Example: Open models with two types of agents

The next example is an open market or sector model with two types of agents. By reinterpreting the number of agents as measured in some basic units, we may translate the results in terms of the number into results in terms of sizes of agents such as firms.

7.1.1 Equilibrium distribution

Assume that all transition rates are time-homogeneous. Suppose that the entry probability intensity is given by

for n_i ≥ 0, and the exit transition rate is specified by

for n_i ≥ 1, i = 1, 2. In addition, agents change from type 1 to type 2 with probability intensity

and likewise from type 2 to type 1 with the coefficient λ₂₁.

The detailed-balance conditions hold, and it is easy to see that the equilibrium probability distribution is given by

with

i = 1, 2, where c_i is the normalizing constant and the constants art and provided

Exercise 1. Verify that with the expression for the equilibrium distributions, the detailed-balance conditions are satisfied.

7.1.2 Probability-generating-function method

The master equation is given by

Wherewesupressthetimeargumentof n₁ and n₂, and where z₁ and z₂ below are dummy or auxiliary variables of the generating function to extract appropriate probabilities from the function.

Define the probability generating function by

7.1.3 Cumulant generating functions

Often, we are interested in the time response patterns of the first few moments such as the mean, variance, and skewness. Now, by using the device in Cox and Miller (1965, p. 159), we derive the ordinary differential equations for the mean and variance.

Given a random variable X(t), its probability generating function is changed into the moment generating function by settingand defining

Next introduce the cumulant generating function by

Denoting the rth cumulant by κ_r, we extract it fromas the coefficient

7.2