The phase space structure of the oligopoly dynamical system by means of Darboux integrability
TThe phase space structure of the oligopoly dynamicalsystem by means of Darboux integrability
Adam Krawiec a,e , Tomasz Stachowiak b,c , Marek Szyd(cid:32)lowski d,e a Institute of Economics, Finance and Management, Jagiellonian University, Lojasiewicza 4,30-348 Krak´ow, Poland b Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik´ow 32/46,Warszawa, Poland c Department of Applied Mathematics and Physics, Graduate School of Informatics, KyotoUniversity, 606-8501 Kyoto, Japan d Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Krak´ow, Poland e Mark Kac Complex Systems Research Centre, Jagiellonian University, Krak´ow, Poland
Abstract
We investigate the dynamical complexity of Cournot oligopoly dynamics ofthree firms by using the qualitative methods of dynamical systems to studythe phase structure of this model. The phase space is organized with one-dimensional and two-dimensional invariant submanifolds (for the monopoly andduopoly) and unique stable node (global attractor) in the positive quadrantof the phase space (Cournot equilibrium). We also study the integrability ofthe system. We demonstrate the effectiveness of the method of the Darbouxpolynomials in searching for first integrals of the oligopoly. The general methodas well as examples of adopting this method are presented. We study Darbouxnon-integrability of the oligopoly for linear demand functions and find firstintegrals of this system for special classes of the system, in particular, rationalintegrals can be found for a quite general set of model parameters. We showhow first integral can be useful in lowering the dimension of the system usingthe example of n almost identical firms. This first integral also gives informationabout the structure of the phase space and the behaviour of trajectories in theneighbourhood of a Nash equilibrium.
1. Introduction
In the economic study of imperfect markets a special place is held by theoligopoly. It is a market structure where a substantial market share is held by asmall number of firms. The behaviour of firms in the market, studied first byCournot [1], consists in the firms producing a homogeneous product and fixing
Email addresses: [email protected] (Adam Krawiec), [email protected] (Tomasz Stachowiak), [email protected] (MarekSzyd(cid:32)lowski)
Preprint submitted to Elsevier a r X i v : . [ q -f i n . E C ] A ug he price taking into account that any change of price by a rival firm influencesits profit. There is an equilibrium in which these firms maximize profits (it is aNash equilibrium). The special cases of two and three firms are called a duopolyand triopoly, respectively.From the economic point of view the study of oligopoly dynamics is importantas we want to know a mechanism governing the market dynamics. The existenceof equilibrium of the market, its genericity, its stability and the dynamicalbehaviour in the neighbourhood of the equilibrium are crucial for understandingthe formation and evolution of markets. In modern economics, the studiesof imperfect markets, especially the oligopoly model, is important as it is theprevailing market structure. Accordingly, the oligopoly has been a subject ofintensive studies [2]. Many researchers have developed different variants ofthe classical Cournot model. These investigations pursued both discrete andcontinuous time scales using the methods of dynamical systems. These studiesconcentrated on the asymptotic stability. The summary of earlier results can befound in Okuguchi [3] while Bischi et al. presented comprehensive results fordynamics of discrete oligopoly systems [4].In our analysis of the oligopoly, we consider two main problems. First,we study the structure of the phase space, by modelling the oligopoly as anautonomous n dimensional system of ordinary differential equations. The motiva-tion for such investigation is looking for dynamical complexity of the trajectories’behaviour in the phase space. Exploration of the phase structure seems tobe interesting because our knowledge about trajectories describes the possible(long-term) behaviour of the firms and equilibria which they reach.The second is the problem of the existence of first integrals of a differentialequation systems. It has been studied in many dynamical systems of physicaland biological origin. However, in economic theory the problem of finding firstintegrals of dynamical systems is almost absent. Among the few examples thereare the production/inventory model [5] and the multiplier-accelerator modelof business cycle [6]. In this paper we consider this problem for the oligopolymodel.To this aim, we propose to apply methods of the Darboux polynomials infinding first and the so called “second” integrals. We then search for rationaland algebraic, time-dependent first integrals by means of combinations of theDarboux polynomials. Given enough such quantities, we can also obtain time-independent first integrals of the original oligopoly system. The knowledge oftime-independent and functionally independent first integral yields importantinformation about phase space structure of any dynamical model.If the system is two dimensional (duopoly) then the existence of one firstintegral means that the system is completely integrable and thus phase spacestructure is completely characterized. If the first integral does not dependqualitatively on the special value of the model’s parameters, then its phaseportrait is completely determined on the phase plane. In this case one can easilystudy periodic orbits, limit cycles etc.The advantages of dynamical system methods is that they gives us thepossibility of geometrization of the dynamical behaviour and thus visualisation2f dynamics. The evolutional paths of a model are represented by trajectoriesof a system. The phase space contains all trajectories for all admissible initialconditions. Thus, the global information on dynamics is obtained. In the phasespace, trajectories can behave in a regular way or exhibit complex behaviour[8]. The phase space has a structure organised by fixed points, periodic orbits,invariant submanifolds, etc. The main aim of this paper is reveal this structure inthe context of oligopoly dynamics. To this aim, we construct 3-dimensional phaseportraits representative for the problem and discuss in details integrability of theoligopoly systems. In the discussion of problem of integrability we pay attentionto Darboux integrability. The analysis of integrability gives us information aboutexistence or not-existence of first integral. For visualization we construct thelevel sets of the first integral.The problem of complex oligopoly dynamics was investigated in [9] for discreteversion of oligopoly. In the paper we consider smooth dynamical systems withcontinuous time. The existence of first integrals gives us information on non-existence of complex chaotic behaviour. Although it is not possible to formulatesuch conclusions in full generality, it holds for sufficiently different firms. Thegeneric nature of the critical points points towards preservation of the regularphase space structure in the presence of small perturbations. When they exist.For 3-dimensional dynamical systems the information that there is firstintegral means that solutions are restricted to the level sets of such a function. Ifan n -dimensional system admits a time independent first integral (as for examplein the case of oligopoly dynamics of n almost identical firms), then the firstintegral can be used to lower the dimension of the original system by one (ofcourse if we can effectively solve the conservation law for one of the variables, sothat it can be excluded from the system). We will illustrate how for 3 almostidentical firms the first integral can be used to lower the dimension of the systemto dimension two.
2. The oligopoly model
Let us consider the oligopoly market with three firms. These three playersproduce a homogeneous product of the total supply Q ( t ) = q ( t ) + q ( t ) + q ( t ).The price of the good is p and the demand is given by a linear inverse demandfunction p ( Q ( t )) = a − bQ ( t ) (1)where a and b are positive constants. The former is the highest market price ofthe good. We assume that the firms’ cost function has the quadratic form C i ( q i ) = c + d i q i ( t ) + e i q i ( t ) , i = 1 , , , (2)where c i is the positive fixed cost of firm i , and d i , e i are constants. As themarginal cost of the firm must be less than the highest market price we havethe condition d i + 2 e i q i < a , i = 1 , , i -th firm isΠ i ( q ( t ) , q ( t ) , q ( t )) = q i ( t )( a − bQ ( t )) − ( c + d i q i ( t )+ e i q i ( t )) , i = 1 , , , (3)and its marginal profit is ∂ Π i ( q , q , q ) ∂q i = a − bQ ( t ) − bq i ( t ) − d i − e i q i ( t ) , i = 1 , , . (4)We assume that all the firms have imperfect knowledge of the market. There-fore, they follow a bounded rationality adjustment process based on a localestimate of the marginal profit ∂ Π i /∂q i [7, 10]. It means that the firm increasesits production as long as the marginal profit is positive. When the marginalprofit is negative the firm reduces its production. This adjustment mechanismhas the form dq i ( t ) dt = α i q i ( t ) ∂ Π i ( q ( t ) , q ( t ) , q ( t )) ∂q i , i = 1 , , , (5)where α i is a positive speed of adjustment. The three-firm oligopoly is given bythe system of differential equations˙ q ( t ) = α q ( t )[ a − bQ ( t ) − bq ( t ) − d − e q ( t )] , ˙ q ( t ) = α q ( t )[ a − bQ ( t ) − bq ( t ) − d − e q ( t )] , (6)˙ q ( t ) = α q ( t )[ a − bQ ( t ) − bq ( t ) − d − e q ( t )] , or ˙ q ( t ) = α q ( t )[ a − b q ( t ) − bq ( t ) − bq ( t )] , ˙ q ( t ) = α q ( t )[ a − bq ( t ) − b q ( t ) − bq ( t )] , (7)˙ q ( t ) = α q ( t )[ a − bq ( t ) − bq ( t ) − b q ( t )] , where a = a − d , a = a − d , a = a − d ,b = b + e , b = b + e , b = b + e . (8)The variables q i , i = 1 , , Let the cost function be linear and identical for all three firms C ( q i ) = c + dq i , c > , d > α is the same for all threefirms so without loss of generality we take α = 1, then the dynamical systemhas the form ˙ q ( t ) = q ( t )[¯ a − bq ( t ) − bq ( t ) − bq ( t )] , ˙ q ( t ) = q ( t )[¯ a − bq ( t ) − bq ( t ) − bq ( t )] , (10)˙ q ( t ) = q ( t )[¯ a − bq ( t ) − bq ( t ) − bq ( t )] , a = a − d > , a > , d > b > . Now the cost function is quadratic C ( q i ) = c + dq i + eq i , c > , d > , e > . (11)Additionally, we assume that the speed of adjustment α is the same for all threefirms so without loss of generality we take α = 1, then the dynamical systemhas the form ˙ q ( t ) = q ( t )[¯ a − bq ( t ) − bq ( t ) − bq ( t )] , ˙ q ( t ) = q ( t )[¯ a − bq ( t ) − bq ( t ) − bq ( t )] , (12)˙ q ( t ) = q ( t )[¯ a − bq ( t ) − bq ( t ) − bq ( t )] , where ¯ a = a − d > b = b + e > . Now the cost function is quadratic and different for each firm C i ( q i ) = c i + dq i + e i q i , c i > , d i > , e i > α is the same for all threefirms and without loss of generality we take α = 1, then the dynamical systemhas the form ˙ q ( t ) = q ( t )[ a − b q ( t ) − b q ( t ) − b q ( t )] , ˙ q ( t ) = q ( t )[ a − bq ( t ) − b q ( t ) − bq ( t )] , (14)˙ q ( t ) = q ( t )[ a − bq ( t ) − bq ( t ) − b q ( t )] , where a i = a − d i > b i = b + e i > ,a > , d i > , b > , e i > .
3. Analysis of the model’s dynamics
In this section we present the general analysis of dynamical system (7)˙ q ( t ) = α q ( t )[ a − b q ( t ) − bq ( t ) − bq ( t )] = h ( q , q , q ) , ˙ q ( t ) = α q ( t )[ a − bq ( t ) − b q ( t ) − bq ( t )] = h ( q , q , q ) , (15)˙ q ( t ) = α q ( t )[ a − bq ( t ) − bq ( t ) − b q ( t )] = h ( q , q , q ) , a = a − d , a = a − d , a = a − d ,b = b + e , b = b + e , b = b + e . (16)To find a critical points of system (7) we solve the system h i ( q ∗ ) = 0, i = 1 , ,
3, for ( q ∗ , q ∗ , q ∗ ). Here, we obtain multiple solutions: E = (0 , ,
0) (17) E = (cid:18) a b , , (cid:19) (18) E = (cid:18) , a b , (cid:19) (19) E = (cid:18) , , a b (cid:19) (20) E = (cid:18) a b − ba b b − b , a b − ba b b − b , (cid:19) (21) E = (cid:18) , a b − ba b b − b , a b − ba b b − b (cid:19) (22) E = (cid:18) a b − ba b b − b , , a b − ba b b − b (cid:19) (23) E = (cid:18) a b b + ( − a + a + a ) b − a b + a b ) b b b b + 2 b − b + b + b ) b , a b b + ( a − a + a ) b − a b + a b ) b b b b + 2 b − b + b + b ) b , a b b + ( a + a − a ) b − a b + a b ) b b b b + 2 b − b + b + b ) b (cid:19) (24)All these are points in the phase space which represent stationary states of thesystem. The critical point E is trivial without any supply in the market. Thecritical points E , E and E correspond to a monopoly market with firm “1”,firm “2” or firm “3”, respectively. The critical points E , E and E correspondto a duopoly market with firms “1” and “2”, firms “1” and “3”, or firms “2” and“3”, respectively. The last critical point E corresponds to a three firms oligopoly.In the critical point E all three firms choose their supply to be optimalin such a way that their profit is maximized. It corresponds to the Cournotequilibrium for three firms, and is also a Nash equilibrium as no firm wants tochange (increase or decrease) its supply as this would lead to a decrease in itsprofit.There also exists a generic correspondence between the critical points andsets invariant with respect to the flow. The point E usually lies outside invariantsubmanifolds, the monopolies E , E , E lie on one-dimensional submanifolds,the duopolies E , E , E lie on two dimensional submanifolds, and E could be6aid to lie on a zero-diemnsional one, but this is, strictly speaking, true for anycritical point.From (17) we have that in the Cournot equilibrium the total supply is Q ∗ = q ∗ + q ∗ + q ∗ (25)and the price is P ∗ = a − b ( q ∗ + q ∗ + q ∗ ) . (26)Let us try local stability analysis of the critical points represented the Cournotequilibrium ( E ). Following the Hartman-Grobman theorem [11] the dynamics inthe neighbourhood of this non-degenerate critical point is approximated throughlinear part of the system. The linearization at q ∗ is ddt q − q ∗ q − q ∗ q − q ∗ = M q − q ∗ q − q ∗ q − q ∗ (27)where M is the Jacobian calculated at q ∗ M = ∂h ∂q (cid:12)(cid:12) q = q ∗ ∂h ∂q (cid:12)(cid:12) q = q ∗ ∂h ∂q (cid:12)(cid:12) q = q ∗ ∂h ∂q (cid:12)(cid:12) q = q ∗ ∂h ∂q (cid:12)(cid:12) q = q ∗ ∂h ∂q (cid:12)(cid:12) q = q ∗ ∂h ∂q (cid:12)(cid:12) q = q ∗ ∂h ∂q (cid:12)(cid:12) q = q ∗ ∂h ∂q (cid:12)(cid:12) q = q ∗ (28)= − α b q ∗ + α g − α bq ∗ − α bq ∗ − α bq ∗ − α b q ∗ + α g − α bq ∗ − α bq ∗ − α bq ∗ − α b q ∗ + α g (29)where g = g ( q ∗ , q ∗ , q ∗ ) = a − b q ∗ − bq ∗ − bq ∗ g = g ( q ∗ , q ∗ , q ∗ ) = a − bq ∗ − b q ∗ − bq ∗ g = g ( q ∗ , q ∗ , q ∗ ) = a − bq ∗ − bq ∗ − b q ∗ . The stability of the critical point as well as its character depends on theeigenvalues of the linearization matrix which are the solutions of the characteristicequation det[ M − λ I ] = λ − tr M λ + ((tr M ) − tr M ) λ + (tr M ) + 2 tr M − M tr M = m λ + m λ + m λ + m = 0 . (30)Let us consider critical point E . From the economic point of view this pointis the equilibrium for oligopoly of three firms. The Jacobi matrix calculated atthis critical point is M = − α b q ∗ − α bq ∗ − α bq ∗ − α bq ∗ − α b q ∗ − α bq ∗ − α bq ∗ − α bq ∗ − α b q ∗ (31)7nd the characteristic equation isdet[ M − λ I ] = m λ + m λ + m λ + m = λ + 2( α b q ∗ + α b q ∗ + α b q ∗ ) λ +[ α α q ∗ q ∗ (4 b b − b ) + α α q ∗ q ∗ (4 b b − b ) + α α q ∗ q ∗ (4 b b − b )] λ +2 α α α q ∗ q ∗ q ∗ ( b + 4 b b b − b b − b b − b b ) = 0(32)The sign of the discriminant ∆ of the characteristic equation determineswhether eigenvalues are real or complex. If the discriminant is negative, thenall eigenvalues are real. In turn, from the Routh-Hurvitz stability criterion wehave that if all coefficients of characteristic equation are positive a i > a a > a a is fulfilled, then the critical point is stable. The firstcondition of the criterion is always fulfilled because b i > b , i = 1 , , c i , i = 0 , , , E which give additional inequalities for the parameters constraints • there is a critical point inside the positive quadrant of the phase space(point E ) case I q ∗ i > i = 1 , , . • there is no critical point inside the positive quadrant of the phase space;the critical point E is located on a 2-dimensional invariant manifoldcase IIa q ∗ i > , q ∗ j > , q ∗ k = 0 , i (cid:54) = j (cid:54) = k (cid:54) = i • there is no critical point inside the positive quadrant of the phase; thecritical point E is located on a 1-dimensional invariant manifoldcase IIb q ∗ i > , q ∗ j = q ∗ k = 0 , i (cid:54) = j (cid:54) = k (cid:54) = i. For case I, the phase portrait of system (7) is presented in fig. 1 with conditions q ∗ > a b b + ( − a + a + a ) b − a b + a b ) b > ,q ∗ > a b b + ( a − a + a ) b − a b + a b ) b > ,q ∗ > a b b + ( a + a − a ) b − a b + a b ) b > . (33)The stable node E is inside the positive quadrant of the phase space. Underthe above conditions, the set of initial conditions which lead to this point is thewhole positive quadrant. This point represents the Cournot equilibrium wherethree firms, maximazing profits, coexist in the market. This is the unique pointin the positive quadrant which is a global attractor.For case IIa, the phase portrait of system (7) with conditions q ∗ > a b b + ( − a + a + a ) b − a b + a b ) b > ,q ∗ > a b b + ( a − a + a ) b − a b + a b ) b > ,q ∗ = 0 i.e. 4 a b b + ( a + a − a ) b − a b + a b ) b = 0 . (34)8
123 1 2 30123 q q q AEF HGD CB
Figure 1: Left panel: the phase portrait for the system (7) with the condition 4 a b b +( − a + a + a ) b − a b + a b ) b >
0, 4 a b b + ( a − a + a ) b − a b + a b ) b > a b b + ( a + a − a ) b − a b + a b ) b >
0. Right panel: the eigenvectors for thesecritical points. is presented in fig. 2. In this case there is no critical point in the positive quadrantof the phase space. The set of initial conditions constituing the positive quadrantleads to the critical point located on the two-dimensional invariant submanifold q = 0. It means that the marginal cost of one of the firms (paramater d ) issufficiently greater than marginal costs of other firms, such that as time goes toinfinity the firm reduces its production to zero.This case describes evolutional scenario that one of the firms graduallywithdraws from the market due to costs higher than for the competition.Case IIb is a variant of case IIa, where two firms have higher marginal coststhan the third firm. Both these firms gradually withdraw from market and onlyone firm, the monopolist, is left.For deeper understanding of the phase portrait, the additional map of eigen-values calculated at the critical points is presented in both previous figures.These vectors span the plane tangent to the stable and unstable manifolds.To show the explicite implication of Routh-Hurvitz criterion, let us consider asimplier case of identical firms with a linear cost function and speed adjustmentto market is equal 1, i.e. α = α = α = 1 , b = b = b = b (35)and the critical point E has the coordinates q ∗ = q ∗ = q ∗ = q ∗ = ¯ a a = a − d and d = d = d = d . Now the characteristic equation has theform λ + 32 ¯ aλ + 916 ¯ a λ + 116 ¯ a = c λ + c λ + c λ + c = 0 . (37)9
123 1 2 3012 q q q AEF GD C B
Figure 2: Left panel: The phase portrait for the system (7) with the conditions 4 a b b +( − a + a + a ) b − a b + a b ) b >
0, 4 a b b + ( a − a + a ) b − a b + a b ) b and4 a b b + ( a + a − a ) b − a b + a b ) b = 0. Right panel: the eigenvectors for thesecritical points. Let us state the theorem describing two possibilities for eigenvalues in thesystem (10).
Theorem 1.
For system (10) equilibrium point E is a stable node. The discriminant of the third order polynomial (37) is∆ = q p
27 = 0 (38)where p = −
316 ¯ a , q = 132 ¯ a . Both p and q are different from zero so then there isa) one real eigenvalue of multiplicity oneb) and one real eigenvalue of multiplicity two.Let us check the Routh-Hurvitz criterion of stability of the critical point. Allthe coefficients of the characteristic equation (37) are positive. The condition( m m > m m ) is also fulfilled for any parameter ¯ a .Hence, the critical point (36) is a stable node. So far, the dynamics of the system was considered inside the phase space { ( q , q , q ) : q i > , i = 1 , , } . Now, let us consider the two-dimensional planesin this 3-dimensional phase space defined as { ( q , q ) , q = 0 } , { ( q , q ) , q =0 } , and { ( q , q ) , q = 0 } . The system (27) possesses at least three invariantsubmanifolds on which it assumes the form of two-dimensional autonomousdynamical system. 10or the first invariant submanifold the system has the form˙ q ( t ) = 0 , ˙ q ( t ) = α q ( t )[ a − bq ( t ) − b q ( t ) − bq ( t )] , (39)˙ q ( t ) = α q ( t )[ a − bq ( t ) − bq ( t ) − b q ( t )] , For the second invariant submanifold the system has the form˙ q ( t ) = q ( t )[ a − b q ( t ) − bq ( t ) − bq ( t )] , ˙ q ( t ) = 0 , (40)˙ q ( t ) = α q ( t )[ a − bq ( t ) − bq ( t ) − b q ( t )] , For the third invariant submanifold the system has the form˙ q ( t ) = q ( t )[ a − b q ( t ) − bq ( t ) − bq ( t )] , ˙ q ( t ) = α q ( t )[ a − bq ( t ) − b q ( t ) − bq ( t )] , (41)˙ q ( t ) = 0 , where a i = a − d i and b i = b + e i .These exist regardless of the values of the parameters, but we will be able tofind other linear submanifolds under some general assumptions – we defer thisdiscussion to the section on Darboux polynomials.Let us choose the following values of the parameters α = 1 , a = 10 , a = 20 , a = 30 , b = 0 . , e i = 0 . (42)to present the dynamics on the invariant submanifolds.Let us consider the first system on invariant submanifold (39) and performthe dynamical analysis for it. This system has four critical points E = (0 ,
0) (43) E = (cid:18) a b , , (cid:19) (44) E = (cid:18) , a b , (cid:19) (45) E = (cid:18) a b − ba b b − b , a b − ba b b − b , (cid:19) (46)The phase portrait for this system is presented on the plane { ( q , q ) , q = 0 } inFig. 3.In the positive quadrant there is only one critical point: the stable node,which corresponds to the duopoly equilibrium. Only two firms exist on themarket and they reach the stable equilibrium with fixed production level. Thecondition for the existence of this point is2 a b − ba > a b − ba > . (47)11or the chosen parameter values (42) we have 2 a b − ba = 0. In this casethere is no critical point inside the positive quadrant as well as on the respectivesubmanifolds, which are the planes { ( q , q ) , q = 0 } and { ( q , q ) , q = 0 } .Markets with initial conditions in those planes tend towards monopolies on axes q or q as seen in fig. 3, the exception being the q axis which is an invariantsubmanifold itself. For the system (41) we will look for the Lapunov function. We assume thatboth conditions 2 a b − ba > a b − ba = 0 are fulfilled and q ∗ > q ∗ >
0, so we consider the critical point (46).Let us take the Taylor expansion of the right-hand sides of (41) in theneighbourhood of the critical point ( q ∗ , q ∗ ) given by (46) and write the systemin the form [12, p. 206-207] ˙ x = A × x + g ( x ) (48)where x is a vector of components x = q − q ∗ and x = q − q ∗ . The matrix A and the vector g ( x ) has the form A = (cid:20) − α b q ∗ − α bq ∗ − α bq ∗ − α b q ∗ (cid:21) = (cid:20) a a a a (cid:21) (49)where p = tr A = − α b q ∗ + α b q ∗ ) < q = det A = α α q ∗ q ∗ (4 b b − b ) > b > b and b > b , and g ( x ) = (cid:20) − α (2 b x + bx x ) − α (2 b x + bx x ) (cid:21) . (52)We define the Lapunov function [12, p. 206-207] V ( x ) = x T Kx (53)where K is a 2 × A T K + KA = − I × in the form K = m ( A T ) − A − + nI × . It takes place if m = − q/ p and n = − / p . Hence, we find that K = − pq (cid:20) a + a + q − a a − a a − a a − a a a + a + q (cid:21) (54)And the Lapunov function V = − ( a x − a x ) + ( a x − a x ) + q ( x + x )2 pq > x (cid:54) = 0(55)12 A B CE D q q Figure 3: The co-existence of three two-dimensional submanifold of system () representing theplanes of { ( q , q ) , q = 0 } , { ( q , q ) , q = 0 } and { ( q , q ) , q = 0 } .
13s positive definite.Given that the Lapunov function is of the form (53) and matrix K (54) weget ˙ V = − x − x + 2 g T Kx (56)where2 g T Kx = 1 | p | q {− α (2 b x + bx x )[( a + a + q ) x − ( a a + a a ) x ] }− α (2 b x + bx x ) { ( − a a − a a ) x + ( a + a + q ) x } . (57)For any p and q , the first two terms − x − x predominate over the thirdterm 2 g T Kx in the neighbourhood of the origin, and ˙ V is negative definite.Therefore, system (41) has a strict Lapunov function.
4. The first integral analysis
For the purpose of the subsequent sections it will be convenient to adopt aredefined set of parameters. As the system has the general form dq i dt = α i q i ( a − d i − ( b + 2 e i ) q i − bQ ) , i = 1 , , , (58)where Q := q + q + q , one can see that b = 0 makes the equation decoupleand the system completely solvable. It thus makes sense to rescale the time by t → t/b and introduce new parameters: f i := α i b ( a − d i ) ,(cid:15) i := 2 α i b ( b + e i ) . (59)We can then make use of the following matrix notation dq i dt = q i f i + (cid:88) j =1 β ij q j , (60)where β := − (cid:15) α α α (cid:15) α α α (cid:15) . (61) In a polynomial system it is natural to look for polynomial first integrals,which are a special case of analytic ones, but even the latter class is veryrestrictive. In particular, if one demands that the conserved function I be14nalytic in all variables, then each hyperbolic critical point provides additionalnecessary conditions. For around it, the system can be linearized to be˙ q i = λ i q i + O ( q ) , i = 1 , . . . , , (62)assuming the point is at q = 0. The derivative of the lowest monomial in theexpansion of I around that point is thendd t ( q n q n q n ) = ( λ n + λ n + λ n )( q n q n q n ) + O ( q n + n + n +1 ) . (63)It follows that for ˙ I to vanish, λ i must be linearly dependend over N . The generalstatement, due to Poincar´e, is that the same holds when λ i are the eigenvaluesof the linearization matrix around any critical point, even if there are Jordanblocks.In our case this immediately leads to the stringent result: Theorem 2.
If the system has a first integral analytic at the origin, the quanti-ties f , f and f must be linearly dependent over the non-negative integers. In particular this means that f i cannot all have the same sign bceause n f + n f + n f cannot vanish for any positive integers n i . The cases and are thus precluded from having an analytic integral.The same can be done around all the other points, the most straightforwardones being E , E and E , for which we have the following sets of eigenvalues (cid:26) − f i , f j − α j f i (cid:15) i , f k − α k f i (cid:15) i (cid:27) , (64)where ( i, j, k ) are cyclic permutation of (1 , , N for there to exist a first integral analytic at E , E and E ,respectively.The drawback of this approach is that the analysis has to be done separatelyat each point, and the sets of integer coefficients n i is different every time. Also,the requirement of analyticity is rather restrictive and we shall see in the nextsection that under fairly general assumptions there can still exist rational firstintegrals.
5. Darboux Polynomials
The analysis of rational integrals for homogeneous systems relies on the socalled Darboux polynomials F , which are defined by the following property˙ F = (cid:88) i =0 V i ∂ x i F =: P F, (65)where P is called the cofactor, and is also necessarily a polynomial. Thepolynomials F are sometimes called partial or second integrals, since they are15ot conserved in general but each of them is conserved on the set defined by F = 0 as seen above. If the initial conditions are such that the system starts insuch a set it will remain there for all later times. If the cofactor is zero, then thepolynomial is just a first integral.The first fundamental fact that we will use here is that if the system has arational integral R = R /R with R and R relatively prime, they must bothbe Darboux polynomials with the same cofactor. This is immediately seen fromdirect differentiation.Secondly, any product of Darboux polynomials is itself a Darboux polynomial ddτ (cid:32) n (cid:89) k =1 F γ k k (cid:33) = (cid:32) n (cid:88) k =1 γ k P k (cid:33) n (cid:89) k =1 F γ k i , γ k ∈ N . (66)Note that if the domain of motion is such that the above functions are welldefined, any real γ k are admissible in a construction of first integrals, although itmight not be rational or even algebraic then. In other words, if there are enoughDarboux polynomials, so that one can find numbers γ k such that the cofactor in(66) vanishes, a first integral can be found.Let us now apply the above to our system. It is immediately visible thateach of the dependent variables q i is itself a Darboux polynomial with cofactorsgiven by f i + Σ j β ij q j =: f i + L i . Their linear dependence is a major factor indetermining integrability.Since these are three polynomials in three variables, they will generically belinearly independent. In fact, dependence does not occur for economically viablevalues of the parameters, but we include this case nevertheless for the sake ofcompleteness – the result applies to any system of the prescribed form. det( β ) = 0If it so happens that − det( β ) α α α = 2 − (cid:15) α − (cid:15) α − (cid:15) α + (cid:15) (cid:15) (cid:15) α α α = 0 , (67)a linear combination of L i will vanish and this will lead to at least a time-dependent first integral. To see this, let us notice that when the determinant of β is zero, it has a null left eigenvector γ k which is not the zero vector (cid:88) k =1 γ k β kj = 0 , j = 1 , . . . , . (68)The coefficients γ k might not be integer, but in any case the following functionwill be defined for positive q j : I := (cid:89) j =1 q γ j j , (69)16hose cofactor, by (66) will be P = (cid:88) j =1 γ j (cid:0) f j + Σ k =1 β jk q k (cid:1) = (cid:88) j =1 γ j f j . (70)Because this is constant, the equation ˙ I = P I can be immediately integratedto yield the time-dependent integrale − P t I = const . (71)It might additionally happen that P = 0, so there is no time dependence, orthat there are two zero eigenvalues of β such that two independent null vectors γ k , θ k can be found. In the second case there will be two first integrals andtime can be eliminated from at least one of them, because both depend on itexponentially.To illustrate the above, let us look at the case when all α i are equal. Thisis not a crucial assumption in this case, since they must all be non-zero, sotheir values do not change the condition that det( β ) = 0. The other parametersmust satisfy some constraint for the condition to be true, and there are manypossibilities due to the large number of parameters, but equation (68) can alwaysbe solved, For example, in the generic case of γ k (cid:54) = 0, the following parametersgive vanishing determinant: e i = − b γ + γ + γ + γ i γ i , (72)The corresponding the first integral is I = e − ( f γ + f γ + f γ ) t q γ q γ q γ . (73)After taking logarithm both sides and differentiating with respect to time weobtain that the rates of growth are linearly dependent and satisfy the followingcondition f γ + f γ + f γ = γ ˙ q q + γ ˙ q q + γ ˙ q q . (74)The rates of growth of firms’ production are strictly related as a given firmneed to adjust its production rate of growth with respect to production rates ofgrowth of two other firms. This relation is a dynamical constraint imposed onthe production rates of growth at any moment. It is analogous to the reactioncurves analysis [2].In a special case of constant rate of growth there is a solution q ( t ) ∝ e f t , q ( t ) ∝ e f t , q ( t ) ∝ e f t . (75)
6. The case det( β ) (cid:54) = 0. We can next proceed to the generic, and more difficult, case when det( β ) (cid:54) = 0,noting first that any cofactor of our system must be of the form P = p + (cid:88) i =1 p i q i , (76)17ecause the right-hand sides of the system are quadratic.We will first try to find all linear Darboux polynomials F = w + (cid:88) i =1 w i q i , (77)which do not reduce to just q i or a constant. Calculating ˙ F − P F and equatingcoefficients of different monomials to zero, gives system of polynomial equationsfor w i and p i : q : w p = 0 , (78) q : w p + w ( p − f ) = 0 ,w p + w ( p − f ) = 0 ,w p + w ( p − f ) = 0 , (79) q : w ( p + (cid:15) ) = 0 ,w ( p + (cid:15) ) = 0 ,w ( p + (cid:15) ) = 0 ,w ( α + p ) + w ( α + p ) = 0 ,w ( α + p ) + w ( α + p ) = 0 ,w ( α + p ) + w ( α + p ) = 0 , (80)In the generic situation all w i are non-zero, which then leads to a homogeneouslinear system: α − (cid:15) α − (cid:15) α − (cid:15) α − (cid:15) α − (cid:15) α − (cid:15) w w w = 0 , (81)whose non-zero solution requires the determinant to vanish D := ( α − (cid:15) )( α − (cid:15) )( α − (cid:15) ) + ( α − (cid:15) )( α − (cid:15) )( α − (cid:15) ) = 0 . (82)If a solution of the system exists, the set of equations (79) gives threeconstraints f i = p − w w i (cid:15) i , (83)which have to take into account that w p = 0. So either w = 0 and p = f i ,which must all be equal; or w = 1 ( F is always defined up to a constant factor), p = 0 and f i w i = (cid:15) i .The remaining exceptional cases occur when one of w i is zero, and becausethe system has complete symmetry with respect to the indices, it is enoughto consider w = 0. Direct computation then shows that the only non-trivialsolution is F = ( α − (cid:15) ) q + ( (cid:15) − α ) q , P = f − α q − (cid:15) q − (cid:15) q , (84)18ith the additional constraints of f = f , α = α . (85)All the other solutions (and constraints) can be obtained by cyclic permutationsof the indices: 1 → → → n . Direct computation shows thatthere are no new second or third order polynomials, other than products of allthe linear ones. The authors have been unable to proceed with the proof for thegeneral case, i.e., when all the parameters are independent. At the same time,imposing some restrictions immediately produces linear Darboux polynomialsand first integrals, so it seems reasonable to formulate the following Conjecture 1.
All Darboux polynomials of the system in question for generalvalues of parameters are generated by the linear polynomials.
So far we have not made use of the det( β ) (cid:54) = 0 condition, and if Darbouxpolynomials can be found via the procedure above, each of them defines aninvariant set F = 0 by itself. But now that the matrix β is not singular, the linearforms L i form a basis, so the linear part of the cofactor P can be decomposedin full analogy with the previous section: L = (cid:88) i p i q i = (cid:88) i γ i L i , γ i = (cid:88) j p j [ β − ] ji . (86)Then the function I := F (cid:89) i x − γ i i , (87)has a constant cofactor ˙ I = (cid:32) p − (cid:88) i f i γ i (cid:33) I =: P I , (88)and a time-dependent first integral is, as before, I = e − P t I .Coming back to the simpler cases of the system (7), we can consider all firmsidentical i.e. their cost function are the same and/or the speed of adjustment tomarket α i are the same. The first two subcases, 2.2 and 2.4, of identical firms can in fact be treatedtogether. That is, regardless of whether the cost function is linear or quadraticwe can find first integrals.Taking all α i to be the same, and the cost function C ( q i ) = c + dq i + eq i , c > , d > , e > , (89)19he parameters (17) have the form f = ¯ ab(cid:15) = 2 b ( b + e ) (90)The matrix β in this case isdet β = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:15)
11 1 (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − (cid:18) b + eb (cid:19) + 6 b + eb − (cid:54) = 0 (91)if b > e > b + ee > D vanishes and all f i are equal.There are thus three additional linear Darboux polynomials, together with theircofactors they are: F = q − q , P = f − q − q − q ,F = q − q , P = f − q − q − q ,F = q − q , P = f − q − q − q . (92)According to the general treatment of section 8, we thus have three time depen-dent integrals I = e ft/ q ( q − q )( q q q ) − / ,I = e ft/ q ( q − q )( q q q ) − / ,I = e ft/ q ( q − q )( q q q ) − / , (93)and time can be eliminated from two of them, to yield a first integral I = q ( q − q ) q ( q − q ) , (94)and we note that cyclic permutations of indices produce also first integrals, butthey are all functionally dependent. This is a subcase of the situation of 2.4, and could be called “almost identical”firms, as the parameters coincide in the linear parts ( d i and α i are equal), butthe quadratic terms of the cost functions are different for each company. Theprevious two subcases are just restrictions of this one, and we state here thegeneral results regarding invariant submanifolds and stability.The parameters f and (cid:15) i (17) now have the form f i = a − db ,(cid:15) i = 2 b ( b + e i ) . (95)20he matrix β in this case isdet β = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:15)
11 1 (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − (cid:15) (cid:15) (cid:15) + ( (cid:15) + (cid:15) + (cid:15) ) − . (96)Because b > e i > (cid:15) i > β < F = ( (cid:15) − q − ( (cid:15) − q , P = f − (cid:15) q − (cid:15) q − q ,F = ( (cid:15) − q − ( (cid:15) − q , P = f − q − (cid:15) q − (cid:15) q ,F = ( (cid:15) − q − ( (cid:15) − q , P = f − (cid:15) q − q − (cid:15) q . (97)All the time dependent first integrals can then be obtained by cyclic indexpermutations from the following one I = e − P t q (( (cid:15) − q − ( (cid:15) − q ) q − γ q − γ q − γ , (98)where γ i = (cid:15) i − β (cid:18) − (cid:15) (cid:15) (cid:15) (cid:15) i (cid:19) , (99)and P = ( (cid:15) − (cid:15) − (cid:15) − β f. (100)The elimination of time yields an ordinary first integral of I = q (( (cid:15) − q − ( (cid:15) − q ) q (( (cid:15) − q − ( (cid:15) − q ) . (101)As mentioned before, each Darboux polynomial defines an additional invariantsubmanifold F i = 0, which in this case are planes through the origin. In fact allthree intersect along a line because F = F + F and their normal vectors arenot independent. Its equation is q = (cid:18) q (cid:15) − , q (cid:15) − , q (cid:15) − (cid:19) , q ∈ (0 , ∞ ) . (102)The Nash equilibrium E lies exactly on this line and is an attracting node.We can say even more thanks to the explicit formula for I and the remainingtwo integrals. In the positive quadrant q q q is nonzero, while P < I = const.we immediately obtain from (98) that F → t → ∞ , and similarly for theother two.This means that trajectories are attracted by all the invariant submanifolds(planes) and hence also by the line. On the line itself it is straightforward tocheck that˙ q = q (cid:18) f − (cid:18) (cid:15) − (cid:15) − (cid:15) (cid:15) − (cid:19) q (cid:19) = f q (1 − q /q ∗ ) , (103)21 igure 4: The level sets of the first integral I , for (cid:15) = 2 . (cid:15) = 2 . (cid:15) = 2 . where q ∗ is the position of E in this coordinate. The Nash equilibrium isthus seen to be the attractor of two trajectories on the line, which in turn areattracting trajectories for the whole positive quadrant.The level sets of I , as seen in Figure 4 are surfaces also intersecting at thecentral line. Taking the first integral as one coordinate, the system becomesjust two dimensional on any such leaf, by solving I = C , eliminating one of theoriginal variables, e.g. q = C ( (cid:15) − q q (1 − (cid:15) ) q + ( (cid:15) − C ( (cid:15) − q =: Q ( q , q ) , (104)and substituting into the original system:˙ q = h ( q , q , Q ( q , q ))˙ q = h ( q , q , Q ( q , q )) . (105)22 .3. Extension to n -dimensions The special shape of the Darboux polynomials and first integrals suggests astraightforward generalisation to n dimensions, and indeed it is easy to check,that for n almost identical (in the above sense) firms, one has a family of Darboux n ( n − / F ij = ( (cid:15) i − q i − ( (cid:15) j − q j , P ij = f − (cid:15) i q i − (cid:15) j q j − (cid:88) k (cid:54) = i,j q k . (106)Like before, there exist time-dependent first integrals I ij = e − P t F ij (cid:89) k q − γ kij k , (107)where γ kij is the k -th component of the vector γ ij = − [1 , . . . , , (cid:15) i , , . . . , , (cid:15) j , , . . . , β − , (108)and it so happens that for all pairs ( i, j ), the sum of the components is the samegiving P = f − f (cid:88) k γ kij = f det β (cid:89) k ( (cid:15) k − . (109)The ratios of these integrals are thus time-independent, but again not all arefunctionally independent. E.g., for I := I /I and I = I /I we have I = 1 − (cid:15) + I (1 − (cid:15) ) I ( (cid:15) − . (110)Finally, because det β < F ij tend to zero exponentially with time, so alltrajectories approach the line through the Nash equilibrium, as before.
7. Conclusions
We used the qualitative methods of dynamical systems to study the phasestructure of the oligopoly model. Exploring the example of three firms oligopolywe analyse the complexity of the model dynamics. The phase space is organizedwith one-dimensional and two-dimensional invariant submanifolds (on whichthe system reduces to monopoly and duopoly) and a unique stable node (globalattractor, Cournot and Nash equilibrium) in the positive quadrant of the phasespace.Its inset is the positive quadrant of the phase space { ( q , q , q ) : q > , q > , q > } . The boundaries of this quadrant are three two-dimensional sub-manifolds where the dynamics of duopolies is restricted to. Trajectories fromthe bulk space depart asymptotically from unstable invariant 2-dimensionalsubmanifolds. In a generic case they reach the Nash equilibrium point. Thedynamics of monopolies is restricted to the axes of coordinates systems. On this23ine there exits stable critical points (A, C, F in Fig. 1) they are a equilibriumpoints of monopolies.From our dynamical analysis of the system one can derive general conclusionthat the system under consideration is structurally stable, because its phaseportrait in the generic case contains saddle and nodes. It seems to be importantin the economic context because its structural stability means that dynamicscannot be destroyed by small perturbations [11].The problem of integrability of the model was also addressed using Darbouxpolynomials, since this is the natural setting for polynomial systems. Becauseof the large number of parameters, the full analysis was not possible, butsome general cases of interest were shown to posses additional linear Darbouxpolynomials and also time-dependent conserved quantities. This allowed to givequalitative analysis of the asymptotic behaviour and dimensional reduction inthe general case of firms different at the quadratic level.The following points of interest were also addressed: • We formulate a criterion of asymptotic stability of a Cournot equilibriumwhich indicate that phase space interior of positive octant in R . Thephase space structure is organised as follows: the global attractor andthree repelling invariant submanifolds. They form the boundary of thisoctant. From the economic point of view they represent the dynamics ofduopolies. • The Cournot equilibrium is a global attractor in the phase space for genericclass of initial conditions and model parameters. • We found the new algebraic interpretation of reaction curves for someforms of oligopoly system admitting time dependent first integral. Thisconstraint assumes the form of linear combination of production rates ofgrowth. • We demonstrate how the dimension of the system can be lowered by onedue to existence of first integral in the case of firms with different quadraticterm in cost function. • When a first integral exists it extends our knowledge of invariant submani-folds beyond the linear case (planes) as depicted in Fig. 4. Additionally itpoints to the lack of chaotic behaviour in the system. • The relations between growth rates and the variables themselves obtainedthanks to the first integrals provide immediate observable constraints thatcan be tested against data.
Acknowledgments
This work has been supported by the grant No. DEC-2013/09/B/ST1/04130of the National Science Centre of Poland. The authors thanks Franciszek Humiejafor comments and remarks. 24 ibliographyReferencesibliographyReferences