Consider an oligopolistic market with N=3 identical firms, all three making a homogeneous product. The inverse demand for this product is
p = 75 3Q
where Q is the market quantity. The marginal cost of production is equal to the average cost and is identical for all firms and given by c = 33.
(a) Calculate the Nash equilibrium output, price and profits of each firm using quantity as the strategic variable.
(b) Compute the Lerner index for each firm.
(c) Calculate the Nash equilibrium output, price and profits of each firm using price as the strategic variable. What is the Lerner index under this equilibrium?
Now suppose that (N + 1) firms are in this market. One of the firms produces at a low marginal cost of 20. The other N firms produce at marginal cost 33. There are still no fixed costs. The firms play a Nash-in-quantities game.
(d) Re-solve for the best response functions for both types of firms. (You might find it helpful, here and in part (e), to order firms so that firm 1 has c = 20 and firms 2; 3; :::; N + 1 have c = 33.)
(e) Solve for the optimal quantities for both types of firms.