AnswerKeyExercise1-Publicgoods

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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 
 
Public Good Exercise 
 
 
Suppose that the demand function of each individual for fresh water quality (which has 
characteristics of non-excludability and non-rivalry), in Clearwater City is: 
 
𝑝𝑑 = 45 − 3𝑞𝑑 
 
While social marginal costs of supplying the environmental good are: 
 
𝑝𝑠 =
5
8
𝑞𝑠 − 3000 
 
(a) If total population is 1500, find the aggregate demand function for this public 
good. 
 
Total social marginal benefits (aggregate demand) are 
 
𝑝𝑑 = (45 − 3𝑞𝑑)1500 
 
𝑝𝑑 = 67500 − 4500𝑞𝑑 
 
(b) Determine the socially optimal provision of this public good. 
 
Socially optimal provision of this public good is where social marginal benefits = 
social marginal costs, that is 
 
67500 − 4500𝑞∗ =
5
8
𝑞∗ − 3000 
𝑞∗ = 15,66 
 
 
Note that 
 
𝑝𝑠 = 𝑝𝑑 = −2990,21 
 
Graphically we can see that 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Therefore, each individual would not consume 15,66 since marginal benefits are 
negative. 
 
The maximum consumption would be where 
 
𝑝𝑑 = 67500 − 4500𝑞𝑑 = 0 
 
𝑞𝑑 = 15 
 
Thus optimal consumption is 𝑞𝑑 = 15. 
 
(c) Determine the consumer surplus, the producer surplus and Net Annual Benefits. 
 
Consumer surplus is the area under demand up to the optimal consumption (lets 
consider 15). In this case area a in the above graph. 
 
𝐶𝑆 = 𝑎 = 
(67500)(15)
2
= 506250 
 
There is no producer surplus since a public good is not offered in a market setting. 
 
Net Annual Benefits (NAB) will be 
 
𝑁𝐴𝐵 = 𝐶𝑆 − 𝑐(𝑞∗) 
 
In this case the cost of supplying a quantity of 15 units of the public good is 
 
q 
p 
15,66 
−2990,21 
a 
67500 
Optimal consumption = 15 
𝑐(𝑞∗) =
5
8
𝑞∗ − 3000 = −2990,625 
 
Which we know cannot be. (This occurred because of a mistake in the problem 
design). 
 
𝑁𝐴𝐵 = 𝐶𝑆 − 𝑐(𝑞∗) = 36740,627 
 
This makes no economic sense since NAB > NB = CS 
 
Because of a neighboring city’s overpopulation, the government is evaluating a 
migration policy, which consists in increasing Clearwater City’s population by 30%. 
 
 
(d) Analyze the new socially optimal provision of the good, given this migration 
policy (Use a graph). 
 
This policy leads to a higher aggregate demand function: 
 
𝑝𝑑 = (45 − 3𝑞𝑑){(1500)1,3} = 87750 − 5850𝑞𝑑 
 
In principle 𝑞∗∗ is determined such that social MB = social MC; i.e. 
 
87750 − 5850𝑞∗∗ =
5
8
𝑞∗ − 3000 
𝑞∗∗ = 15,51 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
However, due to the error in the problem, optimal consumption would be 
 
𝑝𝑑 = 87750 − 5850𝑞∗∗ = 0 
q 
p 
15,66 
−2990,21 
a 
67500 
87750 
15,51 
 
𝑞∗∗ = 15 
 
 
(e) (10 points) Determine the new consumer surplus, producer surplus and Net 
Annual Benefits. 
 
New CS is 
𝐶𝑆 =
(87750)(15)
2
= 658125 
 
As before, PS = 0. We will not calculate net benefit due to the error.