Two factories, 1 and 2, in a small town both emits some hazardous air pollution.

Question

Two factories, 1 and 2, in a small town both emits some hazardous air pollution. Both factories simultaneously choose the quantity of outputs, qi for i = 1, 2, which bring them more profit for i, i = ln(qi) + 2, as well as more hazardous pollution that hurts both factories, hi(qi , qi) = qi + qi , where subscript i denotes the other player. We assume that each factory’s utility is the profit net of of the disutility from pollution, i.e. their payoffs are vi(qi , qi) = i(qi) hi(qi , qi).

1.For factory i, find the best response q ' i given the belief that the other factory produces qˆi .

2.Is there a dominant strategy for each factory? Briefly explain.

3. (N.E. case) Identify the N.E. of this game, i.e. how much does each firm produce in the equilibrium?

4.(central planning case) If there is a town government whose interest is to maximize the joint payoff of both factories, v1 + v2. And assume that the government can enforce an output restrictions on how much each firm produce, what is the optimal output levels q m 1 and q m 2 that the government will mandate?

5.Compare the N.E. outcome with the central planning outcome, specifically 1. how much total pollution is generated in the N.E case vs. the central planning case? 2. how much are the payoffs for each factory under the N.E. case vs. the central planning case?

6.Briefly explain, what did you learn from this question, if anything?

Explanation / Answer

Profit for firm 1 = ln(q1) +2 - q1 - q2

Maximize this w.r.t q1,

1/q1 -1 =0 => q1=1(Firm2's reaction function)

Profit for firm 2 = ln(q2) + 2 - q1 - q2

Maximize this w.r.t q2

1/q2 - 1 = 0 => q2=1 (Firm2's reaction function)

2. (q1,q2) = (1,1) is the dominant strategy.

Firm1 will produce 1 unit irrespective of what firm2 produces as firm2's quantity does not figure in the reaction function of firm1.

Similarly firm2 will produce 1 unit irrespective of what firm1 produces.

3. (q1,q2) = (1,1) is the nash equilibrium as it is the dominant strategy of each firm.

4. Maximize v1+v2 = ln(q1) +2 - q1-q2 + ln(q2) + 2 - q1 - q2

Maximize w.r.t q1

1/q1 - 2 = 0

q1 = 1/2

Maximize v1+v2 w.r.t q2

1/q2 - 2 =0

q2 = 1/2

Government will mandate (qm1,qm2) = (1/2 , 1/2)

5. Payoffs for firm1 with (q1,q2)=(1,1) : ln(1) +2 -1-1 = 0

payoff for firm1 with (qm1,qm2) = (1/2,1/2) = ln(1/2) - 2 - 1 = 0.31

Payoffs for firm2 with (q1,q2)=(1,1) : ln(1) +2 -1-1 = 0

payoff for firm2 with (qm1,qm2) = (1/2,1/2) :  ln(1/2) - 2 - 1 = 0.31

Pollution with (1,1) as solution : 2

Pollution with (1/2,1/2) as solution : 1

6. This question explains that by maximizing social profit is beneficial for each firm as well as for the society as a whole as it reduces the amount of pollution and at the same time increases the profit of the firms.

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