The master equation, probability generating function and cumulant equations
Consider a system composed of two economies of size n1 and n2 with probability p (n1 n2). If we take di and ci to be the death rate and birth rate, respectively, of block i, the net growth of this block becomes gi = ci — di. We assume that g2 > g1, and denote the rate of migration of agents from block 2 to block 1 by 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 (n1, n2) into
In deriving the above expressions we must collect terms with the coefficients of the same powers of z1 and z2.
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, Ki are the expected values of n, i = 1,2, and the covariance matrix elements are given by the cumulant k1 1, k2 2, and k12, which are the covariance matrix elements. See Cramer (1949, p. 186), Cox and Miller (1965), Aoki (2002), or Aoki and Yoshikawa (2007, p. 37).