# Speculative Bubbles and Excess Demand Functions. 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))

and

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)

Global bifurcations:

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.

Speculative bubbles:

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.

# Conjuncted Again: Noise Traders, Chartists and Fundamentalists: The In-Betweeners Interesting questions are whether the rational traders (fundamentalists) will drive irrational traders (chartists and noise traders) out of the market, or whether the irrational traders (chartists and noise traders) will derive the rational traders (fundamentalists) out of the market. As in many studies on heterogeneous interacting agent models, it may seem natural that switching between different trading strategies plays an important role. Now the question is: how a trader makes his choice between the fundamentalist and chartist strategies. The basic idea is that he chooses according to the accuracy of prediction. More precisely, the proportion of chartists κt is updated according to the difference between the squared prediction errors of each strategy. Formally, we write the dynamics of the proportion of chartists κ as

κt = (1 – ξ)/(1 + exp(Ψ(Ect – Eft)) —– (1)

Ect = (pt – pct)2>, Eft = (pt – pft)2

where ψ measures how sensitively the mass of traders selects the optimal prediction strategy at period t. This parameter was introduced as the intensity of choice to switch trading strategies. Equation 1 shows that if the chartists’ squared prediction error Ect is smaller than that of fundamentalists Eft , some fraction of fundamentalists will become chartists, and visa versa.

# Stocks and Fundamentalists’ Behavior Let us consider a simple stock market with the following characteristics. A large amount of stock is traded. In the market, there are three typical groups of traders with different strategies: fundamentalists, chartists, and noise traders. Traders can invest either in money or in stock. Since the model is designed to describe stock price movements over short periods, such as one day, the dividend from stock and the interest rate for the risk-free asset will be omitted for simplicity. Traders are myopic and bent on maximizing utility. Their utility depends on the price change they expect, and on their excess demand for stock rather than simply their demand. Their excess demand is derived from utility maximization.

Let Ytf be the amount of money that a fundamentalist holds at a time t and Xtf be the number of shares purchased by a fundamentalist at time t. Let pt be the price per share at time t. The fundamentalist’s budget constrain is given by

Ytf + ptXtf = Yt-1f + ptXt-1f —– (1)

or equivalently

ytf + ptxtf = 0 —– (2)

where

ytf = Ytf – Yt-1f

denotes the fundamentalist’s excess demand for money, and

xtf = Xtf – Xt-1f

his excess demand for stock. Suppose that the fundamentalist’s preferences are represented by the utility function,

u = α(ytf + pt+1fxtf + βxtf – (1 + βxtf) log (1 + βxtf) —– (3)

where pt+1f denotes the fundamentalist’s expectation in period t about the price in the following period t + 1. The parameters α and β are assumed to be positive. Inserting (2) into (3) the fundamentalist’s utility maximization problem becomes:

maxxtf  u =  α(pt+1– pt)xtf  βxtf – (1 + βxtf) log (1 + βxtf) —– (4)

The utility function u satisfies the standard properties: u′ (|xtf|) > 0, u′′(|xtf|) < 0 ∀ |xf|t ≤ |xf*|, where |xf*| denotes the absolute value of xf producing a maximum utility. Thus, the utility function is strictly concave. It depends on the price change expected by fundamentalists (pt+1– pt) as well as fundamentalist’s excess demand for stock xtf. The first part α(pt+1– pt)xtf implies that a rise in the expected price change increases his utility. The remaining part expresses his attitude toward risk. Even if the expected price change is positive, he does not want to invest his total wealth in the stock, and vice versa. In this sense, fundamentalists are risk averse. β is the parameter that sets the lower limitation on excess demand. All excess demand for stock derived from the utility maximization is limited to -1/β. When the expected price change (pt+1– pt) is positive, the maximum value of the utility function is also positive. This means that fundamentalists try to buy stock. By analogy, when the expected price change (pt+1– pt) is negative, the maximum value of the utility function is negative, which means that they try to sell. The utility maximization problem (4) is solved for the fundamentalist’s excess demand,

xtf = 1/β(exp(α(pt+1– pt)/β) – 1) —– (5)

Excess demand increases as the expected price change (pt+1– pt) increases. It should be noticed that the optimal value of excess supply is limited to -1/β, while the optimal value of excess demand is not restricted. Since there is little loss of generality in fixing the parameter β at unity, below, we will assume β to be constant and equal to 1. Then let us think of the fundamentalist’s expectation formation. We assume that he form his price expectation according to a simple adaptive scheme:

pt+1f = p+ ν(p* – pt) —– (6)

We see from Equation (6) that fundamentalists believe that the price moves towards the fundamental price p* by factor ν. To sum up fundamentalists’ behavior: if the price pt is below their expected price, they will try to buy stock, because they consider the stock to be undervalued. On the contrary, if the price is above the expected value, they will try to sell, because they consider the stock to be overvalued.