Civil Systems Planning
Benefit/Cost Analysis
Scott Matthews
12-706 / 19-702
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Admin Issues
Please try to install Decision Tools Suite
ASAP (in case there are problems)
Installation in CEE cluster continues
(Group) Project 1 due Friday
Quick demo/recap of TopRank plugin
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Why these Lectures?
Very important to know who the benefits,
costs accrue to in public (policy) analysis
Benefit-cost analysis a simple and useful
framework to assist with this
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Efficiency Definitions/Metrics
Allocative - resources are used at highest value
possible
But welfare economics uses another..
An allocation of goods is Pareto efficient if no
alternative allocation can make at least one
person better off without making anyone else
worse off.
Inefficient if can re-allocate to make better without
making anyone else worse
Assumed that decisions made with this in mind?
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A Pareto Example
Try splitting $ between 2 people
Get total ($100) if agree on how to split
No agreement, each gets only $25
Pareto efficiency assumptions:
More is better than less
Resources are scarce
Initial allocation matters
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$100
?
Given this graph, how can
We describe the ‘set of all
Possible splits between 2 people
That allocates the entire $100?
0
$100
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$100
Line is the ‘set of all
possible splits that
allocates the entire $100,
Also called the potential
pareto frontier. Is the
line pareto efficient?
0
$100
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$100
No. Could at least get the
‘status quo’ result of (25,25)
if they do not agree on
splitting. So neither person
would accept a split giving
them less than $25. Is status
quo pareto efficient?
$25
0
$100
$25
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$100
No. They could agree on
splits of (25, 30) or (30,
25) if they wanted to all the way to (25,75) or
(75,25). All would be
pareto improvements.
Which are pareto
efficient?
$75
$25
0
$25
$75
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$100
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$100
The ‘pareto frontier’ is
the set of allocations that
are pareto efficent. Try
improving on (25,75) or
(50,50) or (75,25)…
We said initial alloc.
mattered - e.g. (100,0)?
$25
0
$100
$25
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Pareto Efficiency and CBA
If a policy has NB > 0, then it is possible
to transfer value to make some party
better off without making another worse
off.
To fully appreciate this, we need to
understand willingness to pay and
opportunity cost in light of CBA.
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Willingness to Pay
Example: how much would everyone pay to
build a mall ‘in middle of class’
Near middle may not want traffic costs
Further away might enjoy benefits
Ask questions to find indifference pts.
Relative to status quo (no mall)
E.g. middle WTP -$2 M, edges +$3 M
Edges ‘pay off’ middle , still better off
Only works if Net Benefits positive!
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Opportunity Cost
Def: The opportunity cost of using an
input to implement a policy is its value in
its best alternative use.
Measures value society must give up
What if mall costs $2 M?
Total net WTP = $1M, costs $2M
Not enough benefits to pay opp. cost
Can’t make side payments to do it
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Wrap Up
As long as benefits found by WTP and
costs by OC then sign of net benefits
indicated whether transfers can make
pareto improvements
Kaldor-Hicks criterion
A policy should be adopted if and only if
gainers could fully compensate losers and still
be better off
Potential Pareto Efficiency (line on Figure)
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Three Legs to Stand On
Pareto Efficiency
Make some better / make none worse
Kaldor-Hicks
Program adopted (NB > 0) if winners COULD
compensate losers, still be better
Fundamental Principle of CBA
Amongst choices, select option with highest
‘net’ benefit
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Welfare Economics
Concepts
Perfect Competition
Homogeneous goods.
No agent affects prices.
Perfect information.
No transaction costs /entry issues
No transportation costs.
No externalities:
Private benefits = social benefits.
Private costs = social costs.
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(Individual) Demand Curves
 Downward Sloping is a result of diminishing marginal
utility of each additional unit (also consider as WTP)
 Presumes that at some point you have enough to
make you happy and do not value additional units
Price
A
Actually an inverse
demand curve (where
P = f(Q) instead).
B
P*
0
1
2
3
4
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Q*
Quantity
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Social WTP (i.e. market demand)
Price
A
B
P*
0
1
2
3
4
Q*
Quantity
‘Aggregate’ demand function: how all potential
consumers in society value the good or service
(i.e., someone willing to pay every price…)
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This is the kind of demand
curves we care about18
Total/Gross/User Benefits
Price
A
P1
B
P*
0
1
2
3
4
Q*
Quantity
Benefits received are related to WTP - and
approximated by the shaded rectangles
Approximated by whole area under demand:
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triangle AP*B + rectangle
0P*BQ*
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Benefits with WTP
Price
A
B
P*
0
1
2
3
4
Q*
Quantity
 Total/Gross/User Benefits = area under curve or
willingness to pay for all people = Social WTP = their
benefit from consuming = sum of all WTP values
 Receive benefits from consuming this much
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regardless of how much they pay to get it
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Net Benefits
Price
A
A
B
P*
B
0
1
2
3
4
Q*
Quantity
Amount ‘paid’ by society at Q* is P*, so total
payment is B to receive (A+B) total benefit
Net benefits = (A+B) - B = A = consumer
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surplus (benefit received
- price paid)
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Consumer Surplus Changes
Price
CS1
A
P*
B
P1
0
1
2
Q*
Q1
Quantity
New graph - assume CS1 is original consumer
surplus at P*, Q* and price reduced to P1
Changes in CS approximate WTP for policies
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Consumer Surplus Changes
Price
A
CS2
P*
B
P1
0
1
2
Q*
Q1
Quantity
CS2 is new cons. surplus as price decreases
to (P1, Q1); consumers gain from lower price
Change in CS = P*ABP1 -> net benefits
and 73-359
Area : trapezoid =12-706
(1/2)(height)(sum
of bases) 23
Consumer Surplus Changes
Price
A
CS2
P*
B
P1
0
1
2
Q*
Q1
Quantity
Same thing in reverse. If original price is P1,
then increase price moves back to CS1
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Consumer Surplus Changes
Price
A
CS1
P*
B
P1
0
1
2
Q*
Q1
Quantity
If original price is P1, then increase price
moves back to CS1 - Trapezoid is loss in CS,
negative net benefit
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Elasticity - Some Formulas

q
q
p
p

pq
qp
Point elasticity = dq/dp * (p/q)
For linear curve, q = (p-a)/b so dq/dp = 1/b
Linear curve point elasticity =(1/b) *p/q =

(1/b)*(a+bq)/q =(a/bq) + 1
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Maglev System Example
Maglev - downtown, tech center, UPMC,
CMU
20,000 riders per day forecast by
developers.
Let’s assume price elasticity -0.3; linear
demand; 20,000 riders at average fare of
$ 1.20. Estimate Total Willingness to Pay.
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Example calculations
We have one point on demand curve:
1.2 = a + b*(20,000)
We know an elasticity value:
elasticity for linear curve = 1 + a/bq
-0.3 = 1 + a/b*(20,000)
Solve with two simultaneous equations:
a = 5.2
b = -0.0002 or 2.0 x 10^-4
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Demand Example (cont)
Maglev Demand Function:
p = 5.2 - 0.0002*q
Revenue: $1.2*20,000 = $ 24,000 per day
TWtP = Revenue + Consumer Surplus
TWtP = pq + (a-p)q/2 = 1.2*20,000 + (5.21.2)*20,000/2 = 24,000 + 40,000 = $ 64,000
per day.
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Change in Fare to $ 1.00
 From demand curve: 1.0 = 5.2 - 0.0002q, so q becomes
21,000.
Using elasticity: 16.7% fare change (1.2-1/1.2), so q would
change by -0.3*16.7 = 5.001% to 21,002 (slightly different value)
 Change to Revenue = 1*21,000 - 1.2*20,000 = 21,000 24,000 = -3,000.
 Change CS = 0.5*(0.2)*(20,000+21,000)= 4,100
 Change to TWtP = (21,000-20,000)*1 + (1.2-1)*(21,000-20,000)/2
= 1,100.
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BCA Part 2: Cost
Welfare Economics Continued
The upper segment of a firm’s marginal cost curve corresponds
to the firm’s SR supply curve. Again, diminishing returns occur.
Price
At any given price, determines
how much output to produce to
maximize profit
Supply=MC
AVC
Quantity
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Market Supply Curves
• Producer surplus is similar to CS -- the amount over and
Above cost required to produce a given output level
• Changes in PS found the same way as before
Supply=MC
Price
P*
PS*
P1
PS1
TVC1
Producer Surplus = Economic Profit
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TVC*
Q1
Q*
Quantity
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Example
Demand Function: p = 4 - 3q
Supply function: p = 1.5q
Assume equilibrium, what is p,q?
In eq: S=D; 4-3q=1.5q ; 4.5q=4 ; q=8/9
P=1.5q=(3/2)*(8/9)= 4/3
CS = (0.5)*(8/9)*(4-1.33) = 1.19
PS = (0.5)*(8/9)*(4/3) = 0.6
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Social Surplus
Social Surplus = consumer surplus + producer surplus
Is difference between areas under D and S from 0 to Q*
Losses in Social Surplus are Dead-Weight Losses!
P
S
P*
D
Q*
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Q
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Allocative Efficiency
Allocative efficiency occurs when MC = MB (or S = D)
Equilibrium is max social surplus - prove by considering Q1,Q2
Price
S = MC
b
P*
D = MB
a
Q1
Q*
Q2
Is the market equilibrium Pareto efficient?
Yes - if increase CS, decrease PS and vice versa.
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Quantity
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Further Analysis
Price
A
CS1
P2
P*
0
1
2
C
B
Q2
Q*
Old NB: CS2
New NB: CS1
Change:P2ABP*
Quantity
Assume price increase is because of tax
Tax is P2-P* per unit, tax revenue =(P2-P*)Q2
Tax revenue is transfer from consumers to gov’t
To society overall , no effect
Pay taxes to gov’t, get same amount back
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But we only get yellow
part..
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Deadweight Loss
Price
A
CS1
P2
B
P*
0
1
2
Q*
Q1
Quantity
Yellow paid to gov’t as tax
Green is pure cost (no offsetting benefit)
Called deadweight loss
Consumers buy less than they would w/o tax
(exceeds some people’s WTP!) - loss of CS
There will always12-706
be and
DWL
when tax imposed
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Net Social Benefit Accounting
Change in CS: P2ABP* (loss)
 Government Spending: P2ACP* (gain)
Gain because society gets it back
Net Benefit: Triangle ABC (loss)
Because we don’t get all of CS loss back
OR.. NSB= (-P2ABP*)+ P2ACP* = -ABC
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