Transcript Welfare Economics Demand Estimation

Civil Systems Planning
Benefit/Cost Analysis
Scott Matthews
Courses: 12-706 and 73-359
Lecture 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 others
Make rough Net Benefits estimate
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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
4
Market Demand
Price
A
A
B
B
P*
P*
0
1
2
3
4
Q
0
1
2
3
4
5 Q
If above graphs show two (groups of) consumer
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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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 about 7
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) 12
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
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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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 = 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 (1.2-1/1.2), 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 + (1.21)*(21,000-20,000)/2 = 1,100.
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Estimating Linear Demand
Functions
As above, sometimes we don’t know demand
Focus 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).
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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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