# For which number of firms in a Cournot oligopily a merger is profitable

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.

Your calculations are correct. Salant, Switzer and Reynolds (1983) show that merger of $$m+1$$ firms (the 'insiders') increases the profits of the insiders if and only if $$g(n,m)>0$$ where (using your notation with $$b=1$$):

$$g(n,m)=(a-c)^2\left[\frac{1}{(n-m+1)^2}-\frac{m+1}{(n+1)^2}\right]$$

You are looking at the case where $$m=1$$. We have

$$g(n,1)=(a-c)^2\left[\frac{1}{n^2}-\frac{2}{(n+1)^2}\right]$$

and the condition $$g(n,1)>0$$ requires

$$\frac{1}{n^2}>\frac{2}{(n+1)^2}.$$

This is the same as the inequality you found.

The explanation given by Salant, Switzer and Reynolds (1983) for why merger can reduce profits of the merging firms (insiders) is as follows

• A merger causes the equilibrium output of the insiders to contract and the output of the outsiders to expand.

• For any given level of production by outsiders, the aggregate profits of the insiders can only increase (since they can always run their plants so as to mimic the premerger equilibrium).

• But as the outputs of the outsiders increase, the profits of the insiders decrease.

• Hence the possibility arises that the increase in production by outsiders following the merger will reduce insider profits by more than the increase in profits that would have occurred had outsider production remained constant.

• Thus, it is the output expansion of the outsider firms that can in principle cause a reduction in profits for the merging firms.

In the case of monopoly, there are no outsider firms and so the merger can only increase profits.