Transcript Welfare Economics Demand Estimation

Civil Systems Planning
Benefit/Cost Analysis
Scott Matthews
Courses: 12-706 and 73-359
Lecture 3 - 9/4/2002
1
What about Other Goals, nonEfficiency?
Multigoal Analysis
Economic performance
Social performance
Environmental performance
Technological performance
Flexibility
We’ll come back to this later in course
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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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Demand Curves
Downward Sloping is a result of diminishing
marginal utility of each additional unit.
Price
A
B
P*
0
1
2
3
4
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Q*
Quantity
4
Social WTP
Price
A
B
P*
0
1
2
3
4
Q*
Quantity
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
Price
A
P1
B
P*
0
1
2
3
4
Q*
Quantity
Benefits received are related to WTP - and
equal to the shaded rectangles
Approximated by whole area under demand:
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triangle AP*B + rectangle
0P*BQ*
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Gross Benefits with WTP
Price
A
B
P*
0
1
2
3
4
Q*
Quantity
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
Price
A
B
P*
0
1
2
3
4
Q*
Quantity
A “price discriminator” could collect A0Q*B for
output level Q*. But only one price is charged in the
market, so consumers pay P*0Q*B.
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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 the
total payment is B to get A+B 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 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 P*ABP1 = gain =
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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 gov’t
To society, no effect (we get taxes back)
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
P*
B
P1
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!)
There will always12-706
be and
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
and 73-359 market’
consumer surplus12-706
of entire
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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 > 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 < 1
If P inc. by 1%, demand dec. by less than 1%.
P
P
Q
Q
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Elasticities of Demand
P
Necessities, demand is
Completely insensitive
To price
Perfectly
Inelastic
P
Q
Perfectly
Elastic
A change in price causes
Demand to go to zero
(no easy examples)
Q
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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)*(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 (1.21/1.2), so q would change by -0.3*16.7 =
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 = 1*21,000 1.2*20,000 = 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 = a*p^b 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 = ap^b. 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 eq’n to generate unit costs
Average Total: ATC = TC/q
Variable: AVC = VC/q
Marginal: MC = [TC]/  q = DTCDq
but  F/  q = 0, so MC = [VC]/  q
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