Homework 4. Public Projects
1. A city government is considering two types of town-dump sanitary systems, Design
A requires an initial outlay of $400,000, with annual operating and maintenance costs
of $50,000 for the next fifteen years; design B calls for an investment of $300,000,
with annual operating and maintenance costs of $80,000 per year for the next fifteen
years. Fee collections from the residents would be $85,000 per year. The interest rate
is 8%, and no salvage value is associated with either system.
(a) Using the benefit cost ratio, which system should be selected?
(b) If a new design (design C) requiring an initial outlay of $350,000 and annual
operating and maintenance costs of $65,000 is proposed, would your answer in (a)
change?
(a) B/C analysis
Design A:
NPW of investment, I = $400,000
NPW of costs, C' = $50,000 (P/A, 8%, 15) = $427,974
NPW of benefits, B = $85,000 (P/A, 8%, 15) = $727.56
Design B:
NPW of investment, I = $300,000
NPW of costs, C' = $80,000 (P/A, 8%, 15) = $684,758
NPW of benefits, B = $85,000 (P/A, 8%, 15) = $727.56
Incremental analysis :Fee collections in the amount of $85,000 will be the same
for both alternatives. Therefore, we will not be able to compute the classical B/C
ratio. When this happens, we may select the best alternative based on either the
least cost (I+C') or the incremental modified (net) B/C ratio. Using the modified
(net) B/C ratio,
modified B/C: A-B = (BA - BB - (C'A- C'B) ) / (IA - IB)
= (0 - (427,974 - 684,758))/(400,000-300,000) = 2.56
Select Design A
(b)
Incremental analysis (A-C)
modified B/C: A-C = (BA - Bc - (C'A- C'c) ) / (IA - Ic)
= (0 - (427,974 -556,366))/(400,000-350,000) = 2.57 > 1
Select Design A
2. A government river authority is considering modifying a particular stretch of river.
The river is now a wilderness rapid, home to various wild animal and bird species and
the favorite recreational location of fishermen, hikers and canoeists. The problem
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with the river is that during the spring of the year, the town downstream of the river is
often covered with floodwater. The town has potential for growth with new industry,
but there is a shortage of a reliable water supply. Farms in the region grow crops that
do not require much water because of the hot dry summers; however, they could grow
much more valuable crops if irrigation water were available.
Three plans have been proposed. Plan A is to install a small flood control project at a
cost of $5,000,000. This would have the effect of reducing spring flooding, thus
saving those who live and do business in town $1,000,000 per year. This option
would last forever, and annual maintenance cost of the project would be $100,000 per
year.
Plan B is install a larger water reservoir that would store water for urban industrial
use. This would cost $15,000,000. Operating and maintenance costs would be
$300,000 per year. In addition to the flood control benefits of Plan A, the benefits to
the region would be growth in business valued at 1,000,000 in the first year. The
business benefits would increase by $500,000 per year for each of the next ten years.
The business benefit will then be $6,000,000 in the eleventh year. After that time the
annual business benefits will stay constant at $6,000,000 per year. Because of silting
problems the reservoir will become useless for all purposes after 50 years. At that
point operating and maintenance costs and all benefits will terminate.
The third plan would build a much larger reservoir for $30,000,000. The annual
operating costs for this plan are $600,000 per year. In addition to the benefits for the
first two plans, this plan will provide agricultural benefits equal to $2,000,000 per
year. Again after 50 years the reservoir becomes useless after 50 years because of
silting, and all operating and maintenance costs and all benefits will terminate.
If any of the plans are adopted, the natural uses of the river stretch will be lost
forever. A study has determined that these are worth $500,000 per year.
Do a benefit/cost analysis to determine which, if any, of the three plans should be
adopted. The discount rate to be applied for future benefits and cost is 12% per year.
Do an incremental benefit cost analysis. We compute the NPV of the net future
benefits less future costs for an infinite lifetime.
All numbers are in thousands of dollars.
The first increment is the flood control project over the wild river.
The net annual benefits for the first plan over the wild river are 1,000 - 500 - 100
or 400 per year. The annual cost at 12% interest is 5,000*.12 = 600
The B/C ratio is 400/600 = 0.67. Reject this increment.
Using PW, the NPW of benefits less cost: 400/0.12 = 3333.33
The B/C ratio is 3333/5000 = 0.67 (note that the ratio is the same as for NAW.
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The next increment is the urban storage project over the wild river.
The incremental investment of the second plan over the first is 15,000. The PW
of the benefits are
NPW(B) = 2,000(P/A, .12, 50) - 300(P/A, .12, 50) + 500(P/G, .12, 10)
+ 5,000(P/A, .12, 40)(P/F, .12, 10) - 500(1/.12)
NPW(B) = 2,000(8.304) - 300(P/A, .12, 50) + 500(20.254)
+ 5,000(8.244)(0.3220) - 500/.12
NPW(B) = + 10127 +13,273 - 4167 = 33,350
The B/C ratio is 33,350/15,000 = 2.2233
The project is justified.
Compare the third increment ot the second.
The incremental difference in benefits is the 2,000 gain in agricultural productivity
less the increased maintenance costs of 300. This lasts for 50 years.
NPW(C) = 1,700(P/A, .12, 50) = 1700*8.304 = 14116.800
The B/C ratio is 14117/15000 = 0.941. The larger reservoir is not justified.
Select the 15,000 reservoir.
3. The book suggests three ways to compute a benefit-cost ratio.
Benefit-cost ratio = (All Benefits)/(All Costs)
Benefit-cost ratio = (Public Benefits - Public Costs)/ (Government Costs)
Benefit-cost ratio = (Net Public Benefits - Government Maintenance
Costs)/(Government Investment Cost)
a. For a given project, all of these definitions may yield different values. For a given
project, what general true statement can you make about the values obtained by the
three methods?
b. For most government projects, how will an increased value of the interest rate affect
the value of the benefit-cost ratio?
a. If one is greater than 1, they all will be greater than 1.
If one is less than 1, they all will be less than 1.
b. The ratio will go down because most of the benefits are in the future while the
costs are in the present.
4.
a. A government wastewater treatment project has an investment cost of $2,000,000.
We estimate a benefit to the public of $140,000 per year. The project has a very long
life, so we use infinity for an approximation. What is the benefit cost ratio for this
project? Using this measure should we accept or reject it? The government uses an
MARR of 6%,
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The benefit cost ratio is: NPW(Benefits)/NPW(Costs)
= 100(P/A, i, inf.)/2000 = 140(1/.06)/2000
= 2333/2000= 1.167
Annualized: NAW(Benefits)/NAW(Costs) = 100/2000(.06) = 140/120
= 1.167
Should accept the project since Benefit/Cost > 1
b. There are three alternatives for building a bridge over a river. The government has
listed the initial cost of the projects, the PW of benefits and the PW of operating costs
in the table below. The operating cost figure does not include the initial cost. All
economic figures are in millions of dollars. All the projects have the same lives. Use
the benefit cost ratio method to select the best of the three projects.
A
B
C
Initial Cost
19
18
10
Present Worth of Benefits
31
28
20
Present Worth of Operating Costs
7
7
6
All the options have B/C > 1. Must do incremental analysis
Compare B – C: ratio = .889 so reject B-C
Compare A – C: ratio = 1.1 so accept A-C
Choose A.
5. A reservoir project in Texas will be primarily financed by the Federal Government.
The following financial and social data has been gathered concerning the costs and
benefits of the project. For simplicity assume that all costs and benefits are expended
or realized on January 1 for the years given. The dam will begin operations on
January 1, 2009. All costs and benefits are in millions of dollars. Neglect all benefits
after 2029.
INVESTMENT COSTS
Date
2003
2005
2007
2009
Item
Cost
Purchase and Clear Land 23
Build Dam
20
Install Turbines
20
Fill and Begin Operation 3
OPERATING COSTS
Date
2010 - 2029
Item
Operating Cost
Cost per Year
2
BENEFITS
Date
2010 - 2029
Item
Flood Control
Benefit per Year
2
4
2010 - 2029
2015 - 2029
Electric Power
Recreation
5
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Determine the benefit cost ratio as of the opening date, January 1, 2009, using an interest
rate of 6%. Is this a desirable investment?
Solution
2003
2005
2007
2009
Purchase and Clear Land
Build Dam
Install Turbines
Fill and Begin Operation
23
20
20
3
Total I
NPW (2009)
32.63
25.25
22.47
3.00
83.35 (I)
OPERATING COSTS
Date
2010 - 2029
Item
Operating Cost
Cost per Year
2
22.94 (C’)
Benefit per Year
2
5
4
Total B
22.94
57.35
29.03
109.32 (B)
BENEFITS
Date
2010 - 2029
2010 - 2029
2015 - 2029
Item
Flood Control
Electric Power
Recreation
$83.35 Investment
$22.94 Op. Cost
$109.32 Benefits
B/(C’+I)
(B-C’)/I
1.03
1.04
Accept Project
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