Dynamics of binary choice models

As the first look at the dynamics or derivation of nonstationary probability distribution for the state vector, let us suppose that in a small time interval ∆t the probability thatthe number of agents with choice 1 increases by one from k to k + 1 is given by r_k ∆t + o(∆t), and the probability that the number decreases by one from k to k — 1 is given by l_k ∆t + o(∆t).

We do not stop here to discuss how r_k and l_k may depend on k and possibly on other parameters in the model. We have seen some examples earlier. For the simple binary choice model under consideration, each of k agents may change his or her mind independently of the others at the rate μ, and each of n — k agents may do so at the rate υ over a small interval of time. For this illustrative example, we postulate

For the purpose of this simple model analysis, assume that μ and λ are constant. In more realistic or complex models, they may be functions of states (the so-called state-dependent models).

We derive the differential equation for the probability distribution that there are k agents with choice 1 at time t. Denote this by P (k, t).

Keeping track of inflows and outflows of probability flux, we obtain

from which the master equation in the limit of letting ∆t go to zero is

for positive k. We need the boundary condition for k = 0,

There are many references to this kind of derivation of the equation. On the backward Chapman-Kolmogorov equation, see Taylor and Karlin (1994, p. 325) for example.

When the total number of agents is fixed at n, then we need another boundary condition

¹ In closed binary models, the state space is the set {0, 1,..., n}.

² This can be deduced in some cases. See CJinlar (1975), for example.

5.2