MEASURING BENEFITS OF VALUE OF TIME USING OBSERVABLE
DEMAND
Jane Romero1) and Hisa Morisugi2)
1) Graduate School of Information Sciences, Tohoku University
Aoba 06 Aoba-ku 980-8579 Sendai, Japan
Email: jane@plan.civil.tohoku.ac.jp
2) Graduate School of Information Sciences, Tohoku University
Aoba 06 Aoba-ku 980-8579 Sendai, Japan
Email: morisugi@plan.civil.tohoku.ac.jp
Abstract Value of time is one of the most important variables for both trip demand
forecasting and transportation project appraisal because its value is a key factor in the
generalized travel cost. In this paper, an approach to measure the benefits of value of
time in terms of observable demand is developed. As utility is usually not easily
observed, it is necessary to find a way to observe the manifestations how the changes
in utility are valued. Our approach is to begin with the observed market demand and
then derive the unobserved indirect utility function and expenditure function. The
indirect utility function and expenditure function lead to the exact benefit
measurement of the value of time using the equivalent variation method.
Key words: value of time, cost benefit analysis, observable demand, transport
benefits
1
1.
INTRODUCTION
Over the years there has been increasing recognition of the importance of time
valuation to the measurement of economic contribution of proposed projects and
policies. In transport research, value of time is one of the most important variables for
both trip demand forecasting and transportation project appraisal because of its
significant contribution in the user benefit and its value is a key factor in the
generalized travel cost. Needless to say, measurements of value of time and transport
benefits are critical in cost-benefit analysis.
Cost-benefit analysis is the name of the game in the field of public finance, and in
particular public spending, both in developed and developing countries. Investment in
infrastructure is a major indicator that separates the developed from the developing
ones. But even in developed countries, because of scarce public funding, social rates
of return between alternative projects must be computed and compared among
alternatives, and infrastructure, particularly transport infrastructure is the one of the
fields of application of these cost-benefit techniques considering how capital intensive
is the requirements for funding of transport projects. Although much is left to be
desired on the empirical implementation of the theoretical construct behind costbenefit analysis as its mechanistic application may overlook factors such as
governance and other behavioral responses that often led to unrealistic evaluations, it
is still the litmus test in project feasibility studies and evaluation. This study, in
general, aims to improve the process of cost-benefit analysis by introducing a new
approach in determining realistic transport benefits ex post through measurement of
benefits using observable market demand.
2
Since the early attempts to incorporate time into consumer theory, such as those by
Becker (1965), previous approaches primarily relied on the wage rate to provide
information about a person’s willingness to trade time with money. In applied
economic research, there have been criticisms on the use of the wage rate as the
relevant opportunity cost of time. Recent approaches, for example Bockstael et al
(1987) and Feather and Shaw (2000) have explicitly recognized that people might not
be able to freely choose their work hours, as such have allowed the shadow value of
time to deviate from the wage rate. Even with the recent developments that clarified
more explicitly the nature of the value of time, there is still no consensus yet on a
workable method for estimating the exact value of time.
Although the framework of how time should enter the structure of demand has
become clear, it remains vague how to determine its value in practice. Basically, in
practice, value of time is derived either by the wage rate approach or by mode choice
analysis using revealed preference (RP) data or stated preference (SP) data. Both
methods neglect the changes induced to the value of time derived from the changes in
economic factors such as prices, travel time, travel costs, among others. Other than
the current discrete choice models approach, the concept of measuring the value of
time in terms of observable demand from market data is yet under explored. Although
this concept is already popular in valuing non-market goods in environmental
economics, its application in the context of value of time analysis is not yet tapped.
There are series of papers by Larson and Shaikh (2001, 2003, 2004) that mentioned
the observability of demand in the context of valuing leisure time. However, their
analysis only considered VOT as a resource and made no mention of the case of VOT
as a commodity.
3
Larson and Shaikh (2001) modified Becker’s full price approach to introduce the twoconstraints model for recreation choice wherein time is a binding constraint. With this
approach, it is possible to extend it to evaluate the general case of person trips and
carry out exact welfare measurement that is consistent to the utility theoretic model.
They first mentioned the observability of VOT as a resource and applied it to derive
recreation demand. However, there is no approach yet wherein the components of
VOT can be specifically derived based on actual observable demand. Moreover,
measurement of transport benefits in terms of observable demand is a novel approach
that has not been introduced to the value of time research yet. Further thorough
analysis of this approach will be of use to establish the theoretical basis for future
empirical applications.
Basically, this study attempts to fill the research gaps mentioned above vis-à-vis in the
definition and welfare measurement of the value of time and transport benefits.
Generally, as the title implies, it aims to

measure the value of time and its components using observable demand

measure the time savings benefits in terms of observable demand
De Serpa (1971) concluded that the value of saving time consists of value of time as a
resource (i.e., value of reassignment of time to other activities) and value of time as a
commodity (i.e., value of time allocated to a certain activity). Jara-Diaz (2003)
proposed the third component that value of time changes in the consumption pattern.
Although there have been a number of previous studies measuring the value of time,
few studies were concerned with measuring its actual components primarily because
4
the components are viewed as largely theoretical constructs. The components of the
value of time also vary in different ways which adds to the confusion on how to
analyze it. For example, value of time as a resource is more familiar as it is
synonymous to time savings that can be re-assigned to do other activity. However,
there is still an effect in the value of time as commodity although there is no time
savings when there is an improvement in the traveling conditions. Therefore,
measuring the components of the value of time separately leads to better
understanding of the factors that affects it which also contributes to better estimation
reliability.
This paper will only focus on the case of the value of time as a resource for nonbusiness person trips. The discussion is highly on the theoretical framework on how to
measure the value of time as a resource using observable demand. The application of
the theory is still a work in progress. The paper will be organized as follows. After the
introduction is the discussion on the non-business person trips. Specifically, it will
measure the value of time as a resource in terms of observable Marshallian demand.
After expressing the value of time in terms of observable demand, what follows is the
measurement of time savings benefits by equivalent variation approach. Finally, the
paper ends with the conclusion.
2.
NON-BUSINESS PERSON TRIPS
Value of time is one of the most important variables for both trip demand forecasting
and transportation project appraisal because its value is a key factor in the generalized
travel cost. The value of time for household is defined as the change in travel fee
5
corresponding to the change in travel time with the utility level kept constant, i.e., the
willingness to pay for marginal change in travel time. Value of time for firms is
defined as the same with the profit kept constant. Mathematically, it is defined as
VOT  
dp
dt u or  const.
(1)
where p is transport cost, t is necessary transport time, u is utility and π is profit. The
subjective value of time is the marginal rate of substitution between travel time and
travel cost under constant utility or profit level. It is commonly referred to as the
monetary appraisal of value of time or the willingness to pay for savings in travel time.
To express the value of time in terms of observable demand for non-business person
trip, the indirect utility function and expenditure function are derived based on the
given demand. This is different from the more common approach of starting from a
specified utility function like trans-log then estimating the unknown parameters from
the derived market functions. However, the analysis on how to choose the appropriate
demand function for the given demand will not be discussed here. In other words, it is
assumed that market demand is observable, and that the preference is not observed.
Instead, the focus is on the next step, on the indirect utility function and expenditure
function, on how to express the value of time in terms of observable demand.
The process applied here is an extension of the Larson and Shaikh’s (2001) approach,
which they introduced to measure recreation demand. With the extension, it is then
possible to estimate the general case of an endogenous value of time jointly with
6
observable behavior in a utility-consistent framework, which utilizes full price – full
income specification subject to two binding constraints (i.e., income and time
constraints) on demand parameters. This approach, with the indirect utility function
and expenditure function, lead to exact welfare measurement derived consistently
from observable demand for person trips.
2.1
Value of time as a resource
2.1.1
Modeling individual behavior
First, to formulate the VOT, think of an individual on a non-business trip, which is,
either commuting or shopping. Let u z, x, l  be the individual’s preference function.
The utility of an individual is assumed to be a function of the demand for commodity
goods z whose price is normalized to 1, transport service demand x and leisure time l.
This formulation assumes neutrality in time t where disutility induced by the trip
duration such as fatigue, inconvenience and discomfort are not included in the
analysis.
Also, assume that working time is fixed then referring to (2) and (3), the individual
maximizes utility subject to income y and total available time T. The indirect utility
function, V 1, p, t , T , y  , is given by
V 1, p, t ,T , y   max uz, x, l 
(2)
z , x,l
s.t.
z  px  y,
tx  l  T
(3)
7
The utility function is assumed to be differentiable and quasi-concave in x , and the
constraints are differentiable and linear in x and in both money and time prices and
money and time budgets.
Graphically, the utility maximization problem subject to the income constraint and
budget constraint is shown as,
Figure 1
Graph of individual choice maximization
The individual’s choice is represented by plane a-b, which is better illustrated in
Figure 2.
Figure 2
Graph of individual’s choice
The Lagrangian function of (2) and (3) is represented by
V 1, p, t ,T , y   max uz, x, l     y  z  px   T  l  tx
(4)
z, x,l
The Lagrange multipliers  and  represent the shadow values of money and time,
respectively. Intuitively,  is the marginal utility of income and  is the marginal
utility of time.
By applying envelope theorem to (4), the following are derived:
8
Vy   1, p, t , T , y  ,
(5a)
VT   1, p, t , T , y  ,
(5b)
V p   1, p, t , T , y  x1, p, t , T , y  ,
(5c)
Vt   1, p, t , T , y  x1, p, t , T , y 
(5d)
where Vy ,VT ,V p ,Vt are partial derivatives wrt the subscript
Using the envelope results, the Marshallian demand x1, p, t ,T , y  can be recovered
through two separate versions of Roy’s identity,
2.1.2
V p  Vy x
(6a)
Vt  VT x
(6b)
VOT as a resource expressed by observable Marshallian demand
Value of time is defined as the marginal substitution rate for utility between price p
and duration time t . Recalling (1) and applying Roy’s identities above, the value of
time can be expressed as
VOT  
V
 VT x VT 
dp
 t 


dt v  const. V p  V y x V y 
(7)
where notation with subscripts indicate partial derivatives wrt the subscript
9
To express (7) in terms of observable demand, the approach is to take the derivative
of (5a) and (5b) with respect to t and p to express the VOT in terms of x,


V pt   V y x  Vty x  V y xt   VT x  y x  V y xt  VTy x 2  VT x y x  V y xt
t
(8)

(9)
T
Vtp   VT x  p  V pT x  VT x p    V y x x  VT x p  V yT x 2  V y xT x  VT x p
Note that (8) and (9) are equal, so equating the two equations, it can derive


VT x p  xxy  Vy xt  xT x 
(10)
Therefore, it follows from (10) that VOT as a resource can be expressed in terms of
observable demand x as
V
x  xxT
VOT v const.  T  t
V y x p  xxy
(11)
Analyzing (11), value of time under the assumption of constant utility can be
measured from observable changes in the Marshallian demand x where xt is the
change in demand with respect to change in time, x p is the change in demand with
respect to change in transport cost, xT is the change in demand with respect to change
in total available time and x y is the change in demand with respect to change in
income.
The denominator in (11) is the price substitution effect while the numerator is the time
substitution effect. Looking at it, it shows that VOT is observable just by examining
10
only the price and time substitution effects of transport demand. Also, it shows at the
same time that VOT is a function of p, t , T , y and other individual characteristics.
3. MEASUREMENT OF TIME SAVINGS BENEFITS USING OBSERVABLE
DEMAND
The dual approach to the problem is to consider the associated minimization problem,
which defines the expenditure function


e 1, p, t , T , u  min h0  ph
s.t.
(12)
h0 ,h
th  h2  T ,
uh0 , h, h2   u
(13)
where hi i  0,2 represents the demand for commodity goods z and leisure l
respectively, h is the compensated transport demand, t is transport time, T is total
available time and u is given level of utility.
The expenditure function may be expressed as the minimization of the Lagrange
function,


e  L  h0  ph   T  th  h2    u  uh0 , h, h2 
(14)
Then the following are derived from the envelope theorem applied to (14):
ep  h
(15a)
11
et  h  eT h
(15b)
eT  
(15c)
3.1 Compensated VOT and Time-extended Slutsky Equations
The compensated VOT is defined as the marginal substitution rate for expenditure
function between price p and time t , which is equal to the marginal value of
available time T . The compensated VOT in terms of the expenditure function is
given as,
compensated VOT  
dp
e
e h
 t  T  eT  
dt e const.
ep
h
u  const.
(16)
where  eT  can be interpreted as the compensated value of time
Now, to express the compensated VOT in terms of the compensated demand h, take
the derivative of the e p  h with respect to t and et  h  eT h with respect to p,
which results to
e pt  ht
(17a)
etp   eT h p  e pT h  eT h p  hT h  eT h p
(17b)
Then equating (17a) and (17b), the compensated VOT is equal to
12

ht  hhT 
hp
(18)
Note that the compensated VOT, i.e. (18), is not equal to the Marshallian VOT given
V
x  xxT
by (11), which is VOT v const.  T  t
.
V y x p  xxy
This compensated VOT can be expressed by the compensated demand function. The
compensating demand function is always equal to the usual demand function at
income equal to e1, p, t , T , u  , which can be expressed as,
h1, p, t, T , u   x1, p, t, T , e1, p, t, T , u 
(19)
Taking the derivative of (19) with respect to p, t and T, and recall that when u is at
V 1, p, t ,T , y  then it follows that e1, p, t,T , v1, p, t,T , y,V   y and accordingly, it
can be derived that xe  x y , h  x and VOT    eT , then it can be shown that
h p  x p  xxy
(20a)
ht  xt  VOT xxy
(20b)
hT  xT  VOT x y
(20c)
13
From (20b) and (20c), this relationship can be derived showing how ht can be
expressed in terms of hT ,
ht  xhT  xt  xxT
ht  xt  xxT  xhT
(20d)
Notice that (20a)-(20d) show the relationships between the observable uncompensated
Marshallian demand and the unobservable Hicksian demand, that is, the changes in
 
compensated demand h with respect to travel cost h p , travel time ht  and total
available time hT  can be expressed in terms of the changes in observable
Marshallian demand x. These expressions lead to evaluating the compensated demand
in terms of observable demand. These relationships of the uncompensated demand to
observable demand have not been expressed this way before. These formulations will
be more beneficial in the succeeding texts where transport benefits will be derived
based on observable demand.
Following the same logic, the compensating VOT is always equal to the usual
Marshallian VOT at income equal to e1, p, t , T , u  , which can be expressed as,
 1, p, t,T , u   VOT 1, p, t,T , e1, p, t,T , u  . Accordingly, the compensated VOT can
be expressed in terms of observable VOT, which take the form similar to a Slutsky
equation as follows:
 p  VOT p  xVOT y
(21a)
14
 t  VOT t  VOT xVOT  y
(21b)
T  VOT T  VOT VOT  y
(21c)
It is also necessary to calculate the combined compensated demand and VOT. With
h 1, p, t,T , u   xVOT 1, p, t,T , e1, p, t,T , u  , the derivatives of hp with respect to p
and t can be derived as,
 h  p   xVOT  p  x  xVOT  y
(22a)
 h t   xVOT t   xVOT  xVOT  y
(22b)
h T  xVOT T  VOT xVOT y
(28c)
These extended Slutsky formulations of the compensated VOT in terms of observable
VOT will come in handy in the measurement of welfare changes as it is only the
Marshallian VOT that is observable and yet the compensated VOT is the vital
component that needs to be measured.


Finally, at A  1, p A , t A , T A , y A , comparing the Marshallian VOT to the Hicksian
h  hhT  .
V
x  xxT
VOT, recall (11), VOT v const.  T  t
, and (18),   t
V y x p  xxy
hp
Referring to (20a), where h p  x p  xxy , then we could say that the denominators of
the Marshallian VOT and HIcksian VOT are equal.
15
Looking at the numerator of  
ht  hhT 
hp
and referring to (20b) and (20c), which
are ht  xt  VOT xxy and hT  xT  VOT x y respectively, then from (20b) and
(20c), this equation can be derived relating ht  xt  xxT  xhT . Substituting that
formulation of ht to the numerator of  
ht  hhT  , given that
hp
x  h , then it
V
x  xxT
becomes equal to the numerator of VOT v const.  T  t
. Therefore, with
V y x p  xxy
the derived Slutsky equations, it is possible to relate the observable Marshalian VOT
to the compensated Hicksian VOT.
3.2 Estimation of time savings benefits using observable demand
The indirect utility function and the expenditure function provide the theoretical
structure for welfare estimation. The theoretical method is to use information
available about the expenditure function or its accompanying Hicksian demand curves
for market goods to obtain compensating or equivalent variation measures of a change
in a nonmarket good.
The compensating variation, CV, is the amount of money that must be given to the
loser, or taken from the gainer, in order to keep the individual on the initial reference
curve. In conventional references with price change and no time constraint, it is

 



expressed by CV  e p B ,V A  e p A ,V A where the term e p A ,V A is the total


expenditure level in the initial situation, denoted by m A and e p B ,V A is the
16


expenditure function for p B ,V A . On the other hand, the equivalent variation, EV,
is the amount that the individual would be prepared to pay, in the new situation, to

 

avoid the price change, that is, EV  e p B ,V B  e p A ,V A , where the term


e p B ,V B is the total expenditure after the price change, denoted by m B . In the
partial equilibrium context, only the prices change, so m A  m B .
Applying the concept of equivalent variation to the value of time problem, it is not
only the price change that needs to be evaluated. As the definition of value of time
implies, it is the willingness to pay for savings in travel time so the changes that we
will consider to measure the exact welfare changes between the before (A) and after
(B) scenarios are time change t A  t B  , price change  p A  pB , income change
 y A  yB  and change in utility v A  v1, p A , t A , T , y A   v B  v1, p B , t B , T , y B .
It is possible that these changes may occur individually or simultaneously depending
on the given circumstances or the need of evaluation but to measure the exact welfare
changes, we will consider a whollistic approach including all the changes.
Two expressions of equivalent variation, EV, will be considered. It will be evaluated
with respect to the Marshallian demand with adjustment of the marginal ratio of
income, e y x , and with respect to the compensated demand, h .
3.2.1
Exact welfare measurement formula wrt e y x
First, looking at the reference point A, EV can be expressed as
17
 
   
 e1, p A , t A , T ,V B   e1, p A , t A , T ,V A 
EV  e A,V B  y A  e A,V B  e A,V A
(29a)
Another way of expressing EV is by
 
 
 eA,V B   eB,V B   y B  y A
 e1, p A , t A , T ,V B   e1, p B , t B , T ,V B   y B  y A
EV  e A,V B  y A  e A,V B  y B  y B  y A
(29b)
Now, to evaluate EV in terms of Marshallian demand and applying Roy’s identity
where V p  Vy x, Vt  VT x, VOT  VT Vy ,

EV  e A,V
B
 eA,V  
A
VB
Aev dV
V

 eV V p dp  eV Vt dt  eV Vy dy
A B

 eV Vy  x dp  eV Vy VT
(29c)

V y  x dt  eV V y dy
A B

 e y  A,V  xdp  xVOTdt  dy 
A B
Looking at (29c), it is necessary to express the term ey  A,V  in terms of demand x.
Evaluating ey  A,V  , it can be shown that,
e y  A,V   ev  A,V V y 1, p, t , T , y  
V y 1, p, t , T , y 

V y 1, p A , t A , T , y

(29d)
18
So e y may be called as the marginal utility ratio of income. Note that
e A,V  A, y   y , so e y  A,V  A, y   y at any y , therefore e yy  0 .
3.2.2
Exact welfare measurement formula wrt h
Another way of expressing EV is in terms of compensated demand h by applying
Sheppard’s lemma e p  h, et  h such as,

 
 

  e p dp  et dt  y B  y A 
B A
  h1, p, t , T ,V B dp  h1, p, t , T ,V B  1, p, t , T ,V B dt  y B  y A 
EV  e 1, p A , t A , T ,V B  e 1, p B , t B , T ,V B  y B  y A
(30)
B A
In order to calculate and express (29a) and (30) in terms of consumer’s surplus with
respect to the Marshallian and compensated demand functions x, h and to multiply by
marginal utility ratio of income e y x the first case.
3.2.3
Linear approximation for the marginal utility ratio of income, e y x
By linear approximation of the demand function, that is, second-order approximation
of EV, (29a) can be re-written as
19

 
EV  e A,V B  e A,V A

 e y  A,V  xdp  xVOTdt  dy 
A B




 

1 A A
1
e y x  e yB x B p A  p B  e yA x AVOT A  e yB x BVOT B t A  t B
2
2
1 A
 e y  e yB y B  y A
2



(31)



since e yA  1 , it needs to express only e yB  e y A, v B in terms of observable demand.
So e yB is linearly approximated and to be expressed in terms of observable demand,
A
A
 pB  p A   e ytA t B  t A   e yy
 yB  y A 
e yB  e yA  e yp
(32)
A
where e yy
 0,

 

A
A
A
e yp
 e py
 eVA  A,V V p   e y  A,V x A   x yA
y
y
A
A
A
e yt
 etyA  eV Vt A
y   eV VT x y  xVOT y
Simplifying, it becomes,
e yB  1  x yA  pB  p A   xVOT A
y t B  t A 
(33)
From (33), it shows that e yB can be approximately expressed in terms of observable
demand as defined in (11).
20
The above formula, (31), is the well-known trapezoidal formula for area with
modification of x B by e yB x B where e yB is an addition by income effect. This is not
similar to Willig’s consumer’s surplus approach wherein fixed income is a valid
assumption. However, if the income effect e yB is negligible, the modified demand
function could be equal to the Marshallian demand function.
3.2.4
Graphical illustration of the changes in  p, t , y 
Referring to (29a), the following figures illustrate the changes in  p, t , y  . The first,
second and third terms can be expressed by the consumer’s surplus due to price, time
and income change equal to the shaded area with respect to the adjusted demand
function, adjusted time values demand function and adjusted marginal ratio of income
function.
Figure 3
Change in p
Figure 4
Change in t
Figure 5
Change in y
From Figure 4, also note that the generalized cost (full price) approach is misleading
unless the project does not result to a change in VOT.
3.2.5
Linear approximation for compensated demand, h
21
From (30), it can be shown that by linear approximation for h ,

 

EV  e 1, p A , t A , T ,V B  e 1, p B , t B , T ,V B  y B  y A

 hdp  hdt  y
B
 yA
B A
   
 y B  y A 
 


1
1
  h A,V B  h B p B  p A  h A,V B  A,V B  h B  B t B  t A
2
2
(34)

 
 B   B,V B   VOT B
where h B  h B,V B  x B




Referring to the equation above, terms h A,V B and h A,V B   A,V  can be linearly
approximated around h B and  h   , respectively.
B

 



h A,V B  h B,V B  h pB  p A  pB   htB t A  t B 


(35a)


h A,V B  A,V B  h B  B  h Bp p A  p B  h tB t A  t B

(35b)
Substituting Slutsky equations for h p , ht , h  p and h t ,






B B B A
B B
A
B
B
B
h A,V B  x B  x B
p  x x y p  p  xt  VOT  x x y t  t




h A,V B  x BVOT B  xVOT Bp  x B xVOT By p A  p B




(36a)

B


 xVOT tB  x BVOT xVOT  y  t A  t B



(36b)
22




Thus h A,V B and h A,V B can be expressed by observable demand and
observable Marshallian VOT as defined in (11). Note also the application of the
extended Slutsky equations in the derivations of (36a) and (36b).
3.2.6
Relationship between e y x and h
The difference between the two methods become clear by looking only at the price
change indicated as the first term in (31) and (34) as illustrated in Figure 6.
Figure 6
Relationship between e y x and h
Equation (31) is the expression of EV= trapezoid p A AEp B while (34) is the
expression of EV= trapezoid p A DBp B . Time change can be expressed graphically
the same except by expressing demand in terms of xVOT  or h  .
The trapezoid bounded by p A AEp B derived from the modified demand function is
equal to the trapezoid bounded by p A DBp B from the compensated Hicksian demand.
To measure the welfare change, referring to Figure 6, it is necessary to establish the
Hicksian demand line DB or the modified demand line AE. However, the
compensated Hicksian demand nor the modified demand cannot be observed in the
market whereas only the Marshallian demand indicated by line AB is observable. So
basically, the approach discussed can be summarized as the approach on how lines
23
DB and AE can be calculated in terms of line AB, which is in terms of the observable
Marshallian demand.
4
CONCLUSIONS
This study develops an approach for welfare measurement of the value of time based
on observable demand. While the current practice derives the value of time assuming
specified utility and demand functions, analysis thru observable demand is possible
and is consistent with utility-theoretic model. It is an important finding as this
approach expresses the VOT in terms of Marshallian observable demand and even
capable to analyze the components of the value of time.
It has been known that the use of compensated demand curves lead to appropriate
welfare measures. However, as Hausman (1981) commented, ‘While this result has
been known for a long time by economic theorists, applied economists have only a
limited awareness of its application’. True enough, its application to the benefit
measurement of the value of time has not been shown, how VOT can be derived and
its benefits be measured from observable demand. This study showed that the indirect
utility function and expenditure function provide the appropriate compensated
demand curve and thus the appropriate welfare function.
VOT as a resource assumes utility neutrality in travel time. For the exact welfare
measurement of VOT, two approaches by equivalent variation considering time
change, price change, income change and utility change are demonstrated. These two
methods resulted from the transformation of the compensated demand and VOT in
24
terms of Marshallian demand and VOT by applying the extended Slutsky equations.
The first method is by expressing the marginal utility ratio of income in terms of
Marshallian demand and VOT while the second method is by expressing the
compensated demand and VOT in terms of Marshallian demand and VOT.
REFERENCES
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Brockstael , N.E., Strand, I.E. and W.M. Hanemann (1987) ‘Time and recreation
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Cornes, R. (1992). Duality and Modern Economics (New York: Cambridge
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De Donnea, F.X. (1971) ‘Consumer behavior: transport mode choice and value of
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De Serpa, A.C. (1971) ‘A theory of the economics of time’, The Economic Journal,
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Gonzales, R.M. (1997) ‘Value of time: a theoretical review’, Transport Reviews,
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25
Hanemann, M.W. (1991) ‘Willingness to pay and willingness to accept: how much
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26
McConnell, K.E. and I.E. Strand (1981) ‘Measuring the cost of time in recreation
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27
LIST OF FIGURES
Figure 1
Graph of individual choice maximization
Figure 2
Graph of individual’s choice
Figure 3
Change in p
Figure 4
Change in t
Figure 5
Change in y
Figure 6
Relationship between e y x and h
where:
AB: Marshallian demand function, x
AC,DB: Hicksian demand at v A and v B , h
AE= e yA  A, v x : modified demand function
28
b
z
u  z , x, l   preference function
l
tx  l  T
A
a
t
p
z  px  y
x
29
u  z , x, l 
b
zy
A
pT
z  y
t
z  y  px
z  px  y
a
T
t
l0
x
x, l 
such that
x0
l T
30
p
e y x1, p, t , T , y 
pA
pB
ey x
31
t
e y xVOT 1, p, t ,T , y 
tA
tB
e y x VOT 
32
e y 1, p, t , T , y 
33
p
x
pA
A
D
h
ey x
pB
C
xA

 
h pB ,u A h p A,u B
B

xB
E
x, h
34