Question

There are two main company(A, B) in a market. Their products are identical. MC = $1/1 product.

Qd = -1000P + 15000 (Demand curve)

What is a payoff matrix by finding two company’s profit if two company make a cartel and split the market equally?

What is a payoff matrix by finding two company’s profit if company A makes 1000 more product while company B keeps the previous agreement?

What is the Nash equilibrium in the two situation? The cartel is good or not for the two company? Why?

Answer 1

Qd = -1000P + 15000
or, 1000P = 15000 -Qd
or. P = 15 - .001Qd
or, PxQd = 15Qd - .001Qd^2
Therefore, MR = 15 -.002Qd
Now the profit maximizing quantity can be found by equating MR with MC.
So, we get, 15-.002Qd = 1
or, 14/.002 = Qd = 7000
So, P = 15-7 = 8
So if both companies share Qd equally, each will produce 3500 units.
Profit for each will be : 3500 x 8 - 3500 = 24500
So if the two companies form a cartel and split the market equally,
Payoff for each company will be = 24500

If A makes 1000 more than B while B keeps the previous agreement,
Qd = 8000 , P = 7
Profit of A = 4500 X 7 - 4500 = 27000
Profit of B = 3500 x 7 - 3500 = 21000

If both increases by 1000,
Qd = 9000, P = 6
Profit for A = 4500 x 6 - 4500 = 22500
Profit for B = 4500 x 6 - 4500 = 22500

Now we can formulate the payoff matrix

A,B	honor agreement	dishonor agreement
honor agreement	24500,24500	21000,27000
dishonor agreement	27000,21000	22500,22500

So, it seems that both the compant has an incentive to break the agreement and produce more in order to earn more profit. However, the dilemma is, profit is higher if only one of them dishonors the agree ment and the other doesnt but if both does, then they both earn profits lower than what they would have earned while keeping the agreement.
This game is akin to Prisoner's Dilemma game. So the cartel is actually good for the companies but their Nash equilibrium strategy is to break the cartel and thus have a sub optimal equilibrium and earn a lower amount of profit.