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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 312574, 14 pages
http://dx.doi.org/10.1155/2015/312574
Research Article

The Asymptotic Behavior in a Nonlinear Cobweb Model with Time Delays

1Department of Economics, International Center for Further Development of Dynamic Economic Research, Chuo University, 742-1 Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan
2Department of Applied Mathematics, University of Pécs, Ifjúság Útja 6, Pécs 7624, Hungary

Received 24 March 2015; Accepted 29 June 2015

Academic Editor: Peng Shi

Copyright © 2015 Akio Matsumoto and Ferenc Szidarovszky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 Section 2, 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 are adjusted to be proportional to excess demand:where is the adjustment coefficient, is the expected price, and and 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:where for and is the delayed price or the price realized at time . Again for the sake of simplicity, demand and supply functions are assumed to be linear:The equilibrium price and quantity satisfy the conditions of and and are obtained as where, for positivity of the equilibrium price, is assumed. This is a natural assumption requiring that the maximum demand exceeds the minimum supply.

Substituting (3) and (4) into (1), taking , and then multiplying both sides of the resultant equation by yield a nonlinear price adjustment equation:The equilibrium price is also a stationary point.1 To examine stability of the equilibrium price, we denote the right-hand side of (6) by and linearize it around : or where and . Its solution is Since , the equilibrium price is always locally stable meaning that the market price converges to the equilibrium price as 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 where the expected price at time is the market price realized at time with .

Assumption 1. Consider .

Accordingly, the supply function is modified asSubstituting (10) into (6) presents a delay price adjustment equation:which is a first-order nonlinear delay differential equation. It can be confirmed that is also a unique positive stationary state of (11). If denotes the right-hand side of (11), then a linearized equation in a neighborhood of the stationary point is orIntroducing the new variable and the new parameters, and , we obtain the following form of the linearized equation:where is the only stationary point. Assuming an exponential solution and substituting it into (14) give the corresponding characteristic equationWithout delay , the stationary point is locally asymptotically stable. If stability of the trivial solution of (14) switches to instability at , then (16) must have a pair of pure conjugate imaginary roots. It is then assumed, without loss of generality, that with is a root. Substituting it into (16) breaks down the characteristic equation to the real and imaginary parts:Moving the constant terms to the right-hand sides and adding the squares of the resulting equations give If , then there is no , implying that the delay is harmless.2

Theorem 2. If , then the positive steady state of (6) is locally asymptotically stable for any positive values of .

On the other hand, if , then we can define as It is substituted into (17) to obtain threshold values of :3In order to determine the direction of the stability switch, we can think of the root of (16) as a continuous function of the delay . Then differentiating (16) with respect to and arranging terms yieldThusHenceThis inequality implies that all roots that cross the imaginary axis at cross from left to right as increases. So at point with stability is lost and cannot be regained later.

Further substituting into (21) determines the critical value of the delaywithIn 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 and , increasing the value of and decreasing the value of 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 where the parameters are specified as follows.

Figure 1: Numerical results on one-delay model (11).

Assumption 3. Consider , , , , and .

Notice that the maximum demand is larger than the minimal supply , 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 , we run delay system (11) for . To take away the initial disturbance, we discard the data of for and plot the local maximum and local minimum of for against the value of . The value of is increased with an increment and then the same procedure is repeated until arrives at . Under these circumstances, when , the threshold value of denoted by is obtained as It is seen in Figure 1(a) that the equilibrium is stable for and unstable for . 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. Given , the positive steady state of (11) is locally asymptotically stable if , loses stability at , and bifurcates to a limit cycle if , where

4. Cobweb Model with Two Delays

The price expectation is formed naively under Assumption 1. Convergence to the stationary point occurs for . The producers with delayed price information can eventually arrive at the stationary state. On the other hand, since cyclic oscillation arises for , 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, and . 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. Consider with .

Accordingly the supply function is modified asand then the price adjustment is governed by a two-delay differential equation:It is clear that the equilibrium price is a unique positive stationary point of (29). To examine local dynamics, we let be the right-hand side of (29). The linear approximation in the neighborhood of the stationary point is orAs before, the last form can be reduced towith

We now turn our attention to the delay feedback with which price adjustment equation (1) is modified as where is an adjustment coefficient and is a coefficient of the feedback. This can be rewritten as a differential equation with two delays:If we assume and denotethen 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 :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 and introducing the new functionssimplify the left-hand side of (37):Substituting a possible solution with into the two equations of (38) results inTheir absolute values are

We can consider the three terms in (39) as three vectors in the complex plane with the magnitudes , , and . The solution of means that these vectors form a triangle if we put them head to tail. That is, solving algebraically is equivalent to constructing a triangle geometrically with the following three conditions:Substituting the absolute values of (41) and (42) converts these three conditions to the following conditions:that can be rewritten asIt is clear that the second inequality does not hold if . Hence there is no , implying that the delays are harmless in the two-delay dynamic model.

Theorem 6. If , then the positive steady state of (29) is locally asymptotically stable for any positive values of and .

On the other hand, if , then conditions (45) hold for in the interval , wherewithwhere the subscripts “” and “” mean the starting point and the end point, respectively.

We will next find all the pairs of satisfying . Let be the base of the triangle and then denote an angle between and by and an angle between and by . By the law of cosine, the angle’s magnitudes are expressed in terms of the model parameters:From (40), we obtain the argumentsSince the triangle can be located above and under the real axis, the following two equations hold for and : which yield the threshold values of the delaysHere and are nonnegative integers such that and . Thus for any , , and , we can define the pairs of constructing the stability switching curve as follows.

Theorem 7. Given , the stability switching curve is described by with , where The segments and have the same starting point whereas the segments and have the same end point.

Proof. It can be verified thatwhich leads toLet the starting points of and be denoted by Substituting and into (51) and (52) yieldswhich imply that . In the same way, let the end points of and be denoted byThen substituting and into (51) and (52) yieldswhich imply that . This completes the proof.

Under Assumption 3 with and , the stability switching curve is illustrated in Figure 2 in which the red segments show and the blue segments show for . Notice the segment shifts upward as the value of increases and to the right as the value of increases.4 The lowest red and blue segments are and where they are connected to each other at point . The middle red and blue segments are and where is connected to at . As increases the two segments are connected in the same way to construct the continuous stability switching curve, . 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)For , the system is locally asymptotically stable irrespective of the values of , implying that delay is harmless.(ii)For , stability loss and gain repeatedly occur when increases from zero.(iii)Depending on the value of , two different dynamic phenomena are seen when the value of 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 changing , keeping the value of at some positive value. In Figure 3(a), the bifurcation diagram of with respect to is illustrated. The value of is fixed at and the value of is increased along the vertical dotted line at in Figure 2. For each value of , the dynamic system runs for and discards the data for to get rid of the transients. The local maximum and minimum obtained from the remaining data are plotted against the value of . The value of is increased with and then the same procedure is repeated until arrives at . If the resultant bifurcation diagram has only one point against the value of , 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 crosses the stability switching curve five times. We denote the values of of the green intersection points by   ,   ,   ,   , and    in the ascending order. The bifurcation diagram in Figure 3(a) indicates the following dynamics: after stability is lost at , a limit cycle emerges for and its amplitude first expands and then shrinks to zero at when stability is regained. The same process is repeated for larger values of .

Figure 3: Bifurcation diagrams showing stability losses and gains.

We next draw attention to the effect caused by changing the value of . With similar procedure, Figure 3(b) illustrates the bifurcation diagram with respect to along the dotted horizontal line at shown in Figure 2 in which the line crosses the stability switching line three times denoted by three black dots at   ,   , and    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 by and . To simplify the analysis, the demand and supply functions in the corn market are assumed to be linear: and those in the hog market are also linear: where all parameters , , , , and delay are positive. In each market, the demand for the commodity depends on its current price observed at time . 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 because they decrease the quantity of hogs when the corn price increases. Equilibrium prices satisfying and arewhere and 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. Consider and with and and .

The price adjustment system is given by a two-dimensional system of delay differential equations: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 parametersyield the linearized systemSupposing exponential solutions and with and and substituting them into (65), we see that nontrivial solutions for and exist if and only ifThe corresponding characteristic equation iswhich implies that the two delays are independent.5 Notice that the delay corn price does not affect local dynamics. Equation (67) can be divided into two independent equations:Before proceeding we check the no delay case (i.e., ) in which, from (67), the characteristic roots are real and negative: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 and into (68) and (69) yields Dividing each equation into the real and imaginary parts and solving them for for , we haveThere are several combinations of and according to whether and are positive or not. To simplify the analysis, we assume the following.

Assumption 9. Consider and

These conditions mean that supplies increase faster than demands decrease when market or expected price increases.

Under Assumption 9 we can determine the threshold values of the delays6At for , the stationary point loses stability. Hence the line and the line form the stability switching curve. Returning to the original notation of the delays and using these lines, we divide the plane into four parts as shown in Figure 4. The stationary point is locally asymptotically stable in the yellow region in which and and unstable otherwise (in the union of the white regions and the blue region) where and . 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. Consider , , , , , , , , and .

Notice that under these conditions maximum demands are larger than minimum supplies, so the equilibrium is positive. Furthermore Assumption 9 is also satisfied, so stability switches can be observed.

Under Assumption 10, the threshold values of the delays areFixing the value of at in the first simulation and in the second simulation, we investigate the delay effect caused by changing the value of . The simulations are performed in the same way as before. The value of is increased from to with an increment of and the delay model (63) is run for for each value of . The results obtained are summarized in Figures 5(a) and 5(b) in which the bifurcation diagrams of are illustrated. In the first simulation, the stationary state is locally stable for while at 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 from . 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 since . The bifurcation diagram in Figure 5(b) indicates that a limit cycle already emerges for , it is distorted for close to , and further increasing generates complicated dynamics.

Figure 5: Bifurcation diagrams with respect to .

Figure 6 provides two phase diagrams of and for points and in Figure 4. It is seen in Figure 5(b) that the vertical dotted line at 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 . The value of is then increased to while 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 points and .

Further we examine the effects caused by changing the value of . For this purpose, fixing the value of at in the first simulation and at in the second simulation, we increase the value of from to with increment and run delay model (63) for each value of . When the value of 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 and gives rise to a limit cycle after stability is lost for . 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 to .

We 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 in Figure 4 and run the system for . We eliminate the price data for as transitory dynamics and plot the remaining data in the plane. It can be seen that the prices behave in a very complicated way. In Figure 8(b), the value of is increased to . 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 points and .

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 Assumption 3 and ; 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.

Appendix

In this Appendix, we apply the method developed by Lin and Wang [14] to solve a two-delay differential equation to our model. To this end, we first rewrite the left-hand side of (67):whereand and for are defined accordingly. Substituting (A.3) and (A.2) into (A.1) and expanding it yield the characteristic equation where , , , and . Since is not a solution of , we look for a pair of the delays for which the characteristic equation has purely imaginary roots. Since roots of a real function come in conjugate pairs, we can assume that and . Substituting this into , we have two different forms: We introduce new functions:where

Following Lin and Wang [14], we haveUsing (A.2) and (A.3), we obtainWe then finally arrive at the following forms:It is clear that if   and , then (A.10) and (A.11) hold for any and , respectively. Further the conditions and can be rewritten asNotice that (A.12) is the same as (72). Let and be solutions of and . These are equivalent to and . Hence we obtain

On the other hand, if or , then from (A.10),implying that so meaning that in this case and is arbitrary. A similar argument shows that if or , then and is arbitrary. So there are no additional stability switching points besides the two lines and .

In summary the stability switching curves are given by the two line segments as depicted in Figure 4:

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to anonymous referees for their helpful comments. They also highly appreciate the financial supports from the MEXT-Supported Program for the Strategic Research Foundation at Private Universities 2013–2017, Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 24530202, 25380238, and 26380316), and Chuo University (Joint Research Grant). The usual disclaimers apply.

Endnotes

  1. Although dynamic equation (6) has also zero as stationary point, we ignore it in this study.
  2. The stability condition in a continuous-time model is the same as the one in a discrete-time model. Assuming , we can construct a discrete-time cobweb model:where is the stability condition including a cyclic solution.
  3. It is possible to substitute it into (18) to obtain the same value in a different form:
  4. Since the stability switching curve with is located in the second quadrant of the plane and the curve with is in the fourth quadrant, they are not depicted in Figure 2.
  5. In the Appendix, we apply a more general method to solve the two-delay differential equation for this particular case and obtain the same result.
  6. We can have the different form for the same threshold value:

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