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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 1 
 
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 Tuesday, August 28 before 10:00 and another printed copy must be handed 
in by 10:00 on the same date. 
 
Show all your calculations 
 
1. (10 puntos) Can we safely say that history has refuted the Malthusian hypothesis? What 
main factors have worked against Malthus’s perspective? How might that perspective 
still be relevant today? 
 
History has proved the simple Malthusian hypothesis wrong: both population and living 
standards in Europe rose rapidly throughout the two centuries following Malthus’s Essay. 
Malthus’ model depends on assumptions about technological progress and feedback 
patterns among the variables in the model. A more optimistic view considers increased 
efficiency, pollution control, and a transition to alternative, more sustainable technologies. 
 
But if we consider a more sophisticated argument, that a growing human population and 
economic system will eventually outrun its biophysical support systems, the debate turns 
out to have strong current relevance. As John Stuart Mill proposed, growth is a race of 
technological change and innovation vrs decreasing yields. 
 
 
 
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. (10 puntos) Explain the importance of the question of the substitution possibilities 
between human-made and natural capital in defining sustainable development. 
 
There are multiple definitions of sustainability. These depend on the assumptions of 
substitution possibilities between human-made and natural capital. If one allows for 
substitution possibilities between human-made and natural capital, then we have weak 
sustainability. Under this definition, sustainability requires that total capital stock should at 
least be constant (clearly it can grow). 
 
However, if we do not allow for substitution possibilities between human-made and natural 
capital, then sustainability id feasible only if natural capital is at least constant over time. 
This is the definition of strong sustainability. 
 
Clearly, this has important implications on policy design. 
 
3. Consider an economy with two goods (𝑥, 𝑦), and two agents - Ann and Bob. Ann and 
Bob wish to trade with one another in order to maximize their individual utilities. 
Suppose Ann is endowed with one unit of 𝑥 and half a unit of 𝑦, i.e. 𝑒𝐴𝑛𝑛 =
(𝑒𝐴𝑛𝑛
𝑥 , 𝑒𝐴𝑛𝑛
𝑦
) = (1,
1
2
) and Bob is endowed with 1 unit of 𝑥 X and 1.5 units of 𝑦, i.e. 
𝑒𝐵𝑜𝑏(𝑒𝐵𝑜𝑏
𝑥 , 𝑒𝐵𝑜𝑏
𝑦
) = (1,
3
2
). Additionally, suppose their utility functions are given by: 
 
𝑈𝐴𝑛𝑛 = 𝑥𝑦 
𝑈𝐵𝑜𝑏 = 𝑦 + 2𝑥 
 
a. (10 puntos) Draw an Edgeworth box indicating the endowment and preferences of 
this problem.1 
 
 
1 Recall marginal utility of 𝑥 is 𝑈𝑥 = 𝜕𝑈 𝜕𝑥⁄ and 𝑅𝐶𝑆𝑥𝑦 = 𝑈𝑥 𝑈𝑦⁄ . 
 
 
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. (15 puntos) Find the set of Pareto Optimal Allocations in this economy and depict 
these in the Edgeworth box. What is this set of points called? 
 
The contract curve is the set of points in which moving away causes, at least one of the 
individuals to be worse off. The efficiency condition states that these points are where 
each individual’s indifference curves are tangent. That is 
 
𝑅𝐶𝑆𝑥𝑦
𝐴𝑛𝑛 = 𝑅𝐶𝑆𝑥𝑦
𝐵𝑜𝑏 
𝑅𝐶𝑆𝑥𝑦
𝐴𝑛𝑛 =
𝑦𝐴𝑛𝑛
𝑥𝐴𝑛𝑛
= 𝑅𝐶𝑆𝑥𝑦
𝐵𝑜𝑏 = 2 
𝑦𝐴𝑛𝑛
𝑥𝐴𝑛𝑛
= 2  𝑦𝐴𝑛𝑛 = 2𝑥𝐴𝑛𝑛 
 
However, this efficiency condition is only valid for 𝑥𝐴𝑛𝑛 ≤ 1,since the endowment is 
such that 𝑥 = 𝑦 = 2. For all 𝑥𝐴𝑛𝑛 > 1, the contract curve is the set of points on the top 
edge of the box. 
 
 
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 
 
 
 
c. (20 puntos) Find the equilibrium consumption of 𝑥 and 𝑦 for Ann and Bob in this 
economy and determine the price ratio that supports this equilibrium. (Hint: Ann or 
Bob’s income is determined by the amount they would have if they sold their entire 
endowment at given prices; ie 𝑝𝑥𝑒𝑖
𝑥 + 𝑝𝑦𝑒𝑖
𝑦
= 𝑝𝑥𝑥𝑖 + 𝑝𝑦𝑦𝑖 ∀ 𝑖 = Ann, Bob. 
Additionally, it is very helpful to normalize one of the prices to 1, let 𝑝𝑥 = 1.) 
Recall that the competitive equilibrium allocation in this economy will satisfy 2 
conditions2: 
 
1. Both Ann and Bob are maximizing their utility subject to their budget 
constraint. 
2. The final allocations of (𝑥, 𝑦) satisfy the resource constraint (i.e. they do not add 
up to more than 2) 
 
For Anne, condition 1 implies: 
 
𝑅𝐶𝑆𝑥𝑦
𝐴𝑛𝑛 =
𝑦𝐴𝑛𝑛
𝑥𝐴𝑛𝑛
=
𝑝𝑥
𝑝𝑦
=
1
𝑝𝑦
 
and 
 
𝑒𝐴𝑛𝑛
𝑥 + 𝑝𝑦𝑒𝐴𝑛𝑛
𝑦
= 𝑥𝐴𝑛𝑛 + 𝑝𝑦𝑦𝐴𝑛𝑛  1 + 𝑝𝑦
1
2
= 𝑥𝐴𝑛𝑛 + 𝑝𝑦𝑦𝐴𝑛𝑛 
 
2 You can also solve for the optimal allocations with restricted optimization, maximizing each one’s 
Langrange function. 
 
 
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 
 
 
Solving these two conditions for Ann leads to 
𝑥𝐴𝑛𝑛 =
2 + 𝑝𝑦
4
 
𝑦𝐴𝑛𝑛 =
2 + 𝑝𝑦
4𝑝𝑦
 
 
For Bob, condition 1 implies: 
 
𝑅𝐶𝑆𝑥𝑦
𝐵𝑜𝑏 = 2 =
𝑝𝑥
𝑝𝑦
=
1
𝑝𝑦
 
and 
 
𝑒𝐵𝑜𝑏
𝑥 + 𝑝𝑦𝑒𝐵𝑜𝑏
𝑦
= 𝑥𝐵𝑜𝑏 + 𝑝𝑦𝑦𝐵𝑜𝑏  1 + 𝑝𝑦
3
2
= 𝑥𝐵𝑜𝑏 + 𝑝𝑦𝑦𝐵𝑜𝑏 
 
These conditions are not sufficient to solve for (𝑥𝐵𝑜𝑏 , 𝑦𝐵𝑜𝑏)3 
 
To solve for optimal consumption and relative prices we use condition 2; i.e. 
 
𝑥𝐴𝑛𝑛 + 𝑥𝐵𝑜𝑏 = 2 
𝑦𝐴𝑛𝑛 + 𝑦𝐵𝑜𝑏 = 2 
 
Thus, the equilibrium consumption of 𝑥 and 𝑦 for Ann and Bob and the price ratio that 
supports this equilibrium is 
𝑝𝑦
𝑝𝑥
=
1
2
 
(𝑥𝐴𝑛𝑛, 𝑦𝐴𝑛𝑛) = (
5
8
,
10
8
) 
(𝑥𝐵𝑜𝑏 , 𝑦𝐵𝑜𝑏) = (
11
8
,
6
8
) 
 
d. (5 puntos) Will this equilibrium allocation be Pareto Efficient? Explain 
Recall that consumption bundles along the contract curve are Pareto Efficient. This 
consumption bundle satisfies 
 
 
3 There is only one equation and two unknowns. 
 
 
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 
 
𝑦𝐴𝑛𝑛 = 2𝑥𝐴𝑛𝑛 
 
and thus, is a Pareto Efficient consumption bundle. 
 
e. ( 10 puntos) Suppose the government decides that the competitive equilibrium is not 
a good allocation and they would prefer for Ann to consume (
3
4
,
3
2
) and Bob to 
consume (
5
4
,
1
2
). 
i. Is this acompetitive equilibrium? 
Yes because it satisfies the contract curve. 
 
ii. Is it attainable from the initial endowment? Why or why not? 
It is not attainable from the original endowment since at equilibrium prices 
(𝑝𝑦 =
1
2⁄ ), Ann does not satisfy her budget constraint; her budget is (
5
4⁄ ) and 
the consumption costs are (6 4⁄ ). 
 
4. Suppose a simple two-period model where we know in period 1 that due to 
technological change, the demand for the resource will decrease in period 2. Hence, 
there are different demand functions for each period. In particular, inverse demand 
functions (marginal benefits) for the two periods are: 
 
𝑝1 = 8 − 0.4𝑞1 
𝑝2 = 6 − 0.4𝑞2 
where 𝑝𝑡 is the price in period t = 1,2 and 𝑞𝑡 is the resource extraction in period t = 
1,2. Consider that the discount rate, , is 
 
𝛿 =
1
1 + 𝑟
 
 
a. (15 puntos) What is the dynamic efficient optimal quantity of resource extraction in 
the two periods? If initial marginal extraction costs are 2 for each time period (ie 
𝑀𝐶𝑡 = 2 ∀ 𝑡 = 1,2), initial endowment is 20, and r = 10%. 
 
 
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 
 
Dynamic efficiency conditions are that the present value of marginal net benefits 
(𝑃𝑉𝑀𝑔𝑁𝐵𝑡) are equal over time
4; i.e. 
 
𝑃𝑉𝑀𝑔𝑁𝐵𝑡 = 𝑃𝑉𝑀𝑔𝑁𝐵𝑡+1 ∀ 𝑡 
Graphically 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Thus, 𝑞𝑡
∗ is determined such that 
 
8 − 0.4𝑞1 − 2 =
6 − 0.4𝑞2 − 2
1 + 0.1
 
and 
𝑞1 + 𝑞2 = 20 
 
6 − 0.4𝑞1 = 3,63 − 0.36𝑞2 
𝑞1 =
2.4 + 0.36𝑞2
0.4
 
Replacing 𝑞1 in the endowment equation implies 
 
2.4 + 0.36𝑞2
0.4
+ 𝑞2 = 20 
1.9𝑞2 = 20 − 6.0 
 
4 This is Jevon’s Equimarginality principle in a dynamic framework. 
𝑀𝑔𝑁𝐵1 
𝑀𝑔𝑁𝐵2
1 + 𝑟
 
O1 O2 𝑞1
∗
 
MUC1 
20 
 
 
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 
 
𝑞2
∗ = 7.3 
𝑞1
∗ = 12.7 
 
Note that marginal user cost in time period 1 is 𝑀𝑈𝐶1 = 2.92. 
 
b. (15 puntos) How does the efficient solution change if the discount rate is greater 
than 10%? Explain (use graphs). 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
When interest rates increase, the present value of net benefits of period 2 decreases, 
that is the preference for present consumption increases and thus 𝑞1
∗′ > 𝑞1
∗ and 𝑞2
∗′ <
𝑞2
∗. 
 
𝑀𝑔𝑁𝐵1 
𝑀𝑔𝑁𝐵2
1 + 𝑟
 
O1 O2 𝑞1
∗
 
MUC1 
20 
𝑀𝑔𝑁𝐵2
1 + 𝑟′
 
𝑞1
∗′

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