Learning in Monopolies with Delaye Price Information Akio Matsumoto y Chuo University Ferenc Sziarovszky z University of Pécs February 28, 2013 Abstract We call the intercept of the price function with the vertical axis the maximum price the slope of the price function the marginal price. In this paper it is assume that a monopoly has full information about the marginal price its own cost function but is uncertain on the maximum price. However, by repeate interaction with the market, the obtaine price observations give a basis for an aaptive learning process. It is also assume that the price observations have xe elays, so the learning process can be escribe by a elaye i erential equation. In the cases of one or two elays, the asymptotic behavior of the resulting ynamic process is examine, stability conitions are erive the occurrence of Hopf bifurcation is shown at the critical values. It is also shown that the nonlinear learning process can generate complex ynamics when the steay state is locally unstable the elay is long enough. Keywors: Boune rationallity, Monopoly ynamics, Fixe time elay, Aaptive learing, Hopf bifurcation JEL Classi cation : C62, C63, D21, D42 The authors highly appreciate the nancial supports from the Japan Society for the Promotionof Science (Grant-in-Ai for Scienti c Research (C) 24530202) Chuo University Grant for Special Research Joint Research Grant 0981). y Professor, Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, Japan; akiom@tamacc.chuo-u.ac.jp z Professor, Department of Applie Mathematics, Universeity of Pecs, Ifjusag, u 6, H-7624, Pecs, Hungary; sziarka@gmail.com 1
1 Introuction This paper uses the familiar monopoly moel in which there is only one rm having linear price cost functions. Its main purpose is to show how cyclic erratic ynamics can emerge from quite simple economic structures when uncertainty, information elays behavioral nonlinearites are present. Implicit in the text-book approach is an assumption of complete instantaneous information availability on price cost functions. In consequence, the monopoly can choose its optimal choices of price quantity to maximize pro t. Moreover the monopoly moel becomes static in nature. The assumption of such a rational monopoly is, however, questionable unrealistic in real economies, since there is always a time elay in collecting information etermining optimal responses, in aition, function relations such as the market price function cannot be etermine exactly base on theoretical consieration observe ata. Getting closer to the real worl improving the monopoly theory, we replace this extreme but convenient assumption with the more plausible one. Inee, the rm is assume, rst, to have only limite knowlege on the price function, secon, to obtain it with time elay. As a natural consequence of these alternations, the rm gropes for its optimal choice by using ata obtaine through market experiences. The moi e monopoly moel becomes ynamic in nature. In the literature, there are two i erent ways to moel time elays in continuous-time scale: xe time elay continuously istribute time elay ( xe elay continuous elay henceforth). The former is applicable when an institutionally or socially etermine xe perio of time elay is presente for the agents involve. The latter is appropriate for economic situations in which i erent lengths of elays are istribute over heterogeneous agents or the lengthof the elay is rom epening on unforeseeable circumstance. The choice of the type of elay has situation-epenency results in the use of i erent analytical tools. In the cases of xe elay, ynamics is escribe by a elay i erential equation whose characteristic equation is a mixe polynomialexponential equation with in nitely many eigenvalues. Burger (1956), Bellman Cooke (1956) Cooke Grossman (1992) o er methoology of complete stability analysis in such moels. Fixe elay ynamics has been investigate in various economic frameworks incluing microeconomics (i.e., oligopoly ynamics) macroeconomics (i.e., business cycle). On the other h, in the cases of continuous elay, Volterra-type integro-i erential equations are use to moel the ynamics. The theory of continuous elays with applications in population ynamics is o ere by Cushing (1977). Since Invernizzi Meio (1991) have introuce continuous elays into mathematical economics, its methoology is use in analysing many economic ynamic moels. This paper continues monopoly ynamics consiere by Matsumoto Sziarovszky (2012) where the monopoly oes not know the price function xe time elays are introuce into the output ajustment process base on 2
the graient of the expecte pro t. 1 It also aims to complement Matsumoto Sziarovszky (2012a,b) where uncertain elays are moele by continuous elays when the rm wants to react to average past information instea of suen market changes. Graient ynamics is replace with an aaptive learning scheme base on pro t maximizing behavior. Although there is price uncertainty the price information is elaye, the monopoly is still able to upate its estimate on the price function via the usage of price observations its optimal price beliefs. We will consier the cases of a single elay two elays, respectively, then emonstrate a variety of ynamics ranging from simple cyclic oscillations to complex behavior involving chaos. This paper evelops as follows. After the mathematical moel is formulate, the case of a single elay is examine, when the rm uses the most current elaye price information to form its expectation about the maximum price. Then it is assume that the rm formulates its price expectation base on two elaye observations by using a linear preiction scheme. We will consier three cases here with slightly i erent asymptotic behavior. In all cases complete stability analysis is given, the stability regions are etermine illustrate. The occurrence of Hopf bifurcation is shown at the critical values of the bifurcation parameter, which is the length of the single elay or one of the two elays. The last section o ers conclusions further research irections. 2 The Mathematical Moels Consier a single prouct monopoly that sells its prouct to a homogeneous market. Let q enote the output of the rm, p(q) = a bq the price function C(q) = cq the cost function. 2 Then the pro t of the rm is given as = (a bq)q cq: Since p(0) = a j@p(q)=@qj = b; we call a the maximum price b the marginal price. There are many ways to introuce uncertainty into this framework. In this stuy, it is assume that the rm knows the marginal price but oes not know the maximum price. In consequence it has only an estimate of it at each time perio, which is enote by a e (t). So the rm believes that its pro t is e = (a e bq)q cq 1 In the framework with iscrete-time scale, Puu (1995) shows that the bounely rational monopoly behaves in an erratic way uner cubic em function prouction elay. Naimzaa Ricchiuti (2008) also show that complex ynamics can arise in a nonlinear elay monopoly with a rule of thumb. Naimzaa (2012) exhibits that elay monopolistic ynamics can be escribe by the well-known logistic equation when the rm takes a sspecial learning scheme. More recently Matsumoto Sziarovszky (2013) consier ynamics of a bounely rational monopoly show that the monopoly equilibrium unergoes to complex ynamics through either a perio-oubling or a Neimark-Sacker bifurcation. 2 Linear functions are assume only for the sake of simplicity. We can obtain the same learning process to be e ne even if both functions are nonlinear. It is also assume for the sake of simplicity that the rm has perfect knowlege of prouction technology (i.e., cost function). 3
consequently its best response is believe to be q e = ae c : 2b Further, the rm expects the market price to be p e = a e bq e = ae + c : (1) 2 However, the actual market price is etermine by the real price function p a = a bq e = 2a ae + c : (2) 2 Using these price ata, the rm upates its estimate. The simplest way for ajusting the estimate is the following. If the actual price is higher than the expecte price, then the rm shifts its believe price function by increasing the value of a e ; if the actual price is the smaller, then the rm ecreases the value of a e : If the two prices are the same, then the rm wants to keep its estimate of the maximum price. This ajustment or learning process can be moele by the i erential equation, _a e (t) = k [p a (t) p e (t)] where k > 0 is the spee of ajustment. Substituting equations (1) (2) reuces the ajustment equation to a linear i erential equation with respect to a e as _a e (t) = k [a a e (t)] : (3) In another possible learning process, the rm revises the estimate in such a way that the growth rate of the estimate is proportional to the i erence between the expecte actual prices. Replacing _a e (t) in equation (3) with _a e (t)=a e (t) yiels an another form of the ajustment process _a e (t) a e (t) = k [a ae (t)] or multiplying both sies by a e (t) generates the logistic moel _a e (t) = ka e (t) [a a e (t)] (4) which is a nonlinear i erential equation. If there is a time elay in the estimate price, then we write the estimate price market price at time t base on information at time t as p e (t; t ) = a e (t ) bq e (t; t ) p a (t; t ) = a bq e (t; t ) 4
where q e (t; t ) is the elay best reply, q e (t; t ) = ae (t ) c : 2b Then equations (3) (4) have to be moi e, respectively, as _a e (t) = k [a a e (t )] (5) _a e (t) = ka e (t) [a a e (t )] (6) If the rm uses two past price information, then the elay ynamic equations turn to be _a e (t) = k [a!a e (t 1 ) (1!)a e (t 2 )] (7) _a e (t) = ka e (t) [a!a e (t 1 ) (1!)a e (t 2 )] (8) where 1 2 enote the elays in the price information. If the rm uses interpolation between the observations, then 0 <! < 1, if it uses extrapolation to preict the current price, then the value of! can be negative or even greater than unity. Notice that in the cases of! = 0! = 1, equations (7) (8) reuce to equations (5) (6) while in the case of interpolation, the cases of! 1=2 are the same as! 1=2 because of the symmetry of the moel between 1 2. Similarly, in the case of extrapolation, if! < 0; then 1! > 1: So the cases of! < 0! > 1 are also equivalent. Therefore in the case of moels (7) (8), we will assume that! 1=2! 6= 1: By introucing the new variable z(t) = a e (t) a; equation (5) the linearize version of equation (6) are written as _z(t) + z(t ) = 0 (9) where = k or = ak: By the same way, equation (7) the linearize version of equation (8) are moi e as _z(t) +!z(t 1 ) + (1!)z(t 2 ) = 0: (10) In the following sections, we will examine the asymptotic behavior of the trajectories of equations (9) (10). 3 Single Fixe Delay If there is no elay, then = 0 equation (9) becomes an orinary i erential equation with characteristic polynomial +: So the only eigenvalue is negative implying the global asymptotic stability of the steay state z = 0 if the original equation is linear the local asymptotic stability if nonlinear. The steay 5
state correspons to the true value of the maximum price. If > 0; then the exponential form z(t) = e t u of the solution gives the characteristic equation, + e = 0: (11) Since the only eigenvalue is negative at = 0, we expect asymptotical stability for su ciently small values of a loss of stability for su ciently large values of : If the steay state becomes unstable, then stability switch must occur when = i: Since if is an eigenvalue, then its complex conjugate is also an eigenvalue, we can assume, without any loss of generality, that > 0: So i + e i = 0 by separating the real imaginary parts we have cos = 0 Therefore sin = 0: cos = 0 sin = with = leaing to in nitely many solutions, = 1 2 + 2n for n = 0; 1; 2; ::: (12) The solution with n = 0 forms a ownwar-sloping curve with respect to, = 2 with = k or = ak: Applying the main theorem in Hayes (1950) or the same result obtaine ifferently in Sziarovszky Matsumoto (2012c), we can n that this curve ivies the non-negative (; ) plane into two subregions; the real parts of the roots of the characteristic equation are all negative in the region below the curve some roots are positive in the region above. This curve is often calle a partition curve separating the stability region from the instability region. Notice that the critical value of ecreases with ; so a larger value of cause by the high spee of ajustment /or the larger maximum price makes the steay state less stable. We can easily prove that all pure complex roots of equation (11) are single. If is a multiple eigenvalue, then + e = 0 1 + e ( ) = 0 6
implying that or 1 + = 0 = 1 which is a real negative value. In orer to etect stability switches the emergence of a Hopf bifurcation, we select as the bifurcation parameter consier as function of ; = (). By implicitly i erentiating equation (11) with respect to, we have implying that + e = 2 1 + as = i; its real part becomes 2 Re = Re 1 + i = = 0 2 1 + () 2 > 0: At the critical value of ; the sign of the real part of an eigenvalue changes from negative to positive it is a Hopf bifurcation point of the nonlinear learning process (6) with one elay that has a family of perioic solutions. Thus we have the following results. Theorem 1 (1) For the linear ajustment process (5), stability of the steay state is lost at the critical value of ; = 2k cannot be regaine for larger values of >. (2) For the logistic ajustment process (6), stability of the steay state is lost at = 2ak limit cycles appear through Hopf bifurcation for > : An intuitive reason why stability switch occurs only at the critical value of with n = 0 is the following. Notice rst that the elay i erential equation has in nitely many eigenvalues secon that their real parts are all negative for <. When increasing arrives at the partition curve, then the real part of one eigenvalue becomes zero its erivative with respect to is positive 7
implying that the real part changes its sign to positive from negative. Hence the steay state loses stability at this critical value. Further increasing crosses the (; ) curve e ne by equation (12) with n = 1: There the real part of another eigenvalue changes its sign to positive from negative. Repeating the same arguments, we see that no stability switch occurs for any n 1. Hence stability is change only when crosses the partition curve. Theorem 1 is numerically con rme. In Figure 1(A), three cyclic trajectories generate by the linear elay equation (5) are epicte uner a = 1 k = 1: The initial functions are the same, (t) = 0:5 for t 0 but lengths of elay are i erent. The blue trajectory with = 0:1 shows ampe oscillation approaching the steay state, the black trajectory with = converges to a limit cycle the re trajectory with = + 0:1 cyclically iverges away from the steay state. On the other h, in Figure 1(B), one limit cycle generate by the logistic elay equation (6) is illustrate uner the same parametric conitions, the same initial function = + 0:05. By comparing these numerical results, it is quite evient that nonlinearity of the logistic equation can be a source of persistent uctuations when the steay state loses its stability. (A) linear learnng (B) nonlinear learning 4 Two Fixe Delays Figure 1. Cyclic uctuations In this section, we raw attention to asymptotic behavior of i erential equation (8) with two elays. Its characteristic equation is obtaine by substituting the exponential form q(t) = e t u into equation (10) arranging the terms: +!e 1 + (1!)e 2 = 0 8
or +!e 1 + (1!)e 2 = 0 (13) where = =; 1 = 1 2 = 2 : If 1 = 2 = 0 (or 1 = 2 = 0); then the steay state is asymptotically stable, since the only eigenvalue is negative. In orer to n stability switches, we assume again that = i with > 0: Then equation (13) becomes i +!e i 1 + (1!)e i 2 = 0 separating the real imaginary parts yiels! cos 1 + (1!) cos 2 = 0 (14)! sin 1 (1!) sin 2 = 0 (15) when! 1=2! 6= 1 are assume. We rst examine two bounary cases, one with 1 = 0 the other with 2 = 0, to obtain the following two results. Theorem 2 If 1 = 0; then the steay state is locally asymptotically stable for all 2 > 0: Proof. In the case of 1 = 0; equation (14) is reuce as cos 2 =! 1! If! = 1=2; then cos 2 = 1 so sin 2 = 0 implying that = 0; which is a contraiction. If 1=2 <! < 1; then!=(1!) < 1; so no solution exists. If! > 1; then!=(1!) > 1; so there is no solution either. Therefore equation (13) has no pure imaginary roots that cross the imaginary axis when 2 increases from zero. This conclues Theorem 2. Putting this result i erently, we can say that for 1 = 0; elay 2 is harmless implying that the steay state is locally asymptotically stable regarless of the values of 2 : We now procee to the other case. Theorem 3 If 2 = 0; then the steay state is locally asymptotically stable for 0 < 1 < 1 locally unstable if 1 > 1 where the critical value of 1 is e ne as 1 = cos 1 1!! p : 2! 1 Proof. In the case of 2 = 0; (14) (15) are reuce to (1!) +! cos 1 = 0! sin 1 = 0: 9
Moving 1! to the right h sies, squaring both sies aing the resulte equations yiel 2 = 2! 1: A positive solution of is = p 2! 1 where! > 1=2: Substituting this value into the rst equation above solving the resultant equation, we have 1; the critical value of 1. In the same way as in proving Theorem 1, we arrive at the stability result by applying Hays theorem or our result in Sziarovszky Matsumoto (2012c). We now examine the general case of 1 > 0 2 > 0: By introucing the new variables x = sin 1 y = sin 2 ; equation (14) implies that or from (15),! 2 (1 x 2 ) = (1!) 2 (1 y 2 )! 2 x 2 + (1!) 2 y 2 = 1 2! (16) By combining (16) (17), we get an equation for x; implying that from (17), y =!x 1! : (17) 2!x! 2 x 2 + (1!) 2 = 1 2! 1! x = 2 + 2! 1 2! y = 2 2! + 1 2(1!) : These two equations provie a parametric representation in the ( 1 ; 2 ) plane: sin 1 = 2 + 2! 1 2! sin 2 = 2 2! + 1 2(1!) : (18) The feasibility of solutions requires that 1 2 + 2! 1 2! 1 2 2! + 1 2(1!) 1 (19) 1: (20) 10
Consier rst conition (19), which is a pair of quaratic inequalities in ; 2 + 2! + 2! 1 0 (21) 2 2! + 2! 1 0 (22) The roots of (21) are 1 1 2!; the roots of (22) are 2! 1 1. Conition (20) is also a pair of quaratic inequalities, 8 < 0 if! < 1 2 + 2(1!) + (1 2!) (23) : 0 if! > 1 8 < 0 if! < 1 2 2(1!) + (1 2!) : 0 if! > 1 (24) with 1 2! 1 being the roots of (23) 1 2! 1 being the roots of (24). So if! < 1; then the value of has to satisfy the following relation: 2! 1 1: Similarly, if! > 1; then the value of has to satisfy relation 1 2! 1: Since the feasibility omains for! < 1! > 1 are i erent, we iscuss these cases separately. As it was explaine earlier, we have assume that! 1=2: The remaining part of this section is ivie into three subsections. First, the non-symmetric case (i.e., 1 >! > 1=2) is examine in Section 4.1, then the extrapolation case (i.e.,! > 1) in Section 4.2 nally we will turn our attention to the symmetric case (i.e.,! = 1=2) in Section 4.3. 4.1 The Case of 1 2 <! < 1 In this subsection, we assume that 1=2 <! < 1. Since from (14), we see that the signs of cos 1 cos 2 are i erent, equation (18) imply that 8 >< 1 = 1 sin 1 2 + 2! 1 + 2k (k 0) 2! L 1 (k; n) : >: 2 = 1 (25) sin 1 2 2! + 1 + 2n (n 0) 2(1!) or 8 >< 1 = 1 L 2 (k; n) : >: 2 = 1 sin 1 2 + 2! 1 + 2k 2! sin 1 2 2! + 1 + 2n 2(1!) 11 (n 0) (k 0) (26)
which gives two sets of parametric curves in the ( 1 ; 2 ) plane. The omain of! is the interval [2! 1; 1]. At the initial point = 2! 1; we have 2 + 2! 1 2! at the en point = 1; we have 2 + 2! 1 2! = 1 2 2! + 1 2(1!) = 1 2 2! + 1 2(1!) = 1 = 1: Therefore the initial en points of L 1 (k; n) are 1 I 1 (k; n) = 2! 1 2 + 2k 1 3 ; 2! 1 2 + 2n E 1 (k; n) = 2 + 2k; 2 + 2n : similarly, the initial en points of L 2 (k; n) are I 2 (k; n) = 1 2! 1 2 + 2k ; 1 2! 1 2 + 2n E 2 (k; n) = 2 + 2k; 2 + 2n : Notice that E 1 (k; n) = E 2 (k; n) I 1 (k; n) = I 2 (k; n+1); that is, L 1 (k; n) L 2 (k; n) have the same en points L 1 (k; n) L 2 (k; n + 1) have the same initial points. Hence the segments (L 2 (k; 0); L 1 (k; 0); L 2 (k; 1); L 1 (k; 1); L 2 (k; 2); L 1 (k; 2); :::) starting at I 2 (k; 0) passing through points E 2 (k; 0) = E 1 (k; 0); I 1 (k; 0) = I 2 (k; 1); E 2 (k; 1) = E 1 (k; 1); ::: form a continuous curve. Figure 2 illustrates the loci L 1 (k; n) L 2 (k; n) with the value of varying from 2! 1 to unity for n = 0; 1; 2 k = 0. The parameter value! = 0:8 is selecte. The re curves are L 1 (k; n) the blue curves are L 2 (k; n). The re blue curves shift upwar when n increases rightwar when k increases. There the initial point I 2 (0; 0) is infeasible, L 2 (0; 0) is feasible only for p 2! 1. Notice that I 1 (0; n) = I 2 (0; n + 1) at point I n for n = 0; 1 E 1 (0; n) = E 2 (0; n) at point E n for n = 0; 1; 2: m 1 ' 1:493 is the minimum 1 -value of the segment L 1 (0; n) while M 1 ' 2:733 is the maximum 1 -value of the segment L 2 (0; n). It can be checke that 0 1 ' 2:354 is the 1 -value of the intersection of the segment L 2 (0; 0) with the horizontal axis is ientical with 1 given in Theorem 3. The partition curve stays within the interval [ m 1 ; M 1 ] 12
when k = 0 but its shape coul be i erent for a i erent value of!: It will be shown that the steay state is locally asymptotically stable in the yellow region. Figure 2. Stability-switch curves with k = 0 n = 0; 1; 2 Apparently m 1 < 0 1 implying that the steay state is locally asymptotically stable for 0 < 1 < m 1 2 = 0 ue to Theorem 3. Since the segment L i (0; n) is in the interval [ m 1 ; M 1 ], there are no eigenvalues changing the sign of the real part when 2 increases. In other wors, the real parts of the eigenvalues are negative for 1 < m 1 any 2 > 0: Hence the steay state is locally asymptotically stable. Such elays of ( 1 ; 2 ) o not a ect asymptotic behavior of the steay state thus are harmless. We summarize this result in the following theorem: Theorem 4 If k = 0 0 < 1 < m 1 ; then elay 2 is harmless, so the steay state is locally asymptotically stable. We now move to the case of 1 > m 1 examine the irections of the stability switches by selecting 1 as the bifurcation parameter with xe value of 2. So we consier the eigenvalue as a function of the bifurcation parameter, = ( 1 ). By implicitly i erentiating equation (13) with respect to 1 ; we have +!e 1 1 1 + (1!)e 2 1 2 = 0 1 implying that! = e 1 1 1! 1 e 1 (1!) 2 e : (27) 2 13
Since from (13) (1!)e 2 =!e 1; the right-h sie of equation (27) can be rewritten as If = i; then! = e 1 1 1 + 2 +!( 2 1 )e : 1 i! (cos = 1 i sin 1 ) 1 1 + i 2 +!( 2 1 ) (cos 1 i sin 1 ) : (28) Its real part is! [sin Re = 1 + 2 cos 1 ] 1 (1 +!( 2 1 ) cos 1 ) 2 + ( 2!( 2 1 ) sin 1 ) 2 ; (29) As is alreay shown in Theorem 3, the steay state is asymptotically stable for 1 = 0 any 2 > 0. Graually increasing the value of 1 with xe value of 2 ; the horizontal line crosses either L 1 (0; n) or L 2 (0; n). Consier rst the intercept with the segment L 1 (0; n): Since both sin 1 cos 1 are positive for 1 2 (0; =2), Re > 0: 1 This inequality implies that as 1 increases, stability is lost when 1 crosses the segment L 1 (0; n). In the case of the linear learning process, local instability leas to global instability. However this is not necessary true if the learning process is nonlinear. To con rm global behavior, two bifurcation iagrams generate by the elay logistic equation (8) are illustrate. 3 In particular, we vary 1 from m 1 to M 1 in 0:01 increments, calculate 1000 values for each value of 1 use the last 300 values to get ri of the transients. The local maximum minimum values of the trajectory are plotte against 1 to construct a bifurcation iagram of a e with respect to 1. In Figure 3(A), 2 is xe at 4 1 increases along the lowest horizontal otte line of Figure 2. The stable steay state loses stability at P 1 ' 1:691; the intersection with the segment L 1 (0; 0). 4 The bifurcation iagram in Figure 3(A) inicates that trajectories have one maximum one minimum for 1 > P 1 : In other wor, the steay state bifurcates to a limit cycle. It is further seen that the limit cycle exps 3 The linear equation (7) with two elays generates the same simple ynamics as the linear equation (3) with one elay. So no further consierations are given to it. 4 Using the secon equation of (25), we solve equation 4 = 1 sin 1 2 2! + 1 2(1!) for then substitute the solution into the rst equation of (25) to obtain the value of P 1 : 14
as 1 becomes larger. In Figure 3(B), 2 is change to 12 1 increases along the thir highest horizontal otte line of Figure 2. The stability is lost at P 1 ' 1:80 where the otte line crosses the segment L 1 (0; 1) from the left. 5 As 1 gets larger than P 1, erratic (chaotic) behavior emerges via a perio-oubling bifurcation. Further increasing 1 suenly reuces complex ynamics to simple perioic oscillations for 1 being close to M 1. Only the values of 2 are i erent between these two iagrams. So a larger 2 can be a source of erratic uctuations of a e (t). The results are numerically con rme thus summarize as follows: Proposition 1 The steay state loses local stability when increasing 1 crosses the segment L 1 (0; n) from left for the rst time its global behavior has a variety ranging from perioic to aperioic oscillations epening on the value of 2 : (A) 2 = 4 (B) 2 = 12 Figure 3. Bifurcation iagram I Assume next that 1 crosses a segment L 2 (0; n) where cos 1 < 0 as 1 2 (=2; ). It is clear from (29) that Re = 1 3! cos 1 @ 2 (1 +!( 2 1 ) cos 1 ) 2 + ( 2!( 2 1 ) sin 1 ) 2 @ (30) where, by implicitly i erentiating the secon equation in (18) using (14), we have @ 2 @ = 1 2 sin 1 + 2 cos 1 cos 1 : 5 See footnote 4 for etaile arguments to etermine P 1. 15
Since the rst factor with the minus sign of (30) is positive, stability is switche to instability when @ 2 =@ > 0 instabiliy might be switche to stability when @ 2 =@ < 0: Although it is possible to con rm analytically the sign of the erivative @ 2 =@ on the segment L 2 (0; n); we numerically check it. 6 We also examine responses of the nonlinear learning as a function of 1 for two i erent xe values of 2, 2 = 3 2(2! 1) 2 = 9 2 : We start with 2 = 3=(2(2! 1)) ' 7:85. Although it is not clear in Figure 2, the segment L 2 (0; 1) takes a convex-concave shape. So the otte horizontal line at 3=(2(2! 1)) coul have multiple intersects with L 2 (0; 1): The secon equation of (26) etermines a value of 2. So solving the following equation 3 2(2! 1) = 1 sin 1 2 2! + 1 + 2 2(1!) with! = 0:8 for yiels three solutions, a = 1; b ' 0:84, c = 0:6; each of which is then substitute into the rst equation of (26) to obtain three corresponing values of 1 ; a 1 = 2 ' 1:57; b 1 ' 0:84, c 1 = 2(2! 1) ' 2:62: Notice that a 1 c 1 are the 1 -values of the points E 1 I 0 in Figure 2. Fixing the parameters 2 at 3=(2(2! 1)) a = k = 1, we perform simulations of equation (8) for i erent 1 values numerically con rm two ynamic results; one is the appearance isappearance of a limit cycle the other is multi-stability (i.e., coexistence of attractors). Characterization of local bifurcation along occuring along the otte line is given. Start with Figure 4(A). The steay state is locally stable for 1 < a 1; loses stability for 1 = a 1 at which @ 2 =@ > 0 the real part of an eigenvalue becomes positive. With increasing 1 ; it bifurcates to a limit cycle which rst exps, then shrinks nally merges to the steay state to regain stability for 1 = b 1 at which 2 =@ < 0 the real part of the same eigenvalue becomes negative again. For b 1 < 1 < c 1,the steay state is surely locally asymptotically stable. In orer to examine global behavior we present two bifurcation iagrams with slightly i erent initial functions e ne for t 0, one is '(t) = 0:9 for the re iagram the other is '(t) = 0:1 for the blue iagram, taking other parameter values xe. The re trajectory is convergent while the blue trajectory exhibits oscillatory behavior. Depening on a choice of the initial function, the same elay equation generates i erent global ynamics, stable re one perioic blue one, inicating coexistence of stable steay 6 See Matsumoto Sziarovszky (2012) for analytical examinations of the sign of the erivative. 16
state, a stable limit cycle unstable limit cycle that is not observable. The steay state with '(t) = 0:9 becomes locally unstable for 1 = c 1 further increasing 1 > c 1 generates a limit cycle that is ientical with the limit cycle with '(t) = 0:1. In Figure 4(B), 2 is increase to 9=2 ' 14:14 the horizontal otte line crosses the segment L 2 (0; 2) twice at the points a 1 = 2 ' 1:57 b 1 ' 1:78: Stability is lost at 1 = a 1 for which the otte line crosses L 2 (0; 2) with @ 2 =@ > 0 regaine at 1 = b 1 for which the otte line crosses L 2 (0; 2) with @ 2 =@ < 0: The otte line also crosses the segment L 1 (0; 1) with @ 2 =@ > 0 at c 1 ' 1:93 for which stability is lost again. The appearance isappearance of a limit cycle for a 1 < 1 < b 1 is also observe coexistence of multiple attractors is also notice in the small interval whose right en point is b 1: It is further observe that complex behavior emerges via a perio oubling bifurcation for P 1 < 1 < M 1 : Comparing the two iagrams where only the values of 2 are i erent leas to the same conclusion that a larger 2 can be a source of erratic oscillations of a e (t): Proposition 2 When the horizontal line of 1 crosses the segment of L 2 (0; n) for the rst time from left, then three i erent ynamics emerge; (1) the steay state bifurcates to a limit cycle that exps, shrinks then merges to the steay state for a 1 < 1 < b 1; (2) epening on a choice of the initial functions, it becomes locally asymptotically stable or bifurcates to a perioic oscillation for b 1 < 1 < c 1; (3) it procees to chaos via a perio-oubling cascae for c 1 < 1 < M 1 : 17
(A) 2 = 3 2(2! 1) (B) 2 = 9 2 Figure 4. Bifurcation iagram II In the next simulation, 2 is increase to 17 then kept xe. 1 is increase along the highest horizontal otte line of Figure 2. As can be seen, the horizontal line has three intercepts: the rst one with L 1 (0; 2) at which the steay state becomes unstable; the secon one with L 1 (0; 1) at which the real part of one more eigenvalue becomes positive; at the thir intercept with L 2 (0; 2) at which only one real part of the two eigenvalues changes back to negative. However, in this case no stability switch occurs since stability cannot be regaine. Accoring to Figure 5, the steay state is replace with a perioic oscillation just after it becomes unstable there is a very short perio-oubling cascae to chaos. As we can see further, interesting ynamics begins; complex ynamics suenly isappears a perioic oscillation appears unergoes 18
a perio-oubling bifurcation cascae to chaos again. Figure 5. Bifurcation iagram III Let us summarize the main point that has been mae so far. Proposition 3 Given k = 0; the bounary of the stable region consists of the envelop of the segments L 1 (0; n) L 2 (0; n) for n 0: We can also show that at stability switches only one eigenvalue changes the sign of its real part, that is, the pure complex eigenvalues are single. Assume not, then = i solves both equations +!e 1 + (1!)e 2 = 0 (31) from which we have 1! 1 e 1 (1!) 2 e 2 = 0 (32) e 1 = 1 + 2 ( 1 2 )! e 2 = 1 1 ( 1 2 )(1!) : (33) If = i, then by comparing the real imaginary parts, we conclue that sin 1 + 2 cos 1 = sin 2 + 1 cos 2 = 0 or tan 1 + 2 = tan 2 + 1 = 0: (34) 19
Let L i (k; n) enote the segment containing ( 1 ; 2 ), then this point is a common extremum of L i (k; n) with respect to 1 2 ; which is impossible, since L i (k; n) is a i erentiable curve. Consier now a point ( 1; 2) with positive coorinates, consier the horizontal line 2 = 2 its segment for 0 1 2: There are nitely many intercepts of this horizontal segment with the set L = [ 1 n=0 [ 1 k=0 fl 1 (k; n) [ L 2 (k; n)g : Let s( 1; 2) enote the number of intercepts where stability is lost g( 1; 1) the number of intercepts where stability can be regaine. With ( 1; 2) the system is asymptotically stable if g( 1; 2) s( 1; 2) unstable otherwise. The stability region is illustrate as the shae omain in Figure 2. 4.2 The Case of! > 1 From (14) we see that cos( 1 ) cos( 2 ) must have the same sign, so in this case we have also two types of segments, 8 >< 0 1 = 1 sin 1 2 + 2! 1 + 2k (k 0) 2! L 1 (k; n) : >: 0 2 = 1 (35) sin 1 2 2! + 1 + 2n (n 0) 2(1!) 8 >< L 2 (k; n) : >: 0 1 = 1 0 2 = 1 sin 1 2 + 2! 1 + 2k 2! sin 1 2 2! + 1 + 2n 2(1!) (k 0) (n 0) (36) where 1 2! 1. As in the previous case, at = 1; at = 2! 1; 2 + 2! 1 2! 2 + 2! 1 2! = 1 2 2! + 1 2(1!) = 1 2 2! + 1 2(1!) Therefore the initial en points of L 1 (k; n) are I 1 (k; n) = 2 + 2k; 2 + 2n = 1 = 1: 20
E 1 (k; n) = 1 2! 1 2 + 2k 1 ; 2! 1 those of L 2 (k; n) are I 2 (k; n) = 2 + 2k; 2 + 2n E 2 (k; n) = 2 + 2n 1 2! 1 2 + 2k 1 3 ; 2! 1 2 + 2n : Figure 3 shows the loci L 1 (k; n) L 2 (k; n) with the value of varying from unity to 2! 1. The parameter value! = 1:2 is selecte. The re curves are the L 1 (0; n) segments the blue curves are the L 2 (0; n) loci. If n = 0; then E 1 (k; n) is infeasible L 1 (k; 0) is feasible only for p 2! 1. In this case I 1 (k; n) = I 2 (k; n) E 2 (k; n) = E 1 (k; n + 1); so L 1 (k; n) L 2 (k; n) have the same initial point E 1 (k; n + 1) E 2 (k; n) have the same en point. Hence segments (L 1 (k; 0); L 2 (k; 0); L 1 (k; 1); L 2 (k; 1); L 1 (k; 2); L 2 (k; 2); :::) starting at E 1 (k; 0) passing through points I 1 (k; 0) = I 2 (k; 0); E 2 (k; 0) = E 1 (k; 1); I 1 (k; 1) = I 2 (k; 1); E 2 (k; 1) = E 1 (k; 2); ::: form a continuous curve. If 1 = 0; then the steay state is asymptotically stable for any 2 0 accoring to Theorem 2. If 2 = 0; then the steay state is locally asymptotically stable if 0 < 1 < 1 locally unstable if 1 > 1 accoring to Theorem 3. We now procee to the general case of 1 > 0 2 > 0: With xe value of 1 > 0 k = 0; the vertical line with graually increasing values of 2 crosses L 1 (0; n) L 2 (0; n) in nitely many times. We numerically con rm that stability switch coul occur at some of those intersections. Consier stability switches rst along the vertical otte line at a 1 = 1:2 then along the 21
vertical otte line at b 1 = 1:5 as shown in Figure 6. Figure 6. The stability-switch curve with! > 1 The rst numerical result is illustrate in Figure 7(A) where the steay state repeats stability-loss stability-gain four times nally becomes unstable. More precisely, stability is lost at the intercepts with L 2 (0; n) for n = 0; 1; 2; 3 while it is regaine at the intercepts with L 1 (0; n) for n = 0; 1; 2; 3: The system generates only perioic oscillations. The secon numerical result is illustrate in Figure 7(B). It can be seen that the system gives rise to perioic oscillations as well for a length of elay approximately less than 10 to more complex oscillations for a larger value of the elay. The special case of! = 1 is ientical to the case of a single elay, which was alreay iscusse. 22
(A) 1 = a 1 (B) 1 = b 1 Figure 7. Bifurcation iagrams with i erent values of 1 4.3 The Symmetric Case of! = 1 2 If! = 1=2; then equations (14) (15) become cos( 1 ) + cos( 2 ) = 0 1 2 (sin( 1) + sin( 2 )) = 0 the segments L 1 (k; n) L 2 (k; n) are simpli e as follows: 8 >< 1 = 1 sin 1 () + 2k (k 0) L 1 (k; n) : >: 2 = 1 sin 1 () + 2n (n 0) (37) 8 >< 1 = 1 sin 1 () + 2k (k 0) L 1 (k; n) : >: 2 = 1 sin 1 () + 2n (n 0): (38) Clearly has to be in the unit interval in orer to have feasible solutions. The segments L 1 (k; n) L 2 (k; n) for small values of k n are illustrate as the re the blue curves in Figure 8. When k = n = 0; these segments construct a hyperbolic curve passing through the point (=2; =2) which is the common 23
point of L 1 (0; 0) L 2 (0; 0): It ivies the rst quarant of the ( 1 ; 2 ) plane into two subregions: in the yellow region uner the curve, the steay state is locally asymptotically stable in the white region above, it is locally unstable. Note that the curve is symmetric with respect to the iagonal asymptotic to the line i = 1 for i = 1; 2. This implies that any elay i > 0 is harmless if j 1 for i; j = 1; 2 i 6= j: Figure 8. Stability switch curves with! = 0:5 Two numerical simulations are one with two i erent values of 1 : In the rst simulation epicte in Figure 9(A), we increase the value of 2 from zero to 20 along the vertical otte line at 1 = =2. The steay state loses stability at 2 = =2 bifurcates to cyclic oscillations with nite number of perioicity as 2 becomes larger. As far as the simulations are concerne, only perioic cycles can be born. In the secon simulation shown in Figure 9(B), we change the value of 1 to (=2) + 1: A limit cycle emerges after stability is lost at 2 = a 2 when the increasing value of 2 crosses the hyperbolic curve, then exhibits aperioic 24
oscillations for 2 being about 14 returns to perioic oscillations afterwars. (A) 1 = 2 (B) 1 = 2 + 1 5 Conclusion Figure 9. Bifurcation iagrams with! = 1 2 An aaptive learning process is introuce when the monopoly knows its cost function, the marginal price uncertain about the maximum price. It is able to upate repeately its belief of the maximum price by comparing the actual preicte market prices. It is assume that the rm s preiction is either the most current elaye price information or it is obtaine by interpolation or by extrapolation base on two elaye ata. The asymptotical stability of the resulte ynamic learning process is examine. If it is asymptotically stable, then the beliefs of the rm about the maximum price converge to the true value, so successful learning is possible. Stability conitions are erive, the stability regions are etermine illustrate. The global behavior of the trajectory is examine by using simulation. The ynamic moel (5) (7) are linear, when local asymptotical stability implies global asymptotical stability. However (6) (8) are nonlinear, where only local asymptotic stability can be guarantee uner the erive conitions. In our future research further nonlinearities will be introuce by selecting nonlinear price cost functions. 25
References [1] Bellman, R. Cooke, K. (1956), Di erential-di erence Equations, Acaemic Press, New York. [2] Burger, E. (1956), On the Stability of Certain Economic Systems, Econometrica, 24, 488-493. [3] Cooke, K. Grossman, Z. (1992) Discrete Delay Distribute Delay Stability Switches, Journal of Mathematical Analysis Applications, 86, 592-627. [4] Cushing, J (1977), Integro-Di erence Equations Delay Moels in Population Dynamics, Springer-Verlag, Berlin/Heielberg/New York. [5] Hayes, N. D. (1950), Roots of the Transcenental Equation Associate with a Certain Di erence-di erential Equation, Journal of the Lonon Mathematical Society, 25, 226-232. [6] Invernizzi, S. Meio, A. (1991), On Lags Chaos in Economic Dynamic Moels, Journal of Mathematical Economics, 20, 521-550. [7] Matsumoto, A. Sziarovszky, F. (2013), Discrete-Time Delay Dynamics of Bounely Rational Monopoly, forthcoming in Decisions in Economics Finance. [8] Matsumoto, A. Sziarovszky, F. (2012a), Dynamic Monopoly with Multiple Continuously Distribute Time Delays, IERCU DP #184 (http://www2.tamacc.chuo-u.ac.jp/keizaiken/iscuss.htm), Chuo University. [9] Matsumoto, A. Sziarovszky, F. (2012b), Boune Rational Monopoly with Single Continuously Distribute Time Delay, IERCU DP #180 (http://www2.tamacc.chuo-u.ac.jp/keizaiken/iscuss.htm), Chuo University. [10] Matsumoto, A. Sziarovszky, F. (2012c), An Elementary Stuy of a Class of Dynamic Systems with Single Time Delay, forthcoming in CUBO: A Mathematical Journal. [11] Matsumoto, A. Sziarovszky, F. (2012), Nonlinear Delay Monopoly with Boune Rationality. Chaos, Solitons Fractals, 45, 507-519. [12] Naimzaa, A. (2012), A Delay Economic Moel with Flexible Time Lag, forthcoming in Chaos, Solitons Fractals. [13] Naimzaa, A. Ricchiuti, G. (2008), Complex Dynamics in a Monopoly with a Rule of Thumb, Applie Mathematics Computation, 203, 921-925. [14] Puu, T. (1995), The Chaotic Monopolist, Chaos, Solitons Fractals, 5, 35-44. 26