transport lecture11
TRANSCRIPT
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Summary Ch 2 Essays
Transportation Economics:
Analysis of Demand II
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II Discrete Choice
Suppose that consumers n=1,,N are
choosing over a discrete number of
items j=1,,J (e.g. modes of transport) As a researcher, we do not exactly know
consumer tastes -- tastes are random
across consumers How should we model utility and
demand?
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Conditional utility:
Conditional on choosing alternative j,
utility is:
Xj are the characteristics of good j
Sn is characteristics of the decision maker
is (unknown) parameters of utility
is random utility from consuming item j
is systematic utility
jnnjjn );S,X(VU +=
jn
);S,X(VV njjn =
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Systematic portion of utility:
For example, utility obtained from item j:
where, cjn = costs of transport, 1
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Random portion of utility:
Prob. that consumer n chooses item i:
This will depend on the probability
distribution for
]ijallforUU[obPrP jninin >=
]ijallforVV[obPr
]ijallforVV[obPr
jnininjn
ininjnjn
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1
2
2- 1V1
-
V2(choose mode 2)
2- 1=V1-V2
Random portion of utility - graph
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Probit distribution:
For example, suppose there are only two items i=1,2,
and that the utility difference is normally distributed
Denote the density function for the standard normal
distribution as
where is the cumulative standard normal distribution
2/x
2
112
2e]xProb[(x)
===
12
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Probit Distribution (contd):
V1
-V2
0 2
-1
Buy
good 2
Buy
good 1
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Estimation of Probit (contd):
where, DT=1 for transit, =0 for auto
tj=time in travel,
Distj=distance,
cj=cost of travel
Incn=income, wn=wage
)DFemale(057.0)DAge(0255.0c0186.0
)DDistInc(0254.0tw00759.0D08.2V
Tn
Tnj
Tnjn
Tjn
+
=
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Demand Curve:
Notice that utility declines in cost:
So the integral under the normal curve
is re-computed, and we obtain adownward sloping demand!
0186.0c
V
j
jn =
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Value of Time:
We can also compute the value of time as:
so that time spent in commuting is valued at
41% of the individuals wage. But with more than two alternatives, Probit
becomes difficult to estimate, so consider...
n
n
jjn
jjn
jn w41.00186.0
w00759.0
c/V
t/V
VOT =
=
=
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Multinomial Logit distribution:
Suppose that the j=1,,J random terms
are distributed as extreme value:
With = 1, the probability of choosing
item i is:
=
=J
1j
VVin
jnin eeP
)eexp(]xProb[ xj =
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Estimation of Logit (contd):
Example, from a sample of San Francisco commuters choosing
between:
(1) auto alone,
(2) bus + auto access,
(3) bus + walking access,
(4) carpool (this was before BART)
The estimated systematic utility function is:
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Estimation of Logit (contd):
where, Dj=1 for mode j, =0 otherwise
tj=in-vehicle time in travel,
tjout
=out of vehicle travel time cj=cost of travel
wn=wage of traveler
4
)25.0(
3
)24.0(
1
)26.0(
outj
)0070.0(j
)0072.0(nj
)0054.0(jn
D15.2D78.1D89.0
t0531.0t0201.0)w/c(0412.0V
=
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Demand Curve:
Expected demand for mode i is:
So,
Again, we obtain downward sloping demand!
Slope of demand will be smaller for the mode
of transport with higher-wage travelers.
= ====
N
1n
J
1j
VVN
1nini
jnin ee)(PX
=
i
i
c
X)P1(P inin
N
1nnw
0412.0
=
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Value of Time:
We can also compute the value of time as:
so that time spent in commuting is valued at 49% of
the wage, but time spent out of the vehicle is
valued at (or higher) than the wage!
nn
jjn
jjnjn w49.0
0412.0
w0201.0
c/V
t/VVOT =
=
=
nn
jjn
outjjnout
jn w29.10412.0
w0531.0
c/V
t/VVOT =
=
=
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Problem with the Logit:
The relative probability of choosing items i
and k is:
So Pi/P
kis independent of other alternatives
This is the irrelevance of independentalternatives (IIA) property
E.g., the red bus, blue bus problem
knin VV
knin eeP/P =
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Red bus-blue bus problem:
Suppose the relative probability of choosing
the bus over a car is 2:1 (this is the odds
ratio). Initially only a redbus is available.
Then another bus comes available, which is
blue, but otherwise identicalto the red bus. It
seems sensible that the relative probability
of choosing anybus over a car is 2:1, so therelative probability of choosing each bus over
the car should be 1:1
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Red bus-blue bus problem (contd):
But for the logit, the relative probability Pi/P
kis
unchangedwhen a new alternative become
available, so the red bus still has 2:1 odds
ratio over a car.
Therefore, the blue bus also has a 2:1 odds
of being chosen (as it is identical to the red).
So the relative probability of taking anybus
over a car becomes 4:1 when the blue bus is
added.
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Red bus-blue bus problem (contd):
Why does this problem with the logit arise?
Because we have assumed that the random
terms in utility are independent!
Contrast with the probit, where we assumed
that the difference was normally
distributed, so that these errors are correlated
(but probit is hard to estimate for more thantwo modes of transport)
Solution: use nestedlogit
j
21
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Nested Logit: e.g. Vacation Choice
Each type of choice is modeled as logit
New
YorkSF
Bus Car Air
Miami
Rail
Destination
choice
Mode choice
Local transport
choice
Chic-ago
Rent
car
Do not rent
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Typical Elasticities (Table 2-2)
Market, model, elasticity
type
Elasticity estimates
Freight Rail Truck
Aggregate model splitmodel
a
Price -0.25 to 0.35 -0.25 to 0.35Transit time -0.3 to 0.7 -0.3 to 0.7
Aggregate model fromtranslogcost function
b,c
Price -0.37 to 1.16d
-0.58 to 1.81e
Disaggregate modechoice model
b,f
Price -0.08 to 2.68 -0.04 to 2.97Transit time -0.07 to 2.33 -0.15 to 0.69
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Typical Elasticities (Table 2-2, contd)
Market, model,
elasticity type
Elasticity estimates
Urban passengerg
Auto Bus Rail
Price -0.47 -0.58 -0.86
In-vehicle time -0.22 -0.60 -0.60
Intercity passengerh
Auto Bus Rail Air
Price -0.45 -0.69 -1.20 -0.38
Travel time -0.39 -2.11 -1.58 -0.43
Automobileutilization
iOne-vehiclehousehold
Two-vehiclehousehold
Short-run operating
cost
-0.228 -0.059
Long-run operating
cost
-0.279 -0.099
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Value Time Elasticities (Table 2-3)
Transportation mode Estimate of value of time
Freighta
Rail Truck
(As percentage of dailyshipment value)
Total transit time 6-21 8-18Urban work trips
bAuto Bus
(As percentage of aftertax wage rate)
In-vehicle time 140 76
Walk access time ... 273Transfer wait time ... 195Intercity passenger
cAuto Bus
dRail
eAir
(As percentage ofpretax wage rate)
Total travel time 6 79-87 54-69 149