Civil Systems Planning Benefit/Cost Analysis

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Civil Systems PlanningBenefit/Cost Analysis

• Scott Matthews
• Courses 12-706 and 73-359
• Lecture 3 - 9/4/2002

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What about Other Goals, non-Efficiency?

• Multigoal Analysis
• Economic performance
• Social performance
• Environmental performance
• Technological performance
• Flexibility
• Well come back to this later in course

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Welfare EconomicsConcepts

• 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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Demand Curves

• Downward Sloping is a result of diminishing
  marginal utility of each additional unit.

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Social WTP

• An aggregate demand function how all potential
  consumers in society value the good or service
  (i.e. there is someone willing to pay every
  price)

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Gross Benefits
P1

• Benefits received are related to WTP - and equal
  to the shaded rectangles
• Approximated by whole area under demand triangle
  APB rectangle 0PBQ

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Gross Benefits with WTP

• Total/Gross Benefits area under curve or
  willingness to pay for all people Social WTP
  their benefit from consuming

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Price Discrimination
A price discriminator could collect A0QB for
output level Q. But only one price is charged
in the market, so consumers pay P0QB.
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Net Benefits
A
B

• Amount paid by society at Q is P, so the
  total payment is B to get AB benefit
• Net benefits (AB) - B A consumer surplus
  (benefit received - price paid)

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Consumer Surplus Changes
Price
A
CS1
P
B
P1
0 1 2 Q
Q1
Quantity

• New graph
• Assume CS1 is the original consumer surplus at
  P, Q

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Consumer Surplus Changes
Price
A
CS2
P
B
P1
0 1 2 Q
Q1
Quantity

• CS2 is the new consumer surplus when price
  decreases to (P1, Q1)
• Change in CS Trapezoid PABP1 gain positive
  net benefits

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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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Further Analysis
Price
A
CS1
P
B
P1
0 1 2 Q
Q1
Quantity

• Assume price increase is because of tax
• Tax is P-P1 per unit, revenue (P-P1)Q
• Is a transfer from consumers to govt
• To society, no effect (we get taxes back)
• Pay taxes to govt, get same amount back
• But we only get yellow part..

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Deadweight Loss
Price
A
CS1
P
B
P1
0 1 2 Q
Q1
Quantity

• Yellow paid to govt as tax
• Green is pure cost (no offsetting benefit)
• Called deadweight loss
• Consumers buy less than they would w/o tax
  (exceeds some peoples WTP!)
• There will always be DWL when tax imposed

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Market Demand
Price
A
A
B
B
P
P
0 1 2 3 4
Q
0 1 2 3 4
5 Q

• If the above graphs show the two groups of
  consumers demands, what is social demand curve?

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Market Demand
P
0 1 2 3 4
5 6 7 8 9 Q

• Found by calculating the horizontal sum of
  individual demand curves
• Market demand then measures total consumer
  surplus of entire market

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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. Thus the
  advanced methods in this course

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Elasticities of Demand

• Measurement of how responsive demand is to some
  change in price or income.
• Slope of demand curve Dp/Dq.
• Elasticity of demand, e, is defined to be the
  percent change in quantity divided by the percent
  change in price. e (p Dq) / (q Dp)

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Elasticities of Demand
Elastic demand e gt 1. If P inc. by 1,
demand dec. by more than 1. Unit elasticity e
1. If P inc. by 1, demand dec. by
1. Inelastic demand e lt 1 If P inc. by
1, demand dec. by less than 1.
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Elasticities of Demand
Necessities, demand is Completely insensitive To
price
Perfectly Inelastic
Perfectly Elastic
A change in price causes Demand to go to zero (no
easy examples)
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Elasticity - Some Formulas

• 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)(abq)/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.
• Lets 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 - 0.0002q
• Revenue 1.220,000 24,000 per day
• TWtP Revenue Consumer Surplus
• TWtP pq (a-p)q/2 1.220,000
  (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 -0.316.7 5.001 to 21,002
  - slightly different result.
• Change to TWtP (21,000-20,000)1
  (1.2-1)(21,000-20,000)/2 1,100.
• Change to Revenue 121,000 - 1.220,000
  21,000 - 24,000 -3,000.

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Estimating Linear Demand Functions

• Ordinary least squares regression used
• minimize the sum of squared deviations between
  estimated line and 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).
• Use Excel/other software to do the hard work
• Can have multiple linear terms.

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User cost versus Price

• Some circumstances - better to estimate demand
  function and willingness-to-pay versus user cost
  rather than just price.
• Price is only one component of user cost.
• Classic example travel demand, in which travel
  time is major user cost.
• Second example equipment requirements, such as
  computers for AOL.

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User Cost Versus Price

• For travel, can define demand function and
  performance functions with respect to travel
  time.
• Alternative can value all aspects of user cost
  in amounts. For example, what is value of time
  for congestion delays?

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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

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Log-linear Function

• q apb 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 apb. From above, b -0.3, so if p 1.2
  and q 20,000, then 20,000 a(1.2)-0.3 and a
  21,124.
• If p becomes 1.0 then q 21,124(1)-0.3
  21,124.
• Linear model - 21,000

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Making Cost Functions

• Fundamental to analysis and policies
• Three stages
• Technical knowledge of alternatives
• Apply input (material) prices to options
• Relate price to cost
• Obvious need for engineering/economics
• Main point consider cost of all parties
• Included labor, materials, hazard costs

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Types of Costs

• Private - paid by consumers
• Social - paid by all of society
• Opportunity - cost of foregone options
• Fixed - do not vary with usage
• Variable - vary directly with usage
• External - imposed by users on non-users
• e.g. traffic, pollution, health risks
• Private decisions usually ignore external

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Commentary - Externalities

• External costs SHOULD be included
• Measurement difficult, maybe impossible
• Typically no market transactions to use
• Proxy cost of eliminating hazard created
• Beware transfers / double counting!
• Example Construction disrupts commerce
• business not lost - just relocated in interim

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Functional Forms

• TC(q) F VC(q)
• Use TC eqn to generate unit costs
• Average Total ATC TC/q
• Variable AVC VC/q
• Marginal MC ?TC/ ? q ?TC??q
• but ? F/ ? q 0, so MC ?VC/ ? q