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Corso di Logistica Territoriale A.A. 2019-2020

Surplus variation withDescriptive Demand Model Approach

Adapted from: Ennio Cascetta

Transportation System Engineering: Theory and Methods- Kluwer

CORSO DI

LOGISTICA TERRITORIALE

A.A. 2019–2020

Prof. Ing. Antonio Comi

Ing. Antonio Polimeni


Introduction

Computation of Users’ Surplus Changes

The impacts perceived by users can be calculated as a change in net

perceived utility (or surplus) associated with the travel choices

made in the project and nonproject situations.

Either of two different calculation approaches can be applied,

depending on whether the underlying demand model is a

behavioral random utility model or a descriptive model.

The descriptive approach is analyzed in the following section.

2


Generalised travel cost

A “demand function” relates the number of users undertaking trips

with given characteristics to the travel price, here average

generalized travel cost (and other explanatory variables).

This cost is defined, as a (linear) combination of the amount of

resources spent by the user on a trip (time, money, etc.), with

weights reflecting the user’s travel behaviour.

The cost parameters (weights) may vary according to trip purpose

and socioeconomic category (i.e., user class).

3


Travel cost

A simplified specification of the generalized cost g is:

where t and mc are, respectively, the travel time and the monetary

cost.

The coefficients here have been explicitly associated with

user class i.

1 2( ) ( ) ( )i i

odmk odmk odmkg i i t i mc

4


Surplus variation

To introduce the method for calculating surplus variation with

descriptive demand models, consider first a simple system consisting

of a single O-D pair connected by a single mode and a single path.

Assumption: all users belong to one class (i.e. they have the same

behavioral parameters).

In this case, the demand model can be formally written as

dod = dod(god)

o d

5


Travel demand function

di

o d

Ui

god

si=Ui-gigi

Representation od the relationship dod(god)

si: Traveller Surplus

(all users belong to one class)

6


Traveller Surplus

The demand curve, in its traditional interpretation, represents the

ordering of individual trips by users on the basis of the generalized

cost that they are willing to pay to undertake the trip; this is a

measure of the utility of the trip to the user.

In other words, the marginal trip corresponding to each point on the

horizontal axis has a total utility (or willingness to pay) equal to the

corresponding value of the generalized cost on the vertical axis.

An increase in the cost would discourage this marginal user from

making the trip and therefore reduce the value of the demand dod.

7


Traveller Surplus

All users of a given class incur the same generalized cost.

Let the be this cost and be the number of users traveling

in the non-project situation.

For all trips undertaken, except the marginal one, there is a net

utility, or surplus, given by the difference between the amount that

the user would be willing to pay to make the trip, and the cost that is

actually paid.

NP

odg ( )NP

od odd g

8


Surplus change

If as a result of project P the generalized cost is reduced to , the

number of users traveling changes to .

To calculate the total surplus change resulting from project P, a

distinction should be made between:

1) trips already undertaken in the situation NP and

2) the new trips that are only undertaken because of the cost

reduction (trips generated or induced by the project).

P

odg

( )P

od odd g

9


Surplus change: trips undertaken in the situation NP

For a trip by user i in the first group, the surplus change will be

given by:

that is, by the difference between the generalized cost in the non-

project and project situations.

( ) ( )i P i NP NP P

od od od odDS U g U g g g

10


Surplus change: trips already undertaken in the

situation NP

The total surplus change DS'p for all the trips/users of this group is

therefore:

' ( ) ( )NP NP P

P od od od odDS d g g g

DS'p is represented by the area A

11


Surplus change: new trips undertaken in the situation P

A trip i that is generated by the cost reduction brought about by

project P will have a surplus Ui – gPod in the project situation, and

zero in the non-project situation. The total surplus change for the

p project generated trips is therefore given by

the area B.

(*) Extra trips undertaken because of the effect of the generalized cost reduction are sometimes

called the demand generated or induced by the project P.

* ( ) ( )P NP

od od od od odd d g d g


Typically it is assumed that all generated trips d*od experience

identical utility, given by the average value of the interval :

Therefore the total surplus for the generated demand can be

calculated as:

[ , ]NP P

od odg g

2

NP P

od odi

g gU

* * *1

2 2

NP PP NP Pod od

P od od od od od

g gDS d g d g g

13

Surplus change: new trips undertaken in the situation P


Surplus change: trips undertaken in the situation P

If the demand curve is approximated by a line between

-: the change in surplus for generated trips results

from the formula for the area of the (approximate) triangle B.

NP P

od od od odd (g ) and d (g )

DS*p is represented by the area B

* *1

2

NP P

P od od odDS d g g


Surplus change: total

The total surplus change is given by the sum:

This expression can be interpreted as the product of the average

demand between situations P and NP by the change in the

corresponding generalized cost.

Equivalently, with the linear approximation mentioned above, this

expression can be interpreted as the area of the (approximate)

trapezoid consisting of the two parts A and B.

' * 1( ) ( ) ( ) ( )

2

1( ) ( )

2

NP NP P P NP NP P

P P P od od od od od od od od od od

P NP NP P

od od od od od od

DS DS DS d g g g d g d g g g

d g d g g g

15


Surplus change

The exact expression for the surplus change can be obtained by

calculating the hatched area as the integral of the demand function

d(g):

( )P

NP

g

Pg

DS d g dg

16


Surplus change

The results described still hold if the project increases the generalized

cost (gPod>gNP

od). In this case, there will clearly be a reduction of

surplus and a decrease in the number of trips.

17


Surplus change: generalization

The concept of surplus change and expressions and can be

generalized to the case of multiple cost “dimensions” (e.g., multiple

destinations and/or modes and/or paths).

However, this generalization is neither straightforward nor universal.

18



Consider a case with two possible alternatives, for example, two paths

with costs g1 and g2: the two demand curves can be defined as d1(g1 g2)

and d2(g1 g2). The demand, that is the number of trips on each path, depends

on the cost of both paths.

19



In this case the exact expression for the surplus change is:

1 2

1 2

,

1 2 1 2,

1,2

( , )P P

NP NP

g g

P ig g

i

DS d g g dg dg

20



As stated, the exact expression for the surplus change is:

However, the value of this integral usually depends on the path of

integration between the two limits.

The integral depends only on the extremes of integration if the

Jacobian of demand functions is symmetrical with respect to

generalized path costs:

This condition is seldom, if ever, met by usual demand models.

1 2

1 2

,

1 2 1 2,

1,2

( , )P P

NP NP

g g

P ig g

i

DS d g g dg dg

1 2

2 1

d d

g g

21


Surplus change: average demand method

To determine the surplus change, two heuristic approaches can be

followed, corresponding to two approximate methods for the evaluation of

previous integral.

The first approach, which can be called the average demand method,

calculates the surplus change as

where diNP and di

P are, respectively, equal to di(g1P, g2

P) and di(g1NP, g2

NP).

1,2

1

2 i i i i

NP P NP P

P

i

DS d d g g

22


Surplus change: average demand method

The result of the average demand method corresponds to the sum of the

two hatched areas in figure (it is assumed that cost decreases with project

for path 1 and increases for path 2).

Path 1 Path 2

23


Surplus change: average cost method

The alternative approach, which can be called the average cost method,

reduces the problem to a single choice dimension by considering an average

trip cost തg given by the weighted average of the costs in each dimension:

where p1 and p2 are the demand shares of each dimension:

pi = di/(d1 + d2)

1 2 1 1 21 2 2, ,

P P P P P P Pg p g g g p g g g

1 2 1 1 21 2 2, ,

NP NP NP NP NP NP NPg p g g g p g g g

24


Surplus change: average cost method

In this approach, the demand curve expresses the total demand

dT = d1 + d2

as a function of the average cost തg .

25


Surplus change: average cost methodThe surplus change can therefore be calculated as:

The surplus change can be interpreted as the product of the average of the

total demand between the states P and NP and the change in average cost

between the two states.

1

2

P NP NP PT T

PDS d g d g g g

26


Surplus change: demand model

The partial share demand model can be conveniently expressed as the

product of the demand level and the fraction of trips with given

characteristics:

where:

Note that both dio and pi

dmk/o depend on a vector of socio-economic and

activity system attributes SE, as well as on a vector of level of service

attributes, expressed by the perceived generalized costs for all destinations,

by all modes and on all paths gi.

. /, ,i i i i i i i

odmk o dmk od d p SE g SE g

.

/

is the number of trips from zone undertaken by users of class ,

is the fraction of these trips with the characteris

tics .

i

o

i

dmk o

o i

p dmk

d

27


Surplus change: approach extension (*)

The surplus change for user class i resulting from the passage from state NP

with costs gNPi to state P with costs gPi, can be calculated by extending the

two previous approximate expressions to the general case.

The average demand method yields:

The average cost method yields:

with:

gPi = dmk pidmk/o(g

Pi )gPiodmk

g NPi = dmk pidmk/o(g

NPi)gNPiodmk

1

,2 odmk odmk

i NPi i Pi NPi Pi

P odmk odmk

dmk

DS o i d d g g g g

. .

1,

2

NPi Pii Pi i NPi

P o oDS o i d d g g

g g

-

28


Surplus change: approach extension

Previous expressions (average demand method and average cost method)

are the equivalent of expression for descriptive demand models.

The surplus change for all system users can be calculated by adding the

results of these expressions for all user classes, all zones, and all trip

purposes.

However, because the surplus changes resulting from a project may be

positive for some user classes, zones, or phases of the project and negative

for others, it is helpful to keep these values separate.

29


Surplus change: example

od

k1

k2

k3

k4

Calculation of DSp for users in a single market segment with 4 paths

exp( )( )

exp( )

k

k

k

Vp k

V

Probability

Assumption

• V= -g

30



Path cost

gNPodk1 = 1,2

gNPodk2 = 1,4

gNPodk3 = 2,0

gNPodk4 = 2,0

gPodk1 = 1,0

gPodk2 = 1,0

gPodk3 = 1,2

gPodk4 = 1,4

Not Project (NP) Project (P)

31



Probability

exp( )( )

exp( )

k

k

k

Vp k

V


pNPodk1 = 0,36

pNPodk2 = 0,30

pNPodk3 = 0,17

pNPodk4 = 0,17

pPodk1 = 0,29

pPodk2 = 0,29

pPodk3 = 0,23

pPodk4 = 0,19

32



Demand


dodk1 (gNP) = 100pNPodk1= 36




dodk1 (gP) = 100pPodk1= 29




do. (gNP)= 100 do. (gP)= 100

33



Average demand method

1

2

36 29 1,2 1,0 30 29 1,4 1,00,5

17 23 2,0 1,2 17 19 2,0 1,4

45,10

odk odk

NP P NP P

P odk odk

dk

DS o d d g g

g g

34



Average cost method

. .

1

2

1,2 0,36 1,4 0,30 2,0 0,17 2,0 0,170,5 100 100

1,0 0,29 1,0 0,29 1,2 0,23 1,4 0,19

41,0

NP PP NP

P o oDS o d d g g

g g

35

1 2 1 1 21 2 2, ,

NP NP NP NP NP NP NPg p g g g p g g g

pi = di/(d1 + d2)

1 2 1 1 21 2 2, ,

P P P P P P Pg p g g g p g g g

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