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Civil Systems PlanningBenefit/Cost Analysis
Scott MatthewsCourses: 12-706 and 73-359Lecture 4 - 9/13/2004
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Qualitative CBA
If can’t quantify all costs and benefits
Quantify as many as possible Make assumptions Estimate order of magnitude value of
othersMake rough Net Benefits estimate
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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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Market DemandPrice
P*
0 1 2 3 4 Q
A
B
If above graphs show two (groups of) consumer demands, what is social demand curve?
P*
0 1 2 3 4 5 Q
A
B
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Market Demand
Found by calculating the horizontal sum of individual demand curves
Market demand then measures ‘total consumer surplus of entire market’
P*
0 1 2 3 4 5 6 7 8 9 Q
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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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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
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Commentary
It is trivial to do this math when demand curves, preferences, etc. are known. Without this information we have big problems.
Unfortunately, most of the ‘hard problems’ out there have unknown demand functions.
We need advanced methods to find demand
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First: Elasticities of Demand
Measurement of how “responsive” demand is to some change in price or income.
Slope of demand curve = p/q.Elasticity of demand, , is defined to
be the percent change in quantity divided by the percent change in price. = (p q) / (q p)
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Elasticities of DemandElastic demand: > 1. If P inc. by 1%, demand dec. by more than 1%.Unit elasticity: = 1. If P inc. by 1%, demand dec. by 1%.Inelastic demand: < 1 If P inc. by 1%, demand dec. by less than 1%.
Q
P
Q
P
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Elasticities of Demand
Q
P
Q
P
PerfectlyInelastic
PerfectlyElastic
A change in price causesDemand to go to zero(no easy examples)
Necessities, demand isCompletely insensitiveTo price
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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
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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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Estimating Linear Demand Functions
As above, sometimes we don’t know demandFocus on demand (care more about CS) but can
use similar methods to estimate costs (supply)Ordinary least squares regression used
minimize the sum of squared deviations between estimated line and p,q observations: p = a + bq + e
Standard algorithms to compute parameter estimates - spreadsheets, Minitab, S, etc.
Estimates of uncertainty of estimates are obtained (based upon assumption of identically normally distributed error terms).
Can have multiple linear terms
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Log-linear Function
q = a(p)b(hh)c…..Conditions: a positive, b negative, c positive,... If q = a(p)b : Elasticity interesting =
(dq/dp)*(p/q) = abp(b-1)*(p/q) = b*(apb/apb) = b. Constant elasticity at all points.
Easiest way to estimate: linearize and use ordinary least squares regression (see Chap 12) E.g., ln q = ln a + b ln(p) + c ln(hh) ..
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Log-linear Function
q = a*pb and taking log of each side gives: ln q = ln a + b ln p which can be re-written as q’ = a’ + b p’, linear in the parameters and amenable to OLS regression.
This violates error term assumptions of OLS regression.
Alternative is maximum likelihood - select parameters to max. chance of seeing obs.
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Maglev Log-Linear Function
q = a*pb - From above, b = -0.3, so if p = 1.2 and q = 20,000; so 20,000 = a*(1.2)-0.3 ; a = 21,124.
If p becomes 1.0 then q = 21,124*(1)-0.3 = 21,124. Linear model - 21,000
Remaining revenue, TWtP values similar but NOT EQUAL.
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