Transportation Research Part B - UPHF · not, respectively. By adding scheduling considerations to both the utility function and the time consumption constraint, Small (1982)

Transportation Research Part B 93 (2016) 207–224

Contents lists available at ScienceDirect

Transportation Research Part B

journal homepage: www.elsevier.com/locate/trb

The impact of travel time variability and travelers’ risk

attitudes on the values of time and reliability

Mickael Beaud, Thierry Blayac, Maïté Stéphan

∗

LAMETA Université Montpellier, UFR d’Économie Avenue Raymond DUGRAND - Site de Richter C.S. 79606 34960 Montpellier Cedex 2

France

a r t i c l e i n f o

Keywords:

Value of travel time savings

Value of travel time reliability

Risk attitudes

Reliability proneness

Prudence

Cost benefit analysis of transport

infrastructure projects

a b s t r a c t

In this paper, we derive implementable measures of travelers’ willingness to pay to save

travel time ( vot ) and to improve the reliability ( vor ) of a given trip. We set out a simple

microeconomic model of transport mode choice in which each trip is fully characterized by

its price and the statistical distribution of its random travel time, assuming that travelers

have expected utility preferences over the latter. We then explore how the vot and vor are

affected by the statistical distribution of travel time and by travelers’ preferences towards

travel time variability.

© 2016 Elsevier Ltd. All rights reserved.

1. Introduction

How to improve transport systems is an important public policy issue for virtually any government. In practice, public

decisions on transport infrastructure usually rely on cost benefit analysis of alternative projects. Among the benefits of an

improved transport system, it is now well established that travel time savings and travel time reliability gains are two

important elements. In general, the appropriate appraisal of almost any transport system requires monetary estimates of

both the value of travel time ( vot ) and the value of travel time reliability ( vor ). This is particularly the case for policy

makers who may have to choose between mutually exclusive public transport infrastructure projects that include travel

time savings (e.g., the construction of a new high speed rail) and/or reliability gains (e.g., the construction of a bypass that

increase capacity). Therefore, this study derives theoretically-based and implementable monetary measures of the vot and

the vor .

The vot has a long history in microeconomic theory, dating back at least to the seminal contribution on the optimal al-

location of time by Becker (1965) , where time appears as an unconsumed input to prepare final goods, from which utility is

ultimately derived. Accordingly, the vot would be understood as an opportunity cost, i.e., the cost of not earning money dur-

ing an out-of-work period, and, thus, would simply be given by the wage rate. Then, by explicitly incorporating both work

time and leisure time into preferences, as separate arguments of the consumer’s utility function, Johnson (1966) showed

that the vot was equal to the sum of two terms: the wage rate and a monetary value of the marginal disutility of work

time. Accordingly, Johnson (1966) concludes that the wage rate provides an upward biased estimate of the vot . Soon af-

ter, Oort (1969) reached the same conclusion, and claimed that travel time itself should be added to the arguments of

the consumer’s utility function. Going a step further towards the formal integration of time in standard microeconomic

demand theory, DeSerpa (1971) generalized these frameworks by distinguishing explicitly between the time spent by ne-

∗ Corresponding author.

E-mail address: [email protected] , [email protected] (M. Stéphan).

http://dx.doi.org/10.1016/j.trb.2016.07.007

0191-2615/© 2016 Elsevier Ltd. All rights reserved.


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208 M. Beaud et al. / Transportation Research Part B 93 (2016) 207–224

cessity and the time spent by choice, depending on whether the time consumption inequality constraints are binding, or

not, respectively. By adding scheduling considerations to both the utility function and the time consumption constraint,

Small (1982) introduced the now standard scheduling model with endogenous departure time. 1

Moreover, with the aim of providing a theoretical foundation for the so-called safety margin, Gaver (1968) and

Knight (1974) were among the first to consider travel time variability by relaxing the standard assumption of a certain

travel time in departure time choice models. Subsequently, assuming mean-variance preferences over a random travel time,

Jackson and Jucker (1981) were able to explain data collected from a discrete choice experiment in stated preferences, in

which subjects were asked to choose among risky trips with random travel times. Note that these early contributions only

implicitly rely on the assumption of Von Neumann and Morgenstern (1947) ’s expected utility preferences over random travel

time. In this respect, notable advances resulted from Polak (1987) and Senna (1994) , who derived measures of both the vot

and the vor in the context of a general model of travel choice, and to Noland and Small (1995) and Noland et al. (1998) , who

extended Small (1982) ’s scheduling model to accommodate reliability.There are at least two distinct approaches used in the

literature to incorporate travel time variability in theoretical models: the Bernoulli approach ( Jackson and Jucker (1981) ;

Polak (1987) ; Senna (1994) ; Small et al. (2005) ; Hensher et al. (2011) ; Beaud et al. (2012) ; Devarasetty et al. (2012) ;

Kouwenhoven et al. (2014) ) and the scheduling delays approach (e.g. Noland and Small (1995) ; Noland et al. (1998) ;

Batley (2007) ; Asensio and Matas (2009) ; Fosgerau and Karlstrom (2010) ; Engelson and Fosgerau (2011) ; Fosgerau and En-

gelson (2011) ; Koster and Koster (2015) ), the present paper belonging to the former. 2

However, as observed by Small (2012) , after decades of study, the vot and the vor are still incompletely understood con-

cepts. In particular, the literature is not clear on how the vot and the vor are impacted by the statistical distribution of travel

time or by travelers’ preferences towards travel time variability. In this study, these issues are addressed from the point of

view of the theory of individual choice under financial risk (Bernoulli, 1738; Arrow (1963) ; Pratt (1964) ; Rothschild and

Stiglitz (1970) ). Thus, we try to further bridge the gap between the notion of travel time reliability in transport policy and

the notion of financial risk in microeconomic theory. Using the Bernoulli approach, we set out a simple microeconomic

model of transport mode choice, in which each trip is fully characterized by its price and the statistical distribution of its

random travel time. A traveler’s preferences function is assumed to be separable and quasi-linear, and is the sum of a linear

function of price and the expectation of a non-linear univariate function of travel time. 3 Then, we introduce model-free def-

initions of the vot and the vor , the definition of the vor being new. The vot is defined as the willingness to pay for a given

reduction in travel time, while the vor is defined as the willingness to pay to eliminate all variability in travel time. Hence,

we explore how the vot and the vor , which are functions of travel time rather than values, are affected by the statistical

distribution of travel time and by travelers’ preferences towards travel time variability, i.e., their risk attitudes.

By definition, reliability-prone travelers prefer a fully reliable trip (which has a single travel time outcome) to a risky

trip (which has multiple travel time outcomes) whenever both have the same price and mean travel time. For instance,

reliability-prone travelers prefer a 90 min trip to a risky trip of either 60 min or 120 min, with equal probability. As one

would expect, reliability proneness is equivalent to the concavity of the preferences function with respect to travel time. In

other words, reliability proneness is equivalent to a decreasing marginal utility of travel time, reflecting travelers’ increasing

sensibility in the duration of the journey. 4 An important – and unrecognized – consequence of this is that reliability prone-

ness implies that the vot is increasing with travel time. More generally, we show that, for all reliability-prone travelers, the

vot is increased by any first-order stochastic deterioration in the distribution of travel time. This result implies, for instance,

that reliability-prone travelers are willing to pay less to save 30 min on a 90 min trip than to save 30 min on a 120 min

trip. Furthermore, if the marginal utility of travel time is concave, then, borrowing Kimball (1990) ’s terminology, we say that

travelers are prudent. 5 We then show that if travelers are reliability-prone and prudent, then their vot is increased by any

second-order stochastic deterioration in the distribution of travel time. This result implies, for instance that all reliability-

prone and prudent travelers are willing to pay less to save 30 min on a 60 min trip than to save 30 min on any risky trip

with a mean travel time of 60 min. A key understanding these results is that reliability proneness and prudence character-

ize a general preference for the combination of good (e.g., a decrease in travel time) and bad (e.g., a high or an unreliable

travel time). Roughly speaking, reliability-prone and prudent travelers have an increasing and convex vot , and, therefore,

are willing to pay more to save time for alternatives with an already high and/or unreliable travel time.

1 See Jara-Diaz (20 0 0) for a comprehensive survey of this literature. 2 Lam and Small (2001) and Borjesson et al. (2012) used both approaches. Useful reviews of the literature can be found in Wardman (1998) , Noland and

Polak (2002) , Jong et al. (2004) , Li et al. (2010) , Carrion and Levinson (2012) and Engelson and Fosgerau (forthcoming). Engelson and Fosgerau (forthcoming)

identify a third type of approach: mean-dispersion models. These models are defined directly in terms of statistics of the travel time distribution. They

consist of measures that are linear in the mean travel time and some measure of the dispersion of travel time. 3 The implications of non-separability between price and time for the vot are explored in Blayac and Causse (2001) and Jiang and Morikawa (2004) . 4 Our model treats travelers’ reliability proneness as exogenous. This contrasts with the scheduling preferences model, in which reliability proneness may

be viewed as endogenous. In particular, travelers arriving at their preferred arrival time (PAT) during a fully reliable trip would dislike the introduction

of actuarially-neutral travel time variability (making the trip risky, without affecting the mean travel time), ceteris paribus, because they would then

necessarily suffer schedule delay early and/or schedule delay late in some states of the world. On the other hand, reliability proneness cannot be a general

property of scheduling preferences in that reliability proneness would be obtained for a given trip, while reliability aversion would be obtained for another.

This may viewed by considering travelers who do not arrive at their PAT during a fully reliable trip. Here the risk may become desirable as because the

arrival time would be closer to the PAT in some states of the world. 5 The notion of prudence was introduced by Kimball (1990) to measure the intensity of saving in the face of a future risk affecting wealth.

M. Beaud et al. / Transportation Research Part B 93 (2016) 207–224 209

Furthermore, we consider two complementary measures of travelers’ willingness to pay to escape all travel time vari-

ability. The first measure introduced by Batley (2007) in a scheduling model, is expressed in time units and is called the

reliability premium, analogous to Pratt (1964) ’s risk premium. It is defined as the additional amount of time that a traveler

would be willing to accept to eliminate all variability in travel time. For instance, if a traveler is reliability-prone, he/she

may be indifferent between a 100 min trip and a risky trip of either 60 min or 120 min, with equal probability. In this

example, the reliability premium is equal to 10 min, which is the difference between the certainty equivalent of the risky

trip (100 min) and its mean travel time (90 min). The problem with the reliability premium is that it is expressed in time

units, while the cost benefit analysis of a transport infrastructure typically requires a monetary measure. The second mea-

sure, which is new, is expressed in monetary units and is called the vor . It is defined as the additional amount of money

that a traveler would be willing to pay to eliminate all variability in travel time. For instance, if a traveler is reliability-

prone, he/she may be indifferent between a 90 min trip at a price of $60 and a risky trip of either 60 min or 120 min,

with equal probability, at a price of $50. In this example, the vor is equal to $10, which is the difference between the price

of the reliable trip ($60) and the price of the risky trip ($50), between which the traveler is indifferent. Locally, we show

that the reliability premium is equal to the so-called reliability ratio ( vor / vot ). Of course, the reliability premium depends

positively on both the riskiness of the distribution of the travel time and the degree to which travelers are reliability prone.

On the other hand, their impact on the vor is more subtle. Finally, we show how our theoretical measures can be imple-

mented to provide empirical valuations of the vot and the vor . Furthermore, in light of our theoretical results, we discuss

the implications of the choice of a particular functional form for the preferences function estimated in empirical studies.

The rest of the paper is organized as follows. Section 2 presents the theoretical framework and introduces the concepts of

reliability proneness and prudence. In Section 3 , we define the vot and we establish the link between reliability proneness

and the shape of the vot . In Section 4 , we present and explore the properties of the reliability premium and the vor . The

empirical implementation of the theory is discussed in Section 5 . Section 6 summarizes our results and concludes the paper.

The appendix contains the formal proofs of all propositions.

2. Travelers’ risk attitudes and travel time variability under the Bernoulli approach

2.1. Basic model, notation and assumptions

Consider travelers (i.e., users of a transport system), who wish to reach a destination, and suppose they can choose

among a given number of trips. Each trip i is fully characterized by its cost c i > 0 and its random travel time ˜ t i . Let-

ting F i (t) = Pr [ ̃ t i ≤ t] be the cumulative distribution function ( cdf ) of ̃ t i , and letting the semi-open interval (t min , t max ] ={t ∈ R + : t min < t ≤ t max

}be its bounded support, the mean or expected travel time is

μi = E ̃ t i =

∫ t max

t min

t dF i (t ) (1)

Here, it is also useful to consider the � th central moment of the distribution of travel time:

σ � i = E[ ̃ t i − μi ]

� =

∫ t max

t min

[ t − μi ] � dF i (t) (2)

For instance, selecting � = 2 gives the variance, and selecting � = 3 gives the skewness of travel time.

We assume that travelers’ preferences over trips can be represented by a separable and quasi-linear preferences function

of the expected utility form:

U i = EU(c i , ̃ t i ) = −λc i + Eu ( ̃ t i ) = −λc i +

∫ t max

t min

u (t) dF i (t) (3)

where λ > 0 represents the constant and strictly positive marginal utility of wealth, and where u : R + → R is a differentiable

utility function defined over travel time. This latter function is intended to play the same role as the Bernoulli (1738)’s utility

function in the theory of individual choice under financial risk. The essential distinction is that travel time is assumed to be

a non-desirable good:

u

′ (t) ≤ 0 ∀ t (4)

Hence, it is known that there exists a relationship between the sign of the first derivative of the utility function and a

stochastic dominance order, named the first-degree stochastic dominance ( fsd ) order. In the present context, the fsd order

allows the unambiguous ranking of any two transport alternatives with the same cost. 6

Definition 1. ˜ t j is an fsd deterioration of ̃ t i if and only if

F i (t) ≥ F j (t) ∀ t (5)

6 More generally, an unambiguous ranking is possible whenever the cost of the dominated trip is not smaller.


Fig. 1. Monotony and preference relation over trips.

In other words, the probability of experiencing a smaller travel time than any given one t is always greater with trip i

than it is with trip j . Thus, we could search for the condition, on preferences, under which travelers would unanimously

prefer trip i to trip j whenever ̃ t j is an fsd deterioration of ̃ t i . On the other hand, we could search for the condition, on the

distribution of travel time, under which travelers who dislike more travel time would unanimously prefer trip i to trip j .

Proposition 2 states that these two approaches are equivalent.

Proposition 2. The following two statements are equivalent:

I. All travelers who dislike more travel time prefer ̃ t i to ̃ t j .

II. ˜ t j is an FSD deterioration of ̃ t i .

A direct implication of Proposition 2 is that fsd deteriorations in travel time should be disliked by a wide set of travelers.

It is obvious that all travelers who dislike more travel time also dislike fsd deteriorations. Indeed, fsd deteriorations involve

a transfer of probability mass to the right, that is, from low travel time states of the world to high travel time states of the

world, or, equivalently, by adding some non-negative risk to the travel time. Thus, any fsd deterioration implies a greater

mean travel time. For instance, all travelers who dislike more travel time prefer a reliable 120 min trip to a risky trip with

random a travel time ̃ t 1 = (120 min, 1/2; 150 min, 1/2). In other words, they dislike the addition to a reliable 120 min trip

of the non-negative risk ˜ ε1 = (0 min, 1/2; 30 min, 1/2). More generally, Proposition 2 tells us that all travelers who dislike

more travel time prefer ̃ t 1 to ̃ t 2 = (120 min, 1/4; 150 min, 3/4) and prefer ̃ t 1 to ̃ t 3 = (120 min, 1/2; 150 min, 1/4; 180 min,

1/4), where ̃ t 2 and ̃

t 3 are obtained from ̃

t 1 by adding the non-negative risk ˜ ε1 either to the low (120 min) or to the high

(150 min) travel time realization of ̃ t 1 , respectively. Of course, the fsd order is incomplete, 7 but when it works, it provides

a normatively appealing criterion to rank two transport alternatives ( Fig. 1 ).

2.2. Reliability proneness

To deal with travelers’ attitudes towards travel time variability, we begin with a model-free definition of reliability prone-

ness, which mimics the standard definition of risk aversion in the theory of individual choice under financial risk.

Definition 3. A traveler is reliability-prone whenever he/she always prefers a reliable trip with a certain travel time to any

risky trip with a random travel time, whenever both trips feature the same cost and the same mean travel time.

According to Definition 3 , reliability-prone travelers prefer, for instance, a reliable 135 min trip to a risky trip, with ran-

dom travel time ˜ t 1 = (120 min, 1/2; 150 min, 1/2). In other words, they dislike the addition to a reliable 135 min trip of

the actuarially-neutral risk ˜ ε2 = ( −15 min, 1/2; 15 min, 1/2). In contrast to fsd deteriorations, it is less obvious that all

real-world travelers dislike any actuarially-neutral increase in the variability of travel time. This is because such changes

necessarily make the trip shorter in some states of the world. As a result, many travelers might be reliability-averse. 8 In

7 In particular, ̃ t 2 and ̃ t 3 have the same mean ( μ2 = μ3 = 142.5min) and, thus, cannot be ranked according to the fsd criterion. 8 A traveler is reliability-averse whenever he/she always prefers a risky trip with a random travel time to a reliable trip with a certain travel time,

whenever both trips feature the same cost and the same mean travel time. In addition, reliability-neutral travelers are not affected by travel time variability,

and only consider the cost and the mean travel time of trips.


t̃

I

I

V

this study, we focus on the implication of the reliability proneness assumption, guessing that it is the most widespread risk

attitude among travelers. However, our analysis may also be adapted trivially to obtain the implications of reliability aver-

sion. Thus, our analysis does not predict reliability proneness and is not specific to this assumption. However, as a caveat,

each traveler is assumed to be either reliability-prone or reliability-averse, while real-world travelers might exhibit reliability

proneness at some travel time levels and reliability aversion at others ( Fig. 2 ).

Under the expected utility assumption, it is well-known that financial risk aversion is equivalent to the concav-

ity of Bernoulli (1738)’s utility function. Of course, this result also applies here to reliability proneness. According to

Definition 3 and to the preferences function defined in (3) , reliability proneness is equivalent to u (E ̃ t i ) ≥ Eu ( ̃ t i ) for any

random travel time ̃ t i . By Jensen (1906) ’s inequality, we know that this is true if and only if u is concave over the support of

i . As a result, a traveler is reliability-prone if and only if his/her preferences function is a concave function of travel time: 9

u

′′ (t) ≤ 0 ∀ t (6)

Furthermore, Definition 3 is equivalent to saying that reliability-prone travelers dislike any actuarially-neutral variability

affecting a reliable trip. Therefore, the implication of reliability proneness is therefore that travelers prefer no variability

to actuarially-neutral variability. However, what about travelers’ choices among different risky trips with the same mean

travel time? For instance, it is not evident a priori that reliability-prone travelers prefer ̃ t 1 to ̃ t 4 = (105 min, 1/4; 135 min,

1/4; 150 min, 1/2) and prefer ˜ t 1 to ˜ t 5 = (120 min, 1/2; 135 min, 1/4; 165 min, 1/4), where ˜ t 4 and ̃

t 5 are obtained from

˜ t 1by adding the actuarially-neutral risk ˜ ε2 either to the low (120 min) or to the high (150 min) travel time realization of ̃ t 1 ,

respectively ( Fig. 3 ).

Fortunately, Proposition 5 tells us that reliability-prone travelers dislike more actuarially-neutral variability affecting any

trip, be it initially risky or not. Following Rothschild and Stiglitz (1970) , we consider the following definition of a mean-

preserving increase in risk.

Definition 4. ˜ t j contains more actuarially-neutral variability than ̃

t i if and only if

μi = μ j and

∫ t max

t

F i (s ) ds ≥∫ t max

t

F j (s ) ds ∀ t (7)

Then, Proposition 5 below is obtained by simply adapting the result of Rothschild and Stiglitz (1970) to the case of a

random travel time (rather than random financial wealth). Again, the essential distinction is that travel time is assumed to

be a non-desirable good. 10


II. All reliability-prone travelers prefer ̃ t i to ̃ t j .

V. ˜ t j contains more actuarially-neutral variability than ˜ t i does.

More generally, taking as given that they dislike more travel time, Proposition 7 shows that reliability-prone travelers

unanimously respect the second-degree stochastic dominance ( ssd ) order.

Definition 6. ˜ t j is an ssd deterioration of ̃ t i if and only if ∫ t max

t

F i (s ) ds ≥∫ t max

t

F j (s ) ds ∀ t (8)

Combining any fsd deterioration with any actuarially-neutral increase in variability yields a ssd deterioration. Note that

an ssd deterioration is obtained only if the mean travel time does not decrease, and only if the variance increases (although

the reverse is not necessarily true). Hence, Proposition 7 proves that there exists a link between the ssd order and the sign

of the first two successive derivatives of a travelers’ preferences function with respect to travel time.


V. All reliability-prone travelers who dislike more travel time prefer ̃ t i to ̃ t j .

I. ˜ t j is an SSD deterioration of ˜ t i .

Finally, observe that Arrow (1963) and Pratt (1964) demonstrate that the degree of concavity of Bernoulli (1738)’s utility

function, captured by the absolute risk aversion function, governs the behavior of decision-makers under financial risk and

9 On the other hand, a traveler is reliability-averse if and only if his/her preferences function is a convex function of travel time. 10 In particular, because travel time is a non-desirable good, we integrate from t to t max in (7) , rather than from t min to t , as in the usual definition of an

increase in (financial) risk.


X

˜

allows for comparative risk aversion. Here, the absolute risk aversion function is defined as minus the ratio of its second-

derivative to its first-derivative. In the present context, we define the absolute reliability proneness function as

r(t) =

u

′′ (t)

u

′ (t) (9)

Taking as given that travelers dislike more travel time, this function is positive for all reliability-prone travelers. Moreover,

we define comparative reliability proneness as follows.

Definition 8. Consider two travelers with decreasing utility functions u and v , say traveler u and traveler v . Traveler v is

more reliability-prone than traveler u is if traveler v dislikes any variability affecting travel time that traveler u dislikes.

Hence, Proposition 9 is obtained by adapting Pratt (1964) ’s fundamental theorem on comparative risk aversion. This

Proposition states that the absolute reliability proneness function provides an indicator of the intensity of travelers’ aversion

towards travel time variability.

Proposition 9. The following three statements are equivalent:

VII. Traveler v is more reliability-prone than traveler u is.

VIII. The absolute reliability proneness function is uniformly greater for traveler v than it is for traveler u.

IX. v is a concave transformation of u.

It is useful to recognize that comparative reliability proneness can be applied to a single traveler as well. Indeed, while

all reliability-prone travelers dislike travel time variability, they may find a given risk more or less harmful, depending on

whether it affects a trip with a longer travel time. In other words, reliability-proneness may be increasing, constant, or

decreasing with travel time, depending on the sign of the slope of the absolute reliability proneness function. Thus, at least

three kinds of preferences can be distinguished. Travelers exhibiting increasing absolute reliability proneness ( iarp ), find

any risk more harmful when it affects a longer trip. On the other hand, travelers exhibiting decreasing absolute reliability

proneness ( darp ) find any risk less harmful when it affects a longer trip. Finally, travelers exhibiting constant absolute

reliability proneness ( carp ) find any risk equally harmful, independent of the trip length. As shown in Section 2.3 , the

functional forms commonly used in the transportation literature, such as quadratic and power utility functions, all impose

darp .

2.3. Prudence

Next, we consider the implications of the sign of the third derivative of the preferences function with respect to travel

time. As shown hereafter, this sign is important because it determines the shape of both the vot and the vor . In the

expected utility literature, where utility is defined over wealth, a positive sign of the third derivative of Bernoulli (1738)’s

utility function defines prudence and characterizes downside risk aversion. In contrast, because travel time is a non-desirable

good, prudence is consistent with a negative third-derivative of the preferences function with respect to travel time:

u

′′′ (t) ≤ 0 ∀ t (10)

In addition, prudence characterizes travelers’ aversion towards more upside variability in travel time. By definition, changes

in upside variability preserve both the mean and the variance.

Definition 10. ˜ t j contains more upside variability than ̃

t i does if and only if

μi = μ j and σ 2 i = σ 2

j and

∫ t max

t

∫ t max

s

[ F i (� ) − F j (� )] d �d s ≥ 0 ∀ t (11)

Hence, Proposition 11 states that prudent travelers unanimously dislike any given change in the variability of travel time

if and only if this change takes the form of an increase in upside variability.


X. All prudent travelers prefer ̃ t i to ̃ t j .

I. ˜ t j contains more upside variability than ̃ t i does.

To illustrate the implications of prudence, consider again

˜ t 1 = (120 min, 1/2; 150 min, 1/2) and suppose that the

actuarially-neutral risk ˜ ε2 = ( −15 min, 1/2; 15 min, 1/2) can be added either to the low (120 min) or to the high (150

min) travel time realization of ̃ t 1 . A prudent traveler should prefer the former option. Equivalently, he/she should prefer ̃ t 4 =(105 min, 1/4; 135 min, 1/4; 150 min, 1/2), where ˜ ε2 has been added to the low travel time realization of ̃ t 1 (120 min), to t 5 = (120 min, 1/2; 135 min, 1/4; 165 min, 1/4), where ̃ ε2 has been added to the high travel time realization of ̃ t 1 (150 min).

Observe that ̃ t 4 and ̃

t 5 have the same mean, μ4 = μ5 = 135 min, and the same variance, σ 2 = σ 2 = 337.5 min, but that
4 5


t̃

5 contains more upside variability than ̃

t 4 does. Indeed, the distribution of ̃ t 4 is skewed to the left with a negative third

moment, σ 3 4

−5062 . 5 min < 0 (i.e., its left tail is longer and its mass is concentrated on the right), while the distribution

of ̃ t 5 is skewed to the right with a positive third moment, σ 3 5

= −σ 3 4

> 0 (i.e., its right tail is longer and its mass is concen-

trated on the left) ( Fig. 4 ). Intuitively, reliability-prone and prudent travelers are willing to combine the good (such as a low

travel time) and the bad (such as more travel time variability). 11

In contrast to prudence, darp implies that the actuarially-neutral risk ˜ ε2 is perceived as more harmful if it affects a

120 min trip rather than a 150 min trip. This may help to recognize that darp works against prudence. Indeed, the absolute

prudence function is defined as

p(t) =

u

′′′ (t)

u

′′ (t) (12)

Then, observe that the sign of r ′ is equal to the sign of p− r . Therefore, we have that p ≤ r is equivalent to darp , p ≥ r

is equivalent to iarp , and p = r is equivalent to carp . On the one hand, all travelers exhibiting iarp are prudent, but the

reverse is not true. Hence, prudent travelers may exhibit darp , iarp , or carp . Moreover, all non-prudent travelers exhibit

darp . Intuitively, under darp , travelers are allowed to be prudent, but not too much.

Finally, using the same reasoning as that used to establish Proposition 7 , we have that reliability-prone and prudent

travelers who dislike more travel time unanimously respect the third-degree stochastic dominance ( tsd ) order.

Definition 12. ˜ t j is a tsd deterioration of ̃ t i if and only if ∫ t max

t

∫ t max

s

[ F i (� ) − F j (� )] d �d s ≥ 0 ∀ t (13)

Combining any ssd deterioration with any increase in upside variability yields a tsd deterioration. Thus an increase in

upside variability is a particular tsd deterioration, just as an actuarially-neutral increase in variability is a particular ssd . 12


XII. All reliability-prone and prudent travelers who dislike more travel time prefer ̃ t i to ̃ t j .

XIII. ˜ t j is a TSD deterioration of ˜ t i .

3. The value of travel time

The vot measures travelers’ willingness to pay for a given reduction in travel time. In the case of a risky trip with a

random travel time, the vot reflects travelers’ willingness to pay to reduce the mean travel time of the trip. However, there

are many different ways in which this can be done. In this study, we suppose that the random travel time is reduced by the

same positive amount in all states of the world. Thus, the vot is defined for a certain travel time saving arising, for instance,

through a reduction in the free-flow travel time.

Definition 14. The vot is the maximum monetary amount that a traveler is willing to pay per unit(s) �t > 0 of total travel

time saved in all states of the world.

According to Definition 14 , the vot , denoted vot i for a given trip i , is implicitly defined by the following equality:

EU(c i , ̃ t i ) = EU(c i + vot i �t , ̃ t i − �t ) (14)

From (3) , we obtain the explicit formula:

vot i = E vot ( ̃ t i ) =

∫ t max

t min

vot (t) dF i (t) (15)

where

vot (t) =

1

λ

u (t − �t) − u (t)

�t (16)

This exact measure can be used to compute the vot for any reduction �t in travel time. An important special case is

obtained when the amount of travel time saved tends to zero, because this gives the following, more familiar, marginal

11 Note that reliability-averse travelers, who prefer combining good with good (and bad with bad), would also be prudent. Indeed, they would prefer

to allocate some zero-mean risk on travel time (a good) to a lower travel time outcome (a good) rather than to a higher travel time outcome (a bad).

Therefore, both reliability-prone and reliability-averse travelers would consistently dislike any increase in upside variability. For more on this interpretation,

see Eeckhoudt and Schlesinger (2006) , Eeckhoudt et al. (2009) , and Crainich et al. (2013) . 12 This approach can easily be extended to higher orders of stochastic dominance.


˜

measure: 13

vot ∗i = lim

�t→ 0 vot i = E vot

∗( ̃ t i ) =

∫ t max

t min

vot ∗(t) dF i (t) (17)

where

vot ∗(t) = lim

�t→ 0 vot (t) = −u

′ (t)

λ(18)

Clearly, the vot is positive for all travelers who dislike both more travel time and less wealth. Furthermore, recalling that

λ represents the marginal utility of wealth, it is apparent that the vot is a monetary measure of a traveler’s welfare gain

from saving travel time. Therefore, the vot is greater for travelers with a low marginal utility of wealth. As a result, if the

marginal utility of wealth is decreasing with wealth, then the vot would be greater for richer travelers, as common sense

suggests.

A less obvious point is whether the vot is decreasing or increasing with travel time. Would a traveler be willing to pay

more or less to save travel time when the mean travel time is greater? Although this is ultimately an empirical question, it

is important to know under which conditions, in terms of the form of a traveler’s preferences function, each property holds.

The following result gives a clear answer to this question.


XIV. For all reliability-prone travelers the VOT (for given �t) is greater for ̃ t j than it is for ̃ t i .

II. ˜ t j is an FSD deterioration of ̃ t i .

Without variability, that is in the particular case where both ̃

t i and ̃

t j are certain, a fsd deterioration is simply equivalent

to more travel time. Therefore, Proposition 15 implies that all reliability-prone travelers are willing to pay more to reduce

the travel time of longer trips. For instance, all reliability-prone travelers are willing to pay more to save 60 min on a

150 min trip than they are to save 60 min on a 120 min trip. Roughly speaking, reliability proneness implies that the vot

is increasing with travel time. This follows from the fact that reliability-prone travelers have a decreasing marginal utility of

travel time. As a result, reliability-prone travelers value more travel time savings at higher levels of travel time.

Moreover, we can determine the condition under which the vot is greater or smaller for a risky trip than for a reliable

trip, whenever both have the same mean. Proposition 16 tells us that the answer to this question depends on the sign of

the third derivative of the preferences function with respect to travel time (i.e., whether or not travelers are prudent).


XV. For all prudent travelers, the VOT (for given �t) is greater for ̃ t j than it is for ̃ t i .

IV. ˜ t j contains more actuarially-neutral variability than ̃ t i does.

According to Proposition 16 , prudent travelers are unanimously willing to pay more to reduce the travel time of more

risky trips. Thus, their vot is increasing with the variability of travel time. For example, prudent travelers would be willing

to pay more to reduce a risky trip ̃

t 1 = (120 min, 1/2; 150 min, 1/2) by 15 min than they would to reduce a reliable trip of

135 min by 15 min. Thus, they would prefer a change from ̃

t 1 to ̃ t 6 = (105 min, 1/2; 135 min, 1/2), where ̃ t 6 is obtained from t 1 by subtracting 15 min from each possible realization of ̃ t 1 , to a change from 135 min to 120 min. To better understand

why this property of the vot is linked to the sign of the third derivative of the utility function, observe from (15) that the

vot is calculated as an expectation. In particular, the vot is greater for a risky trip with random travel time ̃ t 1 than it is for

a reliable 135 min trip if and only if

vot 1 =

1

2

vot (120) +

1

2

vot (150) ≥ vot (135) (19)

By Jensen (1906) ’s inequality, we know that this is true if and only if the vot is a convex function of travel time, which is

equivalent to prudence. This is illustrated in Fig. 5 .

To illustrate further, consider any random travel time ̃ t i , expressed as ̃ t i = μi + k ̃ ε, where μi is the mean travel time, ̃ ε is

an actuarially-neutral risk, and k is a positive scalar representing the size of the risk. Furthermore, from (2) , we have σ � i

=k � E ̃ ε� . As a result, if the size of the risk is multiplied by a factor k , then the � th moment is multiplied by a factor k � . Using

a Taylor series expansion of the vot around k = 0 , we obtain an instructive approximation: 14

vot ∗i vot

∗(μi ) −1

σ 2 i

u

′′′ (μi ) (20)

2 λ

13 Of course, whenever the utility function is linear, the marginal measure involves no approximation. Otherwise, if the utility function is decreasing

and concave (resp. convex), that is if the traveler dislikes more travel time and is reliability-prone (resp. reliability-averse), then the marginal measure

overestimates (resp. underestimates) the vot . 14 Let f (k ) = E vot

∗(μi + k ̃ ε) denote the vot as a function of the size of the risk. Differentiating with respect to k yields the � th derivative f (� ) (k ) =

−E ̃ ε� u (� +1) (μi + k ̃ ε) /λ. Using the remainder form of the Taylor series expansion of f ( k ) around k = 0, we obtain f (k ) = f (0)+

∑ n � =1 k

� f (� ) (0) /� !+ R n +1 ,

where R n +1 = k n +1 f (n +1) (k ∗) / [ n + 1]! , with k ∗ lying between 0 and k . Observe that f ′ (0) = 0, because E ̃ ε = 0, by assumption. Substituting σ � i

= k � E ̃ ε� and

f (0) = vot ∗( μi ), we obtain vot

∗i

= vot ∗(μi ) −

∑ n � =2 σ

� i

u (� +1) (μi ) /� ! λ+ R n +1 . Finally, selecting n = 2 and neglecting the remainder yields the result.


This formula makes explicit the separate effects on the vot of the mean travel time, captured by the first term, and of the

variability of travel time, captured by the second term. The first term is simply the vot , calculated for a certain travel time

equal to the mean travel time of the alternative. The second term is minus half the product of the variance of travel time

and the third derivative of the utility function (evaluated at the mean travel time), divided by the marginal utility of wealth.

Then, it is clear from (20) that the variance of travel time increases the vot of prudent travelers. In general, the � th moment

of the distribution of travel time increases the marginal measure of the vot for all travelers with a utility function of travel

time exhibiting a negative [ � + 1] th derivative.

Finally, recall that an ssd deterioration in the distribution of travel time can be obtained by combining any fsd deteriora-

tion with any actuarially-neutral increase in variability. As a result, Proposition 17 follows directly from Propositions 15 and

16 .

Proposition 17. The following statements are equivalent:

XVI. For all reliability-prone and prudent travelers, the VOT (for given �t) is greater for ̃ t j than it is for ̃ t i .

VI. ˜ t j is an SSD deterioration of ̃ t i .

4. Two measures of the value of travel time reliability

4.1. The reliability premium

We first consider a non-monetary measure of the value of travel time reliability, which is equivalent to that proposed by

Batley (2007) . Because it is similar to the Arrow–Pratt risk premium, it is called the reliability premium. Observe that, in

contrast to the vor defined hereafter, the reliability premium is expressed in time units and, therefore, is a non-monetary

measure of the value of travel time reliability.

Definition 18. The reliability premium π is the maximum amount of additional travel time that a traveler is willing to

accept to escape all variability in travel time.

According to Definition 18 , the reliability premium π i for trip i is implicitly defined by the following equality:

EU ( c i , ̃ t i ) = U(c i , μi + πi ) (21)

Because the reliability premium is expressed in time units, it is not possible to obtain an explicit formula. Using (3) , the

above condition is equivalent to

Eu ( ̃ t i ) = u (μi + πi ) (22)

By Jensen (1906) ’s inequality, we know that Eu ( ̃ t i ) ≤ u ( μi ) if and only if u is concave over the support of ̃ t i . Therefore, it is

clear from (22) that the reliability premium is positive for all reliability-prone travelers who dislike more travel time. This

is illustrated in Fig. 2 . In addition, the reliability premium is increasing with any actuarially-neutral increase in travel time

variability.


XVII. For all reliability-prone travelers who dislike more travel time, the reliability premium is greater for ̃ t j than it is for ̃ t i .


As above, consider any random travel time ˜ t i , expressed as ˜ t i = μi + k ̃ ε, where μi is the mean travel time, ˜ ε is an

actuarially-neutral risk, and k is a positive scalar representing the size of the risk. Thus, by considering the case of a small

risk, we obtain a local approximation of the reliability premium, which is identical to Pratt (1964) ’s famous local approxi-

mation of the risk premium: 15

πi π ∗i =

1

2

σ 2 i r(μi ) (23)

The reliability premium is approximately half the product of the variance of travel time and of the absolute reliability

proneness function (evaluated at the mean travel time). As demonstrated by Proposition 19 , it is apparent from (23) that

the reliability premium is increasing with the variability of travel time. In addition, ( 23 ) reveals that the reliability pre-

mium i increasing with the traveler’s degree of reliability proneness, measured by the absolute reliability proneness func-

tion. The more a traveler is reliability-prone, the greater is his/her reliability premium. Using Pratt (1964) ’s terminology,

15 Let g ( k ) represent the reliability premium as a function of the size of the risk. By definition, we have Eu (μi + k ̃ ε) = u (μi + g(k )) . Observe that g(0) =

0. Differentiating with respect to k yields E ̃ εu ′ (μi + k ̃ ε) = g ′ (k ) u ′ (μi + g(k )) . Evaluating at k = 0, we obtain g ′ (0) = 0, because E ̃ ε = 0, by assumption.

Differentiating again with respect to k yields E ̃ ε2 u ′′ (μi + k ̃ ε) = g ′′ (k ) u ′ (μi + g(k ))+

[g ′ (k )

]2 u ′′ (μi + g(k )) . Evaluating at k = 0 yields g ′′ (0) = r(μi ) E ̃ ε2 .

Finally, using the remainder form of the Taylor series expansion of g ( k ) around k = 0, we obtain g(k ) =

1 2

k 2 g ′′ (0)+ R 3 , where R 3 =

1 3!

k 3 g ′′′ (k ∗) , with k ∗

lying between 0 and k . Thus, substituting σ 2 i

= k 2 E ̃ ε2 and neglecting the remainder yields the result.


Fig. 2. Utility function and travelers’ reliability attitudes.

Fig. 3. Reliability-proneness and preference relation over trips.


Fig. 4. Prudence and preference relation over trips.

Fig. 5. Travelers’ risk attitudes and the shape of the vot .

Proposition 20 proves that this is true in the small and in the large, and gives an additional equivalent statement to

Proposition 9 .

Proposition 20. The following statements are equivalent:

VII. Traveler v is more reliability-prone than is traveler u.

XVIII. Traveler v’s reliability premium is always greater than that of traveler u.

As a corollary, we have that the reliability premium is decreasing with travel time for all travelers exhibiting darp .

Indeed, under darp , additional travel time reduces the traveler’s degree of reliability proneness, and may be viewed as


a change in preferences from v to u . Applying Proposition 20 , we have that additional travel time reduces the reliability

premium.

4.2. The monetary value of travel time reliability

One significant limitation of the reliability premium is that it is expressed in time units. Indeed, in practice, the cost-

benefit analysis of transport infrastructure projects requires a monetary value of travel time variability, that is, the vor .

Definition 21. The vor is the maximum monetary amount that a traveler is willing to pay to escape all variability in travel

time.

According to Definition 21 , vor i for a given trip i is implicitly defined by the following equality:

EU(c i , ̃ t i ) = U(c i + vor i , μi ) (24)

From (3) , we obtain an explicit formula:

vor i = E vor ( ̃ t i ) =

∫ t max

t min

vor (t) dF i (t) (25)

where

vor ( ̃ t i ) =

u (μi ) − u ( ̃ t i )

λ(26)

Thus, according to Jensen (1906) ’s inequality, it is clear that the vor is positive for all reliability-prone travelers whose

preferences function is concave with travel time. More generally, we have the following result.


XIX. For all reliability-prone travelers, the VOR is greater for ̃ t j than it is for ̃ t i .


Proposition 22 simply tells us that the vor is increasing with the variability of travel time. However, the vor is not

necessarily increased by any ssd deterioration in the distribution of travel time. The reason is that ssd deteriorations (which

include fsd deteriorations) may strictly increase the mean travel time, and this has a negative impact on the vor . In addition,

in contrast to the reliability premium, the vor is not governed by the absolute reliability proneness function. To illustrate

this point, consider a random travel time ̃ t i and any two reliability-prone travelers with identical marginal utility of wealth

and utility functions u and v over travel time. We have that the vor of traveler v is greater if and only if v (μi ) − u ( μi )

≥ E[ v ( ̃ t i ) − u ( ̃ t i )] , which is equivalent to the convexity of v − u , that is, v ′ ′ ≥ u ′ ′ everywhere. Thus, the values of travelers’

marginal utility of travel time are not involved.

Finally, consider, once again, any random travel time ̃ t i expressed as ̃ t i = μi + k ̃ ε, where μi is the mean travel time, ε̃is an actuarially-neutral risk and k is a positive scalar representing the size of the risk. Thus, by considering the case of a

small risk, we have: 16

vor i vor

∗i =

1

2

−u

′′ (μi )

λσ 2

i = π ∗i vot

∗(μi ) (27)

In other words, the local approximation of the vor is equal to the product of the local approximation of the reliability

premium and that of the marginal measure of the vot . This makes sense, because the vot may be viewed as a money-

metric function that transforms time into money. Indeed, the marginal measure of the vot gives the traveler’s willingness

to pay per (small) unit of travel time saved, and its dimension is money per unit of time. On the other hand, the reliability

premium gives the number of units of travel time that the traveler is willing to accept to escape all variability in travel

time. Thus, the product of these two terms gives the traveler’s willingness to pay to escape all variability in travel time. We

can also observe from (27) that the so-called reliability ratio ( vor / vot ) is locally equal to the reliability premium.

5. Functional forms of travelers’ utility

In empirical studies providing estimations of the vot and the vor , a functional form is consistently given to travelers’

preferences functions. Therefore, we need to distinguish between the implications derived from the researcher’s choice of

a particular utility function and the implications derived from empirical facts. We briefly study the three main functional

forms used in the empirical literature using the Bernoulli approach: quadratic, negative exponential, and power utility func-

tions.

16 Let h (k ) = Evor (μi + k ̃ ε) represent the vor as a function of the size of the risk. Successive differentiation with respect to k yields its � th derivative

h (� ) (k ) = −E ̃ ε� u (� ) (μi + k ̃ ε) /λ. Using the remainder form of the Taylor series expansion of h ( k ) around k = 0 , we obtain h (k ) = h (0) +

∑ n � =1 k

� h (� ) (0) /� ! +

R n +1 , where R n +1 = k n +1 h (n +1) (k ∗) / [ n + 1 ] ! , with k ∗ lying somewhere between 0 and k . Observe that h (0) = 0 and that h ′ (0) = 0 , because E ̃ ε = 0 , by

assumption. Hence, substituting in σ � i

= k � E ̃ ε� , we obtain vor i = − ∑ n � =2 σ

� i

u (� ) (μi ) /� ! λ + R n +1 . Finally, substituting in ( 23 ), (9) , and (18) , selecting n = 2 ,

and neglecting the remainder yields the result.


5.1. Quadratic utility functions

Following the theory of finance, the quadratic utility function has been the most widely used in pioneering empirical

studies on the vot and the vor , such as Jackson and Jucker (1981) . The quadratic utility function of travel time takes the

following form:

u (t) = αt + βt 2 (28)

This function is strictly decreasing and concave if α < −2 tβ and β < 0. Substituting (28) into (3) yields a mean-variance

preferences function that only depends on the first two moments of the distribution of travel:

U i = −λc i + αμi + βμ2 i + βσ 2

i (29)

This preferences function appears in Senna ( 1994 , eq. 15, p. 211). In Jackson and Jucker (1981) , the square of the mean travel

time μ2 i

is omitted. Substituting (28) into (15) and (17) we have:

vot i = −α + β[ 2 μi − �t ]

λand vot

∗i = −α + 2 βμi

λ(30)

Clearly, under reliability proneness ( β < 0), the vot is increasing with the mean travel time of the trip, as predicted by

Proposition 15 . In addition, because the third derivative of quadratic utility functions is zero, travelers with such preferences

are neither prudent nor non-prudent. Their vot is an affine function of travel time and, therefore, is unaffected by travel

time variability. This illustrates Proposition 16 . Moreover, the absolute reliability proneness function is

r(t) =

2 β

α + 2 βt (31)

Clearly, travelers with a quadratic utility function of travel time exhibit darp . According to Proposition 20 , we know that

their reliability premium is decreasing with travel time. Using the approximation of the reliability premium in (23) we

obtain

π ∗i =

βσ 2 i

α + 2 βμi

(32)

which is clearly decreasing with the mean travel time of the trip. Finally, by substituting (28) into (25) and (27) , we obtain:

vor i = vor

∗i = −β

λσ 2

i (33)

Observe that the approximation of the vor in (27) is exact in the quadratic case. In contrast to the reliability premium, the

vor here is independent of the mean travel time.

5.2. Negative exponential utility functions

Other specifications than the quadratic utility function have been used in the transportation literature. In particular,

Polak (1987) proposed the following negative exponential form:

u (t) = − exp (βt) (34)

This function has the property that the sign of its � th-derivative is the sign of −β� . Thus, if β > 0, travelers dislike more

travel time, are reliability-prone, and prudent. In fact, absolute reliability proneness and absolute prudence are equal to the

same constant, namely r(t) = p(t) = β . Substituting (34) into (3) yields the following preferences function, which depends

only on the first two moments of the distribution of travel:

U i = −λc i − E exp (β˜ t i ) (35)

Substituting (34) into (15) and (17) , we obtain:

vot i =

1 − exp (−β�t)

λ�t E exp (β˜ t i ) and vot

∗i =

β

λE exp (β˜ t i ) (36)

As predicted by Proposition 17 , under reliability proneness, the vot is an increasing and convex function of travel time

and, therefore, must be greater for longer and more risky trips. Moreover, the negative exponential utility generates carp ,

as well as constant absolute prudence. Thus, an important consequence of carp is that travelers’ attitudes towards travel

time variability are unaffected by the trip length. According to Proposition 20 , we know that their reliability premium is

independent of travel time. Using the approximation of the reliability premium in (23) , we obtain:

π ∗i =

1

2

σ 2 i β (37)

which is clearly independent of the mean travel time of the trip. Finally, by substituting (34) into (25) and (27) , we obtain:

vor i =

E exp (β˜ t i ) − exp (βμi ) and vor

∗i =

1

σ 2 i

β2

exp (βμi ) (38)
λ 2 λ


In contrast to the reliability premium, which is constant, the vor is, in this case, strictly increasing with the mean travel

time of the trip.

5.3. Power utility functions

Finally, consider a power utility function known as the Box-Cox transformation:

u (t ) =

α

1 + γ[ t 1+ γ − 1] (39)

This function is strictly decreasing if α < 0, and concave if αγ < 0. Therefore, if α < 0 and γ > 0, the traveler dislikes more

travel time and is reliability-prone. In addition, if γ > 1, then the traveler is also prudent. On the other hand, if α < 0 and

γ < 0, the traveler dislikes more travel time, is reliability-averse, and (necessarily) prudent. 17 Substituting (39) into (3) , we

obtain the following preferences function:

U i = −λc i +

α

1 + γ[ E ̃ t

1+ γi

− 1] (40)

Substituting (39) into (15) and (17) , we obtain:

vot i =

α

λ[1 + γ ]�t E [

[̃ t i − �t

]1+ γ −˜ t 1+ γi

] and vot ∗i = −α

λE ̃ t

γi

(41)

As predicted by Proposition 17 , under reliability proneness, the vot is an increasing and convex function of travel time and,

therefore, must be greater for longer and more risky trips ( Table 1 ).

Moreover, the absolute reliability proneness and prudence functions are:

r(t) =

γ

t and p(t) =

γ − 1

t (42)

Power utility functions all exhibit darp (and decreasing absolute prudence if they are also prudent). According to

Proposition 20 , we know that the reliability premium is decreasing with the mean travel time of the trip. As above, us-

ing (22) and (23) yields:

πi = [ E ̃ t 1+ γi

] 1

1+ γ − μi and π ∗i =

1

2

γσ 2

i

μi

(43)

Finally, by substituting (39) into (25) and (27) , we obtain:

vor i =

α

λ[1 + γ ] [ μ1+ γ

i − E ̃ t

1+ γi

] and vor

∗i = −1

2

αγ

λ

σ 2 i

μ1 −γi

(44)

6. Summary and conclusions

Using the Bernoulli approach, this study tackles the theoretical issue of assessing travelers’ willingness to pay to save

travel time ( vot ) and to improve its reliability ( vor ), for a given trip. We set out a simple microeconomic model of transport

mode choice, in which each trip is fully characterized by its price and the statistical distribution of its random travel time,

assuming that travelers have expected utility preferences over the latter. The vot is defined as the willingness to pay for a

given reduction in travel time, while the vor is defined as the willingness to pay to eliminate all variability in travel time.

We consider two complementary measures of travelers’ willingness to pay to escape all travel time variability. The first

measure introduced by Batley (2007) in a scheduling model, is expressed in time units and is called the reliability premium,

analogous to Pratt (1964) ’s risk premium. The problem with this measure of the value of reliability is that it is expressed

in time units while the cost benefit analysis of a transport infrastructure typically requires a monetary measure. Thus, we

establish a new measure expressed in monetary units, defined as the additional amount of money that a traveler would be

willing to pay to eliminate all variability in travel time.

A contribution of our study is that it provides theoretical foundations for the mean-variance model. Another contribution

is that it demonstrates how the vot and the vor , which are functions of travel time rather than values, are affected by both

the statistical distribution of travel time and travelers’ preferences towards travel time variability (i.e., their risk attitudes).

Hence, for example, we show that for a reliability prone and prudent traveler, the vot is increasing and convex with mean

travel time. However, for a reliability averse and non-prudent traveler, the vot is decreasing and concave. In the same

examples, the vor is positive and negative, respectively.

In empirical studies using the centrality-dispersion approach to provide estimations of the vot and the vor , a functional

form is consistently given to travelers’ preferences functions. Therefore, we first need to distinguish between the implica-

tions derived from the researcher’s choice of a particular utility function and those derived from empirical facts. In this

17 The first three successive derivatives of the utility function are: u ′ (t) = αt γ , u ′′ (t) = αγ t γ −1 , and u ′′′ (t) = αγ [ γ − 1 ] t γ −2 .


Table 1

Travelers’ risk attitudes, functional forms of utility and the properties of the vot and the vor .

Functional form Negative exponential Quadratic Box Cox

Utility function u (t) = − exp (βt) u (t) = αt + βt 2 u (t ) =

α1+ γ [ t 1+ γ − 1]

Monotony ( u ′ < 0) β > 0 α + 2 βt < 0 α < 0

Reliability attitudes Prone Averse Prone Averse Prone Averse

( u ′ ′ < 0) ( u ′ ′ > 0) ( u ′ ′ < 0) ( u ′ ′ > 0) ( u ′ ′ < 0) ( u ′ ′ > 0)

β > 0 Inconsistent β < 0 β > 0 γ > 0 γ < 0

Prudence Prudent – – Prudent Non-prudent Prudent

γ > 1 0 < γ < 1 γ < 0

vot Increasing – Increasing Decreasing Increasing Increasing Decreasing

Convex Affine Affine Convex Concave Concave

vor Positive – Positive Negative Positive Positive Negative

Increasing Constant Constant Increasing Decreasing Increasing

U

context, we show that the power utility function should be preferred, because it imposes the least restrictions on a trav-

eler’s preference towards travel time reliability. This is not the case for the negative exponential utility function, in which

travelers are assumed to be reliability prone and prudent: the vot is increasing and convex with mean travel time, and the

vor is positive and increasing.

Finally, the analytical formulae established in this study can be implemented easily by way of discrete choice experiments

in order to provide empirical valuations of the vot and vor . Such valuations would be of primary interest to policy makers

who have to conduct cost benefit analyses of public transportation infrastructure projects.

Acknowledgment

We thank the editor and the three anonymous referees for their comments which have helped us improve the paper.

Appendix

Proof of proposition 2

ii ⇒ i . Define δ(t) = F i (t) − F j (t) and observe that δ(t min ) = δ(t max ) = 0 . By definition, ii is equivalent to δ ≥ 0 every-

where, while i is equivalent to U i − U j =

∫ t max

t min u (t) dδ(t) ≥ 0 for all u with u ′ ≤ 0 everywhere. Integrating by parts, we get

i − U j = − ∫ t max

t min u ′ (t) δ(t) dt . It is then clear that ii is sufficient for i .

i ⇒ ii . We prove the necessity of ii by contradiction. Suppose that ii is not true, so that δ( t ∗) < 0 for some t ∗. Hence, we

can always find a decreasing utility function u such that i . would be violated. For instance, take u (t) = k > 0 for t ≤ t ∗ and

u (t) = 0 for t > t ∗. With this decreasing utility function, U i − U j = k ∫ t ∗

t min dδ(t) = kδ(t ∗) < 0 , which contradicts i . �


iv ⇒ iii . Define η(t) =

∫ t max

t δ(s ) ds and observe that η(t max ) = 0 . By definition, iv is equivalent to η ≥ 0 everywhere

with μi = μ j . Then note that η(t min ) = 0 is equivalent to μi = μ j . Indeed, by definition, η(t min ) =

∫ t max

t min δ(t) dt = 0 , while

μi − μ j =

∫ t max

t min t dδ(t ) . Integrating by parts, we get μi − μ j = −η(t min ) . Thus, iv is equivalent to η ≥ 0 everywhere with

η(t min ) = 0 , while iii is equivalent to U i − U j =

∫ t max

t min u (t) dδ(t) ≥ 0 for all u with u ′ ′ ≤ 0 everywhere. Integrating by parts

twice we get U i − U j = − ∫ t max

t min u ′′ (t) η(t) dt . It is then clear that iv implies iii .

iii ⇒ iv . We prove the necessity of iv by contradiction. Suppose that iv is not true, so that η( t ∗) < 0 for some t ∗. Hence,

we can always find a concave utility function u which is linear above and below t ∗ and strictly concave in the neighborhood

of t ∗, such that iii would be violated. �


v ⇔ vi . We have proved in proposition 2 that all travelers who dislike more travel time prefer ̃ t i to ̃ t j if and only if ̃t jis a fsd deterioration of ̃ t i . Besides, we have proved in proposition 5 that all reliability-prone travelers prefer ̃ t i to ̃ t j if and

only if ̃ t j contains more actuarially-neutral variability than ̃

t i . Hence, the proof of proposition 7 follows by observing that

combining any fsd deterioration with any actuarially-neutral increase in variability yields a ssd deterioration. �


viii ⇔ ix . Taking as given that the utility functions u and v are strictly decreasing, there always exists a strictly increasing

function ψ : R → R defined as ψ(u ) = v . In this context, ix. is equivalent to ψ

′ ′ ≤ 0 everywhere. By chain rule, we get v ′ =


˜

ψ

′ (u ) u ′ and v ′′ = ψ

′′ (u )[ u ′ ] 2 + ψ

′ (u ) u ′′ . Combining these two relations yields v ′′ / v ′ − u ′′ /u ′ = u ′ (t) ψ

′′ (u (t)) /ψ

′ (u ) . Thus

viii is equivalent to ix .

ix ⇒ vii . Consider a random travel time expressed as t + ̃

ε, where t is a positive constant and

˜ ε is any risk that is

disliked by traveler u , i.e. any risk ˜ ε satisfying Eu (t + ̃

ε) ≤ u (t) . According to definition 8 , vii is equivalent to Eu (t + ̃

ε) ≤u (t) ⇒ Ev (t + ̃

ε) ≤ v (t) . Substituting ψ(u ) = v , vii. is also equivalent to Eu (t + ̃

ε) ≤ u (t) ⇒ Eψ(u (t + ̃

ε)) ≤ ψ(u (t)) . On the

other hand, by Jensen (1906) ’s inequality, ix is equivalent to Eψ(u (t + ̃

ε)) ≤ ψ(Eu (t + ̃

ε) . Because ψ is increasing, we

also have Eu (t + ̃

ε) ≤ u (t) ⇒ ψ(E(u (t + ̃

ε))) ≤ ψ(u (t)) . Thus, ix implies that Eu (t + ̃

ε) ≤ u (t) ⇒ Eψ(u (t + ̃

ε)) ≤ ψ(E(u (t +˜ ε))) ≤ ψ(u (t)) .

vii ⇒ ix. We prove the reverse by contradiction. Suppose that ix is not true so that there exists an interval in the image

of u where ψ

′ ′ > 0. Suppose also that the random travel time t + ̃

ε has its support on that interval and a utility function

u satisfying Eu (t + ̃

ε) = u (t) . By Jensen (1906) ’s inequality, ψ

′ ′ > 0 implies Eψ(u (t + ̃

ε)) > ψ(Eu (t + ̃

ε) or, equivalently,

Ev (t + ̃

ε) > v (t) , which contradict vii . �


xi ⇒ x . Define κ( t ) =

∫ t max

t η(s ) ds and observe that κ(t max ) = 0 . Thus, xi is equivalent to κ ≥ 0 everywhere with μi =μ j and σ 2

i = σ 2

j . Recall that μi = μ j is equivalent to η(t min ) = 0 (see the proof of proposition 5 ). Furthermore, if μi =

μ j , then σ 2 i

− σ 2 j

=

∫ t max

t min t 2 dδ(t) . Integrating by parts twice, we get σ 2 i

− σ 2 j

= −2 ∫ t max

t min t dδ(t ) = −2 κ(t min ) showing that

σ 2 i

= σ 2 j

is equivalent to κ(t min ) = 0 . Hence, xi is equivalent to κ ≥ 0 everywhere with η(t min ) = κ(t min ) = 0 , while x is

equivalent to U i − U j =

∫ t max

t min u (t) dδ(t) dt ≥ 0 for all u with u ′ ′ ′ ≤ 0 everywhere. Integrating by parts three times we get

U i − U j = − ∫ t max

t min u ′′′ (t ) κ( t ) dt . It is then clear that xi is sufficient for x .

x ⇒ xi . We prove the necessity of xi by contradiction. Suppose that xi is not true, so that κ( t ∗) < 0 for some t ∗. In this

context, we can always find a utility function u with its slope u ′ being linear above and below t ∗ and strictly concave in the

neighborhood of t ∗, such that x would be violated. �


xii ⇔ xiii . We have already proved in proposition 7 that all reliability-prone travelers who dislike more travel time prefer t i to ̃ t j if and only if ̃ t j is a ssd deterioration of ̃ t i . In addition, we have proved in proposition 11 that all prudent travelers

prefer ̃ t i to ̃ t j if and only if ̃ t j contains more upside variability than ̃

t i . Hence, proposition 13 directly follows from these two

results by observing that combining any ssd deterioration with any increase in upside variability yields a tsd deterioration.

�


ii ⇒ xiv . By definition, ii is equivalent to δ ≥ 0 everywhere (see the proof of proposition 2 ), while xiv is equivalent

to vot i − vot j =

1 λ�t

∫ t max

t min [ u (t − �t) − u (t )] dδ(t ) ≤ 0 for all u with u ′ ′ ≤ 0 everywhere. Integrating by parts, we get vot i −vot j =

−1 λ�t

∫ t max

t min [ u ′ (t − �t) − u ′ (t)] δ(t) dt . Because by assumption λ > 0 and �t > 0, it is clear that ii is sufficient for xiv .

xiv ⇒ ii . We prove the necessity of ii by contradiction. Suppose that ii is not true, so that δ( t ∗) < 0 for some t ∗. In this

context, we can always find a utility function u which is linear above and below t ∗ and strictly concave in the neighborhood

of t ∗, such that xiv would be violated. �


iv ⇒ xv . By definition, iv is equivalent to η ≥ 0 everywhere with η(t min ) = 0 (see the proof of proposition 5 ), while xv

is equivalent to vot i − vot j =

1 λ�t

∫ t max

t min [ u (t − �t) − u (t )] dδ(t ) ≤ 0 for all u with u ′ ′ ′ ≤ 0 everywhere. Integrating by parts

twice, we get vot i − vot j =

−1 λ�t

∫ t max

t min [ u ′′ (t − �t) − u ′′ (t)] η(t) dt . Because by assumption λ > 0 and �t > 0, it is clear that

iv implies xv .

xv ⇒ iv . We prove the necessity of iv by contradiction. Suppose that iv is not true, so that η( t ∗) < 0 for some t ∗. In this

context, we can always find a utility function u which is linear above and below t ∗ and non-linear with a strictly negative

third-derivative in the neighborhood of t ∗, such that xv would be violated. �


xvi ⇔ vi . We have already proved in proposition 15 that for all reliability-prone travelers, the vot is greater for ̃ t j than

for ̃ t i if and only if ̃ t j is a fsd deterioration of ̃ t i . In addition, we have proved in proposition 16 that for all prudent travelers,

the vot is greater for ̃ t j than for ̃ t i if and only if ̃ t j contains more actuarially-neutral risk than ̃

t i . Finally, proposition 17 is

proved by observing that combining any fsd deterioration with any actuarially-neutral increase in variability yields a ssd

deterioration. �



xvii ⇔ iv . We know from proposition 5 that all reliability-prone travelers prefer ̃ t j to t i if and only if iv is true. Thus, iv

is equivalent to U i − U j = u (μi + πi ) − u (μ j + π j ) ≥ 0 for all u with u ′ ′ ≤ 0 everywhere. Using the fact that μi = μ j = μ, iv

is also equivalent to u ( μ + πi ) ≥ u ( μ + π j ) for all u with u ′ ′ ≤ 0 everywhere. Finally, iv is equivalent to π i ≤ π j for all u

with u ′ < 0 and u ′ ′ ≤ 0 everywhere. �


vii ⇔ xviii . Whenever v ′ < 0 everywhere, xviii is equivalent to Eu ( ̃ t i ) = u (μi + πi ) ⇒ Ev ( ̃ t i ) ≤ v (μi + πi ) . The l.h.s. of the

implication simply says that π i is the reliability premium of traveler u . The r.h.s. of the implication says that the reliability

premium of traveler v is greater than the one of traveler u . Because any random travel time ̃ t i may be written as μi + ̃

ε,

where ˜ ε is an actuarially-neutral risk, xviii is equivalent to Eu (μi + ̃

ε) = u (μi + πi ) ⇒ Ev (μi + ̃

ε) ≤ v (μi + πi ) . Defining t =μi + πi and

˜ ε = ̃

ε − �i , the implication can be rewritten as Eu (t + ̃

ε ) = u (t) ⇒ Ev (t + ̃

ε ) ≤ v (t) which says that traveler v

dislikes any variability affecting travel time that traveler u dislikes. We know from definition 8 and proposition 9 that this

is true if and only if vii is true. �


iv ⇒ xix . By definition, iv is equivalent to η ≥ 0 everywhere with η(t min ) = 0 which is equivalent to μi = μ j (see the

proof of proposition 5 ), while xix is equivalent to vor i −vor j =

1 λ

[ u (μi ) − u (μ j ) −

∫ t max

t min u (t ) dδ(t ) ]

≤ 0 for all u with u ′ ′

≤ 0 everywhere. Taking as given that ̃ t i and ̃

t j have the same mean and integrating by parts twice we get vor i −vor j =1 λ

∫ t max

t min u ′′ (t) η(t) dt . It is then clear that iv is sufficient for xix .

xix ⇒ iv . We prove the necessity of iv by contradiction. Suppose that iv is not true, so that η( t ∗) < 0 for some t ∗. In this

context, we can always find a utility function u which is linear above and below t ∗ and strictly concave in the neighborhood

of t ∗, such that xix would be violated. �

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Further reading

Bates, J. , Polak, J. , Jones, P. , Cook, A. , 2001. The valuation of reliability for personal travel. Transp. Res. Part E 37, 191–229 .

Bernoulli, D. , 1954. Exposition of a new theory on the measurement of risk (specimen theoriae novae de mensura sortis, 1738). Econometrica 22 (1), 23–36 .Brownstone, D. , Small, K.A. , 2005. Valuing time and reliability: assessing the evidence from road pricing demonstrations. Transp. Res. Part A 39, 279–293 .

Jong, G.C.de , Kouwenhoven, M. , Kroes, E. , Rietveld, P. , Warffemius, P. , 2009. Preliminary monetary values for the reliability of travel times in freight transport.Eur. J. Transp. Infrastruct. Res. 9 (2), 83–99 .







































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Transportation Research Part B - UPHF · not, respectively. By adding scheduling considerations to both the utility function and the time consumption constraint, Small (1982)