Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 312574, 14 pages

http://dx.doi.org/10.1155/2015/312574

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

^{1}Department of Economics, International Center for Further Development of Dynamic Economic Research, Chuo University, 742-1 Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan^{2}Department 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 :^{3}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 . 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.