Deterministic share dynamics

The rate of changes of the share of group A is

We assume that individual firms' rates of change of shares are proportional to their deviations from the share-weighted average over all firms of some variable denoted as Xj, j = 1, 2,..., n, such as the price charged by firm j or unit cost of firm j:

where

The growth of the share of group A can be written as

by expressing the sum over group B in terms of that over A and δ, and substi­tuting it back into the original expression for the rate of change.

This last equation shows that the difference between the subgroups, δ, drives the dynamics for the group share, S_a. S_a (t) converges to 1 if δ remains positive, and to 0 if δ remains negative. It is only with δ = 0 that the shares of groups stabilize.

8.5