Question

Four firms in an industry are considering forming a cartel. They meet in a smoke-filled hotel room to discuss the terms of the cartel and to weigh their options. Each firm has marginal cost given by MC=20+2q. Demand for the market is given by Q=400-2p.

1. What is the supply for each firm? For all four firms?

2. Suppose the firms compete. What would be the resulting market prices and quantities under competition? How much would each firm produce and what would be their producer surplus?

3. Now suppose that the firms successfully form a cartel. They agree to divide the cartel quantity equally among the four firms. How much will each firm produce under the cartel? What will be the producer surplus of each firm?

4. One of the firms is considering cheating on the cartel agreement. Discuss how the firm could cheat on the cartel agreement and what benefits the firm gets from cheating. [Hint: show with equations or a graph how the firm’s profits would change, assuming that the other three firms stick to the cartel agreement.

Answer 1

1) Supply is nothing but inverse marginal cost.

P=20+2q

q=0.5p-10, individual firm supply

Market supply is sum of individual supply

Market supply or four firm supply:

Q=4*(0.5p-10)=2p-40

B) competitive equilibrium at where demand= supply

400-2p=2p-40

P=440/4=110

Q=440-2*110=180

Each firm production= equilibrium quantity/ number of firms=180/4=45

Producer surplus( each firm)=1/2*45*(110-20)=45*45=2025

C) Q=2p-40

Aggregate MC=20+0.5Q

Cartel equilibrium at ,MR= MC

Inverse demand:P=200-0.5Q

MR=200-Q

Equilibrium:

200-Q=20+0.5Q

Q=180/1.5=120

P=200-0.5*120=200-60=140

Each firm output=120/4=30

Each firm MC at output=20+2*30=80

Producer surplus=(140-80)*30+1/2*30*(80-20)=1800+30*30=2700

D) inverse demand:P=200-0.5Q=200-0.5q1-0.5q2-0.5q3-0.5q4

Let say firm 1 is considering to cheat, and assume others three combined output is Z.

MR1=200-q1-0.5Z

MC1=20+2q1

200-q1-0.5Z=20+2q1

Q1=60-(Z/6) , best output function of firm 1.

Given cartel output ,so Z=3*30=90

So best output for firm 1,

Q1=60-(90/6)=60-15=45

So by cheating firm 1 will produce 45 , instead of 30.

New Market output=30*3+45=90+45=135

P=200-0.5*135=132.5

MC of firm 1 at new output=20+2*45=110

Producer surplus of firm1=(132.5-110)*45+1/2*45*(110-20)=22.5*45+45*45=3037.5

So by cheating firm 1 increased their producer surplus or profit.