Aggregate dynamics: Dynamics for the mean of the fraction
The master equation is
where lk is the transition rate for the leftward move from state k, and rk is that of the rightward move.
This is essentially the birth-death equation we have discussed earlier in Chapters 3 through 5. The only difference arises from the fact that the leftward move involves a step of two units, not a single unit, because a matched pair of agents - that is, two agents - change their status.In Section 9.2, we have described the transition rate for the rightward move to be
and that for the leftward move rate to be
Since this equation cannot be solved exactly, we proceed as in Chapters 3 and 4. See also Aoki (1996a, p. 123). The master equation is expanded in Taylor series, and terms of the same order of magnitude are collected. Change the variable as
The variable φ is the expected fraction of employed, and ξ represents random fluctuations about the mean. This scaling implies that fluctuations are expected
In the Taylor series expansion, after substituting the change of the variable, we match the left-hand side of order
with the terms of the same order on
4
That this is the correct order is indicated by the fact that the coefficients of the Fokker-Planck equation for ξ, to be derived below, are independent of n.
the right-hand side. We derive the aggregate dynamic equation for φ as
This is in agreement with the dynamical equation for e in Diamond (1982, (1)).
Here we define Φ(φ) as above because it is convenient to introduce a natural time unit in terms of the arrival rate of the production opportunity a and define 
to rewrite the dynamical equation for the mean as
In other words, we measure the arrival rate for trading opportunity relative to that of the production opportunity by defining b = b/a as shorthand, because this grouping of terms arises several times below.
9.4