Question

There are two oil-producing countries: A and B. Each can choose their own level of oil production: low L or high H. At low production, a country produces 2 million barrels of oil per month, while at high production, a country produces 4 million barrels of oil per month. Costs of production for A are $8 per barrel; for B, $16 per barrel The price of oil each month depends on the total output from these two countries onto the world market. If together they produce 4 million barrels, the price per barrel is $100. If together they produce 6 million barrels, the price is $60 per barrel. If together they produce 8 million barrels, the price per barrel is $40 (a) (3 marks) Calculate the payoffs to cach country for every possible strategy profile. Write payoffs in units of millions of dollars (b) ( marks) Using the payoffs from part a), fill in the payoff matrix for this game. Find all Nash equilibria for this game (c) (3 marks) Suppose this game is repeated every month, indefinitely. Consider the following trigger strategy, which we will call T: "Choose a low level of production. However, if either country ever chooses a igh level of production, even once, then choose high production forever." We suppose that cach country discounts future earnings at a monthly discount rate o (where 0 < 8 < 1). Find a condition on 6 such that (T, T) is subgame perfect

Answer 1

Here,

Cost of production for A = $8 per barrel,

Cost of production for B = $16 per barrel

a) Now, profit, ∏ = Price*Quantity – Cost*Quantity

Denoting pay-off by ∏(Q_A,Q_B)

where, Q_A = Quantity produced by Country A

            Q_B = Quantity produced by Country B

We get:

For (Q_A, Q_B) = (2, 2); ∏_A = 100*2 – 8*2 = 184; ∏_Ab = 100*2 – 16*2 = 168

For (Q_A, Q_B) = (2, 4); ∏_A = 60*2 – 8*2 = 104; ∏_B = 60*4 – 16*4 = 176

For (Q_A, Q_B) = (4, 2); ∏_A = 60*4 – 8*4 = 208; ∏_B = 60*2 – 16*2 = 88

For (Q_A, Q_B) = (4, 4); ∏_A = 40*4 – 8*4 = 128; ∏_B= 40*4 – 16*2 = 128

b)                                       PAY-OFF MATRIX:

	Low	High
Low	(2,2)	(2,4)
High	(4,2)	(4,4)

           

                             

                       

            

Note: Here Country A is row player and country B is column player

By definition, Nash Equilibrium is that strategy from which no player has no unilateral incentive to deviate.

• If Country A chooses Low, Country B chooses High [as ∏_B = 176 > 168],
• If Country A chooses High, Country B chooses High [as ∏_B = 128 > 88],
• If Country B chooses Low, Country A chooses High [as ∏_A = 208 > 184],
• If Country B chooses High, Country A chooses High [as ∏_A = 128 > 104]

Thus, (High, High) is the Nash Equilibrium.

c) Assuming discount rate of ẟ, we get:

• Countries do not corporate: 2 + 4ẟ + 4ẟ² + ……. + upto infinite rounds = 2 + 4ẟ(1 + ẟ+ ……. + upto infinite rounds) = 2 + 4ẟ/(1 - ẟ)
• Countries do corporate: 2 + 2ẟ + 2ẟ² + ……. + upto infinite rounds = 2(1 + ẟ + ẟ² + ……. + upto infinite rounds) = 2/(1 - ẟ)

Thus, corporation is desirable iff: 2 + 4ẟ/(1 - ẟ) ≥ 2/(1 - ẟ) or, ẟ ≥ 0