Behavior of market excess demand
We derive approximate expression for the market excess demands with two large fractions x and y, which approximately sum to 1. In the previous section, we have shown that about 95 percent of the total market participants belong to the largest two subgroups of agents by types, when the parameter θ is about 0.4.
With the largest two clusters, there are two regimes: one in which a cluster of agents with strategy 1 is the largest share, and the other in which a cluster of agents using strategy 2 is the largest. For ease of comparison, we use the same formulas for the individual excess demand functions as in Day and Huang (1990). The agents with strategy 1 have the excess demand
and the agents with strategy 2 have the excess demand
with h(P) = [(P - m)(M - P)]-1 /2, where we have set a = b = 1 in Day ;md Huang's specification and set u = (M + m )/2 without loss of generality. In the language of Day and Huang, agents with strategy 1 are fundamentalists and those with strategy 2 are chartists. We note that the two excess demands are of opposite sign, i.e., the two types of agents are on opposite sides of the market.
Let P denote the price of the shares, and let dx (P) denote the individual excess demand of the type that happens to have fraction x. Similarly for dy (P).
11.4.1 Conditions for zero excess demand
The market excess demand D is then given by summing over individual excess demands:
We have divided the excess demand by the total number of market participants so that we can express it in terms of fractions.
Set the right-hand side equal to zero to find the critical prices at which the condition of zero market excess demand is realized. In the case where the agents in the largest cluster are using strategy 1, there are three prices at which the market excess demand is zero if the inequality (M - m)/2 ≥ (x/y) holds. One is P = u, and the other two are given by the roots of
Denote them by P', and P+, where P. < u < P*. These critical values depend on x and y, although we omit the arguments not to clutter the notation.
11.4.2 Volatility of the market excess demand
Watterson (1976) and WattersonandGuess (1977) have shownhowto derive the 
This condition, if satisfied, will determine γ by
where
4
Blank and Solomon point out an error in Gabaix (1998).
11.4.3.1 Simulation experiments
We have established that for a small value of θ, there are two or three groups in a market, and the largest two occupy about 95 per cent or more of the market share. For example, We have seen that
θ = 0.4, where x and y are largest two cluster sizes.
Having seen this, we conduct a simulation experiment in which only two groups of agents are in the market, and agents of the two types arrive at Poisson rates m ι and m2. We pick m1 = 5 and m2 = 1. We assume that prices are adjusted by a trading specialist who buys and sells at prices he posts. He attempts to maintain the market excess demand near zero. When the price deviates beyond the limits he sets, he adjusts the posted price by the rule
where I(∙) is the indicator function, and where L is the limit, which is taken to be the same for upper and lower limits.
Returns are defined by xt = ln(Pt /Pt-1). In the range where the specialist adjusts prices, they are governed approximately by the stochastic difference equation
Depending on the initial conditions, there may be one or three equilibrium points of the dynamics. Conditions under which stationary distributions exist as solutions of x∞ = Ax∞ + B for some A, B pair have been examined in several papers; see Brandt (1986), de Haan, et al. (1989), Letac (1986), Goldie (1991), and Vervaat (1979), among others. de Haanet al. discuss the case where the A's and B's are uncorelated and i.i.d. Vervaat and Letac discuss the existence of stationary solutions. Goldie gives a nice discussion of power laws. Brandt does not require the i.i.d. condition.
where y is the standardized return, positive in the log-log plot.
Figures 11.1 and 11.2 are the results of the 100 simulations of the last 100 time steps of a total of 300 time steps. This is done to ensure that the returns
Fig.
11.1. Tail distribution, with that of normal distribution superimposed (dashed line).
Fig. 11.2. Left upper panel: Plots of ln P vs ln ∖dP|, left lower panel that of standard normal for comparison.
Right upper panel: Plot of Xa(Jrequency) vs dP, where dP is standardized. Left lower panel that of normal distribution for comparison. Note the differences in scale.
are approximately stationary. Figure 11.1 plots the tail distribution with that of the standard normal distribution superimposed for comparison.
The right-hand panel of Fig. 11.2 compares the simulation result with the normal distribution. It shows thattails are much larger in the returns. Its left-hand panel indicates the exponent of the power law as the negative slope of the top plot. It appears that γ is definitely less than 2, about 1.7 or 1.8.
The bottom panel shows the normal distribution for comparison. Various choices of adjustment-speed parameters affect the rate of convergence and the choice of local equilibria, but do not affect the tail behavior. These simulation runs are not conclusive, but are indicative of the tail behavior of the model.