Discussion Paer No.247 Heterogeneous Agents Model of Asset Price with Time Delays Akio Matsumoto Chuo University Ferenc Szidarovszky University of Pécs February 2015 INSTITUTE OF ECONOMIC RESEARCH Chuo University Tokyo, Jaan
Heterogeneous Agents Model of Asset Price with Time Delays Akio Matsumoto y Ferenc Szidarovszky z Abstract This study constructs a heterogeneous agents model of a nancial market in continuous time framework. There are two tyes of agents, fundamentalists chartists. The former follow the traditional e ciency market theory have a linear dem function whereas the latter exerience delays in the formation of rice trends ossess a S-shaed dem function. The main feature of this study is a theoretical investigation on the e ects caused by two time delays in a rice adjustment rocess. In articular, two main results are demonstrated: one is that the stability switching curves are analytically derived the other is that the stability losses gains can reeatedly occur when the shae of the curves are meering. These imly that multile delays might generate rice deviations from the equilibrium value. Keywords: Heterogeneous agents model, Two time delays, Limit cycle, Hof bifurcation, Stability switching curve The authors highly areciate the nancial suorts from the MEXT-Suorted Program for the Strategic Research Foundation at Private Universities 2013-2017, the Jaan Society for the Promotion of Science (Grant-in-Aid for Scienti c Research (C) 24530202, 25380238 26380316). The usual disclaimers aly. y Professor, Deartment of Economics, Senior Researcher, International Center for further Develoment of Dynamic Economic Research, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Jaan; akiom@tamacc.chuo-u.ac.j z Professor, Deartment of Alied Mathematics, University of Pécs, Ifjúság u. 6, H-7624, Pécs, Hungary; szidarka@gmail.com 1
1 Introduction In a nancial market ersistent volatility or deviations of the asset rice from the equilibrium value is often observed. Traditionally the e cient market hyothesis is adoted to analyze the behavior of fundamentals, however, it fails to exlain the discreancy between the observed market rice the fundamental value of the asset. Recently various heterogeneous agents models are roosed to describe such uctuations of asset rice time evaluation. There are two tyes of traders, fundamentalists who follow the traditional e cient market hyothesis determine the dem based on the di erence between the observed the fundamental values of the asset chartists who base their decision on the ast attern on rice changes. It is shown that these heterogeneities with some nonlinear behavior can cause dynamic henomena that the fundamental analysis is unable to exlain, that is, various dynamics ranging from cyclic uctuations to erratic behavior. See Day Huang (1990), Brock Hommes (1998), Chiarella He (2003), Chiarella et al. (2002, 2006), to name only a few. These models are considered in a discrete-time framework. In addition, continuous-time models with heterogeneous agents are also develoed. Chiarella (1992) extends the model of Beja Goldman (1980) shows that it is caable of generating a rich form of rice dynamics including sustainable uctuations around the fundamental value when the chartist dem ossesses some nonlinearities. Only recently Dibeh (2005) alies the continuous model to a case in which detection of rice movements is less than erfect resents the one-asset model with time delays in the rice trends. This study extends the scoe of Dibeh s model one ste further. His asset rice dynamics is described by a general di erential equation with n-di erent delays. Since the general model is mathematically intractable, numerically seci ed cases are examined in Dibeh (2005). In articular, convergence of threedelay models with various seci ed values of the delays the birth of limit cycle in a one-delay model are numerically illustrated. Qu Wei (2010) rovide theoretical foundations for the one-delay model. This study constructs a theoretical underinning of a two-delay model concerns how the delays a ect rice time evolutions. The main results to be obtained are (1) The stability switching curve is theoretically derived in the two delays arameter lane. (2) Limit cycles (i.e., ersistent divergence of the market rice from the corresonding fundamental value) emerge via Hof bifurcation when stability of the equilibrium rice is lost. (3) Stability losses gains can reeatedly take lace as one delay becomes larger the other delay is xed at some length. This aer is organized as follows. Section 2 recaitulates the basic elements of the one delay model derives a benchmark dynamics of the one delay solution. Section 3 constructs a two delay model theoretically derives stability 2
switching curves on which the birth death of limit cycles occur. Section 4 resents some numerical simulations concerning the theoretical results. Section 5 contains concluding remarks further research directions. 2 Model with One Delay In this section we brie y summarize a time delay model of seculative asset rices roosed by Dibeh (2005). There are two tyes of agents, the fundamentalists the chartists, the time evolution of the asset rice is controlled by a nonlinear delay di erential equation, _(t) (t) = (1 m)dc ((t); (t )) + md f ((t)) (1) where D f () D c () are the dem functions of the fundamentalists the chartists, resectively, m 2 (0; 1] is the market fraction of the fundamentalists. The functions are seci ed as follows: D f ((t)) = ((t) ); D c ((t); (t )) = tanh[(t) (t )]: (2) The fundamentalists base their dem formation on the di erence between the actual rice of the asset, the fundamental rice of the asset, : Since fundamentalists believe that asset rices must converge instantaneously to ; they will sell (buy) the asset when > (<): The chartists base their decisions of market articiation on the rice trend of the asset. If chartists believe that rices will rise (fall), then their dem for the assets will increase (decrease). The model becomes _(t) = (1 m) tanh [(t) (t )] (t) m [(t) ] (t) (3) that is a nonlinear delay di erential equation where the rst term is bounded from both sides. Clearly = (t) = (t ) = is a unique ositive equilibrium rice. If RH 1 ((t); (t )) denotes the right h side of equation (3), then the linear aroximation in a neighborhood of an equilibrium = (; ) is _ (t) = @R1 H (t) + @R1 H (t ) @(t) @(t ) where = ; @RH 1 @(t) = (1 2m) @RH 1 @(t ) = (1 m): Therefore the linearized equation has the form _ (t) = (1 2m) (t) (1 m) (t ) (4) 3
Substituting an exonential solution into (4) arranging the terms yield the corresonding characteristic equation, For = 0; equation (5) is reduced to with a unique negative root (1 2m) + (1 m)e = 0: (5) (1 2m) + (1 m) = 0 = m < 0: So the model with no delay is locally asymtotically stable. Further the model with a ositive but short delay might be stable. However a longer delay could destabilize the model. It is clear that = 0 is not a solution of (5). Hence if increasing the length of the delay changes stability to instability, then the real art of an eigenvalue must be zero. Suose now that the eigenvalue is urely imaginary, = i! with! > 0: 1 Substituting it into (5) breaking down the resultant characteristic equation into the real imaginary arts resent two equations, (1 2m) + (1 m) cos! = 0; (6)! (1 m) sin! = 0: Moving the constant terms to the right h side adding the squares of the two equations give the relation! 2 = 2 m(2 3m): If 2 3m 0; then there is no ositive!; imlying that there is no solution such as = i! with! > 0: Hence the negative real arts of the eigenvalues do not change their sign to ositive for any value of therefore no stability switches occur. On the other h, if 2 3m > 0; then there is a ositive solution,! 0 = m(2 3m): Substituting it into the rst equation of (6) solving for determines the values of for which the characteristic equation has a urely imaginary solution, given v; m k; k (m; ) = 1 1 2m [cos 1 + 2k] for k = 0; 1; 2; :::! 0 1 m Hence the stability switch might occur when increases becomes any of these threshold values, 0 = 0 (m; v): 1 We will have the same results even if! < 0. 4
To check the direction of stability switches, we assume = () di erentiate equation (5) with resect to to obtain, d d + (1 d m)e d = 0 that is solved for the inverse of the derivative d d 1 = 1 [ + (1 2m)] where the relation obtained from the characteristic equation is used, (1 m)e = + (1 2m): At = i!; the real art of the derivative is " d # 1 1 Re = Re d i! [ i! + (1 2m)] =i! So we have the following result: = 1! 2 + (1 2m) 2 2 > 0: Theorem 1 The system is stable with m 2=3 for all > 0: If m < 2=3, then stability is lost at the smallest stability switch curve, 1 2m cos 1 1 m 0 (m; ) = m(2 3m) stability cannot be regained later. At = 0 (m; ) Hof bifurcation occurs with the ossibility of the birth of limit cycles. Grahical reresentations are given in Figure 1. The yellow region in Figure 1(A) is the stable region. The uward-sloing black curve is the locus of (m; ) satisfying = 0 (m; ) asymtotic to the dotted line at m = 2=3: Thus this gure con rms rst that the solution of (3) is locally asymtotically stable for any 0 if m 2=3 second that it is locally stable for < 0 (m; ) unstable for > 0 (m; ) if m < 2=3. Since @ 0 (m; )=@ < 0; increasing the fundamental rice shifts the stability switching curve downward thus has a destabilizing e ect in the sense that the stability region becomes smaller. Figure 1(B) deicts the bifurcation diagram with resect to in which the local maximum minimum of the trajectories are lotted against : It shows that the trajectory converges to the stationary oint v for < 0 (m; ); this stability is lost at = 0 (m; ) a limit cycle having one maximum one minimum 5
emerges for > 0 (m; ): 2 It is also seen that the limit cycle is exing as the length of the delay becomes larger. (A) Stability switching curve (B) Bifurcation diagram Figure 1. Dynamics of the one delay model 3 Model with Two Delays We next extend the model one ste further by introducing a second delay in the rice trend. The rice dynamic equation with two delays is where _(t) = (1 m) tanhff([(t); (t 1 ); (t 2 )]g(t) m [(t) ] (t) (7) f[(t); (t 1 ); (t 2 )] = [(t) (t 1 )] + (1 ) [(t 1 ) (t 2 )] with 2 [0; 1]: Let R H be right h side of equation (7). The linear aroximation of equation (7) evaluated at the stationary rice vector = (; ; ) is _ (t) = @R H @R H (t) + @R H (t 1 ) + (t 2 ) @(t) @(t 1 ) @(t 2 ) where @R H @(t) = =: [(1 m) m] ; (8) @R H @(t 1 ) = 1 =: (1 m)(1 2) (9) 2 In articular, we take m = 1=2 = 5; which lead to 0 (1=2; 5) ' 0:628: If m is increased to 0:62 as in Dibeh s examle, 0 (0:62; 5) ' 1:5304: 6
@R H @(t 2 ) = 2 =: (1 m)(1 ) < 0: (10) Although 2 > 0 always, the signs of 1 deend on values of m in the following way, R 0 according to R m 1 m 1 R 0 according to Q 1 2 : The linearized equation with these new arameters is rewritten as _ (t) = (t) + 1 (t 1 ) 2 (t 2 ) which is, with (t) = e t u; further converted into the characteristic equation, 1 e 1 + 2 e 2 = 0: (11) This is essentially the same equation that Gu et al. (2005) investigate. Following their method, we determine the location of the eigenvalues. To this end, we divide the left h side of (11) by denote the result as a(): with new functions a() := 1 + a 1 ()e 1 + a 2 ()e 2 (12) a 1 () = 1 a 2 () = 2 : In the secial case of = 1=2; 1 = 0 imlying that a 1 () = 0; so equation (11) becomes the one-delay equation + 2 e 2 = 0 or 1 2 (1 3m) + 1 2 (1 m)e 2 = 0: This equation does not deend on 1 ; so it can be arbitrary. Similarly to Theorem 1 we can rove that system is stable for all 2 as m 1=2. If m < 1=2; then stability is lost at the smallest stability switching curve, 1 3m cos 1 1 m 2 (m; ) = m(1 2m) stability cannot be regained later. Furthermore at 2 = 2 (m; ) Hof bifurcation occurs. If 6= 1=2; then both 1 2 are nonzero, so equation (11) is a two-delay equation. 7
For = i! with! > 0; a 1 () a 2 () are written as Their absolute values are a 1 (i!) = 1 2 +! 2 + i 1! 2 +! 2 (13) a 2 (i!) = 2 2 +! 2 ja 1 (i!)j = i 2! 2 +! 2 : (14) j 1 j 2 +! 2 (15) ja 2 (i!)j = 2 2 +! 2 : (16) We can consider the three terms in a() as three vectors in the comlex lane. If i! is a solution of a() = 0; then utting these vectors head to tail forms a triangle. Solving a(i!) = 0 analytically is equivalent to solving the following three inequality conditions grahically, each of which means that the length of the sum of two adjacent line segments of the triangle is not shorter than the length of the remaining line segment, (i) 1 ja 1 (i!)j + ja 2 (i!)j ; (ii) ja 1 (i!)j 1 + ja 2 (i!)j ; (iii) ja 2 (i!)j 1 + ja 1 (i!)j : Using the absolute values in (15) (16), we rewrite these three conditions in terms of m. First, substituting the absolute values into condition (i) yields 2 +! 2 j 1 j + 2 (17) where the right-h side has two exressions according to whether the value of is less than 1=2 or not, 8 >< (1 m)(2 3) if < 1 2 ; j 1 j + 2 = >: (1 m) if 1 2 which is nonnegative in both cases. So condition (17) is equivalent to! 2 1 (m; ) := (j 1 j + 2 ) 2 2 : (18) If 1 (m; ) 0, then there is no! > 0, imlying no stability switch for any delays. We then look for conditions under which 1 (m; ) > 0 or jj < j 1 j + 2 : 8
If < 1=2; then yields two inequality conditions, j(1 m) mj < (1 m)(2 3) < f 1 (m) := 2 3m 2(1 m) (19) < f 2 (m) := 2 m 4(1 m) (20) where the rst condition is violated with all > 0 for m 2=3; the second condition is ine ective as f 2 (m) > 1=2 for m 2 [0; 1]: If 1=2; then j(1 m) mj < (1 m) generates two inequality conditions, one is > f 3 (m) := m 2(1 m) (21) the other is m > 0; which is already assumed above. A grahical descrition of conditions (19) (21) is given in Figure 2(A). The vertically-stried region is surrounded by the curve of f 2 (m) = 1=2 while the horizontally-stried region is constructed by the curve of f 1 (m) = 1=2: Hence 1 (m; ) > 0 in the union of these stried regions 1 (m; ) 0 in the gray region including its boundaries where m 2=3 or f 1 (m) f 3 (m) hold. Denoting the stried region by U; U = U 1 [U 2 where U 1 = (m; ) j < 1 2 < f 1(m) U 2 = (m; ) j 1 2 > f 3(m) : We summarize the result concerning condition (i) as follows: Lemma 1 If (m; ) 2 U; then 1 (m; ) > 0 condition (i) holds for! <! f = 1 (m; ) which is violated for! >! f ; also for all (m; ) =2 U. Second, condition (ii) is rewritten as j 1 j 2 2 +! 2 (22) 9
where the left-h side has two exressions according to whether the value of is less than 1=2 or not, 8 >< (1 m)( ) if < 1 2 ; j 1 j 2 = >: (1 m)(2 3) if 1 2 : In the rst case j 1 j sign, 2 < 0 always in the second case, it can be of either 8 >< 0 if 2 3 ; j 1 j 2 >: < 0 if < 2 3 : We see that if < 2=3; then the left-h side of (22) is negative imlying that the inequality is satis ed. On the other h, if 2=3; then (22) is equivalent to! 2 2 (m; ) := (j 1 j 2 ) 2 2 : (23) If 2 (m; ) 0, then any! > 0 satis es (22). To have 2 (m; ) > 0, we then look for a condition for jj 1 j 2 j > jj : (24) Since (22) is satis ed for < 2=3 with any! > 0; we can focus on the case of 2=3: Then (24) can be rewritten as or j(1 m) mj < (1 m)(3 2) (1 m)(3 2) < (1 m) m < (1 m)(3 2) which is two inequalities. The left h side is the right h side is So let A = > f 2 (m) (25) > f 1 (m): (26) (m; ) j 2 3 ; > f 1(m) > f 2 (m) ; which is the vertically stried region in Figure 2(B). Hence we have the following result. Lemma 2 If (m; ) 2 A; then 2 (m; ) > 0; condition (ii) holds if! >! g = 2 (m; ); violated if!! g : If (m; ) =2 A; then condition (ii) holds for all! > 0. 10
Lastly, condition (iii) is rendered to 2 j 1 j 2 +! 2 : (27) It can be shown that 2 j 1 j is nonositive if 2=3 ositive otherwise. Thus (27) is satis ed if 2=3: So we focus only on the case of < 2=3. On the other h, if the left h side of (27) is ositive, then it is required to satisfy the equivalent inequality,! 2 2 (m; ) = ( 2 j 1 j) 2 2 (28) which is the same as (23). If ( 2 j 1 j) 2 2 0; then (27) holds for all! > 0. So we need to nd condition such that 2 ( 2 j 1 j) 2 : If < 1=2, then this relation can be written as (1 m) < (1 m) m < (1 m): The right h side always holds, the left h side can be rewritten as > f 3 (m): (29) Notice that The region B = f 3 (m) < 1 2 if m < 1 2 : (m; ) j < 1 2 ; > f 3(m) is the horizontally-stried region in Figure 2(B). Assume next that 1=2 < 2=3. Then 2 ( 2 j 1 j) 2 holds if (1 m)(2 3) < (1 m) m < (1 m)(2 3): The left h side can be rewritten as the right h side as De ne the region C = < f 1 (m) < f 2 (m): (m; ) j 1 2 < 2 3 ; < f 1(m); < f 2 (m) which is diagonally-stried in Figure 2(B). Our result can be summarized as follows. Lemma 3 If (m; ) 2 B [ C, then 2 (m; ) > 0; condition (iii) holds if! >! g = 2 (m; ) violated if!! g : If (m; ) =2 B [ C; then condition (iii) holds for all! > 0. 11
Notice that 2 (m; ) < 1 (m; ) for all values of m : Lemmas 1, 2 3 imly the ful llment of conditions (i), (ii) (iii),! 2 1 (m; ) 0! 2 2 (m; ) 0: Theorem 2 If (m; ) 2 A [ B [ C; then 2 (m; )! 2 1 (m; ) for! 2 [! g ;! f ] whereas if (m; ) 2 U A [ B [ C; then 0! 2 1 (m; ) for! 2 [0;! f ]: (A) 1 (m; ) > 0 (B) 2 (m; ) > 0 Figure 2. The triangle conditions (i), (ii) (iii) We will next nd all airs of delays ( 1 ; 2 ) satisfying a(i!) = 0: Treating the three terms in (12) as three vectors constructing a triangle, we suose that j1j is its base denote the angle between j1j ja 1 (i!)j by 1 the angle between j1j ja 2 (i!)j by 2 : Then by the law of cosine, these angles are 1 (!) = cos 1 2 +! 2 + 2 1 2 2 2 j 1 j 2 +! 2 2 (!) = cos 1 2 +! 2 + 2 2 2 1 : 2 2 2 +! 2 Since the triangle may be located above below the horizontal axis in the comlex lain, we have two ossibilities, farg a 1 (i!)e! 1 ] + 2kg 1 (!) = 12
farg a 2 (i!)e! 2 ] + 2ng 2 (!) = : Solving these equations for 1 2 yields the threshold values of the delays, 1 (!; k) = 1! farg a 1(i!) + (2k 1) 1 (!)g (30) 2 (!; n) = 1! farg a 2(i!) + (2n 1) 2 (!)g : (31) Notice that articular values of arg a 1 (i!) arg a 2 (i!) deend on the choice of the arameter values. Let be the interval of! seci ed in Theorem 2. Then for! 2 we can nd the airs of ( 1 ; 2 ) constructing the stability switching curves consisting of two sets of arametric segments, L 1 (k; n) = + 1 (!; k); 2 (!; n) (32) L 2 (k; n) = 1 (!; k); + 2 (!; n) (33) where the arameter! runs through interval. 4 Examles In this section, we secify the arameter values erform numerical simulations to con rm the shae of the stability switching curve. As seen in (13) (14), the values of arg(a 1 (i!)) arg(a 2 (i!)) deend on the signs of 1 ; the signs of which, in turn, deend on the values of m. The (m; ) lane is divided into four subregions by the loci of = 0 1 = 0. Accordingly we select four di erent combinations of m ; keeing = 5 in the following examles. Examle 1: m = 1=4, = 2=5 = 5 The seci ed oint (m; ) is in the set B: Under these seci cations, > 0 1 > 0 so arg a 1 (i!) = tan 1! arg a 2 (i!) = + tan 1! : Substituting them into (30) (31) gives 1 (!; k) = 1! 2 (!; n) = 1! n tan 1! o + (2k 1) 1 (!) (34) n tan 1! o + 2n 2 (!) : (35) 13
The stability switching curve is shown in Figure 3(A) with n = 0 k = 0; 1; 2; 3. We see that L 1 (k; 0) L 2 (k; 0) for k = 0; 1; 2; 3 are the red blue segments of the curve shift rightward as k increases. The segment L 2 (0; 0) is located in the second quadrant in which 1 (!; 0) is negative thus is not deicted. So L 1 (0; 0) is the left-most red segment, starts at the initial oint ( + 1 (! s; 0); 2 (! s ; 0)) arrives at the end oint ( + 1 (! e; 0); 2 (! e ; 0)) as! increases from! s to! e where! s = q (j 1 j 2 ) 2 2! e = q (j 1 j + 2 ) 2 2 : L 2 (1; 0) is the left-most blue segment, starts at the initial oint ( 1 (! s ; 1); + 2 (! s; 0)) arrives at the end oint ( + 1 (! e; 1); 2 (! e ; 0)): It can be checked that + 1 (! s; 0) = 1 (! s ; 1) ' 0:947 + 2 (! s; 0) = 2 (! s ; 0) ' 0:947: The segments L 1 (0; 0) L 2 (1; 0) have the same initial oint located on the diagonal. The end oint of the segment L 1 (1; 0) is given by ( + 1 (! e; 1); 2 (! e ; 0)) where + 1 (! e; 1) = 1 (! e ; 1) ' 1:548 2 (! e ; 0) = 2 (! e ; 0) ' 0:495: Hence the segments L 1 (1; 0) L 2 (1; 0) have the same end oints. In this way, one segment is connected with the other segment the connecting segments for n = 0 k = 0; 1; 2; 3 form the stability switching curve as shown in Figure 3(A). More generally, the following can be shown: Proosition 1 Given the arameter seci cations resulting in > 0 1 > 0; the segments L 1 (k; n) L 2 (k +1; n) have the same initial oint whereas the segments L 1 (k; n) L 2 (k; n) have the same end oint for any ositive integer values of k n: Fixing the value of 2 at 2 = 0:5 gradually increasing the value of 1 from zero to 8; the horizontal line at 2 = 2 crosses the stability switching curve ve times with alternating stability losses gains. The crossing oints are denoted by the black dots their 1 -coordinates are labelled by a 1(' 1:548); b 1(' 2:654); d 1(' 3:650) d 1(' 5:443); resectively, in the ascending order. 3 The bifurcation diagram of Figure 3(B) shows the stable the unstable regions for 1 where two-cycles occur, the magnitude of the oscillations increases then decreases back to the stable steady state when stability 3 The fth oint e 1 (' 5:443) is not labelled in order to avoid grahical conjestion. 14
is regained. (A) Stability switching curve (B) Bifurcation diagram Figure 3. Illustration of Examle 1 Examle 2: m = 2=5, = 7=12 = 5 The seci ed oint (m; ) is in the set C makes < 0 1 < 0: Therefore arg a 1 (i!) = + tan 1! arg a 2 (i!) = + tan 1! : Substituting these exressions into (30) (31) give 1 (!; k) = 1! n tan 1! o + (2k + 1) 1 (!) (36) 2 (!; n) = 1 n tan 1! o + (2n + 1) 2 (!) : (37)! Figure 4(A) shows the stability switching curve. With xed value of 2 = 1:5; we gradually increase 1 from 0 to 8. The horizontal line crosses the stability switching curve three times with two stability losses one stability regain. As the bifurcation diagram in Figure 4(B) shows, two-cycles emerge in the unstable regions of 1 similarly to Examle 1. It is ossible to obtain the general result 15
similar to Proosition 1. (A) Stability switching curve (B) Bifurcation diagram Figure 4. Illustration of Examle 2 Examle 3: m = 2=5, = 5=6 = 5 The seci ed oint (m; ) belongs to the set C: Under these seci cations, > 0 1 < 0 so arg a 1 (i!) = tan 1! arg a 2 (i!) = + tan 1! : Substituting these exressions into (30) (31) give 1 (!; k) = 1 n tan 1! o + 2k 1 (!)! 2 (!; n) = 1 n tan 1! o + 2n 2 (!) : (39)! The stability switching curve is shown in Figure 5(A) with k = 0 n = 0; 1; 2: As in Examle 1, L 1 (0; n) L 2 (0; n) are the red blue segments of the curve shift uward as n increases. L 1 (0; 0) is the lowest red segment crossing the horizontal line at 1 ' 0:785: It starts at the initial oint ( + 1 (! s; 0); 2 (! s ; 0)) arrives at the end oint ( + 1 (! e; 0); 2 (! e ; 0)) as! increases from! s to! e : 4 L 2 (0; 0) is the lowest blue segment starting at the initial oint ( 1 (! s ; 0); + 2 (! s; 0)) arriving at the end oint ( 1 (! e ; 0); + 2 (! e; 0)): It can be checked that the end oint is on the diagonal with + 1 (! e; 0) = 1 (! e ; 0) ' 0:559 (38) 4 In this examle,! s ' 1:414! e ' 2:449. Since 2 (!; 0) 0 for! 2 [! s; 2] where ( + 1 (!s; 0); 2 (!s; 0)) ' (0:870; 1:351) ( + 1 (2; 0); 2 (2; 0)) ' (0; 0:785); L 1(0; 0) is deicted in Figure 5(A) for! 2 [2;! e] 16
2 (! e ; 0) = + 2 (! e; 0) ' 0:559: These equalities imly that segments L 1 (0; 0) L 2 (0; 0) have the same end oint. The starting oint of L 1 (0; 1) is given by ( + 1 (! s; 1); 2 (! s ; 0)) where + 1 (! s; 1) = 1 (! s ; 0) ' 0:870 2 (! s ; 0) = + 2 (! s; 0) ' 3:092: Hence segments L 2 (0; 0) L 1 (0; 1) have the same initial oint. More generally, the following can be shown: Proosition 2 Given the arameter seci cations resulting in > 0 1 < 0; the segments L 1 (k; n) L 2 (k; n+1) have the same initial oint whereas the segments L 1 (k; n) L 2 (k; n) have the same end oint for any ositive integer values of k n: A art of the stability switching curve is enlarged in Figure 5(B), in Figure 6 two bifurcation diagrams are shown. With xed value of 1 = 0 1, the value of 2 increases from 0 to 4; the corresonding vertical line has three intersections with the stability switching curve with two stability losses one regain, which is well shown in Figure 6(A). If the value of 2 is xed at 2 = 0 2 1 is increased from 0 to 3=2, then three similar intersections are obtained with two stability losses one stability regain. This is also veri ed in Figure 6(B). In all cases two-cycles are found in the cases of instability, where the magnitude increases from 0 to a maximum then decreases back to zero when stability is regained. So the dynamics behavior of the system is relatively simle, limit cycles occur in the unstable regions. (A) Stability switching curve (B) Enlargement Figure 5. Stability switching curves in Examle 3 17
(A) Bifurcation diagram wrt 2 (B) Bifurcation diagram wrt 1 Figure 6. Bifurcation diagrams of Examle 3 Examle 4: m = 2=5, = 2=5 = 5 The seci ed oint (m; ) is in the set B as in Examle 1. Under these seci cations, we have < 0 1 < 0 that are di erent from those in Examle 1. As 1 < 0; arg a 1 (i!) = + tan 1! arg a 2 (i!) = 2 + tan 1! : Substituting these exressions into (30) (31) give 1 (!; k) = 1! n tan 1! o + 2k 1 (!) (40) 2 (!; n) = 1 n tan 1! o + (2n + 1) 2 (!) : (41)! Figure 7(A) shows the stability switching curve Figure 7(B) is the bifurcation diagram where 2 = 1:2 1 increases from 0 to 8: Stability losses 18
gains also reeatedly occur in this examle. (A) Stability switching curve (B) Bifurcation diagram 5 Concluding Remarks Figure 6. Illustration of Examle 4 We develo a heterogeneous agents model in which the chartists estimate the rice trend with delayed information. It is analytically shown that the delay lane is divided into two regions by a stability switching curve the equilibrium rice is stable in the region including the origin of coordinates even if the delays are involved while it is unstable in the other region. Concerning global dynamics, it is numerically shown that xing one delay at a ositive length, the equilibrium rice loses stability when the length of the other delay increases to arrive at the stability switching curve it bifurcates to a limit cycle emerges via Hof bifurcation when the delay is further increased. Stability regains never occur in the one delay model while stability losses regains might reeatedly occur in the two delay model. There is a number of directions for further research. Firstly, we should rovide a theoretical analysis of the numerical examles in three-delay models conducted in Dibeh (2005) in which the convergent damed oscillations are given the following conjecture is addressed: the longer is the time delay, the longer is the convergence time of the asset rice to the fundamental value. Furthermore, the construction of the stability switching surface in the three delay sace is an interesting extension. Secondly, since the model we deal with is an aggregative descritive model, there is a need to construct a microeconomic underinning. Third, based on the results obtained in the one-asst model, its extension to a multi-asset model is necessary as the key factor in nancial theory is ortfolio diversi cation. 19
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