Closed binary choice models
For examples with agents with discrete choice sets, we begin with binary choice models to introduce our notation, basic formulation for the dynamics, and
for k > 1, and
for k ≥ 0, for some positive constants f, c, and d.[5] A model of these transition rates is called a birth-death-with-immigration model, where the constant term represents a constant flow of immigrants.
In the economic interpretation, the constant term may represent some economic inducement for entry, such as excess profit, shortage of production capacity, or the like. In some cases, the constant term may depend on previous history of decisions made by agents. See the business-cycle model in Section 8.6 for an example of such an interpretation.[6] The detailed-balance condition, given by
is a first-order difference equation for the equilibrium probability π (j), and gives
for k ≥ 0, where g = c/d. The ratio of gamma functions is often written as
This model may be used to analyze market shares of two brands of consumer goods, for example. We may model growth of a sector of an economy by associating choices with investment or no-investment decisions of firms, as we later describe. Further, we may model growth in a macroeconomic context by associating agents' choices with relocation decisions in a two-sector economy. For example, models of intersectoral capital reallocation under uncertainty such as that of Dixit (1989) may be redone using the framework of our binary choice model.
4.3