Assume a Cournot oligopoly with N (equal to 2 or higher) symmetric firms. Firms face a demand function P=a-bQ and an average cost c. I need to find the number of firms N for which it will be profitable for the 2 (out of N) firms to merge.
Then each firm has a profit of (a-c)^2/b*(N+1)^2.
Two of N firms merge — now we have a Cournot oligopoly with (N-1) symmetric firms. Each firm will get a profit of (a-c)^2/b*N^2. If we assume that in case of merger each of the “merged” firms gets half of the firm’s profit, the condition for an optimal N should be 1/2N^2 > 1/(N+1)^2. From this I get that N should lie within the interval (-0,4142; 2,4142), which, since N is integer and equal to 2 or bigger, transforms into condition N=2.
This result seems quite counterintuitive to me. I wonder, whether there is a mistake in my logic or computations.