Consider two firms whose marginal costs of production are MC

Consider two firms whose marginal costs of production are MC(Q1) = 10Q1; MC(Q2) = 5Q2: Suppose each unit of output produced, Q, results in one unit of emissions, E. Suppose the two firms sell their output in a perfectly competitive market, with perfectly elastic demand at a price of $90. Furthermore, suppose there is a constant external cost of emissions of $10.

Now suppose the regular decides to implement a cap and trade system to attain the socially optimal level of emissions via the Coase theorem. Firm 2 has more political clout, however, so the regulator distributes all permits to firm 2.

(a) How many permits should the regulator allocate to firm 2?

(b) Now suppose firm 2 sells permits to firm 1 such that they get to the socially optimal allocation of emissions across the two rms. How many permits would firm 2 sell to firm 1?

(c) What is the minimum price firm 2 would be willing to accept for the permits?

(d) What is the maximum price firm 1 would be willing to pay for these permits?

Solution

a) Since each unit of output produced, Q, results in one unit of emissions, E. and there is a constant external cost of emissions of $10

therefore, MC2 = 5Q2 + 10Q2 = 15Q2

A firm is in equilibrium where MC = MR

Since AR is constant at Rs. 90

MR is also equal to 90

to be in equilibrium,

MC = MR

5Q2 + 10Q2 = 90

15Q2 = 6 units

therefore, 6 permits must be allocated to firm 2.

b) For this the two firms will act as monopoly and their combined profit must be maximum.

20Q1 + 15 Q2 < or equal to 90

Q1 + Q2 < or equal to 6

on solving,

4Q1 + 3 Q2 < or equal to 18

Q1 + Q2 < or equal to 6

Q2 = 6

Q1 = 0

Since the marginal cost of firm one is double the marginal cost fo firm 2. demand curve for both the firms is identical. therefore, firm 2 will not sell any permits ot firm 1.

c) If at all the firm 2 would be selling permits, it would accept a price of 90 which it can earn by selling the product.

d) If at all the firm 1 would be buying permits, it would give a maximum price fo Rs. 90 which it can earn by selling the product.