# The Asymptotic Behavior in a Nonlinear Cobweb Model with Time Delays

Authors: Akio Matsumoto; Ferenc Szidarovszky
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We study the effects of production delays on the local as well as global
dynamics of nonlinear cobweb models in a continuous-time framework. After
reviewing a single delay model, we proceed to two models with two delays.
When the two delays are used to form an expected price or feedback for price
adjustment, we have a winding stability switching curve and in consequence
obtain repetition of stability losses and gains via Hopf bifurcation. When
the two delays are involved in two interrelated markets, we find that the
stability switching occurs on straight lines and complicated dynamics can
arise in unstable markets.


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## Body

## 1. Introduction

It is now well known that the cobweb model or cobweb theory has been developed in various directions since the pioneering work of Kaldor [1]. It explains why and how certain types of markets give rise to fluctuations in prices and quantities. Since it mainly focuses on the agricultural markets in which producers determine their outputs before observing market prices and a delay between planting and harvesting is inevitable, its key issues are an expectation formation of price and a production delay. In early stage, the models are essentially linear and constructed in discrete-time scales in which production delay is incorporated from the beginning. Thus the main question is on how the expectation formations such as naive, adaptive, and rational expectations are responsible for the emergence of fluctuations. During the last two decades, increasing attention has been given to nonlinear dynamics. A summary of recent developments in nonlinear oligopolies is given in Bischi et al. [2] with a wide variety of the extensions of the classical Cournot model. More recently, Shirai and Amano [3] examine a production process with nonlinearity of the rate of return on sales. Their mathematical model is based on a Van der Pol differential equation. Nonlinear models such as discrete-time cobweb models can generate a wide spectrum of dynamic behaviors involving chaos. See Dieci and Westerhoff [4] and Hommes [5] as additional references, to name a few.It is, however, less known that a continuous-time cobweb model with fixed time delay is also developed with the same problem consciousness as early as in the 1930s. In particular, Haldane [6] found the similarity between the effects caused by the rise in the birth rate in biology and the ones caused by a rise in commodity price in economics and built a simple economic model to examine the fluctuations in price and the rate of production, coaxing the idea from theoretical biology. Independently from Haldane, Larson [7] presents a linear continuous-time model in which a hog cycle is described as a harmonic motion. It is assumed that realized production has 12-month delay from planned production and the rate of production change is proportional to the deviation of price from equilibrium. Mackey [8] gives a nonlinear price adjustment model with production delay and rigorously derives a stability switching condition for which the stability of equilibrium is lost. Furthermore, it is shown that a Hopf bifurcation takes place and thus the stable equilibrium bifurcates to a limit cycle after the loss of stability. Recently Gori et al. [9] propose a delay cobweb model with the profit-maximizing behavior to characterize production cycles. Although the delay models have been an object of study for a long time, these are subject to a single delay and little is known about multiple delay models.The theory of delay differential equations is well known from the mathematical literature. The classical book of Bellman and Cooke [10] summarizes the earlier results. A simple analytic method of examining systems with a single delay is given, for example, in Matsumoto and Szidarovszky [11], and it is extended to a special class of models with two delays in Matsumoto and Szidarovszky [12]. A more general geometric approach is introduced in Gu et al. [13], which can be used to find the stability switching curves and the directions of the stability switches on these curves. Their method is further improved by Lin and Wang [14] giving the tool for examining more complex systems.The purpose of this study is, based on Mackey’s formulation, to investigate how multiple delays affect cobweb price dynamics, applying the recent mathematical developments to characterize the stability of two-delay differential equations developed by Gu et al. [13] and Lin and Wang [14]. Two main results demonstrated in this paper are the following:(i)
Simple dynamics emerge but stability losses and gains are repeatedly taking place in a single market with two time delays.(ii)
No stability gain occurs but complex dynamics can arise when two markets with two delays are unstable.This paper is organized as follows. In Section2, a continuous-time nonlinear price adjustment model is presented as a basic model. In Section 3, a single production delay is introduced to review how the delay affects dynamics. In Section 4, the model with two production delays is constructed and the stability switching curve is analytically and numerically derived. In Section 5, two markets’ models with two delays are considered to develop the conditions under which the two markets are stable or unstable. It is shown that various dynamics arise when the two markets are unstable. In the final section, concluding remarks and further research directions are given.


## 2. Basic Cobweb Model

As in Mackey [8], we consider price dynamics in a continuous-time framework in which relative variations in market price p(t) are adjusted to be proportional to excess demand:(1)p˙tpt=KDpt-Spet,where K>0 is the adjustment coefficient, pe is the expected price, and D(p) and S(pe) are the demand and supply functions of commodity to be considered. Following the tradition, it is assumed that demand negatively depends on price while supply positively depends on the expected price. For the sake of analytical simplicity it is also assumed that consumers and producers make their decisions based only on the price information appearing in the good market. This assumption is taken away in Section 5. The expected price is formed based on the past observed prices:(2)pet=Fpt-τ1,pt-τ2,…,pt-τn,where τi>0 for i=1,2,…,n and p(t-τi) is the delayed price or the price realized at time t-τi. Again for the sake of simplicity, demand and supply functions are assumed to be linear:(3)Dt=d1-d2ptwithd1>0,d2>0,(4)St=s1+s2petwiths1>0,s2>0.The equilibrium price and quantity satisfy the conditions of p∗=pe(t)=p(t) and q∗=D(p∗)=S(p∗) and are obtained as (5)p∗=d1-s1d2+s2,q∗=d1s2+d2s1d2+s2,where, for positivity of the equilibrium price, d1>s1 is assumed. This is a natural assumption requiring that the maximum demand exceeds the minimum supply.Substituting (3) and (4) into (1), taking pe(t)=p(t), and then multiplying both sides of the resultant equation by p(t) yield a nonlinear price adjustment equation:(6)p˙t=Kptd1-s1-d2+s2pt.The equilibrium price p∗ is also a stationary point.1 To examine stability of the equilibrium price, we denote the right-hand side of (6) by G1(p(t)) and linearize it around p=p∗: (7)p˙δt=dG1ptdptp=p∗pδtor (8)p˙δt=-kd2+s2pδt,where pδ(t)=p(t)-p∗ and k=Kp∗. Its solution is (9)pt=p∗+p0-p∗e-kd2+s2t.Since k(d2+s2)>0, the equilibrium price is always locally stable meaning that the market price converges to the equilibrium price as t→∞ if the initial price is close enough to the equilibrium.


## 3. Cobweb Model with a Single Delay

A production time delay is introduced into basic model (6). Concerning the expectation formation, we start with the simplest form of F(·) where the expected price at time t is the market price realized at time t-τ with τ>0.Assumption 1.
Considerpe(t)=p(t-τ)>0.Accordingly, the supply function is modified as(10)St=s1+s2pt-τ.Substituting (10) into (6) presents a delay price adjustment equation:(11)p˙t=Kptd1-s1-d2pt-s2pt-τwhich is a first-order nonlinear delay differential equation. It can be confirmed that p∗ is also a unique positive stationary state of (11). If G2(p(t),p(t-τ)) denotes the right-hand side of (11), then a linearized equation in a neighborhood of the stationary point p2∗=(p∗,p∗) is (12)p˙δt=∂G2∂ptp2∗pδt+∂G2∂pt-τp2∗pδt-τor(13)p˙δt=-kd2pδt-ks2pδt-τ.Introducing the new variable x(t)=pδ(t) and the new parameters, α=kd2 and β=ks2, we obtain the following form of the linearized equation:(14)x˙t+αxt+βxt-τ=0,where x∗=x(t)=0 is the only stationary point. Assuming an exponential solution (15)xt=eλtuand substituting it into (14) give the corresponding characteristic equation(16)λ+α+βe-λτ=0.Without delay τ=0, the stationary point is locally asymptotically stable. If stability of the trivial solution x(t)=0 of (14) switches to instability at τ=τ¯, then (16) must have a pair of pure conjugate imaginary roots. It is then assumed, without loss of generality, that λ=iω with ω>0 is a root. Substituting it into (16) breaks down the characteristic equation to the real and imaginary parts:(17)α+βcos⁡τω=0,(18)ω-βsin⁡τω=0.Moving the constant terms to the right-hand sides and adding the squares of the resulting equations give (19)ω2=k2s2+d2s2-d2.If s2≤d2, then there is no ω>0, implying that the delay isharmless.2Theorem 2.
Ifs2≤d2, then the positive steady state of (6) is locally asymptotically stable for any positive values of τ.On the other hand, ifs2>d2, then we can define ω¯>0 as (20)ω¯=ks2+d2s2-d2.It is substituted into (17) to obtain threshold values of τ:3(21)τ¯=1ω¯cos-1⁡-d2s2+2nπn=0,1,2,….In order to determine the direction of the stability switch, we can think of the root of (16) as a continuous function of the delay τ. Then differentiating (16) with respect to τ and arranging terms yield(22)dλdτ-1=1-τβe-λτλβe-λτ.Thus(23)dRe⁡λdτλ=iω-1=Re⁡1-τβe-λτλβe-λτλ=iω=Re⁡-1λλ+αλ=iω=Re⁡ω+iαωω2+α2=1ω2+α2.Hence(24)dRe⁡λdτλ=iω>0.This inequality implies that all roots that cross the imaginary axis at iω cross from left to right as τ increases. So at point τ¯ with n=0 stability is lost and cannot be regained later.Further substitutingω¯ into (21) determines the critical value of the delay(25)τ¯d2,s2=cos-1⁡-d2/s2Kp∗s2+d2s2-d2with(26)∂τ¯∂d2=s21-d2/s22s22-d22+s221-d2/s22cos-1⁡-d2/s2Kd1-s1s2-d2s22-d223/2>0,∂τ¯∂s2=-d21-d2/s22s22-d22+s221-d2/s22cos-1⁡-d2/s2Kd1-s1s2-d2s22-d223/2<0.In Figure 1(a), the stability switching curve is depicted as a hyperbolic curve on which the real parts of the eigenvalues are zero. The equilibrium price is stable in the shaded region below the curve and unstable in the white region above. Since ∂τ¯/∂d2>0 and ∂τ¯/∂s2<0, increasing the value of d2 and decreasing the value of s2 shift the stability switching curve upward, implying that those parameter changes enlarge the stability region and thus have stabilizing effects. Figure 1(b) illustrates the bifurcation diagram with respect to τ where the parameters are specified as follows.Figure 1
Numerical results on one-delay model (11).
(a)
Stability region
(b)
Bifurcation diagramAssumption 3.
ConsiderK=1, d1=4, s1=1, d2=1, and s2=2.Notice that the maximum demand(d1) is larger than the minimal supply (s1), so the nonzero equilibrium is positive. Furthermore the condition of Theorem 2 is violated, so stability switch can be observed. The diagram is constructed in the following way. With a fixed value of τ, we run delay system (11) for 0≤t≤1000. To take away the initial disturbance, we discard the data of p(t) for t≤950 and plot the local maximum and local minimum of p(t) for 950≤t≤1000 against the value of τ. The value of τ is increased with an increment 0.01 and then the same procedure is repeated until τ arrives at 3. Under these circumstances, when K=1, the threshold value of τ denoted by τ¯ is obtained as (27)τ¯d2,s2=2π33≃1.209.It is seen in Figure 1(a) that the equilibrium is stable for τ<τ¯ and unstable for τ>τ¯. The two branches of the diagram given in Figure 1(b) indicate two issues; one is that a trajectory has one maximum and one minimum, implying the birth of a limit cycle which is confirmed by Hopf bifurcation theorem with (24), and the other is that a cycle becomes larger as the length of the delay increases. Theorem 4 summarizes the results.Theorem 4.
Givens2>d2, the positive steady state of (11) is locally asymptotically stable if 0<τ<τ¯, loses stability at τ=τ¯, and bifurcates to a limit cycle if τ>τ¯, where τ¯=τ¯(1,2)≃1.209.


## 4. Cobweb Model with Two Delays

The price expectation is formed naively under Assumption1. Convergence to the stationary point occurs for τ<τ¯. The producers with delayed price information can eventually arrive at the stationary state. On the other hand, since cyclic oscillation arises for τ>τ¯, the producers sooner or later realize that their expectations are systematically wrong. As a natural consequence, even though they are assumed to be boundedly rational, the producers may change somehow the way they form expectations. One possible way is to utilize more price information obtained in the past. To make it simpler, the producers are assumed to use different prices at times, t-τ1 and t-τ2. There are at least two different ways to employ these two prices. One is to make the price expectation as the weighted average of these prices and the other is to use the difference between these two prices as delay feedback. We first make the following assumption which is a direct extension of Assumption 1 when the price expectation formation is the weighted average of the past two prices.Assumption 5.
Considerpe(t)=θp(t-τ1)+(1-θ)p(t-τ2) with 0<θ<1.Accordingly the supply function is modified as(28)Spet=s1+s2θpt-τ1+1-θpt-τ2and then the price adjustment is governed by a two-delay differential equation:(29)p˙t=Kptd1-s1-d2pt-s2θpt-τ1-s21-θpt-τ2.It is clear that the equilibrium price is a unique positive stationary point of (29). To examine local dynamics, we let G3(p(t),p(t-τ1),p(t-τ2)) be the right-hand side of (29). The linear approximation in the neighborhood of the stationary point p3∗=(p∗,p∗,p∗) is (30)p˙δt=∂G3∂ptp3∗pδt+∂G3∂pt-τ1p3∗pδt-τ1+∂G3∂pt-τ2p3∗pδt-τ2or(31)p˙δt=-kd2pδt-ks2θpδt-τ1-ks21-θpδt-τ2.As before, the last form can be reduced to(32)x˙t+αxt+βxt-τ1+γxt-τ2=0with(33)α=kd2,β=ks2θ,γ=ks21-θ.We now turn our attention to the delay feedback with which price adjustment equation (1) is modified as (34)p˙tpt=k1d1-s1-d2pt-s2pt-τ1+k2pt-τ1-pt-τ2,where k1 is an adjustment coefficient and k2 is a coefficient of the feedback. This can be rewritten as a differential equation with two delays:(35)p˙t=k1ptd1-s1-d2pt-s21-k2s2k1pt-τ1-s2k2s2k1pt-τ2.If we assume k2<s2k1 and denote(36)1-k2s2k1=θ,k2s2k1=1-θ,then the differential equation with the delay feedback (35) is identical with (29), the differential equation with the average price. Hence these equations generate essentially the same dynamics, although their economic interpretations are definitely different. In the following we focus on (29) only because the number of the parameters is smaller; however, the same results can be obtained by examining (35).The corresponding characteristic equation of (32) is obtained by substituting an exponential solution x(t)=eλtu:(37)λ+a+βe-λτ1+γe-λτ2=0.Stability of (32) depends on the locations of the eigenvalues of (37) that is investigated by applying the method developed by Gu et al. [13]. Dividing both sides of (37) by λ+a and introducing the new functions(38)a1λ=βλ+a,a2λ=γλ+αsimplify the left-hand side of (37):(39)aλ=1+a1λe-λτ1+a2λe-λτ2.Substituting a possible solution λ=iω with ω>0 into the two equations of (38) results in(40)a1iω=aβα2+ω2-iβωα2+ω2,a2iω=aγα2+ω2-iγωα2+ω2.Their absolute values are(41)a1iω=βα2+ω2,(42)a2iω=γα2+ω2.We can consider the three terms in (39) as three vectors in the complex plane with the magnitudes 1, a1(λ), and a2(λ). The solution of a(λ)=0 means that these vectors form a triangle if we put them head to tail. That is, solving a(λ)=0 algebraically is equivalent to constructing a triangle geometrically with the following three conditions:(43)1≤a1iω+a2iω,a1iω≤1+a2iω,a2iω≤1+a1iω.Substituting the absolute values of (41) and (42) converts these three conditions to the following conditions:(44)β-γ2-α2≤ω2≤β+γ2-α2that can be rewritten as(45)k2s222θ-12-d22≤ω2≤k2s22-d22.It is clear that the second inequality does not hold if d2≥s2. Hence there is no ω>0, implying that the delays areharmless in the two-delay dynamic model.Theorem 6.
Ifs2≤d2, then the positive steady state of (29) is locally asymptotically stable for any positive values of τ1 and τ2.On the other hand, ifs2>d2, then conditions (45) hold for ω in the interval [ωs,ωe], where(46)ωs=0ifθ≤θ0ks222θ-12-d22ifθ>θ0with(47)θ0=121+d2s2that solvess222θ-12-d22=0,ωe=ks22-d22,where the subscripts “s” and “e” mean the starting point and the end point, respectively.We will next find all the pairs of(τ1,τ2) satisfying a(iω)=0. Let 1 be the base of the triangle and then denote an angle between 1 and a1(iω) by θ1 and an angle between 1 and a2(iω) by θ2. By the law of cosine, the angle’s magnitudes are expressed in terms of the model parameters:(48)θ1ω=cos-1⁡ω2+α2+β2-γ22βα2+ω2,θ2ω=cos-1⁡ω2+α2-β2+γ22γα2+ω2.From (40), we obtain the arguments(49)arg⁡a1iω=-tan-1⁡ωα,arg⁡a2iω=-tan-1⁡ωα.Since the triangle can be located above and under the real axis, the following two equations hold for τ1 and τ2: (50)arg⁡a1iωe-iωτ1+2mπ±θ1ω=π,arg⁡a2iωe-iωτ2+2nπ∓θ2ω=πwhich yield the threshold values of the delays(51)τ1±ω,m=1ω-tan-1⁡ωα+2m-1π±θ1ω,(52)τ2∓ω,n=1ω-tan-1⁡ωα+2n-1π∓θ2ω.Here m and n are nonnegative integers such that τ1>0 and τ2>0. Thus for any m, n, and ω∈[ωs,ωe], we can define the pairs of (τ1,τ2) constructing the stability switching curve as follows.Theorem 7.
Givens2>d2, the stability switching curve is described by C1(m,n)∪C2(m,n) with m,n=0,1,2,…, where (53)C1m,n=τ1+ω,m,τ2-ω,n,C2m,n=τ1-ω,m,τ2+ω,n.The segments C1(m,n+1) and C2(m,n) have the same starting point whereas the segments C1(m,n) and C2(m,n) have the same end point.Proof.
It can be verified that(54)ωs2+α2+β2-γ22βα2+ωs2=1,ωe2+α2+β2-γ22βα2+ωe2=1,ωs2+α2-β2+γ22γα2+ωs2=-1,ω2+α2-β2+γ22γα2+ω2=1which leads to(55)θ1ωs=0,θ1ωe=0,θ2ωs=π,θ2ωe=0.Let the starting points of C1(m,n) and C2(m,n) be denoted by (56)C1sm,n=τ1+ωs,m,τ2-ωs,n,C2sm,n=τ1-ωs,m,τ2+ωs,n.Substituting θ1(ωs)=0 and θ2(ωs)=π into (51) and (52) yields(57)τ1+ωs,m=τ1-ωs,m,τ2-ωs,n+1=τ2+ωs,nwhich imply that C1s(m,n+1)=C2s(m,n). In the same way, let the end points of C1(m,n) and C2(m,n) be denoted by(58)C1em,n=τ1+ωe,m,τ2-ωe,n,C2em,n=τ1-ωe,m,τ2+ωe,n.Then substituting θ1(ωe)=0 and θ2(ωe)=0 into (51) and (52) yields(59)τ1+ωe,m=τ1-ωe,m,τ2-ωe,n=τ2+ωe,nwhich imply that C1e(m,n)=C2e(m,n). This completes the proof.Under Assumption3 with θ=0.8 and m=1, the stability switching curve is illustrated in Figure 2 in which the red segments show C1(1,n) and the blue segments show C2(1,n) for n=1,2,3. Notice the segment shifts upward as the value of n increases and to the right as the value of m increases.4 The lowest red and blue segments are C1(1,1) and C2(1,1) where they are connected to each other at point C1e(1,1)=C2e(1,1). The middle red and blue segments are C1(1,2) and C2(1,2) where C1(1,2) is connected to C2(1,1) at C1s(1,2)=C2s(1,1). As n increases the two segments are connected in the same way to construct the continuous stability switching curve, C1(1,n)∪C2(1,n). This curve divides the first quadrant into two parts as shown in Figure 2. One contains the origin and its every point can be reached from the origin via continuous curve not crossing the stability switching curve. At any point in this region, the real parts of the eigenvalues are negative, so the system is locally asymptotically stable. On the other hand, at the points in the complement of this region except the stability switching curve, the system is unstable. Observing Figure 2, we find the following three issues:(i)
Forτ1≤τ1h≃1.188, the system is locally asymptotically stable irrespective of the values of τ2, implying that delay τ2 is harmless.(ii)
Forτ1>τ1h, stability loss and gain repeatedly occur when τ2 increases from zero.(iii)
Depending on the value ofτ2, two different dynamic phenomena are seen when the value of τ1 increases. One is when stability is lost and cannot be regained as in the one-delay model, and the other case is when stability regain can occur.Figure 2
Stability switching curve.We now examine the effect caused by changingτ2, keeping the value of τ1 at some positive value. In Figure 3(a), the bifurcation diagram of p(t) with respect to τ2 is illustrated. The value of τ1 is fixed at τ11.8=1.8 and the value of τ2 is increased along the vertical dotted line at τ11.8 in Figure 2. For each value of τ2, the dynamic system runs for 0≤t≤1000 and discards the data for 0≤t≤950 to get rid of the transients. The local maximum and minimum obtained from the remaining data are plotted against the value of τ2. The value of τ2 is increased with 1/400 and then the same procedure is repeated until τ2 arrives at 10. If the resultant bifurcation diagram has only one point against the value of τ2, then the system is locally stable and that point corresponds to the stationary point. If it has two points, then a limit cycle with one maximum and one minimum emerges. As is seen in Figure 2, the vertical line at τ1=τ11.8 crosses the stability switching curve five times. We denote the values of τ2 of the green intersection points by τ2a  (≃0.714), τ2b  (≃3.489), τ2c  (≃5.342), τ2d  (≃9.028), and τ2e  (≃9.928) in the ascending order. The bifurcation diagram in Figure 3(a) indicates the following dynamics: after stability is lost at τ2a, a limit cycle emerges for τ2∈(τ2a,τ2b) and its amplitude first expands and then shrinks to zero at τ2=τ2b when stability is regained. The same process is repeated for larger values of τ2.Figure 3
Bifurcation diagrams showing stability losses and gains.
(a)
τ
1
=
1.8
(b)
τ
2
=
7We next draw attention to the effect caused by changing the value ofτ1. With similar procedure, Figure 3(b) illustrates the bifurcation diagram with respect to τ1 along the dotted horizontal line at τ2=τ27(=7) shown in Figure 2 in which the line crosses the stability switching line three times denoted by three black dots at τ1A  (≃1.421), τ1B  (≃2.758), and τ1C  (≃3.252) in the ascending order. A limit cycle emerges when stability is lost and stability losses and gains are repeatedly observed.


## 5. Two-Market Model with Two Delays

In this section we examine two-delay price adjustments in the interrelated markets like hog and corn markets. Let us denote corn and hogs byc and h. To simplify the analysis, the demand and supply functions in the corn market are assumed to be linear: (60)Dcpct=δ1-δ2pct,Scpcet=σ1+σ2pcetand those in the hog market are also linear: (61)Dhpht=d1-d2pht,Shphet,pct=s1+s2phet-s3pct-τ,where all parameters δi, σi, di, si, and delay τ are positive. In each market, the demand for the commodity depends on its current price observed at time t. The supply of corn depends only on the expected corn price while the supply of hogs depends on the expected price of hogs and the delay price of corn since hog suppliers are corn demanders who determine their demand decisions observing the current price. It is natural for hog producers to have s3>0 because they decrease the quantity of hogs when the corn price increases. Equilibrium prices satisfying Dc(pc∗)=Sc(pc∗) and Dh(ph∗)=Sh(ph∗,pc∗) are(62)pc∗=δ1-σ1δ2+σ2,ph∗=d1-s1+s3pc∗d2+s2,where δ1>σ1 and d1>s1 are assumed to assure positivity of the equilibrium prices. These conditions can also be interpreted by requiring that maximum demands are larger than minimum supplies. We assume the simplest expectation formations as in Assumption 1.Assumption 8.
Considerpce(t)=pc(t-τc) and phe(t)=ph(t-τh) with τc>0 and τh>0 and τ=τh.The price adjustment system is given by a two-dimensional system of delay differential equations:(63)p˙ct=kcpctδ1-σ1-δ2pct-σ2pct-τc,p˙ht=khphtd1-s1-d2pht-s2pht-τh+s3pct-τh.It can be confirmed that the equilibrium prices are the stationary point of the adjustment system. Linearizing this system in a neighborhood of the stationary point and introducing new variables and new parameters(64)xt=pct-pc∗,yt=pht-ph∗,α=kcpc∗,β=khph∗,τx=τc,τy=τhyield the linearized system(65)x˙t=α-δ2xt-σ2xt-τx,y˙t=βs3xt-τy-d2yt-s2yt-τy.Supposing exponential solutions x(t)=e-λtu and y(t)=e-λtv with u≠0 and v≠0 and substituting them into (65), we see that nontrivial solutions for u and v exist if and only if(66)λ+αδ2+ασ2e-λτx0-βs3e-λτyλ+βd2+βs2e-λτyuv=00.The corresponding characteristic equation is(67)λ+αδ2+ασ2e-λτxλ+βd2+βs2e-λτy=0which implies that the two delays are independent.5 Notice that the delay corn price pc(t-τh) does not affect local dynamics. Equation (67) can be divided into two independent equations:(68)λ+αδ2+ασ2e-λτx=0,(69)λ+βd2+βs2e-λτy=0.Before proceeding we check the no delay case (i.e., τx=τy=0) in which, from (67), the characteristic roots are real and negative:(70)λ=-αδ2+σ2<0,λ=-βd2+s2<0.The stationary point is always stable without delays. Thus stability can be preserved for positive delays as far as the delays are sufficiently small. In order to confirm to what extent the stationary point is stable, we determine the threshold values of the delays for which stability is just lost. Equations (68) and (69) have the same form as (16) and can be solved in the same way. Substituting λ=iωx and λ=iωy into (68) and (69) yields (71)iωx+αδ2+ασ2cos⁡τxωx-sin⁡τxωx=0,iωy+βd2+βs2cos⁡τyωy-sin⁡τyωy=0.Dividing each equation into the real and imaginary parts and solving them for ωj for j=x,y, we have(72)ωx∗=ασ22-δ22,ωy∗=βs22-d22.There are several combinations of ωx and ωy according to whether σ2-δ2 and s2-d2 are positive or not. To simplify the analysis, we assume the following.Assumption 9.
Considerσ2-δ2>0 and s2-d2>0.These conditions mean that supplies increase faster than demands decrease when market or expected price increases.Under Assumption9 we can determine the threshold values of the delays6(73)τ~x,m=1ωx∗cos-1⁡-δ2σ2+2mπ,τ~y,n=1ωy∗cos-1⁡-d2s2+2nπ.At τj=τ~j,0 for j=x,y, the stationary point loses stability. Hence the τx=τ~x,0 line and the τy=τ~y,0 line form the stability switching curve. Returning to the original notation of the delays and using these lines, we divide the (τc,τh) plane into four parts as shown in Figure 4. The stationary point is locally asymptotically stable in the yellow region in which 0≤τc≤τ~c and 0≤τh≤τ~h and unstable otherwise (in the union of the white regions and the blue region) where τ~c=τ~x,0 and τ~h=τ~y,0. More precisely, one of the two equations in (65) is unstable in the white regions and the two equations are unstable in the blue region. To see the effects caused by the delays, we perform simulations under the following numerical specification.Figure 4
Stability switching curve.Assumption 10.
Considerδ1=4, δ2=1, σ1=1, σ2=2, d1=4, d2=1, s1=1, s2=3, and s3=2.Notice that under these conditions maximum demands are larger than minimum supplies, so the equilibrium is positive. Furthermore Assumption9 is also satisfied, so stability switches can be observed.Under Assumption10, the threshold values of the delays are(74)τ~c=2π23≃1.209,τ~h=cos-1⁡-1/332≃0.450.Fixing the value of τc at τc1=0.6 in the first simulation and τc2=1.5 in the second simulation, we investigate the delay effect caused by changing the value of τh. The simulations are performed in the same way as before. The value of τh is increased from 0 to 1 with an increment of 1/400 and the delay model (63) is run for 0≤t≤500 for each value of τh. The results obtained are summarized in Figures 5(a) and 5(b) in which the bifurcation diagrams of ph are illustrated. In the first simulation, the stationary state is locally stable for τh<τ~h while at τh=τ~h the real part of one eigenvalue becomes positive from negative and thus stability is lost. It bifurcates to a limit cycle and no stability regain occurs for further increasing τh from τ~h. This result is essentially the same as the one obtained in the single delay model. On the other hand, in the second simulation, we obtain qualitatively different results. The stationary state is unstable even for τh=0 since τc2>τ~c. The bifurcation diagram in Figure 5(b) indicates that a limit cycle already emerges for τh=0, it is distorted for τh close to τ~h, and further increasing τh generates complicated dynamics.Figure 5
Bifurcation diagrams with respect toτh.
(a)
τ
c
=
τ
c
1
=
0.6
(b)
τ
c
=
τ
c
2
=
1.5Figure6 provides two phase diagrams of pc(t) and ph(t) for points A and B in Figure 4. It is seen in Figure 5(b) that the vertical dotted line at τh=τ~h crosses the bifurcation diagram six times. This phenomenon is described from a different viewpoint in Figure 6(a) in which the distorted limit cycle has three local maximums and three local minimums for point A=(τc2,τ~h). The value of τh is then increased to τc3=0.6 while τc is kept at the same value. It is seen in Figure 6(b) that the limit cycle has seven local maximums and minimums if we observe it carefully and thus its time trajectory exhibits fluctuations with more ups and downs.Figure 6
Phase diagrams at pointsA and B.
(a)
τ
c
=
τ
c
2 and τh=τ~h
(b)
τ
c
=
τ
c
2 and τh=τh3Further we examine the effects caused by changing the value ofτc. For this purpose, fixing the value of τh at τh1=0.25 in the first simulation and at τh2=0.6 in the second simulation, we increase the value of τc from 0 to 2 with 1/400 increment and run delay model (63) for each value of τc. When the value of τh is chosen in the yellow (i.e., stable) region, the dynamic system generates simple dynamics as shown in Figure 7(a) in which the system is stable for τc<τ~c and gives rise to a limit cycle after stability is lost for τc>τ~c. As expected, when both equations in (63) are destabilized, more complicated dynamics can arise as shown in Figure 7(b).Figure 7
Bifurcation diagram with respect toτc.
(a)
τ
h
A
=
0.25
(b)
τ
h
B
=
0.6We now shift our emphasis from the bifurcation diagrams to the phase diagrams, especially to see what dynamics arise when both equations of delay system (63) are locally unstable. In Figure 8(a), we choose point C in Figure 4 and run the system for 0≤t≤500. We eliminate the price data for t≤300 as transitory dynamics and plot the remaining data in the (pc,ph) plane. It can be seen that the prices behave in a very complicated way. In Figure 8(b), the value of τh is increased to τh3. It is also seen that the price behavior is also complicated. What these numerical examples make clear is the following:(i)
If one equation of the two-delay system is stable, then the resultant dynamics is essentially the same as in the one-delay system.(ii)
If both equations are unstable, then the two-delay system can generate various dynamics from a simple limit cycle to complicated dynamics having many ups and downs.Figure 8
Phase diagrams at pointsC and D.
(a)
τ
c
=
τ
c
2 and τh=τh2
(b)
τ
c
=
τ
c
3 and τh=τh2


## 6. Concluding Remarks

In this study, we have examined the delay effect on price dynamics in three different models. After reviewing a single delay model in which a limit cycle can emerge via Hopf bifurcation, we proceed to two models with two production delays. When the two delays are used to form an expected price or feedback for price adjustment, we find that the stability switching curve on which stability is changed is winding and as a natural consequence stability losses and gains are repeated when the length of one delay increases. It is numerically confirmed that only simple dynamics such as a limit cycle can emerge when stability is lost. We determine the stability switching curve under Assumption3 and θ=0.8; however other parameter specifications might result in different shapes of this curve. Gori et al. [15] investigated earlier how the structure of a two-delay model changes under variation of model parameters. In our next project we will examine the different shapes of the stability switching curves as the values of the parameters of our model vary.On the other hand, when the two delays are considered in interrelated markets, they affect price dynamics differently. The stability switching curves become straight lines regardless of the values of the model parameters. When one market is stable and the other market is unstable, the resultant dynamics is simple and essentially the same as the one in the single delay model. However, when both markets are unstable, a broad spectrum of dynamic behaviors can be found. In the cobweb literature, the discrete-time model has been considered and it is known that it can give rise to complicated dynamics when behavior nonlinearities get stronger. This study develops a continuous-time model and indicates that it also reasonably explains various dynamic behaviors observed in commodity markets.



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*Source: 312574-2015-07-26.xml*