Transition rates
We mostly use transition rates that are standard in the birth-death or birth-death- with-immigration models in the probability textbooks, with one important modification. In our economic models, not all parameters in the transition rates are exogenously given; some are endogenously determined.
In particular, some parameters in the transition rates are to be determined by the value-maximization processes of agents. We have several examples of this nature in this book. See, for example, the Diamond model in Chapter 9.Initially, however, this aspect is not involved here. We take it up later when we discuss evaluation of alternative choices by agents in Chapter 6.
Rates at which agents enter the model in open models are specified by the entry rates. For example, if an agent of type i enters the model, the number of agents of type i increases by one. This is represented by the vector n becoming n + ei, where ei is the K-dimensional vector with sole nonzero element 1 at component i. This event has the transition rate w (n, n + ei). In discrete-time models, this event has the conditional probability Pr(n + ei |n).
In closed models, changing of decisions by agents may be modeled as entry together with exit. For example, an agent of type i, where type is associated with decision or choice, may become type j by exiting from a group or cluster of agents made up of type i and entering a cluster of agents of type j. This is expressed as having the transition rate w (n, n + ej - ei).
More will be said in Chapter 4 as well as in a section of the Appendix at the end of the book.
2.6