Assignment 4

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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 4 
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 Tueday, October 23 before 10:00 and another printed copy must be handed 
in by 10:00 on the same date. 
 
Show all your calculations 
 
 
Non-renewable Resource Management 
1) (10 points) Explain, with diagrams, why a monopolistic non-renewable resource market is 
biased towards conservation and therefore will increase the ‘life’ of the resource. 
 
 
 
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 
 
 
Under perfect competition, Hotelling’s rule (𝑝(𝑡)) and optimal extraction trajectory (𝑞(𝑡)) can be 
represented in the following graph 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Where 𝐷(𝑞(𝑡)) is the demand function, 𝑘 represents the choke price, 𝑟 the discount rate, 𝑞(𝑡) 
represents the optimal resource extraction path, 𝑝(𝑡) the resource price, 𝑀𝐶(𝑡) and 𝑀𝑈𝐶(𝑡) are 
marginal cost and marginal user cost, respectively, and 𝑇 is the resource’s depletion time. 
 
A monopoly determines optimal extraction when 𝑀𝑅(𝑡) = 𝑀𝐶(𝑡) + 𝑀𝑈𝐶(𝑡); i.e. 𝑞𝑜
𝑚 which is 
lower than 𝑞𝑜, the initial extraction under a competitive market. The monopoly sets initial price 
above that of a competitive market 𝑝𝑜
𝑚 > 𝑝𝑜. Optimal price path changes and starts above the 
initial price path and after a period of time is below the competitive market’s price path, reaching 
the choke price at a 𝑇𝑚 > 𝑇;if the price path grows as the same rate as before, the monopolist 
would leave resource that was not extracted, which would not be optimal. Therefore, the resource is 
depleted at a slower rate and, thus, the monopolist is biased towards conservation and therefore will 
increase the ‘life’ of the resource. 
 
 
 
 
𝑝𝑜 
𝑘 
𝑞𝑜 𝑇 
𝑇 
𝑝(𝑡) 
𝑞(𝑡) 𝑡 
𝑡 
𝑝(𝑡) = 𝑝𝑜𝑒
𝑟𝑡 
𝐷(𝑞(𝑡)) 
𝑀𝐶(𝑡) + 𝑀𝑈𝐶(𝑡) 
𝑀𝑅(𝑡) 
𝑞𝑜
𝑚 
 
 
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 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2) Discuss, with diagrams1, the consequences of the discovery of new copper reserves for 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 Use the graphical representation of solutions to the optimal resource depletion model (Figure 15.3 Perman et al.) 
𝑇𝑚 
𝑝𝑜 
𝑝𝑜 
𝑇 
𝑇 
𝑞𝑜 
𝑝(𝑡) 
𝑞(𝑡) 𝑡 
𝑡 
𝑘 
𝑝(𝑡) = 𝑝𝑜𝑒
𝑟𝑡 
𝐷(𝑞(𝑡)) 
𝑀𝐶(𝑡) + 𝑀𝑈𝐶(𝑡) 
𝑀𝐶(𝑡) + 𝑀𝑈𝐶(𝑡) 
𝑇 
𝑇 
𝑞𝑜 
𝑝(𝑡) 
𝑞(𝑡) 𝑡 
𝑡 
𝑘 
𝑝(𝑡) = 𝑝𝑜𝑒
𝑟𝑡 
𝐷(𝑞(𝑡)) 
𝑞𝑜
𝑚 𝑇𝑚 
 
 
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 
 
 
a) (3 points) The price and output levels for the copper market; 
 
As shown in the graph, an increase in reserves due to discoveries reduces marginal user cost 
and, thus initial price decreases. The rate of growth of prices remains the same but the initial 
price falls. Additionally, initial extraction increases as the marginal user cost has fallen. 
 
b) (2 points) The date of exhaustion of copper reserves. 
 
The time to exhaustion increases as shown in the graph. 
 
c) (5 points) What will be the probable path over time of copper prices if there are frequent 
discoveries of copper? 
 
If there were a sequence of discoveries, prices would show a tendency to decrease as with 
each discovery MUC and 𝑝𝑜 decrease. Graphically: 
 
 
 
 
 
 
 
 
 
 
 
 
 
Forest Resource Management 
3) (10 points) Interpret economically Faustmann’s harvesting rule that determines the optimal 
rotation period, 𝑇: 
 
𝜕𝜋(𝑇)
𝜕𝑇
= 𝑟𝜋(𝑇) + 𝑟𝑉 
 
Where 𝜋(𝑇) represents net profits of harvesting the forest in time T, 𝑉 is the present value of 
profits obtained of all future rotations, and 𝑟 is the discount rate 
 
 
 
𝑝(𝑡) 
𝑡 
 
 
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 
 
 
Faustmann’s harvesting rule is a non-arbitrage condition that states that the optimal harvest 
period is determined when the marginal returns of postponing the harvest (
𝜕𝜋(𝑇)
𝜕𝑇
) are equal to 
the forgone marginal benefits due to marginally postponing harvest (𝑟𝜋(𝑇) + 𝑟𝑉). The forgone 
marginal benefits are the returns that would have been accrued had the forest been harvested 
(𝑟𝜋(𝑇)) and the reduction in the present value of the infinite rotation since all future harvest 
period are marginally postponed ((𝑟𝑉).