Regional Science and Urban Economics II (1981) 573-584. North-HoGand Publishing Company

RETURNS TO SCALE IN THE UNITED STATES INTERCITY BUS INDUSTRY*

Martin WILLIAMS

and

Carol HALL

Northern Illinois University, DeKalb, IL601 15, USA

Received October 1980, final version received February 1981

This paper has taken into account the a priori restrictions available from neoclassical cost theory in evaluating .he relationship between cost and the level of output and input prices for U.S. intercity bus service. A general translog cost function is used which allows tests of the de&l% of returns to scale, homotheticity and non-constant elasticities of substitution among input pairs. Major empirical findings are: (i) the intercity bus Service can be modeled by a homothetic production function, (ii) operators can substitute labor for capital by using vehicles more extensively, (iii) there are potential economies of scale in the provision of intercity bus service, and (iv) the Cobb-Douglas functional form used in earlier studies of the industry is inappropriate.

1. Iatroduetion

The Middle East oil embargo of 1973 brought a new dimension to the problem of energy availability in the U.S. transportation sector. At the height of the crisis major airlines were forced to cancel flights due to the fuel shortages. While the threat of future disruption in supply is relatively small, the outcome of the 1973 embargo has been felt in terms of increased prices of fuel.

When airlines cancel flights because of fuel shortages, or when the price of airline travel is too high relative to the price of other modes of intercity transport, travelers are forced to seek alternative transport modes to reach their destinations. Intercity bus passenger transportation is one alternative mode used by travelers. Intercity bus service connects more towns and cities than rail and airline service. It is estimated that over 15,000 cities and towns are served hy bus transportation, compared to 500 local areas with rail service and 850 airports with airline service.’

*Tnanks are due A&&ir D&xl and an anonymous referee for mOSt Constructive commenti on an earlier version of this paper.

‘Fravel (1978. p. 551).

/)166-0462/81/oooO-oooO/$O2.75 0 1981 North-Holland

574 M. M’illiams mui C. Hall, Re:urns to scale in the U.S. intercity bus industry

Direct government regulation of the industry occurs both at the federal and state levels. Fares for interstate services are regulated by the Interstate Commerce Commission as are route certificates. Service levels and safety requirements are determined by both the Interstate Commerce Commission and state agencies. We evidence to date provides insufficient information to conclude that intercity bus regulation is necessary. The only existing empirical study of the industry has not provided conclusive evidence towards reso!tring the question of whether there are increasing returns to scale. Although Fravel (1978) has found some evidence of slightly increasing returns, a more complete analysis of the existence of economies of scale is important to a thorough understanding of the regulatory process. This study attempts to improve on the. previous research by incorporating more variables which can influence intercity bu3 costs, and by employing a very general functional form for the cost function specification. Fravel’s model failed to take account of the prior restrictions available from neoclassical theory in evaluating the relationships between these intercity bus systems’ cost conditions and the level of service and prices of inputs. Rather, the pioneering study by Fravel (1978) has adopted the Cobb-Douglas cost function approach which is a restrictive functional form, ruling out an exploration of the nature of the production technology. For example, the basic properties of the functional form we have chosen allow changing scale economies, and do not impose a priori restrictions on the elasticities of substitution between input nails [in a Cobb-Douglas formulation the

Elasticity of substitution (0) between factor inputs equals one]. The general functional form used in this study also allows us to test for the Cobb- Douglas and other specifications through parametric restrictions on it.

Section 2 discl.sses the production model of intercity bus service. The third section discusses the data. In section 4 we present and discuss the empirical results. The last section summarizes our findings and considers their implications.

2. Model of intercity bus costs

Apart from the single study by Fravel (1978), there is ii!tle knowledge of the nature of the cost structure of intercity bus transl;;ortation service. Intzrcity bus transportation inputs are classified into labor, capital (vehicles) and fueL2 Cost (C) is total operating expenses for labor, and fuel and capital exlrcnditures. Carrier produce levels of service (Q) which are determined by the regulatory agency over some predetermined region. Fares are also determined by the agnncy. Fuel prices, wages and the opportunity costs of borrowing funds for new vehicles are exogenously determined. Thus, for

‘We ignore inputs such as garages, terminal operations and other inputs used at check point stations, since the data are sketchy and incomplete.

M. Williams umf C. Hall, Returns to scale in the U.S. inrercity bus industry 575

intercity bus operations it does not seem unreasonable to assume that intercity operators attempt to minimize cost of producing the required output, subject to a production function and the exogenously determined prices they pzy for the inputs.

The cost of producing output depends on input prices. This relationship can be brought together by specifying the cost function as (l),

(1)

where C is total cost, Q is output, and P,, PK and PF are prices for labor,

capital and fuel respectively. Output is measured as revenue passenger miles,

since this measure takes into account the fares paid by passengers and tht. 1eng;h of trip.3

Given the assumption of cost minimization, a neoclassical cost function

can be used to estimate the characteristics of the production technology by invoking the Shephard (1953), McFadden (1979) duality theorems.4 The translog cost function specification is a consequence of the minimization process. This apmoach offers distinct advantages over the other approaches in that an explicit theoretical model serves as a basis for specification. It allows a high degree of generality, since the Allen partial elasticities of substitution, as well as homotheticity, can be allowed to vary in this general format.

The general specification of the translog cost function can be expressed as in eq. (2). We assume rhat it is cn exact representation of the minimum cost of producing output with given input prices.’ That IS:

%Yass I intercity carriers also proouce freight and package services, but data for these measures are inconqlete. Our sketchy da&a suggest that the volumes of these services are so small that they can be disregarded without affecting the results.

‘For a theoretical development of duality theory and the dual correspondence between cost and production see Shephard (1953); for a discussion on choice of specification see Spady and Friedlaender (1978).

‘There are precedents in the literature for this approach in tile empirical implementation of the translog functional iorm. See as examples Berndt and Christensen (197%, 1973b). Rerndt and Wood (1975). and Christensen and Greene (1976:. An altsrnative approach would be to use the translog as a second-order approximation a! a point to an arbitrary cost function. The distinction between the approximate and the exact functional form is discussed extensively by ‘Jenny and Fuss (1977). They point out thnt the use of the approximare form permits a separable technology to be represented as a flexible second-order approximation, whereas if the exact functional form is used, the set of constraints which must be imposed for a PeparabIc technology tr. exist involve additional restrictions which do not permit the fun&on ta be viewed as a flexible second-order approximation. Kowzver, as they indicate, the disadvantnye of usin.? the approximate functional form is that test results regarding separability can orJy be interpreted as holding at a point of approximation and not globally. [Simnlnns rtrd W&rts (1979) make a similar point in the context of the utility functicq.) Furthermore. Denny an4 F-~ss (1977) note that for this reason the loss in using an approximation rather than an exact Iorm may not be trivial.

576 M. Williams und C. Hall, Returns to scale in the U.S. infercity bus industry

lnC=a,+~~lnQ+~~a(lnQ)2+CBilnPi i

+ l/2 Cx Sij In Pi In fj + 1 yiu lnPi In Q. @) ij i

where c is total cost, Q is output, and Pi is the price of the ith input. A minimum requirement for the cost function (2) to be well-behaved is

:hat it be positive, and that it be homogeneous of degree one in prices; that is, for a fixed level of output, total cost must increase proportionately when factor prices increase proportionately. This implies the following restrictions on the parameters;

i= I i=l _

3 3 3 3

C 'ij=C dij=C C fiijZoa i=l j= 1 i= 1 jz 1

A convenient property for the cost function is that the derived &mand equations for Factor inputs can be easily computed by partialiy differentiating

(3)

(4)

(2) with respect to th, logarithms of the factor prices. The result is known as Shephard’s lemma, and

logarithmic form for the translog cost function as (S), can be written in

where Oi is the cost share of the ith factor input, and X;C is the cost minimizing input level. Note that the cost shares sum tr: unity.

The Allen (1938) partial elasticities of substitution in terms of the parameters of the translog cost function are expressed as

Oij I= (Sij + ~iOj)/Oi~j~ i+j, (6)

bii = [;i,j + Ui(Ui - 1 )]/C’iz, (7)

where Oi is factor i’s ccst share. The partial elasticities of substitution are invariant with respect to the

ordering of’ the factor inputs. Therefore Cii=Dji. The speckation of t.he translog knction (2) also allows us to express the

extent of scale ecanomies as dependent on the elasticity of cost with respect to output. In othr;r words, technological economies of scale arise if a car-ier

M. Williums and C. Hull, Returns to scde in the U.S. intercity bus industry 577

doubles its output (revenue passe-ger miles) and cost is less than doubkd. Technological disecanomies of scale arise if a carrier doubles its output and cost is more than doubled. We follow Christensen and Green (1976) and define scale economies as one minus the elasticity of cost with respect to output, as in eq. (8).

s = - 1 !T- (iInQ

1 E Q -= -aQ'c' (8)

The expression will clearly be positive for increasing returns and negative for decreasing returns. For the translog function (2),

SInC --==Q+2aQ,InQ+ i YiQInPi. C7lnQ i= 1

Scale economies (S) will thus depend on the value: of Q and the values of the prices.

3. Data

The data used in examining the characteristics of the US. ktercity bus industry were developed from the I.C.C. Bureall of Accounts for a cross- section of class I carriers in ten regions over the period 1971 to 1975. Thus our sample has 50 observations. Measures of costs, revenue- and other detailed operations were obtained from the F’inunciul and Opclretiq Sruristics

Chss 1 MO~OFS Carriers o$ Passengers Statement no. 750. The statistics pertain to class I carriers at the regional level. All variables

described relate to the average carrier in the regionm6 Three inputs are used: labor, owned equipment and fuel. Expenditures on each of these variables arc obtained from the I.C.C. reports -.-. Transport Statistics in the United States. Purr 2: Motor Carriers. Prices for these inputs are derived as follows: The price of labor is average annual compensation for hourly paid workers divided by the average number of man-hours. The price of capital services was constructed in terms of a Jorgensonian rental price of capital based :)n the equation

P,=P,(r+Sh, (10)

hWe have followed the examples of Griliches (1967). and ZarembKa (1970) for 0111 study of

the characteristics of the intercity bus industry, by examining the performance of I& average carrier in the region and assuming that th’: correctly depicts the behavior of the ir,dustry as a whole.

578 M. Wll1ium.s and C. Hall, Peturns to scale in the U.S. intercity bus industry

where P, is the amaunt spent on new buses, r is an intereet rate and d a rate of depreciation on buses.’ Fuel consumption was obtained from Table 1-A of I.C.C. Financial Shtement 750. However, because the method of reporting changed in 1974, the quantities of fuel consumed were interpolated from 1974 to the previous years based on the total number of intercity bus miles driven for the given year. * Sverage fuel prices are calculated on a regional basis by dividing the fuel expenditures by the quantity of fuel consumed. The output variable is revenue passenger miles. This is a preferable measure of output over vehicle miles since passenger miles account for passenger trips, bus miles and revenue.’

4. Estimation and empirical results

The translog cost function given in (2) can be respecified for a three-input, one-output technology with 15 parameters to be estimated. The logarithmic differentiation of thz total cost function with respect to factor prices, yields the relevant cost sharti for each factor as a function of input prices and outputs in logs. This yields three cost-share equations.

The approach used to estimate the unknown parameters of the translog cost function is to estimate two cost-share equations jointly with the total ccst function. An error term is included in each of these equations because of random errors in cost minimizing behavior. Since the cost shares sum to unity this implies that their associated errors cannot be mutually independent. This restriction implies that their covariance structure is singular and one equation must be dropped for joint estimation. Thus the cost share relating to ,.apital (K) was deleted. Moreover, the error structure is assumed constant across carriers. lo Following convention [Christensen and Greene (1976)3, a non-linear iterative Zeliner (1962) estimation procedure was used to estimate the parameters of (2) and to test the parameter restrictions. This procedure assures that the estimates will be

‘The rate of depreciation (6) us,d in this study is 0.034. This estimate is established by Jorgenson and Stephenson (1967). We assume an interest rate (r) of 6 percent on investment in new vehicles.

‘Quantity or fuel consumed in

Quantity consumed,,,, x total vehicle miles, yr, = - --- --

Tc;al vehicle miles,,,, ----.--- .

‘in thia study we exclude all information dealing with the Greyhound Company. Data 0’1 Greyhound service arc not applicable since the company is not required to provide information on a regicnal basis.

“We have assumd that the disturbances in the model are classically well-behaved. When we des.! with a pooled sample it is often necessary to account iot heteroskedasticity and autoregression in the disturbances. Unfortunately, in our model the appropriate tests for either autocorrelation or heteroskedadticity sre not very powerful in distinguishing these sources of non-sphericity in the error structure. See Kopp and Smith (1980) who clearly point out that there is no simple mechanism available for detecting amendments in the error structure in these models.

M. Williams and C. Hall, Returns to scale in rhe U.S. inter&y bus industry 579

invariant to which equation is deleted [Barten (1969)J. Kmenta and Gilbert (1968) have shown that iteration of the Zellner procedure until convergence yields maximum likelihood estimates.

Table 1 lntercity bus industr.,: Cost function parameter estimates (numbers in

paren;Ceses are student r-statistics). -

Models

Parameters 1 2 3

a0

a0

%Q

S‘

BK

s LF

8 LK

s FK

SLL

6 FF

sKK

YQP

YQL

YQK

Log tit likelihood function

IS.890 (2.84)

- 2.206 (-2.12)

0.1171 (2.43)

0.1998 (0.88)

a.2098 (2.02)

0.5904 (2.27)

- 0.0449 (-4.05)

- 0.0767 (2.87)

0.0094 (0.80)

0.1216 (4.35)

0.0543 (6.70)

0.0673 (2.Z)

-O.OOlC ( - 0.090)

0.0347 (1.45)

c.0337 1.27)

-51.921 - 53.05 - 7wti

14.29’1 I (2.61)

- I.865 ( - 1.85)

0.0989 (2.15)

0.5219 (11.73)

0.2020 (5.93)

0.2701 (5.39)

- 0.0452 ( - 5.02)

- 0.0456 ( - 2.84)

0.0090 11.52)

0.0908 (0.50)

0.0362 (4.70)

0.0366 (2.08)

2.513 (5.85)

0.2820 (7.15)

0.5908 (42.97)

0.0966 (14.12)

0.3;25 (24.80)

“Key: Q ;; revenue passenger miles, L is labor, F is fuel, and K is capital.

Table 1 presents the joint estimates af the translog cost function and factor shares equations. The numbers in parentheses beneath the estimated ccefficients are associated t-values. The log likelihood function is also

580 M. Williabnv and C. Hall, Returns to scale in the U.S. intercity bus industq

reported. The inputs labor, fuel and capital are denoted by the paran !eter subscripts L, P’ and # respectively.

Model I presents estimates of the cost function corresponding to a non- homothetic production technology with homogeneity in factor prices and symmetry restrictions of cross-price terms imposed. In model 2 we assume

Table 2

Test statistics for homotheticity and unitary elasticity of substitution restrictions on model 1.

__ --. -~-

Hypothesis - 2( In & - In Lu )” No. of restrictions x2 (0.01)

Homot heticity 2.26 2 9.21

Unitary elasticity and homogeneity (Cobb-Douglas) 34.06 4 16.81

‘InL, IS the log ol the calculated maximum for the likelihood function without res!ristions and InL, is the log of the calculated maximum likelihood function with restrictions imposed. For each hypothesis tested, homogeneity in tactor prices and symmetry in cross-price terms are also imposed.

homotheticily in output. That is, the trarrslog cost function has been restrict4 so that the coefficients ;‘iQ=O. M~LA 3 presents the cost function estimates corresponding to a Cobb-Douglas technology. TO obtain these, we

asriunle that the homogeneity restriction holds, and that we have unitary elasticities OF substitution. The later assumption requires that the parameters (irJ = 0. In model 1. out of the six Sij parameter estimates, five are statistically different from zero at the 95 percent level of confidence. These significant interaction terms indicate that reductiou to a Cobb-Douglas function specificazion would be inappropriate. We proceed further to test the validity of the restrictions imposed on models 2 and 3. We employ test statistics based on the likelihood ratio, defined to be the ratio of the maximum value of the restricted likelihood functions (&) to the maximum value of the unrestricted likelihood function (L,). Under the null hypothesis, -210g of the liikelihcod ratio test statistic is asymptotically distributed as chi-squared (~‘i with the number of degrees of freedom equal to the number of rcslriclions 6o be tested.”

The set of restrictions on the parameters of the translog cost function implied by homotheticity cannot be rejected at the 1 percent significance level. However+ we are able to reject the hypothesis that tte cost function exhibits properties consist,nt with Cobb-Douglas technology.

Some further interpretation of the parameter estimates are provided by analyzing patterns of factor substitution in intercity bu, service. in order to

“Kendall and Smart (1967, p. 230).

M. Williams and C. Hall, Returns to scale in the U.S. intercity bus industry 581

provide some evidence we computed estimates of the Allen partial elasticities of substitution among factors at the sample means of the cost shares for models 1 and 2 respectively.12

Table 3 Estimdtcs of the Allen-Partial elasticities of

substitution implied by the traalslog cost functions.’ _--

Non-homotMic Homot hetic Easticities model model

@LF 0.2113 0.2060 (0.1950) (0.1581)

OLK 0.5848 0 7532 (0.1456) (0.0866)

OliF 1.165 1.158 (0.3888) (0.2592)

‘The number in parentheses below the estimated elasticities are asymptotic standard errors. The computiq formula of SE[aij) = SE(Bi,)/BitYj; where S, is th* relevant parameter estimate and B,O, are the resmtive mean cc& shares.

The elasticities are given in table 3 for each model. The numbers in parentheses below these estimates are asymptotic standard errors. A negative

elasticity of substitution between factors i and j implies that the factors are complements, while a positive elasticity of substitution denotes that the factors are substitutes. The estimates seem reasonable and are not in conflict with what we would expect. All inputs are substitutes for each other. For example, capital and labor are substitutes. If the price of capital increased relative to the price of labor and operators use one bus csn a given route which formerly required tw o buses to pro*;ide the same service, they are substituting labor for capital through more extensive use of the veniclc. Capital and fuel are substitutes. If the price of fuel increases (ceteris parihs) operators may reschedule service, so that buses are required to stand idle for longer periods as fuel is conserved [or use on runs with higher loads. A smell elasticity of substitution exists between labor and fuel. Holvever, the standard errors of these input pairs are larpe, signifying that they are not significant.

Estimates of scale economies are also derived from the translog cost function. Recall t”at we have already rejected the non-hotnothetic and

12A derivation of the Allen partial elasticities of substirution in term!: of the cross deriqakives of the cost function is

cq, lx- r’c l?T bij = _.__I. where (‘i = , Cj=_ ._._, Cij= . .._..

c,.c, lPi iJP, dP, ’ r?P,

582 M. Wiliiums and C. Hd1. Returns to scale in the U.S. intercity bus industry

Cobb-Douglas functional forms. In other words, since we accept that the production structure is homothetic, therefore our discussion on the characteristics of returns to scale will be limited to this specification. Eq. (9)

now reduces to

iJlnC ---=aQ+?uQQInQ. 2 In Q

(11)

Now. the average cost curve will be U-shaped if the value of ii In C/a In Q changes from less than unity to greater than unity as output increases. It is clear from (11) that this will be the case if aQ < 1 and QQQ > 0 (where we are operating in the relevant range with cSln C/8 In Q ~0). Since our results yield Zy= - 1.865 and aQQ = 0.0989, it is clear that a U-shaped ae erage cost curve is implied.

Recall that our scsle measure, S, is defined as unity minus the elp ,ticity of total cost with respwt to output; S>O implies positive scale economies and S < 0 implies diseconomies of scale. F’or the homothetic specification, the cost elasticity varies at different output levels. The meln estimates of scale economies are given in table 4. The evidence implies that there are increasing

Table 4

Estimated scale economies (standard errors in parentheses).

-

Model S

N3n-homothctir 0.701 (0.161)

Homothetic 0.695 (0.152)

- ---

returns to scale.13 Thus, as carriers expand service, the per unit cost of production decreases. The present regulatory restrictions limit entry and operating rights on routes that carriers can travel. Our findings indicate that firms currently have the incentive to obtain new operating rights so as to expand service and realize the observed economies of scale.

5. Couclusions

This paper has taken ;nto account the a priori restrictions available from neoclassical cost theory in evaluating the relationship between cost and the Icvel of output and input j:,rices for intercity bus service. A general translog

“Our findmgs runforce those 14 Fra*vrl (1978).

M. Williams and C. Hall, Returns to scale in the U.S. intercity bus industry 583

cost function is used which allo-Jvs tests of the degree of retwns to sea!e,

homotheticity and non-constant elasticity of substitu5on between input pairs.

We find that intercity bus service can be modeled by a homothetic production function. The acceptance of homotheticity permits the investigation of the degree of r&urns to scale. There are potential economies of scale in the provision Ji &srcity bus service. This finding indicates that under the present reguWry policy which limits routes and operating rights, that firms currently have 1 clear incentive to seek to obtain new operating routes md expand serGc2 Our kindings also indicate that all inputs are substitutes for each cLner. F?r example, operators can substitute labor for capital through more extensive use of vehicles.

Our results also suggest that the restrictive Cobb-Douglas functional form used in the past is inappropriate. The more flexible function form used here is able to shed more lighr on the production technology of the intercity bus industry.

In closing we may note that to the best of our knowledge there has been no study which attempts to simultaneously estimate returns to scale and the elasticity of substitution for carriers in the intercity bus industry. Our findings are not conc!usive. Further work needs to be undertaken to estimate cost functions for broader samples. As better data bases become available the additional research efforts should provide more comparative evidence on the production technologies of the intercity bus industry.

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