MOVEMENT OF EQUILIBRIUM OF COURNOT DUOPOLY AND THE VISUALIZATION OF BIFURCATIONS OF ITS ADJUSTMENT DYNAMICS

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1 The Regionl Economics Applicions Loror (REAL) is cooperive venure eween he Universi of Illinois nd he Federl Reserve Bnk of Chicgo focusing on he developmen nd use of nlicl models for urn nd regionl economic developmen. The purpose of he Discussion Ppers is o circule inermedie nd finl resuls of his reserch mong reders wihin nd ouside REAL. The opinions nd conclusions epressed in he ppers re hose of he uhors nd do no necessril represen hose of he Federl Reserve Bnk of Chicgo, Federl Reserve Bord of Governors or he Universi of Illinois. All requess nd commens should e direced o Geoffre J.D. Hewings, Direcor, Regionl Economics Applicions Loror, 607 Souh Mhews, Urn, IL, , Phone (7) , F (7) We pge: MOVEMENT OF EQUILIBRIUM OF COURNOT DUOPOLY AND THE VISUALIZATION OF BIFURCATIONS OF ITS ADJUSTMENT DYNAMICS Michel Sonis nd Chokri Dridi REAL 03-T-0 M, 003

2 MOVEMENT OF EQUILIBRIUM OF COURNOT DUOPOLY AND THE VISUALIZATION OF BIFURCATIONS OF ITS ADJUSTMENT DYNAMICS Michel Sonis,c nd Chokri Dridi,c. Br-Iln Universi, Isrel. Deprmen of Agriculurl nd Consumer Economics, Universi of Illinois Urn-Chmpign, USA c. Regionl Economics Applicion Loror, Universi of Illinois Urn-Chmpign, USA Asrc: This pper dels wih he nlicl nd grphicl represenion of he ifurcions ppering from he djusmen dnmics of -pler Courno duopol, proposed Puu (997). We eslish dmissiili condiions on he iniil se of he djusmen dnmics nd visulize he dnmics in he spce of oris. Kewords: Bifurcion surfces, Chos, Courno, Duopol, Nonliner dnmics, Sili.. INTRODUCTION. Courno (838) undersood nd clerl sed h he quni compeiion in noncoopering duopol leds o quni oucome h is sle nd h n firm deviing from i, self-ineres lone, will e rough ck o equilirium in sequence of djusmens. While no using he sme ec erms, Courno lso sed in reference o he sili of he oucome h he duopol equilirium is n rcor while he cooperive monopol equilirium poin is repeller herefore, while mos fvorle o producers, i is unsle wihou selfenforcemen or n enforcing mechnism. Friedmn (983) considers mopic duopol where ech firm mimizes is profi in given period given wh is rivl(s) produced in he previous period, oviousl such model cn e criicized on mn poins u for us one slien filure, menioned he uhor, is he inili of firms o djus heir oupus o he oher firms' oupu level. In his pper, we consider n djusmen mechnism where firm's oupu in period + Corresponding uhor. E-mil ddresses: sonism@mil.iu.c.il (M. Sonis) nd cdridi@uiuc.edu (C. Dridi)

3 is he oupu in period djused frcion of he oupu ecess/shorge in he previous period (Puu, 997). This pper ims o nlicll descrie he criicl ifurcion curves, while oher uhors did recognize he ifurcion effecs using ril nd error pproch (Agiz, 998), here we derive he ec forms of ifurcion sed on prese vlues of coordines of Courno equilirium chosen s ifurcion prmeers nd visulize he dnmics moving he equilirium poin long given ph. In he ne secion, we descrie he djusmen dnmics nd chrcerize is sili, in he hird secion we sud he ifurcion emerging from he djusmen dnmics, in secion four we represen ifurcion perns of ineres nd pu condiions on dmissile iniil poins, nd secion five concludes he pper.. MODEL Puu (997, Ch.5) inroduced n ierive process, which leds wo oligopolies o he Courno- Nsh equilirium vi he following ierive process: where (, ) + + F HG F HG = + λ = + µ re he supplies ime of wo compeiors in duopol; nd re heir consn mrginl coss, nd λ nd µ re he djusmen speeds such h { λµ } I KJ I KJ () 0,. The firms produce n undifferenied produc sold mrke price P =. The fied poin + (, ) of his ierion dnmics sisfies he ssem of lgeric equions: F HG F HG = + λ = + µ I KJ I KJ () wih non-zero soluion:

4 = ; = (3) + g + g which is he Courno-Nsh equilirium. The deiled nlsis of he srucure of he domin of rcion of his equilirium nd he ifurcions of is djusmen dnmics in () cn e descried nlicll s follows (Sonis, 000, pp ). The Jcoi pproimion mri for he dnmics in () is: J, + λ λ = µ -µ (4) A he Courno equilirium, his mri ecomes J ( - ) λ λ = µ ( ) -µ (5) Therefore, g g g g TrJ = λ + µ ; de J = λ µ + λµ (6) 4 I is well known (Hsu, 997) h he domin of rcion of ever -dimensionl discree dnmics hs form: ± TrJ < dej < (7) I is es o see h for he djusmen dnmics in () we lws hve: ± TrJ < dej (8) Hence he domin of rcion of he Courno equilirium is defined he inequli de J <, which gives: λ g µ g g + λµ < (9) 4 3

5 or + g 4 < + λ µ (0) Le us inroduce he rio of he mrginl coss; F HG = k = I () KJ nd le + = Λ () λ µ Oviousl we hve k > 0 nd Λ. Then from (0)-(), he domin of rcion of he Courno equilirium is: or k g < Λ (3) + k 4k g (4) k Λ + < 0 This implies h he following inequli represens he domin of rcion for he Courno equilirium: k < k < k (5) where k, k > 0 re he (posiive) roos of (4) when rnsformed ino n equli: g g k, = Λ Λ Λ (6) Oviousl, hese roos re reciprocl, since we cn use he reciprocl rio k =. The chrcer of ifurcions in hese roos re he sme nd is defined he vlue of α (Sonis, 000, p. 34): α = TrJ = λ + µ > 0 g (7) 4

6 If he quni Ω= α rccos is rionl numer p π q hen he djusmen dnmics will e q- periodic; if Ω is irrionl, hen he djusmen dnmics will e qusi-periodic. Few emples re of specil ineres; if he djusmens ( λ, µ ) re unir, i.e. α = 0 hen we hve 4-period ccle sring he Feigenun doule-periodic w o chos; if λ + µ = hen α = nd we hve 6-period ccle ifurcions. Moreover, from (7) α > 0, he duopol djusmen cnno hve -periodic ifurcion (wih α = ) nd 3-period ifurcion (wih α = ), u i cn hve he 5-period ccle corresponding o α = , i.e., λ + µ = (see figure for he 5-periodic ccle), ec. 3. MOVEMENT OF COURNOT EQUILIBRIUM AND CORRESPONDING BIFURCATIONS OF ADJUSTMENT DYNAMICS IN THE SPACE OF ORBITS In formule (3), if he mrginl coss nd re chnging, i.e. he poin (,) is moving in he spce of mrginl coss, hen he Courno equilirium (,) lso chnges is posiion in he spce of oris (, ). For emple, if he poin (,) is moving on he srigh line = s+ r in he spce of mrginl coss hen he Courno equilirium (,) is moving on he curve = r+ s + g in he spce of oris. I is possile o see, h equion (3) implies h = ; = (8) + g + g This mens h we cn consider he coordines of he Courno equilirium s pling he role of ifurcion prmeers. In such w we re replcing he se of ifurcion prmeers ( λ, ;, µ ) new se ( λ, ;, µ ). Fiing he speeds of chnge (, ) λ µ we cn visulize in he sme spce of oris he domin of rcion of Courno equilirium, is movemen, nd he ifurcions of cul djusmen dnmics. This is convenien ecuse he oundries of he domin of rcion in he spce of oris depend on he prmeers ( λ, µ ) lone. Indeed, fer he susiuion of (8) ino (5), he nlicl descripion of he domin of rcion oins he form: 5

7 (Λ ) Λ( Λ ) < < (Λ ) + Λ( Λ ) (9) Geomericll, his inequli defines he ngle in he spce of oris wih he vere in he origin of coordines nd he sides defined he srigh lines: Since Λ hen = ( Λ ) + Λ Λ = ( Λ ) Λ Λ g g (0) ( ) (Λ ) + Λ Λ 3+ ( ) (Λ ) Λ Λ 3 () This mens h he domin of sili of Courno equilirium in he spce of oris lws includes he ngle (see figure ): 3 3+ () << inser figure here >> If he Courno equilirium lies on he oundries (0) of his domin he djusmen dnmics undergoing he ifurcion re defined he vlues of Λ = λ + µ (see eq. (7)); if he quni Λ Ω = π rccos is rionl numer p q hen he djusmen dnmics will e q-periodic; if Ω is irrionl, hen he djusmen dnmics will e qusi-periodic. 4. VISUALIZATION OF BIFURCATIONS OF ADJUSTMENT DYNAMICS The rnsfer from he spce of mrginl coss o he spce of oris, ogeher wih he immovili of he oundries of he domin of rcion in he spce of oris when he Courno equlirium is chnged wih consn speeds of chnge provides he possiili o visulize ll dmissile quliive feures of he ehvior of he djusmen dnmics, for he 6

8 equilirium ner nd on he oundries of he rcion domin. The movemens of he equilirium in he spce of oris on he segmens of srigh lines nd he crossing of he oundries of he rcion domin revel he plehor of possile ws from sili, periodici, Arnold horns nd qusi-periodici o chos. 4.. Admissile iniil ses of he djusmen dnmics The Courno-Nsh equilirium is found solving ssem of recion funcions deermining he oupu of one firms s funcion of he oupu of he oher firm, such recion funcions mus led o non-negive levels of oupu, his rnsles ino hving; 0 0 (3) nd provides he condiion on he iniil poin ( ) We cll he iniil se (, ) 0 0 produces he non-negive ses (, ), 0, 0,. 0 0 dmissile if he djusmen dnmics ech sep. The condiions of non-negivi cn e presened in he following form: Lemm. Le for some fied he following condiions hold 0 0 hen µ λ 4 Proof. Condiions () impl h (4) (5) 0; 0 (6) 7

9 Therefore, + = λ + ( λ) 0 + = µ + ( µ ) 0 (7) Furher nd λ + = λ + ( λ ) 4 λ λ + = + λ 4 4 (8) Lemm. If hen ech iniil se (, ) µ + = µ + ( µ ) 4 µ µ + = + µ 4 4 such h 0 0 (9) 4 (30) ; 0 0 (3) is dmissile nd he djusmen dnmics () converges o he rcor poin = ( + ) ; = ( + ). Proof. Condiions (30) impl h 0 4 nd 4 0, herefore, from (8) nd (9) we hve + + λ nd + + µ

10 For ech, he se (, ) is non-negive, i.e. ech iniil se sisfing (3) is dmissile. Condiion (30) geomericll represens he ngle 4 which lies inside he domin of 4 rcion defined (0) of he djusmen dnmics nd he fied poin = ; = is he rcor poin for his dnmics. + g + g Remrk. If condiion (5) does no hold, hen for visulizion of he djusmen dnmics one should chose he dmissile iniil se of he form, for emple, 0 0 suile smll ε > Visulizion of djusmen dnmics = + ε; = + ε wih The numericl procedure of he descripion of such phenomen includes he consrucion of spil ifurcion digrms in which he ifurcion prmeer is he equilirium iself. The consrucion of wo-dimensionl ifurcion digrm, i.e. he visulizion of he movemens of equiliri in he spce of oris on he segmens of srigh lines cn e ccomplished in he following w. Le us fi he speeds of chnge ( λ, µ ) nd he sme iniil se ( for, ) djusmen dnmics () ll ifurcion seps. Furher le us chose he numer of ifurcion seps S nd he segmen of he srigh line of movemen of equilirium ( ) ( ) ( ) ( ) ( 0, 0 ),( S, S ). I is possile o prmeerize he segmen of he srigh line eween ( ( 0 ), ( 0 )) nd ( ) ( ) ( S, S ) s: j j ( j) = ( 0) + ( S), S S j j ( j) = ( 0) + ( S), S S j = 0,,..., S (3) where j is ifurcion prmeer nd S is numer of ifurcion seps. g g Formule (7) helps o consruc he mrginl coss j, j for ech ifurcion sep. Choosing hese mrginl coss we cn clcule wih he help of () he ori of corresponding djusmen dnmics. The wo-dimensionl ifurcion digrm is he presenion of he "ils" 9

11 ( = P, P+,... T) of oris he djusmen dnmics (, ) on ll ifurcion seps. The usul one-dimensionl ifurcion digrm cn e oined from (3) presening for ech ( P, P,... T) = + he digrm of chnge of coordines nd seprel gins he ifurcion prmeer j. The imporn conclusion is h hese seps provide he visulizion of djusmen dnmics wih prese of quliive properies. Le us sr he visulizion of he djusmen dnmics wih he cse of rnsfer from he rcion o 4-periodic ccle (see figure ). The 4-periodic ccle corresponds o he miml speeds of djusmen λ = µ =. If he Courno equilirium is crossing over he criicl srigh lines = 3 ± he djusmen dnmics undergoes he ifurcion from he rcion o Courno equilirium o he rcive 4-periodic ccle. << inser figure here >> The 5-perido ifurcion digrms is provided in figure nd he following ifurcion digrms represen he djusmen dnmics, corresponding o he 9-period dnmics (figure 3). << inser figure 3 here >> 5. CONCLUSION In his pper he nlicl nd geomericl consrucions of he domin of rcion of - pler Puu djusmen dnmics ws chieved choosing he coordines of Courno equilirium s ifurcion prmeers. Such choice of ifurcion prmeers llows o descrie nlicll he se of ll possile ifurcion phenomen nd o visulize he prese ifurcion evens. In Sonis (000), nlogicl mehod ws elored for 3-pler Courno gme in heir spce of oris while Agiz (998) sudies he sili of 3-pler nd 4-pler Courno gmes 0

12 nd chrcerizes heir ifurcion in he spce of oris. Vriions inroduced o he sud of he djusmen dnmic of he Courno model involve heerogeneous epecions in Agiz nd Elsdn (00) where he uhors showed h vring he djusmen speed of he ounded rionl pler leds o unsle ehvior, or cpci consrin (see Puu nd Norin, 003) REFERENCES Agiz, N. H "Eplici Sili Zones wih for Courno Gme wih 3 nd 4 Compeiors", Chos, Solions nd Frcls 9,, Courno, A Recherches sur les Principes Mhémiques de l Théorie des Richesses, (English rns. 995), Jmes & Gordon Pulishers, Cliforni, Ch. 7. Friedmn, J Oligopol Theor, Cmridge Universi Press, Cmridge, Ch.. Hsu, C. S "On Non-liner Prmeric Eciion Prolem", Advnces in Applied Mhemics 7, M. Sonis, 000. "Criicl Bifurcion Surfces of 3D Discree Dnmics", Discree Dnmics in Nure nd Socie 4, pp Puu T Nonliner Economic Dnmics. 4 h ed., Springer-Verlg, Berlin. Puu T. nd A. Norin 003. "Courno Duopol when he Compeiors Opere under Cpci Consrins", Chos, Solions nd Frcls (forhcoming).

13 Figure : Domin of rcion of he Courno equilirium in he spce of oris nd 5-period dnmics ( λ = µ = , Time: 5000, Fied poin: 000)

14 Figure. Two-dimensionl ifurcion digrm descriing he 4-periodic ccle 3

15 Figure 3. Two-dimensionl ifurcion digrm descriing he 9-periodic ccle 4