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Civil Systems PlanningBenefit/Cost Analysis

Scott Matthews12-706 / 19-702

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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

$1000

Given this graph, how canWe describe the ‘set of all Possible splits between 2 peopleThat allocates the entire $100??

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$100

$1000

Line is the ‘set of all possible splits that allocates the entire $100, Also called the potential pareto frontier. Is the line pareto efficient?

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$100

$1000

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

$25

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$100

$1000

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?

$25

$25

$75

$75

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$100

$1000

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

$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 MEdges ‘pay off’ middle , still better offOnly 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 offPotential 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 EconomicsConceptsPerfect 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

Quantity

P*

0 1 2 3 4 Q*

A

B

Actually an inverse demand curve (whereP = f(Q) instead).

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Social WTP (i.e. market demand)

Price

Quantity

P*

0 1 2 3 4 Q*

A

B

‘Aggregate’ demand function: how all potential consumers in society value the good or service (i.e., someone willing to pay every price…)

This is the kind of demand curves we care about

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Total/Gross/User BenefitsPrice

Quantity

P*

0 1 2 3 4 Q*

A

B

Benefits received are related to WTP - and approximated by the shaded rectangles

Approximated by whole area under demand: triangle AP*B + rectangle 0P*BQ*

P1

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Benefits with WTPPrice

Quantity

P*

0 1 2 3 4 Q*

A

B

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 regardless of how much they pay to get it

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Net BenefitsPrice

Quantity

P*

0 1 2 3 4 Q*

A

BA

B

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 surplus (benefit received - price paid)

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Consumer Surplus Changes Price

Quantity

P*

0 1 2 Q* Q1

A

BP1

CS1

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

Quantity

P*

0 1 2 Q* Q1

A

BP1

CS2

CS2 is new cons. surplus as price decreases to (P1, Q1); consumers gain from lower price

Change in CS = P*ABP1 -> net benefitsArea : trapezoid = (1/2)(height)(sum of

bases)

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Consumer Surplus Changes Price

Quantity

P*

0 1 2 Q* Q1

A

BP1

CS2

Same thing in reverse. If original price is P1, then increase price moves back to CS1

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Consumer Surplus Changes Price

Quantity

P*

0 1 2 Q* Q1

A

BP1

CS1

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

Point elasticity = dq/dp * (p/q)For linear curve, q = (p-a)/b so dq/dp

= 1/bLinear curve point elasticity =(1/b)

*p/q = (1/b)*(a+bq)/q =(a/bq) + 1

€

ε =Δqq

Δpp

= pΔqqΔp

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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 +

(5.2-1.2)*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: CostWelfare Economics Continued

The upper segment of a firm’s marginal cost curve correspondsto the firm’s SR supply curve. Again, diminishing returns occur.

Quantity

Price

Supply=MCAt any given price, determineshow much output to produce tomaximize profit

AVC

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Market Supply Curves

Quantity

Price Supply=MC

P1

Q1 Q*

• 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

P*

PS1

PS*

TVC1TVC*

Producer Surplus = Economic Profit

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Example

Demand Function: p = 4 - 3qSupply function: p = 1.5qAssume equilibrium, what is p,q?In eq: S=D; 4-3q=1.5q ; 4.5q=4 ;

q=8/9P=1.5q=(3/2)*(8/9)= 4/3CS = (0.5)*(8/9)*(4-1.33) = 1.19PS = (0.5)*(8/9)*(4/3) = 0.6

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Social SurplusSocial Surplus = consumer surplus + producer surplusIs difference between areas under D and S from 0 to Q*Losses in Social Surplus are Dead-Weight Losses!

Q

P

Q*

P*

S

D

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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

Q*

P*

S

D = MB

= MC

Q1 Q2

a

bPrice

Quantity

Is the market equilibrium Pareto efficient?Yes - if increase CS, decrease PS and vice versa.

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Further Analysis

Assume price increase is because of taxTax is P2-P* per unit, tax revenue =(P2-

P*)Q2Tax revenue is transfer from consumers to

gov’t To society overall , no effect Pay taxes to gov’t, get same amount back

But we only get yellow part..

Price

Quantity

P2

0 1 2 Q2 Q*

A

BP*

CS1

C

Old NB: CS2

New NB: CS1

Change:P2ABP*

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Deadweight Loss

Yellow paid to gov’t as taxGreen 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 always be DWL when tax imposed

Price

Quantity

P2

0 1 2 Q* Q1

A

BP*

CS1

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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