Consider a market with the following demand function: P = A - Q/N
where P is the product price, Q the quantity demanded, A a constant, and N the number of consumers in the market. The total cost function for all firms is identical and as follows:
Ci = F + cqi
where C is the total cost, F the fixed cost, c the marginal cost and q output. Subscript i denotes the ith firm. Firms in the market behaves as in the Cournot model (quantity setting) and there is no restriction on entry and exit, i.e., the equilibrium profit is equal to zero.
a) Find the equilibrium number of firms (n), and the Herfindahl index (HI) as a function of the number of consumers in the market. In other words, derive the following f and h functions:
n = f(N;A,F, c) HI = h(N; A, F, c)
where N and HI are the number of firms and the Herfindahl index, N the number of consumers, and A, F and c are parameters (constants).
b) What is the relationship between market concentration and market size?