Problem Set 3
Environmental Valuation
1.
Arturo derives utility from a composite good X and indoor air quality, Q such that 𝑈𝑈 = 𝑋𝑋𝑋𝑋.
Indoor air quality depends on pollution levels outside, P, and defensive expenditures, D,
such that 𝐷𝐷 = 𝑃𝑃𝑄𝑄 2 . Arturo’s income is Y and the price of the composite good is equal to 1.
1.1 Write Arturo’s objective along with any constraint(s) that he faces.
1.2 Suppose Y=10 and P=1. What are the optimal amounts of X and Q for Arturo? Show
your work by writing the Lagrangian equation along with the relevant first order
conditions.
1.3 What happens to X and Q if P increases to 2? What happens to X and Q if Y increases to
20 (assuming P is back to 1)?
1.4 It is observed that low income individuals are exposed to more pollution. Is your answer
to part (c) consistent with this observation?
1.1
max 𝑋𝑋𝑋𝑋
𝑋𝑋,𝑄𝑄
1.2
𝑠𝑠. 𝑡𝑡. 𝑌𝑌 = 𝑃𝑃𝑄𝑄 2 + 1𝑋𝑋
ℒ = 𝑋𝑋𝑋𝑋 + 𝜆𝜆( 𝑌𝑌 − 𝑋𝑋 − 𝑃𝑃𝑄𝑄 2 )
Lagrange is
FOCs
𝜕𝜕ℒ
= 𝑄𝑄 − 𝜆𝜆 = 0
𝜕𝜕𝜕𝜕
𝜕𝜕ℒ
= 𝑋𝑋 − 𝜆𝜆2𝑃𝑃𝑃𝑃 = 0
𝜕𝜕𝜕𝜕
𝜕𝜕ℒ
𝜕𝜕𝜕𝜕
= 𝑌𝑌 − 𝑋𝑋 − 𝑃𝑃𝑄𝑄 2
=0
Using the first two FOCs we have the equilibrium, 𝑃𝑃𝑄𝑄 2 = 𝑋𝑋/2. Substitute the value of 𝑃𝑃𝑄𝑄 2 into
the last FOC. We derive, 𝑌𝑌 − 𝑋𝑋 −
𝑋𝑋
2
= 0. So, 𝑋𝑋 ∗ =
2𝑌𝑌
.
3
2
If Y=10, X* is 6.67. Substitute the
expression for X* into the equilibrium condition 𝑃𝑃𝑄𝑄 = 𝑋𝑋/2 and solve for Q*. We find, 𝑄𝑄∗ =
𝑌𝑌
1.3
0.5
�3𝑃𝑃�
. If Y=10 and P=1 we have Q* is 1.826.
If P increases to 2, we find X* still equal to 6.67. But now, Q* is equal to 1.29.
If Y increases to 20, we find X* still equal to 13.33. But now, Q* is equal to 2.58.
1.4
Yes because their demand for Q* is lower which means the exposed level of pollution is higher.
2. A housing developer is deciding to purchase a large piece of land currently used as a park in
order to convert the park into subdivisions. You are tasked by the government to see if the
benefits of selling the land outweigh the benefits from its current use as a park. At the
moment, there is no admission fee charged and 4075 people visit the park daily. You have
interviewed visitors and delineated 6 geographical zones where the visitors originate. The
following table summarizes the travel cost of visitors to the park:
2.1 Using the travel cost method calculate, calculate the demand schedule illustrating the total
number of visitors at different admission fee levels for fees from $1 to $5. Show your
solution for at least one fee level. (Hint: try to find the equation relating travel cost to visitors
per population).
Answer: Need to find visitor equation: V/pop = 0.06-0.01*C. Using this equation and adjusted
levels of travel cost, we can calculate the predicted level of visitors with a fee of $1:
Zone
1
2
3
4
5
6
Total
Travel cost ($)
1+1
2+1
3+1
4+1
5+1
6+1
Population
50000
25000
12500
6000
8000
10000
Visitor/population
0.04
0.03
0.02
0.01
0.00
0.00
Visitors
2000
750
250
60
0
0
3060
So now we get 3060 visitors when an admissions fee of $1 is imposed. Do this for fees from 1 to
6 and we arrive at the following demand schedule:
Price
Total
Visits
0
4075
1
3060
2
2125
3
1250
4
500
5
0
2.2 Given the results from 2.1, you estimate the inverse demand curve as P = 6 – (1/815)*Q
where P is the admission fee and Q is the number of visitors. Using this fact and the
information above calculate the daily value of keeping the land as a park.
You can still get an approximation of surplus by solving for each level visitors per day when the
entrance fees are 0, 1, 2, 3, 4 and 5 . This is what I found:
P
5
4
3
2
1
500 1250 2125 3060 4075
Here, an approximation of the area under this curve is simply:
$4075+$3060+$2125+$1250+$500 = $11,010 per day.
Alternatively, you could have plugged in values of P = 0, 1, 2, 3, etc to get this demand schedule:
Price
Total
Visits
0
4890
1
4075
2
3260
3
2445
4
1630
5
815
6
0
Note that your results will differ slightly depending on which method you use but both methods
are correct.
Here, an approximation of the area under this curve is simply:
$4890+$4075+$3260+$2445+$1630+$815 = $17,115 per day.
2.3 Using an interest rate of 5%, please compute the present value of the benefits the park
delivers from year 1 on (Hint: the present value of an annualized benefit in perpetuity is
annual return divided by the interest rate). You may assume that each year's benefits are
365 times the daily benefit and arrive in a single payment: e.g., the annual value for year
1 is 365 times the daily value and arrives in one payment at t=1. That is, you do not need
to worry about the timing of daily payments within the year.
Answer: You will get: 365 days x $11,010 per day = $4,018,650 per year. So, the annualized
value is: $4,018,650 / 0.05= $80,373,000. Note this is an underestimate since we calculated an
approximation
If you use the provided equation, you get, 365 days x $17,115 per day = $6,246,975 per year. So,
the annualized value is: $6,283,475 / 0.05= $124,939,500. Note this is an underestimate since
we calculated an approximation
1.4 If a developer would be willing to pay $28 million for the land, and conversion would
eliminate the benefits of the park starting in year 1, what should be done? Explain.
Answer: The present value of the land as a park is larger than the value that a developer would
give. Therefore development is not recommended. Our analysis is also an underestimate so we
are undervaluing the land. Note that since we use TCM, we may even be undervaluing the park
because non-use value is not included.
3. The Bighorn Sheep population is dwindling. A group would like to put it in the
endangered species list but they need to attach an economic value to the species. If you
were tasked to derive the value of the Bighorn Sheep, what is the most appropriate
valuation technique given that you have a large amount of budget and sufficient time to
conduct a thorough study? Describe in detail how this valuation technique is conducted
for this particular species. Enumerate the advantages and disadvantages of this technique
for this case.
The best technique to use is Contingent Valuation Method. This technique entails constructing a
hypothetical market situation to obtain willingness to pay of respondents to preserve the Bighorn
Sheep population. A survey needs to be conducted to establish the willingness to pay to increase
the Bighorn Sheep population in a particular locality. The survey may be a mail-in, internet,
telephone or face to face survey. If we have unlimited budget, I recommend a face to face
survey. The survey should contain a few important elements: (1) clear definition of the market
and (2) clear representation of the elicitation method. The market should be defined such that we
are talking clearly about increasing the population of Bighorn Sheep. Information regarding what
Bighorn sheep is, what it looks like and what it does in the ecosystem should be presented. We
need to specify the payment mechanism which I propose to be a monthly contribution to an
organization that is specifically tasked to be a sanctuary and research center to study population
preservation of the sheep. Pictures showing different population clusters during the current time
and what it will be afterwards can be provided so that individuals know what this actually
entails. A referendum choice should be presented to minimize bias. Once the data has been
gathered, determine average WTP for preserving Bighorn Sheep. Then determine the marginal
effect of Bighorn Sheep population on WTP using a regression analysis. After which, use the
estimated regression to determine total consumer surplus from the Bighorn Sheep population.
Main advantage: This technique allows us to capture nonuse value of Bighorn Sheep which other
techniques cannot do.
Disadvantages: Respondents may give hypothetical answers. Embedding bias may also occur if
people have had past preconceived notions of other sheep species. Strategic bias may also occur
if there are respondents who are extremely passionate on this issue one way or the other.