Problem Set 3
Environmental Regulations
1. A small town has a total population of 1000 individuals. The town has two paper manufacturers that
emit sulfur dioxide. The total damage for each individual in the town is equal to (1/200,000)S2 where S is
the amount of sulfur dioxide emitted. An economist estimates the total savings for one of the
manufacturers to be TS1 = 4.4 S1 – (1/500)S12 and the other total savings to be TS2 = 6 S2 – (1/300)S22.
1.1 Derive the aggregate damage and marginal savings functions. Draw the individual marginal savings
for each firm, the aggregate marginal savings and aggregate marginal damage functions. Identify the
equilibrium quantity and price for sulfur dioxide emissions.
Aggregate damage is a vertical summation of individual damages. This means it is equal to individual
damage multiplied by total number of individuals. Therefore,
TD=(1/200) S2
Marginal damage is the derivative of total damage so,
MD=(1/100) S
To derive aggregate marginal savings, first derive individual marginal savings. In this case,
MS1 = 4.4 – (1/250)S1
MS2 = 6 – (1/150)S2
To derive aggregate marginal savings, a horizontal summation of individual marginal savings is
conducted,
Re-arranging each MS, we find
S1 = 1100 – 250MS1
S2 = 900 – 150MS2
Thus, SA= S1 + S2 = 1100 – 250MS1+900 – 150MS2 = 2000-400MSA. Re-arranging,
MSA = 5-(1/400) SA This is aggregate MS when MS <4.4
MSA = 6 – (1/150) SA This is aggregate MS when MS >=4.4
Graphically,
$
MD
MS2
6
4.4 MS1
P*
MSA
S* 900
1100
2000
S
1.2 Calculate the Pigouvian tax and aggregate standard in this market. Show your solution.
We need
MSA = MD
5-(1/400) SA = (1/100) SA
S*=400 So this is the standard
Plug S* into either equation,
P*=4. This is the Pigouvian tax
1.3 Which regulatory mechanism is the most efficient based on total savings, the Pigouvian tax or
equitable standard (each firm allowed to emit half of aggregate standard)? Prove your answer by
calculating total savings under each mechanism.
With a tax, each firm will produce
S1* = 1100 – 250(4) = 100
S2* = 900 – 150(4) =300
Pigouvian tax is more efficient. Total savings with a tax is,
TSt = TS1 + TS2= 4.4 (100)– (1/500)(100)2 + 6 (300) – (1/300)(300)2 = 1920
Total savings with a standard means each firm is only allowed 200 units of pollution each. So,
TSs = TS1 + TS2= 4.4 (200)– (1/500)(200)2 + 6 (200) – (1/300)(200)2 = 1866.67
2. The Fireyear and Goodstone Rubber Companies are two firms located in the rubber capital of the world.
These factories produce furnished rubber and sell that rubber into a highly competitive world market at
the fixed price of $60 per ton. The process of producing a ton of rubber also results in a ton of air pollution
that affects the rubber capital of the world. There is a one to one relationship between rubber output and
pollution is fixed and immutable at both factories. The total cost for Fireyear from producing QF units of
tires is 300+2Q2F while the total cost for Goodstone from producing QG is 500+Q2G. Total pollution
emissions generated are EF + EG = QF + QG. Marginal damage from pollution is equal to $12 per ton of
pollution.
2.1 In the absence of regulation, how much rubber would be produced by each firm? What is the profit
for each firm?
max 60𝑄𝑄𝐹𝐹 − (300 + 2𝑄𝑄𝐹𝐹 2 )
𝑄𝑄𝐹𝐹
FOC is 60 − 4𝑄𝑄𝐹𝐹 = 0. This means 𝑄𝑄𝐹𝐹 = 15.
Plug 15 into the profit equation and profit becomes 150.
max 60𝑄𝑄𝐺𝐺 − (500 + 𝑄𝑄𝐺𝐺 2 )
𝑄𝑄𝐺𝐺
FOC is 60 − 2𝑄𝑄𝐺𝐺 = 0. This means 𝑄𝑄𝐺𝐺 = 30.
Plug 30 into the profit equation and profit becomes 400.
2.2 The local government decides to impose a Pigouvian tax on pollution equal to $12 per ton of pollution
or $12 per output. What are the post-regulation levels of rubber output and profits for each firm?
The problem now becomes,
max 60𝑄𝑄𝐹𝐹 − �300 + 2𝑄𝑄𝐹𝐹 2 � − 12 𝑄𝑄𝐹𝐹
𝑄𝑄𝐹𝐹
FOC is 60 − 4𝑄𝑄𝐹𝐹 − 12 = 0. This means 𝑄𝑄𝐹𝐹 = 12.
Plug 12 into the profit equation and profit becomes -12.
max 60𝑄𝑄𝐺𝐺 − �500 + 𝑄𝑄𝐺𝐺 2 � − 12𝑄𝑄𝐺𝐺
𝑄𝑄𝐺𝐺
FOC is 60 − 2𝑄𝑄𝐺𝐺 − 12 = 0. This means 𝑄𝑄𝐺𝐺 = 24.
Plug 24 into the profit equation and profit becomes 76
2.3 Suppose instead of the emission tax, the government decides to offer a subsidy to each firm for each
unit of output reduced equal to the value of the Pigouvian tax. Calculate the levels of output and profit for
each firm.
The problem now becomes,
�𝐹𝐹 − 𝑄𝑄𝐹𝐹 �
max 60𝑄𝑄𝐹𝐹 − �300 + 2𝑄𝑄𝐹𝐹 2 � + 12 �𝑄𝑄
𝑄𝑄𝐹𝐹
�𝐹𝐹 = 15 from (2.1).
FOC is 60 − 4𝑄𝑄𝐹𝐹 − 12 = 0. This means 𝑄𝑄𝐹𝐹 = 12. Here, 𝑄𝑄
Plug 15 into the profit equation and profit becomes 168.
�𝐺𝐺 − 𝑄𝑄𝐺𝐺 �
max 60𝑄𝑄𝐺𝐺 − �500 + 𝑄𝑄𝐺𝐺 2 � + 12�𝑄𝑄
𝑄𝑄𝐺𝐺
�𝐺𝐺 = 30 from (2.1).
FOC is 60 − 2𝑄𝑄𝐺𝐺 − 12 = 0. This means 𝑄𝑄𝐺𝐺 = 24. Here, 𝑄𝑄
Plug 24 into the profit equation and profit becomes 436
2.4 Compare the output and profits for the two firms given a tax versus a subsidy. Comment on the
differences, if any, and the possibility of one or both of the firms dropping out of the market.
In the short run, output levels are equal under a tax or subsidy. However profit is lower under a tax.
Fireyear may even exit the market.
In the long run, more firms enter under a subsidy compared to a tax. Thus more pollution occurs in the
long run.
3. Suppose that the inverse demand curve for paper is p = 300 – 2Q, the private total cost is Cp = 50Q +
Q2. During the production of paper, pollution is generated and the total external cost from pollution is Ce
=0.5 Q2.
3.1 What is the unregulated competitive equilibrium?
max 𝑝𝑝𝑝𝑝 − (50𝑄𝑄 + 𝑄𝑄 2 )
𝑄𝑄
FOC is 𝑝𝑝 − (50 + 2𝑄𝑄) = 0. Substituting 𝑝𝑝 = 300 − 2𝑄𝑄 and solving for Q yields, Q = 62.5. The price is
then, p=175.
3.2 What is the social optimum? What specific tax (per unit of output) results in the social optimum if a
perfectly competitive market exists?
To maximize social welfare we solve the following problem,
max 𝑝𝑝𝑝𝑝 − (50𝑄𝑄 + 𝑄𝑄 2 ) − 0.5 𝑄𝑄 2
𝑄𝑄
FOC is 𝑝𝑝 − (50 + 2𝑄𝑄) − 𝑄𝑄 = 0. Substituting 𝑝𝑝 = 300 − 2𝑄𝑄 and solving for Q yields, Q = 50. The price
is then, p=200.
To solve for the tax, solve the following problem,
max 𝑝𝑝𝑝𝑝 − (50𝑄𝑄 + 𝑄𝑄 2 ) − 𝑡𝑡 𝑄𝑄
𝑄𝑄
FOC is 𝑝𝑝 − (50 + 2𝑄𝑄) − 𝑡𝑡 = 0. Substituting 𝑝𝑝 = 300 − 2𝑄𝑄 and knowing that Q = 50 is the target level,
we can solve for t=50.
3.3 Assume that a monopoly market structure exists. What is the unregulated monopoly equilibrium?
What specific tax (per unit of output) results in the social optimum?
The monopoly’s problem is,
max(300 − 2𝑄𝑄)𝑄𝑄 − (50𝑄𝑄 + 𝑄𝑄 2 )
𝑄𝑄
FOC is 300 − 4𝑄𝑄 − (50 + 2𝑄𝑄) = 0. Solving for Q yields, Q = 41.67. The price is then, p=216.67.
To solve for monopoly’s tax,
max(300 − 2𝑄𝑄)𝑄𝑄 − (50𝑄𝑄 + 𝑄𝑄 2 ) − 𝑡𝑡𝑡𝑡
𝑄𝑄
FOC is 300 − 4𝑄𝑄 − (50 + 2𝑄𝑄) − 𝑡𝑡 = 0. Knowing that we want Q =50, the tax becomes t = -50. A
subsidy!!
Submission instructions:
Deadline: October 2, 2015. Late problem sets will be penalized by 10% of the total grade every 30
minutes.