Question

An industry has three firms each of which produces output at a constant unit cost of $10 per unit. The demand function for the industry is q= 200-p. Firms compete in price in a Bertrand game

1. Calculate the minimum  that can sustain collusion if firms use a trigger strategy.
2. Suppose that two firms merge, calculate the minimum  that can sustain collusion after the merger.
3. After the merger, the two firms reach a cross share holding agreement. Each of the two firms owns a share αof its rival. The share is small enough for each firm to keep full control of its activities and decision. The rival is minority shareholder and just receives a share α of the firm’s profits.  What is the minimum  that can sustain collusion if firms use a trigger strategy?Is the likelihood of collusion affected by this cross share holding agreement?

Answer 1

P = 200 - 2Q Where Q is total industry output.

The market is occupied by two firms, each with constant marginal costs equal to $10

Calculate the equilibrium price and quantity assuming the two firms compete in quantities.

Now, we need to actually go through the steps: Assume that firm one’s cost is $10.

P = 200 - 2q2 - 2q1

Total revenues equal price times quantity.

2 Pq1 = 200 - 2q2 q1 - 2q1

Marginal revenue is the derivative with respect to quantity.

MR = 200 - 2q2 - 4q1

Set marginal revenue

equal to marginal cost and solve for quantity. To get market price, remember there are two firms.

(200-2q2)-4q1=10

Q1 = (190-2q2)/4 = 47.5-0.5q2

This is firm one’s best response function. Note that firm two has the same problem to solve, but with MC =$8

Q1 = 47.5-0.5q2

Q2 = 48 – 0.5q1

Solving these two,

Q1 =31.3

Q2 = 32.3

P = 78.8

Note that firm 2 exploits firm one’s cost increase by grabbing market share

Assuming the competition is in prices rather than quantities, with identical products, both firms charge a price equal to marginal cost (in this case, $8) and profits are zero. If firm one’s costs rise to $10, firm 2 charges a price equal to $9.99, takes the entire market and earns profits equal to ($9.99 - $8)(95) = $189.