Chapter 7 Exercises
1. The initial cost of constructing a permanent dam (i.e., a dam that is expected to last
forever) is $425 million. The annual net benefits will depend on the amount of rainfall:
$18 million in a “dry” year, $29 million in a “wet” year, and $52 million in a “flood”
year. Meteorological records indicate that over the last 100 years there have been 86
“dry” years, 12 “wet” years, and 2 “flood” years. Assume the annual benefits,
measured in real dollars, begin to accrue at the end of the first year. Using the
meteorological records as a basis for prediction, what are the net benefits of the dam if
the real discount rate is 5 percent?
1. The first step is to calculate the expected value of the annual net benefits:
(.86)($18 million) + (.12)($29 million) + (.02)($52 million)
= $20 million
The second step is to find the present value of the stream of annual net benefits. As the dam
is assumed to be permanent, the formula for the present value of a perpetuity can be used:
PV = ($20 million)/(.05) = $400 million.
The final step is to subtract the cost of construction from the present value of the annual
benefit stream to obtain the overall present value of expected net benefits (PVENB):
PVENB = $400 million - $425 million = -$25 million. Thus, the dam does not pass the net
benefits test.
2. Use several alternative discount rate values to investigate the sensitivity of the present
value of net benefits of the dam in exercise (1) to the assumed value of the real discount
rate.
2. The following table shows the present value of expected net benefits for
different real discount rates:
Real Discount Rate Present Value of Expected Net Benefits
(millions of dollars)
.01 1,575.00
.02 575.00
.03 241.67
.04 75.00
.05 -25.00
.06 -91.67
.07 -139.29
.08 -175.00
, .09 -202.78
.10 -225.00
The "breakeven" value of the discount rate, dBE, can be found by solving for the
rate at which the present value of the stream of expected annual net benefits just equals the
cost of construction:
($20 million)/dBE = $425 million
dBE = .047
Thus, the discount rate would have to be no larger than .047 for the present value of expected
net benefits for the dam to be positive.
3. The prevalence of a disease among a certain population is .40. That is, there is a 40
percent chance that a person randomly selected from the population will have the
disease. An imperfect test that costs $250 is available to help identify those who have
the disease before actual symptoms appear. Those who have the disease have a 90
percent chance of a positive test result; those who do not have the disease have a 5
percent chance of a positive test. Treatment of the disease before the appearance of
symptoms costs $2,000 and inflicts additional costs of $200 on those who do not actually
have the disease. Treatment of the disease after symptoms have appeared costs $10,000.
The government is considering the following possible strategies with
respect to the disease:
S1. Do not test and do not treat early.
S2. Do not test and treat early.
S3. Test and treat early if positive and do not treat early if negative.
Find the treatment/testing strategy that has the lowest expected costs for a
member of the population.
In doing this exercise, the following notation may be helpful: Let D indicate
presence of the disease, ND absence of the disease, T a positive test result, and NT a
negative test result.
Thus, we have the following information:
P(D) = .40, which implies P(ND) = .60
P(T|D) = .90, which implies P(NT|D) = .10
P(T|ND) = .05, which implies P(NT|ND) = .95
This information allows calculation of some other useful probabilities:
P(T) = P(T|D)P(D)+P(T|ND)P(ND) = .39 and P(NT) = .61
P(D|T) = P(T|D)P(D)/P(T) = .92 and P(ND|T) = .08
P(D|NT) = P(NT|D)P(D)/P(NT) = .07 and P(ND|NT) = .93
3. First notice that the strategies being considered by the government are not
exhaustive. For example, one could test and then treat no matter what the result. Obviously,
with costly testing this strategy would be dominated by S2. Similarly, testing and then not
treating no matter what the result would be dominated by S1. All the logical possibilities
could be discovered by displaying a decision tree with chance and decision nodes. In this
problem, only S1, S2, and S3 need to be considered.
, Now, calculate the expected cost of each strategy:
E(cost of S1) = (.4)($10000)+(.6)(0) = $4,000
E(cost of S2) = (.4)($2000)+(.6)($2000 + $200) = $2,120
As the expected cost of strategy S2 is less than the expected cost of strategy S1,
early treatment should be given in the absence of testing. Thus, the best testing strategy, S3,
must have expected costs less than $2,120 to be chosen over not testing.
E(cost of S3) = $250
+ P(T)[P(D|T)($2000)+P(ND|T)($2000+$200)]
+ P(NT)[P(D|NT)($10000)+P(ND|NT)($0)] = $1,436
As S3 has a lower expected cost than either S1 or S2, it is the optimal strategy.
4. In exercise (3) the optimal strategy involved testing. Does testing remain optimal if
the prevalence of the disease in the population is only .05? Does your answer suggest
any general principle?
4. Using the same procedures as in exercise 3:
E(cost of S1) = $500
E(cost of S2) = $2190
E(cost of S3) = $530
If the prevalence of the disease in the relevant population is only .05, the optimal strategy is
S1, which is simply to treat the disease after symptoms appear.
The general point is that the optimal testing strategy depends on the prevalence
of the disease. Thus, if the probability of having the disease is low in the general population,
tests may not be cost-effective for the general population. However, they may be cost-
effective for subsets of the population sharing specific risk factors that give them a higher
probability of having the disease.
5. (Use of a spreadsheet recommended for parts a through e and necessary for part f)
A town with a population of 164,250 persons who live in 39,050 households is
considering introducing a recycling program that would require residents to separate
paper from their household waste so that it can be sold rather than buried in a landfill
like the rest of the town’s waste. Two major benefits are anticipated: revenue from the
sale of waste paper and avoided tipping fees (the fee that the town pays the owners of
landfills to bury its waste). Aside from the capital costs of specialized collection
equipment, household containers, and a sorting facility, the program would involve
higher collection costs, inconvenience costs for households, and disposal costs for paper
that is collected but not sold. The planning period for the project has been set at eight
years, the expected life of the specialized equipment.
The following information has been collected by the town’s sanitation
department:
1. The initial cost of constructing a permanent dam (i.e., a dam that is expected to last
forever) is $425 million. The annual net benefits will depend on the amount of rainfall:
$18 million in a “dry” year, $29 million in a “wet” year, and $52 million in a “flood”
year. Meteorological records indicate that over the last 100 years there have been 86
“dry” years, 12 “wet” years, and 2 “flood” years. Assume the annual benefits,
measured in real dollars, begin to accrue at the end of the first year. Using the
meteorological records as a basis for prediction, what are the net benefits of the dam if
the real discount rate is 5 percent?
1. The first step is to calculate the expected value of the annual net benefits:
(.86)($18 million) + (.12)($29 million) + (.02)($52 million)
= $20 million
The second step is to find the present value of the stream of annual net benefits. As the dam
is assumed to be permanent, the formula for the present value of a perpetuity can be used:
PV = ($20 million)/(.05) = $400 million.
The final step is to subtract the cost of construction from the present value of the annual
benefit stream to obtain the overall present value of expected net benefits (PVENB):
PVENB = $400 million - $425 million = -$25 million. Thus, the dam does not pass the net
benefits test.
2. Use several alternative discount rate values to investigate the sensitivity of the present
value of net benefits of the dam in exercise (1) to the assumed value of the real discount
rate.
2. The following table shows the present value of expected net benefits for
different real discount rates:
Real Discount Rate Present Value of Expected Net Benefits
(millions of dollars)
.01 1,575.00
.02 575.00
.03 241.67
.04 75.00
.05 -25.00
.06 -91.67
.07 -139.29
.08 -175.00
, .09 -202.78
.10 -225.00
The "breakeven" value of the discount rate, dBE, can be found by solving for the
rate at which the present value of the stream of expected annual net benefits just equals the
cost of construction:
($20 million)/dBE = $425 million
dBE = .047
Thus, the discount rate would have to be no larger than .047 for the present value of expected
net benefits for the dam to be positive.
3. The prevalence of a disease among a certain population is .40. That is, there is a 40
percent chance that a person randomly selected from the population will have the
disease. An imperfect test that costs $250 is available to help identify those who have
the disease before actual symptoms appear. Those who have the disease have a 90
percent chance of a positive test result; those who do not have the disease have a 5
percent chance of a positive test. Treatment of the disease before the appearance of
symptoms costs $2,000 and inflicts additional costs of $200 on those who do not actually
have the disease. Treatment of the disease after symptoms have appeared costs $10,000.
The government is considering the following possible strategies with
respect to the disease:
S1. Do not test and do not treat early.
S2. Do not test and treat early.
S3. Test and treat early if positive and do not treat early if negative.
Find the treatment/testing strategy that has the lowest expected costs for a
member of the population.
In doing this exercise, the following notation may be helpful: Let D indicate
presence of the disease, ND absence of the disease, T a positive test result, and NT a
negative test result.
Thus, we have the following information:
P(D) = .40, which implies P(ND) = .60
P(T|D) = .90, which implies P(NT|D) = .10
P(T|ND) = .05, which implies P(NT|ND) = .95
This information allows calculation of some other useful probabilities:
P(T) = P(T|D)P(D)+P(T|ND)P(ND) = .39 and P(NT) = .61
P(D|T) = P(T|D)P(D)/P(T) = .92 and P(ND|T) = .08
P(D|NT) = P(NT|D)P(D)/P(NT) = .07 and P(ND|NT) = .93
3. First notice that the strategies being considered by the government are not
exhaustive. For example, one could test and then treat no matter what the result. Obviously,
with costly testing this strategy would be dominated by S2. Similarly, testing and then not
treating no matter what the result would be dominated by S1. All the logical possibilities
could be discovered by displaying a decision tree with chance and decision nodes. In this
problem, only S1, S2, and S3 need to be considered.
, Now, calculate the expected cost of each strategy:
E(cost of S1) = (.4)($10000)+(.6)(0) = $4,000
E(cost of S2) = (.4)($2000)+(.6)($2000 + $200) = $2,120
As the expected cost of strategy S2 is less than the expected cost of strategy S1,
early treatment should be given in the absence of testing. Thus, the best testing strategy, S3,
must have expected costs less than $2,120 to be chosen over not testing.
E(cost of S3) = $250
+ P(T)[P(D|T)($2000)+P(ND|T)($2000+$200)]
+ P(NT)[P(D|NT)($10000)+P(ND|NT)($0)] = $1,436
As S3 has a lower expected cost than either S1 or S2, it is the optimal strategy.
4. In exercise (3) the optimal strategy involved testing. Does testing remain optimal if
the prevalence of the disease in the population is only .05? Does your answer suggest
any general principle?
4. Using the same procedures as in exercise 3:
E(cost of S1) = $500
E(cost of S2) = $2190
E(cost of S3) = $530
If the prevalence of the disease in the relevant population is only .05, the optimal strategy is
S1, which is simply to treat the disease after symptoms appear.
The general point is that the optimal testing strategy depends on the prevalence
of the disease. Thus, if the probability of having the disease is low in the general population,
tests may not be cost-effective for the general population. However, they may be cost-
effective for subsets of the population sharing specific risk factors that give them a higher
probability of having the disease.
5. (Use of a spreadsheet recommended for parts a through e and necessary for part f)
A town with a population of 164,250 persons who live in 39,050 households is
considering introducing a recycling program that would require residents to separate
paper from their household waste so that it can be sold rather than buried in a landfill
like the rest of the town’s waste. Two major benefits are anticipated: revenue from the
sale of waste paper and avoided tipping fees (the fee that the town pays the owners of
landfills to bury its waste). Aside from the capital costs of specialized collection
equipment, household containers, and a sorting facility, the program would involve
higher collection costs, inconvenience costs for households, and disposal costs for paper
that is collected but not sold. The planning period for the project has been set at eight
years, the expected life of the specialized equipment.
The following information has been collected by the town’s sanitation
department:
