Chapter 7: Cost Benefit Analysis
The best way to evaluate a government
policy is through the social welfare
function W=f(U1, U2…)
BUT, this function is difficult to accurately
calculate.
A more practical way to evaluate public
expenditure is through COST-BENEFIT
ANALYSIS.
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Chapter 7: Cost-Benefit Analysis
Present Value Calculations and Alternatives
Public Sector Discount Rate
Calculating Public Costs and Benefits
Cost-Benefit Analysis Errors
Distribution
Uncertainty
Math- Compound Interest
Investment: $100
Interest rate: 2%
Derived Formula:
S = P (1+r)t
S = value after t years
P = principle amount
r = interest rate
t = years
Year
Calc.
1
Amount
100
100.00
2 100*1.02
102.00
3 100*1.022
104.04
4 100*1.023
106.12
5 100*1.024
108.24
Compound Interest Example
The City loans $2 million to a local sports
team, to be paid back in 10 years at 3%
interest. How much will the City get?
S = P (1+r)t
S = $2 million (1+0.03)10
S = $2.69 million
Math - Present Value
How much do I have to invest now to have a
given sum of money in the future?
How much is a future amount worth now?
PV = S/[(1+r)t]
PV = present value (money invested now)
S = sum needed in future
r = real, compound interest rate
t = years
Present Value Example
If the City wins their bid for the Video Games
Olympics (VGO) in 5 years, they will gain
$10 billion in economic activity. What is the
present value of that gain if interest rates
are 4%?
PV = S/[(1+r)t]
PV = $10 billion/[(1+0.04)5]
PV = $8.22 billion
Theory – Discount Rate
Since the future value is “discounted”
according to the interest rate r, r is often
referred to as the DISCOUNT RATE.
Similarly, (1+r)t is often referred to as the
DISCOUNT FACTOR
This PV formula changes if a stream of income
occurs:
Math - Present Value of a stream
If an investment today yields future returns of
Rt, where t is the year of the return, then
the present value becomes:
R1
R2
RT
PV  Ro 


...

2
T
(1  r ) (1  r )
(1  r )
Note that failing to use present values to
discount future benefits can overestimate
the value of a project.
Theory - Inflation
Typically, all prices in an economy change
(typically increase) with inflation.
A $100 DVD player today with 3% inflation
will cost $103 next year. A $50,000 salary
tied to inflation will be $50,150 next year.
NOMINAL VARIABLES are the “sticker
price” variables, the variables/prices actually
faced in that year. (ie: $103 DVD player)
REAL VARIABLES are variables after
inflation has been factored out (ie: $100
DVD player)
Inflation Rule
When doing financial calculations, the general
rule is that
ALL VARIABLES MUST BE IN NOMINAL TERMS
OR ALL VARIABLES MUST BE IN REAL TERMS.
-Therefore if future revenues/costs are in nominal
terms, simply use the going interest rate r
-If future revenues/costs are in REAL terms, the
nominal interest rate must be converted into
the real interest rate through:
rreal  rnom  inflation
Math - Private Firm Evaluation
-In order to evaluate public projects, it is best
to begin with private projects
-If a firm is considering project X with costs
and benefits Ctx and Btx in year t, present
value is calculated:
X
X
X
X
X
X
(
B

C
)
(
B

C
)
(
B

C
X
X
X
1
1
2
2
T
T )
PV  ( B0  C0 ) 

 ... 
2
(1  r )
(1  r )
(1  r )T
Where r is the firm’s rate of return; or its
opportunity cost of funds.
Theory - Present Value Criteria
A firm will undertake projects based on the
following:
1) A firm will only undertake projects whose
PV is positive
2) If given a choice between many mutually
exclusive projects, it will chose the one with
the highest PV
Private Firm Evaluation
Note that r should represent a firm’s accurate
opportunity cost of funds
-a high r favors projects with a quick return
(therefore an incorrectly high r discriminates
against long projects)
-a low r allows for project with a longer return
(therefore an incorrectly low r discriminates
against short projects)
Math - Benefit-Cost Ratio
-One limited way to evaluate projects is using
a cost benefit ratio
-Given a present value of costs and benefits:
X
X
X
C
C
C
2
T
C X  C0X  1 

...

(1  r ) (1  r ) 2
(1  r )T
X
X
X
B
B
B
2
T
B X  B0X  1 

...

(1  r ) (1  r ) 2
(1  r )T
-The BEFEFIT-COST ratio is B/C
Benefit-Cost Ratio
-A project can be undertaken if B/C>1
-This implies that B-C>0, similar to the present
value calculation considered earlier
-The benefit-cost ratio is poor for comparing
between projects, as it doesn’t consider
the amount of PV:
-$3 net PV could have a B/C value of 1.7
while $2,000,000 net PV could have a B/C
value of only 1.1
Theory - Public Sector Discount Rate
In the private sector, the discount rate, r, is
the opportunity cost of funds
-ie: the highest return on another
investment or through the bank
For public projects, the public sector
discount rate is harder to calculate
-Is the project funding through reducing:
1) Private investment or
2) Private consumption
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1) Private Investment Funding
Assume that funding a public project
decreases private investment.
Ie: A new road costs $10,000, but the
private sector would have used that
$10,000 to make $2,000 – 20% return.
In this case the public sector discount rate
is the private sector’s before-tax return.
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2) Private Consumption Funding
Assume that funding a public project
decreases private consumption.
Ie: If the private sector consumes $10,000,
that could have made 20% return, or
$2,000. If tax is 50%, they are effectively
giving up $1,000 or 10% return.
In this case the public sector discount rate
is the private sector’s after-tax return.
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Public Sector Discount Rate Difficulty
Typically, government funding decreases
both private consumption and private
investment.
Ie: If $10,000 funding decreased
consumption and investment by $5,000
each, r = (20%)/2+(10%)/2=15%
-But it is hard to calculate impact on
consumption and investment, and this
impact changes with each tax
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Theory - Social Discount Rate
Some argue that a SOCIAL RATE OF
DISCOUNT is more appropriate, representing
the value SOCIETY places on decreased
consumption.
SOCIAL DISCOUNT RATE is less than market
rate of return for 3 reasons:
1) Concern for future generations
2) Paternalism
3) Market Inefficiency
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1) Concern for Future Generations
One may argue that the private sector has a high
discount rate, heavily discounting benefits to
future generations
-Therefore the social discount rate should be
lower to take the future into account
BUT
-The government is not fully omniscient and
benevolent towards future generations
-Plus projects with long-term benefits should have
long-term profits, expressed in the rate of
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return
2) Paternalism
Even if people are selfish, they may not be farsighted enough to take the future into account
(“I’ll never get sick – I don’t need to save”)
-The government would use the discount rate
people would use if they knew better
-the government is paternalistic – forcing
decreased consumption now in order to save
for the future
BUT
Should public preference be forced upon people?
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3) Market Inefficiency
Often private investment results in positive
externalities.
-These positive externalities are
UNDERPROVIDED
-A lower social discount rate can correct this
inefficiency
BUT
-How big is this externality?
-Would a subsidy make more sense (chapter 5)
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Public Discount Rate Conclusion
-The arguments for a social discount rate instead
of a market discount rate still fail to provide for
a specific discount rate
-Typical discount rates fall between the before-tax
and after-tax private sector rates of return
-The Treasury Board Secretariat (1976)
“recommended the use of a social discount
rate of 10 per cent, and of 5 and 15 percent
for sensitivity analysis”
-If PV is positive with all these rates, a good
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argument exists for doing the project
Theory - Valuing Public Benefits and Costs
Private benefits and costs are easy to
calculate – revenue and expenditures.
Public benefits and costs are harder to calculate
since SOCIAL costs and benefits may not be
reflected in market prices.
Some approaches and considerations are:
1) Market Prices (SMC)
2) Shadow/Adjusted Market Prices (SMC)
3) Consumer/Producer Surplus (SMB)
4) Inferences from Economic Behavior
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5) Valuing Intangibles
1) Market Prices
If the First Fundamental Theorem of Welfare
Economics holds, then
-P=SMC; we have Pareto Efficiency
-In this case market prices can be used for
project valuation
Even if the Theorem fails, market prices are
often used because:
1) Market prices are simple and straightforward
2) Other measures are:
a) Complicated and questionable
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b) Time consuming and costly to calculate
2) Adjusted Market Prices
The SHADOW PRICE of a commodity is its
underlying social marginal cost in an imperfect
market
SHADOW PRICE relates to MARKET PRICE
depending on how the market reacts to
government intervention.
Ie: a) Monopoly
b) Taxes
c) Unemployment
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2a) Monopoly and Shadow Price
In a monopoly, P>MC
i) If use of a monopoly input increases its
production by the full amount,
MC = SHADOW PRICE
ii) If the project consumes inputs that are
taken directly from the private consumer,
PMonopoly = SHADOW PRICE
iii) If production is increased somewhat,
PMonopoly < SHADOW PRICE < MC
(weighted average)
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2b) Tax and Shadow Price
Taxes force Pconsumer>Pproducer
i) If production of the fully expands,
Pproducer = SHADOW PRICE
ii) If the project consumes inputs that are
taken directly from the private consumer,
Pconsumer = SHADOW PRICE
iii) If production increases somewhat,
Pproducer < SHADOW PRICE < Pconsumer
(weighted average)
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2c) Unemployment and Shadow Price
Employment in government projects can
come from other jobs or the unemployed
i) If a worker is hired away from another job,
WagePrivate Sector = SHADOW PRICE
ii) If the worker doesn’t leave another job, things
are more complicated:
-is the worker taking someone else’s job?
WagePrivate Sector = SHADOW PRICE
-would the worker have found a job?
-what is the worker’s opportunity cost of
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leisure?
2c) Unemployment and Shadow Price
-Forcasting future employment is difficult
-Fully understanding unemployment is difficult (in
2011 Alberta had both high vacancies and
higher unemployment due to imperfect job
matches)
-Without a major recession, it is assumed that
WagePrivate Sector = SHADOW PRICE
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3) Consumer/Producer Surplus
Typically, private firms have little effect on the
price of a commodity in the market
The government, however, can have a huge
impact on the price of a commodity in a
market
-How should the project be valued?
-Pre-project prices?
-Post-project prices?
-Somewhere in between?
-A good alternative is to calculate change in 32
producer and consumer surplus
P
Consumer Surplus
S
Consumer
Surplus
Equilibrium
Or market
Price
P*
D
Q
Q*
Originally, consumer surplus is calculated as the area
below demand and above equilibrium price
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P
Consumer Surplus
S’
S
P*
Consumer
Surplus
P’
D
Q*
Q’
Q
If a government policy increases supply, consumer surplus
also increases.
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Producer Surplus
P
S
Equilibrium
Or market
Price
P*
Producer
Surplus
D
Q
Q*
Originally, producer surplus is calculated as the area above
supply and below equilibrium price
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P
Producer Surplus
S’
S
P*
P’
Producer
Surplus
D
Q*
Q’
Q
If a government policy increases supply, producer surplus
also changes (depending on who is producing the good).
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Old Surplus
P
S’
S
Consumer
Surplus
P*
P’
Producer
Surplus
D
Q*
Q’
Q
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New Surplus
P
S’
S
P*
P’
Consumer
Surplus
Producer
Surplus
D
Q*
Q’
Q
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P
Gained Surplus
S’
S
P*
P’
D
Q*
Q’
Q
Note that since producer surplus is lost, businesses are hurt,
possibly having negative consequences (unemployment?)
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4) Inferences from Economic Behavior
If no market exists for a commodity, other
strategies must be used to estimate
costs or benefits such as:
a) The value of time
b) The value of life
i) Lost earnings
ii) Probability of Death
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4a) The Value Of Time
Often public projects save time, or
(temporarily or permanently) cost time:
ie: Edmonton LRT saves time compared to the
bus
ie: Edmonton bridge construction slows traffic
until its done; closing of Keillor Road
increases travel time
And “time is money”
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4a) The Value Of Time
The theory of leisure-income choice states that
people will work up until the point where their
wage is equal to their value of one extra hour
of leisure.
If AFTER-TAX WAGE > Leisure value, the
person will work
If AFTER-TAX WAGE < Leisure value, the
person will relax
Therefore one could value time saved as the
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AFTER-TAX WAGE RATE.
4a) The Value Of Time
BUT:
1) Some people can’t choose how many hours
they work (ie: as involuntary unemployment)
2) Not all time away from a job is equal
-ie: someone who likes the bus may have a
lower opportunity cost of time
-ie: someone who hates the morning commute
may have a higher opportunity cost of time
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4a) The Value Of Time
Alternately, one could compare the cost of slow
and fast travel
ie: (Cost of a taxi)-(Cost of the bus) = value of
time saved by using a taxi
BUT
Other things affect choice of travel (fear of taxis,
income, environmental concerns)
Typically, travelling time cost is estimated at
about 50% of before-tax wage rate
(Small, 1992)
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4b) The Value Of Life
Life is priceless
But
Resources ($) are limited
Therefore
Life has a dollar value attached
A community hospital has $1 million in funding.
Does it spend it on an MRI to catch obscure
diseases and save 4 young adult lives, or on
flu clinics to save 10 elderly lives?
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4b) The Value Of Life Question
Approximately 200 people die of the flesh-eating
disease (Necrotizing Fasciitis) each year.
Assume that if all the resources spent on the
internet were redirected to this disease
(internet would shut down), these deaths
would be prevented.
Is it worth it?
What if only youtube shut down?
Itunes?
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4b) The Value Of Life
Unfortunately, as an aggregate society we
need to quantify the value of life.
Two methods of assigning finite value to
human life are:
i) Lost Earnings
ii) Probability of Death
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4bi) Lost Earnings
Law courts often consider the present value of a
person’s lifelong earnings when assigning
compensation to relatives of fatal accidents.
This is one way to estimate the value of life
BUT
It assigns a value (benefit) of ZERO to the old
and retired.
Since everyone in society carries a cost, cost
benefit analysis would execute the old and
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retired. This is unacceptable.
4bii) Probability of Death
A better value of life is determined through
varying the probability of death. (As most
projects affect the probability of death; they
don’t guarantee avoiding death forever).
People often accept a probability of death to save
time or make money in their everyday lives
(driving without snow tires, driving a small car,
taking a dangerous job).
The change in probability of death and money
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value can be used to calculate value of life.
4bii) Probability of Death
Studies have used this device to put the value of
a life between $4 million US and $10 million US
(Viscusi, 2006)
-This is a great range, but still allows reasonable
projects (ie: safety rails) and argues against
“unreasonable” projects (ie: Asbestos removal:
$100 million per life saved)
-This approach is still controversal. A new cancer
treatment reducing probability of death by
0.01% takes on new meaning if you need the
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treatment.
5) Valuing Intangibles
Some costs and benefits seem impossible to
value (ie: pride at winning Olympics)
3 issues arise from intangibles:
a) Intangibles can subvert the entire cost-benefit
exercise.
b) Cost-benefit analysis can force planners to put
a limit on the value of intangibles
c) Even if intangibles can’t be measured,
alternative measures be examined to find the
cheapest possible
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5a) Intangibles can ruin everything
-if you complain or believe that an intangible cost
or benefit is big enough, you can approve or
deny many projects
ie: “Losing the Oilers would be unbearable, we
should build them a new arena!”
“The piece of mind is worth it, buy the
warrantee!”
“PS3 - $300. Giving your boyfriend the best
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birthday gift ever – priceless.”
5b) Intangible Value Limit
Decisions can reveal the value of an intangible:
Ie: If, for the Oiler’s Arena,
(Cost-Benefits)=$50 million, and the city
DOES build the arena, the Oiler’s are worth at
least $50 million
Ie: If the cost of a laptop is $1000, with a $200
warrantee, and the laptop failure rate is 5%,
(Cost-Benefits)=$200-$50=$150, and
someone DOESN’T buy the warrantee, laptop
peace of mind is worth less than $150
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5c) Cheapest Method to achieve
Intangibles
Cost-effectiveness analysis can be used to
chose an option to achieve a benefit that can’t
be measured.
ie: If you can’t put a value on health, you don’t
need a gym membership ($600/year), when
you can walk or run outside or at a mall
($0/year).
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Theory - Cost-Benefit Analysis Errors
We’ve already discussed a variety of issues with
cost-benefit analysis.
Tresch (2002) noted the following common errors
in cost-benefit analysis:
1) The Chain-Reaction Game
2) The Labour Game
3) The Double-Counting Game
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1) The Chain-Reaction Game
Often proposals consider SECONDARY profits
arising from their benefits.
ie: A downtown arena produces $1 million profit
itself and increases local business profit by
$4 million
BUT, if you consider secondary benefits, one
must also consider secondary COSTS
ie: Moving the arena from the Northeast would
hurt Northeastern businesses by $X million.
Often secondary benefits are simply transfers
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across individuals.
2) The Labour Game
Often a project calls “creating employment” a
benefit.
BUT wage rates are a COST.
As mentioned earlier, for the truly involuntary
unemployed, social cost is less than wage.
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3) The Double-Counting Game
One can’t count the benefit of two mutually
exclusive uses of a commodity (ie: selling
and farming on farmland)
ie: If the government provides housing for low
income families, the benefit is either:
1) Housing low income families
Or
2) Selling the house on the market
Not both.
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Theory - The Problem of Distribution
In PRIVATE projects, distribution is irrelevant
-if the present value is positive, it is assumed
that the winners COULD compensate the
losers
-The POTENTIAL PARETO
IMPROVEMENT CRITERION
-The HICKS-KALDOR CRITERION
-Public projects can be based on this
POTENTIAL gain in social welfare
-Some can endure cost as long as others
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gain more benefits
The Added Problem of Distribution
ON TOP of the cost-benefit calculation, one could
also be concerned about DISTRIBUTION.
We can AVOID this problem by assuming that:
1) The government can costlessly transfer
between winners and losers
2) This transfer is nondistortionary
 This assumption is difficult to justify
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The Added Problem of Distribution
We can ADDRESS this problem by assuming that
certain members of society are especially
deserving
-A dollar benefit to these members is greater
than a dollar benefit to others
BUT:
1) Who is more deserving (poor, disabled, hard
workers, women, men)?
2) How much more than a dollar ($1.10, $2)?
POLITICS now complicates the issue.
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Theory – The Problem of Uncertainty
Often the costs and benefits of a project are
UNCERTAIN.
(ie: Vaccines MAY prevent the swine flu, or
they may go to people who were at no risk.)
Costs and Benefits must therefore be converted
to CERTAINTY EQUIVALENTS – the amount
of certain cost or benefit that a person would
trade for an uncertain cost or benefit scheme.
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Uncertainty Example 1
Many game shows give an opportunity for the
winner to keep their winning or RISK IT ALL
(ie: pick what’s in case #2).
Assume a suitcase can hold $0 with a probability
of 0.5 and $20,000 with a probability of 0.5
E($)=$0(0.5)+$20,000(0.5)=$10,000
If a contestant would take the chance UNTIL his
winnings increased to $4,000, that would be
his CERTAINTY EQUIVALENT.
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Uncertainty Example 2
Jonah wants a laptop. He can buy a Whale
laptop worth $1000 or a Ninva laptop worth
$600 with a 4-year warrantee. (Same price)
From Gartner (2006) we can infer a 25% failure
rate on laptops over 4 years.
Ninva: Benefit=$600
Whale: Benefit=$1000(0.75)+0(0.25)=$750
If Jonah is indifferent between laptops, he has a
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CERTAINTY EQUIVALENT of $600
Chapter 7 Summary
 Since costs and benefits occur over time, their
PRESENT VALUE must be calculated
 Typically, the public sector discount rate (r) is:
before tax return>r>after-tax return
 Alternatively, social discount rates argue for
less than public sector discount rates
 Costs and benefits can be measured at
market prices or shadow prices
 Labour costs are calculated considering
unemployment and chance to remain
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unemployed
Chapter 7 Summary
 Consumer and Producer surplus can be
effective ways of measuring costs and benefits
 The prices of some nonmarket commodities
(time, life) have to be inferred from observing
behavior
 Some intangible benefits cannot be measured
 In cost benefit analysis, never: count
secondary benefits without secondary costs,
view wages as a benefit, or double-count
 Distributional issues and uncertainty further
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confuse cost-benefit analysis