Stochastic business-cycle model
In this section, we consider a stochastic model in which the gap between demands and supplies of firms in subgroups drives the dynamics.
Let N denote the total number of firms, assumed to be fixed.
The number of firms of group A is denoted by n. There are N - n firms in group B, which is assumed to be less productive than those of firms in group A. Thus the total output of the economy is
where c1 ≥ c2 > 0,andtime arguments are suppressed from y and n. We express this in terms of the fraction
Denote the share of demand for group As goods by i. That for group B is then 1 - i. Both efficiency of production and shares could be functions of x in our analysis.
The gap between demand and supply for group A is
In a special case in which c1 = c2 := c, we have
In a deterministic model, one might postulate some adjustment mechanism that increases x if this gap is positive, and decreases x if it is negative, with 
being the equilibrium share of group A. Instead, we use the framework of birth-death processes, similar to the one in Aoki (1996a, Chap. 5), and postulate the transition rate for the number of firms in group A as 
and 
with
where h(g) is an increasing function of the gap g.
It may include some adjustment or moving-cost component as well. The parameter β plays a crucial role. As in our earlier applications, β incorporates the effects of uncertainty, incomplete information, ignorance, and so on. In the simple case of c1 = c2 = c, g is positive when i > x and η1 is bigger than 1 /2, and is less than 1 /2 when i < x.If we treat i, c1, and c2 as fixed parameters, then g is linear in x. If we assume that h is linear in g, then h is linear in x. With h nonlinear in g, or g made nonlinear by assuming that share the i, or efficiency of production, is a function of x, we could have situations in which h is cubic in x.In this case we know from models discussed in Aoki (1996a, Sec. 5.10) that there may be three critical points to the aggregate dynamics, two of which may be locally stable. All depends on the value of β introduced in the transition rates to embody uncertainty or lack of information on the future streams of profits that firms face. We show next how to use the master equation, the aggregate dynamics, and the Fokker-Planck equation to gain information on the fluctuations.
The master equation is 
Change time to τ = t/N. Then, the left-hand side of the master equation becomes
Equating the terms on the two sides of the largest order in N, we arrive at the aggregate dynamic equation
with
Stationary solutions are obtained as the zeros of the function α(φ) = 0; by substituing the expressions for the transition rates, we find that the zeros are the solutions of 
As we remarked earlier, there are at most three solutions when h is cubic in φ.
Depending on the magnitude of β, a unique locally stable φ, two locally stable φ's, or a single unstable φ is found in the range 0 < φ < 1. See Aoki (Γ996a, Chap. 5; 1998a). We also know that these critical points correspond to those of a double-well potential
with H(φ) = -φ ln φ — (1 — φ) ln(1 — φ) the entropy.
The remainder of the master equation yields
where rn and ln are evaluated at the equilibrium value φe of the aggregate equation, when we set the left-hand side equal to zero to obtain the stationary Fokker-Planck equation. We obtain a Gaussian distribution for ξ, with mean zero and variance
where the prime indicates differen
tiation with respect to φ, and evaluated at locally stable φ-values.
8.6