Estimating the inefficiency in the Norwegian bus industryfrom stochastic cost frontier models

FINN JØRGENSEN, PÅL ANDREAS PEDERSEN & ROLF VOLDEN1

Bodø Graduate School of Business, Norway

Accepted 17 February 1997

Key words: cost frontier models, Norwegian bus industry, ownership structure

Abstract. A stochastic cost frontier function based on data from 170 of the 175 Norwegiansubsidized bus companies is estimated under two alternative presumptions regarding the distri-bution of the inefficency among the bus operators. When the inefficiency is assumed to behalf-normally distributed, the average inefficiency in the industry is estimated to be 13.7 per cent.This calculated value is nearly halved (7.2 per cent) when the exponential distribution is applied,while the ranking of the companies according to inefficiency is unchanged. By regressing theestimated inefficiency values for each company on some exogenous variables describing itsownership structure and the subsidy policy which it faces, it is seen that inefficiency of thecompanies which negotiate with the public authorities over the subsidy amounts is slightlyhigher than the inefficiency of the companies which face a subsidy policy based on costnorms. Our analysis gives, however, no significant differences in the efficiency between privatelyowned bus companies and publicly owned bus operators, and shows only minor economies ofscale.

1. Introduction

The Norwegian bus operators have experienced significant changes in theirworking conditions since the early 1980s. Before 1981, each bus company heldlicences and the state was responsible for subsidies through balancing oftheir accounts. The state controlled the bus companies’ fares and to a largeextent the most relevant aspects of their supply. Hence, the bus operatorshad relatively few incentives to run their business efficiently. From 1981,however, fixed net subsidy contracts between bus operators and county councilswere introduced; the size of the subsidy to an operator for the coming yearwas partly based on standardised cost norms figured by the public authori-ties and partly based on negotiations between the operator and the countycouncil (Jørgensen, Pedersen & Solvoll 1995). During the 1980’s, standard-ised cost norms became increasingly more important under the subsidydistribution process. This led to the situation where the possibilities for the buscompanies to influence their subsidy levels worsened. In 1991 a process started

Transportation 24: 421–433, 1997 1997 Kluwer Academic Publishers. Printed in the Netherlands.

to change the regulatory regime by removal of licences and with openingfor competitive tendering on certain routes: in 1994 the first bus routes weretendered.2

When the issue of competitive tendering in the Norwegian bus industrybecame an important issue, the opponents referred to the industry’s signifi-cant productivity improvement during the 1980s; this development indicatedthat the potential productivity gains from competitive tendering were likelyto be small. Moreover, other political means to increase the industry’s pro-ductivity were possible such as influencing its ownership structure and furtherchanges of the subsidy system. The protagonists of tendering on the other side,asserted, of course, that the bus industry was still far from as effective as itcould be and that increased competition between the bus operators was themost effective way to improve their productivity.

The objective of this paper is to discuss the above parties’ points of viewby estimating, firstly, a stochastic cost frontier model based on 1991 datafrom 170 of the 175 Norwegian bus operators. This model enables us toestimate the average inefficiency of the industry as well as the inefficiencyfor each bus operator. Secondly, the amount by which a bus company lies mar-ginally above its cost frontier (productive inefficiency) is tested against itsownership structure and against the subsidy policy it faces. Even though theNorwegian bus industry has been the subject of previous research analyses (seefor example Jørgensen, Pedersen & Solvoll 1995), its potential productivitygains and the distribution of inefficiency between its operators have not beeninferred on the basis of either a deterministic or a stochastic frontier approach.It is worth noting that the analysis is carried out using data for the year 1991;i.e. only before competitive tendering became the order of the day. If the modelresults show significant inefficiency which cannot be explained by the own-ership structure of the bus industry or by the subsidy policy, the protagonistsof competitive tendering were right. If not, the opponents were right; theinefficiency is either small or it can be improved by other political meansthan introducing tendering.

The further arrangement of this paper is as follows: In section 2, we brieflydescribe the econometric models and outline some a priori assumptions aboutthe signs and the magnitudes of the models’ parameters. Further, in section3 we describe our data sources and present the estimated results. Finally, insection 4, we summarize the main conclusions and emphasize the most relevantweaknesses of the paper.

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2. Models’ specifications

The stochastic frontier approach

Following the seminal works of Aigner, Lovell and Schmidt (1977) andMeusen and Broeck (1977) on estimation of efficiency in an industry byusing a stochastic frontier approach, we here specify the cost structure ofthe Norwegian bus industry by a parametric function of the type:

Yi = F(

φ; Xi) + εi (1)

where i indexes companies, Y and X denote some transformation of costs pervehicle-km and a vector of (transformed) observable exogenous variables influ-encing costs, respectively, φ is a vector of parameters and εi is an error termfor observation i.

The error term is composed of two independent components such thatεi = Vi + Ui. The first component, Vi, is a symmetrically distributed variablewhich captures the effects of measurement error, other statistical noise andrandom shocks outside the companies’ control. Traditionally, the V’s aresupposed to be independently drawn from the same normal distribution withzero mean and the same standard devation, σv. The second component, Ui,independent of Vi, is a non-negative, one sided stochastic variable measuringthe inefficiency of company i. Hence, the composite error term, εi is posi-tively skewed and has a non-zero mean. The value of Ui can be regarded asthe company’s X-inefficiency since it measures the difference between actualcost and minimum attainable cost (see Hay & Morris 1991).

The above model formulation is a stochastic frontier approach becausethe possible costs of company i are bounded below by the stochastic frontiercosts, (F (φ; Xi) + Vi) and the inefficiency of the company, Ui, is measuredagainst its unique frontier. As outlined by Førsund, Lovell & Schmidt (1980)and adapted to our problem, the stochastic frontier (F (φ; Xi) + Vi) can be inter-preted as an “absolute” cost frontier, given the factors exogenous to the buscompanies such as available technology of the buses, the quality of the roadsand the current input prices. The “absolute” cost frontier is never actuallyreached by any of the bus companies; none of the companies are 100% effec-tive. The estimated F(φ; Xi) function denotes expected minimum costs whenthe values of the X’s are known. The characteristics of the composite error term(εi) imply that the majority of observations are above the F(φ; Xi)-function.If, for example, company i’s observed costs (Yi) are on or below the F(φ; Xi)-function, this must be interpreted as a very lucky event, measurement erroror another form of statistical noise such that the random error, Vi, is negativeand –Vi ≥ Ui.

3

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To be able to estimate the F(φ; Xi)-function and the inefficiency for thebus industry, some explicit assumptions regarding the distribution of theone-sided inefficiency term, Ui, have to be made. We find it difficult to justifyany specific choice of probability distribution of inefficiency in the busindustry. Since the early works of Aigner, Lovell and Schmidt (1977) andMeusen and Broeck (1977), it has often been assumed in empirical studies thatthe U’s are independently drawn from either a half-normal or an exponentialdistribution. In Jondrow, Lovell, Materov and Schmidt (1982) a proceduremaking it possible to identify the inefficiency of each of the companies (Ui)is found when one of the above mentioned distributions are presumed. Lateron, statistical data programs are developed for these two distributions in par-ticular (Green 1995). Since we find it difficult to form clear presumptionsregarding whether the inefficiency between the Norwegian bus companies ishalf-normal or exponential distributed, we will run the empirical analysis usingboth of these two distributions. The main difference between them is, broadlyspeaking, that the former one describes fewer “almost” efficient and fewer“extremely” inefficient companies than the latter one.

Choice of functional forms

As often found in empirical works on cost structure of the bus industry, wesuppose that F(φ; Xi) can be specified by the following simple translog costfunction or what might be called a modified Cobb-Douglas function:

ln Y = α0 + α1 ln X + α2(ln X)2 + β1 ln Z1 + β2 ln Z2 + d1D1 + d2D2 (2)

where Y is total costs per vehicle-kms in the bus company, X is the numberof vehicle-kms produced by the company, Z1 is the average bus size of thecompany, measured by the sum of the seating capacity and standing places,Z2 is the number of passengers boarding the buses of the company per vehicle-km. D1 describes whether the bus operator is also engaged in sea transport(D1 = 1) or not (D1 = 0) and, finally, D2 describes whether the company operatesin a coastal area (D2 = 0) or not (D2 = 1). When the V’s and the U’s havethe statistical properties which are outlined above and the inefficiency differsamongst the bus operators (σu ≠ 0), the ordinary least squares procedure (OLS)will produce unbiased and consistent estimates of all the coefficients but theconstant α0 (Liu 1995). Thus, even though this model differs slightly fromthe model presented in Jørgensen, Pedersen and Solvoll (1995) where an OLS-estimation procedure was chosen on the same data, it is reasonable to assumethat the signs of the parameters here are similar to their estimations; that isα1 < 0, α2 > 0, 0 < β1, β2 < 1, d1 > 0, d2 < 0. Although the simple translogfunction in (2) does not place any restrictions on whether economies of scale

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are present or not, the presumptions of the signs of the parameters describea long-range average cost curve which is U-shaped.

Given the inefficiency estimates for each of the companies, Ui, the nextstep in our analysis is to test whether the values of the U’s can be explainedby different subsidy procedures and different ownership structure. These cir-cumstances are, as previously emphasized, relevant Norwegian transport policymeans. The test is made by regressing the inefficiency estimates on thefollowing dummies; i.e:

U = h0 + h1H1 + h2H2 + h3H3 (3)

where U is the estimated inefficiency of the bus company, H1 describes whetherthe bus company is publicly4 owned and faces a subsidy policy based oncost norm (H1 = 1) or not (H1 = 0), H2 describes whether it is privately ownedand has the ability to negotiate with the county council over the size of thesubsidy (H2 = 1) or not (H2 = 0), H3 describes whether the company isprivately owned and faces a subsidy policy based on cost norms (H3 = 1) ornot (H3 = 0). Consequentely, if H1 = H2 = H3 = 0 the company is publicly ownedand negotiates with the county council. Finally, the h’s are parameters whichare to be estimated.

When the subsidy transfers to different bus operators are based on stan-dardised cost norms, it is reasonable to expect that the operators have lessability to influence the size of the subsidy than when the transfers are basedon negotiations alone. This results from the fact that the standardised costnorms give the county councils more knowledge of the bus industry cost struc-ture. Thus, it is reasonable to expect that, irrespective of ownership structure,the inefficiency of a bus company facing cost norms is less than for a nego-tiating company; that is h1 < 0 for publicly-owned companies and (h3 – h2)< 0 for private companies. From the general theory of public enterprises, seefor instance Rees (1984) and Hay and Morris (1991), it is reasonable to assumethat privately owned bus companies have stronger profit incentives than thepublic ones. Even though empirical studies which compare the efficiency ofprivately owned bus companies to public ones show some conflicting results(Jørgensen, Pedersen & Solvoll 1995), we here suppose, irrespective of subsidypolicy, that the inefficiency is higher for publicly owned companies than forprivate companies such that h2 < 0 for negotiating companies and (h3 – h1)< 0 for companies facing cost norms. Obviously, a private company facing costnorms is assumed to have less X-inefficiency than a public company whichis negotiating; that is h3 < 0. It is also worth noting that by the model spec-ification in (2), the value of 100Ui denotes company i’s percentage distancefrom its unique cost frontier; if, for example, Ui = 0.1, this may be inter-preted such that company i’s costs are 10% above its minimum possible cost.

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In other words, when assuming that all companies are facing the same factorprices, bus company i uses 10% more resources than necessary.

3. Data, empirical methods and results

Data

The data are derived from official reports from the bus companies to the countycouncils. All of the 175 subsidised Norwegian bus companies providing localbus services in 1991 are contained in the data base. However, five compa-nies were discarded; they all appeared to be outliers as they had studentizedresiduals with absolute values greater than 3. Hence, the estimations below arebased on 170 observations where the size of the companies, measured in yearlyvehicle-kms, varies from 4500 kms to 24.3 mill kms. The average size ofthe bus companies in the data set is 1.6 mill kms.

Generally, “extreme observations” can be explained by inaccurate reportingbehaviour from the companies, imprecise functional presumptions or omittedrelevant exogenous variables. A thorough investigation of the outliers, indicatethat three of them must have reported inaccurate data. The remaining twocompanies operate under extreme working conditions which are not sufficientlycaptured by our model description. One of them is the main bus operator inOslo, which runs its business under extreme urban conditions compared withother Norwegian bus companies. The other one is a very small company havingextremely low costs because a number of routes are operated by hired taxicabs.

Empirical methods

All the estimations were made by the statistical program “LIMDEP 6.0” (seeGreene 1995) which, amongst other things, is based on the estimationprocedures described by Jondrow, Lovell, Materov and Schmidt (1982). Fromthe theoretical model it follows that the residual ε is positively skewed inthe presence of inefficiency. Therefore, before estimating the stochastic frontiercost function, we first checked that the OLS estimates of the ε’s werepositively skewed. Secondly, we calculated the unbiased and consistent MLestimates for the parameters in the model described in equation (1) and (2)under the two presumptions regarding the distribution of the inefficiency term.This gave us two different estimates of the average inefficiency as well astwo estimates of the inefficiency of each company in the bus industry. Finally,based on the two calculated distributions of the inefficiency within theNorwegian bus industry, the respective versions of equation (3) were estimated.

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Results

Table 1 presents the estimates of the stochastic cost frontier models describedby equation (1) and (2) above. The first column shows the estimated para-meters resulting when the OLS procedure is used, the second column presentsthe ML-estimates when the inefficiency term is supposed to be half-normallydistributed (ML1), and, finally, the third column gives the ML-estimateswhen it is presumed that the U’s are exponentially distributed (ML2). Plotsof the residuals give no clear indication which of the two assumptions, con-cerning the distribution of inefficiency among the bus companies, are thebest.

As anticipated, from the fact that all the methods give unbiased slopeestimates of the parameters in equation (2) (Liu 1995), we find that the actualempirical slope values (α1, α2, β1, β2, d1, d2) are nearly equal for all methods.Since the mean of the U’s is positive, however, the OLS estimate for theconstant α0 will be positively biased. Empirically, this is verified by thehigher estimated value of α0 when OLS is used compared with the estimatesproduced by the two ML-regressions, which give unbiased (and consistent)estimates of α0.

Furthermore, it is seen that the estimates of α1 and α2 are significantlydifferent from 0. Since α1 is negative and α2 is positive, it follows that theaverage cost function is U-shaped. This means that economies of scale arepresent up to a certain production level, then diseconomies of scale occur. Fromthe estimated values of α1 and α2 it follows, however, that the average costcurve is rather flat indicating small productivity gains by changing the sizedistribution of the bus companies.5

Moreover, it is seen from the cost frontier function presented in Table 1 thatthe parameter β1 is evidently higher than 0 and less than 1, and the pointestimate tells us that the costs increase by about 0.22 per cent when the bussize increases by 1 per cent. Also the estimate of β2 is significantly above 0and less than 1, and the estimated value indicates that the costs increase byabout 0.11 per cent when the number of passengers per vehicle-km increasesby 1 per cent. The estimated value of parameter d1 is 0.11 and is signifi-cantly above 0; the companies which are engaged in both sea transport and bustransport produce their bus services at 11 per cent higher costs than thosecompanies which produce bus services only. This indicates that no economiesof scope exist between bus transport and sea transport; at least that sea trans-port implies no cost advantages in bus operations. Finally, as supposed above,d2 is significantly negative. This tells us that bus companies operating in coastalregions seem to have “harder” exogenous production conditions resulting in7%–8% per cent higher costs than companies operating in interior parts ofNorway. It is worth noting that the bus companies which are engaged in both

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bus transport and sea transport all operate in coastal areas while the com-panies which are engaged in bus transport only, operate in all parts of thecountry. Hence, the D2 variable captures the main differences in exogenousworking conditions between these two groups of bus companies.

As far as the average inefficiency within the Norwegian bus industry is con-cerned, EU, it is seen that ML1 and ML2 produce different estimates. Whenthe one-sided error term is supposed to be half-normally distributed, the pointestimate (0.137) indicates that industry could reduce costs by 13.7 per centif all companies companies moved to the cost frontier. The correspondingestimate when the inefficiency term is exponentially distributed is 0.072,leading us to the conclusion that the potential cost savings could only be 7.2per cent of the actual costs in 1991. This means that ML1 produces an estimateon potential relative cost savings which is nearly twice as high as the estimatein ML2. Unlike Liu (1995), who estimates the efficiency of British ports,our results indicate that the actual choice of a particular one-sided distribu-tion influences the average inefficiency estimate for an industry.

From the fact that σu/σv = 1.27 > 1 in ML1, it follows, in the case wherethe estimation is based on the assumption that the U’s are half-normal dis-tributed, that the stochastic inefficiency term (U) is more important than thesymmetric distributed error term (V) in determining the performance of ofNorwegian bus companies. However, in ML2, σu/σv = 0.469 < 1, indicatingthat the conclusion above is reversed when the U’s are presumed to beexponentially distributed. The economic interpretation of this is that if theassumption of a half-normal distribution of the one-sided error term holds, itis more important, in order to explain a company’s actual costs, to find out

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Table 1. Stochastic cost frontier function of Norwegian bus companiesa (standard errors in paren-theses).

OLS ML1 ML2

α0 –4.5397 (0.8193) –4.3893 (0.8500) –4.4674 (0.8462)α1 –0.4087 (0.1347) –0.4059 (0.1393) –0.4080 (0.1381)α2 –0.0149 (0.0051) –0.0151 (0.0054) –0.0149 (0.0054)β1 –0.2269 (0.0599) –0.2141 (0.0547) –0.2269 (0.0555)β2 –0.1105 (0.0232) –0.1108 (0.0234) –0.1105 (0.0236)d1 –0.1111 (0.0476) –0.1155 (0.0608) –0.1111 (0.0595)d2 –0.0828 (0.0285) –0.0721 (0.0285) –0.0876 (0.0289)EU –0.1373 –0.0723σv –0.1351 –0.1542σu/σv –1.2741 –0.4685σε –0.1740 –0.2188 –0.1703R2 –0.3685

a All estimated parameters are significantly different from 0 at 10 per cent level or better.

the relative efficiency of the operator than whether he has been lucky or not.However, in the case where the U’s really are drawn independently from anexponential distribution, the value of V, describing the company’s “luck”,would be more important to reveal than his inefficiency value if one intendsto explain the operator’s costs.

Moreover, using the procedure of Jondrow, Lovell, Materov and Schmidt(1982), we have identified and calculated the inefficiency of each of the com-panies, given the two different assumptions of the distribution of the U’s.The Spearman’s correlation coefficient, measuring the correlation of therankings of the bus companies based on the estimated inefficiencies in ML1and ML2, is equal to 0.9974. This tells us that the estimated inefficiencyranking of the bus companies is nearly independent of whether the U’s aresupposed to be exponentially or half-normally distributed.

The OLS-estimated versions of equation (3), where we intend to explain thetwo calculated values of inefficiency by variations in ownership structureand differences in subsidy policy, are presented in Table 2. Since the lefthand side variable (U) is exponentially or half-normally distributed, we triedalternative estimation techniques, like “least absolute deviation regression”,“least trimmed squares regression” and “a ML-regression built on a Γ-distributed error term”. These techniques produced, however, only negligibleand insignificant differences in estimated values, indicating that the OLS resultsare quite robust. The first column in Table 2 shows the estimated parametersof equation (3) based on the calculated inefficiencies for each of the com-panies when ML1 is used (the half-normal distribution), and the second columnpresents the estimates when the calculated one-sided error term is derived fromML2 (the expontential distribution).

Table 2 reveals that the signs of the regression estimates (h1, h2, h3) andof the deduced estimates ((h3 – h2), (h3 – h1)) are equal in ML1 and ML2and in line with our prior assumptions. However, only the estimates of h1

and h3 are significantly negative at the 10% level or better. The multiple cor-

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Table 2. Explained inefficiency in the Norwegian bus industry (standard errors in parantheses).

u(ML1) u(ML2)

h0 –0.1584 (0.0135) –0.0836 (0.0071)h1 –0.0353 (0.0207) –0.0187 (0.0109)h2 –0.0035 (0.0162) –0.0031 (0.0085)h3 –0.0372 (0.0154) –0.0189 (0.0081)R2 –0.2590 –0.2414(h3 – h2) –0.034 (0.0294) –0.016 (0.0154)(h3 – h1) –0.002 (0.0321) –0.0002 (0.0169)

relation coefficient (R2) is about 0.25 for both regressions. The estimationsin Table 2 give rise to several economic interpretations:

Firstly, we notice from both regressions that the dummies describing thevariation in subsidy policies and ownership structures together only explainabout 25 per cent of the total variation in inefficiency amongst the Norwegianbus companies (R2 ≈ 0.25). According to our model, this indicates that about3/4 of the variation in inefficiency cannot be explained by differences inownership structure and subsidy policy. Our results, therefore, to some extend,contradict the opponents of competitive tendering in Norway; a rather smallproportion of X-inefficiency within the Norwegian bus industry can be removedby actual transport political means aiming to change its ownership structureand the subsidy system (see section 1).

From the point estimates of h0, it can be seen that negotiating publicly-owned companies (H1 = H2 = H3 = 0) in the regressions based on ML1 andML2 have an average inefficiency of 15.8 per cent and 8.4 per cent, respec-tively. As one could expect, from the prior hypotheses in section 2 whichtell us that the above mentioned companies have lowest profit incentives,both estimates of h0 are above the average inefficiency estimates (EU).

Furthermore, it is seen from Table 2 that h1 = –0.035 when the U’s arehalf-normally distributed and h1 = –0.019 when the U’s are exponentiallydistributed. Since h1 is significantly negative in both cases, this means thatindependent of inefficiency distribution, publicly owned companies will reduceinefficiency when the subsidy policy changes from negotiations to cost norms.The point estimates indicate, however, that the estimated magnitude of thechange in inefficiency is dependent on the distribution assumption. Table 2also shows that the deduced estimates of (h3 – h2) are about the same as theestimates of h1, both when the U’s are half-normally and exponentially dis-tributed. The interpretation of this is that the above mentioned changes inthe subsidy system will have nearly the same influence on X-inefficiency inprivately-owned bus companies as in publicly-owned bus companies. Theestimated values of (h3 – h2) are, however, only significant at a 20 per centlevel.

In both cases when the U’s are assumed to be half-normally and expo-nentially distributed , neither the values of h2 nor the values of (h3 – h1) aresignificantly different from 0 at a reasonable statistical level. This meansthat independent of subsidy system, the analyses give no significant supportto the hypothesis that privately-owned bus companies behave more efficientlythan publicly owned bus companies. Even though this result is in conflictwith common theory of public enterprises, it is not surprising in the light ofempirical studies which compare the efficiency of privately-owned bus com-panies to public ones; as mentioned previously these results show no clearpicture.

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Finally, as expected, we notice that the value of h3, measuring the influ-ence on inefficiency of moving from a negotiating publicly-owned companyto a private company facing cost norms, is significant in both cases at a 5per cent level or better. The point estimates indicate a reduction in inefficiencybased on ML1 and ML2 of 3.7 percentage points and 1.9 percentage points,respectively.

4. Conclusions

The main conclusions in our analysis are, firstly, that the average cost curveis slightly U-shaped. Thus, a policy which aims to change the size distribu-tion of the bus companies will have a minor influence on the efficiency ofthe Norwegian bus industry.

Secondly, given the size distribution of the Norwegian bus operators in 1991and relevant measures of their working conditions, the potential gain of costsis estimated to be about 7.2 per cent and 13.7 per cent if the inefficiencyterm is exponentially distributed and half-normally distributed, respectively.It is worth noting that the above results show that the average inefficiencyestimate is about twice as large when the inefficiency term is assumed to behalf-normal distributed as when it is assumed to be exponentially distexoge-nousributed. Consequently, since it is normally difficult to decide from a priorireasoning alone which of these two distributions are the best, one shouldshow some scepticism to inefficiency estimates for industries which rest uponone distribution only. As commented on previously, the inefficiency rankingof the bus companies is, however, invariant to the choice of distribution ofthe inefficiency term.

Thirdly, by regressing the inefficiency estimates for each bus company onvariables indicating its ownership structure and the subsidy system which itfaces, we find that inefficiency was not influenced by whether the bus companywas privately-owned or not; a policy which aims to privatise all the Norwegianbus companies will, therefore, have minimal influence on productivity. Theintroduction of standardised cost norms, seems, however, to improve efficiency.From Table 2 it follows that inefficiency is reduced by about 3.5 percentagepoints and 1.7 percentage points when the inefficiency term is half-normallyand exponentially distributed, respectively.

If one really believes that implementing competitive tendering means thatall bus operators were forced to move to the cost frontier, our analysis supportsthe protagonists of competitive tendering; regardless of what kind of assump-tion of the distribution of inefficiency among the bus companies, changingthe ownership structure of them or the subsidy system towards cost norms,would have far less positive influence on efficiency than competitive tendering.

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The critical question is, however, whether tendering would lead to a realiza-tion of the potential cost savings. It will, among other things, be conditionalon the number of bidders and the tendering procedure. It is only after tenderinghas been practiced for the Norwegian bus industry that it will be possible toidentify the precise efficiency improvements of such a policy.

It should be noticed, however, that the above estimated cost savings bymoving to the cost frontier are broadly in line with the efficiency improve-ments caused by practicing competitive tendering in other countries, see forexample Jansson and Wallin (1991) studying the effects of introducingcompetitive tendering in Sweden, and Mackie, Preston and Nash (1995), Whiteand Tough (1995) or White (1995) analysing the practice of competitivetendering in the UK.

Finally, it should be remembered that our results – in line with all empir-ical cost studies, have several weaknesses due to erroneous data reports,imprecise cost model description, omitted variables indicating the bus com-panies’ working conditions and inaccurate presumptions of the distributionof the inefficiency among the bus operators. Nevertheless, the analysis devel-oped here must be seen as the first attempt to construct stochastic cost frontiermodels for the Norwegian bus industry and to explain inefficiency by own-ership structure and subsidy system.

Notes

1. The authors thank the Editor and anonymous reviewers for helpful comments and sugges-tions and Gisle Solvoll at Nordland Research Institute for helpful data assistance.

2. Still in 1996 only a small proportion of the Norwegian bus routes are tendered, but thisproportion is likely to increase rapidly in the coming five years.

3. From (1) it follows that Yi ≥ (<) F(φ; Xi) if εi ≥ (<) 0 ⇔ –Vi ≤ (>) Ui. Since the V’s areidentically distributed as N(0, σv) and the U’s are positive, the majority of observations areabove the F(φ; Xi)-function.

4. A bus company is defined as publicly owned if 50 per cent or more of the shares are heldby the state, county councils or local communities.

5. From equation (2) it follows that: ElXY = α1 + 2α2 lnX. Using the estimates of α1 and α2 inTable 1, it is, for example, easily seen that the value of ElXY increases from –0.045 to 0.024when X increases from 200000 kms to 2000000 kms. Thus, the absolute value of ElXY is smallfor the most relevant values of X.

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