ECON-4210/6210-01 Cost-benefit Analysis Practice problems for final
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ECON-4210/6210-01
Cost-benefit Analysis
Practice problems for final
1. Happy Valley is the only available camping area in Rural County. It is owned by the county, which allows free access to campers. Almost all visitors to Happy Valley come from the six towns in the county.
Rural County is considering leasing Happy Valley for logging, which would require that it be closed to campers. Before approving the lease, the county executive would like to know the magnitude of annual net benefits that campers would forgo ifHappy Valley were to be closed to the public.
Travel costs play an important role in the net benefits ofHappy Valley camping. An analyst for the county has collected data for a travel cost study. On five randomly selected days, he recorded the license plates of vehicles parked overnight in the Happy Valley lot. (As the camping season is 100 days, he assumed that this would constitute a 5 percent sample.) With cooperation from the state motor vehicle department, he was able to find the town ofresidence of the owner of each vehicle. He also observed a sample ofvehicles from which he estimated that each vehicle carried 3.2 persons (1.6 adults), on average. The following table summarizes the data he collected:
Miles from Number of Estimated Number
Happy
Town Valley
Population (thousands)
Vehicles in
Sample
ofVisitors for
Season
Visit Rate (Visits per 1,000 People)
|
50.1 34.9 15.6 89.9 98.3 60.4 |
146 85 22 180 73 25 |
3,893 2,267 587 4,800 1,947 666 |
77.7 65.0 37.6 53.4 19.8 11.0 |
Total 14,160
In order to estimate the correlation between visits to Happy Valley and the admission fee, we need more data points in addition to the current one ($0, 14,160). To find more data points, we need to predict the reduction in the number of campers from each town as price is increased. Suppose the analyst learned about the estimated regression equation between the visit rate (VR) and the travel cost (TC) of a vehicle visit from prior studies. Specifically, the estimated linear regression is as follows (standard errors in parentheses):
VR = 93.14 – 2.88 * TC.
(11.15) (-0.647)
Consider an increase of $10 admission fee. Find out the total number of predicted visits given the $10 admission fee.
(Hints: This is done in three steps at each price: First, use the coefficient of TC from the regression to predict a new VR for each town. The new VR is equal to the original VR plus the
predicted change as a result of the admission fee. Note that, the increase in admission fee is the only change in travel costs in this context; nothing else in travel costs have changed. Second, multiply the predicted VR of each town by its population to get a predicted number ofvisitors. Third, sum the visitors from each town to get the total number of predicted visits. In case the predicted visit rate becomes negative, let predicted visits be zero.)
KEY:
Consider, for example, the impact of a $10 admission fee. The following table summarizes the calculation procedure:
Town New Visit Rate Predicted Visits
A
B
C
D
E
77.7-2.88*10=48.9
65.0-2.88*10=36.2
37.6-2.88*10=8.8
53.4-2.88*10=24.6
19.8-2.88*10=-9.0
2,451
1,264
138
2,213
0
F
Total
11.0-2.88*10=-17.8 0
6,065
Note that a $10 admission fee leads to a prediction of negative visit rates for Towns E and F. As visit rates cannot be negative, we set these predicted visit rates to zero. Summing the predicted number of visits for the towns gives a total 6,065 visits.
2. Consider a government training program that provided low-skilled men job-specific training. The program lasted for one year. To evaluate this program, members of the target population were randomly assigned to either a treatment group that was eligible to receive services under the program or to a comparison group that was not. Using this evaluation design, the following information was obtained:
• Members ofthe treatment group were found to remain in the program for the entire year. During this year, they received no earnings, but were paid a tax-free stipend of $4,000 by the program to help them cover their living expenses. During the program year, the average annual earnings of members of the control group were $10,000, on which they paid taxes of $1,000. During the program year, the welfare and unemployment compensation benefits received by the two groups were virtually identical.
• Program operating costs (not counting the stipend) and the cost of services provided by the program were $3,000 per trainee. These costs were paid by the government, not the trainees.
• During the four years after leaving the program, the average annual earnings ofmembers ofthe treatment group were $20,000, on which they paid taxes of $2,000. During the same period, the
average annual earnings of members of the control group were $15,000, on which they paid taxes of $1,500.
• During the four years after leaving the program, the average annual welfare payments and unemployment compensation benefits received by members of the treatment group were $250. During the same period, the average annual welfare payments and unemployment compensation benefits received by members ofthe control group were $1,250.
Using a 5 percent discount rate and a zero decay rate, compute the present value of the net gain (or loss) from the program from the trainee perspectives. In doing this, ignore program impacts on leisure. Assume that all benefits and costs accrue at the end ofthe year in which they occur.
(Hint: The time horizon in the calculation is five years, one year for the program and four years after the program. Since the treatment and control groups were divided by randomization, the net benefits are simply the differences between the treatment and control groups. Calculate the differences in annual benefits and costs between both groups during the first year (i.e., the program year) and each of the (four) years after the program, respectively. Then compute the net present value across all the five years.)
KEY:
Trainee perspective: Since members of the control group received $9,000 in after tax earnings during the program year, on average, it is reasonable to presume that this was the amount of earnings forgone by an average trainee while undergoing training. However, this cost to trainees was partially offset by the $4,000 stipend they received. Thus, trainees incurred a net cost of $5,000 during the program year. During each ofthe next two years, trainees received after tax earnings of $18,000, while members ofthe control group received after tax earnings of only $13,500. Thus, the program's impact on the after-tax earnings of trainees during each ofthe two post-training follow-up years was $4,500. However, during each ofthese years, members ofthe control group received $1,000 more in transfer benefits than members ofthe treatment group. Hence, the program's net impact on the average income of trainees was $3,500 during each ofthe two follow-up years. Using this information, the present value of the net benefits received by the trainees (PVNBT) can be computed for the five-year time horizon as follows:
PVNBT = -$5,000/(1+.05) + $3,500/(1+.05)^2 + $3,500/(1+.05)^3 + $3,500/(1+.05)^4 +
$3,500/(1+.05)^5 = $7,058
3. A government data processing center has been plagued in recent years by complaints from employees of back pain. Consultants have estimated that upgrading office furniture at a net cost of $425,000 would reduce the incidence and severity of back injuries, allowing the center to avoid medical care that currently costs $68,000 each year. They estimate that the new furniture would also provide yearly benefits of avoided losses in work time and employee comfort worth $18,000. The furniture would have a useful life of five years, after which it would have a positive
salvage value equal to 10 percent of its initial net cost. The consultants made their estimates of avoided costs assuming that they would be treated as occurring at the beginning of each year. In its investment decisions, the center uses a nominal discount rate of 9.5 percent and an assumed general inflation rate of 3 percent.
(a) Suppose medical care prices will rise at the general rate of inflation, should the center purchase the new furniture? (Hint: First calculate the real discounting rate. In this case, the benefits of avoided medical care would grow at the same rate as other benefits.)
(b) Now suppose the inflation rate for medical care is 5 percent, i.e., the cost of medical care will rise 2 percent faster. In this case, should the center purchase the new furniture? (Hint: In this case, the benefits of avoided medical care would grow at another two percent each year, in addition to the general inflation.)
KEY:
(a) Working in real dollars, the first task is to convert the nominal discount rate to a real discount rate:
NPV = (-$425,000 + $68,000 + $18,000)
+ ($68,000 + $18,000)/(1+.063)1
+ ($68,000 + $18,000)/(1+.063)2
+ ($68,000 + $18,000)/(1+.063)3
+ ($68,000 + $18,000)/(1+.063)4
+ ($42,500)/(1+.063)5
= -$339,000 + $80,903 + $76,108 + $71,598 + $67,354 + $31,313
= -$11,724.
(b) If, instead, it is assumed that the price of medical care inflates 2 percent faster than the general price level, then the net benefits are calculated as follows:
NPV = (-$425,000 + $68,000 + $18,000)
+ ($69,360 + $18,000)/(1+.063)1
+ ($70,747 + $18,000)/(1+.063)2
+ ($72,162 + $18,000)/(1+.063)3
+ ($73,605 + $18,000)/(1+.063)4
+ ($42,500)/(1+.063)5
= -$339,000 + $82,183 + $78,539 + $75,063 + $71,744 + $31,313
= -$158
In this case, the purchase would not pass the net benefits test if it were assumed that medical care prices will rise at the general rate of inflation, or if it were assumed medical care prices will rise 2 percent faster than the general rate of inflation. Though medical care prices have appeared to rise substantially faster than other prices in recent years in the United States, some ofthe faster
increase may result because medical care price indexes do not take full account of improvements in the quality of care.
4. A proposed government project in a rural area with 101 unemployed persons would require the hiring of20 workers. The project would offer wages of $12 per hour. Imagine that the reservation wages ofthe 101 unemployed are uniformly distributed between $2 and $20, which suggests that the “shadow” reservation wage will be average ofthe two.
a. Estimate the opportunity cost of the labor required for the project assuming that the government makes random offers to the 101 unemployed until 20 of them accept jobs. Is it larger or smaller than the budgetary outlay?
b. Estimate the opportunity cost of the labor required for the project assuming that the government can identify and hire the 20 unemployed with the lowest reservation wages.
Assume that the reservation wages ofthe one-hundred unemployed are evenly distributed between $2 and $20. In other words, if we rank the 101 workers in ascending order ofthe reservation wages, the first worker has a reservation wage $2, followed by $2.18, $2.36, …, and $20.
KEY:
3.a. Since the offers are randomly made among all the 100 unemployed, we use the “shadow” wage, i.e., the average wage: ($20 + $2 ) / 2 = $11.
3.b. The total average wage between the first 20 workers is $74.2.
5. Use the information in Chapter 4 in the textbook to answer this question. Consider the example presented in Figure 4.3. Compute the annual loss in consumer surplus for the price increase from $1.25 to $1.75.
a. Assume a linear demand curve as per equation (4.7)
KEY:
1.a. The loss in consumer surplus is the sum of the loss to consumers on trips they continue to take ($1.75-$1.25)(13.1 million) = $6.55 million plus the deadweight loss equal to 0.5($1.75-1.25)(14.5 million -13.1 million) = $0.35 million for a total loss of $6.9 million.
1.b. Assuming a constant-elasticity demand curve, the quantity demand only falls to 13.6 million trips per year. The total reduction in social benefits equals the area under the constant elasticity demand curve, which is given by
()1/ b1
q1 - q0 |
p |
where ß0 = 15.2, ß1 = -0.2 and ρ = [1+ 1/ß1] = -4. Using a calculator enables us to find the area under the demand schedule to be $1.34 million. The deadweight loss is thus $1.34 million - ($1.25)(14.5 million-13.6 million) = $0.216 million. The additional cost of the trips that consumer continue to take (13.6 million) times the added cost per trip ($0.5) = $6.8 million. Thus the total loss in consumer surplus is $6.8 million plus $0.216 million or $7.016 million.
2021-12-20