Question

Two firms can reduce emissions of a pollutant at the following marginal costs:

MC1 = $24q1

MC2 = $12q2

where q1 and q2 are, respectively, tons of emissions reduced by the first and second firms. Total pollution-control cost functions for the two firms are:

TC1 = $12 + $12(q1)2

TC2 = $10 + $6(q2)2

Assume that with no control at all, each firm would be emitting 10 tons of emissions (for aggregate emissions of 20 tons), and assume that there are no significant transaction costs.

1. Assuming these are the only two firms in the industry, what are the total industry costs of pollution control if a uniform (quantity) emission standard is utilized to achieve an aggregate reduction (for both firms combined) of 4 tons of emissions?

2. What are the marginal costs of pollution control for firm #1 and for firm #2 under the uniform standard described in part a?

3. Compute the cost-effective allocation of control responsibility if a total reduction of 4 units of emissions is necessary, i.e. how many units of emissions will each firm reduce under a cost-effective allocation (assume tons are divisible)?

4. What are the total industry costs of pollution control (for both firms combined) with a cost-effective allocation of control responsibility?

5. Suppose firm #1 is freely allocated 7 tons of emissions permits and firm #2 is freely allocated 9 tons of emissions permits. If trading of permits is allowed, what will be the equilibrium allocation of permits? Assume permits are divisible. How will the equilibrium allocation be affected by a change in the initial allocation?

6. If the pollution control authority chose to reach its objective of 4 tons of aggregate reduction with an emission charge, what per-unit charge should be imposed? How much government revenue will the tax system generate, if the tax is levied on every unit of emissions?

7. Assuming the same target for aggregate emission reductions in each case, which policy instrument—taxes, tradeable permits, or a uniform standard—would you expect private industry to prefer? Why?

Answer 1

1. If uniform standard is followed, both firms will reduce 2 tons. So, q1=q2= 2 tons.

TC1 = 12 + 12*q1^2 = 12 + 12*2*2 = 60

TC2 = 10 + 6*q2^2 = 10 + 6*2*2 = 34

Total Industry Cost = 60+34 = 94

2. MC1 = 24*q1 = 24*2 = 48

MC2 = 12*q2 = 12*2 = 24

3. q1+q2 =4

TC = 22+ 12*q1^2 + 6*q2^2 = 22+ 12*q1^2 + 6*(4-q1)^2

= 22 + 12*q1^2 + 6*(16+q1^2 - 8*q1)

= 118 + 18*q1^2 - 48*q1

for cost effectiveness d(TC)/dq1 = 0

=> 36*q1 -48 = 0

=> q1 = 48/36 = 4/3

consequently, q2 = 4- q1 = 4-4/3 = 8/3

(q1,q2) = (4/3, 8/3)

4. Total Industry cost = 118+ 18*q1^2 -48*q1

= 118 + 18*16/9 - 48*4/3

= 118 + 32 - 64

= 118-32

= 86

Cost for Firm 1 = 12 + 12*q1^2 = 12+ 12* 4/3 * 4/3 = 12 + 21.34 = 33.34

Cost for Firm 2 = 10 + 6*q2^2 = 10 + 6* 8/3 * 8/3 = 10+42.67 = 52.67