irunspn Rrs B Vol. 17B, No Printed in Great Bntam
6, pp. 463470,
1983
01912615/83 mu 1983 Pergamon
$3.00 + IlO Press Ltd.
ROUTE CHOICE AND THE VALUE OF COMMUTING TIME: AN ECONOMIC MODEL OF DICHOTOMOUS CHOICE JOHN F. MCDONALD? Department
of Economics, (Received
University 20 April
of Illinois
at Chicago
1981; in revised form
Circle,
Chicago,
10 December
IL 60680, U.S.A.
1982)
AbstractThis paper presents an economic model of generalized travel cost and provides an empirical study of the parameters of the cost function. The routechoice model that is estimated combines McFadden’s theory of qualitative choice behavior with a function for the value of travel time in which total trip time and the income level are assumed to influence the marginal value of time. The empirical results indicate that, for a sample of commuters in the Chicago metropolitan area in 1972, the value of time is a positive function of total trip time, but is not a function of income. 1. INTRODUCTION
AND
BACKGROUND
reductions in travel time are the primary user benefits of public investments in transportation facilities. For example, in the classic costbenefit analysis of the Victoria Line in London by Foster and Beesley (1963), the value of travel time savings constituted 48% of the total benefits of the project. Since the original study by Beesley (1965), an enormous number of empirical studies have been done which reveal a valuation for travel time savings. These studies have been reviewed by Bruzelius (1979) Hensher and Dab (1978) and Hensher and Stopher (1979), among others. However, most studies are not based firmly on a formal economic model of consumer choice. A major reason for this failure stems from the fact that the decision which consumers make is qualitative (or discrete) in nature; a route or a travel mode is chosen to the exclusion of one or more possible alternatives. As Small (1978) has pointed out, the work begun by McFadden (1973) permits the construction of formal economic models which can be applied to the route or modal choice decision. The formulation and estimation of McFaddentype models can lead to estimates of the value of travel time savings which are grounded in formal economic theory. However, as is discussed below, the models estimated by McFadden (1974) Domencich and McFadden (1975) and others have failed to exploit fully the possible connections between formal economic models and empirical testing.? In particular, the traveler’s generalized trip cost function should be grounded in economic theory. The purpose of this paper is to estimate such a model using data on actual route choice (tollway or freeway). In this paper it is assumed that the marginal cost of travel time may depend upon the traveler’s income and the length of the trip. This approach is in contrast to the position taken by Bruzelius (1979, p. 16) that “. . . in regard to revealed preference approaches and the problems of obtaining data for such methods, models based on constant marginal values of time constitute the only feasible approach.” An alternative approach to the questions examined in this paper is the use of psychometric attitudinal surveys, such as those used in recent research by Morisugi et al. (198 1) and Horowitz (1978). While these studies have found that a hypothetical value of time varies with several factors, these models have not been grounded in a formal economic model of the value of time. The crucial issue in the development of the formal economic model to be estimated is its level of complexity. The analyst must determine the extent to which the decision under The
tMcFadden has, of course, continued to develop the theory of probabilistic choice. McFadden’s survey paper (1981) discusses the aggregation of preferences and cosumers’ surplus in random utility models, choice of functional form in probabilistic choice systems (e.g. logit, probit, dogit, etc.), and preference trees and sequential estimation. McFadden’s theoretical work has not been concerned with the value of time. 463
J. F. MCDONALD
464
study is subject to constraints, and a level of complexity of the generalized cost function must be assumed. In the case of route choice decisions, the constraints in the problem are relatively straightforward. It is assumed that (1) the individual must make the trip in question (to work, for example), (2) the origin and destination of the trip are given, and (3) the mode of travel is already determined. The question for the traveler may be formulated as the choice between a slower, cheaper (in money terms) route and a faster, more expensive route. In this paper the choice examined is between a faster tollway and a slower route on free streets and highways for auto drivers on the trip to work. It is assumed that the choices of residence and workplace locations and the choice of hours of work are independent of the fact that the particular route choice options under study exist.? Given that the constraints in the routechoice problem can be specified so easily, the focus can shift to the issue of the assumed complexity of the generalized cost function for the traveler. A number of theoretical contributions such as Bruzelius (1979) McDonald (1975), and Small (1978) have suggested that the marginal value of travel time may vary with income and the length of the trip. A simplified model with these implications is presented in the next section. As indicated above, the studies which provide the impetus for this paper are McFadden (1974), and Domencich and McFadden (1975). These studies examined modal choice with the purpose of making predictions of modal split. While the primary objective of these studies was not to estimate a value of time, values of time were generated as a byproduct of the estimation of the model. The model of work modal choice estimated by Domencich and McFadden (1975, pp. 15%165) assumes that there is a single value of invehicle time (and a single value of transit walking time) for the members of the sample studied. This assumption may be true, but as is shown in the next section, it is not the assumption that is normally embedded in the theory of consumer choice. The later study by McFadden (1974) made use of the assumption that the value of time is proportional to the commuter’s wage rate and thus estimated the invehicle value of time as 32% of the wage. While this formulation is better than the assumption of a single value of time, the value of time need not be a proportion of the wage.1 This is a proposition which can be tested empirically. 2. A MODEL
OF GENERALIZED
TRAVEL
COST
In order to conduct a simple derivation of a commuter’s generalized travel cost function, consider a household that consists of only one individual who works at a center of employment. Assume the individual has a utility function which can be written U = U(X, L, N, C, H), where X is a composite good, L time, and His time spent at work. to provide a simple rationale commuting distance. The utility differentiable. Utility is maximized is written
is land consumption, N is leisure time, C is commuting Land consumption and land rent are added to the model for the existence of benefits derived from additional function is ordinal, continuous, real valued and twice subject to money and time constraints. The Lagrangian
L* = U(X, L, N, C, H) + II(wH X
 r(u)L
 t(u)) + Y/(T  N  C  H),
where w is the after tax wage, i and r] are Lagrange multipliers, u is travel distance r(u) is the rental price of land, and t(u) is the money cost of commuting.
(2)
to work,
tThe analysis of modal choice makes the same assumptions, except for the assumption that the choice is between alternative modes of travel. However, the analysis of modal choice is complicated by the fact that the features of the two modes, other than time and money cost, may be quite different from the consumer’s point of view. For example, the consumer must drive the auto himself, but can leave the driving to the bus driver. Modespecific features probably play an important role in the choice decision. See Hensher (1979) for a recent attempt to control for “alternativespecific variables.” The analysis of route choice provides a simpler context in which to study the value of time. fThe more recent studies by Hensher (1977, 1978, 1979) also make use of this assumption.
Route
choice
and the value of commuting
time
465
The firstorder conditions found by differentiating with respect to X, L,N,H, and u are
u,n
=Q,
(3)
u,  Lr(u) = 0,
(4)
UNV =o,
(5)
u,+2w
(6)
_r =o,
and
respectively. Simple manipulations
of these equations yield
(
 r’(u)L = t’(u) +fg
w+
u,u
UC x
1
(8)
and

ac u,u,
r’(u)L = r’(u) +x
(
u
X
) .
(9)
These conditions state that the marginal benefit of added commuting distance r'(u)L equals the marginal cost of distance. The marginal cost of distance consists of the marginal money cost t’(u) plus the marginal value of time times K/au, the time cost of a marginal unit of distance. The marginal value of time can be expressed either as w + [(U, U,)/U,] or as (UN U,)/U,. The former expression is the net cost of giving up a marginal unit of work time, while the latter is the net cost of giving up a unit of leisure time. If we assume that commuting time and work time do not produce utility, then the marginal value of time reduces to w = U,/U,. A large number of empirical studies of the value of commuting time contradict the conclusion that the aftertax wage is the correct value. Estimates reviewed by Bruzelius (1979) tend to be less than 50% of the beforetax wage rate. In the context of the model under consideration here, this result has three possible explanations. Commuting may be a good, meaning that UC/U,> 0;there may be a constraint imposed on hours of work which makes w > (UN U&U,; or work time may produce disutility meaning that U,lU,c0. However, the hypothetical constraint on hours worked implies that workers wish to work more than they do currently. It is not clear why workers should not be able to work more hours if they so wish. Also, it is possible that there is disutility in work at the margin, but it should be realized that U, I 0 means that the individual prefers to have his total time budget (his life span) reduced to working the marginal hour at no pay. Obviously the individual would rather have leisure than work at no pay, but it is doubtful that the individual would opt to have his life shortened. Thus, if we assume that all uses of time contribute to utility in the relevant range, we are left with the notion that UC/U, > 0 is the best explanation for the observed empirical results. It thus seems reasonable to consider the marginal value of time to be (U, UJU,. This formulation has important implications for the form of the generalized travel cost function. As commuting time increases, the marginal value of time can be expected to increase because it is likely that UN/U,, the marginal value of leisure, rises and UC/U,, the marginal value of commuting time, falls as C increases. Also, an increase in income can be expected to increase U,jU,and have an ambiguous effect on UC/U,. (It is possible that higher income people own more expensive autos which are more “fun” to drive.) In summary, the basic model of time allocation for commuters presented in this section leads to the hypothesis that the marginal value of travel time may increase with the length of the trip and income.
466
J. F. MCDONALD
The basic theory of generalized cost and the value of time developed in this section is similar to the theoretical underpinnings for empirical models developed recently by other researchers. The study by Train and McFadden (1978) is based upon the assumption that leisure time and expenditures on goods are the only arguments in the commuter’s utility function. Uti!ity is maximized subject to a time constraint (7’  N  C  H = 0) and a money constraint (wH  X  t(u) = 0). In such a model the money value of a marginal reduction in required commuting time is simply equal to the aftertax wage rate, which is the value of additional leisure time. A more general model has been developed by Truong and Hensher (1982). A simplified version of their model would modify the utility function of eqn (1) to be
U = U(X L, T,, i21, where T, is not an assumed standard
(10)
and T, are times spent in travel on modes 1 and 2. In their model work time argument in the utility function, and no consumption good such as land is to have a price which varies over space. Utility is maximized subject to the money and time constraints wH p,j,
pzj,
 X = 0
(11)
and TLT,TzH=O,
(12)
where p, and p2 are the money costs per trip on modes 1 and 2, and j, and_& are the numbers of trips (journeys) taken on the two modes. In addition, Truong and Hensher (1982) add four constraints of the form T,  t,j, = 0
(13)
Mi  p,j, = 0,
(14)
and
where t, is the time needed for a trip on mode i and M, is the total amount of money spent on trips on mode i (i = 1 or 2 in this case). However, eqns (14) play no role in the further development of the TruongHensher model, so they are dropped from this discussion. The Lagrangian to be maximized is thus
L* = u(x, L, T,, TJ +
i(wH
p,j,
p2j2
 X) + q(T  L  T,  T,) + &CT,  tlj,) + &(T,  tzjJ,
where K, and K, are additional Lagrange multipliers. Maximization respect to X, L, T, and T2 produces the firstorder conditions
(15)
of eqn (15) with
u,  i = 0,
(16)
u,r/ =o,
(17)
U,,
‘1 +
K, = 0,
(18)
and U,r/+K,=O. From
these conditions,
(19)
one can derive
(K/u,) = (u,lux>  WT,P.J
(20)
and (&/Ur)
= (U,lU,)
 (U,lU,).
(21)
Route choice and the value of commuting time
461
Here K, and K2 are shadow prices of time; the additional amounts of utility which would result if the amount of time needed to takej, and j, journeys, respectively, were reduced by one unit. Equations (20) and (21) state that the money values of reductions in travel times equal the money value of leisure at the margin minus the money value at the margin of time spent in travel. This result is the same as the result for the value of a reduction of commuting time in eqn (9). Truong and Hensher (1982) focus attention on the hypothesis that K,/U,may not be equal to K,/U,(and that neither is equal to zero) because of an imperfect allocation of time due to the presence of technological constraints on travel time. Indeed, if there were no constraints on the travel times which the individual may choose, the net value of a reduction in travel time and corresponding increase in leisure time would be zero. For such a person the last minute spent driving the car to work (or riding the bus) is as enjoyable as the last minute of leisure time. Few such people exist. This assumption in their model performs essentially the same function that the inclusion of land consumption and land rent performs in the present model. In their empirical work, Truong and Hensher (1982) estimate modespecific values for a reduction in travel time. However, they do not consider the possibility that K,/U,and K,/lJ, both may vary systematically across individuals, or for the individual that K,/U,(or KJU,) vary may as a function of exogenous variables. The hypothesis examined in this paper is that the value of a reduction in travel time for a single mode may vary for a single individual and across individuals. 3. A STOCHASTIC
MODEL
OF INDIVIDUAL
CHOICE
This section presents a simple version of the stochastic binary choice model as developed by Domencich and McFadden (1975). Assume that the generalized cost of the commuting trip on route i for the jth person is
(22) where M, is the money cost, C, is commuting time, (V,,/C,) is the average value of commuting time (with V, = the total value of commuting time), and uii is the random component in generalized cost. As Hausman and Wise (1978) have discussed, there are two possible explanations for the random component. It may be that individuals behave randomly so that the individual does not make the same choice when repeatedly faced with the same choice set. A better explanation is to argue that uil represents variables that are not observed by the researcher but influence the generalized cost. For the problem at hand the choices for the commuter are only two; the commuter may select to drive on a free street or highway, or to drive on a tollway. If the tollway is chosen, the commuter saves time at the expense of paying a toll. The commuter will choose the tollway if GCYj > GC, where f and t refer to freeway from eqn (22) produces
The probability
that the tollway
and tollway,
is chosen
respectively.
(23) Substituting
for GCf, and GC,j
can thus be written
P,=Prob[u,ju,j>(M,jMI,)+[(~)C,,(~)C,j]],
(25)
or
(26)
468
J. F. MCDONALD
where F[ ] is the cumulative distribution function of (~4,~ u,~). In this problem F[ ] is assumed to be a normal distribution with mean zero. Because 1  F(x) = F(  x), eqn (26) becomes
(27) Now consider the functional form for (V,,/C,), the average value of commuting time. It is assumed that the average value of time is a linear function of commuting time and income, or
v, = 00 + 01c,,+ u*Yij,
0
Cl,
where Y, is income of the jth person if the ith route is chosen, and uo, u, and v2 are the parameters of the function. This function implies that the marginal value of time is v; = Ug+ 2u,cij + u* Yij. Substitution
of eqn (28) for the tollway
and freeway
(29)
choices
into eqn (27) yields
Now variables (M,,  M,,) and (Crj  C,,> are simply the differences in the money and time costs of the two routes; the variables that are conventional to this type of analysis. The variables ((Cc  C’i.) and ( Y,Ctj  Y,jC’J are the additions to the specification implied by the hypothesis that the average value of time varies with trip time and income. Given that F[ ] is the normal distribution, probit analysis can be used as the estimation technique. The convention in probit analysis is to assume that the underlying error term (or threshold level) is distributed as a standard normal variable (unit variance). Thus, the distribution function to be estimated is that of (ufi  u,,>/u instead of (ur,  uIj), where 0 is the standard error of (Us,  u,,). Equation (30) becomes
PI= F [
kt”t]

Mfj)

:
(Ctj
where F[ ] now has a unit variance.
4. THE
DATA

Cf,)  : (Cs  Cj,)  % (Y,C,,  Y,C,,)
This equation
AND
,
(31)
1
is estimated.
EMPIRICAL
RESULTS
The data used involve a choice situation for commuters who all drive a private auto to work and who do not carry any passengers. The commuters have the choice of taking a tollway (a faster trip) or using free streets and highways (a slower trip). Individuals who face this choice are not hard to find because, assuming people value their time, tollways should be less congested than freeways. Commuters filled out the questionnaire at their suburban places of work. If the commuter uses the tollway, he was asked to specify the amount of time saved. If the commuter uses a freeway, he was asked to specify the amount of toll he avoids (if a tollway is relevant to him) and how much time is added to the trip to avoid the toll. Some 900 persons at 17 different firms filled out the questionnaire, and 115 persons indicated that they face the tollwayfreeway (and responded to the other questions). The sample of 115 is used for the results reported here. The survey was conducted in May 1972 for the Northwest Conference of Mayors, a group of mayors from towns in the northwestern area of Cook County, Illinois (Chicago metropolitan area). An earlier analysis of these data was done by the author (McDonald, 1975). The variables used in the analysis are: C,, = trip time for the journey to work if tollway is chosen (min); C,, = trip time for the journey to work if the freeway is chosen (min);
Route
choice
and the value of commuting
469
time
Mf,) = the toll on the tollway (cents); Y/ = household income (cents/day); and household income minus the toll on tollway (cents/day).? The analysis is conducted assuming that the toll is the only difference in monetary expenditures between the tollway and freeway choices. This assumption is reasonably accurate because the tollway users purchase a somewhat faster trip, but not a shorter trip. In the area of the study there are always parallel free streets and highways in the close vicinity of the tollway. Also, the toll of 30 cents or more was, in 1972, large in relationship to possible differences in vehicle operating costs associated with differences in speed. Furthermore, drivers probably do not perceive these small differences in operating costs. Means and standard deviations for the basic variables are shown in Table 1. Estimates of alternative formulations of eqn (31) are shown in Table 2. In column 1 only the toll on the tollway and (C,,  C,,), negative the amount of time saved on the tollway, are included as independent variables. Both variables are highly statistically significant, and the value of a reduction in commuting time is estimated to be 3.27 cents/min, or $1.96/hr. However, the addition of the other variables implied by (28) for the average value of time, the results of which are shown in columns 2 and 3, significantly increases the log of the likelihood functi0n.S In particular, from the results in columns 2 and 3, it is clear that the only additional variable that makes a significant contribution is (Cb  Cf,). The other variable which incorporates the hypothesis that the average value of time is a function of income is not at all significant. From eqn (29) the estimates in column 2 can be used to form the expression for the marginal value of commuting time, which can be written
(II~,~ Y,, =
v:,=
 0.048 1 + 0.0024 C,, 0.0213
cents/min.
Evaluated at the mean of C, of 48.15 min, the marginal value of time is 3.17 cents/min. More importantly, perhaps, the marginal value of time increases by 0.11 cents/min as commuting time increases by 1 min. In other words, two commuters who travel 30 or 60 min for the journey to work value a Imin reduction in commuting time at 1.12 and 4.40 cents, respectively.
Table
I. Route
choice
data:
variables,
means
and standard
deviations Standard Deviation
Mean .40
Tollway user Trip time if freeway is chosen (minutes)
48.15
18.47
Toll if tollway iS chosen (cents)
40.04
20.91
Household income (dollars/year )
15,330
$5758.
Time saved if tollway is chosen
10.64
8.04
115
Sample size
tThe income variables used in the analysis are not net of vehicle operating costs because these data are not available. However, the correlation between income level and trip time on the freeway (one measure highly correlated with trip length in miles and operating costs) is low (R2 = 0.008). Thus the failure to adjust income for vehicle operating costs essentially creates a minor problem of random measurement error in a variable that already is probably subject to measurement error. fThe likelihood ratio test is calculated  2 In i = 2[ This statistic
is distributed
Xsquare
with 2 degrees
 70.16 + 60.82]= of freedom,
18.7
and is significant
at the 99% level.
470
J. F. MCDONALD Table
2. Probit
analysis
of route
choice
Independent variable Toll on tollway (MtjMfj) Time saved on tollway times 1
(dependent
variable
(1)
= 1 for tollway; (2)
= 0 for freeway)
(3) .0259
.0213 (4.07)
.0259 (4.47)
.0697
.0481 (1.38)
.0474 (1.03)
.0012 (3.97)
.0012 (3.97)
(3.86)
(4.46)
Cc,jmcfj)
(C:jC:j)
(ytjctjyfjcfj) Log of likelihood function
___
__
. 000000~ (0.02)
70.1594
60.8174
60.8171
Unsigned t values are in parentheses.
5. SUMMARY
Previous studies of the value of commuting time have not been based upon a form of the commuter’s generalized cost function that is sufficiently general. The purposes of this paper were to show that an improved model can be developed and to present an empirical example. The model combines a McFaddentype model of route choice for auto drivers and a generalized cost function in which the value of time is a function of commuting time and income. The route choice decision studied is the choice between a faster tollway trip and a slower freeway trip for the journey to work. Probit analysis is used for empirical estimation, and the results for a sample of commuters in the Chicago metropolitan area for 1972 indicate that the value of of commuting time increases with the length (in mins) of the trip to work. This empirical finding should help to provide more precise estimates of the benefits of road improvements. More importantly, the model developed in this paper may lead to additional studies of the value of time that are more rigorously connected to the economic theory of time allocation. Acknowledgements~The author wishes to thank Pat Phillippi who organised referees provided helpful comments for which the author is grateful.
the collection
of the data.
The
REFERENCES Beesley M. (1965) The value of time spent travelling: some new evidence. Economica 32, 174185. Bruzelius N. (1979) The Value of Travel Time: Theory and Measurement. Croom Helm, London. Domencich T. and McFadden D. (1975) Urban Travel Demand. North Holland, Amsterdam. Foster C. and Beesley M. (1963) Estimating the social benefit of constructing an underground railway in London. J. Roy. Statistical Sot. Series A 126, 4692. Hausman J. and Wise D. (1978) A conditional probit model for qualitative choice : discrete decisions recognizing interdependence and heterogeneous preferences. Econometrica 46, 403426. Hensher D. (1977) Valuation of interurban travel time savings: a study in choice of route. Int. J. Trans. Economics 4, 6980. Hensher D. (1978) Valuation of commuter travel time savings. J. Trans. Economics and Policy 12, 9097. Hensher D. and Dalvi M. (Eds.) (1978) Determinants of Travel Choice. Saxon House, Teakfield. Hensher D. (1979) Individual choice modelling with discrete commodities: theory and application to the Tasman Bridge reopening. Economic Rec. 55, 243260. Hensher D. and Stopher P. (Ed%) (1979) Behavioral Travel Modelling. Croom Helm, London. Horowitz A. J. (1978) The subjective value of time spent in travel. Transpn Res. 12, 385393. McDonald J. (1975) Variations in the value of reductions in commuting time. J. Urban Economics 2, 265.277. McFadden D. (1973) Conditional logit analysis of qualitative choice behavior. In Frontiers in Econometrics (Edited by P. Zarembka). Academic Press, New York. McFadden D. (1974) The measurement of urban travel demand. J. Public Economics 3, 303328. McFadden D. (1981) Econometric models of probabilistic choice. In Structural Analysis of Discrete Datu with Econometric Applications (Edited by C. Manski and D. McFadden), Chap. 5. M. I. T. Press, Cambridge, Mass. Morisugi H., Miyatake N. and Katoh A. (1981) Measurement of road user benefits by means of a multiattribute utility function. Papers, Reg. Sci. Assoc. 46, 3143. Small K. (1978) Studies of the value of commuter travel time saving: a comment. J. Trans. Economics and Po1ic.y 12, 8&89. ’ Train K. and McFadden D. (1978) The goods/leisure tradeoff and disaggregate work trip mode choice models. Transpn Res. 12, 349353. Truong T. and Hensher D. (1982) The value of travel time savings form a discretechoice model. Mimeo, unpublished, Macquarie University, North Ryde, Australia.