SCHOOL OF FINANCE AND ECONOMICS

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1 SCHOOL OF FINANCE AND ECONOMICS UTS:BUSINESS WORKING PAPER NO. 11 DECEMBER, 1991 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays Carl Charella ISSN:

2 c:\wpdata\carl\research\91_07_16.cc1 THE BIRTH OF LIMIT CYCLES IN COURNOT OLIGOPOLY MODELS WITH TIME DELAYS CARL CHIARELLA SCHOOL OF FINANCE AND ECONOMICS UNIVERSITY OF TECHNOLOGY, SYDNEY PO BOX 123, BROADWAY NSW 2007 AUSTRALIA Abstract We consder the fate of output n the Cournot olgopoly model when the equlbrum s locally unstable. We dscuss types of nonlneartes whch may be present to bound the moton and ntroduce tme lags n producton and nformaton whch may serve as bfurcaton parameters. In the case of dentcal frms we apply the Hopf bfurcaton theorem to determne condtons under whch lmt cycle moton s born. Paper Presented at 13 IMACS World Congress on Computaton and Appled th Mathematcs. Trnty College Dubln, Ireland, July 22-26, 1991.

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4 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays 1. Introducton The lterature on the stablty of the Cournot olgopoly model, n both dscrete and contnuous tme, s one of the rchest n economc dynamcs. Okuguch [1976] gves a thorough dscusson of many of the key results and generalsatons to mult-product frms are contaned n Okuguch and Szdarovsky [1990]. Typcally stablty condtons relate to a frm's margnal costs and margnal revenues, tme lags n producton and nformaton processng as well as ndustry structure. The focus of research has been to enlarge as far as possble the regons of stablty of the Cournot model, wth regons of nstablty beng consdered as regons to avod as t was assumed (or hoped) that no economy would operate there. The neglect of the regons of nstablty of the Cournot model was also due to a lack of approprate techncal tools. However, the advances n recent decades n the qualtatve theory of nonlnear dfferental equatons and the theory of nonlnear dynamcal systems as expounded n Arnold [1978], Guckenhener and Holmes [1983] and Jackson [1989] have gven the economc theorst both the nclnaton and the tools to analyse the behavour of the model n the regons of local nstablty. Once the model enters the regon of local nstablty we would expect that nonlneartes and tme lags n the governng economc relatonshps operate so that quanttes and prces converge to some stable (possbly qute complcated) attractor. For the economc theorst wshng to analyse the Cournot model n ts regon of local nstablty the task s twofold. Frstly, to pnpont the relevant nonlneartes and tme lags. Secondly, to be able to say as much as possble about the nature of the attractor, partcularly relatng ts salent Page 1

5 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays features to the underlyng economc relatonshps. The tools whch the modern theory of nonlnear dynamcal systems puts at our dsposal for ths task are, (a) bfurcaton theory, whch enables us to dentfy regons where certan types of self-sustanng oscllatng attractors are born, (b) centre-manfold theory, whch allows us to reduce the dmensonalty of the dfferental system under consderaton to a manfold or whch the attractor les, and (c) the method of averagng whch allows us to relate, albet n an approxmate manner, behavour of the attractor to underlyng economc quanttes. An early analyss of the nstablty of the Cournot model s gven by Seade [1980], who establshes condtons under whch the Cournot soluton to the olgopoly problem s ether unstable or a saddle pont wth an unstable manfold of dmenson at most 1. Al- Nowah and Levne [1985] gve suffcent condtons for nstablty when the adjustment process s contnuous and show that these may be sgnfcant when the number of frms s small. Furth [1986] gves an extensve dscusson of the stablty and nstablty of the Cournot model. In contrast to Seade he gves examples of equlbra whose the unstable manfold has dmenson 2. He also consders the queston of whether there s a unque nonstable nteror equlbrum and shows that whenever there s a unque nonstable nteror equlbrum there s at least one stable boundary equlbrum. In ths paper we wsh to ntate a study of the fate of output n the Cournot model when the equlbrum s locally unstable. To ths end we dscuss the types of nonlneartes whch could come nto play to bound the tme paths of output. We also ntroduce tme lags whch act as bfurcaton parameters n the brth of lmt cycle moton of the tme path of output. Our use of tme lags s nspred by the studes of Howroyd and Russell [1984] and Russell, Rckard and Howroyd [1986]. Essentally they ntroduce three types of lag - a producton Page 2

6 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays lag n adjustment of frms' output to ts desred output; an nformaton lag n the recept of nformaton about rval frms' output; an "own" nformaton lag about frms' own output. We focus on the frst two of these wthn the framework of the Cournot olgopoly model and, for the specal case of dentcal frms, apply the Hopf bfurcaton theorem to consder condtons under whch lmt cycle moton s born. Ths nvolves a relatonshp between the tme lags and cost and demand functons n the model. We do not at ths stage address the ssue of the stablty of the lmt cycle whch requres an explct consderaton of the nonlnearty n the model and applcaton of some approxmaton technque such as the method of averagng. 2. Some Possble Nonlnear Mechansms To obtan sustaned output fluctuatons n duopoly models we wll need to ntroduce tme lags and nonlneartes nto the standard formulaton. Tme lags can be ntroduced as n Chapter 7 of Okuguch [1976] or as n Howroyd and Russell [1984] and Russell, Howroyd and Rchard [1986]. Typcally frms adjust ther output to desred output wth a lag (due to the producton lag referred to n the prevous secton) and use an adaptve expectatons scheme to form ther expectaton of the other frms' output (due to lags n nformaton about the rvals' output or about ther own output). A range of nonlneartes s possble and nclude:- (a) ntroducng cubc and hgher order terms nto the cost functon, and/or, ntroducng quadratc and hgher order forms nto the demand functon; Page 3

7 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays * (b) ntroducng constrants on output adjustment. Suppose for example that x s the desred output of frm and that the frm adjusts ts actual output x to the * desred level wth a lag accordng to: x = k (x - x ), k >0. However, t may be costly for the frm to make large output changes, ndeed output changes above or below certan levels may be mpossble. Ths effect could be captured by assumng a relatonshp between x and x as shown n Fgure 1. 3 (c) An alternatve way to capture the effect n (b) would be to nclude an x term n the cost functon X Slope=K X * X Fgure 1 3. Modellng the Lag Structure Page 4

8 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays In order to perform an ntal analyss, let's assume a nonlnearty of the type (a) above. In the stuaton where there are n frms, suppose demand s gven by p a b n 1 x, (b>0). (1) Defnng Q j x j the cost functon for frm s assumed to be gven by C (x,q ) x [c h (Q )] d x 2 2 x 3 3, (2) where c 0,d >0, 0 and no partcular assumpton s made about h (Q ). The proft maxmsng condton for frm s a 2bx bq (e) C x (x,q (e) ), (3) where s frm 's expectaton of rvals' output Q. Q (e) Equaton (3) mples a reacton functon relatonshp between x and Q (e) whch we wrte Page 5

9 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays x g [Q (e) ], ( 1,...,n). (4) In order to focus our attenton on the smplest possble case we shall assume the cost and demand functons satsfy condtons that ensure g [Q (e) ]<0,, (5) and that the system of equatons x g (Q ) 0, ( 1,...,n), (6) has a unque soluton. In fgure 2 we llustrate the reacton functons and the unque equlbra n the case of duopoly. We assume that each frm adjusts to the desred level of output wth a lag as dscussed n secton 2,.e. x k [g (Q (e) ) x ], (7) where the k (>0) ( = 1,...,n) are speed of adjustment parameters. Page 6

10 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays X 1, X 2 g 2 g 1 ( ) X, X 1 2 ( ) Fgure 2 Russell, Howroyd and Rckard [1986] ntroduce a tme lag (T ) that frm experences n obtanng nformaton about rvals' output as well as a lag (S ) n frm obtanng or mplementng nformaton about ts own output. In terms of our current notaton ther adjustment process would be wrtten x (t) k [g (Q (e) (t T )) x (t S )]. (8) Equaton (8), beng a dfferental-dfference equaton, presents the techncal dffculty that ts egenvalue spectrum s nfnte and hence applcaton of the Hopf bfurcaton theorem may become an analytcally ntractable problem. We must therefore address the problem of fndng some practcal way of modellng tme lags. The smplest way seems to be to adopt contnuously dstrbuted lags. Thus frm 's expectaton of rvals' output could be wrtten Page 7

11 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays Q (e) (t) t 0 W (t s)q (s)ds, (9) where W (t) s a weghtng functon that frm apples to rval frms' prevous output. An approprate class of weghtng functon would be W (m) (t) 1 m! ( m ) m 1 t m e mt/t T,m 1, 1 e t/t T,m 0, (10) where T >0 and m s an nteger. Ths class has the followng propertes:- (a) for m=0, weghts are exponentally declnng wth most weght beng gven to most recent output; (b) for m 1, zero weght s assgned to most recent output, rsng to maxmum weght at t=t and declnng exponentally to zero thereafter; (c) the area under the weghtng functon s unty T, m; (d) the functon becomes more peaked around t=t as m ncreases. Indeed for m=4 the functon may for all practcal purposes be regarded as very close to a Drac delta functon (e. a unt mpulse functon) centred at t=t e. t 0 W (t s)q (s)ds Q (t T ), (11) s not an unreasonable approxmaton for m 4. (e) as T 0, the functon tends to a Drac delta functon e. lm T 0 W (m) (t s) (t s), Page 8

12 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays so that lm T 0 t 0 W (m) (t s)q (s)ds Q(t). (12) ( m) W (t) m=0 m 2 m > m 2 1 m 1 T (m) t Fgure 3 The Weghtng Functon W (t) (m) The weghtng functon W (t) s llustrated n fgure 3 for varous values of m. Under the assumpton that frm 's own nformaton lag S s zero, the dynamcs of the economy are then governed by the ntegro-dfferental equaton system x k [g (Q (e) ) x ], (13a) Q (e) (t) t W (m) (t s)q (s)ds. (13b) 0 Page 9

13 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays By dfferentatng (13b) (m+1) tmes we are able to reduce the system (13) to a set of n(m+2) ordnary dfferental equatons. For example, for m=1, the system (13) reduces to x k [g (Q (e) ) x ], (14a) Q (e) 1 T [z Q (e) ], (14b) z 1 T [Q z ], (14c) whch s a system of 3n ordnary dfferental equatons. For the remander of ths paper we shall focus on the specal case whch the n frms are dentcal, so that x x, Q (n 1)x,, (15a) k k, T T, g g,. (15b) The system (13) then reduces to x k[g((n 1)x (e) ) x], (16a) x (e) (t) t W (m) (t s)x(s)ds, (16b) 0 (e) where x (t) s one frm's expectaton at tme t of the output of a typcal rval frm. Page 10

14 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays The equlbrum pont (x, x (e) ) of the ntegro-dfferental system (16) s gven by x g[(n 1)x (e) ], (17a) x (e) x. (17b) As we have stated earler we assume that ths equlbrum s unque. 4. Analyss of the Dynamcs In order to analyse the local dynamc behavour of the ntegro-dfferental system (16) (e) around the equlbrum (17) we consder the lnearsed system. Usng x(t), x (t) to now denote devatons of these varables from ther equlbrum levels, the lnearsaton of (16) may be wrtten x kx k (n 1) t W (m) (t s)x(s)ds, (18) 0 where g ((n 1)x). We could obtan the characterstc equaton of the ntegro-dfferental equaton (18) by reducng t to the equvalent system of (m+2) ordnary dfferental. An alternatve approach, whch s more convenent for our purposes, s to apply the technques outlnes n Mller [1972] for determnng the characterstc equaton of lnear Volterra ntegro- Page 11

15 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays dfferental systems. Applcaton of these technques ndcate that the characterstc equaton of (18) s k k (n 1) 0 W (m) (s)e s ds, (19) whch reduces to ( k) k (n 1) ( T/m 1) m 1, (20) for m>0, and to ( k) k (n 1) ( T 1), (21) for m=0. Consder frst the case m=0 when the weghtng functon s exponentally declnng. The egenvalues are easly calculated as the roots of the quadratc equaton 2 (k 1 T ) k [1 (n 1)] 0. T (22) It s a straghtforward matter to verfy that the real part of the roots of (22) are negatve, gven the assumed sgns of k, and T. A graphcal analyss of (20) (see fgure 4) ndcates that for m odd, there s only one real root, whch s negatve. For m even there are at most two real roots, both of whch must be negatve. Page 12

16 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays m even + -m/t ( -1) ( T/m+1) m+1 m odd Fgure 4 Analyss of the characterstc equaton (20) for general m seems dffcult, so we concentrate on the case m=1, whch reduces (20) to ( k)( T 1) 2 k (n 1) 0. (23) To determne parameter values for whch (23) may have pure complex roots we seek such that ± satsfes (23). Substtutng nto (23) and equatng real and magnary parts we fnd that must smultaneously satsfy 2 2kT 1 2 k(1 (n 1) ) and. T 2 T(2 kt) (24) Page 13

17 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays Equatng the two expressons for 2 we fnd the combnaton of parameter values for whch pure complex roots are possble, namely, (kt) 2 (2 (n 1) ) (kt) (25) If we regard as a bfurcaton parameter then (25) shows that the real part of the complex roots change sgn at the crtcal value * gven by 1 (n 1) [ 1 kt 2kT 2], (26) * 1/ 2 T -(2+ 2) (n-1) Fgure 5 whch s graphed as a functon of kt n fgure 5. By dfferentatng (23) mplctly wth respect to and evaluatng the dervatves Page 14

18 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays at ± (wth gven by (24)) we fnd that R( d d ) k(2kt 1) 2T 2 [(2kT 1) 2 k 2 T 2 (1 (n 1) ) 2 ] < 0. (27) Thus by the Hopf bfurcaton theorem (see eg. Guckenhemer and Holmes [1983]) we can assert the exstence of a lmt cycle for n the neghbourhood of = *. Snce the R( ) decreases as ncreases through * the model would dsplay the dynamc behavour llustrated n Fgure 6, f the lmt cycle were stable. To analyse the queston of the stablty of the lmt cycle we need to explctly consder the partcular type of nonlnearty whch s operatng. Ths analyss we leave for further study. > * < * Fgure 6 Page 15

19 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays 5. Concluson We have seen that by ntroducng nonlneartes nto frms' cost functons and allowng for a suffcently hgh order lag structure n the formaton of expectatons, t s possble for the olgopoly model wth dentcal frms to exhbt fluctuatng output n the form of a lmt cycle. The condton for the bfurcaton to a lmt cycle relates the slope of the reacton functon at equlbrum wth the rato of nformaton tme lag to the producton tme lag. The requrement on the slope of the reacton functon s least strngent when ths rato s around 0.7. Bfurcaton to a lmt cycle s less lkely when these lags dffer greatly. We also observe that the slope of the reacton functon at the bfurcaton pont becomes less steep as the number of frms n the ndustry ncreases. Further research can proceed n two drectons. Frstly to determne how the condton for bfurcaton to a lmt cycle s affected when we allow frms to dffer. An analytcal analyss of ths problem may prove dffcult gven the degree of the characterstc equaton, though some progress should be possble by use of numercal smulaton and algabrac manpulaton computer packages. Secondly an analyss of the stablty of the lmt cycle as well as the relatonshp of ts ampltude to the basc economc relatonshps n the model needs to be undertaken. Technques approprate to ths task are outlned n Guckenhemer and Holmes [1983]. Page 16

20 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays References Al-Nowah, A. and Levne, P.L. [1985], "The Stablty of the Cournot Olgopoly Model: A Reassessment", Journal of Economc Theory, 35, pp Arnold, V.I. [1978], Ordnary Dfferental Equatons, MIT Press. Cushng, J.M. [1977], "Integrodfferental Equatons and Delay Models n Populaton Dynamcs", Lecture Notes n Bomathematcs, Vol.20, Sprnger-Verlag. Furth, D. [1986], "Stablty and Instablty n Olgopoly", Journal of Economc Theory, 40, pp Guckenhemer, J. and Holmes, P. [1983], Nonlnear Oscllatons, Dynamcal Systems and Bfurcatons of Vector Felds, Sprnger-Verlag, New York. Howroyd, T.D. and Russell, A.M [1984], "Cournot Olgopoly Models wth Tme Delays", Journal of Mathematcal Economcs, 13, pp Jackson, E.A. [1989], Perspectves of Nonlnear Dynamcs, Vols 1&2, Cambrdge Unversty Press, Mller, R.K., [1972], "Asympotc Stablty and Perturbatons for Lnear Volterra Integrodfferental Systems", n Delay and Functonal Dfferental Equatons and ther Applcatons, edted by K.Schmtt, Academc Press, New York. Page 17

21 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays Okuguch, K. [1976], "Expectatons and Stablty n Olgopoly Models", Lecture Notes n Economcs and Mathematcal Systems, Vol. 138, Sprnger-Verlag. Page 18

22 The Brth of Lmt Cycles n Cournot Olgopoly Models wth Tme Delays Okuguch, K. and Szdarovsky, F. [1990] "The Theory of Olgopoly wth Mult-Product Frms", Lecture Notes n Economcs and Mathematcal Systems, Vol. 342, Sprnger-Verlag. Seade, J. [1980], "The Stablty of Cournot Revsted", Journal of Economc Theory, 23, pp Russell, A.M., Rckard, J. and Howroyd, T.D. [1986], "The Effects of Delays on the Stablty and Rate of Convergence to Equlbrum of Olgopoles", Economc Record, 62, pp Page 19