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THEORETICAL ANALYSIS ON VALUATION OF TIME SAVING

by

Mobinul HuqDepartment of Economics

University of SaskatchewanSaskatoon Canada S7N 5A 5Telephone: 306 966 5204

Fax: 306 966 5232E Mail: [email protected]

February 2 1

would like to thank Professor Joseph S. DeSalvo at the University of South Florida for introducing me to this research area and numerous exchanges on the topic .



T H E O RE T IC L N L Y SI S ON V LU T IO N O F TIM E S V ING

by

M ob in ul Huq

ST R C T

The v alu e of t im e V OT is def ine d as the max imu m amo u nt of m one y peop le ar e w il lin g to

pay for an ad d iti o nal unit o f ti m e, wh ile the value of tim e sa v ing VT S is de fin ed as the m axi

m um am ou n t of m one y peop le are w ill in g to p ay for a real lo cat ion of tim e betw een two al ternativ e ac tiv it ies . This p aper sho w s th at the sign and am ount of th e VTS d ep ends on the m agni tude

o f th e dif fere n tial eff ec ts o f tim e-u se on an individuaUs mo ney co nstr a int, W hile the ex istin g

m ode ls of va lu e of t r i l ti m e savi n g VTTS’ h in ge on the pre sen ce of a bind ing min imu m travelreq u ire m ent . this pa per p rese n ts a ch oice -the oreti c model wh er e a po sitiv e V hS is de rive d w ith

ou t im po sing a bi ndin g m inim um travel requ irem ent. T h is resea rch su gg ests alte rnat iv e ways o f

esti m ati ng V hS by exam ining market beh aviou r w he re pe op le are obse rved to trade travel timefor m on ey. One w ay is est imat ion o f the de c rea se in ho usin g ex p end itur e s wi t h travel tim e:

o ther is estim atio n o f the mo n ey- time trad e -of f in trave lling b etw een tw o fix ed p oin ts in s imil ar

trav e l circ u ms ta nc es.

Key W o rd s: V alu e of ti m e V OT . va lue of tim e savin g V TS , va lu e of trave l ti m e sav in g

V h S .

JE L C las si fic at ion s: D l, J2, R2, R4.



INTRODUCTION

In the goods-market context, the economic value of a commodity is defined as themaximum amount of money an individual is willing to p ay for an additional unit of that commodity. Applying this definition to the time-allocation problem, the value of t ime VOT is defined as the maximum amount of money an individual is willing to pay for an additional unit oftime. Given the fac t tha t total amount of time available is fixed, which implies that a ny t im esaved in an activity must be allocated to some other activity ies , the value of time saving VTSis defined as the maximum willingness to pay for reallocation of time between two alternativeactivities.

The traditional model of labour-leisure choice, formulated to explain hours of laboursupplied to work, defines leisure as all activities o ther than market work. In addition, it is assumed that the marginal utility of market w or k tim e is identically zero, which means marketwork does no t involve any satisfaction or dissatisfaction. On e implication of these assumptionsis that, when evaluated at the optimum point, there is a unique value of time which is equal to themarket wage rate. In addition, the value of work time saving is equal to the wage rate which isalso equal to the monetary value of the marginal utility of leisure.

In economic models dealing w ith th e allocation of time a mo ng m ore t ha n two alternatives, specifically in the valuation of travel time sav ing, the term value of time has been used indifferent contexts to mean different ideas. Recognizing the lack of well-defined concepts of va lues of time in the context of transportation, Small 1992, p. 36 wrote, “Among the most important quanti ties inferred from travel demand s tu die s ar e th e money values that people place onsaving various forms of travel time. Loosely known as the value of time this concept encom

passes many specific measures

The f ir st purpose of this paper is to review the theoretical literature on value of time ingeneral and in the context of travel-time saving. Distinctions are made between three differentconcepts commonly used in relation to the valuation of time use; the utility or process effect,the consumption o r monetary effect and the net o r total benefit. It is argued that in the literature, VO T concept relates to the n et benefits f ro m t im e use an d VTS refers to difference in theutility process effects of two alternative activities. To explore these relationships, a generalizedtime and goods allocation model is presented in the third se cti on . A t th e optimized point whilethe monetary value of total benefits from t ime use to all activities are equalized at the VOT, the

monetary value of utility effects are n ot equalized. This later result generates n on z er o benefitsfrom time reallocation between alternative activities.

3



In the travel time context, through a review of the literature it is shown that the value of

travel-time saving VTTS) critically hinges on the assumption of a ‘binding minimum traveltime’ constraint that requires individuals to t ravel more than they would choose to travel without

the constraint. It is not obvious, however, whether travel time is a constraint or a choice Handy

et a ., 2005 : Or y et a . , 2004) . In most situations, it is possible to reduce travel time by spendingmore money on travel goods. such as taking a taxi instead of walking, although that might no t bethe optimal choice. In the absence of a binding minimum travel-time requirement. VTTS’ is zeroin the existing models. Th e latter part of this paper presents a model of travel time in a choice-

theoretic framework by combining the standard urban housing location model with mode choice

treated as a continuous variable. One implication of the proposed model is that there is a positive

V]T even when there is no binding minimum travel requirement. Th e main argument is that.although leisure is preferred to travel people travel to obtain better terms of trade. This modeltreats commuting time as a choice variable in the same wa y as the standard literature treats work

time as a choice variable

Th e paper is organized as follows. Th e next section reviews the VO T in terms of the la

bour leisure model and the following sections presents a generalized model. Th e fourth section

presents a discussion of the theoretical literature on VTTS’ Th e fifth section presents the proposed residential choice model with mode choice and discusses its implications fo r travel-time

valuation research. The paper ends with a concluding section.

2. VALUATION OF TIME

The Traditional Time Allocation ModelTh e traditional labour-supply model in its simplest form examines the problem of best

allocation of total available time, M, between market work W, and leisure, L, assuming that anindividual is f ree to choose to work for any number of hours at a fixed wage rate, w. The quasiconcave utility function is defined as u = u X.L), where X stands for market goods measured interms of dollars. One notable feature of the model is that working time is no t included in theutility function. An individual faces a time constraint, M = L + W, as well as a budget constraintwhich states that the total expenditure on market goods X is equal to the sum of labour and non-labour income. v X = wW + v

This constrained optimization problem in Lagrangean form can be written as

A=u X,L +2 wW+ v—X +,u M —W—T)

By the envelope theorem, the Lagrange multiplier p A / represents the marginal utility of

time M, while 2 = A / ay represents the marginal utility of money. The value of time VOT can

be defined as A M I A/v = p / 2 From the first-order conditions f or th is constrained

optimization problem treating W as a choice variable , on e can derive the value of time VOfl

as

VOT = LL. =

lix 2

4



The issue of exclusion of work time from the utility function has been questioned by anumber of researchers. Among others, Johnson 1966 , Oort 1969 , and Evans 1972 arguedthat there is no reason to assume tha t the marginal utility of work t ime is zero. Therefore, by in

cluding W as an additional variable in the utility function, the utility function would be definedas u X, L,W , The value of time in this formulation can be derived as

VO T = = w +2 u

Thus the value of time is no longer equal to the wage rate, but it would exceed fall short of thewage rate when positive negative utility is derived from work time.

Alternatively, the u X,L function can be interpreted as the one where the time constrainthas been substituted for W, that is v X, L, 14’D=v X, L, M-L = u X, L . As correctly pointed out byFlemming 1973 , this will imply,

— = M R S M R SU V V

W he n m ore t ha n one time use activities are included in the utility function, such as L andW in the preceding, it is important t ha t o ne k ee ps th e following time-related concepts distinct.

The monetary value of the marginal utility of time spent in activity i, /3

For an utility function with n alternative activities, u X , T, , this concept can be defined as

U 3 = M R S = i=1,..,n.

ux

In this paper, this MRS is referred to as the utility process effect of time-use and is denoted by

/ fo r activity i DeS erpa 1971 defined this concept as the value of time as a commodity VTC .

In a model with work time incorporated into the utility function,

/JL = M R S = and 3g = M R S =

Ux ‘Ix

These utility effects are activity-specific and in general likely to vary across alternativeuses of time at the optimal point.

The consumption monetary effect of time spent in activity i, In addition to the direct effect o n the utility level, time u se also h as a potential effect on

utility through its effect on the budget constraint. The obvious case is the market work time

which results in an increase in money income by an amount equal to the wage rate, resulting inan expansion in consumption possibilities, hence increased satisfaction. Among others, Jara



Diaz 2003) examined this effect in the travel-time context , and Jiang and Morikaa 2005 referred to it as the value of c o n s u m p t i o n sav ing Vt’S . In this paper, this effect is referred to asthe consumption mo,leWn) efj’ct of time-use in an acthity and is denoted b ?‘ This effects

a re l ikely to \ary across alternative time-use activities. Theoretically, howeer, th e s ig n of thiseffect is indeterminate, which implies that it might not he an inflow but an outflow of money.

In other words, the consumption effect of the ith activity refers to the effect of acti ity ion the availability of the numéraire good X . This effect ca n be derived by differentiation thebudget constraint in the form X = X T Tfl In the standard labour leisure choice model, X =wW + v gives.

71 = — = 0 and y

These effects are independent of the preference of an individual and can be derived frommarket data.

The net total) benefit f r o m t ime s p e n t in act iv i ty i,

a is defined as th e n et or total) benefit generated from a change in time allocated to anactivity i. In other words, it is the rate of change in utility with respect to T measured in terms ofmoney. Th e ne t benefit of time use is th e su m of the utility effect and the consumption effect,namely a,= ,8 + y ,

In the context of the labour-leisure choice model, these effects are

U uwa,

=—

and a , = wux us

On e property of the optimal point is th at th e n et benefits from all alternative time-use activities are equalized = a , and there is no room fo r further u ti li ty gain through t ime reallocation. An y addition to total available t im e c an be assigned to any of the available activities, andthe resulting uti li ty gain w ill be the sa me a nd e qu al to a . Hence a can be interpreted as thevalue of time VOT, which b y th e first-order conditions is equal to the rat io of the two Lagrangemultipliers, p12)). DeSerpa 1971 defined this as the value qf time as a resource VTR .

Willingness to pay to reallocate time from activity i to], p .

Mathematically this value ca n be derived in the following way. Let the utility function beu=u X, .. T, and the time constraint be M ZT, =0 where 1 denotes time allocated to activ

ityi=l,.....n.

Totally differentiating the utility function, assumingM constant, produces

du=dX --t1TdX d7 dT ,

6



S ett in g d u= dT =0 f o r all k = 1.... . i-I. j+ I, ..n , i.e. k o rj an d dT, = dT, sin ce M is a co n

sta n t, g ive s

O =u d X + —dT

IV ii —u

= — — — R S =p

In the com mu ting tim e con text , D eS erpa 19 7 ) defin ed such a di ffere nce be twee n the

m arg in al u tili ty o f lei su re and of trav e l tim e , as the weltie o t trac e -t im e s ai ’in g VT 1S

V1TS =

= P Tu x

N ot e tha t the m arg in al ra te of subs titut ion b etw een tw o a c tivi ties, MRS = h i a di f

fe ren t c o nce p t, repre sent s th e ra te at w h ich an in div idua l is w illi n g to ex cha nge tim e betw een

tw o alte rnat ive acti v itie s, and j. If and o n ly if = suc h a tim e real loca tion w ill sa tisfy the

ti m e co nstr a int. Ot herw ise u ti lity c onst ant tim e tra d e-o ff be twe en two ac ti vit ie s will v iola te the

ti m e co ns tra m t.

3. A GEN ERA LIZ ED M OD E L

An indi vidu al’s u tilit y is assu med to de pend on c onsu mp tion o f g +l) m arke t go o ds th e

n ume raire go o d wi t h price =1 is den oted by X and n +l) alter nati ve ti m e use ac tivit ie s a ba se

or re fer ence activ ity is de note d by 7 .

u = u X

X ,. .. ...,X 1

The mo neta ry value o f the m arg in al utili ty of tim e use in ac tivi ty f3 , is defi n ed as,

,8 = M R S =— i= O. 1 ,..., n

The tim e co n str ai nt is g ive n as.

A = 2

F ollo w in g Ev ans 1 9 72 , it is assum ed that any activ ity m ay be asso ciat ed wi th mon etar y

re mun erat ion in fl ow ) or m one ta ry co mm itme n t o ut flo w ), such as w orkin g fo r pay or pa y ing to

go to a m ovi e re sp ec ti vel y . This amo unt is de n ote d by A po s itiv e v alue i m pl ies tha t an in

d ivi du al r ecei ves R am ou nt p er un it of ti me spe nt in acti vi ty w hile a neg ative v a lue im pl ie s

R exp endi ture pe r unit of time.

7



F o llo w ing urb an econ omi cs li ter at ur e, good s/ho usin g price is ass ume d to be a ffec te d by time

use and m ar ke t go od s f o r exa m ple , more comm utin g time and co m mu ting goods lead to lowerho us e price . n the g ene raliz ed case an y price may be affe c ted by lime use and goo d s such as

m o ne y back re bate by mail needs time and mo ney /post age. That is P tX j X T,.

The mo ney co nstr aint can he wr itt en as

— X O 3

v stands for non—wage income.

The cons ump tion mon etar y ef fe ct of ti me use in ac ti vity I y can be ob ta ine d by di ff er

enti a ting the follo w in g bud ge t con stra int.

X0 =[y R — P 1 7 XjJ I

ii

j

T he net t ota l ben efit o f tim e us e in acti v ity i aj , are

Ua = 13 y = _ L R — -X .V i = O L 2 . ... n

his m od el can be ex tend ed fu rthe r w he re R are a ffec te d by ti m e use fo r e xam ple ,

comm utin g fu rther fo r hig her pa id jo b , o r allo wing for h om e pr o duc tion o f so m e goods.

An in di vid ual’ s opt imiz ation p rob lem is to m axim ize 1 sub je ct to 2 and 3 ,

A =u X A i v R —X — P Xj I it

Th e first-o rder cond ition s fo r ma ximi zatio n are,

— 2 = 0

= u x , — PX = O ,j = l, 2 , ..., g



— — = u

v R T P X z 0

= 0

So me Impl icati o ns of the FO C s:

T he M on et ary V alue o f the M arg ina l U tilit y o f Ti m e Use 13

D iv id e the th ird FOCs b y the fir st one ,

= - = p / A — R = p / — .. i = O .1 .2 ,. . ..n

n+ 2 un know ns, j3, j i/ w ith n+ 1 equ at ion s . S in ce a util ity fun ctio n is in v ari an t

to any mo noto nic tr ansf orm a tion , th ese sub ject ive v alu es and can on iy be est imat ed in te rm s of

dev iatio n fro m an arb itrary or igi n. W hen T is con sider ed to be the b ase/ refer ence nu rn ér air e

ac ti vit y,

— — ,V i =j 2 n

Sp ecia l cas e I: Us e of T with u- 0 as the base /refe renc e acti v ity i m pl ie s

,8, = y — y Vi = l 2 n

Sp ecia l c ase 2: Use of T w ith y 0 as the ba se/re fere n ce acti vi ty imp li es

E qua lize d N et T o tal B enef it o f Tim e Use in all A ctiv ities a

a . = f i+ y ,

= p / 2 -R +I? = C u/2

9



WHlingness to Pay to Reallocate Time from Activity ito k p

P /I /— R k + — ‘- XJ / A + R , __L X

C j=l

p = R. — - - — X — R — - — -X y —y[-I c’

Factorial a possible p values can be defined for n+l activities. However, the number can he

reduced to ii values. p , ,by using To as the base/reference activity,

Pi-o flo = R + X X, = 7 -7

Special case: When the reference activity has zero effect on availability of goods, that is yo =0,

Oj = -/3, R _ X 1=’

The Value of Time VOfl

An increase in time availability can be assigned to any activity, bu t the net total) benefit

will be the same and equal to the VOT,

VOT=a, R — X = M R S +7 for any i=0,1,...,n.

Ux

Since this includes a M R S t erm, which is a subjective value, the VO T cannot be estimated

from observed market data. However, the assumption of marginal utility of an activity equal to

zero will imply that VO T is equal to the measurable consumption effect of that activity.

4. VALUE OF TRAVEL TIME SAVING VITS’ ’

commuting Time in the Simple Labour-Leisure Model

In the simplest versions of this model Troung and Hensher, 1985; Bates. 1987; Train andMcFadden, 1978 , it is assumed that, while commuting requires time 7 and commuting goods C’, there is no direct satisfaction derived from either. In other words, neither T nor C enters the

utility function but is only present in the constraint as a parameter.

Reviews ut the literature can be fiund in Jara-Diaz. 2000. Mackie ci at.. 200 . and Jiang and Marikawa. 2004.

10



The o ptim iz a ti on pro b lem ca n he w ritte n as

A= u Xi +,1 w W+ v— C — X + u M- -W— L- -T

wh ere T and C ar e par amet e rs.

Tre atin g Wa s a choic e va riab le, the fi rst -o rde r c ond ition s are

A =u — 2= 0

= a — p = 0

A = 2w — p =0

A A = w W v— C X =0

1 = M— L— T— W= 0

fro m wh ich the follo win g can be ob tain ed

J L =— - — =W J = — = a n d — =

u 2

U p

=— - — = w, = W 111 1 = 0u k 2

U 1 P U w 4

0 P T L — 2 — — n an 2

In this for mul a tion , c omm utin g ti m e and mo neta ry co s t o n ly af fe ct the tim e and b ud g et

con st rai nt s. As such, T 0 and C = 0 w ou ld be the best cho ice for any ind iv id ua l. G iv en that a

T-C co m bin atio n m us t be ch ose n, ho w eve r, the best ch oi ce w oul d always be the lea s t-c o st one

w her e tim e is val u ed at the w ag e rate , tha t is, m inim um C w i . To see this, afte r subst ituti ng

for W fro m the tim e cons train t into the bu dg et co nstr a int, on e ca n rew rite the opti miz a tion pr ob

lem as: ma xim iz e u = u X. L su b jec t to X wL = wM C w i . The cho ic e o f C and T

that m in imiz es the last te rm in the const raint w ou ld leave the m axim um am oun t of re so ur ce s for

allo catio n b etw een util ity-g ener ating X and L.

Tre a ting C and T as conti n uou s v ar iab le s, the b es t ch oic e will cor resp ond to a s it uat io n

w here the tr ade -off be twe en t ra ve l tim e an d tr av el co st is eq u al to the w ag e rate. T rea ting C and

T as as socia ted wi th a di scre te m ode ch o ice , an ind iv id ua l wo u ld be bett er o ff all ocati n g m o re

l es s t im e to c omm utin g wh en add ition al com mut ing time leads to a trave l cost sa v ing in ex ce ss

l ess of the wage rate. The m ode- choi ce de cisio n is ind epen den t o f pr e fl re nc es in this fra me

work.

W he n travel tim e T ent er s the ut ili ty fun ct ion e.g., O or t 1969: Evans , 1972 and T is

con side red a cho ice va ria b le, the m odel is equiv a len t to the ge n era l m o de l pres ente d ab ov e,

w here o ne of the act iv iti es is tra v el time. As we have alrea d y seen, an im pli catio n of that m ode lis that at the op tim al po int the re is no gain in u ti lity from reallo cati on o f time am ong non -wo rk



activities; hence, gain from travel-time saving at the margin is zero. As Evans 1972, p. 9 noted,if the consumer’s allocation of time is optimal, a reduction in the time spent travelling would

make him no better off and no worse off .”

As shown in the previous section. a positive value of travel-time saving V17S impliesthat at the optimal point the marginal utility of leisure time is greater than the marginal utility of

travel time so that there is a utility g ain f ro m transferring travel time to leisure time. In thetransportation economics literature, there are three different ways in which researchers derived

this gap between the two marginal utilities: I by adding a binding minimum time constraint DeSerpa, 1971: Evans. 1972 . 2 by adding a binding minimum consumption constraint Evans.1972; Jara-Diaz, 2003 , and 3 by introducing etjctive leisure time concept Train and McFadden, 1987; Small, 1992 .

Binding Minimum Trai el- Time Constraint

DeSerpa 1971 and Evans 1972 assumed the presence of a binding minimum time re

quirement that prevents an individual from reaching the unconstrained optimum level of travel

t ime. This assumption makes the marginal utility of travel time lower than the marginal utility of

leisure. Reduction in travel time allows reallocation of t ime from the lower utility-generating

activity, travel, to the higher utility-generating activity, leisure. Th e Lagrange multiplier corresponding to the minimum time constraint shows the difference between the marginal utility ofthese tw o time uses, which in turn equals the willingness to pay to reallocate time VTTS .Technically, with addition of minimum travel-time constraints, given as 7 with corre

sponding Lagrange multipliers, K DeSerpa derived the following result

U UT

2 2 2 2

This represents the willingness to pay to reallocate time from binding travel time, to nonbinding 7 , leisure.

Without a minimum travel-time constraint, VTTS would be zero. In reality, it is difficultto argue that an individual is allocating the minimum required time to travel activity. In the

short-run, it is always possible to reduce commuting time by adopting a faster mode, such as using a taxi instead of walking. In the long run, o ne c an reduce travel time by changing one’s jobandJor residential location. In other words, it can he argued that an individual’s choice of traveltime is not consistent with a binding minimum travel-time constraint.

Binding Minimu,n Consumption ‘onstraint

Another line of argument is that an individual is constrained to spend more than the op

timal amount on travel. In addition to a minimum time constraint, Jara-Diaz 2003 introduced a

binding minimum consumption constraint, given by C g 7 where g >0 . Given this con

straint. Jara-Deaz derived a relationship among VTTS and the monetary value of relaxing theconstraint as well as the monetary value of other marginal utility terms .

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As oppo sed to this view, one can view trav el-t ime sa vin g as a llo w ing an individual tom ov e clo ser to the utility—maximizing level of travel expen d itu re s. As with D eS er pa ’s mi n imu m

travel—time const rain t, an imp osed co nstra int is the so u rce o f posit iv e I’T’fS. no t the solu ti on to

an o p tim izatl o n pro b lem . In m os t ca ses it is poss ible to re du ce the m o ne ta ry por tio n of t ra v e l

costs y’ sp end ing more tim e tr avel li ng. e.g .. by w al ki ng ins te ad of takin g a taxi, alt ho ugh thatmigh t not be the o p tim al ch o ice . The poi nt is tha t how mu ch m on ey one sp end s on com m ut ing

is a ch oi ce , n o t con strai ned by so m e b ind ing m in im u m requ ir em ent.

f ecti v e isu r Time

M os t emp irica l stud ie s o f travel tim e sav in gs are b ased on travel ch o ice in a dis c ret e-

cho ic e se ttin g , the theo retic a l fou ndat io n o f w hi ch was firs t in tro du ce d by T rain and M cFa d den

1978 . Alt h oug h Tra in and Mc F add en d id not emp hasi ze ViT h , th e ir model may be us ed to i l

lu st rate the appro ach to es tima ting VTTS in di scre te-ch oice m o de ls , In d iscu ssing the va lue o fti me , Small 19 92 , pp. 40—43 also uses this m od e l.

Let ii u XL wh ere X =v+ wW —C an d L =M — W — 1, and wh ere the v ari

ables are as pre v iou sly def ine d, exc ept that subs crip ted v aria bl es are defin ed for the ith m o de ,

ii Max imiz ation o f u v + w W — c —W—7 w ith re sp ec t to W p ro du ce s a f irst- orde r

co ndit ion th a t im plies u y w. So lv in g the fi rst-o rder con diti o n for the ut ility -max imiz ing

va lue of W pr o duc es 7 = W * c j,7 ,w ,v ,M , S ubsti tutin g this into the u tili ty fun ction g iv es

y + w147 — — 14 7 _i; v c7 ; w y M wh ic h is the ind irec t u til it y fun ctio n that

d iscre te-c h oic e m od el s e sti m ate . M o de is cho sen if and onl y if r> vj for all j=l ii; i . In

this co n tex t, VTTS is de fined as iC / T for the uti lity- max imiz ing mode. E mpi rical ly, an

equa ti on suc h as a r is estim ated , so that VT JS —dC/AT t ‘

Th eore tical ly, h o we v er, w ith o ut furt h er ass u mp tions , 777’S w in this model, wh ich c an

be seen as fo llo w s

VTTS P r w

lix ‘Ix

orc u T u L — 1 _ u L

u UX —1 Ux

For pur po se s o f est imat ion. T ra in and M cF add en f oll ow ed by S ma ll in tro duc e the c o ncept of cf/’ c tiie leisu re. It is the int ro du c tion of e ffect ive leis ur e that all ow s the m ar gi na l ut ility

o f le is ur e to he g reat er than the m ar gi na l u tility o f travel tim e and V HS to be a po sitiv e fr actio n

o f the wage rate. To sh ow this, a s im p lifi ed ver si on of e ffe c tive le is ure is used here.

13



As su m in g no u tilit y from w or k tim e and on ly o ne ca teg or y of trave l time. e lfec ti ve lei

sure time, m a y he d ef ine d as I L 6ff where is assum ed to he c o ns t an t b et w e e n and Th e n u =u X .1 ,a n d

M R S

,0 0

Th is imp lies th at u > an d th at 1/TI’S> 0. Fu rther m or e ] L U T t I U lO 9 U L

x u x lix

Thus, if 6 is a cons tant b etw een 0 and I. the V77S is p osit ive and a con stant f racti on of the wage

rate. This re sult isin d epe nden t of the spec ific f o rm o f the u tility fun ct ion .

In concl u sio n , all theo retic al m o de ls und erlyi ng the v a lue of trave l-tim e sav ing are base d

on som e b indin g co n str a int on trav e l time and/ o r travel cost. None of the m od els add ress es the

que stion why peopl e tra vel in the first place . Han dy et a . 200 5 and Or y et a . 2 004 argu ed

that in most cases trav e llin g is a choice not a req uire m en t. Th e next sec tion deve lops an d pr e

sents a model that p redi c ts a p osit ive VTTS w ithou t impo sing an y m inim um re qu ir em ents for

travel time or tra vel cost.

5. R E SID E N T IAL LO CAT ION CH O IC E W ITH E ND O GE NO U S T RA V EL T IM E

Th is sec tion pres ents a model of h ouse hold choi ce in w h ich resi dent ial lo catio n and trave l

ti m e are s im u ltan eou sl y cho sen with out a bin d ing m inim um trav e l-ti m e rest ricti on. In the a b

sence of such a co nstr a int, the mot ivati n g fo rce behi nd the all o cat ion of time to travel is the r e

du c tion o f ho u sin g exp endi tures . T h is is sh ow n by inc o rpo ratin g a tr avel -dist ance pro duct io n

funct ion into the st anda rd r e sid entia l locat ion c hoice mo del.

The M od e l

The u tilit y fu nct io n. Th e ho useh o ld’s qu asi-c o nc av e ut i li ty f unct ion is

u= u X H ,L.W .T 4)

whe re H is hou sing c on su m pt ion and the othe r vari able s are as pre viou sly de fine d . M arg inal

utili ties are pos itive fo r X. H, and L. hut they may be po sitiv e or negati ve fo r W and T

In terms of Train and McFadden, let I and de n ote 1—61 0. In terms o f Sm all let 0 and denote

6. Th e s s ump t on th at 0< 6<

ensures that trav el tim e is not

wors e than wo rk

0 > 0 and n ot bett er

th an leis ur e < 1).

13



Prices. It is assumed that P = Pt K and PK < 0. where K is distance from an urban center.

This assumption does not reclude a multicentric city, only that accessibility to any center or

suhcenter is sullicientlv desirable that housing price falls away from it. Empirical studies support this relationship Alonso. 1964. p. 172; Muth. 1969, pp. 192. 237: Wieand, 1973: Coulson.

199 I . As before. X is the nurneraire, so its price is unity.

The distance production Junction. Distance travelled depends on the amount of time

spent travelling and the amount of travel goods, X measured in terms of money

K=K XT,T) 5)

where K > 0 and K 0. XT is assumed to have no consumption value; therefore, it is not a

component of the utility function.

The budget constraint. It is assumed that an individual is free to choose towork

forany

number of hours at a fixed wage rate. The budget constraint states that the total expenditure ongood X is equal to the sum of labour and non-labour income

w W + y = X + P K H + X T 6)

Th e lime c o n s t r a i n t . The time constraint implies that the sum of time allocated to leisure,

market work, and travel equals total available time

M = L + W + T 7

The o p t i l n i z a t i o n problem. The individual’s problem is to choose X. H, X L. T and W tomaximize 4 subject to 5 , 6 and 7 . After substituting 5 into 6 , this problem can be set

up in Lagrangean form as

A=u X,H,L,W.T)+AIwW+v—X—P XT,T)H—XT)+p M—L-W—T)

First-order conditions for maximization are

A =u v—A=0

AH =1111 AP—0

A1 =it — t =

A, =u —p=0

A7 = ii, + AF K TK H t=0

= — PKKVH + = 0

A2 =wW y—X—PK XT H — X O

Af l =M—L—T—W=0

15



O p t im w n a1 1oci iion iif m on ey Since 0, im pl i es

— PK K H 8

T h e le ft ha n d side of sho w s the ma rg ina l ben e fit w hile the right hand side shows the ma rginal

co st of a dol la r sp e n t on trav e l goo ds Xj’,. Th e ma rgina l be ne fi t is the re d ucti on in h ous ing

pe nditu res from tr avel ling far the r and the reby o btai ning a lo we r p rice o f h ous ing. The ma rgin al

cost is the add it ion al d ol lar of expe ndit u re on tr av e l g oo ds . This eq uili b riu m con d itio n th ere fo re

im p lie s that a do lla r al lo ca te d to tra v el g oo ds ge nera te s a do lla r’ s worth o f be nef i ts in hous ing

ex pen ditu re re du ct io n.

From the first tw o FOCs we ha v eU ’

P

T hus the op tim al a lloc ation o f i nc om e bet w ee n go od s X and H is such th at m ar gin al be ne fit

from expe nditu re on the tw o g oo ds is equ alize d and tha t th es e m arg in al ben e fits are equal to the

m ar gin al u ti lit y of m on ey 2.

O pt im um allo cati o n o f tu ne. F ro m the th ird fourt h an d fifth FOCs w e hav e

2 2 l

2

T his ut il ity m axim izati on co nditi on st a tes tha t the m argi n al be nefi t of time sp ent

inleis ur e wo rk.

and trave l is equa lized and that it is equal to the va lue o f time. In o th er w or d s at the op ti ma l

p o int th ere is no ro om fo r u tilit y in c rea se thr ou gh rea lloca tion of time. N o te t h at the m arg in al

be ne fit of le is ure tim e d eriv es so lel y fr om u ti lity gain, u ; the m ar gin al benef it of w o rki ng tim e

de riv es p ar t ly fro m u tili ty gain or loss, u w and pa rt ly from m on e tary gain, w ; and the ma rg in al

ben e fit of tr av el tim e der ive s p ar tl y from u tility gain or loss, u and pa rt ly fro m m one tary gai n

P T H the la ter bei ng the re duc tion in hou si ng ex p end iture fro m tr avell ing far ther and ther eby

obt a inin g a lo we r ho using price.

T he tim e a lloca tion con cept s de fined ab ov e ma y be de rive d as fol lo ws

L an d = L P K H

ux 2 ‘x 2 U 2

a an d = — P K Ki H 2 u A

/1 ttm UT r r

Pw_,. u P T i fl >

and P T w

w T H

16



Conclusion. Important implications of this formulation are that there is a positive VTTSfrom reallocation of travel time to leisure, and that V173’ is equal to an amount that can be measured from observed data without estimating an utility function.

Housing Price Gradient and VTTSAs noted above, the value of travel-time saving V77’S), which is the willingness to pay to

reallocate travel time to leisure, is

K KT H >0

As long as housing price falls with the distance between a household’s residence and its em

ployment center, then, at the optimum, tij must exceeduT

Th e right-hand side showsthe amount of money the individual can save in terms of lowerhousing co st b y commuting one additional unit of time. At equilibrium this is the amount theindividual is paying bynot commuting the extra unit of t ime; hence, it reflects the willingness topay to save travel time.

1TI’S in this framework will, in general, vary withresidential location,PK house/lot size,H, different modes du e to variations in travel speed, KT, and different transportation expenditures, XT Someone occupying a larger house, living closer to the center of employment andlorusing a faster or more expensive mode willhave a higher V77’S.

An implication of this result is that the housing-price gradient in term of travel timemight be a fruitful wayof estimating the value of t rave l t ime saving. A hypothetical examplemight help explain this point. Suppose tha t by travelling 12 additional minutesby bus samefixed fare) every work day an individualcan save 20,000 on housing expenditure. Assuming a4 percent financing cost, this saving is equivalent to a saving of 800 per year, while additionalcommuting time is equivalent to 50 hours per year 12 minutes t imes 250working d ays). T hisgives a VTIS of S 6 per hour.

Coulson 1991) has used an approach similar to this. Coulson runs a regression of lions—

ing sales price on other ‘ ariables. including distance from the central husiness district CR1 .

U sin g t he estimated coeflicient on distance of S2.633.4 the assumption of 500 oiC—\. a’ tripsper ear. and a capital ital ion rate of 10 percent. Coulson calculates the implied cost of t ranspor

tation as 0. 10 > 26.33.4 1/500) 50.527 per mile. Based on data from the I Statistical Ah—

stract. he estimates the money cost of tr a ci as 50.36 p er m il e. Thus, the time cost of ransporla—

lion is SD 7 p er m il e.

.lJone —I/inc irudc—o// iii iim and VJTS

Lc.iuation 5 nnpl ies that for an given distance. K. an indi idual faces a trade—off he—

tv’ een tra\ ci lime. T and travel cost. V a mode choice prohlem Th e implications of this se c

tion s model for m o de c ho ic e can he ex.ainmc l in terms of the filTh and sixth FOC’s. 1 i idiiigthe second FOC h\ the third, using the fourth FOC. produces

17



’ 9)

T hi s is th e stand a rd co st -m in im iz at io n c o ndi tion: th e n eg at ive of th e sl op e o f the “d istan ce i s o

quan t” the m ar gi na l rate of tec hn ic al su bsti tutio n of travel g oo d s for travel time in dis tanc e p ro

ducti on is equal to the rela tive p ric e of the two inputs, travel ti me and travel goods. The p rice of

travel tim e is the w age rate pl us the di ff ere n tial utility ef fe ct of time re al loc a tion , w h il e travel

cos t is one do llar per unit o f travel goods.

The follo w in g rel a tion ship is d eriv ed from the th ird fourth and fifth F O C s and 9

‘=— PK KTH

U U T=V

2 2

In ad diti on to al lo wi n g the estim atio n o f VTTS as — P this sh o ws t ha t o n e c an a lso e stim ate

VTTS as M R T S Fo r ex amp le , if trav ellin g fro m po int to po int B on a toll ro ad en ta ils a 4

toll but saves 15 m inu tes ove r us in g s urf ace roads for the trip, then V77S is 16 per hour. When

comp aring V T TS ’ for al terna tive T and C com bina tion s , it is nec essa ry to co nt ro l for o rigi n and

d estin ation P K) , mo de K T, an d ho using char acte r istic s H .

6. C ON C LU SIO N S

The es tim a tion of the value of travel ti me sa vi ng is one of the m o st w id ely stu di ed to pi cs

in tr ansp orta tion econo mic s , yet at the th eore ti cal level it is one o f the leas t under st oo d. In the

li te rat u re, the val u e-o f-tim e c once p t has be en us ed by di ffere nt r e sea rc he rs to mea n d iffer ent

th i ng s Th ro ug h a rev ie w o f the li te rat u re on ti me allo cati on , the fi rs t par t of this pa p er m ake s

expl icit dis tinc ti ons am on g three con ce pt s: the m one tary value o f the m arg in al u ti lity of tim e

use, the net be nefi ts o f tim e use, and the w illi n gne ss to pa y for ti m e r eallo catio n b etw een two

al te rna tive act iv iti es . This pa per shows that the value of tim e VOT) refers to the seco nd term,

wh ile the v alu e o f trav el tim e sa vi ng V TTS) re fer s to the last term.

T he sec on d part of the p ape r shows that the reas o n for a po sitiv e V T S ’ in ex istin g m od el s

is the im po siti on o f a mi n im u m tra ve l-t im e con st rai n t. No n e of the ex ist in g m od els a ddre sses theque stion why pe ople travel in the firs t place. This p ape r pr opos es an alt erna tive m od el in wh ich

there is no m ini m um tr ave l-tim e r equi re m en t, yet the r e is a posi tive V T I at the op ti ma l poi nt .

T he m os t po pula r app roac h to estim atin g 1/iTS is ba se d on the speci fi cat io n an d e s tim a

tio n of an ind ir ec t u tili ty fun ct io n. The ba sis for this app ro ac h d iffe rs con side rably fr om th at

u nde r lyin g e cono m ic value in go od s m arke ts as un ders tood by e con o mis ts. Alt h ou g h pe o ple ’s

p refe renc es for a pa rticu lar g oo d m ay di ffe r, the market nev e rthe less p la ce s an econo mic va lu e

on the go od eq u al to the go od ’s m ark e t pr i ce T h us, it is unn e ces sary for the rese arch er to s ti

m ate a util it y fu nc tio n to o btai n the econo mic valu e o f the good. Th e sa me pri n cip le can be ap

p li ed to the val ua tio n o f trave l- tim e saving. By st udyi n g m ark et b ehav iour where pe o ple are o ser ve d to trade travel tim e fo r m on ey the resea rche r can esti m ate VTTS w ith out e stim ating a ut il

18



ity function. This paper provides two s of estimating ‘ TTS On e way is to estimate thevariation of housing price with travel time. In this approach. VTTS will vary by mode of co mmuting. residential location transport expenditures and house siLe Another wa y is to estimate

the money time trade off between alternatives for travelling between two fixed points under

similar travel circumstances such as a faster. tolled expressway vs a slower untolled city street.

9



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Mackie, P. J. Sergio R. Jara-Diaz, an d A S. Fowkes. 2 00 1. “ Th e Value of Travel Time Savingsin Evaluation,” Transportation Research, Part E 37 9 1—106.Muth, Edwin 1969. Cities and Housing. Chicago: University of Chicago Press.Oort C. J. 1969. “The Evaluation of Travelling Time,” Journal of Transport Economics and

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Train, Kenneth, and Daniel McFadden. 1978. “The Goods/Leisure Tradeoff an d DisaggregateWork Trip Mode Choice Models,” Transportation Research, 12 349—353.Truong, P. Truong, an d David A. Hensher 1985. “Measurement of Travel Time Values an d

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St. Louis,” Journal of Regional Science 12 469-474.

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