stability conditions may beimmediately deduced by (4) by putting in evidence one ofthe parameters

Corollary 6 For the dynamical equation in (21) the steadystate 119902⋆2 is locally asymptotically stable if

120574 lt4 (1198861 + 1198862) (1 + 120596119873)

11988611198862 (1 + 119873) (26)

Such result shows that as expected reactivity 120574 has adestabilizing role We illustrate such property in Figure 1 inwhichwe consider an oligopoly of119873 = 50 firms with120596119873 = 5rational firms with an economy characterized by 119886 = 30 and119887 = 01 The initial output level is set equal to 11990220 = 6 Inagreement with (26) equilibrium loses its stability for 120574 asymp

06275


120596 gt (120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) 1119873

(27)

Hence increasing 120596 that is the fraction of rationalplayers has a stabilizing role However if for instance

(120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) gt 1 (28)

or equivalently

119873 lt81198861 + 81198862 minus 11988611198862120574

11988611198862120574 (29)

all the heterogeneous oligopoly configurations are stableFor the simulation reported in Figure 2 we considered an

oligopoly of 119873 = 22 firms set in an economy characterizedby 119886 = 24 and 119887 = 02 The initial strategy is 11990220 = 5The oligopoly composition is studied for 120596 isin [122 02] Inthe boxes we highlight the regions corresponding to 1 2 3and 4 rational firms Condition (27) predicts that the Nashequilibrium is stable for 120596 gt 01506 and this means thatat least 4 rational firms (ie 120596 = 01818) are needed toguarantee stabilityWe remark that with a reduced number ofrational firms we have both periodic and chaotic dynamics

Moreover we may put 119873 in evidence in (24) finding thefollowing

Corollary 8 For the dynamical equation in (21) the followingconclusions hold true

(1) if 12057411988611198862 minus 4(1198861 + 1198862) gt 0 then the steady state 119902⋆

2 isnever stable

(2) if 12057411988611198862 minus 4120596(1198861 + 1198862) lt 0 then 119902⋆

2 is always locallyasymptotically stable

(3) if 12057411988611198862 minus 4(1198861 + 1198862) lt 0 lt 12057411988611198862 minus 4120596(1198861 + 1198862) then119902⋆

2 is locally asymptotically stable for

119873 lt4 (1198861 + 1198862) minus 12057411988611198862

12057411988611198862 minus 4120596 (1198861 + 1198862) (30)

Hence we can conclude that increasing the populationsize 119873 has overall a destabilizing effect More preciselyCondition (1) says that for sufficiently large reaction speedsall the considered families of oligopolies independently oftheir size and composition have an unstable equilibriumConversely there are oligopoly compositions (with 120596 gt

(12057411988611198862)[4(1198861 + 1198862)] according to Condition (2)) whichhave a stable equilibrium independently of the oligopolysize We stress that this is in contrast with the literatureof homogeneous oligopolies as there the oligopoly size hasalways a destabilizing role In the present case the neutralrole of 119873 is possible thanks to the presence of rationalplayers on condition that their fraction does not change Inthe remaining situations oligopoly size has a destabilizingeffect (see Condition (3)) We show this last situation inFigure 3 setting 119886 = 20 119887 = 01 and considering oligopolycompositions with 120596 = 19 The initial strategy is 11990220 = 622As predicted by (30) only for oligopolies with size 119873 lt

106364 the equilibrium is stable so just the oligopoly with9 best response firms and 1 rational firm has a stable steadystate In oligopolies with 119873 = 18 27 36 firms and 2 3 4rational firms respectively we observe cyclic trajectorieswhile for 119873 = 45 and 5 rational firms the dynamics arechaotic

Finally putting in evidence 1198861 or 1198862 in (24) we obtain thefollowing result


0 02 04 06 08 14

6

8

10

12

14

120574

q1

0 02 04 06 08 1

48

5

52

54

56

58

6

120574

q2

Figure 1 Bifurcation diagrams of quantities 1199021 and 1199022 with respect to the speed of reaction 120574 which always has a destabilizing role

005 01 015 020

5

10

15

20

005 01 015 02

35

4

45

5

55

q1

q2

120596 120596

Figure 2 Bifurcation diagrams of quantities 1199021 and 1199022 with respect to the oligopoly composition 120596 for119873 = 22 119886 = 24 119887 = 02 Increasingthe number of best response firms leads to instability

10 15 20 25 30 35 40 450

5

10

15

20

N

10 15 20 25 30 35 40 450

5

10

15

20

N

q1

q2

Figure 3 Bifurcation diagrams of quantities 1199021 and 1199022 with respect to the oligopoly size119873 for 119886 = 20 119887 = 01 and keeping fixed the oligopolycomposition 120596 = 19 Increasing the oligopoly size has a destabilizing effect



(1) if (1+119873)1198862120574 minus 4(1 +120596119873) lt 0 then the steady state 119902⋆2is always stable

(2) if (1 + 119873)1198862120574 minus 4(1 + 120596119873) gt 0 then 119902⋆

2 is stable for

1198861 lt41198862 (1 + 120596119873)

(1 + 119873) 1198862120574 minus 4 (1 + 120596119873) (31)

(3) if (1+119873)1198861120574minus4(1+120596119873) lt 0 then 119902⋆

2 is always stable(4) if (1 + 119873)1198861120574 minus 4(1 + 120596119873) gt 0 then 119902

⋆

2 is stable for

1198862 lt41198861 (1 + 120596119873)

(1 + 119873) 1198861120574 minus 4 (1 + 120596119873) (32)

Hence from Conclusions (1) and (2) we can infer thatincreasing the upper bound 1198861 has overall a destabilizingeffect In particular an increase in 1198861 is destabilizing whenminus1198862 is negative enough (see Figure 4 for a graphic illustrationof the case with 119873 = 15 120596 = 13 1198862 = 3 119886 =

60 119887 = 01 11990202 = 3706) Similarly from Conclusions (3)

and (4) it follows that decreasing the lower bound minus1198862 hasoverall a destabilizing effect In particular a decrease in minus1198862is destabilizing when 1198861 is large (see Figure 5 for a graphicillustration of the case with 119873 = 15 120596 = 13 1198861 = 3 119886 =

60 119887 = 01 11990202 = 3706) We stress that the conclusionswe have drawn from Corollary 9 may also be read in terms ofchaos control as they say that decreasing the output variationpossibilities has a stabilizing effect

4 Endogenous Switching Mechanism

We now assume that agents in any time period may switchbetween the groups of rational players and best response play-ers This requires introducing an evolutionary mechanismfor the fractions in which we will denote by 120596

119905the variable

describing the fraction of rational players To such end weassume that the decisional rules are commonly known aswell as both past performances and costs We assume thatthe firms first choose their decisionalmechanism on the basisof the previous period performances and then accordingto such choice they decide their strategy Several switchingmechanisms can be adopted In the present work in order tobetter model a form of competition in which each player isopposed to any other player we adopt the logit choice rulebased on the logit function (see [34])

119911 (119909 119910) =exp (120573119909)

exp (120573119909) + exp (120573119910) (33)

where 119909 and 119910 denote the previous period performances bythe two groups of firms and 119911(119909 119910) describes the evolutionof the fraction of firms whose past profits are given by119909 Moreover the positive parameter 120573 is usually referredto as the ldquoevolutionary pressurerdquo or ldquointensity of choicerdquoand regulates the switching between the different decisionalmechanisms in the sense that as 120573 increases it is more likely

that firms switch to the decisional mechanism with highernet profits Due to the supplementary degree of rationalityof rational firms we suppose that rational players incur costsfor information gathering We take into account such costsconsidering nonnull informational costs119862 gt 0 so that the netprofits of rational firms are actually given by1205871119905 = 1199021119905119901(119876119905)minus11988811199021119905 minus 119862

Sincewe are interested in studying heterogeneous compe-titions only we have to slightly modify the logit function sothat the switching variable assumes values just in (1119873 (119873 minus

1)119873) rather than in [0 1] In this way we always have atleast a firm for each kind of decisionalmechanism Hence weare led to consider the following two-dimensional dynamicalsystem1199022119905+1 = 1199022119905

+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596

119905119873)))1199022119905+(1198882minus119886minus119873120596119905(1198881minus1198882))(2119887(119873120596119905+1))) + 1198862

minus 1)

120596119905+1 =

1119873

+119873 minus 2119873

11 + 119890120573Δ120587119905

(34)

where the net profit differential is

Δ120587119905= 1205872119905 minus1205871119905

= 1199022119905 (119886 minus 119887119876119905minus 1198882) minus 1199021119905 (119886 minus 119887119876

119905minus 1198881) +119862

(35)

with

1199021119905 =119886 minus 1198881 minus 119887 (1 minus 120596

119905)1198731199022119905

119887 (120596119905119873 + 1)

(36)

and thus

119876119905=

119873 (120596119905(119886 minus 1198881) + 1199022119905119887 (1 minus 120596

119905))

119887 (120596119905119873 + 1)

(37)

We stress that in this way we are assuming that the rationalplayers also know which will be the next period oligopolycomposition 120596

119905+1 while the best response players expect itto remain the same as in the previous period

Notice that when 120573 = 0 the evolutionary mechanismreduces to 120596

119905+1 = 12 and thus for any initial partitioning ofthe population the players do not consider nor compare theperformances but they immediately split uniformly betweenthe two decisional mechanisms Conversely as 120573 rarr infinthe logit function becomes similar to a step function sothat the firms are inclined to rapidly switch to the mostprofitable mechanism and 120596

119905+1 approaches 1119873 or 1 minus 1119873In particular if the net profit differential is positive (ie bestresponse firms achieve the best profits) the number of bestresponse firms will be larger than that of the rational players(120596119905+1 gt 12) and vice versa (120596

119905+1 lt 12) when the profitdifferential is negative

We start by analyzing the simpler case in which themarginal costs 1198881 and 1198882 coincide and then we will investigatethe general framework with 1198881 = 1198882

(i) The Case with 1198881

= 1198882 In this particular framework

in which we assume that the marginal costs for all firms


1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1

Figure 4 Bifurcation diagrams of quantities 1199021 and 1199022 with respect to the upper bound of production variation 1198861 Increasing 1198861 leads toinstability

2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2

Figure 5 Bifurcation diagrams of quantities 1199021 and 1199022 with respect to the lower bound of production variation minus1198862 Increasing 1198862 namelydecreasing the lower bound minus1198862 leads to instability

coincide and we set 1198881 = 1198882 = 119888 it is possible to analyticallyfind the unique solution to system (34) Indeed since by (5a)and (5b) it follows that

119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)

then in equilibriumΔ120587lowast= 120587lowast

2 minus120587lowast

1 = 119902lowast

2 (119886minus119887119876lowastminus119888)minus119902

lowast

1 (119886minus119887119876lowastminus 119888) + 119862 = 119862 and thus

120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)

Notice that the expression for 119902lowast

1 = 119902lowast

2 in (38) does notdepend on 120596 and that such steady state values are positivefor 119886 gt 119888 (we stress that a large enough value for 119886

makes the steady state values for 1199021 and 1199022 positive also inthe general case with possibly different marginal costs (see(5a) and (5b))) Hence we will maintain such assumption inthe remainder of the paper

The stability analysis requires computing at the steadystate (119902

lowast

2 120596lowast) the Jacobian matrix for the bidimensional

function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)

whereΔ120587 = 1205872 minus1205871 = (1199022 minus 1199021) (119886 minus 119887119876minus 119888) +119862 (41)


with 1199021 = (119886 minus 1198881 minus 119887(1 minus 120596)1198731199022)(119887(120596119873 + 1)) and thus 119876 =

119873(120596(119886 minus 119888) + 1199022119887(1 minus 120596))(119887(120596119873 + 1))It holds that

119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast

2 120596lowast) is not essential in view of

the subsequent computations Indeed the standard Juryconditions (see [44]) for the stability of our system read as

1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)

Notice that just the second one is not always satisfied but itleads to (24) in which we have to insert the expression for 120596lowastfrom (39) Then it may be rewritten as

119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)

Since 120573119862 gt 0 the latter relation is never satisfied when theright hand side is smaller than 1 that is when

4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)

which is fulfilled for 120574 lt 120574 and for 120574 gt 120574 with

120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)

Notice that 120574 gt 120574 when 119873 gt 2 When 119873 = 2 (45) reads asminus2 lt 0 which is true for any value of 120574

In the remaining cases that is when the right hand sideof (44) is larger than 1 the latter is fulfilled for

120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)

The above considerations say that when the marginalcosts are identical the evolutionary pressure has a destabiliz-ing effect Such phenomenon is illustrated by the simulationin Figure 6 in which we considered an oligopoly of 119873 = 100firms in an economy described by 119886 = 120 and 119887 = 01Marginal costs are 1198881 = 1198882 = 02 and the informational cost ofthe rational firm is 119862 = 01The initial strategy is 11990220 = 9 andthe initial fraction of rational firms is 1205960 = 01 Moreover we

set 1198861 = 1 1198862 = 3 and 120574 = 034 Since the steady state valuefor the fraction of rational firms is decreasing in Figure 6firms are likely to switch to the best response rule which giveslarger profits When 120573119862 increases and reaches the value 32the equilibrium loses its stability through a flip bifurcation

(ii) The Case with 1198881

= 1198882 When the marginal costs are

different the expression for the steady state value for 120596 for(34) can not be analytically computed but a simple continuityargument shows that a steady state for system (34) alwaysexists and that it is unique Indeed at any steady state (119902⋆2 120596

⋆)

it holds that similarly to what obtained in (5b) for the casewithout switching mechanism

119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)

Let us then define the continuous map

120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)

where in the expression

Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)

minus 1199021 (120596) (119886 minus 119887119876 (120596) minus 1198881) +119862

(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)

1199021 (120596) =119886 minus 1198881 minus 119887 (1 minus 120596)1198731199022 (120596)

119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)

we wish to underline the dependence of all terms on 120596Since (119873minus 2)(1+119890

120573Δ120587(120596)) ranges in (0 119873minus 2) then 120595(120596)

ranges in (1119873 (119873minus1)119873) Hence120595 is a continuous functionthat maps [1119873 (119873 minus 1)119873] into (1119873 (119873 minus 1)119873) and thusby the Brouwer fixed point theorem there exists (at least one)120596lowastisin (1119873 (119873minus1)119873) such that120595(120596

lowast) = 120596lowast that is120596lowast is (a)

steady state value for system (34) together with 119902⋆

2 in (48)Thevalue of the steady state for 1199021 can finally be found by (36)

Let us now prove that system (34) admits a unique steadystate To such aim it suffices to show that Δ120587(120596) in (50)is monotone with respect to 120596 In particular we will showthat the derivative of Δ120587(120596) with respect to 120596 we denote by(Δ120587)1015840(120596) is always positive Indeed cumbersome but trivial

computations leads to (Δ120587)1015840(120596) = 2119873(1198881 minus 1198882)

2119887(119873 + 1) gt 0

This concludes the proofFinally we report the expression for the Jacobian matrix

at (119902lowast

2 120596lowast) for the bidimensional function associated with


11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596

Figure 6 Bifurcation diagrams of quantities 1199021 and 1199022 and fraction 120596 with respect to the evolutionary pressure 120573 Increasing 120573 leads toinstability

(34) we still denote by 119866 due to the similarity with the casewith equal marginal costs (see (40)) omitting the (tedious)computations leading to its derivation Of course in this casewe do not know the value of the steady state (119902

lowast

2 120596lowast) Hence

we will just use (48) in order to express 119902lowast2 as a function of120596lowast

It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921

= minus120573 (119873 minus 2) (119886 minus 1198882 minus 21198731198881 + 21198731198882 + 31198731198881120596

lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)

or since in equilibrium we have

120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596

lowastminus 1) (119873 minus 119873120596

lowastminus 1) (119886 minus 1198882 minus 21198731198881 + 21198731198882 + 31198731198881120596

lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)

11986922 = minus120573 (119873 minus 2) (1198881 minus 1198882) (21198881 minus 119886 minus 1198882 + 21198731198881 minus 21198731198882 minus 1198731198881120596

lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)


or equivalently using again (55)

11986922

= minus120573 (119873120596


lowastminus 1) (1198881 minus 1198882)

119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)

sdot (21198881 minus 119886minus 1198882 + 21198731198881 minus 21198731198882 minus1198731198881120596lowast+1198731198882120596

lowast)

(57)

The elements of 119869119866(119902lowast

2 120596lowast) are too complicated and

do not allow obtaining general stability conditions Hencein order to investigate the possible scenarios arising whenthe marginal costs are different we focus on the followingexample We set 119886 = 100 119887 = 01 for the economy 119873 =

100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =

3 120574 = 034 for the decisional mechanism and we studywhat happens for different values of the marginal costs ofbest response firms We consider both the frameworks with1198882 lt 1198881 and 1198881 lt 1198882 giving special attention to the lattercase as it describes the economically interesting situation inwhich rational firms have improved technology (ie reducedmarginal costs) at the price of larger fixed costs

We start by considering the setting with 1198882 = 0101 sothat the marginal costs of the best response firms is larger butvery close to that of the rational firms In this case the profitdifferential at the equilibrium is

Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)

We stress that when the net profit differential is positive thepossible equilibrium fractions are only 120596

lowastisin (1119873 12)

Moreover the positivity of (58) implies that increasing 120573120596⋆ decreases and thus the fraction of rational players may

become too small to guarantee stability This conjecture isconfirmed by Figure 7 where we can see that for 120573 asymp 6819the equilibrium loses its stability through a flip bifurcation Infact for 120573 asymp 6519 the equilibrium fraction can be computednumerically and is given by 120596

lowast= 013 (corresponding to

13 rational firms out of 100) which implies the equilibriumstrategy 119902lowast2 = 98897The largest eigenvalue of 119869

119866(119902lowast

2 120596lowast) that

corresponds to this configuration is such that |120582| = 09287which means that the equilibrium is stable Conversely for120573 = 6845 the equilibrium fraction is 120596lowast = 012 which givesthe equilibrium strategy 119902

lowast

2 = 9889789 and the eigenvaluesof 119869119866(119902lowast

2 120596lowast) are 1205821 = 0506 and 1205822 = minus10006 that is

equilibrium has lost its stability

If we further increase the marginal cost of the bestresponse firms setting for example 1198882 = 011 we find thatthe equilibrium is stable independently of the evolutionarypressure In fact in this case we have that the profit differen-tial is given by

Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)

which is negative for all the possible 120596lowast

isin [1119873 1 minus

1119873) In this case since the equilibrium fraction increaseswith 120573 namely the fraction of rational players increases wecan expect the stability not to be affected by evolutionarypressure To prove such conjecture we notice that at theequilibrium we have

120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)

from which we can obtain the evolutionary pressure 120573 towhich the equilibrium fraction 120596

lowast corresponds that is

120573 =log ((99 minus 100120596lowast) (100120596lowast minus 1))

120596lowast505 minus 15029101000 (61)

Now we consider Jury conditions (43) After substituting theparameter values and using (61) the first condition reducesto

119871 (120596lowast)

sdot(100120596lowast minus 1) (99 minus 100120596lowast) (2033049 minus 20000120596lowast)

19600 (100120596lowast + 1) (15029 minus 200120596lowast)

lt 1

(62)

where we set

119871 (120596lowast) = log(99 minus 100120596lowast

100120596lowast minus 1) (63)

Condition (62) is satisfied since 119871(120596lowast) le 0 as the argument ofthe logarithm is not larger than 1 for120596lowast ge 12 and the secondfactor in (62) is indeed positive for 120596lowast isin [12 1 minus 1119873)

The second condition in (43) reduces to

(99 minus 100120596lowast) (120596lowast minus 001) (40000120596lowast minus 4071249) 119871 (120596lowast) + 3959200000120596lowast minus 215330990

19796000 (100120596lowast + 1)gt 0 (64)

and the positivity of the lhs for 120596lowast

isin [12 1 minus 1119873) canbe studied using standard analytical tools We avoid enteringinto details and we only report in Figure 8 the plot of thefunction corresponding to the lhs of (64) with respect to120596lowast which shows its positivity

The last condition in (43) reads as

(minus510000 (120596lowast)2+ 510000120596lowast minus 5049) 119871 (120596

lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573

Figure 7 Bifurcation diagrams of quantities 1199021 and 1199022 and fraction 120596 with respect to the evolutionary pressure 120573 Increasing 120573 has adestabilizing effect for 1198882 slightly larger than 1198881

Like for the second condition we omit the analytical study ofthe lhs of (65) andwe only report the plot of the correspond-ing function in Figure 9 which shows the positivity Theneach stability condition is satisfied for all120573 which proves thatat least for this parameter configuration the equilibrium isstable independently of the evolutionary pressure

Then we consider 1198881 gt 1198882 = 009 case in which as onemight expect evolutionary pressure has a destabilizing effectIn fact we have that the net profit differential

Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)

is positive independently of120596lowastThe resulting scenario is anal-ogous to the first one with bifurcation diagrams qualitativelysimilar to those reported in Figure 7 The flip bifurcationoccurs for 120573 asymp 16252 In this case for 120573 = 1568 theequilibrium fraction can be computed numerically and isgiven by120596lowast = 003 (3 rational firms out of 100) which impliesthe equilibrium strategy 119902

lowast

2 = 9895 The largest eigenvalueof 119869119866(119902lowast

2 120596lowast) is such that |120582| = 0864 lt 1 and thus the

equilibrium is stable

Conversely the equilibrium fraction corresponding to120573 = 1853 is 120596

lowast= 002 (2 rational firms out of 100) which

gives the equilibrium strategy 119902lowast

2 = 9894 The eigenvalues of119869119866(119902lowast

2 120596lowast) are now 1205821 = minus02 and 1205822 = minus24 and thus the

equilibrium is no more stableA question naturally arises from the second example

what happens if the configuration corresponding to 120573 = 0is unstable and the marginal cost is sufficiently favorable tothe rational firms Let us consider for instance the previousparameters for the economy and the firms but 1198861 = 100 1198862 =

3 120574 = 094 for the decisionalmechanism of the best responsefirms We underline that the new value for 1198861 allows a largerincrease in production while the increased 120574 models firmswhich aremore reactive to production variationsThe generalscenario is indeed more ldquounstablerdquo If we take 1198882 = 018the behavior for increasing values of the reaction speed isreported in the bifurcation diagrams of Figure 10 where thestabilizing role of 120573 is evident For 120573 = 0 there is no fractionevolution and we always have 50 rational firms out of 100 butthe dynamics of production levels are chaotic As 120573 increasesthe dynamics of quantities and fractions become qualitatively


05 06 07 08 09 114

15

16

17

18

19

120596lowast

Figure 8 Plot of the lhs of (64) with respect to the possibleequilibrium fractions 120596lowast isin [12 1 minus 1119873)

05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast


more stable and they converge to the equilibrium for 120573 gt

096 In particular for 120573 = 0877 we have that the equilibriumfraction is 120596

lowast= 078 and the equilibrium strategy is 119902

lowast

2 =

9265 for which the eigenvalues of 119869119866(119902lowast

2 120596lowast) are now 1205821 =

minus1032 and 1205822 = 0123 which gives an unstable dynamicConversely for 120573 = 0962 we have that the equilibriumfraction is 120596


lowast

2 =


2 120596lowast) are now 1205821 =

minus099 and 1205822 = 0127 which means that the equilibrium isstable It is interesting to look also at the profit differentialevolution which for small values of 120573 oscillates betweenpositive andnegative values while for larger values it is alwaysnegative

This allows us to say that referring to particular param-eter sets different behaviors with respect to 120573 are possibleIn particular the neutral and stabilizing roles of 120573 are veryinteresting as in the existing literature evolutionary pressureusually introduces instability The discriminating factor is

for us the net profit differential When the marginal costsare identical as in the previous section and in [30] theequilibrium profits are the same and the net profit is favorableto the best response firms due to the informational costsof the rational ones Hence rational firms are inclinedto switch to the best response mechanism and this shiftsthe oligopoly composition towards unstable configurationsespecially when the evolutionary pressure is sufficiently largeProfits of the rational firms are smaller than those of thebest response firms also when 1198881 lt 1198882 but the difference issufficiently small and when 1198881 gt 1198882 Conversely when 1198881 lt 1198882and the difference is sufficiently large we have 1205871(119902

lowast

2 120596lowast) minus

119862 gt 1205872(119902lowast

2 120596lowast) so that the best response firms switch to

the more profitable rational mechanism and this can bothintroduce or not affect stability

5 Conclusions

The general setting we considered allowed us to showthat the oligopoly size the number of boundedly rationalplayers and the evolutionary pressure do not always havea destabilizing role Even if the economic setting and theconsidered behavioral rules are the same as in [30] themodelwe dealt with gave us the possibility to investigate a widerange of scenarios which are not present in [30] Indeedthe explicit introduction of the oligopoly size in the modelallowed us to analyze in an heterogeneous context the effectsof the firms number on the stability of the equilibrium Wefound a general confirmation of a behavior which is wellknown in the homogeneous oligopoly framework that isincreasing the firms number 119873 may lead the equilibrium toinstability (see [2 3] for the linear context and for nonlinearsettings [12 45 46] where isoelastic demand functions aretaken into account) However we showed that the presenceof both rational and naive firms can give rise to oligopolycompositions which are stable for any oligopoly size if thefraction of ldquostabilizingrdquo rational firms is sufficiently largeSimilarly we showed that even if in general replacing rationalfirms with best response firms (ie decreasing 120596) introducesinstability for particular parameter configurations all the(heterogeneous) oligopoly compositions of a fixed size arestable Hence in the linear context analyzed in the presentpaper the role of 120596 is (nearly) intuitive that is increasing thenumber of rational agents stabilizes the system or varying 120596

does not alter the stability On the other hand in nonlinearcontexts this may not be true anymore For instance in theworking paper [18] where we deal with an isoelastic demandfunction we find that increasing the number of rationalfirms may destabilize the system under suitable conditionson marginal costs It would be also interesting to analyzenonlinear frameworks withmultiple Nash equilibria in orderto investigate the role of 120596 in such a different context as well

The results recalled above are obtained in the presentpaper for the fixed fraction framework in which we provedthat stability is not affected by cost differences among thefirms Conversely regarding the evolutionary fraction settingwe showed that cost differences may be relevant for stabilityIn fact even if for identical marginal costs we analytically


0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587

Figure 10 Bifurcation diagrams related to the stabilizing scenario for 120573 of quantities 1199021 and 1199022 of fraction 120596 and of the net profit differentialΔ120587 The initial composition corresponding to 120573 = 0 is unstable but since the marginal costs are sufficiently favorable to the rational firmsincreasing the propensity to switch stabilizes the dynamics

proved the destabilizing role of the evolutionary pressure andinformational costs in agreement with the results in [30]we showed that the stability can be improved or at leastpreserved by the intensity of switching if rational firms areefficient enough with respect to the best response firms Wehave been able to prove this behavior only for particularparameter choices due to the high complexity of the analyticcomputations This last scenario definitely requires furtherinvestigations in order to extend the results to a widerrange of parameters and to test what happens in differenteconomic settings considering for example a nonlineardemand function Other generalizations may concern theinvolved decisional rules taking into account agents withfurther reduced rationality heuristics which are not based ona best response mechanism or evenmore complex scenarioswhere more than two behavioral rules are considered orwhere the firms adopting the same behavioral rules are notidentical also in viewof facing empirical and practicalmarketcontexts

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewer for the helpfuland valuable comments

References

[1] A A Cournot Reserches sur les Principles Mathematiques de laTheorie des Richesses Hachette Paris France 1838

[2] T F Palander ldquoKonkurrens och marknadsjamvikt vid duopoloch oligopolrdquo Ekonomisk Tidskrift vol 41 pp 124ndash250 1939

[3] R D Theocharis ldquoOn the stability of the Cournot solution onthe oligopoly problemrdquoThe Review of Economic Studies vol 27no 2 pp 133ndash134 1960


[4] M Lampart ldquoStability of the Cournot equilibrium for aCournot oligopoly model with n competitorsrdquo Chaos Solitonsamp Fractals vol 45 no 9-10 pp 1081ndash1085 2012

[5] A Matsumoto and F Szidarovszky ldquoStability bifurcation andchaos in N-firm nonlinear Cournot gamesrdquo Discrete Dynamicsin Nature and Society vol 2011 Article ID 380530 22 pages2011

[6] A KNaimzada and F Tramontana ldquoControlling chaos throughlocal knowledgerdquo Chaos Solitons amp Fractals vol 42 no 4 pp2439ndash2449 2009

[7] G I Bischi C Chiarella M Kopel and F Szidarovszky Non-linear Oligopolies Stability and Bifurcations Springer BerlinGermany 2009

[8] H N Agiza and A A Elsadany ldquoNonlinear dynamics in theCournot duopoly game with heterogeneous playersrdquo Physica AStatistical Mechanics and its Applications vol 320 no 1ndash4 pp512ndash524 2003

[9] HN Agiza andAA Elsadany ldquoChaotic dynamics in nonlinearduopoly game with heterogeneous playersrdquo Applied Mathemat-ics and Computation vol 149 no 3 pp 843ndash860 2004

[10] HNAgiza A SHegazi andAA Elsadany ldquoComplex dynam-ics and synchronization of a duopoly game with boundedrationalityrdquoMathematics and Computers in Simulation vol 58no 2 pp 133ndash146 2002

[11] H N Agiza E M Elabbasy and A A Elsadany ldquoComplexdynamics and chaos control of heterogeneous quadropolygamerdquo Applied Mathematics and Computation vol 219 no 24pp 11110ndash11118 2013

[12] E Ahmed and H N Agiza ldquoDynamics of a Cournot game withn-competitorsrdquoChaos Solitonsamp Fractals vol 9 no 9 pp 1513ndash1517 1998

[13] N Angelini R Dieci and F Nardini ldquoBifurcation analysisof a dynamic duopoly model with heterogeneous costs andbehavioural rulesrdquo Mathematics and Computers in Simulationvol 79 no 10 pp 3179ndash3196 2009

[14] G I Bischi and A Naimzada ldquoGlobal analysis of a dynamicduopoly game with bounded rationalityrdquo in Advances inDynamic Games and Applications J Filar V Gaitsgory and KMizukami Eds vol 5 of Annals of the International Society ofDynamic Games pp 361ndash385 Birkhauser Boston Mass USA2000

[15] J S Canovas T Puu and M Ruiz ldquoThe Cournot-Theocharisproblem reconsideredrdquo Chaos Solitons amp Fractals vol 37 no 4pp 1025ndash1039 2008

[16] F Cavalli and A Naimzada ldquoA Cournot duopoly game withheterogeneous players nonlinear dynamics of the gradient ruleversus local monopolistic approachrdquo Applied Mathematics andComputation vol 249 pp 382ndash388 2014

[17] F Cavalli and A Naimzada ldquoNonlinear dynamics and con-vergence speed of heterogeneous Cournot duopolies involvingbest response mechanisms with different degrees of rationalityrdquoNonlinear Dynamics 2015

[18] F Cavalli A Naimzada and M Pireddu ldquoHeterogeneity andthe (de)stabilizing role of rationalityrdquo To appear in ChaosSolitons amp Fractals

[19] F Cavalli ANaimzada and F Tramontana ldquoNonlinear dynam-ics and global analysis of a heterogeneous Cournot duopolywith a local monopolistic approach versus a gradient rule withendogenous reactivityrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 23 no 1ndash3 pp 245ndash262 2015

[20] E M Elabbasy H N Agiza and A A Elsadany ldquoAnalysis ofnonlinear triopoly game with heterogeneous playersrdquo Comput-ers ampMathematics with Applications vol 57 no 3 pp 488ndash4992009

[21] W Ji ldquoChaos and control of game model based on het-erogeneous expectations in electric power triopolyrdquo DiscreteDynamics in Nature and Society vol 2009 Article ID 469564 8pages 2009

[22] T Li and J Ma ldquoThe complex dynamics of RampD competitionmodels of three oligarchs with heterogeneous playersrdquo Nonlin-ear Dynamics vol 74 no 1-2 pp 45ndash54 2013

[23] A Matsumoto ldquoStatistical dynamics in piecewise linearCournot game with heterogeneous duopolistsrdquo InternationalGameTheory Review vol 6 no 3 pp 295ndash321 2004

[24] A K Naimzada and L Sbragia ldquoOligopoly games with non-linear demand and cost functions two boundedly rationaladjustment processesrdquo Chaos Solitons amp Fractals vol 29 no3 pp 707ndash722 2006

[25] X Pu and J Ma ldquoComplex dynamics and chaos control innonlinear four-oligopolist game with different expectationsrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 23 no 3 Article ID 1350053 15pages 2013

[26] F Tramontana ldquoHeterogeneous duopoly with isoelasticdemand functionrdquo Economic Modelling vol 27 no 1 pp350ndash357 2010

[27] J Tuinstra ldquoA price adjustment process in a model of monop-olistic competitionrdquo International Game Theory Review vol 6no 3 pp 417ndash442 2004

[28] M Anufriev D Kopanyi and J Tuinstra ldquoLearning cyclesin Bertrand competition with differentiated commodities andcompeting learning rulesrdquo Journal of Economic Dynamics andControl vol 37 no 12 pp 2562ndash2581 2013

[29] G I Bischi F Lamantia andD Radi ldquoAn evolutionary Cournotmodel with limited market knowledgerdquo To appear in Journal ofEconomic Behavior amp Organization

[30] E Droste C Hommes and J Tuinstra ldquoEndogenous fluctu-ations under evolutionary pressure in Cournot competitionrdquoGames and Economic Behavior vol 40 no 2 pp 232ndash269 2002

[31] J-G Du Y-Q Fan Z-H Sheng and Y-Z Hou ldquoDynamicsanalysis and chaos control of a duopoly game with heteroge-neous players and output limiterrdquo Economic Modelling vol 33pp 507ndash516 2013

[32] A Naimzada and M Pireddu ldquoDynamics in a nonlinearKeynesian good market modelrdquo Chaos vol 24 no 1 Article ID013142 2014

[33] A Naimzada and M Pireddu ldquoDynamic behavior of productand stock markets with a varying degree of interactionrdquo Eco-nomic Modelling vol 41 pp 191ndash197 2014

[34] W A Brock and C H Hommes ldquoA rational route to random-nessrdquo Econometrica vol 65 no 5 pp 1059ndash1095 1997

[35] A Banerjee and J W Weibull ldquoNeutrally stable outcomes incheap-talk coordination gamesrdquoGames andEconomic Behaviorvol 32 no 1 pp 1ndash24 2000

[36] D Gale and R W Rosenthal ldquoExperimentation imitation andstochastic stabilityrdquo Journal of Economic Theory vol 84 no 1pp 1ndash40 1999

[37] X-Z He and F H Westerhoff ldquoCommodity markets pricelimiters and speculative price dynamicsrdquo Journal of EconomicDynamics and Control vol 29 no 9 pp 1577ndash1596 2005


[38] Z Sheng J DuQMei andTHuang ldquoNew analyses of duopolygamewith output lower limitersrdquoAbstract and Applied Analysisvol 2013 Article ID 406743 10 pages 2013

[39] C Wagner and R Stoop ldquoOptimized chaos control with simplelimitersrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 63 Article ID 017201 2001

[40] C Wieland and F H Westerhoff ldquoExchange rate dynamicscentral bank interventions and chaos control methodsrdquo Journalof Economic Behavior amp Organization vol 58 no 1 pp 117ndash1322005

[41] G I Bischi R Dieci G Rodano and E Saltari ldquoMultipleattractors and global bifurcations in a Kaldor-type businesscycle modelrdquo Journal of Evolutionary Economics vol 11 no 5pp 527ndash554 2001

[42] J K Goeree C Hommes and C Weddepohl ldquoStability andcomplex dynamics in a discrete tatonnement modelrdquo Journal ofEconomic BehaviorampOrganization vol 33 no 3-4 pp 395ndash4101998

[43] J K Hale and H Kocak Dynamics and Bifurcations vol 3 ofTexts in Applied Mathematics Springer New York NY USA1991

[44] E I Jury Theory and Application of the z-Transform MethodJohn Wiley amp Sons New York NY USA 1964

[45] H N Agiza ldquoExplicit stability zones for Cournot games with 3and 4 competitorsrdquo Chaos Solitons amp Fractals vol 9 no 12 pp1955ndash1966 1998

[46] T Puu ldquoOn the stability of Cournot equilibrium when thenumber of competitors increasesrdquo Journal of Economic Behaviorand Organization vol 66 no 3-4 pp 445ndash456 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a different attractor To that research strand belong forinstance the contributions by Agiza and Elsadany [8 9] byAgiza et al [10 11] by Ahmed and Agiza [12] by Angelini etal [13] by Bischi and Naimzada [14] by Canovas et al [15]by Cavalli and Naimzada [16 17] by Cavalli et al [18 19]by Elabbasy et al [20] by Ji [21] by Li and Ma [22] byMatsumoto [23] by Naimzada and Sbragia [24] by Pu andMa [25] by Tramontana [26] and by Tuinstra [27]

Ourwork aims to investigate the role of oligopoly size andcomposition for heterogeneous competitions In particularwe consider an economy characterized by linear demandfunction and we assume linear total costs for the firmsThe heterogeneity we study is in terms of the degree ofrationality of the firms as we consider two different kindsof informational endowments Both the firms decide theirproduction level using a best reply mechanism in order tomaximize their profits with respect to the production levelsof all their competitors However rational firms have fullinformational and computational capabilities to solve suchoptimization problem and are endowed with perfect foresightso that they exactly know their competitors next periodstrategies On the other hand naive best response firms arenot able to predict such strategies and thus they assumethat their opponents next time production levels remainthe same as in the current period that is they have staticexpectations Finally we allow each group of firms to havedifferent marginal costs The approach of considering themarket split into two groups of firms identical within eachgroup is common in the oligopoly literature as shown forinstance by the works of Anufriev et al [28] Bischi et al[29] Cavalli et al [18] Droste et al [30] and Matsumotoand Szidarovszky [5] The theoretical rationale for such anapproach is that although it concerns one of the simplestpossible forms of heterogeneity among firms besides keepingthe study analytically tractable it allows obtaining interestingresults for example with respect to equilibrium stability andchaotic dynamics which can be straightforwardly interpretedand linked to the heterogeneity in rationality

In the adjustment mechanism of the less rational bestresponse firms because of capacity and financial constraintswe introduce a constraint on the strategy variation sothat firms can not immediately increase or decrease theirproduction level by arbitrarily large quantities but they adaptit towards best reply strategy with a certain reaction speed Tothis end we introduce a limiting mechanism similar to thatused in [31] in microeconomic frameworks and in [32 33] inmacroeconomic settings On the contrary due to the highdegree of rationality of the rational firms we assume thatthey are able to freely modify their production level in everyperiod

We study two different settings considering the case offixed fractions in which the behavioral rules are exogenouslyassigned and do not change and that of evolutionary frac-tions in which the firms can switch between the differentmechanisms on the basis of the realized performances of thefirms To exploit the competition between all the oligopolistswe adopt a switching mechanism based on the logit rule asproposed by Brock and Hommes in [34] The performancesof the behavioral rules are evaluated taking into account

the profits that the rule allowed achieving in the previousperiod In particular we assign additional fixed informationalcosts to the rational firms as a consequence of their highdegree of rationality The switching between mechanisms isregulated by the so-called evolutionary pressure parameterwhich describes the propensity of a firm to switch to themostprofitable rule

In the existing literature the study of heterogeneousoligopolies of generic size can be found in the works byAnufriev et al [28] Banerjee and Weibull [35] Droste etal [30] and Gale and Rosenthal [36] However only thesetting and the aims of the contribution by Droste et alare similar to those investigated in the present work as inboth papers the economy is characterized by the same lineardemand function and the firms may be both rational andbest response However in [30] it is considered an infinitepopulation of firms that are randomly matched in pairs ateach discrete-time period and that play a symmetric Cournotduopoly game so that the influence of oligopoly size onthe dynamics can not be studied and only the evolutionaryfractions framework is investigatedMoreover no productionlimiters are taken into account and a replicator switchingmechanism is considered which is different from that studiedin our contribution Another substantial difference is thatin [30] firms compute their best response with respect tothe average industry output while in our work all the firmscompete against each other Finally in [30] the firms haveidentical cost functions while in our paper theymay possiblybe different

About the fixed fractionsmodel ourmain results concernthe effects of the oligopoly size and composition and of theevolutionary pressure on the stability of the equilibriumwhile when evolutionary fractions are considered we studythe role of the evolutionary pressure focusing on whathappens with either identical or different marginal costsMoreover for the model with fixed fractions we investigatethe effects of the production limiters and of the reactionspeed In such setting we also show that marginal costs donot affect the local stability of the unique steady state whichcoincideswith theNash equilibrium and that its loss of stabil-ity occurs through a flip bifurcation Furthermore we provethat even if for most configurations the oligopoly size have adestabilizing effect there exist oligopoly compositions whichremain stable as the number of firms increases differentlyfrom the other results about homogeneous oligopolies in therelated literature

For the evolutionary fractions model we prove that thereis a unique equilibrium fraction which in general is notanalytically computable We then show that the effects ofevolutionary pressure can change depending on the marginalcosts differenceWhen the technology of the firms is identicalor when the best response firms are more efficient than therational ones increasing the evolutionary pressure alwaysleads to instability Conversely we provide simulative evi-dence of the fact that if the marginal cost of the rationalfirms is sufficiently smaller than the marginal cost of thebest response firms evolutionary pressure either can haveno effect or may stabilize the dynamics Such results whichare made possible by studying the effect of different marginal









119901 (119876) = 119886 minus 119887119876 (1)

where 119876 = sum119873


119862 (119902119894) = 119888119894119902119894 (2)






120587119894= 119902119894 (119886 minus 119887119876) minus 119888

119894119902119894 119894 = 1 119873 (3)


Proposition 1 Let

119886



119902⋆


119887 (119873 + 1) (5a)

119902⋆

2 =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1) (5b)

where 119902⋆


⋆




(6)




2119887 (7)



2119887 (8)


119902⋆

1119894 = 119902⋆

1

119902⋆

2119896 = 119902⋆

2 (9)

and consequently

119876⋆

1 = 120596119873119902⋆

1

119876⋆

2 = (1minus120596)119873119902⋆

2 (10)



119902lowast

2 if and only if 1198882 gt 1198881





1205871119894 = 1199021119894119905+1 (119886 minus 119887 (1198761minus119894119905+1 + 1199021119894119905+1 +1198762119905+1))

minus 11988811199021119894119905+1(11)




1199021119905+1 =119886 minus 1198881 minus 1198871198762119905+1

119887 (120596119873 + 1) (12)



2119887 (13)





1199022119896119905+1 =119886 minus 1198882 minus 119887119876

minus119896119905

2119887 (14)


lt (119886 minus 1198882)119887



1199021119905 = 1199021119894119905

1199022119905 = 1199022119896119905(15)




119887 (120596119873 + 1) (16)


2119887 (17)



2119887 (18)


1199022119905+1

= minus 1198871199022119905

2119887(minus119887120596 (1 minus 120596)119873

2

119887 (120596119873 + 1)+ (1minus120596)119873minus 1)


2119887

= minus1199022119905

2(minus120596 (1 minus 120596)119873

2

(120596119873 + 1)+ (1minus120596)119873minus 1)


2119887


= minus1199022119905

2(minus120596 (1 minus 120596)119873

2+ ((1 minus 120596)119873 minus 1) (120596119873 + 1)(120596119873 + 1)

)


2119887 (120596119873 + 1)

= minus1199022119905

2((1 minus 120596)119873 minus (120596119873 + 1)

(120596119873 + 1))

+(119886 minus 1198882) + 120596119873 (1198881 minus 1198882)

2119887 (120596119873 + 1)

=1199022119905

2(2120596119873 + 2 minus 1 minus 119873

(120596119873 + 1)) minus


2119887 (120596119873 + 1)

(19)

which gives

1199022119905+1 = 119891 (1199022119905)

= (1minus(1 + 119873)

2 (1 + 120596119873)) 1199022119905


2119887 (120596119873 + 1)

(20)



1199022119905+1 = 1199022119905 +119892 (120574 (119891 (1199022119905) minus 1199022119905)) (21)


119892 (119909) = 1198862 (1198861 + 1198862

1198861119890minus119909 + 1198862

minus 1) (22)





120593 (1199022) = 1199022 +119892 (120574 (119891 (1199022) minus 1199022)) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

(23)






⋆





12057411988611198862 (1 + 119873)

4 (1198861 + 1198862) (1 + 120596119873)lt 1 (24)



1205931015840(119902⋆

2 ) = 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596119873) (25)

Since 1205931015840(119902⋆




⋆

2 = (119886 minus 1198882 +

119873120596(1198881 minus 1198882))(119887(119873 + 1)) when 12057411988611198862(1 + 119873)(4(1198861 + 1198862)(1 +

120596119873)) = 1


lowast







120574 lt4 (1198861 + 1198862) (1 + 120596119873)

11988611198862 (1 + 119873) (26)


06275


120596 gt (120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) 1119873

(27)


(120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) gt 1 (28)

or equivalently

119873 lt81198861 + 81198862 minus 11988611198862120574

11988611198862120574 (29)






2 isnever stable

(2) if 12057411988611198862 minus 4120596(1198861 + 1198862) lt 0 then 119902⋆




119873 lt4 (1198861 + 1198862) minus 12057411988611198862

12057411988611198862 minus 4120596 (1198861 + 1198862) (30)






0 02 04 06 08 14

6

8

10

12

14

120574

q1

0 02 04 06 08 1

48

5

52

54

56

58

6

120574

q2


005 01 015 020

5

10

15

20

005 01 015 02

35

4

45

5

55

q1

q2

120596 120596


10 15 20 25 30 35 40 450

5

10

15

20

N

10 15 20 25 30 35 40 450

5

10

15

20

N

q1

q2





(2) if (1 + 119873)1198862120574 minus 4(1 + 120596119873) gt 0 then 119902⋆

2 is stable for

1198861 lt41198862 (1 + 120596119873)

(1 + 119873) 1198862120574 minus 4 (1 + 120596119873) (31)

(3) if (1+119873)1198861120574minus4(1+120596119873) lt 0 then 119902⋆


⋆

2 is stable for

1198862 lt41198861 (1 + 120596119873)

(1 + 119873) 1198861120574 minus 4 (1 + 120596119873) (32)







119905the variable


119911 (119909 119910) =exp (120573119909)

exp (120573119909) + exp (120573119910) (33)





+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596


minus 1)

120596119905+1 =

1119873

+119873 minus 2119873

11 + 119890120573Δ120587119905

(34)


Δ120587119905= 1205872119905 minus1205871119905


119905minus 1198881) +119862

(35)

with


119905)1198731199022119905

119887 (120596119905119873 + 1)

(36)

and thus

119876119905=

119873 (120596119905(119886 minus 1198881) + 1199022119905119887 (1 minus 120596

119905))

119887 (120596119905119873 + 1)

(37)












1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1


2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2



119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)


2 minus120587lowast

1 = 119902lowast


lowast


120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)


1 = 119902lowast




lowast


function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)





119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119901 (119876) = 119886 minus 119887119876 (1)

where 119876 = sum119873


119862 (119902119894) = 119888119894119902119894 (2)






120587119894= 119902119894 (119886 minus 119887119876) minus 119888

119894119902119894 119894 = 1 119873 (3)


Proposition 1 Let

119886



119902⋆


119887 (119873 + 1) (5a)

119902⋆

2 =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1) (5b)

where 119902⋆


⋆




(6)




2119887 (7)



2119887 (8)


119902⋆

1119894 = 119902⋆

1

119902⋆

2119896 = 119902⋆

2 (9)

and consequently

119876⋆

1 = 120596119873119902⋆

1

119876⋆

2 = (1minus120596)119873119902⋆

2 (10)



119902lowast

2 if and only if 1198882 gt 1198881





1205871119894 = 1199021119894119905+1 (119886 minus 119887 (1198761minus119894119905+1 + 1199021119894119905+1 +1198762119905+1))

minus 11988811199021119894119905+1(11)




1199021119905+1 =119886 minus 1198881 minus 1198871198762119905+1

119887 (120596119873 + 1) (12)



2119887 (13)





1199022119896119905+1 =119886 minus 1198882 minus 119887119876

minus119896119905

2119887 (14)


lt (119886 minus 1198882)119887



1199021119905 = 1199021119894119905

1199022119905 = 1199022119896119905(15)




119887 (120596119873 + 1) (16)


2119887 (17)



2119887 (18)


1199022119905+1

= minus 1198871199022119905

2119887(minus119887120596 (1 minus 120596)119873

2

119887 (120596119873 + 1)+ (1minus120596)119873minus 1)


2119887

= minus1199022119905

2(minus120596 (1 minus 120596)119873

2

(120596119873 + 1)+ (1minus120596)119873minus 1)


2119887


= minus1199022119905

2(minus120596 (1 minus 120596)119873

2+ ((1 minus 120596)119873 minus 1) (120596119873 + 1)(120596119873 + 1)

)


2119887 (120596119873 + 1)

= minus1199022119905

2((1 minus 120596)119873 minus (120596119873 + 1)

(120596119873 + 1))

+(119886 minus 1198882) + 120596119873 (1198881 minus 1198882)

2119887 (120596119873 + 1)

=1199022119905

2(2120596119873 + 2 minus 1 minus 119873

(120596119873 + 1)) minus


2119887 (120596119873 + 1)

(19)

which gives

1199022119905+1 = 119891 (1199022119905)

= (1minus(1 + 119873)

2 (1 + 120596119873)) 1199022119905


2119887 (120596119873 + 1)

(20)



1199022119905+1 = 1199022119905 +119892 (120574 (119891 (1199022119905) minus 1199022119905)) (21)


119892 (119909) = 1198862 (1198861 + 1198862

1198861119890minus119909 + 1198862

minus 1) (22)





120593 (1199022) = 1199022 +119892 (120574 (119891 (1199022) minus 1199022)) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

(23)






⋆





12057411988611198862 (1 + 119873)

4 (1198861 + 1198862) (1 + 120596119873)lt 1 (24)



1205931015840(119902⋆

2 ) = 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596119873) (25)

Since 1205931015840(119902⋆




⋆

2 = (119886 minus 1198882 +

119873120596(1198881 minus 1198882))(119887(119873 + 1)) when 12057411988611198862(1 + 119873)(4(1198861 + 1198862)(1 +

120596119873)) = 1


lowast







120574 lt4 (1198861 + 1198862) (1 + 120596119873)

11988611198862 (1 + 119873) (26)


06275


120596 gt (120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) 1119873

(27)


(120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) gt 1 (28)

or equivalently

119873 lt81198861 + 81198862 minus 11988611198862120574

11988611198862120574 (29)






2 isnever stable

(2) if 12057411988611198862 minus 4120596(1198861 + 1198862) lt 0 then 119902⋆




119873 lt4 (1198861 + 1198862) minus 12057411988611198862

12057411988611198862 minus 4120596 (1198861 + 1198862) (30)






0 02 04 06 08 14

6

8

10

12

14

120574

q1

0 02 04 06 08 1

48

5

52

54

56

58

6

120574

q2


005 01 015 020

5

10

15

20

005 01 015 02

35

4

45

5

55

q1

q2

120596 120596


10 15 20 25 30 35 40 450

5

10

15

20

N

10 15 20 25 30 35 40 450

5

10

15

20

N

q1

q2





(2) if (1 + 119873)1198862120574 minus 4(1 + 120596119873) gt 0 then 119902⋆

2 is stable for

1198861 lt41198862 (1 + 120596119873)

(1 + 119873) 1198862120574 minus 4 (1 + 120596119873) (31)

(3) if (1+119873)1198861120574minus4(1+120596119873) lt 0 then 119902⋆


⋆

2 is stable for

1198862 lt41198861 (1 + 120596119873)

(1 + 119873) 1198861120574 minus 4 (1 + 120596119873) (32)







119905the variable


119911 (119909 119910) =exp (120573119909)

exp (120573119909) + exp (120573119910) (33)





+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596


minus 1)

120596119905+1 =

1119873

+119873 minus 2119873

11 + 119890120573Δ120587119905

(34)


Δ120587119905= 1205872119905 minus1205871119905


119905minus 1198881) +119862

(35)

with


119905)1198731199022119905

119887 (120596119905119873 + 1)

(36)

and thus

119876119905=

119873 (120596119905(119886 minus 1198881) + 1199022119905119887 (1 minus 120596

119905))

119887 (120596119905119873 + 1)

(37)












1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1


2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2



119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)


2 minus120587lowast

1 = 119902lowast


lowast


120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)


1 = 119902lowast




lowast


function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)





119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1205871119894 = 1199021119894119905+1 (119886 minus 119887 (1198761minus119894119905+1 + 1199021119894119905+1 +1198762119905+1))

minus 11988811199021119894119905+1(11)




1199021119905+1 =119886 minus 1198881 minus 1198871198762119905+1

119887 (120596119873 + 1) (12)



2119887 (13)





1199022119896119905+1 =119886 minus 1198882 minus 119887119876

minus119896119905

2119887 (14)


lt (119886 minus 1198882)119887



1199021119905 = 1199021119894119905

1199022119905 = 1199022119896119905(15)




119887 (120596119873 + 1) (16)


2119887 (17)



2119887 (18)


1199022119905+1

= minus 1198871199022119905

2119887(minus119887120596 (1 minus 120596)119873

2

119887 (120596119873 + 1)+ (1minus120596)119873minus 1)


2119887

= minus1199022119905

2(minus120596 (1 minus 120596)119873

2

(120596119873 + 1)+ (1minus120596)119873minus 1)


2119887


= minus1199022119905

2(minus120596 (1 minus 120596)119873

2+ ((1 minus 120596)119873 minus 1) (120596119873 + 1)(120596119873 + 1)

)


2119887 (120596119873 + 1)

= minus1199022119905

2((1 minus 120596)119873 minus (120596119873 + 1)

(120596119873 + 1))

+(119886 minus 1198882) + 120596119873 (1198881 minus 1198882)

2119887 (120596119873 + 1)

=1199022119905

2(2120596119873 + 2 minus 1 minus 119873

(120596119873 + 1)) minus


2119887 (120596119873 + 1)

(19)

which gives

1199022119905+1 = 119891 (1199022119905)

= (1minus(1 + 119873)

2 (1 + 120596119873)) 1199022119905


2119887 (120596119873 + 1)

(20)



1199022119905+1 = 1199022119905 +119892 (120574 (119891 (1199022119905) minus 1199022119905)) (21)


119892 (119909) = 1198862 (1198861 + 1198862

1198861119890minus119909 + 1198862

minus 1) (22)





120593 (1199022) = 1199022 +119892 (120574 (119891 (1199022) minus 1199022)) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

(23)






⋆





12057411988611198862 (1 + 119873)

4 (1198861 + 1198862) (1 + 120596119873)lt 1 (24)



1205931015840(119902⋆

2 ) = 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596119873) (25)

Since 1205931015840(119902⋆




⋆

2 = (119886 minus 1198882 +

119873120596(1198881 minus 1198882))(119887(119873 + 1)) when 12057411988611198862(1 + 119873)(4(1198861 + 1198862)(1 +

120596119873)) = 1


lowast







120574 lt4 (1198861 + 1198862) (1 + 120596119873)

11988611198862 (1 + 119873) (26)


06275


120596 gt (120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) 1119873

(27)


(120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) gt 1 (28)

or equivalently

119873 lt81198861 + 81198862 minus 11988611198862120574

11988611198862120574 (29)






2 isnever stable

(2) if 12057411988611198862 minus 4120596(1198861 + 1198862) lt 0 then 119902⋆




119873 lt4 (1198861 + 1198862) minus 12057411988611198862

12057411988611198862 minus 4120596 (1198861 + 1198862) (30)






0 02 04 06 08 14

6

8

10

12

14

120574

q1

0 02 04 06 08 1

48

5

52

54

56

58

6

120574

q2


005 01 015 020

5

10

15

20

005 01 015 02

35

4

45

5

55

q1

q2

120596 120596


10 15 20 25 30 35 40 450

5

10

15

20

N

10 15 20 25 30 35 40 450

5

10

15

20

N

q1

q2





(2) if (1 + 119873)1198862120574 minus 4(1 + 120596119873) gt 0 then 119902⋆

2 is stable for

1198861 lt41198862 (1 + 120596119873)

(1 + 119873) 1198862120574 minus 4 (1 + 120596119873) (31)

(3) if (1+119873)1198861120574minus4(1+120596119873) lt 0 then 119902⋆


⋆

2 is stable for

1198862 lt41198861 (1 + 120596119873)

(1 + 119873) 1198861120574 minus 4 (1 + 120596119873) (32)







119905the variable


119911 (119909 119910) =exp (120573119909)

exp (120573119909) + exp (120573119910) (33)





+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596


minus 1)

120596119905+1 =

1119873

+119873 minus 2119873

11 + 119890120573Δ120587119905

(34)


Δ120587119905= 1205872119905 minus1205871119905


119905minus 1198881) +119862

(35)

with


119905)1198731199022119905

119887 (120596119905119873 + 1)

(36)

and thus

119876119905=

119873 (120596119905(119886 minus 1198881) + 1199022119905119887 (1 minus 120596

119905))

119887 (120596119905119873 + 1)

(37)












1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1


2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2



119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)


2 minus120587lowast

1 = 119902lowast


lowast


120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)


1 = 119902lowast




lowast


function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)





119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= minus1199022119905

2(minus120596 (1 minus 120596)119873

2+ ((1 minus 120596)119873 minus 1) (120596119873 + 1)(120596119873 + 1)

)


2119887 (120596119873 + 1)

= minus1199022119905

2((1 minus 120596)119873 minus (120596119873 + 1)

(120596119873 + 1))

+(119886 minus 1198882) + 120596119873 (1198881 minus 1198882)

2119887 (120596119873 + 1)

=1199022119905

2(2120596119873 + 2 minus 1 minus 119873

(120596119873 + 1)) minus


2119887 (120596119873 + 1)

(19)

which gives

1199022119905+1 = 119891 (1199022119905)

= (1minus(1 + 119873)

2 (1 + 120596119873)) 1199022119905


2119887 (120596119873 + 1)

(20)



1199022119905+1 = 1199022119905 +119892 (120574 (119891 (1199022119905) minus 1199022119905)) (21)


119892 (119909) = 1198862 (1198861 + 1198862

1198861119890minus119909 + 1198862

minus 1) (22)





120593 (1199022) = 1199022 +119892 (120574 (119891 (1199022) minus 1199022)) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

(23)






⋆





12057411988611198862 (1 + 119873)

4 (1198861 + 1198862) (1 + 120596119873)lt 1 (24)



1205931015840(119902⋆

2 ) = 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596119873) (25)

Since 1205931015840(119902⋆




⋆

2 = (119886 minus 1198882 +

119873120596(1198881 minus 1198882))(119887(119873 + 1)) when 12057411988611198862(1 + 119873)(4(1198861 + 1198862)(1 +

120596119873)) = 1


lowast







120574 lt4 (1198861 + 1198862) (1 + 120596119873)

11988611198862 (1 + 119873) (26)


06275


120596 gt (120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) 1119873

(27)


(120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) gt 1 (28)

or equivalently

119873 lt81198861 + 81198862 minus 11988611198862120574

11988611198862120574 (29)






2 isnever stable

(2) if 12057411988611198862 minus 4120596(1198861 + 1198862) lt 0 then 119902⋆




119873 lt4 (1198861 + 1198862) minus 12057411988611198862

12057411988611198862 minus 4120596 (1198861 + 1198862) (30)






0 02 04 06 08 14

6

8

10

12

14

120574

q1

0 02 04 06 08 1

48

5

52

54

56

58

6

120574

q2


005 01 015 020

5

10

15

20

005 01 015 02

35

4

45

5

55

q1

q2

120596 120596


10 15 20 25 30 35 40 450

5

10

15

20

N

10 15 20 25 30 35 40 450

5

10

15

20

N

q1

q2





(2) if (1 + 119873)1198862120574 minus 4(1 + 120596119873) gt 0 then 119902⋆

2 is stable for

1198861 lt41198862 (1 + 120596119873)

(1 + 119873) 1198862120574 minus 4 (1 + 120596119873) (31)

(3) if (1+119873)1198861120574minus4(1+120596119873) lt 0 then 119902⋆


⋆

2 is stable for

1198862 lt41198861 (1 + 120596119873)

(1 + 119873) 1198861120574 minus 4 (1 + 120596119873) (32)







119905the variable


119911 (119909 119910) =exp (120573119909)

exp (120573119909) + exp (120573119910) (33)





+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596


minus 1)

120596119905+1 =

1119873

+119873 minus 2119873

11 + 119890120573Δ120587119905

(34)


Δ120587119905= 1205872119905 minus1205871119905


119905minus 1198881) +119862

(35)

with


119905)1198731199022119905

119887 (120596119905119873 + 1)

(36)

and thus

119876119905=

119873 (120596119905(119886 minus 1198881) + 1199022119905119887 (1 minus 120596

119905))

119887 (120596119905119873 + 1)

(37)












1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1


2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2



119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)


2 minus120587lowast

1 = 119902lowast


lowast


120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)


1 = 119902lowast




lowast


function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)





119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205931015840(119902⋆

2 ) = 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596119873) (25)

Since 1205931015840(119902⋆




⋆

2 = (119886 minus 1198882 +

119873120596(1198881 minus 1198882))(119887(119873 + 1)) when 12057411988611198862(1 + 119873)(4(1198861 + 1198862)(1 +

120596119873)) = 1


lowast







120574 lt4 (1198861 + 1198862) (1 + 120596119873)

11988611198862 (1 + 119873) (26)


06275


120596 gt (120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) 1119873

(27)


(120574 (1 + 119873) 119886111988624 (1198861 + 1198862)

minus 1) gt 1 (28)

or equivalently

119873 lt81198861 + 81198862 minus 11988611198862120574

11988611198862120574 (29)






2 isnever stable

(2) if 12057411988611198862 minus 4120596(1198861 + 1198862) lt 0 then 119902⋆




119873 lt4 (1198861 + 1198862) minus 12057411988611198862

12057411988611198862 minus 4120596 (1198861 + 1198862) (30)






0 02 04 06 08 14

6

8

10

12

14

120574

q1

0 02 04 06 08 1

48

5

52

54

56

58

6

120574

q2


005 01 015 020

5

10

15

20

005 01 015 02

35

4

45

5

55

q1

q2

120596 120596


10 15 20 25 30 35 40 450

5

10

15

20

N

10 15 20 25 30 35 40 450

5

10

15

20

N

q1

q2





(2) if (1 + 119873)1198862120574 minus 4(1 + 120596119873) gt 0 then 119902⋆

2 is stable for

1198861 lt41198862 (1 + 120596119873)

(1 + 119873) 1198862120574 minus 4 (1 + 120596119873) (31)

(3) if (1+119873)1198861120574minus4(1+120596119873) lt 0 then 119902⋆


⋆

2 is stable for

1198862 lt41198861 (1 + 120596119873)

(1 + 119873) 1198861120574 minus 4 (1 + 120596119873) (32)







119905the variable


119911 (119909 119910) =exp (120573119909)

exp (120573119909) + exp (120573119910) (33)





+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596


minus 1)

120596119905+1 =

1119873

+119873 minus 2119873

11 + 119890120573Δ120587119905

(34)


Δ120587119905= 1205872119905 minus1205871119905


119905minus 1198881) +119862

(35)

with


119905)1198731199022119905

119887 (120596119905119873 + 1)

(36)

and thus

119876119905=

119873 (120596119905(119886 minus 1198881) + 1199022119905119887 (1 minus 120596

119905))

119887 (120596119905119873 + 1)

(37)












1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1


2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2



119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)


2 minus120587lowast

1 = 119902lowast


lowast


120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)


1 = 119902lowast




lowast


function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)





119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 14

6

8

10

12

14

120574

q1

0 02 04 06 08 1

48

5

52

54

56

58

6

120574

q2


005 01 015 020

5

10

15

20

005 01 015 02

35

4

45

5

55

q1

q2

120596 120596


10 15 20 25 30 35 40 450

5

10

15

20

N

10 15 20 25 30 35 40 450

5

10

15

20

N

q1

q2





(2) if (1 + 119873)1198862120574 minus 4(1 + 120596119873) gt 0 then 119902⋆

2 is stable for

1198861 lt41198862 (1 + 120596119873)

(1 + 119873) 1198862120574 minus 4 (1 + 120596119873) (31)

(3) if (1+119873)1198861120574minus4(1+120596119873) lt 0 then 119902⋆


⋆

2 is stable for

1198862 lt41198861 (1 + 120596119873)

(1 + 119873) 1198861120574 minus 4 (1 + 120596119873) (32)







119905the variable


119911 (119909 119910) =exp (120573119909)

exp (120573119909) + exp (120573119910) (33)





+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596


minus 1)

120596119905+1 =

1119873

+119873 minus 2119873

11 + 119890120573Δ120587119905

(34)


Δ120587119905= 1205872119905 minus1205871119905


119905minus 1198881) +119862

(35)

with


119905)1198731199022119905

119887 (120596119905119873 + 1)

(36)

and thus

119876119905=

119873 (120596119905(119886 minus 1198881) + 1199022119905119887 (1 minus 120596

119905))

119887 (120596119905119873 + 1)

(37)












1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1


2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2



119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)


2 minus120587lowast

1 = 119902lowast


lowast


120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)


1 = 119902lowast




lowast


function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)





119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(2) if (1 + 119873)1198862120574 minus 4(1 + 120596119873) gt 0 then 119902⋆

2 is stable for

1198861 lt41198862 (1 + 120596119873)

(1 + 119873) 1198862120574 minus 4 (1 + 120596119873) (31)

(3) if (1+119873)1198861120574minus4(1+120596119873) lt 0 then 119902⋆


⋆

2 is stable for

1198862 lt41198861 (1 + 120596119873)

(1 + 119873) 1198861120574 minus 4 (1 + 120596119873) (32)







119905the variable


119911 (119909 119910) =exp (120573119909)

exp (120573119909) + exp (120573119910) (33)





+ 1198862 (1198861 + 1198862

1198861119890120574(((1+119873)(2(1+120596


minus 1)

120596119905+1 =

1119873

+119873 minus 2119873

11 + 119890120573Δ120587119905

(34)


Δ120587119905= 1205872119905 minus1205871119905


119905minus 1198881) +119862

(35)

with


119905)1198731199022119905

119887 (120596119905119873 + 1)

(36)

and thus

119876119905=

119873 (120596119905(119886 minus 1198881) + 1199022119905119887 (1 minus 120596

119905))

119887 (120596119905119873 + 1)

(37)












1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1


2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2



119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)


2 minus120587lowast

1 = 119902lowast


lowast


120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)


1 = 119902lowast




lowast


function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)





119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 9 1032

33

34

35

36

37

38

39

1 2 3 4 5 6 7 8 9 10

365

37

375

38

385

39

395

40

405

q1 q2

a1 a1


2 4 6 8 10 12

36

37

38

39

40

41

42

43

44

2 4 6 8 10 12

335

34

345

35

355

36

365

37

375

38

385

q1

q2

a2 a2



119902lowast

1 = 119902lowast

2 =119886 minus 119888

119887 (119873 + 1) (38)


2 minus120587lowast

1 = 119902lowast


lowast


120596lowast=

1119873

(1+119873 minus 21 + 119890120573119862

) (39)


1 = 119902lowast




lowast


function

119866 (0 +infin)times [1119873

119873 minus 1119873

] 997888rarr Rtimes(1119873

119873 minus 1119873

)

(1199022 120596) 997891997888rarr (1198661 (1199022 120596) 1198662 (1199022 120596))

1198661 (1199022 120596) = 1199022

+ 1198862 (1198861 + 1198862


minus 1)

1198662 (1199022 120596) =1119873

(1+119873 minus 21 + 119890120573Δ120587

)

(40)





119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119869119866(119902lowast

2 120596lowast) =

[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)0

11986921 0

]]]

]

(42)

where 11986921 = 12059711990221198662(119902lowast



1minus det 119869119866(119902lowast

2 120596lowast) = 1 gt 0

1+ tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1+ 1minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

1minus tr 119869119866(119902lowast

2 120596lowast) + det 119869

119866(119902lowast

2 120596lowast)

= 1minus 1+12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)gt 0

(43)


119890120573119862

lt4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862) (44)


4 (1198861 + 1198862) (119873 + 2) minus 212057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)lt 0 (45)


120574 =8 (1198861 + 1198862)

11988611198862 (1 + 119873)

120574 =2 (1198861 + 1198862) (119873 + 2)

11988611198862 (1 + 119873)

(46)



120573119862 lt log(4 (1198861 + 1198862)119873 minus 12057411988611198862 (1 + 119873)

11988611198862 (1 + 119873) 120574 minus 8 (1198861 + 1198862)) (47)






⋆)


119902⋆

2 =119886 minus 1198882 + 119873120596

⋆(1198881 minus 1198882)

119887 (119873 + 1) (48)


120595 [1119873

119873 minus 1119873

] 997888rarr R

120596 997891997888rarr1119873

(1+119873 minus 2

1 + 119890120573Δ120587(120596))

(49)


Δ120587 (120596) = (1205872 minus1205871) (120596)

= 1199022 (120596) (119886 minus 119887119876 (120596) minus 1198882)


(50)

with

1199022 (120596) =119886 minus 1198882 + 119873120596 (1198881 minus 1198882)

119887 (119873 + 1)


119887 (120596119873 + 1)

(51)

and thus

119876 (120596) =119873 (120596 (119886 minus 1198881) + 1199022 (120596) 119887 (1 minus 120596))

119887 (120596119873 + 1) (52)









2119887(119873 + 1) gt 0


at (119902lowast



11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










11861

118613

118615

118618

11862

3 31925 3385 35775 377120573C

118613

118614

118614

118615

118615

q 2q 1

3 31925 3385 35775 377120573C

3 31925 3385 35775 3770

0075

015

0225

03

120573C

120596



lowast



It holds that

119869119866(119902lowast

2 120596lowast)

=

[[[[

[

1 minus12057411988611198862 (1 + 119873)

2 (1198861 + 1198862) (1 + 120596lowast119873)

12057411988611198862119873(1198881 minus 1198882)

2119887 (1198861 + 1198862) (1 + 120596lowast119873)

11986921 11986922

]]]]

]

(53)

where

11986921


lowastminus 31198731198882120596

lowast) 119890120573Δ120587

119873(1 + 119890120573Δ120587)2(119873120596lowast + 1)

(54)


120596lowast=

1119873

(1+119873 minus 21 + 119890120573Δ120587

) (55)

equivalently

11986921 = minus120573 (119873120596



lowastminus 31198731198882120596

lowast)

119873 (119873 minus 2) (119873120596lowast + 1)


lowast+ 1198731198882120596

lowast) 119890120573Δ120587

119887 (1 + 119890120573Δ120587)2(119873120596lowast + 1) (119873 + 1)

(56)



11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11986922

= minus120573 (119873120596



119887 (119873 minus 2) (119873120596lowast + 1) (119873 + 1)


lowast)

(57)




100 1198881 = 01 119862 = 005 for the firms 1198861 = 1 1198862 =



Δ120587 (119902lowast

2 120596lowast) =

120596lowast

50500+

30510110100000

gt 0 (58)







119866(119902lowast



lowast





Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505minus

15029101000

(59)




120596lowast=

4950 (119890120573(200120596

lowastminus15029)101000 + 1)

+1100

ge12 (60)




120596lowast505 minus 15029101000 (61)


119871 (120596lowast)



lt 1

(62)

where we set


100120596lowast minus 1) (63)




19796000 (100120596lowast + 1)gt 0 (64)





lowast) + 252399000

1960000000120596lowast + 19600000gt 0

(65)


60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










60 65 70 75 80 85 909885

989

9895

99

9905

991

9915

992

9925

60 65 70 75 80 85 9098895

989

98905

9891

98915

9892

98925

9893

98935

60 65 70 75 80 85 900

005

01

015

02

025

03

q 2q 1

120596

120573 120573

120573




Δ120587 (119902lowast

2 120596lowast) =

120596lowast

505+

24931101000

(66)


lowast













05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05 06 07 08 09 114

15

16

17

18

19

120596lowast


05 06 07 08 09 1012

014

016

018

02

022

024

026

120596lowast





lowast

2 =


2 120596lowast) are now 1205821 =



lowast

2 =


2 120596lowast) are now 1205821 =




lowast


119862 gt 1205872(119902lowast



5 Conclusions





0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 158

85

9

95

10

105

11

115

0 05 1 1585

9

95

10

105

11

115

12

0 05 1 1504

05

06

07

08

09

1

0 05 1 15minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

q 2q 1

120596

120573 120573

120573 120573

Δ120587





Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article effects of size, composition, and

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