Symmetric interactions

7.5.1 Stationary state distribution

Suppose we now specify entry transition rates for the two groups in the same form, that is, we replace f₂ in the previous section by

and respecify the transition rate from type 1 to type 2 as

Then both types have negative binomial distributions as their equilibrium distributions:

i = 1, 2.

7.5.2 Nonstationary distributions

We derive the differential equation for the cumulant generating function as in the previous section. The detailed expressions for φ₁ through φ₄ are slightly different, but the general procedure of analysis remains the same. We do not bother with the detailed results.

Rather, we later examine in Chapter 11, after discussing some growth and business-cycle models in between in Chapters 8 and 10, what happens when the number of groups becomes large, as well as the total number of firms. We find a perhaps surprising connection with the Ewens distribution.