John F. Kennedy School of Government
Prof. Robert N. Stavins
Harvard University
API-135/Econ 1661
PROBLEM SET #1
SOLUTIONS
Problem 1: Short Answer Questions
a) Compare and contrast Pareto Efficiency and the Kaldor-Hicks criterion.
1. Pareto Efficiency is achieved under any scenario in which we could not make any
person better off without making someone else worse off. While a useful concept, it is
nearly impossible to use in practice, as almost all policies or changes to welfare make
some people better off and some worse off.
2. The Kaldor-Hicks criterion is a criterion to improve social welfare that we can use in
practice. Since some people will be made better and some will be made worse off by
any policy, this criterion seeks to find the policy that has the highest net benefit
(benefit – cost). It is assumed that the “winners” can then compensate the “losers.”
3. Note that Kaldor-Hicks is required to achieve Pareto Efficiency.
4. Also note that Kaldor-Hicks is the reason why cost-benefit analysis is used for public
policy.
b) Name two problems with benefit-cost ratios.
1. Scale is ignored: the ratio causes all projects that have, for example, double the
benefits as costs to look the same, regardless of how large the net benefit (benefit –
cost) is to society.
2. Benefit-cost ratios are not invariant to shifting of costs and benefits: If we re-classify
costs and negative benefits and look at the NPV of a project, the NPV will not change.
However, the benefit-cost ratio will change. This causes the ratios to be subject to
manipulation.
3. Note that benefit-cost ratios are fine for determining if a single project passes KaldorHicks (by looking at whether B/C > 1). But, they are not adequate for choosing
between projects for the reasons above.
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Problem 2: Maximizing Net Benefits
There are important trade-offs involved in granting "Wild and Scenic River Status" to portions of
a river. How much of this public good, a free-flowing river, should be protected from further
development? As an analyst in the Office of Policy Analysis of the U.S. Department of the
Interior, you are called upon to make a recommendation. Each year, 1,000 people benefit from
the river's various services. A contingent valuation survey carried out by your office has
estimated that each beneficiary has the same demand function for river preservation,
Q = 30 - (0.25)(P)
where P is the price-per-mile which persons are willing to pay (per year) for Q miles of river
preserved. You find that the marginal (opportunity) cost of preservation is $30,000 per mile per
year. [Hint: You need to derive the market (aggregate) demand curve for a public good.]
River preservation is a pure public good and usually will not be allocated efficiently by private
markets. Pure public goods are non-rival, meaning additional consumption does not diminish
the use or enjoyment of others (i.e. the marginal cost of an additional consumer is zero). Pure
public goods are also non-excludable, meaning that consumer access to the good cannot be
denied (i.e., to those who do not pay).
The efficiency rule for a public good is that the marginal costs to society of providing the good
should equal the sum of the marginal benefits for all consumers (the marginal social benefit). To
calculate the marginal social benefit function, we must add up the marginal benefit – or
willingness to pay – of all individual consumers. This is done by vertically summing up the
individual demand curves to get an aggregate demand curve. Note: Recall that for a private good,
aggregate demand is calculated by the horizontal summation of individual demand curves.
a)
How many miles of the river would be preserved in an efficient allocation?
To aggregate the demand curves vertically, first express the individual demand functions
in terms of P; this is the "inverse demand function":
Q = 30 - 0.25P
0.25P = 30 – Q
P = 120 - 4Q
Now multiply the right hand side by 1000, the number of people who benefit from the
River's various services to derive the aggregate marginal benefit curve for the public good:
P = 1,000 [120 - 4Q]
P = 120,000 - 4,000Q
This has the effect of adding prices (which here represent benefits) at given quantities (i.e.,
a vertical summation of prices at a given quantity). Had we been dealing with a private
good, we would have added quantities at given prices, which is to say horizontally summed
the individual demand curves.
The efficient level of the public good that should be provided can now be found by equating
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the marginal cost to price (MC = P). The marginal cost in the problem is $30,000 per mile
per year, so the efficient level of the public good can be found as follows:
30,000 = 120,000 - 4,000Q
4,000 Q = 90,000
Q = 22.5 river miles per year
b)
What is the magnitude of the total (gross), annual benefits associated with this (efficient
allocation) policy?
Total (gross) annual benefits are equal to the area under the demand curve up to Q = 22.5:
TB = (22.5) (30,000) + (0.5) (22.5) (120,000 30,000)
= 675,000 + 1,012,500
TB = 1,687,500
c)
What are the total, annual costs of the policy?
Total annual costs equal the area under the MC curve up to Q = 22.5 miles:
TC = (22.5) (30,000)
TC = 675,000
d)
What is the magnitude of the total (annual) consumers' surplus?
Assuming that those who benefit from the river’s recreational opportunities are not paying
for the right of access, the public in general incurs the opportunity cost of preservation
(foregone electricity generation, for example). Putting aside the small share of this broadly
distributed opportunity cost that is incurred by those who directly benefit from the river’s
recreational opportunities, the total annual consumers surplus is equivalent in this case to
total annual gross benefits, as in part (b) of this question. Remember that the effective
marginal cost to each user of a public good is zero. In this case, MC = $30,000 is not
borne directly by the users and therefore does not affect their consumer surplus. The
general public may be said to be subsidizing the river’s users. But that subsidy is a
transfer, and does not become part of the net benefit calculation in part (e) below.
CS = (22.5) (30,000) + (0.5) (22.5) (120,000 - 30,000)
CS = 1,687,500
e)
How large are net, annual benefits?
Net annual benefits = TB - TC
NB = 1,687,500 – 675,000
NB = 1,012,000
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f)
If it turns out that the marginal cost of preservation is only $20,000 per mile per year, how
many miles of the river would be preserved in an efficient allocation?
If the marginal cost of preservation is only $20,000 per mile and demand remains
unchanged, the efficient preservation level can be found by setting MC=P:
20,000 = 120,000 - 4,000Q
4,000Q = 100,000
Q = 25 river miles per year
g)
Now assume substitute sites are available to beneficiaries, so their demands are
substantially more elastic: their individual demand functions for river preservation are
Q = 30 - (0.5)(P)
In this case, with the original marginal costs of preservation of $30,000 per mile per year,
how many miles of the river would be preserved in an efficient allocation?
If demand were found to be substantially more elastic, due to the availability of
substitutes, we calculate the market aggregate demand curve from the inverse demand
curve as in part (a):
Q = 30 - 0.5P
0.5P = 30 – Q
P = 60 – 2Q
Aggregate demand is given by multiplying this by 1,000 (the number of individuals):
P = 1,000 [60 - 2Q]
P = 60,000 – 2,000Q
The efficient level of the public good that should be provided can now be found by equating
the marginal cost to marginal benefits, which here is given by price (MC = P). The marginal
cost in the problem is $30,000 per mile per year, so the efficient level of the public good
can be found as follows:
30,000 = 60,000 – 2,000Q
2,000Q = 30,000
Q = 15 river miles per year
This result highlights how estimates of demand have important implications for our
resulting estimates of the efficient allocation.
z
4
P
Graphical Representation of Problem 2
P
g) 15
a) 22.5
5
f) 25
30
Q
Problem 3: Sensitivity of NPV to Discount Rates
Consider a household in Cambridge that chooses to invest in solar panels. The solar panels cost a
one-time amount of $10,000 up front, and the benefit of the panels is $1,000 at the end of each
year. Assume a lifetime of 18 years for the solar panels.
This problem can be solved by plugging in the initial capital cost and future stream of benefits in
the NPV formula from class:
NPV = – $10,000/(1+r)0
= – $10,000/1
= – $10,000
+
+
+
$1000/(1+r)1
$1000/(1+r)1
$1000/(1+r)1
+
+
+
$1000/(1+r)2
$1000/(1+r)2
$1000/(1+r)2
+…
+…
+…
a) Is this investment NPV positive under a 3% discount rate? Under a 7% discount rate?
Note: You may use computer software for your calculation. If you do, please print out the NPV
of each year’s annual benefit under each discount rate (i.e., two columns of NPVs). Please be
sure to state any assumptions you make.
Under a discount rate of 3%, this investment will become NPV positive in year 15. This
investment therefore has a positive NPV based on a solar panel lifetime of 18 years.
Under a discount rate of 7%, this investment will become NPV positive in year 20. This
investment does not a positive NPV based on a solar panel lifetime of 18 years.
This problem highlights the sensitivity of NPV analysis to the choice of discount rate. Results
of NPV analysis are also sensitive to assumptions about when precisely the costs and benefits
are realized (e.g., at the beginning or end of a given year), although usually to a lesser extent.
When in doubt, it is important to clearly state any assumptions you make about when these
events occur.
b) The household invests in solar panels but then decides to move out five years later. Does this
affect the NPV analysis? Why or why not? Please state your assumptions. Hint: You do not
need to complete another quantitative analysis, and there is more than one right answer.
With perfectly competitive real estate markets, the household’s decision to move soon after
installing the solar panels will not affect the NPV analysis: in a perfectly competitive real estate
market, the sale price of the house will incorporate the discounted stream of benefits from the
solar panels that have not yet been realized. In reality, real estate markets are unlikely to be
perfectly competitive for a number of reasons, so the sales price may include less (or more)
than the full discounted stream of unrealized benefits. In this case, information on the expected
valuation of the solar panels at the time of sale should be included in the NPV analysis.
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Problem 4: Benefit Estimation
A survey of all recreational fishermen using Lake Chutzik in the Adirondacks (upstate New York)
during the summer of 1990 produced the following information regarding their numbers and their
residences (origins):
Origin
Per Capita Visitation Rate
Total Travel Cost
Number of Users
#1
0.035
$ 30.00
875
#2
0.025
$ 50.00
3,000
#3
0.020
$ 60.00
2,000
TOTAL
5, 875
a)
Plot the participation function on a graph, labeling the axes and the points representing
each origin.
The relevant comparison is per capita visitation rate and the total travel cost, not the
number of users and the travel cost. Demand curves for each origin (or a given population)
can be derived only after observing the general rate at which people participate (or travel),
not on the basis of observed travel expenditures. The equation of the participation function,
plotted below, can be derived from any two sets of points representing the origins:
Slope = (50 - 60) / (0.025 - 0.020) = -2000
Plugging this into the “point-slope” formula and rearranging terms:
TC - 60
TC - 60
TC - 100
Q
=
=
=
=
-2000 (Q + 0.020)
-2000 Q + 40
-2000 Q
0.05 - 0.0005 TC
TC
120
100
80
Origin #3
60
Origin #2 Origin #3
40
Origin #1
20
0
0
0.01
0.02
0.03
0.04
Participation Rate
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0.05
0.06
b)
On a second graph, draw the related demand functions (for each origin) for recreational
fishing services from the lake. Label the axes and the demand functions.
Using the participation function one can now substitute the visitation rate from each origin
(or zone) into this function to estimate the per capita demand curves. The visitation rate
from each zone becomes the constant in a linear demand curve. In doing this you are trying
to answer the question: How many will participate from each zone at various fees?
In the case of zone 1, with a total travel cost of $30.00 the demand equation is:
Q
=
=
=
=
0.05 - 0.0005 (TC + Fee)
0.05 - 0.0005 (30 + Fee)
0.05 - 0.015 - 0.0005 Fee
0.035 - 0.0005 Fee
Changing the fee gives the following schedule:
Fee
Demand
0
0.035
20
0.025
40
0.015
70
0
This function (on which the points above lie) is the per capital demand function for zone 1.
Note that when the fee is zero the visitation rate is 0.035 – precisely the visitation rate we
observed in the table above when there was in fact no fee charged.
The per capita demand curves for the other two origins can be found in a similar manner. In
the case of zone 2 with a total travel cost of $50.00 the demand equation is:
Q = 0.05 - 0.0005 (TC + Fee)
= 0.05 - 0.0005 (50 + Fee)
= 0.05 - 0.025 - 0.0005 Fee
= 0.025 - 0.0005 Fee
In the case of zone 3, with a total travel cost of $60.00 the demand equation is:
Q = 0.05 - 0.0005 (TC + Fee)
= 0.05 - 0.0005 (60 + Fee)
= 0.05 - 0.030 - 0.0005 Fee
= 0.020 - 0.0005 Fee
The per capita demand curves are plotted as follows:
8
80
70
Preice/Fee
60
D1
50
40
D2
30
20
D3
10
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Quantity
c)
On the second graph, draw the appropriate aggregate (market) demand function.
The aggregate total demand function for fishing at Lake Chutzik is shown below. Before
adding the individual demand curves horizontally, we must convert each demand curve from
"per capital demand" to "total demand" for that origin. You might have done this in either of
two ways.
METHOD 1:
First, you may have multiplied each demand curve by the relevant population of each origin.
This is tricky. Note that the number of users is not the same as the population. The
"number of users" represents the number of users for a fee of zero (number of users is
synonymous here with number of visits). So to multiply through by the population, you
must find the population of each origin by dividing the "number of users" from each origin
by the per capita visitation rate:
{Note that: # people = (# visits) / (# visits/person) = (# visits) / (p.c. visitation rate) }
Origin 1
Origin 2
Origin 3
875 / 0.035 = 25,000
3,000 / 0.025 = 120,000
2,000 / 0.020 = 100,000
Total Demand Functions are then found by multiplying the individual per capital demand
functions through by the population of each respective origin:
Origin 1
D = 25,000 [0.035 - 0.0005 Fee]
= 875 - 12.5 Fee
Origin 2
D = 120,000 [0.025 - 0.0005 Fee]
= 3000 - 60 Fee
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Origin 3
D = 100,000 [0.020 - 0.0005] Fee
= 2000 - 50 Fee
METHOD 2:
Alternatively, you might have estimated the individual demand curves from two points that
you knew for each curve. From the per capita demand curves, you knew that if the
maximum fee was charged, there would be no demand. This gave you one point for each of
the three origins:
Origin 1
Origin 2
Origin 3
(Demand, Fee)
(0,70)
(0,50)
(0,30)
If you correctly interpreted the "number of users" as the number of users from an origin if the
fee was zero, which is somewhat intuitive, because there is no fee on recreational fishing –
that’s why we're interested in this problem – then you have a second point for each of the
three origins:
Origin 1
Origin 2
Origin 3
(Demand, Fee)
(875,0)
(3,000,0)
(2,000,0)
Two points for each origin allows you to draw the appropriate total demand curve for each
origin. [You should convince yourself that these two methods arrive at the same answer.]
Now, using either of these two methods we can graph the aggregate demand curve:
Summing horizontally – think about why "horizontally" (we have created a private good) – to
get the Aggregate Demand Curve:
 Curve starts at demand = 0, price = 70; curve follows demand from origin 1 to P = 50,
where origin 2 starts kicking in;
 First kink is at P = 50; Aggregate Demand = 250
 Second kink is at P = 30; Aggregate Demand = 1700 (origin 1 + origin 2)
 At P = 0, Aggregate Demand = 5875 (add the demand from each origin for P = 0)
875 + 3000 + 2000 = 5875
(Check the "number of users" column on the problem sheet)
For those of you who aggregated the per capita demand curves or those who multiplied
through by the "number of users," rather than the population, check carefully through these
calculations. Professor Stavins' example in class had equal populations in each origin and so
he was able to simply rescale the X-axis without changing the relative positions of the 3
origin demand curves.
10
80
70
60
Price/Fee
50
40
30
20
Aggregate Demand Curve
D2
D3
10
D1
0
0
1000
2000
3000
4000
5000
6000
7000
Q
d)
Calculate the consumer surplus for all users from each origin, and the total consumer
surplus for all users (i.e., from all origins).
Consumer surplus for each origin is simply the area under the total demand curve of each
origin. Recall that there is no price.
Origin 1
= (0.5) (70) (875)
= 30,625
Origin 2
= (0.5) (50) (3000)
= 75,000
Origin 3
= (0.5) (30) (2000)
= 30,000
Total Consumer
Surplus
= 30,625 + 75,000 + 30,000
CS (Total) = 135,625
e)
Assuming that the survey data were accurate and exhaustive, does your grand total of
consumers' surplus in part (d) provide an unbiased assessment of the economic benefits of
the lake for recreational fishing? Explain.
There are several reasons why this survey might not have provided an unbiased assessment of
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economic benefits of the lake for recreational fishing, as measured by consumer surplus.
- Omitted Variable Bias. The omission of key variable from a regression can cause bias
if: (1) the omitted variable is correlated with the included variables (i.e. travel cost), and (2) if
the omitted variable has a statistically significant effect on the dependent variable (visitation
rate). For example, if income, or education were found to be positively correlated with
travel cost, and if they also could be shown to affect the visitation rate, then if they were
omitted, we would systematically overestimate the response of visitation to travel cost.
- Substitute Sites. The existence of substitute sites can in some cases lead to a biased
estimate of CS for the site in question, if the travel cost to visit a substitute site was
correlated with the travel cost to visit the lake.
- Trip length. Variability in the trip length of visitors is likely to cause a lot of variance,
although there could also be some bias. Bias could occur if the length of the trip (i.e., an
extended multi-purpose trip) was correlated with travel cost.
- User Value. The travel cost method only captures user value, and cannot be used to infer
option or existence value. This could cause a downward bias on the estimate of CS.
- Non-Users. The simplest model assumes that non-visitors have zero valuation for the
site; this is probably not correct. Below some threshold level of WTP, there will be
individuals who do not show up; thus, the possibility of "truncation bias" arises.
- Multiple purpose trips. It may be the case that some or all users are not only travelling
to a particular site or region to use the site in question, but also for other uses. For example,
a recreational fishing user of Lake Chutzik also may intend to visit her relatives in Montreal
after fishing. The "travel cost" is therefore potentially confounded.
- Congestion effects. When more people are using the site you yourself may derive less
benefits from fishing, and therefore your opportunity cost could be higher than what we've
measured.
You also might have mentioned the problem of measuring opportunity costs for different users.
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