Assignment 3

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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 3 
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, October 11 before 10:00 and another printed copy must be handed 
in by 10:00 on the same date. 
 
Show all your calculations 
 
 
Compensating and Equivalent Variation 
1) Suppose Marceline the Vampire Queen consumes only two commodities: pizza (𝑞𝑝) and beer 
(𝑞𝑏). Furthermore, she always drinks exactly three beers with each pizza. Suppose the price of 
a pizza is $3, the price of a beer is $1 and Irene has $24 to spend in total. 
a) (10 points) How many pizzas and beers will Marceline consume? Graph her equilibrium 
showing her budget constraint and two of her indifference curves. 
 
Irene always consumes 3 beers with each pizza, thus 
(1) 𝑞𝑏 = 3𝑞𝑝 
 
Her budget restriction is 
(2) 𝑝𝑝𝑞𝑝 + 𝑝𝑏𝑞𝑏 = 𝐼. 
 
Substituting (1) in (2) implies 
𝑝𝑝𝑞𝑝 + 𝑝𝑏3𝑞𝑝 = 𝐼 
𝑞𝑝(𝑝𝑝 + 3𝑝𝑏) = 𝐼 
 
 
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 
 
Thus 
(3) 𝑞𝑝 =
𝐼
(𝑝𝑝+3𝑝𝑏)
 
 
And 
(4) 𝑞𝑏 =
3𝐼
(𝑝𝑝+3𝑝𝑏)
 
When I=$24, 𝑝𝑏=$1 and 𝑝𝑝=$3, Marceline’s optimal consumption is 
𝑞𝑝
∗ = 4 
𝑞𝑏
∗ = 12 
 
The graph showing her equilibrium looks as follows: 
 
 
 
 
 
 
 
 
 
 
 
 
 
b) (10 points) Now suppose the government imposes a $3 tax on pizzas so the price of a pizza 
to Marceline rises to $6. How many pizzas and beers will she consume now? Graph her 
new equilibrium. 
 
 
 
 
 
 
 
 
 
 
 
 
 
U1 
24 
12 
qb 
Uo 
𝑞𝑏 = 3𝑞𝑝 
qp 4 8 
8 
24 
8 
qb 
Uo 
𝑞𝑏 = 3𝑞𝑝 
qp 2.67 6 
 
 
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 
 
Replacing the new parameters in (3) and (4) implies that Marceline’s optimal consumption 
is 
𝑞𝑝
∗∗ = 2.67 
𝑞𝑏
∗∗ = 8 
 
c) (5 points) What are the income and substitution effects of the tax in part (2)? 
 
 
 
 
 
 
 
 
 
 
 
 
 
Given that pizza and beer are perfect complements for Marceline, there is no substitution 
between them, thus the substitution effect is 0. The income effect is thus 
𝐼𝑛𝑐𝑜𝑚𝑒 𝑒𝑓𝑓𝑒𝑐𝑡 = 𝑞𝑝
∗ − 𝑞𝑝
∗∗ = 1.33 pizzas. 
 
d) (10 points) Calculate the compensating and equivalent variation (5 points each) of the tax 
in part (2). 
Compensating variation is the income needed for her to maintain constant her original 
utility level at the new prices, less her original income; that is the income needed for her to 
maintain constant her original consumption bundle level at the new prices, less her original 
income 
 
𝐶𝑉 = [(6 ∗ 4) + 12] − 24 = 36 − 24 = 12 
 
Equivalent variation is Marceline’s original income less the income that is needed for her to 
maintain constant her final utility level at the original prices; that is Marceline’s original 
income less the income that is needed for her to maintain constant her final consumption 
bundle level at the original prices 
 
𝐸𝑉 = 24 − [(2.67 ∗ 3) + 8] = 24 − 16.01 = 7.99 
 
 
 
18.6
8 
Uf 
36 
24 
qb 
Uo 
𝑞𝑏 = 3𝑞𝑝 
qp 
 
 
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 
 
 
Travel Cost Valuation 
2) Parque Nacional Conquillio currently charges no admission fee and is visited by 582 people 
each day. A researcher has interviewed a sample of the visitors and has concluded that they 
come from 6 geographic zones. She has collected the following information regarding their 
numbers and their residences (origins): 
 
Zone 
Travel Cost 
($) 
Population 
(N°) 
Per capita 
visitation 
rate 
Visitors 
(N°) 
1 5 300 0.1 30 
3 15 2000 0.06 120 
4 25 8000 0.02 160 
 
a) (5 points) Plot the travel cost as a function of per capita visitation rate on a graph, label the 
axes and the points representing each origin. 
 
 
 
 
b) (10 points) Derive the implicit equation for per capita visitation rate as a function of travel 
cost. 
𝑣 = 𝛼 − 𝛽𝑇𝐶 
Where 𝑣 is the per capita visitation rate, 𝑇𝐶 is travel cost, and 𝛼 and 𝛽 are parameters. 
 
The TC function is 
𝑇𝐶 = 𝑎 − 𝑏𝑣 
𝑏 =
∆𝑇𝐶
∆𝑣
=
(25 − 5)
(0.1 − 0.02)
=
20
0.08
= 250 
𝑇𝐶 = 𝑎 − 250𝑣 
0
5
10
15
20
25
30
0 0,02 0,04 0,06 0,08 0,1 0,12
TC
per capita visitation rate 
 
 
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 
 
Take any pair (𝑣, 𝑇𝐶) and replace in the above equation to obtain 𝑎, for example, (𝑣, 𝑇𝐶) =
(0.02,25) 
𝑇𝐶 = 25 = 𝑎 − 250(0.02) 
𝑎 = 30 
 
Thus 
𝑇𝐶 = 30 − 250𝑣. 
The implicit equation for per capita visitation rate as a function of travel cost is hence 
𝑣 = 𝛼 − 𝛽𝑇𝐶 =
30
250
−
1
250
𝑇𝐶 = 0.12 − 0.004𝑇𝐶 
 
c) (5 points) Derive the per capita visitation rate demand as a function of an entry fee (𝑓)for 
Parque Nacional Conquillio for each origin. That is 
𝑣𝑖 = 𝛼𝑖 − 𝛽(𝑇𝐶𝑖 + 𝑓) 
Origin 1: 
𝑣1 = 0.12 − 0.004(𝑇𝐶1 + 𝑓) = 0.12 − 0.004(5 + 𝑓) = 0.10 − 0.004𝑓 
 
Origin 2: 
𝑣2 = 0.12 − 0.004(𝑇𝐶2 + 𝑓) = 0.12 − 0.004(15 + 𝑓) = 0.06 − 0.004𝑓 
 
Origin 3: 
𝑣3 = 0.12 − 0.004(𝑇𝐶3 + 𝑓) = 0.12 − 0.004(25 + 𝑓) = 0.02 − 0.004𝑓 
 
Where 𝑣𝑖 is the visitation rate from origin i, 𝑇𝐶𝑖 is travel cost of origin i, 𝑓 is the entry fee, 
and 𝛼 and 𝛽 are parameters. 
 
d) (10 points) Calculate the appropriate aggregate visitation demand function for each origin 
v𝑖 = 𝜃𝑖 − 𝛿(𝑓) 
 
The aggregate visitation demand function for each origin is the horizontal sum of the 
individual demand functions. Before adding the individual demand curves horizontally, we 
must convert each demand curve from "per capita demand" to "total demand" for that 
origin. First, we multiply each demand curve by the relevant population of each origin. 
Note that the number of visitors is not the same as the population. The "number of visitors" 
represents the number of visitors for a fee of zero. 
 
Origin 1: 
v1 = (0.10 − 0.004𝑓)300 = 30 − 1.2𝑓 
 
Origin 2: 
v2 = (0.06 − 0.004𝑓)2000 = 120 − 8𝑓 
 
 
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 
 
 
Origin 3: 
v3 = (0.02 − 0.004𝑓)8000 = 160 − 32𝑓 
 
Graphically, aggregate visit demand function for each origin is 
 
 
 
And for the market the aggregate demand is: 
 
⇒ v = v1 + v2 + v3 = {
30 − 1.2𝑓 15 ≤ 𝑓 ≤ 25
150 − 9.2𝑓 5 ≤ 𝑓 ≤ 15
310 − 41.2𝑓 0 ≤ 𝑓 ≤ 5
 
 
 
⇒ 𝑓 = {
25 − 0.833v 0 ≤ v ≤ 12
16.3 − 0.109v 12 ≤ v ≤ 104
7.52 − 0.024v 104 ≤ v ≤ 310
 
 
e) (5 points) Calculate the value of the park (consumer surplus) for all users fromeach origin, 
and the total value for all users (i.e. from all origins). 
 
𝐶𝑆 = 𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑒 
𝐶𝑆 =
(25 − 15)(12)
2
+ (15)(12) +
(15 − 5)(104 − 12)
2
+ (5)(104 − 12) +
(5)(310 − 104)
2
 
 
𝐶𝑆 = 60 + 180 + 460 + 460 + 515 = 1,675 
0
5
10
15
20
25
30
0 50 100 150 200
 
 
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 
 
 
 
 
Production Change Valuation 
3) Suppose that the production of a good, q, depends on water quality as follows 
 
𝑞 = 𝐿0.1𝐾0.4𝐴0.1 
Where 𝑞 is total production in tons, L is labor in hours, K is capital in $, and A is a measure of 
water quality in ppm1. The associated cost function is 
 
𝐶(𝑞, 𝐴) = 0.5𝑞2𝑤𝑙
0.6𝑤𝑘
0.4𝐴−0.2 
 
a) (5 points) Determine the producers supply function of q, considering that product price is 
𝑝𝑞, and input prices are 𝑤𝑙 and 𝑤𝑘. Suppose the producer is a price taker. 
 
Optimal supply is determined by maximizing profits; that is 
 
Max 𝜋 = 𝑝𝑞𝑞 − 𝐶(𝑞, 𝐴) = 𝑝𝑞𝑞 − 0.5𝑞
2𝑤𝑙
0.6𝑤𝑘
0.4𝐴−0.2 
 
The efficiency condition is 
 
𝑝𝑞 = 𝑀𝐶 = 𝑞𝑤𝑙
0.6𝑤𝑘
0.4𝐴−0.2 
 
Where 𝑀𝐶 represents marginal cost. 
 
1 Ppm=parts per million 
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350
f
v
Aggregate Demand
a
b c
d
e
 
 
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 
 
𝑞 = 𝑝𝑞𝑤𝑙
−0.6𝑤𝑘
−0.4𝐴0.2 
 
b) (5 points) Graph your supply function assuming that 𝑤𝑙 = 10, 𝑤𝑘 = 4, and 𝐴𝑜 = 100. 
 
Supply is 
𝑞 = 0.90𝑝𝑞 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
c) (10 points) Estimate the producer’s WTP for an increase in water quality to 𝐴1 = 250, 
assuming an equilibrium price of 𝑝𝑞
𝑒 = 10 
 
The increase in water quality shifts supply to 𝑞 = 1.1 𝑝𝑞. 
 
At an equilibrium price of 10, optimal supplied quantity at original water quality 𝐴𝑜 is 𝑞𝑜
𝑒 = 9. 
When water quality increases to 𝐴1, optimal quantity supplied increases to 𝑞𝑜
𝑒 = 11. 
 
Producer’s WTP for an increase in water quality is 
 
𝑊𝑇𝑃 = 𝑎 =
1
2
(10)(2) = 10 
 
 
d) (5 points) How much is producer’s WTA to forgo the increase in water quality. 
𝑎 
𝑝𝑞
𝑒 
𝑝𝑞 𝑞 = 0.90 𝑝𝑞 
𝑞 
𝑞 = 1.1 𝑝𝑞 
 
 
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 
 
For the case of producer welfare, we know that producer surplus (PS) is an exact measure of 
welfare change and, thus, 
 
𝑊𝑇𝑃 = 𝑃𝑆 = 𝑊𝑇𝐴 
 
WTA is thus 10. 
 
Contingent Valuation Method 
4) The Contingent Valuation Method (CVM) has been criticized due to its potential biases. 
a) (10 points) Identify and explain the most important biases. 
 
The most important biases are 
1. Strategic Bias: This bias occurs when the person that is surveyed responds strategically 
to the valuation question. For example, if the project increases (decreases) her welfare, 
she may respond with a higher (lower) WTP than her true WTP, so as to bias the result 
towards a positive (negative) result. 
2. Starting point bias: This bias presents itself when the valuation question is based on an 
auction format. The final estimated WTP depends on the initial value. 
3. Hypothetical bias: The criticism is that the researcher obtains hypothetical WTP 
estimates due to the hypothetical nature oaf the scenario. In order to minimize this bias 
the scenario must be clearly presented. 
4. Vehicle of payment bias: The proposed payment mechanism (e.g. entrance fees, taxes) 
can influence the person’s answer towards the Bid question. The person might be 
willing to accept the proposed Bid but rejects it because they do not like the payment 
mechanism. 
5. Part-whole bias (embedding bias): The embedding effect, or order bias, presents itself 
when a specific good receives a different valuation when it is valued independently or 
as a component of a good to which it belongs. This effect results from category 
differences (when a good is a component of a larger good), geographic scales (when 
these goods represent different geographic areas or territories), or time scales (when the 
willingness to pay is defined by different temporal payment schemes). 
 
 
b) (10 points) According to the NOAA Panel, how can each of these biases be minimized?2 
 
1. Strategic Bias: This bias occurs when the person that is surveyed responds strategically 
to the valuation question. For example, if the project increases (decreases) her welfare, 
she may respond with a higher (lower) WTP than her true WTP, so as to bias the result 
towards a positive (negative) result. 
 
2 http://www.economia.unimib.it/DATA/moduli/7_6067/materiale/noaa%20report.pdf 
http://www.economia.unimib.it/DATA/moduli/7_6067/materiale/noaa%20report.pdf
 
 
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. Starting point bias: This bias can be minimized by eliciting WTP with closed ended 
valuation questions. 
3. Hypothetical bias: In order to minimize this bias, the scenario must be clearly 
presented, and it should be tested so as to assess if it is clearly understood. 
4. Vehicle of payment bias: This bias can be reduced by proposing an acceptable 
payment vehicle. This may be determined through focus groups or presurvey tests. 
5. Part-whole bias (embedding bias): To reduce this bias the researcher must adopt 
precautions during the survey design and its application, derived from the revision and 
analysis of previous research.