Ergodic Cobweb Chaos

This study augments the traditional linear cobweb model with lower and upper bounds for variations of output. Its purpose is to detect the relationship between the output constraints and the dynamics of the modified model. Due to the upper and lower bounds, a transitional function takes on a tilted z-profile having three piecewise segments with two turning points. It prevents the price (or quantity) dynamics from explosive oscillations. This study demonstrates, by presenting numerical examples, that the modified cobweb model can generate various dynamics ranging from stable periodic cycles to ergodic chaos if a product of the marginal propensity to consume and the marginal product is greater than unity.


INTRODUCTION
This study investigates the traditional cobweb model with upper and lower bounds for output variations.Its purpose is to consider implications of the quantity constraints on dynamic behavior of the agricultural economy.The traditional cobweb model, which has naive expectations and linear demand and supply curves, can produce only three types of dynamics: convergence to an equilibrium, convergence to period-2 cycles or divergence.None of these types, however, is E-mail: eakio@hle.niigata-u.ac.jp.satisfactory to explain observed irregular fluctua- tions of the agricultural goods.Neither the first type nor the third is consistent with observed ups and downs in real economic data, and the second, which implies persistent oscillations in price and quantity, depends on the unrealistic and shaky condition (i.e., supply and demand have exactly the same elasticities).To overcome those limita- tions, the traditional cobweb model had been modified to produce more realistic dynamics with the help of new developments in non-linear economic dynamics.examples, it demonstrates that the modified cob- web model can generate a wide spectrum of dynamic behavior ranging from stable periodic cycles to ergodic chaos.
This study is organized as follows.Section 2 constructs the linear cobweb model with upper and lower bounds for output variations.Section 3 simulates the model.Section 4 makes concluding remarks.

THE COBWEB MODEL WITH FLEXIBLE CONSTRAINTS
The traditional cobweb model is made up of the following four equations in discrete time: qa D(pt) Demand, qS S(Pt), Supply, qt-qdt q, Temporary equilibrium, pe_ Pt-1, Naive expectation, where pt, p, qdt and q are the actual price, the expected price, the quantity demanded and the quantity supplied, respectively, where all are taken for period t.This model can be reduced to a one- dimensional difference equation qt+l S(D-l(qt)). (1) In a simple version of the cobweb model, the demand function as well as the supply function is monotonic and thus the composite map, S(D -l (qt)), is also monotonic.The slope of S(D-I(qt)) evaluated at the equilibrium point characterizes dynamics, since it equals an eigenvalue of the dynamic equation.As long as the slope lies between 0 and -45 , the eigenvalue in absolute value is less than unity, and thus the equilibrium point is stable.The stable trajectories of price or quantity converge to a stationary state, which does not go with the price or quantity dynamics observed in the real-world.As the slope steepens beyond -45 , the eigenvalue in absolute value-is greater than unity and thus the equilibrium point is unstable.The unstable trajectories explosively oscillate, which also contradicts the actual dynamic behavior.When the slope of S(D-l(q)) is equal to -45 , period-2 cycles can appear.
However, the regular cycles are unlike the irregular nature of the actual cycles.Thus such a simple cobweb model has difficulties to explain cyclical or erratic movements observed in statistical data of agricultural goods.
See, for example, the actual fluctuations in agricultural goods provided by Finkenstidt and Kuhbier (1992).
In the real world, it is not surprising that a competitive firm prevents output tomorrow from changing drastically from output today, taking account of capacity constraints, financial con- straints, high costs for changing production levels, demand uncertainty, etc.The intention of this study is, in recognition of this fact, to look for an alternative modeling of a firm's behavior that may explain the appearance of cobweb fluctuations.Returning to the original spirits of "flexible coefficient" in Day (1980), 2 we incorpo- rate cautious behavior of a competitive firm which puts upper and lower bounds on its growth rate.Our specification of the model is as follows: where the last two equations imply that the growth rate constraints have the effect of preventing output in period + from increasing by more than 100a% or decreasing by less than 100/3% from the output of period t.Alternatively put, the upper bound and lower bound on the growth rate are, respectively, a and /3.As a result of these bounds, the dynamic system of qt becomes a piecewise linear map: (1 + a)qt f (qt) where QM and Qm are a local maximizer and a local minimizer.These are calculated, respectively, as bc a bc a Q + a + bd and Qm -/3 + bd" Under the assumptions of positive parameters, bd > O, bc-a > 0, a > 0, and/3 > 0, the map has the tilted-z profile and its non-linearity becomes more pronounced as bd gets larger.Let q* be a stationary state satisfying q* =/(q*) (i.e., q* (bc-a)/(1 + bd)).If bd> 1, it is oscillatory unstable and forces trajectories to move away in its neighborhood.But fluctuations are bounded by the upper and/or lower con- straints of output and thus perpetuated.In order to explain the limiting behavior of bounded fluctuations, we identify three cases that depend on the relative magnitudes between a and/3: (1) (1 + a)(1 -/3) > 1.The corresponding profiles of f(qt) are depicted, respectively, in Figs.l(a)-(c) where a cone spanned by the upper and lower constraints is symmetric with respect to 45 line in the first case and asymmetric in the second and third cases.Considerable progress is found in the study on a map with two piecewise linear segments and a single kinked point like a tent map.However, only the partial results are q(g+l) FIGURE (a) (1 + a) (1-/3) 1.Three profiles of the dyna- mic equation, f(q).
See Day (1980, p. 197) who considers the symmetric upper and lower constraints.
dynamic structure of the modified cobweb model under each of these three parameter constella- " ' , , , , , , , // tions for a and/.

SIMULATION OF THE MODEL
To explore the dynamic behavior of q, we simulate the model under different.values of bd where b is the marginal propensity to consume and d is the marginal product with respect to the price.We focus on those cases in which oscilla- tions are persistent (i.e., bd > 1).Since the dynamic equation, f(q), has the upper and lower q(t) bounds, it induces any trajectories, which are repelled by the unstable equilibrium, bounce back to a neighborhood of the equilibrium point.As it turns out, a combination of a and fl charac- terizes asymptotic behavior of trajectories.

Symmetric Case
In this subsection, we deal with the symmetric case in which (1 +a)(1-/3)--1 holds.It is numerically and analytically verified that the symmetric-constrained cobweb model can gener- ate stable periodic orbit with period-2.
Two bifurcation diagrams are shown if Fig. 2. Simulations are performed under the same initial point, q0--0.1, and different combinations of a and /3: a= and /3=0.5 in Fig. 2(a) and a=4 and /3-0.8 in Fig. 2(b).The bifurcation para- meters 1/bd is varied in decrement of 0.025 from one to zero (or bd increases from one to infinity).For each values of 1/bd, f(qt) is iterated 200 times.Although the last 100 iterates are plotted on the vertical axis, only two points are seen in the bifurcation diagrams.
These numerical examples suggest that the symmetric-constrained cobweb model generates only period-2 cycles.This is analytically verified and summarized as follows.
Thus every trajectory emanating form a complementary interval of a union of Ao, A and q*, A0 t_J A t3 {q* }, enters into Ao t3 A after finite iterations.Therefore, we have stable period-2 cycles in the symmetric case.THEOREM 1.Given bd> 1, the symmetric-con- stra&ed cobweb model generates only stable peri- od-2 cycles.
Proof Let us denote a local minimum of q by Qmin :--f(Qm) and a local maximum by QMAX:=f(QM).Since omin < QM < Om < QMAX holds in the symmetric case, we can divide an 3.2 Lower-Asymmetric Case In this subsection, we deal with the asymmetric case in which (1 + a)(1-fl)< holds.As seen in Fig. l(b), the lower constraint line deviates from the 45 line more than the upper constraint line so that we call this case lower-asymmetric.Both of the upper and lower bounds are not necessarily effective in the lower-asymmetric case.Only the upper bound is effective for smaller deviations of bd from unity and so are both of the bounds for larger deviations.The dynamics generated by f(qt) which is constrained by the upper-bound is qualitatively different from the one by f(qt) which is by the upper and lower bounds.We call the former the one-kinked dynamics and the latter the two-kinked dynamics.
In Figs.3(a) and (b) below, two numerical simulations are performed for the lower-asymmetric conditions; a 0.25 and/ =0.4 and a 2 and /3=0.8.In both bifurcation diagrams, com- plex dynamics is observed.As is seen shortly after, the vertical dot line in each of Fig. 3 passes through a critical value of 1/bd that separates the one-kinked dynamics from the two-kinked  dynamics.In the following, we confine attention, first, to a case of the one-kinked dynamics in which oscillations are perpetuated due to the upper bound, and then to a case of two-kinked dynamics due to both of the upper-and lower- bounds.
Since a trapping interval eventually traps all trajectories, a restriction of f(q) to the trapping interval governs the asymptotic behavior of q.In the lower-asymmetric case, two distinct trapping intervals can be identified which depend on the relation between maximum QMAX and minimizer Qm.One trapping interval is defined in a case where QMAX <= Qm and the other where QMAX> Qm.In particular, subtracting Qm from QMAX yields (bc a)a QMAX Qm (1 + oz + or)(1 -/3 + or) l+a ] 5, (3)   where r := bd for notational simplicity.Since the lower asymmetric condition, (1 + a)(1-/3)< 1, is transformed into a/(1 + a) </3, there is a r > 1, denoted by CrL, such that crL(a/(1--OZ))-/. 4en Eq. ( 3) implies the following relations between QMAX and Qm: QMAX QMAX > Qm for r > CrL.
Let Vl := [f(QMAX), QMAX] wheref(QMAX) bc-a-oQMAX.It is the trapping interval for < cr < rL.Namely, whichever point a trajectory starts with, it will enter the interval, V, after finite iterations and stay there afterwards.
Furthermore, trajectories inside V are con- strained only by the upper bound.The asympto- tic dynamics are governed by the restriction of f(qt) to V: fv(qt) min{(1 + a)qt, bc a-oq}. (S) The conditions, + a > and a > 1, imply that the restricted map fv, (qt) is expansive and thus 5 has an absolutely continuous invariant measure.
5For an expansive map, see Theorem 3 in Day and Pianigiani (1991, p. 45).For dynamic behavior of the expansive map, see Theorem 3 of Day and Schafer (1987, pp.352-353) and (3) in Property 5 of Nusse and Hommes (1990, p. 13).Then we will show in Lemma below that fvl (qt) is equivalent to a unimodal linear map with two parameters A and B, denoted by gA,B(q): Aq + A+s-A.! B fr 0 <q <1 B, for -1 B -<-q -<-1.
Proof Let q f(QMAX) 99(q) QMAX _f(QMAX) to It can be checked 99(q) is a linear isomorphism from Vii (i L, R) onto the unit interval such that 99 0v o 99 -1 gA, B.
According to Theorem 2, the bifurcation scenario of this numerical example is as follows.When 0-is increased from unity but less than 0-1, 23-periodic chaos appears.This means that there are 23 disjoint intervals and every trajectory starting outside these intervals eventually enters into a union of these intervals.Furthermore, the union is a support of an invariant measure of fv (qt).Intuitively speaking, the trajectory peri- odically travels one interval to another but chaotically oscillates within each interval.As 0- increased further, interval halving bifurcations This is a lower part of Fig. 4 in ITN.Definitions of Do, D1 and D* are given in Appendix.occurs for (A, B) E D0(i.e., 23 --+ 22 --2).7 D and D* are chaotic region in which the system exhibits erratic behavior.Similarly, the bifurcation dia- gram in Fig. 3(b) is obtained by performing simulation for a slice through the (l/A, 1/B) space for 1/A 1 / 2 .The switching value of cr in this example is crL-1.2(i.e., crL -0.83 in Fig. 3(b)).
Next, we consider dynamics for r>r at which there is a qualitative change in the dynamics behavior of the system.Let V2 :-[Qmin, QMAX].It is, as indicated by the second Eq. in (4), the trapping interval for cr > re.Since the inequality relations, Qmin < QM < Qm < QMAX, holds, V2 contains two kinked points, QM and Qm, at each of which the switching of the dynamic system from either (1 + c)q or (1-/3)q to S(D-(q)) or vice versa takes place.A restriction of f(qt) to V2 has a tilted z- shape: fv. (qt) max{(1 )qt, min{(1 + a)qt, bc-a-bdqt}}. (6)though DS (see Table I at p. 355) make some characterizations for such a map with two kinked points, only a little is known about its qualitative behavior.We present numerical examples to detect the influence of cr on the two-kinked dynamics.The bifurcation diagrams left to the vertical dot lines in Fig. 3 explore the two-kinked dynamics generated by fly2.a) and (b), chaotic behavior still appears as cr is increased from cr .If a is further increased, periodic chaos with five intervals appears in the example of Fig.

3(b).
Return maps for cr 2 and or--5 are given in Fig. 5 where both of the upper and lower bounds are effective (i.e., trajectories hit the q(t+l) FIGURE 5(a) a=2.FIGURE 5 or=5.Return maps of f(q) for a=2 and /=0.8.

Upper-Asymmetric Case
In this subsection, we deal with the asymmetric case in which (1 + a)(1-/3)> holds.As seen in Fig. l(c), the upper constraint line deviates from the 45 line more than the lower constraint line so that we call this case upper-asymmetric.The following analysis is analogous to that in the lower-asymmetric case.The one-kinked dynamics is considered and then the two-kinked dynamics is followed.The dis-similarity is found in that the dynamics system with one-kinked point is not expansive.We, first, identify two trapping inter- vals which depend on the relation between minimum omin and maximizer QM: one where QM < omin.the other where QM > omin.Subtract- ing Qmin from QM yields QM Qmin (bc-a) (1 + c + or)(1 -/3 + r) [_ I-/] (7) Since the upper-asymmetric condition, (1 + a) (1-/3) > 1, is written as /3/(1-/3) < a, there is a or> 1, denoted by Cru, such that ru(/3/(1-/3)) a. Then the Eq. ( 7) implies the following relations between QM and Qmin: QM <-Qmin for r < ru, QM > Qmin for r > cru.
Lemma 2 suggests that fv, (q) generates quali- tatively the same dynamics as gA,(q) with A < and B> 1. Figure 6 below, which is similar to Fig. 4 in the lower-symmetric case, is a 2parameter bifurcation diagram where the horizo- ntal axis is 1/B and the vertical axis is 1/A. 8In Fig. 7, two numerical simulations are performed along the straight line passing through 1/A 1.2 and one through I/A 1.8 in Fig. 6. 9 Theorem 3 below summarizes the results for the dynamics generated by the latter numerical example.stable period-2 cycles, periodic chaos with 23 from stable period-2 cycles, to periodic chaos intervals, 22 intervals, 2 intervals and then one with 22 intervals, to one with 2 interval and then interval, to truly chaos (i.e., chaos with interval), the last If is further increased from Cru, Qmin < QM of which shrinks to zero as bd goes to infinity.
holds.Then the dynamic system is switched from What implications do the simulations have for (8) to a restriction of f(qt) to a trapping interval the cobweb model with the upper and lower U2 := [Qmin, Qmax].It has three line segments and bounds?bd > is necessary for generations of two kinked points, QM and Qm.The tilted z- chaos in the asymmetric case.Here b is the shaped restricted map governs the quantity marginal propensity to consume which is the evolution and, according to Fig. 7(a), generates decision variables of the demand side while d is aperiodic fluctuations, the marginal product with respect to price which is the decision variable of the supply side.Thus it can be stated that one source of such complex 4 CONCLUDING REMARKS dynamics involving chaos is an interaction be- tween the consumers and the producers.Further- This study investigates the dynamic structure of a more, since the marginal propensity to consume linear cobweb model with upper and lower is expected to satisfy condition 0<b<l, bd> bounds for variations on output.Simulating the requires the strong production response to a model under different values of bd, we have change in price (i.e., the large value of b).This demonstrated that the modified cobweb model study, therefore, implies that agricultural econo- may generate chaotic behavior if the-output mies with the strong production response tend to constraints are asymmetric and that it can exhibit erratic dynamics of the agricultural goods generate stable period-2 cycles whose amplitudes when the upper and lower bounds are imposed depend on the prevailing parameter constellations on the growth rate.
(i.e., choice of initial point, values of bd, etc.) if the output constraints are symmetric.Although Acknowledgement the bifurcation diagrams in asymmetric cases imply the chaotic behavior, the bifurcation sce- nario to chaos is different according to whether the constraints are upperor lower-asymmetric.
In the lower-asymmetric case, the modified cobweb model generates persistent and aperiodic cycles for any values of bd greater than unity.The control system (i.e., a restriction off(q) to a trapping interval) switches from an expansive I would like to thank K. Higuchi, Y. Sakai, A.
Simonovits and M. Sonis for valuable comments and constructive suggestions.The earlier version of the paper benefited from comments received during presentations at the conference on "Comerce, Complexity and Evolution" at University of New South Wales, Sydney, Australia, Febru- ary 12-13, 1996 and the 5th World Congress of map to a tilted z-shaped map as bd is increased, the RSAI at Risshou University, Tokyo, Japan, In consequence, periodic chaos with disjoint May 2-6, 1996.I am grateful to financial intervals and then truly chaos appear as bd support from the Nomura Foundation.Needless to say, all remaining errors are my own.exceeds unity.On the other hand, in the upper- asymmetric case, the control system switches from an asymmetric tent map to a tilted z-shaped map.

FIGURE 4
FIGURE 4 Regime classification of two parameters, A and B.

Figure 6
Figure 6 is an upper half of Fig. 4 in ITN.Since A -/3,/3 in the former example and/3