Complex Dynamics in a Nonlinear Cobweb Model for Real Estate Market

We establish a nonlinear real estate model based on cobweb theory, where the demand function and supply function are quadratic. The stability conditions of the equilibrium are discussed. We demonstrate that as some parameters varied, the stability of Nash equilibrium is lost through period-doubling bifurcation. The chaotic features are justified numerically via computing maximal Lyapunov exponents and sensitive dependence on initial conditions. The delayed feedback control (DFC) method is applied to control the chaos of system.


Introduction
Cobweb models describe the price dynamics in a market of a nonstorable good that takes one time unit to produce [1].In economic modeling, many examples of cobweb chaos have been demonstrated.Some of the most famous examples include [2][3][4][5][6][7][8][9].Hommes [5] applies the concept of adaptive expectations in a cobweb model with a single producer to investigate the occurrence of strange and chaotic behavior.Finkenstädt [3] applied linear supply and nonlinear demand functions.Hommes [4] and Jensen and Urban [6] used linear demand functions with nonlinear supply equations.These findings indicate that the nonlinear cobweb model may explain various irregular fluctuations observed in real economic data.In this study, we go one step further to study the cobweb model with nonlinear demand and supply function.A possible source of such an evolutionary market dynamics is an interaction between government and real estate developer.
Traditional cobweb models usually describe a dynamic price adjustment in agricultural markets with a supply response lag [2].Consider, for instance, the supply of housing.The time of housing construction guarantees a finite lag between the time the production 2 Discrete Dynamics in Nature and Society decision is made and the time the housing is ready for sale.The real estate developer's decision about how many houses should be built and sale is usually based on current and past experience.This principle is the same as that of agricultural product.So it is feasible to introduce cobweb model into real estate market.
The present paper attempts to establish a nonlinear model for the real estate market, and introduce adjustment parameters of housing price and land price into the model, which can denote the game behavior of players.The system stability with the variation of parameters is analyzed.Numerical simulations verify the complexity of system evolvement.Finally, time-delayed feedback control method is used to keep the system from chaos and bifurcation.

Nonlinear models for real estate market
In this paper we assume that all real estate developers in the market are belong to one benefit group and have a common benefit target.Usually the price p is characterized by the nonlinear inverse demand function of p = a − b Q, where a and b are positive constants, a is the maximum price in the market, and Q is the total quantity in the market.This kind of form has been used in other oligopoly models and in the experimental economics dealing with learning and expectations formation (see, e.g., [10][11][12]).The transformation of this formula is as follows: where b 0 , b 1 , b 2 , c 0 , c 1 , c 2 are positive constants, p 1 (t) is the land price at time period t, p 2 (t) is the housing price at time period t, D 1 (t) is the land demand at time period t, and D 2 (t) is the housing demand at time period t.Due to the law of demand that the slope of demand curve is negative, the prices p 1 (t) and p 2 (t) must, respectively, satisfy the inequalies: 2b 2 p 1 (t) − b 1 < 0 and 2c 2 p 2 (t) − c 1 < 0; 4b 2 b 0 − b 2 1 > 0, 4c 2 c 0 − c 2 1 > 0 must hold, thus the signs of demand equations in formula (2.1) are positive.
In this case, the land market and housing market are interrelated.Though the housing market does not directly affect land market, the land price impacts the housing supply which decreases with increasing land price.This rule is the same as that of hog and corn as stated by Waugh [13].Real estate companies adjust the housing supply according to the relative policies and the situation of housing price and land price.The formula of supply can be supposed as follows: Z(p) is excess demand function descending with price, which denotes the gap between demand and supply.When the price is low, excess demand exists and when the price is high, excess supply exists, thus p * that satisfies the equation Z(p * ) = 0 is called equilibrium point.
Substituting (2.1) and (2.2) into (2.4),we obtain Since Z(p) follows the law of demand, the following conditions must hold: α 1 is the adjustment parameter of land price, which denotes the adjustment degree of benchmark land price controlled by government through the land supply plan.α 2 is the adjustment parameter of housing price, the dynamic model of land price and housing price can be established as follows: where α 1 and α 2 are positive parameters.It is clear that the excess functions of land and housing with adjustment parameters are two-dimensional nonlinear map, which can be regarded as a discrete dynamic system.
implies that four fixed points exist in system (3.1) and period-doubling bifurcation appears.
As λ increases, the number of fixed points continues to grow until λ = 3.5699; when α 1 = 2.5699/ (e 1 + b 1 ) 2 − 4(e 2 − b 2 )(e 0 − b 0 ), the value of x(t) is unequal to any point that appeared before, system enters chaos from period doubling bifurcation.
The same argument holds for the second equation in formula (3.1).Let , system enters chaotic state from period doubling bifurcation.

3.2.
Stability analysis.Now we discuss the stability of fixed points of the discrete dynamic system (3.1) through analyzing the eigenvalues of asymptotic linear equation of formula (3.1).
Four fixed points of difference function (3.1) are obtained: provided that: J. Ma and L. Mu 5 Lemma 3.1.The equilibrium is an unstable equilibrium point.
Proof.In order to prove this result, we find the eigenvalues of the Jacobian matrix J.In fact at E 1 , the Jacobian matrix becomes a triangular matrix: whose eigenvalues are given by the diagonal entries.They are: It is clear that when condition (3.5) holds, then Then E 1 is an unstable equilibrium point of the system (3.1).This completes the proof of the proposition.
The stability of other points can also be judged by the above method.

The stable region of equilibrium point.
In this subsection, we analyze the asymptotic stability of the equilibrium point for the two-dimensional map (3.1).We determine the region of stability in the plane of the parameters (α 1 ,α 2 ).The Jacobian matrix at E * (p 1 * (t), p 2 * (t)) takes the form (3.9) The characteristic equation of the matrix (3.9) has the form where "Tr" is the trace and "Det" is the determinant of the Jacobian matrix (3.9) which are given by The stability region is bounded by the portion of hyperbola with positive values of α 1 and α 2 , whose equations are given by the vanishing of the left-hand sides 1 + Tr+Det = 0 and Det−1 = 0.For the values of (α 1 ,α 2 ) inside the stability region (see Figure 3.1), the equilibrium point is stable node and loses its stability through a period-doubling bifurcation.
The bifurcation curve intersects the axes α 1 and α 2 , respectively, whose coordinates are given by

Numerical simulations
In order to study the complex dynamics of system (3.1), it is convenient to take the parameters values as follows: Figure 3.1 shows the region of stability of Nash equilibrium.Equation (3.15) defines the region of stability in the plane of (α 1 ,α 2 ). Figure 4.1 shows the map of f α1,α2 .xcoordinate is p 1 and y-coordinate is f α1,α2 (p 1 ).Dynamics of land price in the cobweb model is given by system p 1 (t) = f α1,α2 (p 1 (t − 1)) with two model parameters.A graphical analysis in Figure 4.1 shows that the map f α1,α2 is nonmonotonic with one critical  point, where the graph has a (local) minimum, and that initial state p 1 (0) = 1 does not converge to a low periodic orbit.Since the graphical analysis in this case does not converge, it suggests that the dynamical behavior is chaotic.becomes unstable, and one observes complex dynamic behavior occurs such as cycles of higher order and chaos.Also the maximal Lyapunov exponent is plotted in Figures 4.2

Chaos control
Delay feedback control (DFC) method was brought forward by Pyragas [16].The method allows a noninvasive stabilization of unstable periodic orbits (UPOs) of dynamical systems [17].It feeds back part of system output signals as exterior input to the system after a time delay.u(•) is control signal gained by self-feedback coupling between output and input signals in chaotic system.x(t) = f (x(t − 1)) + u(t) is the form of DFC, where u(t) = k(x(t) − x(t − τ)), t > τ, τ is time delay, k is controlling factor.Though delay feedback control is only carried out on one variable, it enables other variables in the system to achieve stability simultaneously.Our goal is to control the system in such way.The system with controlling factor is shown as follows: (5.2) Substituting equilibrium point (0.4, 0.9) into (5.2),we obtain eigenvalues λ 1 = −0.83,λ 2 = (k − 1.7)/(1 + k).So when k > 0.35, absolute values of both eigenvalues are less than 1, which means that the system is stable.
As shown in Figure 4.10 land price is controlled from chaotic state to stable state when k is greater than 0.35, so we select k = 0.4.Housing price and land price are also controlled to equilibrium point (0.4, 0.9) as shown in Figure 4.11.

Conclusion
A nonlinear model for real estate market has been presented based on the cobweb theory.It is a simple dynamic model with nonlinear demand and supply function.From numerical simulations, we deduce that the land supply system has the remarkable influence on real estate market.Therefore, policy makers who intervene in one market should recognize that what they do may also influence other relative markets.We showed that the fast adjustment cause a market structure to behave chaotically.Therefore, the dynamics of market is changed when players apply different adjustment speed.Attempts are also made to stabilize the chaotic system with the delay feedback method.Combining with this method, the land price and housing price evolve from chaotic to stable.
value of p 1 (0) + 0.0001; Δp 2 (t) = p 2 (t) − p 2 (t), and p 2 (t) is the value of housing price at time period t with initial value of p 2 (0) + 0.0001.In both figures, initial condition of one coordinate differs by 0.0001, the other coordinate keeps equal.At the beginning, the difference is indistinguishable but after a number of iterations the difference between them builds up rapidly.From Figures 4.8 and 4.9, we show that the time series of the system (3.1) is sensitive dependence on initial conditions, that is, complex dynamics behaviors occur in this model.