Now, we shall indicate how local and global properties of the deterministic dynamics vary when the parameter α is allowed to vary. α measures the strength of the non-linearity of the excess demand functions
pt+1 = pt + θn[(1−κt−ξ)(exp(αν(p∗ − pt))−1) + κt(exp(α(1−µ)(pct − pt)) − 1) + ξγεt],
pft+1 = pt + ν(p∗ − pt), pct+1 = pct + µ(pt − pct) —– (1)
where κt = 1/(1 + exp(ψ((pt − pct)2 − (pt − pft)2)) .
The first of the equations above represents the adjustment of the price performed by the market maker. The second and third equations show the formation of the expectations of fundamentalists and chartists, respectively. The fourth equation represents the movement of the chartist fraction. Throughout this section, we assume that there exists no noise traders (ξ = 0). Thus, the dynamics of the system is deterministic. As can be checked easily, the dynamic system has as an unique fixed point: P ≡ (p¯, p¯f, p¯c) = (p∗, p∗, p∗).
The local stability conditions:
To investigate the dynamics of the model, we shall first determine the local stability region of the unique equilibrium point P. The local stability analysis of the equilibrium point P is performed via evaluation of the three eigenvalues of the Jacobian matrix at P. Let us denote by
c(λ) = λ3 − Tλ2 + Dλ —– (2)
the associated characteristic polynomial of the Jacobian matrix at P, where
D = (2−µ−0.5θnα(1−µ+ν))
T = (1−µ−0.5θnα(1+ν)(1−µ)).
An eigenvalue of the Jacobian is 0, and the other roots λ1 and λ2, satisfy the relation
λ2 − Tλ + D = 0 —– (3)
Thus, the stability of the equilibrium point is determined by the absolute values of λ1 and λ2. The eigenvalues λ1, λ2 are λ1,2 = T/2 + ± √ ∆/2 where ∆ ≡ T2 − 4D. As is well known, a sufficient condition for local stability consists of the following inequalities:
(i) 1−T +D > 0,
(ii) 1+T +D > 0, and
(iii) D < 1,
giving necessary and sufficient conditions for the two eigenvalues to be inside the unit circle of the complex plane. Elementary computations lead to the condition:
α < (3 − 2µ) / (θn(1 + ν − µ(1 + 0.5ν))) —– (4)
The above local conditions demonstrate that while increasing α starting from a sufficiently low value inside the stability region, a loss of local stability of the equilibrium point P may occur via a flip bifurcation, when crossing the curve
α = (3 − 2µ) / (nθ(1 + ν − µ(1 + 0.5ν))) —– (5)
What happens as α is further increased? We will see that first, a stable 2-cycle appears; secondly, the time path of the price diverges; and thirdly, the time evolution of pt displays a remarkable transition from regular to chaotic behavior around an upward time trend. It is important that the nonstationary chaos can be transformed to a stationary series by differentiation once ∆pt = pt−pt−1. The price series fluctuates irregularly around an upward price trend, and the series of the price change fluctuates chaotically within a finite interval. Below, this non-stationary chaotic pricing will be referred to as a speculative bubble.
Why does a speculative bubble or non-stationary chaos appear when α increases? Unfortunately, it is difficult to prove mathematically the existence of the strange attractor. However, as we shall see, it is possible to give an intuitive interpretation of speculative bubbles. On a phase plot, the stretching and folding is the root of the chaotic price behavior. The next question is why speculative bubbles occur when α is large. The price adjustment equation suggests that a price trend can occur only when excess demand aggregated over two types of traders is positive on average. Why would average excess demand be positive as α become large? For an answer, let us recall the traders’ excess-demand functions. xt denotes the traders’ excess demand, and pet+1 denotes the expected price. The partial derivative of the excess-demand function with respect to the parameter α is obtained as
∂xt/∂α = (pet+1 − pt) exp(α(pet+1 − pt)), —– (7)
where we assume β = 1. This equation suggests a rise in α increasing excess demand exponentially, given the expected price change (pet+1 − pt) is positive. As shown in the preceding paragraph, the maximum value of the excess demand increases freely when α increases, while the maximum value of the excess supply never can be lower than −1/β, regardless of the value of α. Assume that, in the initial state, the price pt exceeds the fundamental price p∗ in period t, and that the chartists forecast a rise over the next period. The fundamentalists predict a falling price. Then, chartists try to buy and fundamentalists try to sell stock. When α is sufficiently large, one can safely state that chartist excess demand will exceed fundamentalist excess supply. The market maker observes excess demand, and raises the price for the following period. If the rise is strong, chartists may predict a new rise, which provokes yet another reaction of the market maker. Once the price deviates strongly from the fundamental price, the fundamentalists become perpetual sellers of stock, and when α is sufficiently large, the chartists’ excess demand may begin to exceed the fundamentalists’ excess supply on average. Furthermore, fundamentalists may be driven out of the market by evolutionary competition. It follows that when the value of α is sufficiently large, average excess demand is positive and speculative bubbles occur.