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The value of time: a theoreticalreviewRosa Marina González aa Departamento de Economía y Dirección de Empresas,Facultad de CC Económicas y Empresariales , University ofLa Laguna , Camino de la Hornera s/n, Guajara Campus,38071 La Laguna, Tenerife, SpainPublished online: 13 Mar 2007.
To cite this article: Rosa Marina González (1997) The value of time: a theoreticalreview, Transport Reviews: A Transnational Transdisciplinary Journal, 17:3, 245-266, DOI:10.1080/01441649708716984
To link to this article: http://dx.doi.org/10.1080/01441649708716984
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TRANSPORT REVIEWS, 1997, VOL. 17, No. 3, 245-266
The value of time: a theoretical review
By ROSA MARINA GONZÁLEZ
Departamento de Economía y Dirección de Empresas, Facultad de CC Económicas yEmpresariales, University of La Laguna, Camino de la Hornera s/n, Guajara Campus, 38071
La Laguna, Tenerife, Spain
We review the main models postulated to study the subjective value of non-working time emphasizing the most important aspects behind the microeconomicformulation of time allocation models: among others we consider the behaviouralassumptions, the role of constraints in the maximization process implicit in thistask, and the relevant variables in the utility function. We analyse first thepioneering models of Becker (1965), De Serpa (1971, 1973) and Evans (1972), whodevelop a general theoretical framework which may be applied to modal choicemodels. We then go on to consider Train and McFadden's model (1978) wherethe formulation incorporates the theory of random utility in order to analyse thechoice of individual transport services. This analysis is further developed in Batesand Roberts (1986), Jara-Díaz and Farah (1987), and Jara-Díaz et al. (1988), whoextend and perfect the analysis to establish what may be considered as the actualposition of research on the subject, and to which the most recent contributions ofJara-Diaz (1994) are to be added.
1. IntroductionThe evaluation of travel time savings plays an important role in the appraisal of
investment projects in the transport sector, if we are to consider that the timevariable is a key component of the so-called generalized cost. Traditionally, the basicdistinction for the analysis of time values has been to divide time into two categories:working and non-working time. Trips made during the working day are part of whatis called working time, whereas other travel activities are included under the labelnon-working time (i.e. trips to work, shopping trips, study trips and so on). Thederivation of working time values is based on the cost saving approach, whose mainprinciple is that the employer engages labour until the marginal value of the productis equal to the wage. Therefore, the value of working time is usually approximated bythe gross wage rate for each work category.
On the other hand, the value of leisure (non-working) time, which is our maininterest here, has been derived from studies based on microeconomic principles ofconsumer behaviour. In transport economics this value is obtained from the analysisof the choices made by individuals when faced with a variety of travel modes orroutes which have different characteristics. The generally accepted method forestimating a subjective value of time (SVT) consists in finding the marginal rate ofsubstitution between travel time and travel cost, typically from disaggregate modelsof discrete choice based on the random utility theory (see for example, Gaudry et al.1989). The interpretation given to a value obtained in such a way is that of thewillingness to pay in order to reduce travel time by one unit.
2. The microeconomic theory of value of timeThe theory of consumer choice which incorporates the time variable (time
allocation theory) appears in microeconomics with the purpose of explaining the
0144-1647/97 $12·00 © 1997 Taylor & Francis Ltd.
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supply of working hours by individuals. Nevertheless, research done in other areaspointed to the need for a more general theory that could explain other economicaspects of the individual's use of time; for example, advances in transport economicshave been made with the aim of determining the exact value of travel time savings, asit was observed that the results of cost-benefit studies vary greatly with the value oftime (Tipping 1968).
The valuation of travel time is based on the theory that time is an economicresource which all individuals have in the same fixed quantity. Individuals mayallocate this resource to each activity in different quantities in such a way that eachtime allocation will have different consequences in the individual's budget and utilitylevel. A basic assumption of the theory is that individuals choose the time allocationthat maximizes their personal utility, subject to the fact that time, unlike money,cannot be stored and as such can only be transferred between different activitieswhich may be interchanged at a particular moment. Another basic assumption of thetheory is that the different allocations of time among activities for an individual havedifferent values which may be measured in money terms. For example, an individualmight be willing to increase travel time if this means paying a lower fare.
The analytical approach followed to obtain travel time values in mode choicemodels often takes the theories of Becker (1965) and/or De Serpa (1971) as startingpoints. This allows for the justification of the presence of time as an explanatoryvariable. Following these authors, other approaches have since appeared which willbe dealt with below.
3. Theoretical review of time allocation modelsThe first important research on this subject dates from 1965 when Becker
proposed a model based on the systematic incorporation of non-working time andwhere utility depends on the consumption of basic commodities (Z,). These basiccommodities have no interchange market available and their production requiresboth market goods (x,) and time (T,) in such a way that the commodities may beexpressed as
Z, =/,(*,, 7-,) (1)In this formulation Tt is a vector where each component refers to different aspects oftime. The hours used during the day may be distinguished from those used at night,and the same distinction may be made between week and week-end hours.
In this context the individual is both a producer and a consumer, and combinestime and market goods through a household production function in order toproduce commodities Z,-. Moreover, combination of these commodities is chosen tomaximize utility; the utility function to be maximized is of the type
U=U(Zu...,Zm) = U(fu...,fm) = U(xu...,xm;Tu..., Tm) (2)
subject to a budget constraint of the form
g(Zi, . . . ,ZM) = Z (3)
where g is an expenditure function of Z,- and Z is the maximum quantity ofresources. One objective is to find measures for g and Z that facilitate the empiricalapplications. In this sense the most direct approach is to consider the mazimizationof the utility function in (2), given the production function (1), subject to expenditureconstraints on market goods and time constraints, where working hours are avariable (W). These constraints are
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The value of time: a theoretical review 247
YjPixi = IF+Ww (4)i
m
Y,Ti = Tc = T-W (5)i
where pt is a vector of prices of the xt, Tc a vector giving the total time spent inobtaining the commodities (Becker calls this 'consumption time', but does notdistinguish between consumption time and time devoted to household production),T the total time available, IF the fixed (non-salary) income, and w the wage rate perhour. As Becker points out, these constraints are not independent (pp. 496-497)'...because time can be converted into goods by using less time at consumption and more atwork". In fact if the production function is reformulated such that
Ti = t,Z, (6)
xi = b,Z, (7)where bt and tt are the requirements of market goods and time per unit of Zt, PFfrom(5) may be substituted in (4) yielding:
£)(?,*,+ /,w)Z, = / f+r.w (8)i
The right-hand side of this equation is identified as full income and represents themaximum quantity of money that the individual could obtain if all time weredevoted to work. This income is spent on commodities (Z,) both directly throughexpenditure on goods (S/?,-6,Z,) and indirectly through the forgoing of income as aresult of using time on consumption rather than on work (Ztjwz,). Therefore, thistheory establishes that a re-allocation of time implies a simultaneous re-allocationof goods (xt) and commodities (Z,); thus these three decisions are intimatelylinked.
Becker's model introduced a series of new features with respect to the traditionaltheory which have been summarized by Pollak and Wachter (1975):
It develops a framework of analysis for the choice of goods and time within thehousehold; the market goods and time are not wanted by themselves but as inputs inthe production of commodities. The role of household technology and tastes asdeterminants of behaviour is emphasised. This allows the possibility of attributingvariations in household behaviour to changes in technology, changes in the price ofgoods (inputs) or in income, as well as changes in tastes. It focuses the attention onthe commodities (Z,) which are the output of household technology, while thetraditional theory ignores commodities and only considers market goods (xi).
A classic example in which Becker's model is applied is offered by Gronau (1970).He characterizes travel mode choice as the result of a decision to minimize totaltravel cost (i.e. time plus money). In this work empirical evidence is offered showingthat the individuals with higher values of time are those that choose the time-savingmodes. However, as Juster and Stafford (1991) point out, this example serves toshow the difficulty in defining the 'Z,' and the inputs (xt, T,); the question arises as towhether the journey itself is a basic commodity (Z,) or, on the contrary, anintermediate input to produce the actual Z,-.
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When Pollak and Wachter (1975) analyse Becker's model, they argue that exceptin special cases this theory does not provide a satisfactory model of commoditydemand and of time allocation. They argue that the application of the householdproduction function requires strong assumptions about the structure of householdtechnology, in particular the presence of constant returns to scale and the absence ofjoint production. Therefore, if these assumptions are not satisfied, the commodityprices depend on the patterns of household consumption and thus fail to play theirtraditional role in consumer theory.
The commodities which involve time as an input are a special problem, as thetime given over to some production activities is a direct utility resource as well as aninput to the provision of commodities. This indicates a type of joint production andsuggests that prices do not act independently of the quantity of commoditiesconsumed. Thus when joint production exists or when the returns are not constant toscale, these authors suggest that commodities demand should be analysed in terms ofthe price of market goods (x,).
On the other hand, Becker's assumption that 'time can be converted intocommodities' is only tenable in a context of analysis similar to that proposed, whereworking hours may be chosen freely by the individual and where, moreover, these donot appear as arguments of the utility function. This affects the interpretation thatshould be made of the results, as Jara-Diaz (1994) notes; furthermore there areimportant restrictions in the model, such as not considering the time required for theconsumption of commodities, which have been taken into account in laterspecifications (De Serpa 1971, Evans 1972).
In 1966 Lancaster presented a study that brought an important change in theapproach to the theory of consumer behaviour that was prevalent at the time,although Quandt (1956) had presented a similar approach ten years before. Thisauthor considers it to be the characteristics and properties of the goods that deriveutility for an individual; therefore it is assumed that consumption is an activity inwhich goods act as an input and that what is really valued by an individual are thecharacteristics of these goods. This assumption gave rise to new formulations of themodels of consumer behaviour, although it may be noted that Becker's model (1965)has a similar approximation in the sense that goods (x,) are not the direct input in theutility function.
In 1971 De Serpa proposed a model which overcomes some of theshortcomings of Becker's model by considering that both the time necessary forthe consumption of market goods and the amount consumed affect the utility.This allows the use of time to be seen in a different light depending on theactivity in which it is spent. Budget and time constraints are included while at thesame time De Serpa adds what may be identified as the first set of 'technical'constraints, which indicate the minimum times required for the consumption of aset of given market goods.
In De Serpa (1971, 1973) the utility function considers a vector x of consumptionof the different goods and a vector t of times allocated to different potentialactivities. Moreover, a variable tw is included which represents the amount ofworking time. The individual maximizes the utility function:
max ,,. . .„.: U(xh x2,... xm; tu t2,..., tm;t^) (9)
x, t
subject to various constraints.
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The value of time: a theoretical review 249
The first constraint is a budget constraint which expresses that the expenditure onconsumption of goods (px) cannot exceed the available income (wt^+y), w being thewage rate per hour and y the non-earned income
wty, + y >px[X] (10)
where the multiplier (1) represents the marginal utility of income.The second constraint considers that the total sum of the times allocated to each
activity cannot exceed the total time available
$ > + 'w<rM (li)
where the multiplier (/i) is the marginal utility of time as a scarce resource.The third constraint supposes that each good or service consumed requires a
minimum time proportional to the amount consumed. Moreover, there is aminimum duration of the working day /w° that is considered in the fourth constraint(13):
tt>atx, Vi[*4] (12)
'w > C [*] (13)where *P, is the marginal utility of reducing the time necessary to consume i, and maybe interpreted as the marginal utility of a travel time saving, and O is the marginalutility obtained if the minimum amount of working hours is reduced by one unit.These last two constraints allow us to see that in this model the amount of timedevoted to each activity is derived as much from an individual choice as from anecessity.
The solution of this maximization problem yields the following first-orderconditions:
H-•*?*-*«* = o (14)
^ - / * * , - 0 (15)
— + \w-n+$ = 0 (16)
U > aiXi (17)
Starting from these conditions different measurements of the value of time can beobtained, each one of them representing different hypothesis and concepts about therole of the time variable. The expression ((8U/8t,)/A) can be taken as the marginalvaluation of time on activity /, what De Serpa calls the value of time as a commodity,the value of which depends on whether time generates utility or (dis)-utility whenconsumed. The value of time as a resource corresponds to the marginal rate ofsubstitution between time and income (ji/X) and indicates what is the monetary valueof having an additional unit of time (7). However, as it is impossible to increase thetotal time available to the individuals, this definition for the value of time makes littlesense at an empirical level. QPt/X) would represent the marginal rate of substitutionbetween the time spent on activity / and money. It considers the propensity of theindividual to pay for saving a unit of time on activity i; this is what is generallydefined as the value of time savings and may be expressed as:
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%/X = n/\ - (6U/Stt)/X (18)Thus in those activities, such as leisure, where constraint (12) does not come intoplay (that is, the individual devotes more time than the minimum required for theconsumption of a good) the dual variable will be zero (*F, = 0) and therefore thevalue of a time saving in the consumption of that activity will also be zero. In thiscase the following equality applies (5U/St,)/A = nil) which indicates that the utilityderived from the time used in the consumption of good i is equal to the value oftime as a resource. The goods that fit this condition are termed goods of 'pureleisure'.
On the other hand, when the amount of time spent in the consumption of acertain good is greater than the consumer feels fit, the transfer of time between theconsumption of that good to one of pure leisure would increase the utility level. Inthis case, Q¥i/l) would be the value of a time saving in activity / transferred to theconsumption of an activity of pure leisure. This concept is what is normallyunderstood in transport as the value of time.
There are strong differences between Becker's theory and that of De Serpa, asTruong and Hensher (1985a) point out. Becker's theory leads to the concept ofshadow price, or opportunity cost, of travel time that is uniform in all activities andcircumstances (given that time is considered an infinitely divisible and homogeneousresource) which give rise to the value of time as a scarce resource. On the contrary,De Serpa's work postulates the theory of time allocation as a discrete choicesituation where the value of time is allowed to differ from a uniform shadow price.According to these authors the difference between De Serpa's value and Becker'sshadow price represents the value of 'transferring' time between alternative activities.When this value is zero or is the same for two activities there is no reason to transfertime between them. On the other hand, when this value differs between activitiesthere is a loss or a net benefit derived from the transfer of time between them. Thedifferences between both theories will be developed further below when consideringTruong and Hensher's approach (1985a) in more detail.
Alternative formulations to those proposed by De Serpa (1971, 1973) are foundin Bruzelius (1979). In these models the consumer maximizes the utility functionwhich has certain mathematical properties, subject to a budget constraint, a timeconstraint and different types of time allocation constraints. The formulation leadsto demand functions where the time requirements are explicitly recognized.
In 1972 Evans proposed a model in which the only argument of the utilityfunction is the time allocated to the different activities. Thus for example, as theauthor points out, the utility function could be said to depend on the time spent atthe cinema instead of the number of visits to the cinema. This model seemsreasonable since some activities, in particular work, must be measured in units oftime given the absence of any other suitable measure. Thus pure time activities canexist as a specific case. The consumer chooses the best combination of activitiessubject to time and money constraints. The utility function of the individual isexpressed as follows:
U=U(t,) (19)
where ?,• denotes the number of units of time that the individual spends in activity /.This theory is static in as much as it assumes that the time considered is small enoughto ignore the problems of capital and interest. The constraints considered in this caseare:
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The value of time: a theoretical review 251
T=J2*i M (20)i.
5 > t i = o W (21)1
where the cost per hour rt may be positive if the individual pays for the activity,negative if it is paid for (i.e. if it is work hours) and equal to zero if the activity is free.The cost per hour of some activities may be equal to the sum of the different costcomponents.
If (19) is maximized subject to the previous constraints n equations result of theform:
U, = n + Xrt ( i = l , 2 , . . . , n) (22)
Thus, the time spent in each activity depends both on the utility or disutilityderived from it, and on the price that the individual receives or pays forundertaking it. Solving (22) for \x allows to observe that if the individual allocatestime optimally among the different activities, a small increase in the time spent inone activity together with an equal decrease in the time spent in another activitywould make the individual no better off and no worse off. However, theempirical evidence in transport studies shows that a reduction in travel timeconstitutes an improvement for the individual, contrary to what this theoryestablishes. Thus, if in this context one defines the marginal rate of substitution(MRS) between the yth and rth activities as the time in the zth activity whichwould just compensate the consumer for the loss of a marginal unit of time in theyth activity, the following equation is obtained:
MRS . - | = | ,23)
Thus, if this equation is substituted in the equilibrium conditions (22) we obtain
^ = ^ i = MRS (24)Uj n + Xrj K '
This equation shows that in equilibrium the ratio of prices is not equal to themarginal rate of substitution:
It is important to note that when the results of empirical studies are compared withthis theory important controversies appear, essentially deriving from the implicitassumption at the theoretical level that the consumer is free to allocate time amongactivities. This made it necessary to introduce a new constraint in the mathematicalformulation of the model, derived from the distinction between the amounts of timethe consumer wants to devote to different activities and those which must actually beused. This new constraint incorporates the relation among the times spent in thedifferent activities and indicates that the time spent in activity / may be technicallyrelated to the time devoted to activity/ In this case the author proposes an examplewhere the individual, when going to the cinema, has to travel, thus relating traveltime (tt) with the time spent at the cinema (/c) through an inequality, via
h > btc (26)
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where b is a constant usually inferior to one. Thus, it is postulated that the individualmay spend more time travelling than btc but not less. This constraint may also beexpressed as:
btc = tt<0 [k] (27)and adding it to the problem Evans proposed the following model:
Max: u = u(tv/, tt,,tc; u) (28)
s.t. tVf + h + tc + YJti=T
btc -h<0
Mw + rttt + rctc + ̂ 2 rxU = owhere tw, tt, tc, /,• denote the amount of time spent in work, travel, cinema and otheractivities, respectively. The first constraint is a time constraint, the second is the timeallocation for travel and the third is a budget constraint, where rw<0 and rt, r c ^0 .
Furthermore, we can observe that if tw, tu tc are greater than zero the first-orderconditions are
uw = n + Arw (29)
ut = fi + Xrt-k (30)
uc = fi + \rc + bk (31)where A: is a Lagrange multiplier which can be identified with the marginal utilityderived from a relaxation in the travel time allocation constraint. This constraint isan inequality and according to the Kuhn-Tucker condition: fc^O; both if k = 0 or ifbtc—tt = 0. The marginal utility cannot be negative and it will be zero when tt>btc.If this inequality is not operative and the time the consumer wishes to spend ontravelling is greater than the time necessarily spent travelling, any increase ordecrease in the travel time will not alter the time allocation. In this case the monetaryvalue of a small relaxation in the constraint related to the time the individual mustuse in travelling is given by (k/X) and is obtained by dividing equation (29) byequation (30).
k_ ut-n , .— — rt r w [iz)A MW — fi
This equation establishes that the marginal valuation assigned by the consumer to adecrease in travel time is equal to the cost of travel time (rt) plus the time savedvalued at the rate rw(i/t—n)/(uw—n) per hour. Thus the monetary value of travel timesavings expressed in (32) may also be formulated as the cost of travel time (rt) minusthe value of the marginal utility of travelling (ut/A) plus the value of the marginalutility of work (MW/A) plus the wage (rw). This is obtained considering that thedenominator (ww — fi) in (32) may be expressed as (Arw), as is clear from the conditiongiven in (29), and that the value of (ji) appearing in the numerator is equal to(liw—Arw).
The value of time is positive since rw is negative. The individual valuation of timewill depend on the preferences for travelling as opposed to other activities. If theindividual wishes to spend more time travelling, the constraint is not effective andthe valuation of travel time will be equal to minus the travel cost. The condition ofequilibrium given by (25) will be
_H = Hl^Ji (33)rw uw - fi
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The value of time: a theoretical review 253
In this situation a small variation will not make the individual any better orworse off.
In 1978 Train and McFadden formulated a model in which the problem studiedis seen as a typically microeconomic decision where the individual must choosebetween leisure (L) and consumer goods (G):
Max: U(G, L) (34)
s.t. G + Cj = w.W+Y
W+L + tj = Twhere Y is unearned income, w is the real wage rate, cj and tj are money and timespent on travelling, T is the total amount of time and W the number of hoursworked. Moreover, j belongs to a set M that represents all available modes.
This problem is solved in two steps: initially the utility is maximized conditionalon values of cj and tj, and secondly it is optimized with no conditional value:
JTM- \M™((»W+Y-c,),T-W-tl)\ (35)
The initial solution produces W* (cj, tj), in such a way that one obtains the indirectutility function whose level depends on the chosen mode:
Uj = U[W(cj, tj)} (36)
Note that in this context the individual can choose the number of hours worked andthus income is changed into an endogenous variable even when one considers thatwages are determined exogenously, which is in many cases a rather unrealisticassumption. The second step is a discrete optimization that requires Uj to becompared to Uk, where bothy and k belong to M. The chosen mode is the one whichproduces the greatest utility.
Starting from here, different functional forms may be specified. In particular, ifone assumes that the utility function has Cobb-Douglas form (U = KG^~®lJ) it ispossible to deduce that the indirect utility function must have the form:
Vt = -K{\ - p)1-f>0fl[(w-fi.cl + w^-V.ti} (37)
where c,- is the fare by mode i, tt is travel time, and w the wage rate.One must emphasize that in the Train and McFadden individual consumer model
(1978) it is supposed that the individual can freely choose the time devoted to work(W) and this implies that the value of working and non-working time is the same andequal to wage rate, as the following equation shows:
Thus, in this model the marginal rate of substitution between goods and leisure(MRSLjG), which is equivalent to the value of time is equal to w. Nevertheless, thestochastic version of this model permits, as Jara-Diaz (1990a) notes, values of timethat differ from the wage rate because the typically used utility function is as follows:
Ut = a + P(c,/Y/) + itt (39)
and the relation (y/P) expresses the subjective value of time as a proportion of thewage rate. This has meant that from the work of Train and McFadden (1978)onwards there has been a tendency to express the value of time as a percentage of the
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254 R. M. Gonzalez
wage rate, though in some specifications of the modal utility function the wage rate isapproximated by income divided by the number of hours worked (see, for example,Ortuzar and Espinoza 1987) or is even replaced by other proxy variables (Swait andBen-Akiva 1987).
Subsequently other theoretical developments have been proposed such as that ofWinston (1982) which postulates a theoretical model where the utility of an activity iat time t is a function of the satisfaction from the activity itself and of the intensitywith which it is carried out. Both satisfaction and intensity may depend on the time t,which leads to the utility of an activity being dependent upon the time or moment ofthe day. Winston (1982) assumes that the utility derived from an activity has twocomponents: the satisfaction of doing something and the satisfaction of achieving it.Alternative assumptions about the distribution of the process and goal achievementutility have been made by Recker et al. (1986).
Truong and Hensher (1985a) specify Becker's model (1965) as the result of thefollowing consumer maximization problem:
Max: u(Gh Lt) (40)
s.t. Gi<M-Ci [A]
Li<T-Ti [fx]where utility is a function of Gt and Lh which represent the goods and leisure time(71,) that remain after having deduced the travel cost (C,) and travel time (T,) of thechosen mode. As Truong and Hensher (1985b) also point out in a subsequent study,(40) means that individuals do not obtain utility or (dis)-utility from the costs ortravel times as such, but from the goods and free time that remain once the travelexpenditure and time are deduced from the available money and time. Thus thecoefficients of cost and time in the modal utility represent the marginal utilities(multipliers) of the constraints for income and time in problem (40). The first-orderconditions for this problem are given by
du/dd = A (41)
du/dLj — \i
where the shadow price of time is defined as
duldLi. =H (42)du/dGr=max A k ;
This approach assumes that the individual can freely allocate Th that is, theindividual does not perceive additional constraints to those of total availability;therefore, if there are no technological constraints and it is assumed that in Becker'sframework the allocation of time is optimal, one may conclude that it is impossibleto get improvements through reallocation of time. However, one might ask about theexistence of the technological constraints considered by De Serpa (1971), which referto the minimum allocation of necessary time associated to an alternative and lead toa different problem formulation.
In order to introduce this last assumption Truong and Hensher (1985a) specifyDe Serpa's model with a utility function of the following form:
Max: U(Gh Lu Tt) (43)
where G, represents commodities (measured in monetary terms), L, is leisure time,
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The value of time: a theoretical review 255
and T{ travel time (both times measured in minutes). Subscript i indicates thedifferent modes of transport that the individual may choose. The utility is maximizedsubject to the following constraints.
Firstly, there is a budget constraint
Gi<M-d (44)
where C, is the cost of transport and M is the income endowment, which is fixed.Secondly, there is a time constraint
Li<T-Tt (45)
where T is the total time available.And thirdly there is a constraint of time available for consumption, which
considers that the individual must assign a minimum amount of time (77* = a,C,-,which is institutionally determined, or by the physical conditions) to travel in each ofthe modes.
Tf < T, (46)
The multiplier associated with this constraint would generate a distortion in the Tt
coefficient. Note that this constraint supports De Serpa's central argument: theindividual must spend the minimum time required but is free to assign more than thenecessary amount. (In value-of-time applications, this minimum time requirementhas been interpreted as the average travel time for each individual).
From the first-order condition to obtain a maximum the value of time savings formode i is as follows:
where the parameters (A, /i, y, ¥,) are the Lagrange multipliers associated with thethree constraints (44), (45) and (46), respectively. Moreover, as the authors pointout, given that Tt now appears in the utility function one can talk about the value oftravel time and refer to what De Serpa called the value of time as a commodity, orwhat Truong and Hensher call the quality-adjusted value of time (this value differsfor different activities — transport modes — because in general the circumstances— what Truong and Hensher call quality — under which the time is spent aredifferent) in contrast to the value of time as resource (/i/A) which is quality-unadjusted.
This value of time may be broken down into a first term (fi/X), which refers to thevalue of time as a scarce resource, and a second component that indicates thedifference between the shadow price of time and its actual value in a specific activityi. This value OF,/A) is labelled the value of saving time by De Serpa and indicates thegain for the individual if technological constraint can be taken close to the value T*t.Likewise, Truong and Hensher (1985a) call this term (T//A) the value of timetransference because they consider that time savings can only imply transfers of timeto other activities, and propose that this term be understood as the value oftransferring time from one activity / to leisure (where *P/A = 0).
Equation (47) may be reordered as follows:
% n du/dTjXX X K '
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256 R. M. Gonzalez
reflecting De Serpa's so-called equilibrium and establishing that the benefits to theindividual derived from allocating less time to the journey in mode i, T,/!, isrepresented by the difference between the marginal value of time as a scarce resource(fi/X) and the marginal value of time spent in the journey by mode i, (dU/ST,)/!.Normally, the value of time varies according to the use made of it; if more time isdevoted than the necessary amount, *P takes on a zero value and therefore the valueof saving time is zero.
Truong and Hensher (1985a) introduce the concept of value of quality-adjusted time ((5U/dtj)/X) and value of transferring time OP/1). These authorspropose a model of discrete choice that allows them to calculate the differencebetween the value of time as a resource (jt/X) and the value of transferring time.For this purpose they consider an individual faced with several transportalternatives. Thus, under the hypothesis that the value of time is the same foreach alternative, there is no reason why the individual should want to transferunits of time between alternatives (*F,-/A = Y//A). On the contrary, the individualwill increase the utility if travel time is transferred to non-transport activities OP,-/A > To/A, where *F0/A represents the value of saving time in a particular non-travelactivity). For example, if *F0//l = 0 for leisure activities and y¥/X>0 for travelactivites, then there is a net benefit gain of welfare Q¥/X — *P0M = *P/A) ontransferring travel time to leisure activities.
In a modal choice model, demand is assumed to be static and in equilibrium, sothat, according to these authors, in this case ¥,•/! must be equal to ^o/A in order toavoid destabilizing effects on the level of demand for travel activities. Moreover,given that there is an ample range of non-travel activities, we must suppose that thetechnological constraint is not binding, that is TQ = 0. Therefore, when it issupposed that the coefficient of time is the same for each alternativeOP, = *Py = x¥0
= 0), this model is consistent with Becker's theory and produces avalue of saving time (in De Serpa) equal to zero, given that what we are reallycalculating is the value of time as a resource (ji/X). On the contrary, if we allow thetime coefficients to be different one might derive gains, in terms of welfare, from thetransfer of time among the alternatives. However, one must note that the existence oftechnological constraints and indivisibilities in the transport market make it doubtfulthat *P, be equal to *P0 when the demand is assumed to be static. Likewise, from thefirst-order conditions one cannot state that Q¥t = *P/) implies necessarily that
Truong and Hensher's (1985a) application of Becker's (1965) and De Serpa's(1971) models, using a formulation of discrete mode choice, shows that theindirect modal utility function should have a specific coefficient for time when thetechnical constraints derived from De Serpa's models are introduced, and thatthis coefficient must be generic if mode choice is derived from the theoreticalframework proposed by Becker. This difference is also due to the fact that traveltime cannot enter the direct utility function of Becker's model, whereas it doesnot explicitly appear in De Serpa's model. Truong and Hensher use a 'goods-leisure' (GiLj) approach for both models. However, as Jara-Diaz (1994) pointsout, given that the goods and 'activities' are explicitly expressed as vectors in DeSerpa, that working time is not adjustable and, moreover, that additional timeconstraints appear, interpreting the arguments of De Serpa's utility function (X,T) as expenditures (goods) and leisure time seems hardly operative in thiscontext.
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The value of time: a theoretical review 257
The framework of analysis proposed by Truong and Hensher (1985a) has beenamended by Bates (1987). This author considers that Truong and Hensher shouldnot regard Becker's model as implying 'P,/^ to be zero. According to Bates, whatBecker's model implies is that the utility from spending time in non-leisure activitiesis zero, because Tt does not enter Becker's utility function. In De Serpa'sformulation, expressed in equation (47), it can be seen how the left hand side ofthat equation is zero in Becker (du/ST,) and that is why the value of transferring timeto leisure is the resource value of time.
According to Bates (1987), Truong and Hensher should have taken into accountthat it is reasonable to expect a negative marginal utility of travel time, which is whythe value of time refers to the value of saving time in transport economics. Theinvalidating argument of Truong and Hensher's model, which is consistent withBecker's framework, is that expressed in the following equation ¥ , / ! = To/A = 0which should be replaced by T,/2 = fi/X. Thus when utility is only expressed in termsof Gt (goods expressed in monetary terms) and Lt (leisure time) in the case of Becker,the following may be assumed:
V/ = a,- + du/dGi.Gi + du/dLi.Lt (49)
Substituting the first-order conditions for a maximum given by (41) yields:
vt = at + \{M - Q) +n(T- Ti) (50)
Moreover, as XM and fiT do not vary between alternatives, this formulation may bereduced to
v,- = on - \.C( - fi.Ti (51)
which is identical with the result of Truong and Hensher (1985a) when consideringBecker's model. However, the reasoning behind it is different.
Likewise, in the case of De Serpa (1971) the utility depends on <?,-, L,- and Tt
(travel time) in such a way that the first-order approximation is given by
v,- = a, + du/dGi.Gi + du/dLi.Li + du/dTh Tt (52)
Substituting from the first-order conditions again we get
v, = a, + \{M - Q) +n{T- Tt) + (/i-*/).T< (53)
and, once again eliminating the invariant elements, we get
v,- = at - X.Q - fi.Ti + {n-9,).Ti = on - X.Q - %.Ti (54)
The only difference between this equation and (51), which is obtained from Becker'smodel, is the substitution of *?,• by [i. Note that \i appears in Becker's formulation asa consequence of the assumption that the marginal utility of travel time is equal tozero.
Bates (1987) comes to the conclusion that the indirect utility function of a modalchoice problem does not in itself refer to the resource value of time, but to the valueof transferring time from activity i to leisure. In this way, negative values are notpossible even when positive utility may be derived by assigning additional time to thetrip because the constraints are not binding; this means that even greater utility couldbe obtained by transferring time to leisure. (Recall that, according to the theory ofLagrange's multipliers, the value of transferring time from any activity to leisure cannever be negative).
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258 R. M. Gonzalez
Bates also points out that the hypothesis of Becker's model cannot be testedempirically unless the resource value of time is known. Furthermore, when DeSerpa's model is contrasted empirically and one finds that the values of transfer timeare not significantly different, this does not guarantee that a resource value of timecan be obtained, because it would be necessary to assume that the marginal utility oftime spent on travelling is zero, rather than merely a constant for all modes.
In the mid-eighties Bates and Roberts (1986), building on the approaches of DeSerpa (1973) and Train and McFadden (1978), presented a model in which eachactivity has a minimum time requirements tp, giving rise to the appearance of twotypes of activities: those in which the individual has to spend a greater amount oftime than desired (among which travelling is of course found) and those where theindividual would like to spend as much time as possible (more than the minimum)and which correspond to 'pure leisure' activities. Minimum time constraints will bebinding in the first case but not in the second.
The assumption of infinite flexibility in choosing the amount of working time israther unrealistic in most cases and constraints found by the individual to determinethe tw value affect mainly the value of time. This is the main reason given by theseauthors to propose a model where working time is an exogenous constant thatindividuals cannot determine as they wish. In this case fw disappears from theconsumer objective function and the total monetary income becomes fixed (Y).Moreover, the total time available is equal to 24 hours daily minus the time requiredfor basic biological necessities and the time spent on work:
T=T-tw (55)
The model formulation considers a utility function such as U(x, q, t\,..., tn), where xis the consumption of a generalized good, q the time spent on a generalized activityand tt the travel times. This utility function is maximized subject to budget and timeconstraints, and a constraint of minimum time for the activities carried out by theindividual (technological constraint):
: U(x, q, th t2,..., tn) (56)x, q, i
s.t. Y = px + J2siCi [A]
i
ti>ti°, Vi [*,]
x, q, t > 0
where c,- is the cost of travelling by mode i, t,° is the minimum time for activity i;and (1, n, *¥,) are the dual variables associated with the problem constraints. Onthe other hand, the 8t variables may only take the values 1 or 0, depending onwhether the corresponding activity is chosen or not. The model formulationassumes that only one of them can be different from zero, as the activities aremutually exclusive (for example, the choice of destination, mode or route for agiven journey). The variable definition must be consistent with the choice of theperiod T. The following first-order conditions define an optimum solution for theconsumer problem:
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The value of time: a theoretical review 259
g-A, (57)dU
dU . . _
(1) Y = px + Yt6icii
(2) T=q + '£(6lt,i
(3) r,>r,° Vi
Moreover if tt>ti° then *P, = 0. Making a linear approximation for the direct
utility function, we get
Replacing the partial derivatives by the values obtained from the first-orderconditions and using the first two constraints in (57) to eliminate the x and qvariables, we obtain a linear approximation of the indirect utility function for mode /:
v,, s a + A( Y - ^2 6,.a) +fiT-J2 WiU (59)
Then, the conditional indirect utility functions, once the terms that do not vary acrossthe alternatives have been eliminated, have the following expression (linearapproximation):
V, s -Xc,-%t, (60)
which corresponds to the deterministic part of the indirect utility function that is usedin discrete choice models (Ortuzar and Willumsen, 1994). It has to be considered thatas a consequence of eliminating working time as a variable of the problem, thecoefficients of time and cost have no relation with the wage rate, contrary to whathappens in Train and McFadden's model (1978).
On the basis of this theory the rate ¥,-/A between the coefficients of discrete traveltime and fare variables may be interpreted as the value of time saved by travelling inalternative /. On the basis of the same theory we might also reasonably suppose thatin general the following proposals hold. Firstly, given that the coefficient of thejourney cost variable (A) is the dual variable of the budget monetary constraint (andtherefore represents an approximation of the marginal utility of income) its valueshould decrease with income. Secondly, given that the coefficient of the travel timevariable Q¥,) represents the difference between the marginal value of time allocatedto travel by alternative / and the value of time as a resource (leisure time), it is logicalto expect that:
(a) its value should increase as the travel conditions worsen.(b) its value should be greater for busier individuals (given the higher alternative
resource value of time).(c) its value should decrease as the constraints on use of travel time saved in
leisure activities increase.
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260 R. M. Gonzalez
Some important limitations, which have been called 'constrained transferabilityof time' in MVA Consultancy et al. (1987), are observed in the theoretical analysis.The theory implies that there are no constraints to transfer time between activities inthe process of utility maximization; however, these constraints do exist in practice. Itmay be observed that on occasions it is necessary to save a minimum amount of timein order to carry out a certain activity; likewise as some activities may only be carriedout in a certain place or occur at a given moment in time it might happen that thetime savings cannot be employed in useful activities. Thus these constraints may beseen as an important limitation to be considered in the model specification, whichaffect the results obtained.
We now turn to analyse the model proposed by Jara-Diaz et al. (1988), who madethe same assumption as Bates and Roberts (1986) by considering work time (tw) asan exogenous variable and in consequence modified the model used by Train andMcFadden (1978), posing the following consumer problem:
max: U = A.G1~0.L/} (61)
s.t. G = wlt + E - B.ct
L=[T-1J)-B.U
where G represents the goods expressed in monetary terms, L is leisure time, c,- and tt
refer to cost and total travel time using mode /, B is the number of journeys madeduring a certain period, w is the wage rate and E is non-salary income. Thereforey = w.ty, + E represents the individual's total income.
Thus, as tw is an exogenous value, the available time that the individual mayfreely allocate is (T— ?w). Given that in this case there is nothing to optimize ifalternative i is determined, one can directly obtain the conditional indirect utilityfunction in (62) using the constraints given in equation (61) and replacing G and L inthe objective function:
V, = A.(Y-B.ci)l-f3.(T-lv, - B.uf (62)
Now making a linear approximation of Vt around (Y, T—tw), defining g = Yj(T—tw) as the expenditure rate (Jara-Diaz and Farah 1987) and eliminatingirrelevant terms, the following expression may be obtained:
v, s a.g-e - u.g'-e (63)
Calculating the value of time as the ratio between the coefficients of time and cost,we find that the value is equal to the expenditure rate (g); this is an approximation tothe value (9F,/67,)/(9F,/8c;) which would be obtained from (62):
VT_dV,/dti (_J_\ ( y-P-ci \ = P-GdV,/dc, \{\-p))\T-twl-Bti) (1 -/?).£ {°V
It is easy to see that an important similarity exists between expressions (37), (60) and(63); they all have the same linear form in the discrete variables c,- and tt but vary inthe value and interpretation of their coefficients.
Expression (37) is derived using a Cobb-Douglas form for the direct utilityfunction and assuming that working time tw is an endogenous variable the value ofwhich is controlled by the individual in order to maximize utility. Therefore, in thiscase the linear form in ct and tt is a result of using a linear approximation for theindividual utility function.
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The value of time: a theoretical review 261
Both Bates and Roberts (1986) and Jara-Diaz et al. (1988) assume that tw isexogenous; however, in the first case the functional form of the utility function is notspecified whilst in the second a Cobb-Douglas function is assumed, as in Train andMcFadden (1978). Thus, the presence of the expenditure rate (g) in (63) and not in(59) is basically a consequence of this last assumption; therefore, in this sense (59)could be viewed as more general than (63), as it is valid for any functional form forthe utility function.
In 1989, Jara-Diaz and Videla proposed a model to detet the existence of anincome effect, where the marginal utility of income depends on the cost of the chosenalternative. In this case the indirect utility function is given by:
V(P, Y- CJ, Qj) = F,(P, Y- cj) + V2(Qj) (65)
where P is the price of goods excluding travel, Y is income level, cj is the cost of usingalternative j , and Qj the other travel attributes. Only Vt is needed to examine theincome effect; for this they use a Taylor expansion of order greater than one around(P, Y), allowing income to appear as an explanatory variable for mode choice:
VX(p, Y- cj) = Vi{P, Y) + £ 1 /P-n(P, Y){-Cj)1 + Rn+i (66)
where V\ indicates the rth derivative of V\ with respect to (Y-cj) evaluated at Y, andRn+i represents the terms of order n+\ and over. If the nth approximation isconsidered sufficient, then Rn+\ tends to zero. Note that when a first-orderapproximation is used the usual linear-in-cost specification is obtained for the utilityfunction.
The choice of transport mode is allowed to depend on income if n is taken ashigher or equal to two. This is so because in that case at least one term of the formV/(P, Y) appears and therefore when V(ch Q,) is compared with V(cj, Qj) differentresults may be obtained for individuals with different income levels. In these cases,the marginal utility of income will be given by
r)V* "~1 iA = w = F!(P) ^ + l ^ r i + 1 { p > *)(-*)' (67)
and it can be seen that, in the linear case (« = 1), X is independent of income.Jara-Diaz and Videla propose a very simple test that consists in introducing a
new term, namely the cost squared, in the linear utility function generally used,studying to see if it is significant or not. In this case the equation of the utilityfunction to be used is
Vi = a0 + aca + - aci c] + V2 (fi/) (68)
In this way, significant coefficients for the squared cost variable estimated fordifferent income segments indicate the existence of a general utility function of thetype V(ch th Y). The value of time in this case will have the following form:
VT=dV/dtL= a,dV/da ac + ac2.Ci
{ '
where a, represents the time coefficient in the indirect utility function, ac is thecost coefficient and ac
2 the coefficient of the squared cost variable. It must benoted that in this case the value of time depends on the cost of the alternative
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262 R. M. Gonzalez
and that this is a direct consequence of using a second-order approximation forthe utility function.
Finally we analyse a model recently proposed by Jara-Diaz (1994). This authorcoincides with Evans in assuming that the basic resource of utility is the time spenton the different activities (including working hours, hours spent sleepingf and soon). In this model it is assumed that goods are both needed to perform the differentactivities and are also the main source of expenses. The model also considers that thetime assigned to each activity is related to that of other activities in two ways:through the direct dependence among times spent on different activities (i.e. theduration of an activity influences the duration of any other activity), and through theshared use of goods.
Taking these considerations into account and establishing that the travel choicemodel can be seen as a problem of time allocation, Jara-Diaz (1994) formulates thefollowing model:
T,W*W,B: W^*V,0 (70)
s.t. (1) Y,Tl+WY + WII + J2Y,Silttl = T
(2) F(X, T)>0
(3) E E PidXid + E E SVCV = IF + WWVid ]-l ieMj
(4) B = B{X)where T is the vector of times Tt dedicated to the different activities in period T; WF
and Wy represent the number of working hours, where the subscripts v and Findicate 'variable' and 'fixed', respectively (it is clear from this that Wv may bechosen by the individual); t is the vector of travel times ty'va period x; B is the numberof journeys in the period; 5tj is equal to 1 if mode / is used in journey j and 0 in othercases; F is a technical transformation function between goods and times; Xid is theamount of good i bought in zone d in the period; PM is the price of good i in zone d;Mj is the set of modes available for journey j ; IF represents fixed income and w is thewage rate.
In this model all activities, even those that the individual needs to carry outwithout wanting it, have a direct impact on utility. Moreover, the goods may be
fThe inclusion of a variable for the time spent on sleeping has not been widely recognizedby all authors; economists have, as Biddle and Hamermesh (1990) point out, made a great effortto study how individuals allocate their time although the time spent on sleeping has beenignored in empirical studies. Biddle and Hamermesh show that at least part of the time givenover to sleeping can be altered when economic circumstances make other uses of time seemmore attractive. Thus, the time the individual spends sleeping is subject to consumer choicewhile also being affected by the same economic variables that influence other uses of time. Onthe other hand, Becker (1965) for example notes that this time is not only necessary forefficiency but is required when the objective is to maximize monetary income; nevertheless, thistime should be determined by its effect on income and not by effects on the utility. However, thelatter appears rather contradictory if one considers that the author himself considers sleeping asan example of commodity (Z,).
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The value of time: a theoretical review 263
bought in different places, with potentially different prices. As the home and place ofwork are given, the number of journeys is only sensitive to the choice of X, which isreflected in the fourth constraint of (70) so this can be viewed as the result of anotherproblem, for example, to determine the optimal number of journeys given aparticular set X.
From this perspective the known variables are WF, IF, ty, Cy, x, P^ and w, whilethe decision variables are Tb 5y, Xic/, Wv and B. The solution for 5 produces the modechoice model. It can be seen that ST,- = L and SP,yZ,v = G. Thus because of thetechnical relation between X and T, there is an implicit relationship between G and Lwhich has a direct interpretation: the consumption of goods requires leisure time (L)and vice versa, an observation not made in the models of Becker (1965) or Train andMcFadden (1978).
In order to study some implications of this model in more depth, Jara-Diaz(1994) examines the mode choice problem in the case of a journey k, assuming allother decisions as given, i.e. number of journeys B, destination (which is one of thedimensions of X) and so on. In this way the model may be written as:
y r r . c W : U(T, WF, Wv, tx, . . . . tik, . . . , tB) (71)X, T, Wv, i £ Mk
s.t. (1) Y,Ti+Wv+WF + Y.tJ + t^ = T
(2) F(X, T)>0
(3) J2 P'X' + Y,CJ + Cik = /F + wWv
i &kin addition to the non-negativity constraints. The relation between B and Zhas beendropped for simplicity, which means that the amount of goods does not affect thenumber of journeys.
Starting from the conditional solutions for variables T, X and W, expressed asTt*, Xf and Wv*, the corresponding indirect utility function is obtained
U(r, WF, Wl, tik, t) = V[r-WF-l- tik, WF, tik, t,~(IF-c- cik)] (72)
where c, t are defined and t is the vector of travel times excepting tik.In this specification time plays a double role: it provides utility, as an attribute of
the utility function in (71), but it also affects the time available for other activities, asa consequence of the first constraint in (71). Neither role can be differentiated whenthe utility function in (72) is a linear approximation because the comparison betweenalternatives should be based upon an approximation such as:
Vt = k + a(r- WF - T- tik) + /3WF + ~<tik + J2e^ + 6~^-'d- cik) (73)
where a, /?, y, 5 are coefficients to be estimated. Taking only the terms that affectmode choice yields
C?,= ( 7 - a ) f t t - « ^ (74)
This equation does not allow us to estimate the coefficients y and a separately.Furthermore, it must be noted that in the approximation given by (73) all the
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264 R. M. Gonzalez
variables with the exception of time and cost are irrelevant in the choice process (forexample, income would play no part). This would not happen if it was consideredthat a suitable approximation for the indirect utility is of second-order, as in thestudies by Jara-Diaz and Videla (1989) and Jara-Diaz (1990b).
The value of time derived from the approximation given by equation (74) has thefollowing form:
dUi/dt* Jri-a)dUt/dcik S/w ^
In this framework the relevant value of w is the hourly wage offered to the individualin exchange for extra work, which represents the real opportunity cost of theactivities undertaken outside the fixed work schedule. According to this, in anempirical study the individual should be asked about work arrangements, so that ifthe salary and the working hours are fixed, the wage rate for additional work shouldbe established and this value entered in the modal utility.
The conditional utility function given by (72) may be interpreted in terms of the'goods and leisure' trade-off; the first term is the total time available to reach T(which has been associated with L) or to keep on working, and the last term is theequivalent time to buy X, for example G/w minus the actual overtime done, Wv.Thus, we can write formally
V, = V(L + Wv, WF, tik, t, | - Wv) (76)
which shows the deviations from the U, (G, L) approximation quite explicitly.
4. Final commentsThe time allocation models discussed here are the basis of a large number of
empirical applications, in particular those referring to the choice of travel mode andthe valuation of time savings. The important economic implications derived from theconsideration of different values of time highlights the usefulness of this type ofmodel. Moreover, given that an observable price of time does not exist, it is essentialto have an adequate theoretical framework for its derivation.
The theoretical assumptions about the relevant variables, the consideration ofwhich are fixed and which are not, and the nature of the constraints that are binding,are all questions affecting the value of time obtained. Thus, it can be seen that if oneworks under the assumption that the individual is free to choose the amount of timespent on work and that the wage rate per hour is fixed and independent from theamount of time worked, one obtains the usual result that the value of time is equal tothe wage rate (Train and McFadden 1978). However, when these behaviouralassumptions are changed the values of time obtained differ from the wage rate andon occasions even bear no relation whatsoever to it.
The review of the different formulations of utility maximization models hashighlighted the existence of great differences between the most relevant articles onthis subject. For example, it can be seen that the consumption of goods is still theessence both for Becker (1965) and for De Serpa (1971, 1973). Becker considers Tasa vector of times required to produce final commodities. For De Serpa, T is part ofthe description of X, and for Evans (1972) X takes on a 'secondary role' (thepurchase of goods is just a means to enjoy T). Likewise, it should be pointed out thatDe Serpa is the first author to note that the value of time savings in an activity is only
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The value of time: a theoretical review 265
positive when the corresponding constraint on consumption time is binding, i.e. theindividual would like to devote less time than required. In this way his studyframework contributes to clarify important aspects of the value of time concept. Onthe other hand, the work of Evans (1972) is the only one that introduces an explicitset of relations among activites (times) supporting the fact that similar commoditiesmay be involved in different activities.
The work of Train and McFadden (1978) must be considered as the connectionbetween time allocation models and mode choice models within the random utilitytheory's framework. Later works following this paradigm (Jara-Diaz and Farah1987, Jara-Diaz and Videla 1989) introduce new behavioural assumptions andconsider specific forms of the utility function giving rise to the appearance of newways of evaluating the existence of an income effect in the choice of travel mode.
Finally, the model proposed by Jara-Diaz (1994) accepts time as the main utilityresource and also postulates that goods must be considered as much a means as anend. To this extent this model goes beyond previous ones in considering that anyminute must be taken into account regardless of the use made of it, which isundoubtedly an important contribution to the formalization of travellers' behaviour.
AcknowledgmentsThis review was undertaken as part of the author's Ph.D. thesis under the
supervision of Professor Juan de Dios Ortiizar at the Pontificia Universidad Catolicade Chile (who kindly helped her with her English) and Dr Nieves Rosa Perez at theUniversity of La Laguna. She is very grateful to both of them for their commentsand encouragement. Thanks are also due to colleagues at the Department ofTransport Engineering, Pontificia Universidad Catolica de Chile, for theirhospitality and warmth, which made producing the draft version of the thesis amuch pleasanter task. Finally, the author would like to thank Professor Sergio Jara-Diaz, at the University of Chile, for having shared some of his knowledge and DrManuel Navarro, at the University of La Laguna, for his comments.
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