Transpn ReL VoL tO, pp. 31-35. Pergamon Press 1976. Printed in Great Britain

MODE CHOICE CHARACTERISTICS AS DETERMINANTS O F INTERURBAN TRANSPORT DEMAND

JOHN KRA~ Federal Energy Administration, Washington, DC 20461, U.S.A.

and

ARTHUR KRAFt Department of Management, The University of Nebraska at Lincoln, Lincoln, NB 68508, U.S.A.

(Received 12 October 1974; in revised [orm 10 September 1975)

Abstract--The ability of individuals to substitute modes of transportation leads to an inability to determine the demand for travel between cities. If one were able to determine ihose characteristics of each mode which influence the choice decision, one would then be able to arrive at more stable demand functions for each of the travel modes. With a more stable demand for travel function policy decisions could better be made concerning the allocation of public funds, to highways, airports, or tail facilities. The question of choosing between modes and the basis of choosing between modes is closely related to the stability of the demand for travel. Baumol and Quandt, Gronau, McGillivray, and Young have all attempted to derive demand equations for travel modes. However, some of the studies attempted to measure the substitutability of the characteristics influencing the demand for travel, rather each focuses on a determination of the quantity of trips undertaken of each mode between origin i and destination j as a function of characteristics. The purpose of this paper is to provide new evidence bearing on the issue of the degree of substituability of the various travel modes. We begin with a utility function rather than a pr/or/specifying the demand equations. In specific we employ a multilevel separable utility function which allows the grouping of various choice characteristics of each mode into subsets or branches from which estimates of the ease of substitution among characteristics of the mode may be obtained. The characteristics which enter the branches for each mode ate proxies for cost, comfort, and convenience. Since the specific function is weakly separable and nonlinear, we will obtain nonlinear demand equations for the mode characteristics. Another important advantage of using this approach is that it allows us to ascertain the preference intensity of individuals for certain mode characteristics which influence the choice of one mode over another. Moreover, the estimated budget parameters will allow us to determine the optimal combinations of characteristics to be preferred by consumers, and thus their model choice preferences.

INTRODUCTION

Over the years many studies have estimated the demand for travel modes on both inter- and intra-city travel networks. Plourde (1968), Quandt and Baumol (1966), Quandt and Young (1967), and Young (1969) used a modified abstract mode approach in the determination of mode travel decisions. Slightly different approaches were employed by McGillivray (1972) and Gronau (1970). The former used a linear probability model and the latter a "Becker" value of travel t ime approach. The present study takes a different approach.

Mode demand is determined by a preference ordering of mode characteristics. The model applies a utility function to inter-city travel in the Northeast corridor of the United States. The demand for any mode is derived from an individual utility funct ion. Demand is a function of not only the mode's price but its other quantifiable characteristics. The model involves a two stage choice procedure. First, the individual determines a preference ordering of quantifiable mode characteristics, such as cost, comfort, and convenience. Individual preferences are then used to order the characteristics of each mode. In the second stage mode travel choices are made based on the preference ordering established in the first stage. Mode selection is accomplished on the relative cost, comfort, and convenience characteristics of each mode.

Mode characteristics are introduced directly into a transportation utility function for each traveler in the

sample. The utility der ived from traveling on a particular mode is a function of i t s attributes. These attributes reflect the cost, comfort, a n d convenience characteristics of each mode. Both t h e monetary and nonmonetary aspects of each mode characteristic are reflected in its demand price. An abs t rac t mode model is developed which explicitly takes a c c o u n t of each traveler's t ime and value of time. A model similar to this was u s e d in an earlier work by Kraft a n d Kraft (1975a). However, the earlier paper had severa l shortcomings: it d id not explicitly take account o f the characteristics o f each mode; it used a less sophisticated model; and it only implicitly utilized a t rave le r ' s estimated travel t i m e and value of time.

THE TRANSI:~OR'I'ATION MODEL

A transportation utility function is maximized subject to a budget constraint to o b t a i n modal demand equations. The utility function def ines individual behavior. Four modes of travel are cons idered: airplane, automobile, bus, and railroad. Thus the utility function has twelve arguments: three characterist ics for each of t h e four modes. The maximization procedure requires a n overall budget constraint which is broken into specific constraints to determine the preference orderings in the first s tage of the maximization procedure . Before turning to t h e choice of a utility function the a t t r ibutes of each mode m u s t be quantified.

31

32 J. KRAFT and A. KRAFT

The data are derived f rom an analysis of thirty-two city pairs in the Northeast corridor of the United States for 1%8. All data apply to the same time period and cover the geographic region from Boston to Washington, D.C.

The cost characteristic for each mode is a function of the number of people who travel on the mode. The number of one way round trips by individuals on each mode reflect the passenger volume for that mode during the time period studied. The passenger volume of the sth mode is Qsv. The price of this argument in the utility function is P,o, which reflects the average cost of a trip in both directions. Not only is the direct cost of traveling on this mode taken into account, but also any indirect costs incurred while traveling to and from the mode's departure and arrival points are added into total costs,

It is assumed that a person is more comfortable the shorter is the time required to go from Point A to Point B. The marginal utility of a trip is inversely related to the travel time of the trip, and the travel time for a mode is a function of the mode's speed. The speed of the sth mode in miles per hour represents the comfort factor (Qs,). The speed of each mode is calculated by dividing the trip distance by the time required to make the journey on that particular mode of transportation. The trip distance includes not only the distance between each city-pair in the sample but also the access and egress distance between the origin and the mode departure point and between the arrival point and the destination. The travel time was also adjusted for the time spent in traveling to and from the mode departure and arrival points. P,, is the price of the comfort attribute for the s th mode. The price or value of the speed of traveling on the sth mode is calculated by using a multinomial logit estimation technique (see Kraft and Kraft, 1974, and Theil, 1%9).

A traveler is less inconvenienced the shorter is the time spent in waiting for a mode to depart. Frequency of departure for the sth mode on an annual basis is used as a proxy for the convenience characteristic (Q,~). The more frequently a mode departs, the less time a traveler waits for a mode to depart and the more convenient it is to travel on that mode. A generalized frequency price (P,g) was constructed for each mode. The traveler's hourly value of time was computed for each mode using a linear logit model (for example, see Kraft and Kraft, 1974). The generalized frequency price of each mode is calculated using the computed hourly value of time and the average waiting time, which is estimated using a bulk queuing model (see Bailey, 1954, and Kraft, 1974).

When selecting a utility function, one should choose a function which allows for both substitutes and comple- ments, as well as one which does not a priori limit the degree of substitutability or complementarity among its arguments. For this reason a separable utility function is employed in this study.

There are, however, several forms for separable utility functions. Let the set of all goods be denoted by Q. That is to say, let

Q = [q , ,q2 . . . . . q.]. (1)

The simplest separable utility function is one which is

additive. A specific function which is additively separable is the following one:

u(o) = ~ U(q,). (2)

This function is based on the Klein-Rubin (1948) linear expenditure system and was first estimated by Stone (1954).

A more complex separable utility function is one derived by Geary (1%0) and Samuelson (1948). It may be written as follows:

n

U ( Q ) = ~ /3 , log(q, - y,), (3) i = I

where q, represents the amount of commodity i con- sumed. Each q, is treated as a Hicksian composite commodity rather than a specific good. Each /3i is a commodity intensity parameter representing the propor- tion of any incremental income spent on the supernumer- ary quantity ( q l - y i), where y~ is defined as the committed level of q~. It is assumed that an individual derives satisfaction only from those commodities con- sumed in excess of their committed levels. For this particular function, the marginal rate of substitution between any two commodities depends only on the quantities of these two commodities.

This utility function (referred to as Stone-Geary) has several limitations which should be noted. All goods are restricted to being substitutes. The own price elasticities are restricted to the range from zero to minus one. A third important limitation is that the elasticities of substitution between all commodities are a priori specified to be unity.

Directly additive utility functions without the Stone- Geary restriction on the elasticity of substitution have been developed by Houthakker (1960) and Sato (1972). The general form for these utility functions is as follows:

U(Q) = [/~,(q,- yt) p + - - ' + fl.(q~ - y. Y]". (4)

For this type of utility function the substitution parameter is p and is equal to 1 - 1/~, where ~ is the elasticity of substitution between all supernumerary quantities.

Thus far, only a single commodity subset has been considered. The commodity subset Q, however, may be partitioned into s disjoint subsets. That is, one may write the expression for Q as follows:

Q = Q, u Q~.. . u Q,, (5)

with Qr N Qs = ~b for r# s. Each of the disjoint subsets in turn consists of n commodities. When the n commodities can be partitioned into s subsets as follows:

U ( Q ) = u[U(Q, ) . . . . . U(Q,)], (6)

the utility function, U(Q), is referred to as a utility tree. A utility tree is weakly separable with respect to the partitioned commodities. As demonstrated by Goldman and Uzawa (1964) and Pollak (1971) weak separability

Mode choice characteristics as determinants of interurban transport demand

occurs when the marginal rates of substitution between any two goods within the same commodity subset or group are independent of the quantities of commodities outside of the group. If q,k denotes the kth good in the rth branch, the marginal rate of substitution between the commodities i and j within branch r is independent of quantities outside this branch.

Research on utility trees has been conducted by Brown and Heien (1972), Leontief (1947), Strotz (1957), (1959), and Gorman (1959). In particular, Brown and Heien's St-branch utility tree is a multilevel function based on Uzawa's (1960) generalization of the CES production function. It may be written as follows:

U = ~,j (q,i - 3',s)°' s ~ l .=

(7)

where all of the parameters may be interpreted in the same way as their counterparts appearing in the functions discussed above. The to, parameter represents the intragroup supernumerary budget share, or the share of the budget allocated to branch s, and is a constant. The St-branch utility tree is a quite general representation of a utility function in that each intrabranch pair of goods can be Hicks-Allen substitutes, complements, or indepen- dents. Furthermore, the own price elasticity of each good is permitted to vary between zero and minus infinity. This generalized function allows the intragroup partial elastic- ity of substitution coefficient between the arguments of the function to be determined empirically rather than specified a priori. Within each branch the commodity

intensity parameters, sum to unity (,~=t/3,~ = t) and the t

intragroup partial elasticity of substitution, tr,, is a function of p,. Specifically, tr, =[l/(1-p,)]. For each interbranch pair of goods, however, the partial elasticity of substitution is restricted to unity (tr = 1).

A further generalization occurs when U(Q) is block additive. In this case, the n commodities in Q may be divided into s disjoint subsets (Q,) as follows:

s

U(Q) = ~ U(Q,). (8) 2)=I

Block additivity occurs when the marginal rate of substitution between two commodities i and j from different subsets, say Qr and Qz, does not depend on the quantities of commodities outside of Qr and Qz (see, for example, Goldman and Uzawa, 1964, and Pollak, 1971). The S2-branch utility function of Brown and Heien, which is written as

S n,

• %,)',J ) , (9)

is an example of a block additive function. Unlike the St-system, the interbranch partial elasticity of substitu- tion (tr) is not necessarily unity. Rather, it is equal to l /(1-p). Note, an additional properly is that the S2-branch utility function satisfies the integrability condi- tion in Pressman's (1970) paper on interdependent

33

demand curves. This is a particularly important condition if one were going to calculate consumer surplus.

The directly additive function (eqn 7) was used in the earlier Kraft and Kraft (1975a) study. In the current study the Srbranch utility function is employed. The parameter estimates of such a function allow one to obtain extensive information regarding the utility of each item. The intragroup partial elasticity of substitution (tr,) varies among different groupings rather than remaining fixed at unity as in the directly additive situation. In addition, the interbranch partial elasticity of substitution (or) is not restricted to unity. The advantages of using such a multilevel function far outweight the additional complex- ity of the function and the additional degrees of freedom used up in estimation.

Having decided on the S2-branch utility system, one must now specify the proper grouping of the utility function. This involves determining the number of branches and deciding which items are allocated to each branch (see Kraft and Kraft, 1975b). At most, there could be twelve branches and at a minimum one. Strotz (1957) mentions that branches correspond to budget categories. An allotment is made for each branch and then spent optimally on the commodities within that branch.

In the present study the utility function represents the satisfaction derived from the various transportation modes by the individuals traveling on modes. Following the lead of Strotz (1957), where the branches correspond to budget categories, each branch represents a transporta- tion mode. The arguments within each branch are the cost, comfort, and convenience characteristics for that mode. Thus, there are four branches (S = 4) with three items (n, = 3) in each branch. The transportation utility function is maximized subject to a transportation budget constraint. The overall income constraint (y) represents that portion of the traveler's budget allocated to the various modes: airplane, automobile, bus, and train. The income constraint for each mode (Y,) is the amount of the traveler's budget allocated for each mode. I1, is the average expenditure for each individual on a particular mode for the sample observations.

Ys = ~ P,jq,~ (10) i ~ s

Y = ~ g~ (11) s= l

In general terms the model becomes:

u = , = , j

:± Y, P,Jq,s for all s = 1 ..... 4. i E s

3

/3,j > o ~'. ~,j : 1 iEs

p , < l ; tr ,= _ (h,~-%i)>0; t r = ~ _ p .

(12)

(13)

As with all multilevel separable utility functions, it is

T.R. 1 0 / l ~

34 J. KRAFr and A. KRAFT

maximized subject t o an income constraint in a two stage process (see Gorman , 1959; Strotz, 1959). First, the utility within g roup s is maximized subject to its constraint (Y,) resu l t ing in the preliminary demand equations:

(~) ' [ - ~ ] [~ ' " . " " ~-', = " + , , , , / • (14)

.iEs

The preliminary d e m a n d eqns (14) are then inserted into eqn (12) giving

where

ns

M , = Y. - ~ P.i%J. iEs

Maximization of (15) subject to total expenditure,

S S ",

v--X Vr=XXm r = l r=l je t

M , = [Ols~Xs(~-I)l(=*-l) l

r=l i 6 r

( ~ $ i ) ~' o" ( (~ - l ) / (~ - l ) ) - I ¢ , = % , + p,_ [a , x~ , ]

]'[ ] X a r t r x ( ( t r - 1 ) l ( e r - I ) ) Y - P~7~ . r=l j e t

The Allen partial elasticities of substitution (cr,,.~) between (q,~ - 7,~) and (q~ - 7r~) are given by

{~ for s = # r, interbranch

~r,i.~ = +-~--~(cr, - ~r) for s = r, intrabranch

where

j ~ s r=l i ~ r

W, is the intragroup supernumerary budget share. A higher value of W, relative to W, for the same utility function implies that the traveler derives more satisfac- tion from the characteristics (q,j) in the branch associated with W,.

e~'OtMETER ESTIMAT~ AND CONCLUSIONS

(15) The stochastic disturbance terms, which are normally distributed with zero mean and constant variance, appear directly in the demand equations. It is assumed that (n, - 1) disturbance terms within each branch have a joint normal distribution (see Pollak and Wales, 1%9, and Parks, 1%9). A nonlinear full-information maximum

(16) likelihood procedure is applied (see Eisenpress and Greenstadt, 1966).

The committed parameter values (7,s) were found to be not significantly different from zero. On the surface this does not appear to be surprising. What would be the

(17) economic meaning of a committed parameter in terms of the cost, comfort, and convenience characteristics of each mode? Surely there is no commitment on the part of people with respec t to mode characteristics. People

(18) derive satisfaction from each mode attribute rather than only those at tr ibutes in excess of a subsistence amount. It would have been extremely difficult to specify target characteristics fo r each mode. For these reasons the

(19) committed parameter values are fixed at zero. Pollak (1960) has previously shown that the other parameter values are unaffected by constraining the subsistence values at zero.

Table 1 contains the estimated parameter values and their corresponding standard errors for the preliminary demand equations (eqn 14). The preferences for each mode attribute a re based upon the relative size of the commodity intensi ty parameters (/3,j) for that mode.

Table 1. Parameter estimates for the preliminary demand equations

Cost Comfort Convenience Substitution Fit

Branch I - Airplane

6 ~ B a I ~2 Ii 12 13

.012541 .725683 .260285 1.2Ol18 .980801 (.002080) (.216928) (.074273) (.197227)

Branch II - Aut~obile

821 822 623 o 2 ~2

.370516 .308215 .321166 1.62019 .984157 (.106385) (.089947) (.097521) (.354B88)

Branch III- Bus

8 6 8 o ~2 31 32 33 3

.683904 .098346 .215407 1.49127 .983325 (.131709) (.005898) (.063077) (.317778)

Branch IV - Railroad

6 8 643 0 ~2 41 42 4

.543409 .246181 .208266 1.83105 .981391 (.149981) (.732750) (.069381) (.287196)

Mode choice characteristics as determinants of interurban transport demand

Table 2. Parameter estimates for the final form of the model

Airplane Automobile Bu___ss Railroad Substitution Fi__~t

Wl W2 W3 W4 ~ ~2

t . 047 336 .401841 .113598 .435834 .810085 .989957

(.008729) (.088638) (.028556) (.110777) (.101997)

35

The parameter estimates f o r the final form of the model are presented in Table 2. A l l estimated values are highly significant. The/~: statistics w e r e computed as one minus the ratio of the variance o f the disturbances to the variance of the dependent va r i ab le (see Pollak and Wales, 1969).

The cost factor is first in the preference ordering of each traveler for railroads followed by comfort and convenience. For bus travel the cost factor ranked first with convenience second a n d comfort last. The attributes for automobile travel had the same ranking in the traveler's preference ordering as those for bus travel. The preference ordering for air t rave l was radically different from that of the previous modes . The comfort factor was first in each traveler's p re fe rence ordering followed by convenience with cost last.

The strong preference for the cost factor in three of the four modes implies that t rave lers feel these modes are economical means of transportation relative to airplane travel. Likewise, the strong dominance of the comfort factor by airplanes indicates that travelers using this mode are interested in moving f r o m their origin to destination point as quickly as possible. The most favorable attribute of the airplane is speed. Comfor t ranked last in the remaining modes except for rail travel. Travelers seemed to feel that the railroads w e r e a fast and comfortable means of transportation be tween origin and destination pairs relative to the automobile and bus. Convenience ranked second for three of t he four modes: airplane; bus; automobile. Travelers seemed to feel that frequency of departure for these modes was more than adequate.

The values of the estimated substitution parameters for each mode are quite high and all significantly greater than unity. Travelers are quite flexible in their substitutability among the characteristics o f each mode. For each of the modes travelers are willing to substitute between the cost, comfort, and convenience factors. They will substitute a change in cost for a change in comfort or vice versa without much difficulty. The same also holds for the other attributes. They would be willing to trade higher costs for increased frequency of departure or increased speed.

When all of the parameters are evaluated and com- pared, the results imply that the travelers in this sample derive the greatest utility from traveling by train as opposed to one of the other modes. These findings would also support the view that rail service has a future in markets of high population density.

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