# The master equation, probability generating function and cumulant equations

Consider a system composed of two economies of size n_{1} and n_{2} with probability *p* (n_{1} n_{2}). If we take *d _{i}* and

*c*to be the death rate and birth rate, respectively, of block i, the net growth of this block becomes

_{i}*g*We assume that

_{i}= c_{i}— d_{i}.*g*and denote the rate of migration of agents from block 2 to block 1 by

_{2}> g_{1},*nu >*0. For this system, the master equation is

where *I* is the inflow and *O* is the outflow of agents, and their expressions are

We define the probability generating function by
and convert the master equation for *P (n _{1},* n

_{2}) into

In deriving the above expressions we must collect terms with the coefficients of the same powers of z_{1} and z_{2}.

See Aoki (2002, ch. 7) on the probability generating functions and master equations.

We solve the above by converting it into the equations for the cumu- lants. The first two cumulants, *K _{i}* are the expected values of

*n, i =*1,2, and the covariance matrix elements are given by the cumulant k

_{1 1}, k

_{2 2}, and k

_{12}, which are the covariance matrix elements. See Cramer (1949, p. 186), Cox and Miller (1965), Aoki (2002), or Aoki and Yoshikawa (2007, p. 37).