I is a constant which measures the sensitivity of the populatim to the differences represented in A~j: if the population is very sensitive, I is high, and vice versa.

The final step in the argument is to modify the price of a unit of k at i, p~, to allow for economies of scale:

k k a , (:, , a and b are suitable constants. In this formulation, the optimum size of a plant at i is given by

above which diseconomies of scale set in. Thus, D~j can be put together using Equations (6.5), (6.7), (6.8), (6.9) and (6.10) as

and D~ can be obtained from (6.6). A quick glance at Equation (6.3) shows that the

equilibrium conditions for S~ could be written as

(6.13) Equation (6.2) shows that the equilibrium condition for the population variable x. can be written

Equation (6.13) has been written out in full to expose all the complexities involved in seeking ani;analytical solution for s~, or indeed any geometrical insights. The main nonlinearities arise in three ways, all difficult to handle: (1) the way in which the s~'s occur in the economies-of-scale term, and the way this term appears in the numerator, thus incl uding every s~, term, i' F i, in each i -k equation; (2) the number of facilities, ni , depends on the pattern of non-zero s~'s, varying over k for each i; (3) each i-k equation involves all the xj's and this provides a strong coupling to each x. equation, each of which involves all the

k J Si 's.