AnswerKeyExercise2-Freeaccesscommongoods

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Pontificia Universidad Católica de Chile 
Facultad de Ciencias Económicas y Administrativas 
Instituto de Economía 
 
EAE 295/AGE 320 
 
ENVIRONMENTAL AND NATURAL RESOURCE ECONOMICS 
 
Free Access Common Good Exercise 
 
 
Consider 2 firms (1 and 2), each one sells its production in two markets A and B. Both 
companies have exactly the same technology, that is, the same total production cost 
𝑐(𝑞𝑖) = 𝐾 + 𝐴𝑞𝑖
2 𝑖 = 1, 2. 
where 𝑞𝑖 is the total amount produced by company i. The quantity sold in market A is 
represented by 𝑦𝑖 and the quantity sold in market B is represented by 𝑥𝑖. Thus 
𝑞𝑖 = 𝑦𝑖 + 𝑥𝑖 𝑖 = 1, 2 
In market A the product price is 𝑝𝐴, constant, independent of the volume of products that 
are sold. On the other hand, in B market, companies sell in a "pool" - as a group - receiving 
a common price that decreases with total sales of production: 
𝑝𝐵 = 𝐵 − 𝐷𝑥𝑇 
where 𝑥𝑇 = 𝑥1 + 𝑥2, is the total sales amount of both firms in the B market. 
The total net benefit of a single company, in terms of 𝑥𝑖 and 𝑞𝑖 (𝑦𝑖 = 𝑞𝑖 − 𝑥𝑖) is 
𝐵(𝑥𝑖) = 𝑝𝐴(𝑞𝑖 − 𝑥𝑖) + [𝐵 − 𝐷(𝑥1 + 𝑥2)]𝑥𝑖 − 𝐾 − 𝐴𝑞𝑖
2 𝑖 = 1, 2 
a. Determine the optimal production allocation that maximizes the group's net income. 
The two companies are equal; hence both have the same optimal decisions; that is, 𝑞1 =
𝑞2 and 𝑥1 = 𝑥2. Total group net benefit, 𝐵𝑔(𝑥1, 𝑥2), is thus 
𝐵𝑔(𝑥1, 𝑥2) = 𝐵(𝑥1) + 𝐵(𝑥2)
= 𝑝𝐴(𝑞1 − 𝑥1) + 𝑝𝐴(𝑞2 − 𝑥2) + [𝐵 − 𝐷(𝑥1 + 𝑥2)]𝑥1
+ [𝐵 − 𝐷(𝑥1 + 𝑥2)]𝑥2 − 𝐾 − 𝐴𝑞1
2 − 𝐾 − 𝐴𝑞2
2 
 
𝐵𝑔(𝑥1, 𝑥2) = 2𝑝𝐴(𝑞1 − 𝑥1) + [𝐵 − 𝐷(2𝑥1)]2𝑥1 − 2𝐾 − 2𝐴𝑞1
2 
The optimum production allocation values are determined by the following first order 
conditions: 
𝜕𝐵𝑔
𝜕𝑞1
= 2𝑝𝐴 − 4𝐴𝑞1
∗ = 0 
𝜕𝐵𝑔
𝜕𝑥1
= −2𝑝𝐴 + 2(𝐵 − 2𝐷𝑥1
∗) − 4𝐷𝑥1
∗ = 0 
Therefore, 
𝑞1
∗ =
𝑝𝐴
2𝐴
 𝑥1
∗ =
𝐵 − 𝑝𝐴
4𝐷
 
and 
𝑞𝑇
∗ = 𝑞1
∗ + 𝑞2
∗ = 2𝑞1
∗ =
𝑝𝐴
𝐴
 
𝑥𝑇
∗ = 𝑥1
∗ + 𝑥2
∗ = 2𝑥1
∗ =
𝐵 − 𝑝𝐴
2𝐷
 
 
b. Calculate the optimal production allocation of each company if they not coordinate. 
In this case, each firm maximizes their own net benefit; that is 
𝐵𝑖(𝑥𝑖) = 𝑝𝐴(𝑞𝑖 − 𝑥𝑖) + [𝐵 − 𝐷(𝑥1 + 𝑥2)]𝑥𝑖 − 𝐾 − 𝐴𝑞𝑖
2 𝑖 = 1,2 
First order conditions for firm 1 are 
𝜕𝐵1
𝜕𝑞1
= 𝑝𝐴 − 2𝐴𝑞1
∗∗ = 0 
𝜕𝐵1
𝜕𝑥1
= −𝑝𝐴 + (𝐵 − 𝐷(𝑥1
∗∗ + 𝑥2)) − 𝐷𝑥1
∗∗ = 0 
Therefore, 
𝑞1
∗∗ =
𝑝𝐴
2𝐴
 (𝑛𝑜 𝑐ℎ𝑎𝑛𝑔𝑒!) 𝑥1
∗∗ =
𝐵 − 𝑝𝐴 − 𝐷𝑥2
2𝐷
 
And because the two companies are equal 
𝑥1
∗∗ =
𝐵 − 𝑝𝐴 − 𝐷𝑥1
∗∗
2𝐷
⇒ 𝑥1
∗∗ =
𝐵 − 𝑝𝐴
3𝐷
 
𝑞𝑇
∗∗ = 𝑞1
∗∗ + 𝑞2
∗∗ = 2𝑞1
∗∗ =
𝑝𝐴
𝐴
 
𝑥𝑇
∗∗ = 𝑥1
∗∗ + 𝑥2
∗∗ = 2𝑥1
∗∗ =
2
3
(
𝐵 − 𝑝𝐴
𝐷
) 
 
c. How much is society’s welfare loss? Consider 𝑝𝐴 = 100, 𝐴 = 2, 𝐵 = 190, 𝐷 = 2, 
and 𝐾 = 20. 
Under coordination 
𝑞𝑖
∗ =
𝑝𝐴
2𝐴
=
100
4
= 25 
𝑥𝑖
∗ =
𝐵 − 𝑝𝐴
4𝐷
=
190 − 100
8
= 11,25 
𝑦𝑖
∗ = 25 − 11,25 = 13,75 
Thus 
𝐵𝑖
∗ = 100(13,75) + (190 − 2(2(11,25))) 11,25 − 20 − 2(252) = 1736,25 
 
 
Under no coordination 
𝑞𝑖
∗∗ =
𝑝𝐴
2𝐴
=
100
4
= 25 
𝑥𝑖
∗∗ =
𝐵 − 𝑝𝐴
4𝐷
=
190 − 100
8
= 15 
𝑦𝑖
∗ = 25 − 15 = 10 
Thus 
𝐵𝑖
∗∗ = 100(10) + (190 − 2(2(15))) 15 − 20 − 2(252) = 1680 
Welfare loss for each firm is 
∆𝐵𝑖 = −56,25 
∆𝑊 = −112,5