Assignment 2

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PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE 
FA C U L T A D D E C I E N C I A S E C O N ÓM I C A S Y A D M I N I S T RA T I V A S 
 
 
Environmental and Natural Resource Economics EAE 295C 
Professor Guillermo Donoso (gdonosoh@uc.cl) 
 
Homework Assignment 2 
General instructions 
1. The assignments must be solved individually. The answers to each question must be 
presented in a single Word file. It is not allowed to scan handwritten tasks to send a 
PDF file to the Buzón de Tareas. 
2. In the development of the answers it is mandatory the use of the Microsoft Equation 
Editor of Microsoft Word (or, alternatively, some other equation editor) in order to 
express mathematically what is requested in each question. 
3. If an attempt of academic fraud is detected, the assignment will be rated with a grade 
of 1.0 both for the student who copied and the one who let them copy their work. 
4. The assignments that do not comply with the indicated instructions will be penalized 
with a cumulative reduction of 30% of the final grade obtained. 
5. An electronic copy of the resolution of the assignment must be sent to the Buzón de 
Tareas on Thursday, September 6 before 10:00 and another printed copy must be 
handed in by 10:00 on the same date. 
 
Show all your calculations 
 
 
Externalities (Coase Theorem1) 
1. In a valley there are two companies, A and B, which are neighbors. Company A, owned 
by the church and operated by nuns, is a heart pacemaker factory for children with heart 
problems and is located west of the valley; and company B, owned by a billionaire, is a 
vineyard east of the valley. The wind blows from west to east, carrying smoke from the 
heart pacemaker factory of company A west of the valley to the vines of company B. 
This smoke causes a reduction in the quality of Mr. B's wine and, therefore, a reduction 
in the price received for his wine in foreign markets. 
 
a. If company B has the right to live in a pollution free environment (have smoke-free 
air), how much is society’s welfare when both parties can negotiate free of cost? 
 
 
1 (Hint: see Pernman et al., Capter 5.10.3.1) 
 
 
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Company A has to compensate the vineyard. Minimum compensation would be equal 
to the impact on the reduction of the profits due to the reduction in the price of wine. 
If it is beneficial to continue producing for company A, (net benefits minus 
compensation is positive), the compensation would be in the form of direct payment 
to company B. If it is not beneficial to continue to produce (net benefits minus 
compensation is less than zero), company A must shutdown. 
 
b. If company A has the right to pollute, how society’s welfare when both parties can 
negotiate free of cost compare to the previous case? 
 
Company B can negotiate with company A and pay so that it does not produce smoke. 
Company B is willing to pay for each unit less produced by company A in proportion 
to the impact of smoke on its profits (through the reduction in price). It will pay as 
long as net benefits minus price is positive. 
 
c. Would the smoke level be approximately equal under the two different assignments of 
property rights over the air quality in the valley? (Explain) 
 
Assuming that the increase in wealth does not affect the receiving individual’s tastes, 
then in both cases, the effective prices to company A must be equal (if the negotiating 
transaction costs are not prohibitive), then the smoke levels in both cases would be 
almost equal. Coase theorem. The only difference is in terms of the welfare of the two 
companies, but the smoke level is the same. 
 
If the assumption that tastes are unaffected by wealth increases is dropped, then the 
outcomes would be different in the two cases2. 
 
2. Two companies, # 1 and # 2, production costs are a function of the emissions of a 
pollutant; formally, 
 
𝑐1 = 1000 − 10𝑒1 + 0.05𝑒1
2 
𝑐1 = 2000 − 20𝑒2 + 0.10𝑒2
2 
 
where 𝑐𝑖 is firm i’s cost and 𝑒𝑖 is firm i’s emissions. 
 
 
2 See Pernman et al. page 138. 
 
 
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FA C U L T A D D E C I E N C I A S E C O N ÓM I C A S Y A D M I N I S T RA T I V A S 
 
a. If pollution is not regulated, what is the optimal emissions level of for each 
company? (The emission level that minimizes costs) 
 
Each firm determines their optimum minimizing costs; i.e. 
Firm 1: 
Min 𝑐1 = 1000 − 10𝑒1 + 0.05𝑒1
2 
First order conditions are3 
 
𝜕𝑐1
𝜕𝑒1
= −10 + 0.1𝑒1 = 0  𝑒1
∗ = 100 
 
Firm 2: 
Min 𝑐2 = 2000 − 20𝑒2 + 0.10𝑒2
2 
First order conditions are4 
 
𝜕𝑐1
𝜕𝑒1
= −20 + 0.2𝑒2 = 0  𝑒2
∗ = 100 
 
b. How much is each one’s production cost? 
Firm 1: 
𝑐1
∗ = 1000 − 10𝑒1 + 0.05𝑒1
2 = 1000 − 10(100) + 0.05(100)2 = 500 
Firm 2: 
𝑐2
∗ = 2000 − 20𝑒2 + 0.10𝑒2
2 = 2000 − 20(100) + 0.10(100)2 = 1000 
c. How much is total cost? 
 
Total cost = 𝑇𝐶∗ = 𝑐1
∗ + 𝑐2
∗ = 1500 
 
Suppose the town’s Emperor wants to reduce pollutant emissions by half; that is to 
(𝑒1
∗ + 𝑒2
∗)/2 where 𝑒𝑖
∗ represents the "optimal" emissions obtained in the previous 
question. To achieve this, a total number of emission permits are issued so as to reach 
the target pollution level. Under this scenario, each firm requires a permit for each 
emission unit; for example, if firm i emits 3 units it requires 3 permits. 
 
 
3 Notice that the cost function is convex in 𝑒𝑖 and, thus, first order conditions are necessary and sufficient. 
4 Notice that the cost function is convex in 𝑒𝑖 and, thus, first order conditions are necessary and sufficient. 
 
 
PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE 
FA C U L T A D D E C I E N C I A S E C O N ÓM I C A S Y A D M I N I S T RA T I V A S 
 
d. Suppose that each firm receives the same amount of emission permits. 
d.1 What is each firm’s production cost? 
 
The target emissions is 
 
(𝑒1
∗∗ + 𝑒2
∗∗) =
𝑒1
∗ + 𝑒2
∗
2
=
100 + 100
2
= 100 
 
where 𝑒𝑖
∗∗ is emissions under the permit regulation and 𝑒𝑖
∗ is emissions when there is 
no regulation. If each firm receives the same number of permits, then each one 
receives permits to emit 50 and, thus costs are 
Firm 1: 
𝑐1 = 1000 − 10𝑒1 + 0.05𝑒1
2 = 1000 − 10(50) + 0.05(50)2 = 625 
Firm 2: 
𝑐2 = 2000 − 20𝑒2 + 0.10𝑒2
2 = 2000 − 20(50) + 0.10(50)2 = 1250 
 
d.2 How much is total cost? 
 
Total cost = 𝑇𝐶∗∗ = 𝑐1
∗∗ + 𝑐2
∗∗ = 1875 
 
d.3 What does this emission reduction cost society? 
 
The emission reduction costs society is 
 
∆𝑇𝐶 = 𝑇𝐶∗∗ − 𝑇𝐶∗ = 1875 − 1500 = 375 
 
e. Alternatively, Robert Haveman and John Mullahy5, the Emperor’s economic 
advisors, propose that it is economically more efficient to allow both firms to pay 
for their permits, allowing each to determine their optimal emissions under this 
regulatory instrument. Assume that permits have a per unit cost of P. 
e.1 How much must each permit cost to achieve the desired level of emissions? 
 
Hint: Notice that total production costs for firm i is 
 
5 Tradable pollution permits were proposed by the economists Robert Haveman and John Mullahy of the 
University of Wisconsin–Madison. 
 
 
PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE 
FA C U L T A D D E C I E N C I A S E C O N ÓM I C A S Y A D M I N I S T RA T I V A S 
 
 
𝑐𝑖 = 𝐹𝑖 − (𝑎𝑖 − 𝑃)𝑒𝑖
∗∗∗ + 𝑏𝑖(𝑒𝑖
∗∗∗)2 𝑖 = 1, 2 
 
Optimal emissions are such that 𝑒𝑖
∗∗∗ minimizes costs of firm i. Thus 
 
Min 𝑐𝑖 = 𝐹𝑖 − (𝑎𝑖 − 𝑃)𝑒𝑖
∗∗∗ + 𝑏𝑖(𝑒𝑖
∗∗∗)2 ∀ 𝑖 = 1,2 
 
First order condition𝑑𝑐𝑖
𝑑𝑒𝑖
= −𝑎𝑖 + 2𝑏𝑖𝑒𝑖
∗∗∗ + 𝑃 = 0 𝑖 = 1, 2 
 
The optimal emissions is 
 
𝑒𝑖
∗∗∗ =
𝑎𝑖 − 𝑃
2𝑏𝑖
 𝑖 = 1, 2 
 
And costs are 
 
𝑐𝑖 = 𝐹 −
(𝑎𝑖 − 𝑃)
2
4𝑏𝑖
 𝑖 = 1, 
 
Emissions are thus 
 
𝑒1
∗∗∗ =
10 − 𝑃
0.1
 𝑒2
∗∗∗ =
20 − 𝑃
0.2
 
 
We want the permit price such that emissions are half of the amount without 
regulation. That is 
 
𝑒1
∗∗∗ + 𝑒2
∗∗∗ =
10 − 𝑃
0.1
+
20 − 𝑃
0.2
= 100 ⇒ 𝑃 =
100
15
= 6.67 
 
e.2 How many permits will each company buy? 
To emit 1 unit firm i requires 1 permit, hence each firm will buy 
 
𝑒1
∗∗∗ =
10 − 6.67
0.1
= 33.33 , 
 
 
PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE 
FA C U L T A D D E C I E N C I A S E C O N ÓM I C A S Y A D M I N I S T RA T I V A S 
 
𝑒2
∗∗∗ =
20 − 6.67
0.2
= 66.67 
 
e.3 What is each firm’s production cost? 
Costs are 
𝑐1
∗∗∗ = 1000 −
(10 − 6.67)2
0.2
= 944.56 
𝑐2
∗∗∗ = 2000 −
(20 − 6.67)2
0.4
= 1555.77 
 
e.4 How much is total cost? 
 
Total cost = 𝑇𝐶∗∗∗ = 𝑐1
∗∗∗ + 𝑐2
∗∗∗ = 2500.33 
 
e.5 What does this emission reduction cost society6? 
 
The emission reduction costs society is 
 
∆𝑇𝐶 = 𝑇𝐶∗∗∗ − 𝑇𝐶∗ + 𝑃(𝑒1
∗∗∗ + 𝑒2
∗∗∗) = 2500.33 − 1500 − 6.67(100) 
∆𝑇𝐶 = 333.33 
 
e.6 Are the Emperor’s economic advisors correct in pointing out that this is a more 
efficient policy? 
Yes since societies cost is now ∆𝑇𝐶 = 333.33, which is 11.11% lower than with 
non-tradeable permits. 
 
Public Goods 
3. Three partners of a community of large farmers want to reduce the likelihood of an 
infestation in their local area. If an infestation occurs in one part of the area, it will 
quickly affect everyone. They have the opportunity to invest in a common pest 
monitoring and control system. Let Q be a measure of the effectiveness of this pest 
monitoring and control system: 0 ≤ Q ≤ 100 (0 means the system is not working and 
100 means a system is perfectly capable of preventing infestations). Due to differences 
 
6 The revenue paid by each firm for the permits is a benefit for the Emperor. 
 
 
PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE 
FA C U L T A D D E C I E N C I A S E C O N ÓM I C A S Y A D M I N I S T RA T I V A S 
 
in their production scales, the partners have different marginal benefits (marginal 
willingness to pay) as functions of the quantity, Q, of this public good: 
𝑀𝐵1 = 10 −
1
10
𝑄 
𝑀𝐵2 = 20 −
2
10
𝑄 
𝑀𝐵3 = 40 −
4
10
𝑄 
 
The total cost function to produce this public good is 
 
𝑐(𝑄) = { 2265 +
3
20
𝑄2 𝑄 > 0
0 𝑄 = 0
 
 
a. How much is the optimal supply of the public good, Q, from the perspective of the 
community? 
 
Aggregate social marginal benefit, 𝑀𝐵𝑆, is 
𝑀𝐵𝑆 = ∑ 𝑀𝐵𝑖(𝑄)
3
𝑖=1
= 70 −
7
10
𝑄 ∀ 𝑄 ≤ 1007 
Marginal cost is 
𝑀𝐶 =
𝑑𝑐
𝑑𝑄
=
3
10
𝑄 
Optimal supply of a public good is where 𝑀𝐵𝑆 = 𝑀𝐶; hence 𝑄
∗is such that 
 70 −
7
10
𝑄∗ =
3
10
𝑄∗ ⇒ 
𝑄∗ = 70 
 
 
7 Note that individual marginal benefits for each individual reach zero at 𝑄 = 100, hence aggregate marginal 
benefit is a continuous function for all 0 ≤ 𝑄 ≤ 100. 
 
 
PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE 
FA C U L T A D D E C I E N C I A S E C O N ÓM I C A S Y A D M I N I S T RA T I V A S 
 
b. Is the total benefit to the community of introducing the pest monitoring and control 
system at the optimal level of question (a) high enough to justify the cost of offering 
the system? 
 
Total benefit of the community is given by the total benefit or consumer surplus 
without considering the cost of the public good: 
𝐵𝑇 = ∫ (70 −
7
10
𝑞) 𝑑𝑞
𝑄∗
0
= (70𝑞 −
7
20
𝑞2)|
0
70
= 3185 
The total cost is 
𝑐(𝑄) = 2265 +
3
20
(70)2 = 3000 
Net benefit is 
𝑁𝐵 = 𝐵𝑇 − 𝑐(𝑄) = 185 
So, yes, the total benefit to the community is high enough to justify the cost of the system. 
 
c. If the three partners must pay the same "tax" for having the public good, who will 
vote favorably to supply optimal amount, if the alternative is not having the public 
good (ie, Q = 0)? Will the public good be offered in this case? 
 
Total Benefit for each producer is: 
𝐵1 = ∫ (10 −
1
10
𝑞) 𝑑𝑞
𝑄∗
0
= (10𝑞 −
1
20
𝑞2)|
0
70
= 455 − 0 = 455 
𝐵2 = ∫ (20 −
2
10
𝑞) 𝑑𝑞
𝑄∗
0
= (20𝑞 −
2
20
𝑞2)|
0
70
= 910 
𝐵3 = ∫ (40 −
4
10
𝑞) 𝑑𝑞
𝑄∗
0
= (40𝑞 −
4
20
𝑞2)|
0
70
= 1820 
Since total costs are 𝑐(𝑄) = 3000, each producer would have to pay a tax of 1000. 
In this case only producer #3 would vote for the public good. Thus, the public good 
would not be offered. 
 
 
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If you do not want to use integration, note that total benefit is the area under the 
aggregate marginal benefits curve between zero and the amount of interest8: 
 𝐵𝑇 =
(𝑎 − 𝑀𝐵∗) ∙ 𝑄∗
2
+ 𝑀𝐵∗𝑄∗ =
𝑎𝑄∗
2
+
𝑀𝐵∗𝑄∗
2
 
For farmer #2 this would be 
𝐵2 =
20 ∙ 70
2
+
(20 − 2
10
70)70
2
= 700 + 210 = 910 
 
 
 
8 See graph 
Q* 
Q 
MB 
a 
a/b 
 
MB* 
0 
MB = a - bQ 
 
 
PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE 
FA C U L T A D D E C I E N C I A S E C O N ÓM I C A S Y A D M I N I S T RA T I V A S 
 
 
 
Free Access Goods 
4. Consider two fishers, Lisa Simpson and Tommy Pickles, each one fishes in a 
common lake. Each one’s cost per kilo of harvested fish for each depends on the 
total amount extracted by both: 
 
𝑐(𝑥1 + 𝑥2) = 𝑐(𝑥𝑐) = 1 + 𝑥𝑐 
 
where 𝑥𝑖 represents fisher i’s total catch and 𝑥𝑐 = 𝑥1 + 𝑥2 is the total extraction by 
the two. Thus, the total extraction cost (in dollars) for each 
 
𝐶(𝑥𝑖|𝑥𝑐) = 𝑥𝑖 ∙ 𝑐(𝑥1 + 𝑥2) = 𝑥𝑖 ∙ (1 + (𝑥1 + 𝑥2)) 
 
Each fisher can sell their harvest at 𝑝 = $13/𝑘𝑔. Hence, each one’s profits are 
 
𝜋𝑖 = 𝑝𝑥𝑖 − 𝐶(𝑥𝑖|𝑥𝑐) 
 
a. How much is the socially optimal harvest when Lisa and Tommy cooperate with 
each other? 
 
The total benefit of a single fisher is 
 
𝐵(𝑥𝑖) = 𝑝𝑥𝑖 − 𝑥𝑖[1 + (𝑥1 + 𝑥2)] 𝑖 = 1, 2 
Both fishers are the same, hence they both have the same optimal decisions; that is, 
is 𝑥1 = 𝑥2. The total benefit of the group is then 
𝐵𝑐(𝑥1, 𝑥2) = 𝐵(𝑥1) + 𝐵(𝑥2)
= 𝑝𝑥1 − 𝑥1[1 + (𝑥1 + 𝑥2)] + 𝑝𝑥2 − 𝑥2[1 + (𝑥1 + 𝑥2)] 
 
Replacing 𝑥1 = 𝑥2 in the previous equation, implies that total benefit is 
 
𝐵𝑐(𝑥1, 𝑥2) = 2𝑝𝑥1 − 2𝑥1[1 + (2𝑥1)] 
Optimal harvest is determined so as to maximize total benefits. 
 
 
 
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Max 𝐵𝑐(𝑥1, 𝑥2) = 2𝑝𝑥1 − 2𝑥1[1 + (2𝑥1)] 
First order conditions are 
𝜕𝐵𝑐
𝜕𝑥1
= 2𝑝 − 2 − 8𝑥1
∗ = 0 
Thus 
𝑥1
∗ =
𝑝−1
4
=
13−1
4
= 3 = 𝑥2
∗ . 
𝑥𝑐
∗ = 𝑥1
∗ + 𝑥2
∗ = 2𝑥1
∗ = 6 
 
b. How much profits (𝜋𝑖) do each obtain under cooperation? 
 
Profits of each fisher are 
𝐵(𝑥𝑖
∗) = 𝜋𝑖 = 13 ∙ 3 − 3[1 + (6)] = 18 𝑖 = 1, 2 
 
c. How much would Lisa and Tommy extract if they do not cooperate with each other; 
that is, under free access? 
 
If they do not cooperate, they decide the optimum catch by maximizing their net 
income individually. The first order condition for fisherwoman # 1 is: 
𝜕𝐵𝑐
𝜕𝑥1
= 𝑝 − 1 − 2𝑥1
∗ − 𝑥2 = 0 
Thus 
𝑥1
∗∗ =
𝑝 − 1 − 𝑥2
2
 
Since both individuals are the same 
𝑥2
∗∗ =
𝑝 − 1 − 𝑥1
2
 
 Solving this system of two equations in two unknowns yields: 
 
 
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𝑥1
∗∗ =
𝑝 − 1 − 𝑥2
∗∗
2
= 
𝑝 − 1 −𝑝 − 1 − 𝑥1
∗∗
2
2
=
𝑝 − 1 + 𝑥1
∗∗
4
⇒ 
𝑥1
∗∗ =
𝑝 − 1
3
= 
13 − 1
3
= 4 = 𝑥2
∗∗ 
𝑥𝑐
∗∗ = 𝑥1
∗∗ + 𝑥2
∗∗ = 8 
 
d. How much profits (𝜋𝑖) do each obtain under free access? 
 
Profits under free access are 
𝐵(𝑥𝑖
∗∗) = 𝜋𝑖 = 13 ∙ 4 − 4[1 + (8)] = 16 𝑖 = 1, 2 
 
e. Calculate society’s welfare loss due to free access. 
 
2𝐵(𝑥1
∗) − 2𝐵(𝑥1
∗∗) = 2(18 − 16) = 4 
 
Bonus 
f. What regulatory instrument would you suggest to prevent this welfare loss? 
 
Catch quotas. For economic efficiency tradable catch quotas.