A dynamic marketing model with best reply and inertia

Size: px
Start display at page:

Download "A dynamic marketing model with best reply and inertia"

Transcription

1 A dynamic marketing model with best reply and inertia Gian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) URL: Urbino, September 18, 14 Urbino, September 18, 14 1 /

2 The Farris s Market Model Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition. Marketing - Journal of Research and Management (05), p x i (t + 1) = (1 λ i )x i (t) + λ i B j =i a j x j (t) a i j =i a j x j (t) Urbino, September 18, 14 2 /

3 The Farris s Market Model Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition. Marketing - Journal of Research and Management (05), p x i (t + 1) = (1 λ i )x i (t) + λ i B j =i a j x j (t) a i j =i It results from standard arguments in marketing modelling: a j x j (t) customers attraction: A i = a i x β i i Urbino, September 18, 14 2 /

4 The Farris s Market Model Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition. Marketing - Journal of Research and Management (05), p x i (t + 1) = (1 λ i )x i (t) + λ i B j =i a j x j (t) a i j =i It results from standard arguments in marketing modelling: a j x j (t) customers attraction: A i = a i x β i i market share: s i (t) = A i(t) n j=1 A j (t) Urbino, September 18, 14 2 /

5 The Farris s Market Model Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition. Marketing - Journal of Research and Management (05), p x i (t + 1) = (1 λ i )x i (t) + λ i B j =i a j x j (t) a i j =i It results from standard arguments in marketing modelling: a j x j (t) customers attraction: A i = a i x β i i market share: s i (t) = A i(t) n j=1 A j (t) profits: Π i (t) = Bs i (t) x i (t) Urbino, September 18, 14 2 /

6 The Farris Market Model The resulting profits are Π i (t) = a i x β i i a i x β i i (t) (t) + j =i a j x β j j (t) x i(t) Urbino, September 18, 14 3 /

7 The Farris Market Model The resulting profits are Π i (t) = a i x β i i a i x β i i (t) (t) + j =i a j x β j j (t) x i(t) Each firm maximizes its own profit function computing its gradient Π i (t + 1) x i = 0 Urbino, September 18, 14 3 /

8 The Farris Market Model The resulting profits are Π i (t) = a i x β i i a i x β i i (t) (t) + j =i a j x β j j (t) x i(t) Each firm maximizes its own profit function computing its gradient Π i (t + 1) x i = 0 Setting β i = 1, this leads to x i (t + 1) = j =i a j x (e) j (t + 1) a i a j x (e) j (t + 1) j =i Urbino, September 18, 14 3 /

9 The Farris Market Model Assuming naïve expectations, x (e) j (t + 1) := x j (t), Farris et al. derive the following Best Response dynamic model with adaptive adjustment: x i (t + 1) = (1 λ i )x i (t) + λ i j =i a j x j (t) a i a j x j (t) j =i Urbino, September 18, 14 4 /

10 The Farris Market Model Assuming naïve expectations, x (e) j (t + 1) := x j (t), Farris et al. derive the following Best Response dynamic model with adaptive adjustment: x i (t + 1) = (1 λ i )x i (t) + λ i j =i a j x j (t) a i a j x j (t) j =i Setting N = 2 and the new (rescaled) variables x = a 1 a 2 x 1, y = a 1 a 2 x 2 we have x ( ) = (1 λ 1 )x + λ 1 a 2 y y Farris : y ( ) = (1 λ 2 )y + λ 2 a 1 x x Urbino, September 18, 14 4 /

11 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 Urbino, September 18, 14 5 /

12 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 new var : x = c 2 q 1, y = c 1 q 2 Urbino, September 18, 14 5 /

13 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 new var : x = c 2 q 1, y = c 1 q 2 Puu : x = (1 λ 1 )x + λ 1 c 2 c 1 ( y y ) y = (1 λ 2 )y + λ 2 c 1 c 2 ( x x ) Urbino, September 18, 14 5 /

14 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 new var : x = c 2 q 1, y = c 1 q 2 Puu : x = (1 λ 1 )x + λ 1 c 2 c 1 ( y y ) y = (1 λ 2 )y + λ 2 c 1 c 2 ( x x ) x ( ) = (1 λ 1 )x + λ 1 a 2 y y Farris : y ( ) = (1 λ 2 )y + λ 2 a 1 x x Urbino, September 18, 14 5 /

15 Similarities with the Puu model ( ) q 1 = (1 λ q2 1) q 1 + λ 1 q 2 c Puu : ( 1 ) q 2 = (1 λ q1 2) q 2 + λ 2 q 1 c 2 new var : x = c 2 q 1, y = c 1 q 2 Puu : x = (1 λ 1 )x + λ 1 c 2 c 1 ( y y ) y = (1 λ 2 )y + λ 2 c 1 c 2 ( x x ) x ( ) = (1 λ 1 )x + λ 1 a 2 y y Farris : y ( ) = (1 λ 2 )y + λ 2 a 1 x x We obtain the Puu from the Farris setting a 2 = 1/a 1 Urbino, September 18, 14 5 /

16 Equilibria The equilibria abscissas follows from the fourth order algebric equation: [ ] (1 a 1 a 2 ) 2 η η (1 a 1 a 2 ) η 2 + a 2 (a 1 + 1) η a 2 = 0 a 1 a 2 where η := x. An analogous equation for ζ := y holds. From Cardano s formula, the number of real solutions is given provided the sign of the discriminant: > 0 1 real solution D(a 1, a 2 ) : = 0 1 real solution and 2 coincident < 0 3 distinct real solutions Urbino, September 18, 14 6 /

17 Equilibria Section of the function D(a 1, a 2 ) with the plane (a 1, a 2, 0) Urbino, September 18, 14 7 /

18 Equilibria Section of the function D(a 1, a 2 ) with the plane (x, y, 0) Urbino, September 18, 14 8 /

19 The General case a 1 = a 2 Proposition Besides E 0 = (0, 0) a non vanishing fixed point always exists in the region S = (0, 1) (0, 1). If a 1 a 2 = 1 then two further distinct fixed points exist in the region S if the following inequality holds D(a 1, a 2 ) = a1 2a4 [ a1 108 (1 a 1 a 2 ) 6 a 2 (4a 1 + 4a 2 18) a1a ] < 0 and if D(a 1, a 2 ) = 0 these two further fixed points are merging, i.e. there are two real coincident solutions of the cubic ( equation. In the ) particular 1 1 case a 1 a 2 = 1 the unique fixed point E =, is get. (a 1 +1) 2 (a 2 +1) 2 Urbino, September 18, 14 9 /

20 Bifurcation path: a 2 = a 1 /9 + 13/ ,8 a2 D < 0 D > a1 0,6 X 0,4 0, a 1 Urbino, September 18, /

21 Bifurcation path: a 2 = a ,8 a2 D < 0 D > a1 0,6 X 0,4 0, a 1 Urbino, September 18, /

22 Bifurcation path: a 2 = a , a 2 = a 1 (gray) 1 6 0,8 a2 D < 0 D > a1 0,6 X 0,4 0, a 1 Urbino, September 18, /

23 Typical scenario (after the final bifurcation) Urbino, September 18, /

24 The symmetric case: a = a 1 = a 2 Fixed poits are: E 1 = ( a 2 (1 + a) 2, a 2 ) (1 + a) 2 = { (x, y) R 2 x = y } Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:

25 The symmetric case: a = a 1 = a 2 Fixed poits are: E 1 = ( a 2 (1 + a) 2, a 2 ) (1 + a) 2 = { (x, y) R 2 x = y } For a 3 the two further fixed points in symmetric positions with respect to : E 2 = E 3 = a 2 2 (a 1) 2 (a + 1) a 2 2 (a 1) 2 (a + 1) ( a 1 + (a + 1) (a 3), a 1 ) (a + 1) (a 3) ( a 1 (a + 1) (a 3), a 1 + ) (a + 1) (a 3) Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:

26 The symmetric case: stability of E 1 Local asymptotic stability for a < a p = 3 Urbino, September 18, /

27 The symmetric case: stability of E 1 Local asymptotic stability for a < a p = 3 Transverse instability (saddle) via pichfork, merging of two further fixed points E 2 and E 3, for a a p Urbino, September 18, /

28 The symmetric case: stability of E 1 Local asymptotic stability for a < a p = 3 Transverse instability (saddle) via pichfork, merging of two further fixed points E 2 and E 3, for a a p Unstability via flip for ( 1 a a f = ) λ 1 λ 2 λ 1 λ 2 Urbino, September 18, /

29 Fixed points and their basin of attraction (a) Codim. 2 bifurcation (b) Just after the pichfork (c) Just after the flip (d) Just after the subcritical flip Urbino, September 18, /

30 The symmetric case: stability of E 2 and E 3 Local asymptotic stability for ( a < a h = ) ( ) 2 16 λ 1 λ 2 λ 1 λ ( ) ( ) 2 16 λ 1 λ 2 λ 1 λ 2 27 Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:

31 The symmetric case: stability of E 2 and E 3 Local asymptotic stability for ( a < a h = ) ( ) 2 16 λ 1 λ 2 λ 1 λ ( ) ( ) 2 16 λ 1 λ 2 λ 1 λ 2 27 Unstability via Hopf for a a h Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:

32 Profits Computing the profits we get: Π 1 > Π 2 (x y)(x + y a 2 ) < 0 Urbino, September 18, /

33 Conclusions: effect of eterogeneities Farris Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:

34 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:

35 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour Urbino, September 18, / ian Italo Bischi - Lorenzo Cerboni Baiardi University of Urbino (Italy) ( URL:

36 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Urbino, September 18, /

37 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Bischi - Cerboni Baiardi Urbino, September 18, /

38 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Bischi - Cerboni Baiardi Stress on the effect of eterogeneities (i.e. differences in parameters) Urbino, September 18, /

39 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Bischi - Cerboni Baiardi Stress on the effect of eterogeneities (i.e. differences in parameters) A rich dynamical scenario is observed Urbino, September 18, /

40 Conclusions: effect of eterogeneities Farris N homogeneous firms (i.e. identical parameters) Study the synchronous dynamics and common behavour N as bifurcation parameter Bischi - Cerboni Baiardi Stress on the effect of eterogeneities (i.e. differences in parameters) A rich dynamical scenario is observed Eterogeneities and initial conditions: asymptotic behavour of the system Urbino, September 18, /

41 Bibliography Bischi, G.I., Gardini, L. and Kopel, M Analysis of Global Bifurcations in a Market Share Attraction Model, Journal of Economic Dynamics and Control, 24 (00) Bischi, G.I., Cerboni Baiardi L. Fallacies of composition in nonlinear marketing models, Communications in Nonlinear Science and Numerical Simulation, DOI: /j.cnsns (14, in press) Farris, P., Pfeifer, P.E., Nierop, E., Reibstein, D. When Five is a Crowd in the Market Share Attraction Model: The Dynamic Stability of Competition Marketing - Journal of Research and Management, 1 (1) (05) Puu T. Chaos in Duopoly Pricing Chaos, Solitons & Fractals, 1 (6) (1991) Urbino, September 18, 14 /

Stable cycles in a Cournot duopoly model of Kopel

Stable cycles in a Cournot duopoly model of Kopel Journal of Computational and Applied Mathematics 18 (008 47 58 www.elsevier.com/locate/cam Stable cycles in a Cournot duopoly model of Kopel W. Govaerts a,, R. Khoshsiar Ghaziani a,b a Department of Applied

More information

Nonlinear dynamics in the Cournot duopoly game with heterogeneous players

Nonlinear dynamics in the Cournot duopoly game with heterogeneous players arxiv:nlin/0210035v1 [nlin.cd] 16 Oct 2002 Nonlinear dynamics in the Cournot duopoly game with heterogeneous players H. N. Agiza and A. A. Elsadany Department of Mathematics, Faculty of Science Mansoura

More information

Oligopoly games with Local Monopolistic Approximation

Oligopoly games with Local Monopolistic Approximation Journal of Economic Behavior & Organization Vol. 62 (2007) 371 388 Oligopoly games with Local Monopolistic Approximation Gian Italo Bischi a,, Ahmad K. Naimzada b, Lucia Sbragia a a Department of Economics,

More information

Quaderni di Dipartimento. Double route to chaos in an heterogeneous triopoly game. Ahmad Naimzada (Università di Milano-Bicocca)

Quaderni di Dipartimento. Double route to chaos in an heterogeneous triopoly game. Ahmad Naimzada (Università di Milano-Bicocca) Quaderni di Dipartimento Double route to chaos in an heterogeneous triopoly game Ahmad Naimzada (Università di Milano-Bicocca) Fabio Tramontana (Università di Pavia) # 149 (06-11) Dipartimento di economia

More information

Analysis of nonlinear duopoly game with heterogeneous players

Analysis of nonlinear duopoly game with heterogeneous players Economic Modelling 24 (2007) 38 48 www.elsevier.com/locate/econbase Analysis of nonlinear duopoly game with heterogeneous players Jixiang Zhang a,, Qingli Da a, Yanhua Wang b a School of Economics and

More information

Chaos in adaptive expectation Cournot Puu duopoly model

Chaos in adaptive expectation Cournot Puu duopoly model Chaos in adaptive expectation Cournot Puu duopoly model Varun Pandit 1, Dr. Brahmadeo 2, and Dr. Praveen Kulshreshtha 3 1 Economics, HSS, IIT Kanpur(India) 2 Ex-Professor, Materials Science and Engineering,

More information

Discussion Paper Series No.51. Complexity yields benefits: An example of a heterogeneous duopoly market. Yasuo Nonaka.

Discussion Paper Series No.51. Complexity yields benefits: An example of a heterogeneous duopoly market. Yasuo Nonaka. Discussion Paper Series No.5 Complexity yields benefits: An example of a heterogeneous duopoly market Yasuo Nonaka October 27 2003 Complexity leads benefits : An example of a heterogeneous duopoly market

More information

Multistability and cyclic attractors in duopoly games

Multistability and cyclic attractors in duopoly games Chaos, Solitons and Fractals 11 (2000) 543±564 www.elsevier.nl/locate/chaos Multistability and cyclic attractors in duopoly games Gian Italo Bischi a, Cristiana Mammana b, Laura Gardini a,c, * a Istituto

More information

Chaotic Dynamics and Synchronization of Cournot Duopoly Game with a Logarithmic Demand Function

Chaotic Dynamics and Synchronization of Cournot Duopoly Game with a Logarithmic Demand Function Appl. Math. Inf. Sci. 9, No. 6, 3083-3094 015 3083 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.1785/amis/090638 Chaotic Dynamics and Synchronization of Cournot

More information

Analysis of Duopoly Output Game With Different Decision-Making Rules

Analysis of Duopoly Output Game With Different Decision-Making Rules Management Science and Engineering Vol. 9, No. 1, 015, pp. 19-4 DOI: 10.3968/6117 ISSN 1913-0341 [Print] ISSN 1913-035X [Online] www.cscanada.net www.cscanada.org Analysis of Duopoly Output Game With Different

More information

Homoclinic tangles associated with closed invariant curves in families of 2D maps

Homoclinic tangles associated with closed invariant curves in families of 2D maps ITERATION THEORY (ECIT 04) W. Förg-Rob, L. Gardini, D. Gronau, L. Reich, J. Smítal (Eds.) Grazer Math. Ber., ISSN 1016 7692 Bericht Nr. 350 (2006), 1 14 Homoclinic tangles associated with closed invariant

More information

DISCRETE-TIME DYNAMICS OF AN

DISCRETE-TIME DYNAMICS OF AN Chapter 1 DISCRETE-TIME DYNAMICS OF AN OLIGOPOLY MODEL WITH DIFFERENTIATED GOODS K. Andriopoulos, T. Bountis and S. Dimas * Department of Mathematics, University of Patras, Patras, GR-26500, Greece Abstract

More information

Heterogenous Competition in a Differentiated Duopoly with Behavioral Uncertainty

Heterogenous Competition in a Differentiated Duopoly with Behavioral Uncertainty Heterogenous Competition in a Differentiated Duopoly with Behavioral Uncertainty Akio Matsumoto Department of Systems Industrial Engineering Univesity of Arizona, USA Department of Economics Chuo University,

More information

A Final Bifurcation in a simple duopoly model

A Final Bifurcation in a simple duopoly model A Final Bifurcation in a simple duopoly model NATASCIA ANGELINI Bologna University MEDAlics, Dante Alighieri University Via del Torrione, Reggio Calabria ITALY natascia.angelini@unibo.it MASSIMILIANO FERRARA

More information

Mathematical Properties of a Combined Cournot-Stackelberg Model

Mathematical Properties of a Combined Cournot-Stackelberg Model ISSN 974-40 WP-EMS Working Papers Series in Economics, Mathematics and Statistics Mathematical Properties of a Combined Cournot-Stackelberg Model Fabio Tramontana (U. Ancona) Laura Gardini (U. Urbino)

More information

INVARIANT CURVES AND FOCAL POINTS IN A LYNESS ITERATIVE PROCESS

INVARIANT CURVES AND FOCAL POINTS IN A LYNESS ITERATIVE PROCESS International Journal of Bifurcation and Chaos, Vol. 13, No. 7 (2003) 1841 1852 c World Scientific Publishing Company INVARIANT CURVES AND FOCAL POINTS IN A LYNESS ITERATIVE PROCESS L. GARDINI and G. I.

More information

Price and Quantity Competition in Dynamic Oligopolies with Product Differentiation

Price and Quantity Competition in Dynamic Oligopolies with Product Differentiation Price and Quantity Competition in Dynamic Oligopolies with Product Differentiation Akio Matsumoto Chuo University Ferenc Szidarovszky University of Arizona Abstract This is a continuation of the earlier

More information

Research Article Complexity Analysis of a Cournot-Bertrand Duopoly Game Model with Limited Information

Research Article Complexity Analysis of a Cournot-Bertrand Duopoly Game Model with Limited Information Discrete Dynamics in Nature and Society Volume, Article ID 877, 6 pages http://dx.doi.org/.55//877 Research Article Complexity Analysis of a Cournot-Bertrand Duopoly Game Model with Limited Information

More information

Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures. 1 Introduction

Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures. 1 Introduction ISSN 749-889 (print), 749-897 (online) International Journal of Nonlinear Science Vol.5(8) No., pp.9-8 Dynamics of a Bertrand Game with Heterogeneous Players and Different Delay Structures Huanhuan Liu,

More information

Nonlinear dynamics in a duopoly with price competition and vertical differentiation

Nonlinear dynamics in a duopoly with price competition and vertical differentiation Nonlinear dynamics in a duopoly with price competition and vertical differentiation uciano Fanti, uca Gori and Mauro Sodini Department of Economics and Management, University of Pisa, Via Cosimo Ridolfi,

More information

arxiv: v1 [q-fin.ec] 15 Oct 2015

arxiv: v1 [q-fin.ec] 15 Oct 2015 Stability and Chaos in a Multi-Market Oligopoly with Economies of Scale arxiv:1510.04550v1 [q-fin.ec] 15 Oct 2015 Marcelo J. Villena and Axel A. Araneda November 20, 2017 Faculty of Engineering & Sciences,

More information

Research Article Chaos Control on a Duopoly Game with Homogeneous Strategy

Research Article Chaos Control on a Duopoly Game with Homogeneous Strategy Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 16, Article ID 74185, 7 pages http://dx.doi.org/1.1155/16/74185 Publication Year 16 Research Article Chaos Control on a Duopoly

More information

MDEF th Workshop MDEF Modelli Dinamici in Economia e Finanza Dynamic Models in Economics and Finance

MDEF th Workshop MDEF Modelli Dinamici in Economia e Finanza Dynamic Models in Economics and Finance MDEF2014 8th Workshop MDEF Modelli Dinamici in Economia e Finanza Dynamic Models in Economics and Finance Dynamics of heterogeneous oligopolies with best response mechanisms F. Cavalli, A. Naimzada, M.

More information

Modeling of Chaotic Behavior in the Economic Model

Modeling of Chaotic Behavior in the Economic Model Chaotic Modeling and Simulation (CMSIM) 3: 9-98, 06 Modeling of Chaotic Behavior in the Economic Model Volodymyr Rusyn, Oleksandr Savko Department of Radiotechnics and Information Security, Yuriy Fedkovych

More information

Routes to Complexity in a Macroeconomic Model Described by a Noninvertible Triangular Map

Routes to Complexity in a Macroeconomic Model Described by a Noninvertible Triangular Map Cubo A Mathematical Journal Vol.05/N ō 03 OCTOBER 2003 Routes to Complexity in a Macroeconomic Model Described by a Noninvertible Triangular Map Roberto Dieci Dipartimento di Studi Economici e Quantitativi,

More information

Cournot Duopoly when the Competitors Operate Multiple Production Plants

Cournot Duopoly when the Competitors Operate Multiple Production Plants ISSN 1974-4110 WP-EMS Working Papers Series in Economics, Mathematics and Statistics Cournot Duopoly when the Competitors Operate Multiple Production Plants Fabio Tramontana, (U. Ancona & U. Urbino) Laura

More information

Economics Discussion Paper Series EDP Dynamic analysis of discontinuous best response with innovation

Economics Discussion Paper Series EDP Dynamic analysis of discontinuous best response with innovation Economics Discussion Paper Series EDP-1708 Dynamic analysis of discontinuous best response with innovation Fabio Lamantia Mario Pezzino Fabio Tramontana February 2017 Economics School of Social Sciences

More information

Dynamical Behavior Analysis of Fixed Points of Investment Competition Model

Dynamical Behavior Analysis of Fixed Points of Investment Competition Model Dynamical Behavior Analysis of Fixed Points of Investment Competition Model Shujuan Guo 1,BingHan, and Chunmei Yuan 1 1 School of Physics and Mathematics, Changzhou University Changzhou 13164, China shujuanguohb@yahoo.cn,

More information

Maps with vanishing denominator explained through applications in Economics

Maps with vanishing denominator explained through applications in Economics Journal of Physics: Conference Series PAPER OPEN ACCESS Maps with vanishing denominator explained through applications in Economics To cite this article: Fabio Tramontana 2016 J. Phys.: Conf. Ser. 692

More information

A COMPETITION GAME WITH KNOWLEDGE ACCUMULATION AND SPILLOVERS

A COMPETITION GAME WITH KNOWLEDGE ACCUMULATION AND SPILLOVERS International Game Theory Review, Vol. 6, No. 3 (24) 323 34 c World Scientific Publishing Company A COMPETITION GAME WITH KNOWLEDGE ACCUMULATION AND SPILLOVERS GIAN-ITALO BISCHI Istituto di Scienze Economiche,

More information

Delayed Dynamics in Heterogeneous Competition with Product Differentiation

Delayed Dynamics in Heterogeneous Competition with Product Differentiation Delayed Dynamics in Heterogeneous Competition with Product Differentiation Akio Matsumoto Department of Economics Chuo University Hachioji, Tokyo, 19-0393, Japan akiom@tamacc.chuo-u.ac.jp Ferenc Szidarovszky

More information

A Special Labor-Managed Oligopoly

A Special Labor-Managed Oligopoly A Special Labor-Managed Oligopoly Ferenc Szidarovszky University of Arizona Akio Matsumoto Chuo University Abstract A special labor-managed oligopoly without product differentiation is considered. The

More information

THEORY OF OLIGOPOLIES: DYNAMICS AND STABILITY OF EQUILIBRIA

THEORY OF OLIGOPOLIES: DYNAMICS AND STABILITY OF EQUILIBRIA ROMAI J., 4, (2008), 47 60 THEORY OF OLIGOPOLIES: DYNAMICS AND STABILITY OF EQUILIBRIA Konstantinos Andriopoulos, Tassos Bountis, Nikos Papadopoulos Centre for Research and Applications of Nonlinear Systems

More information

Global Dynamics in Binary Choice Models with Social Influence Downloaded By: [Bischi, Gian-Italo] At: 17:46 21 September 2009

Global Dynamics in Binary Choice Models with Social Influence Downloaded By: [Bischi, Gian-Italo] At: 17:46 21 September 2009 Journal of Mathematical Sociology, 33:277 302, 2009 Copyright Taylor & Francis Group, LLC ISSN: 0022-250X print/1545-5874 online DOI: 10.1080/00222500902979963 Global Dynamics in Binary Choice Models with

More information

UNIVERSITY of LIMERICK

UNIVERSITY of LIMERICK UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics & Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MS08 SEMESTER: Spring 0 MODULE TITLE: Dynamical

More information

Analysis of a duopoly game with heterogeneous players participating in carbon emission trading

Analysis of a duopoly game with heterogeneous players participating in carbon emission trading 118 Nonlinear Analysis: Modelling and Control, 2014, Vol. 19, No. 1, 118 131 Analysis of a duopoly game with heterogeneous players participating in carbon emission trading Lingrui Zhao, Jixiang Zhang College

More information

Center Bifurcation of a point on the Poincaré Equator

Center Bifurcation of a point on the Poincaré Equator ITERATION THEORY (ECIT 08) A.N. Sharkovsky, I.M. Sushko (Eds.) Grazer Math. Ber., ISSN 1016 7692 Bericht Nr. 354 (2009), 254-279 Center Bifurcation of a point on the Poincaré Equator Iryna Sushko and Laura

More information

Coupled Chaotic Fluctuations in a Model of International Trade and Innovation: Some Preliminary Results

Coupled Chaotic Fluctuations in a Model of International Trade and Innovation: Some Preliminary Results Coupled Chaotic Fluctuations in a Model of International Trade and Innovation: Some Preliminary Results Iryna Sushko a,b, Laura Gardini c, Kiminori Matsuyama d a Institute of Mathematics, NASU, b Kyiv

More information

Fractional Dynamic Oligopoly Model

Fractional Dynamic Oligopoly Model Fractional Dynamic Oligopoly Model Sipang Direkhunakorn Sripatum University 2410/2 Phaholyothin Road, Jatujak, Bangkok 10900 Thailand E-mail: sipang.di@spu.ac.th 2015 Summer Solstice 7th International

More information

Delayed Dynamics in Heterogeneous Competition with Product Differentiation

Delayed Dynamics in Heterogeneous Competition with Product Differentiation Delayed Dynamics in Heterogeneous Competition with Product Differentiation Akio Matsumoto Department of Economics Chuo University Hachioji, Tokyo, 19-0393, Japan akiom@tamacc.chuo-u.ac.jp Ferenc Szidarovszky

More information

INTRICATE ASSET PRICE

INTRICATE ASSET PRICE Chapter 1 INTRICATE ASSET PRICE DYNAMICS AND ONE-DIMENSIONAL DISCONTINUOUS MAPS F. Tramontana, L. Gardini and F. Westerhoff * Department of Economics and Quantitative Methods, University of Urbino, Via

More information

Discussion Paper No.241. Delay Dynamics of a Cournot Game with Heterogenous Duopolies. Akio Matsumoto Chuo University

Discussion Paper No.241. Delay Dynamics of a Cournot Game with Heterogenous Duopolies. Akio Matsumoto Chuo University Discussion Paper No.241 Delay Dynamics of a Cournot Game with Heterogenous Duopolies Akio Matsumoto Chuo University Ferenc Szidarovszky University of Pécs January 2015 INSTITUTE OF ECONOMIC RESEARCH Chuo

More information

NONLINEAR DELAY MONOPOLY WITH BOUNDED RATIONALITY

NONLINEAR DELAY MONOPOLY WITH BOUNDED RATIONALITY NONLINEAR DELAY MONOPOLY WITH BOUNDED RATIONALITY AKIO MATSUMOTO AND FERENC SZIDAROVSZKY Abstract. The purpose of this paper is to study dynamics of a monopolistic rm in a continuous-time framework. The

More information

Hyperchaotic Dynamic of Cournot-Bertrand Duopoly Game with Multi-Product and Chaos Control

Hyperchaotic Dynamic of Cournot-Bertrand Duopoly Game with Multi-Product and Chaos Control Hyperchaotic Dynamic of Cournot-Bertrand Duopoly Game with Multi-Product and Chaos Control FANG WU Group of Nonlinear Dynamics and Chaos School of Management and Economics Tianjin University Weijin Street

More information

A Multiobjective Model of Oligopolies under Uncertainty

A Multiobjective Model of Oligopolies under Uncertainty CUBO A Mathematical Journal Vol.11, N ō 02, (107 115). May 2009 A Multiobjective Model of Oligopolies under Uncertainty Carl Chiarella School of Finance and Economics, University of Technology, Sydney,

More information

Asymptotic relations in Cournot s game

Asymptotic relations in Cournot s game MPRA Munich Personal RePEc Archive Asymptotic relations in Cournot s game Marco Guerrazzi Departmento of Economics, University of Genoa November 0 Online at http://mpra.ub.uni-muenchen.de/476/ MPRA Paper

More information

Basics of Game Theory

Basics of Game Theory Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy {giacomo.bacci,luca.sanguinetti}@iet.unipi.it April - May, 2010 G. Bacci and

More information

On Hotelling s Stability in Competition

On Hotelling s Stability in Competition On Hotelling s Stability in Competition Claude d Aspremont, Jean Jaskold Gabszewicz and Jacques-François Thisse Manuscript received March, 1978; revision received June, 1978 Abstract The purpose of this

More information

Learning to Play Best Response in Duopoly Games

Learning to Play Best Response in Duopoly Games Learning to Play Best Response in Duopoly Games By Jan Wenzelburger Fakultät für Wirtschaftswissenschaften Universität Bielefeld, Postfach 100 131 D-33501 Bielefeld, Germany jwenzelb@wiwi.uni-bielefeld.de

More information

Research Article Chaos and Control of Game Model Based on Heterogeneous Expectations in Electric Power Triopoly

Research Article Chaos and Control of Game Model Based on Heterogeneous Expectations in Electric Power Triopoly Discrete Dnamics in Nature and Societ Volume 29, Article ID 469564, 8 pages doi:.55/29/469564 Research Article Chaos and Control of Game Model Based on Heterogeneous Epectations in Electric Power Triopol

More information

An explicit dynamic model of segregation

An explicit dynamic model of segregation An explicit dynamic model of segregation Gian-Italo Bischi Dipartimento di Economia e Metodi Quantitativi Universita di Urbino "Carlo Bo" e-mail:gian.bischi@uniurb.it Ugo Merlone Dip. di Statistica e Matematica

More information

4. Partial Equilibrium under Imperfect Competition

4. Partial Equilibrium under Imperfect Competition 4. Partial Equilibrium under Imperfect Competition Partial equilibrium studies the existence of equilibrium in the market of a given commodity and analyzes its properties. Prices in other markets as well

More information

A Stackelberg Game Model of Dynamic Duopolistic Competition with Sticky Prices. Abstract

A Stackelberg Game Model of Dynamic Duopolistic Competition with Sticky Prices. Abstract A Stackelberg Game Model of Dynamic Duopolistic Competition with Sticky Prices Kenji Fujiwara School of Economics, Kwansei Gakuin University Abstract We develop the following Stackelberg game model of

More information

DYNAMICAL ANALYSIS AND CHAOS CONTROL IN A HETEROGENEOUS KOPEL DUOPOLY GAME. A. A. Elsadany and A. M. Awad

DYNAMICAL ANALYSIS AND CHAOS CONTROL IN A HETEROGENEOUS KOPEL DUOPOLY GAME. A. A. Elsadany and A. M. Awad Indian J. Pure Appl. Math., 47(4): 617-639, December 016 c Indian National Science Academy DOI: 10.1007/s136-016-006-3 DYNAMICAL ANALYSIS AND CHAOS CONTROL IN A HETEROGENEOUS KOPEL DUOPOLY GAME A. A. Elsadany

More information

Direction and Stability of Hopf Bifurcation in a Delayed Model with Heterogeneous Fundamentalists

Direction and Stability of Hopf Bifurcation in a Delayed Model with Heterogeneous Fundamentalists International Journal of Mathematical Analysis Vol 9, 2015, no 38, 1869-1875 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ijma201554135 Direction and Stability of Hopf Bifurcation in a Delayed Model

More information

Some forgotten equilibria of the Bertrand duopoly!?

Some forgotten equilibria of the Bertrand duopoly!? Some forgotten equilibria of the Bertrand duopoly!? Mathias Erlei Clausthal University of Technology Julius-Albert-Str. 2, 38678 Clausthal-Zellerfeld, Germany email: m.erlei@tu-clausthal.de Abstract This

More information

The dynamic game between imported luxury car and domestic car and competition strategy comparison in Chinese car market

The dynamic game between imported luxury car and domestic car and competition strategy comparison in Chinese car market The dynamic game between imported luxury car and domestic car and competition strategy comparison in Chinese car market FANG WU Tianjin University School of Management Weijin Street 92, 300072 Tianjin

More information

Lecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of:

Lecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of: Lecture 6 Chaos Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of: Chaos, Attractors and strange attractors Transient chaos Lorenz Equations

More information

A Cournot duopoly model with complementary goods: multistability and complex structures in the basins of attraction.

A Cournot duopoly model with complementary goods: multistability and complex structures in the basins of attraction. Fernando Bignami, Anna Agliari A Cournot duopoly model with complementary goods: multistaility and complex structures in the asins of attraction. In this paper we study a Cournot duopoly, proposed y Matsumoto

More information

Chapter 3. Gumowski-Mira Map. 3.1 Introduction

Chapter 3. Gumowski-Mira Map. 3.1 Introduction Chapter 3 Gumowski-Mira Map 3.1 Introduction Non linear recurrence relations model many real world systems and help in analysing their possible asymptotic behaviour as the parameters are varied [17]. Here

More information

Market Equilibrium and the Core

Market Equilibrium and the Core Market Equilibrium and the Core Ram Singh Lecture 3-4 September 22/25, 2017 Ram Singh (DSE) Market Equilibrium September 22/25, 2017 1 / 19 Market Exchange: Basics Let us introduce price in our pure exchange

More information

Transitions to Chaos in a 3-Dimensional Map

Transitions to Chaos in a 3-Dimensional Map Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 5 (2016), pp. 4597 4611 Research India Publications http://www.ripublication.com/gjpam.htm Transitions to Chaos in a 3-Dimensional

More information

Dynamical Behavior of Logistic Equation through Symbiosis Model

Dynamical Behavior of Logistic Equation through Symbiosis Model International Journal of Scientific and Research Publications, Volume 3, Issue 6, June 213 1 ISSN 225-3153 Dynamical Behavior of Logistic Equation through Symbiosis Model M.Sambasiavm Assistant Professor

More information

Mathematical Foundations II

Mathematical Foundations II Mathematical Foundations 2-1- Mathematical Foundations II A. Level and superlevel sets 2 B. Convex sets and concave functions 4 C. Parameter changes: Envelope Theorem I 17 D. Envelope Theorem II 41 48

More information

Additive resonances of a controlled van der Pol-Duffing oscillator

Additive resonances of a controlled van der Pol-Duffing oscillator Additive resonances of a controlled van der Pol-Duffing oscillator This paper has been published in Journal of Sound and Vibration vol. 5 issue - 8 pp.-. J.C. Ji N. Zhang Faculty of Engineering University

More information

Citation for published version (APA): Kopányi, D. (2015). Bounded rationality and learning in market competition Amsterdam: Tinbergen Institute

Citation for published version (APA): Kopányi, D. (2015). Bounded rationality and learning in market competition Amsterdam: Tinbergen Institute UvA-DARE (Digital Academic Repository) Bounded rationality and learning in market competition Kopányi, D. Link to publication Citation for published version (APA): Kopányi, D. (2015). Bounded rationality

More information

On Riddled Sets and Bifurcations of Chaotic Attractors

On Riddled Sets and Bifurcations of Chaotic Attractors Applied Mathematical Sciences, Vol. 1, 2007, no. 13, 603-614 On Riddled Sets and Bifurcations of Chaotic Attractors I. Djellit Department of Mathematics University of Annaba B.P. 12, 23000 Annaba, Algeria

More information

HERD BEHAVIOR AND NONFUNDAMENTAL ASSET PRICE FLUCTUATIONS IN FINANCIAL MARKETS

HERD BEHAVIOR AND NONFUNDAMENTAL ASSET PRICE FLUCTUATIONS IN FINANCIAL MARKETS Macroeconomic Dynamics, 10, 2006, 502 528. Printed in the United States of America. DOI: 10.1017.S136510050605036X HERD BEHAVIOR AND NONFUNDAMENTAL ASSET PRICE FLUCTUATIONS IN FINANCIAL MARKETS GIAN-ITALO

More information

Using MatContM in the study of a nonlinear map in economics

Using MatContM in the study of a nonlinear map in economics Journal of Phsics: Conference Series PAPER OPEN ACCESS Using MatContM in the stud of a nonlinear map in economics To cite this article: Niels Neirnck et al 016 J. Phs.: Conf. Ser. 69 0101 Related content

More information

AN EXTENSION OF THE ANTOCI-DEI-GALEOTTI EVOLUTIONARY MODEL FOR ENVIRONMENT PROTECTION THROUGH FINANCIAL INSTRUMENTS

AN EXTENSION OF THE ANTOCI-DEI-GALEOTTI EVOLUTIONARY MODEL FOR ENVIRONMENT PROTECTION THROUGH FINANCIAL INSTRUMENTS ISSN 1974-4110 WP-EMS Working Papers Series in Economics, Mathematics and Statistics AN EXTENSION OF THE ANTOCI-DEI-GALEOTTI EVOLUTIONARY MODEL FOR ENVIRONMENT PROTECTION THROUGH FINANCIAL INSTRUMENTS

More information

Research Article Nonlinear Dynamics in a Cournot Duopoly with Different Attitudes towards Strategic Uncertainty

Research Article Nonlinear Dynamics in a Cournot Duopoly with Different Attitudes towards Strategic Uncertainty Abstract and Applied Analysis Volume 3, Article ID 339, pages http://dx.doi.org/.55/3/339 Research Article Nonlinear Dynamics in a Cournot Duopoly with Different Attitudes towards Strategic Uncertainty

More information

IERCU. Dynamic Monopoly with Demand Delay

IERCU. Dynamic Monopoly with Demand Delay IERCU Institute of Economic Research, Chuo University 50th Anniversary Special Issues Discussion Paper No.15 Dynamic Monopoly with Demand Delay Akio Matsumoto Chuo University Ferenc Szidarovszky University

More information

FADING MEMORY LEARNING IN THE COBWEB MODEL WITH RISK AVERSE HETEROGENEOUS PRODUCERS

FADING MEMORY LEARNING IN THE COBWEB MODEL WITH RISK AVERSE HETEROGENEOUS PRODUCERS FADING MEMORY LEARNING IN THE COBWEB MODEL WITH RISK AVERSE HETEROGENEOUS PRODUCERS CARL CHIARELLA, XUE-ZHONG HE AND PEIYUAN ZHU School of Finance and Economics University of Technology, Sydney PO Box

More information

Dynamic IS-LM model with Philips Curve and International Trade

Dynamic IS-LM model with Philips Curve and International Trade Journal of Mathematics and System Science 6 (2016) 147-158 doi: 10.17265/2159-5291/2016.04.003 D DAVID PUBLISHING Dynamic IS-LM model with Philips Curve and International Trade Michiya Nozaki Gifu Keizai

More information

DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS

DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University

More information

ANALYSIS AND CONTROLLING OF HOPF BIFURCATION FOR CHAOTIC VAN DER POL-DUFFING SYSTEM. China

ANALYSIS AND CONTROLLING OF HOPF BIFURCATION FOR CHAOTIC VAN DER POL-DUFFING SYSTEM. China Mathematical and Computational Applications, Vol. 9, No., pp. 84-9, 4 ANALYSIS AND CONTROLLING OF HOPF BIFURCATION FOR CHAOTIC VAN DER POL-DUFFING SYSTEM Ping Cai,, Jia-Shi Tang, Zhen-Bo Li College of

More information

Advanced Microeconomics

Advanced Microeconomics Advanced Microeconomics Leonardo Felli EC441: Room D.106, Z.332, D.109 Lecture 8 bis: 24 November 2004 Monopoly Consider now the pricing behavior of a profit maximizing monopolist: a firm that is the only

More information

Oligopoly. Molly W. Dahl Georgetown University Econ 101 Spring 2009

Oligopoly. Molly W. Dahl Georgetown University Econ 101 Spring 2009 Oligopoly Molly W. Dahl Georgetown University Econ 101 Spring 2009 1 Oligopoly A monopoly is an industry consisting a single firm. A duopoly is an industry consisting of two firms. An oligopoly is an industry

More information

Static Models of Oligopoly

Static Models of Oligopoly Static Models of Oligopoly Cournot and Bertrand Models Mateusz Szetela 1 1 Collegium of Economic Analysis Warsaw School of Economics 3 March 2016 Outline 1 Introduction Game Theory and Oligopolies 2 The

More information

CHAOTIC BEHAVIOR IN A TWO-DIMENSIONAL BUSINESS CYCLE MODEL

CHAOTIC BEHAVIOR IN A TWO-DIMENSIONAL BUSINESS CYCLE MODEL CHAOTIC BEHAVIOR IN A TWO-DIMENSIONAL BUSINESS CYCLE MODEL CRISTINA JANUÁRIO Department of Chemistry, Mathematics Unit, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, 1949-014

More information

Oligopoly Theory 2 Bertrand Market Games

Oligopoly Theory 2 Bertrand Market Games 1/10 Oligopoly Theory 2 Bertrand Market Games May 4, 2014 2/10 Outline 1 Bertrand Market Game 2 Bertrand Paradox 3 Asymmetric Firms 3/10 Bertrand Duopoly Market Game Discontinuous Payoff Functions (1 p

More information

Mathematical Methods and Economic Theory

Mathematical Methods and Economic Theory Mathematical Methods and Economic Theory Anjan Mukherji Subrata Guha C 263944 OXTORD UNIVERSITY PRESS Contents Preface SECTION I 1 Introduction 3 1.1 The Objective 3 1.2 The Tools for Section I 4 2 Basic

More information

Oligopoly Notes. Simona Montagnana

Oligopoly Notes. Simona Montagnana Oligopoly Notes Simona Montagnana Question 1. Write down a homogeneous good duopoly model of quantity competition. Using your model, explain the following: (a) the reaction function of the Stackelberg

More information

Monopoly Part III: Multiple Local Equilibria and Adaptive Search

Monopoly Part III: Multiple Local Equilibria and Adaptive Search FH-Kiel University of Applie Sciences Prof Dr Anreas Thiemer, 00 e-mail: anreasthiemer@fh-kiele Monopoly Part III: Multiple Local Equilibria an Aaptive Search Summary: The information about market eman

More information

Stability and Hopf Bifurcation for a Discrete Disease Spreading Model in Complex Networks

Stability and Hopf Bifurcation for a Discrete Disease Spreading Model in Complex Networks International Journal of Difference Equations ISSN 0973-5321, Volume 4, Number 1, pp. 155 163 (2009) http://campus.mst.edu/ijde Stability and Hopf Bifurcation for a Discrete Disease Spreading Model in

More information

PROJECTS on. Multiple Bifurcations of Sample Dynamical Systems

PROJECTS on. Multiple Bifurcations of Sample Dynamical Systems Course on Bifurcation Theory, a.y. 2010/11 PROJECTS on Multiple Bifurcations of Sample Dynamical Systems Students of the Bifurcation Theory course can carry out an individual homework, to be discussed

More information

A model of financial market dynamics with heterogeneous beliefs and state-dependent confidence

A model of financial market dynamics with heterogeneous beliefs and state-dependent confidence A model of financial market dynamics with heterogeneous beliefs and state-dependent confidence Carl Chiarella (carl.chiarella@uts.edu.au) School of Finance and Economics, University of Technology Sydney,

More information

UNBOUNDED SETS OF ATTRACTION

UNBOUNDED SETS OF ATTRACTION International Journal of Bifurcation and Chaos, Vol. 10, No. 6 (2000) 1437 1469 c World Scientific Publishing Company UNBOUNDED SETS OF ATTRACTION GIAN-ITALO BISCHI Istituto di Scienze Economiche, University

More information

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution

More information

Strange Attractors and Chaotic Behavior of a Mathematical Model for a Centrifugal Filter with Feedback

Strange Attractors and Chaotic Behavior of a Mathematical Model for a Centrifugal Filter with Feedback Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 4, Number 2, pp. 179 194 (2009) http://campus.mst.edu/adsa Strange Attractors and Chaotic Behavior of a Mathematical Model for a Centrifugal

More information

Global Bifurcations in a Three-Dimensional Financial Model of Bull and Bear Interactions

Global Bifurcations in a Three-Dimensional Financial Model of Bull and Bear Interactions Global Bifurcations in a Three-Dimensional Financial Model of Bull and Bear Interactions Fabio Tramontana, Laura Gardini, Roberto Dieci, and Frank Westerhoff 1 Introduction In a previous paper Tramontana

More information

epub WU Institutional Repository

epub WU Institutional Repository epub WU Institutional Repository Pasquale Commendatore and Ingrid Kubin and Spiros Bougheas and Alan Kirman and Michael Kopel and Gian Italo Bischi Introduction Book Section (Published) (Refereed) Original

More information

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY. S. H. Saker

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY. S. H. Saker Nonlinear Funct. Anal. & Appl. Vol. 10 No. 005 pp. 311 34 EXISTENCE OF POSITIVE PERIODIC SOLUTIONS OF DISCRETE MODEL FOR THE INTERACTION OF DEMAND AND SUPPLY S. H. Saker Abstract. In this paper we derive

More information

PRICE DYNAMICS IN EQUILBRIUM MODELS

PRICE DYNAMICS IN EQUILBRIUM MODELS PRICE DYNAMICS IN EQUILBRIUM MODELS Advances in Computational Economics VOLUME 16 SERIES EDITORS Hans Amman, University of Amsterdam, Amsterdam, The Netherlands Anna Nagumey, University of Massachusetts

More information

= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F :

= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F : 1 Bifurcations Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 A bifurcation is a qualitative change

More information

SOME NOTES ON APPLYING THE HERFINDAHL-HIRSCHMAN INDEX

SOME NOTES ON APPLYING THE HERFINDAHL-HIRSCHMAN INDEX SOME NOTES ON APPLYING THE HERFINDAHL-HIRSCHMAN INDEX AKIO MATSUMOTO, UGO MERLONE, AND FERENC SZIDAROVSZKY Abstract. The Herfindahl-Hirschman Index is one of the most commonly used indicators to detect

More information

Research on price game process and wavelet chaos control of three oligarchs insurance market in China

Research on price game process and wavelet chaos control of three oligarchs insurance market in China Research on price game process and wavelet chaos control of three oligarchs insurance market in China WENBO REN TianJin University College of Management and economics TianJin, 37 China rw9@6.com JUNHAI

More information

Stability Analysis of Uzawa-Lucas Endogenous Growth Model

Stability Analysis of Uzawa-Lucas Endogenous Growth Model Abstract: Stability Analysis of Uzawa-Lucas Endogenous Growth Model William A. Barnett* University of Kansas, Lawrence and Center for Financial Stability, NY City and Taniya Ghosh Indira Gandhi Institute

More information

Stability and Bifurcation in the Hénon Map and its Generalizations

Stability and Bifurcation in the Hénon Map and its Generalizations Chaotic Modeling and Simulation (CMSIM) 4: 529 538, 2013 Stability and Bifurcation in the Hénon Map and its Generalizations O. Ozgur Aybar 1, I. Kusbeyzi Aybar 2, and A. S. Hacinliyan 3 1 Gebze Institute

More information

Research Article Global Dynamics of a Competitive System of Rational Difference Equations in the Plane

Research Article Global Dynamics of a Competitive System of Rational Difference Equations in the Plane Hindawi Publishing Corporation Advances in Difference Equations Volume 009 Article ID 1380 30 pages doi:101155/009/1380 Research Article Global Dynamics of a Competitive System of Rational Difference Equations

More information