Question

Suppose that demand for carpet-cleaning services in Dirtville is described by P = 200 - 4Q. There are currently 9 identical firms that clean carpet in the area. The unit-cost of cleaning a carpet is constant and equal to $20. Firms in this industry compete in quantities.

a) Show that in the Cournot-Nash equilibrium the profit of each firm is 81.

b) Suppose that one of the carpet-cleaning firms is contemplating merging with another firm. This merged firm would compete on all equal footing with non-merged firms. Show that the profit earned by the merged firm would be insufficient to compensate all the shareholders/owners who owned the original firms and earned profits from them in the pre-merger game.

c) Now suppose that eight of the nine firms have agreed to engage in pairwise mergers (e.g., firm 1 mergers with firm 2, firm 3 merges with firm 4, etc.), reducing the number of firms in the market from nine to five. Would this arrangement be profitable for the merged firms? Which firm benefits the most from this action?

Answer 1

a)

For firm 1,

Revenue,

Profit,

To maximise the profit for firm-1, we need to partially differentiate the profit equation with respect to q₁ and equate to 0

(equation:A)

Similarly, for other firms, we can derive similar equations,

.

.

Adding all these 9 equations, we get,

(Equation B)

Equation A - B yields,

Similarly, every other firm will produce 4.5 units.

Hence, Q = 40.5

P = 38

That is, at equlubrium, the profit for each firm is 81.

b) Let firm-1 and firm-2 merge to form the merged firm

We have to proceed as in the previous answer. The optimization equations for the new firms will be,

For merged firm,

For other firms,

.

.

Solving these 8 equations simultaneously as in previous question, we obtain,

The profit of the merged entity is 100. This is less than sum of the individual profits (89*2=178) earned by the firms earlier

c)

The optimisation equations will be,

Similarly,

.

.

Solving the equations, we obtain,

This is more than the sum of profits earned by the unmerged firms earlier.

Thus all the firms gain from the merger. The unmerged firm (Firm-9) gains the most as its profit jumps from 81 to 225.