Int. J. Production Economics 109 (2007) 149–161

## Firm and time varying technical and allocative efﬁciency:

## An application to port cargo handling ﬁrms

### Ana Rodrı´guez-A´lvarez

a,### , Beatriz Tovar

b### , Lourdes Trujillo

b,1a_{University of Oviedo, Spain}

b_{Research Group Economics of Infrastructure and Transport, EIT, University of Las Palmas de Gran Canaria, Spain}

Received 30 December 2005; accepted 27 November 2006 Available online 9 January 2007

Abstract

In this paper we present an econometric model to calculate both, the technical and the allocative efﬁciency in cargo handling ﬁrms in the port of Las Palmas (Spain). To achieve these aims, we estimate a system of equations consisting of a translog input distance function and cost shares equations. Using this procedure we can check the hypothesis in which, given technology and prices, terminal port inputs are not optimally allocated in the sense that costs are not minimized. The main contribution of this paper is to apply an empirical model that allows the unbiased estimation of allocative inefﬁciency of input use in two ways: an error components approach and a parametric approach. We also avoid the Greene Problem and allow allocative inefﬁciency to be systematic. Both size and trafﬁc mix are ﬁrst shown to be sufﬁciently diverse as to allow for a reliable estimation of a representative ﬂexible (translog) function.

r2007 Elsevier B.V. All rights reserved.

Keywords: Distance function system; Morishima elasticities; Technical and allocative efﬁciency; Spanish ports terminals

1. Introduction

In the last decade, several models have been proposed to estimate time-varying technical efﬁ-ciency. These models could be grouped depending

on the approach chosen to model the inefﬁciency. On the one hand, there are those who model technical inefﬁciency through an error component (see, for example, Kumbhakar, 1990; Battese and Coelli, 1992, 1995; Heshmati and Kumbhakar, 1994;Heshmati et al., 1995orCuesta, 2000). These models involve the disadvantage of making parti-cular distributional assumptions for the one-sided error term associated with technical efﬁciency. On the other hand, there are those who model technical inefﬁciency through the intercept of the function (see, for example, Cornwell et al., 1990; Lee and Schmidt, 1993or Atkinson and Primont, 2002). In this way, these models avoid making particular distributional assumptions. In this paper we can obtain technical efﬁciency indices, which may vary www.elsevier.com/locate/ijpe

0925-5273/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.12.048

Corresponding author. Departamento de Economı´a, Uni-versidad de Oviedo, Campus del Cristo, 33071 Oviedo, Spain. Tel.: +34 985 10 48 84; fax: +34 985 10 48 71.

E-mail address:ana@uniovi.es (A. Rodrı´guez-A´lvarez). 1

A Previous version of this paper has been published like working paper in FUNCAS (Fundacio´n Espan˜ola de las Cajas de Ahorro) n

%

o 201/2005 and were presented at the Permanent Seminar on Efﬁciency and Productivity, Oviedo, April 2004 and at the North American Productivity Workshop, Toronto, Canada June 2004. This research was funded with grants from Gobierno Auto´nomo de Canarias, Project PI2003/188.

through time as well as across ﬁrms following this second approach.

With regard to allocative efﬁciency,Atkinson and Cornwell (1994) present two methods that permit the calculation of allocative inefﬁciency: the para-metric approach and the error component ap-proach. Fa¨re and Grosskopf (1990) and Atkinson and Primont (2002)demonstrate that replacing the usual cost frontier with an input distance function, the main drawbacks of the parametric approach can be overcome by obtaining ﬁrm and time allocative efﬁciency indices. With regard to the error compo-nent approach, the advantages of the distance function are developed inRodrı´guez-A´lvarez et al. (2004) which dealt with a hypothesis in which allocative efﬁciency is time-invariant and only varies across ﬁrms.

In this paper we extend the analysis to the case when the efﬁciency is ﬁrm and time-varying in both approaches. In adition, we can calculate ﬁrm and time-varying technical efﬁciency and, separately, a measure of technical change. To do this, we present a distance system that comprises an input distance function and the associated share cost equations. Finally, we also extend the analysis by calculating Morishima elasticities.

To illustrate our methodology we apply it to panel data using a sample of cargo handling ﬁrms in Spanish ports. The ﬁrst empirical studies that have attempted to measure port efﬁciency based on frontier models appear in the 1990s. There are two different techniques to carry out this type of study. The ﬁrst one, a non-parametric programming method, is called data envelopment analysis (DEA), (i.e. Roll and Hayuth, 1993; Martı´nez Budrı´a et al., 1999; Tongzon, 2001; Valentine and Gray, 2001; Bonilla et al., 2002; Martı´n, 2002; Pestana, 2003;Estache et al., 2004), and the second is through stochastic frontier analysis (SFA), (i.e. Liu, 1995;Ban˜os Pino et al., 1999; Coto Milla´n et al., 2000; Notteboom et al., 2000; Cullinane et al., 2002; Estache et al., 2002; Cullinane and Song, 2003;Dı´az, 2003; Tongzon and Heng, 2005). Both methods allow the derivation of estimates of relative efﬁciency levels for all the operators compared.2

Out of these, only four of them have investigated the efﬁciency of Port Terminals: Notteboom et al. (2000);Cullinane et al. (2002);Cullinane and Song (2003); Tongzon and Heng (2005) and they have several characteristics in common. Firstly, they

estimate a stochastic frontier production function; secondly, they model technical inefﬁciency through an error component thirdly; they only measure technical efﬁciency and ﬁnally, they consider that technical efﬁciency is time-invariant. However, Song et al. (2001) estimate the efﬁciency of the terminal operating companies based on both pooled data set variant) and panel data (time-invariant). Our paper is the ﬁrst one dealing simultaneously with ﬁrm and time varying technical and allocative port terminals efﬁciency, and it is also the ﬁrst one applying a distance function to port terminals.

The paper is organized as follows: Section 2 provides an overview of port terminals and their regulation in Spain. In Section 3 the model is presented. Section 4 concerns itself with the econo-metric model. The data are described and the results are presented in Section 5. The ﬁnal section contains brief concluding comments.

2. Port terminals and their regulation in Spain Although there are cases where several ﬁrms share a port terminal, in our case each cargo handling ﬁrm exclusively operates its own terminal in a consessional basis. The terminals analyzed are typical medium size port ones. Terminal prices are subject to price caps, which are seldom binding, but employment is highly regulated. This is not an unusual situation around the world.

Economic activities within a port are multiple and heterogeneous. On the one hand, among them cargo handling has been one of the most affected by technological changes and by competition among ports on the other. The importance of this activity is evident when realizing that it means from 70% to 90% a vessel’s disbursement account (De Rus et al., 1994). Cargo handling services are usually per-formed in port terminals.

Technological changes have increased the relative importance of speciﬁc terminals within port areas (e.g. multi-purpose,3 containers, liquid and solid bulk). Terminal facilities have now become heavily capital intensive and, depending on port size, more specialized as well, playing a key role in the choice of port by shippers. The role of the port terminals

2_{For more details, see}_{Coelli et al. (1999)}_{.}

3_{A multiple-purpose (MP) terminal is designed to serve}

heterogeneous trafﬁc, including containerized and non-contain-erized cargo. It can be transformed into a specialized one (e.g. containers only) by changing equipment.

within the logistic system makes them key actors of the port industry, playing a central role in the increasing competition within the sector.

In addition, the private sector has become increasingly interested in this type of activity. This has shifted the focus of port authorities competitive strategy design from the port as a whole to the terminals, making them the most important ele-ments within the port industry. As pointed out by Heaver (1995), this change of focus is the main factor explaining the increased competition within the sector.

The cargo handling service is usually viewed as one that has to be directly provided by the public sector or by private ﬁrms through concession contracts. The regulation of the Spanish ports system is based upon a scheme that allows the combination of public ownership of the port infrastructure (docks, land, and so on) with private ownership of the superstructure (warehouses, cranes, etc). The public authority determine the conditions under which the private initiative has to operate by ﬁxing maximum prices, characteristics and length of concessions, and other conditions (for more details seeTovar et al., 2004).

In the production of cargo handling services, the following groups of factors are required: basic infrastructure, superstructure, machines and mobile equipment, as well as labor. Traditionally, labor in a port has been strongly regulated, although changes have taken place worldwide during the last decade or so.

The stevedores or port workers working in Spanish port terminals are divided into two categories: those who are on the payroll (ordinary employment—LC) and those who are not (special employment—LE). These latter can be recruited on a provisional basis by any company to work 6 h shifts. Regulation drives the level and composition of port labor (including the choice between LE and LC) for each terminal.

The possibility of contracting port workers for speciﬁc operations (LE) provides the stevedoring companies4with some ﬂexibility, allowing them to adjust employment to terminal trafﬁc levels. How-ever, this ﬂexibility has its limits, since under the current legislation the operators do not have total

freedom to decide how and to what extent each type of worker has to be used.5 The stevedoring companies have:

The obligation to perform at least 25% of their activities (in tons) with port workers on their payroll (LC). The obligation to use at least one LE port worker in each shift. A global limit on the number of operations they can perform with LC workers of 22 per month; once they have reached this level, they are required to rely on LE workers. Moreover, as we can check in the empirical application, the regulation imposes certain degrees of comple-mentarity between both types of port workers so as to use them jointly.3. Modelling ﬁrm and time-varying technical and allocative efﬁciency

To calculate ﬁrm and time-varying technical and allocative efﬁciency we propose an empirical model, which consists of an input distance function and the associated input cost share equations:

ln 1 ¼ ln Dðy_{pt}; xpt; Kpt; DTÞ þ vptþupt, (1)
xiptwipt
Cpt
¼q ln Dðypt; xpt; Kpt; DTÞ
q ln xipt
þviptþAipt, (2)
where D(ypt,xpt,Kpt, DT) is the short-run input

distance function; y is an output vector, x is a variable input vector, K is a quasi-ﬁxed input, DT is a time year dummy to control for neutral technical change, p denotes port terminal and t time. The vpt

and vipterror components represent statistical noise,

and are assumed to be distributed as multivariate normal with zero mean.

3.1. Measuring technical efficiency

The error component uptX0 in (1) represents the

magnitude of technical efﬁciency (TE). We follow
Cornwell et al. (1990) who speciﬁes a model that
allows us to estimate time-varying technical
efﬁ-ciency levels for individual ﬁrms, without making
strong distributional assumptions for technical
inefﬁciency or random noise. Thus, if the constant
4_{These are ﬁrms that are allowed to handle cargo. In our case,}

the three terminals are also stevedoring companies; conversely they could not do handling cargo by their own, but by contracting a stevedoring company.

5_{A collective agreement is a three-way agreement among the}

State Stevedoring Association, the port workers, and the stevedoring companies.

in (1) is B0, then, it is possible to write:

b_{pt}¼B0þupt¼bpaDpþbpbDpt þ bpcDpt2, (3)
where Dpis a dummy variable for the port terminal

p, bpa; bpb; bpc are parameters to be estimated for

this port terminal, and t is a time trend. However, there is an interpretation problem because there is no easy way to empirically distinguish in (3) between technical inefﬁciency or technical change (for example, there are contradictory interpretations inCornwell et al., 1990orGood et al., 1995).

To solve this problem, and following Heshmati and Kumbhakar (1995)andHeshmati et al. (1995) we have included in (1), the dummy time (DT). In this way, by including a time variable among the regressors, it would be possible to resolve the interpretation problem in the Cornwell et al. (1990) speciﬁcation, since the time variable is associated with technical change and the error component (3) is associated with technical efﬁ-ciency, which is allowed to vary across producers and through time (for details see, for example, Lovell, 1996).

In this sense, the bpa captures time-invariant TE

whereas bpband bpccapture time-varying TE. Thus,

each producer has its own intercept (bpt), which is

allowed to vary quadratically through time at producer-speciﬁc rates. The TE of a port terminal in a time period is obtained from the estimated intercepts as TEpt¼exp(upt) where upt¼bpt

min(bpt). The efﬁciency index constructed in this

way, ranges from 0 to 1. Thus, in each period at least one port terminal is estimated to be 100% technically efﬁcient (with value 1), although the identity of the most technically efﬁcient port terminal may vary through time.

3.2. Measuring allocative efficiency

To calculate allocative efﬁciency two econometric approaches are available (Atkinson and Cornwell, 1994): (a) an error components approach, and (b) a parametric approach

(a) The error components approach In this approach we model allocative inefﬁciency through an error component. Thus in (2) the error compo-nents Aipt4 ¼o0, i ¼ 1,y,n, represent allocative

inefﬁciency, here represented by the difference between actual and stochastic shadow input cost shares form (for more details seeRodrı´guez-A´lvarez and Lovell, 2004). Moreover, if Aipt is positive, the

input i is being over-utilized with regard to other inputs and vice versa.

In our case it is possible to specify allocative inefﬁciency for the input i as

Aipt¼aipaDpþaipbDpt þ aipcDpt2, (4) where Dpis a dummy variable for the port terminal

p and aipa;aipb; aipcare parameters to be estimated

for this port terminal and for the input i and t is a time trend. The aipa captures time-invariant

alloca-tive inefﬁciency, whereas aipb and aipccapture

time-varying allocative inefﬁciency.

(b) The parametric approach: After estimation of systems (1)–(2), shadow price ratios are determined from the dual of Shephard’s Lemma as

qDðy; x; K; DT Þ=qxi qDðy; x; K; DT Þ=qxj ¼w s i ws j , (5)

where ws is the shadow prices vector. If the allocative efﬁciency assumption is satisﬁed, these shadow price ratios coincide with market price ratios. However, if there is allocative inefﬁciency, the two price ratios differ. To study such deviations, a relationship between the shadow prices (obtained through the distance function) and the market input prices is introduced by means of a parametric price correction:

ws_{i} ¼kiwi. (6)

Dividing (6) by the corresponding expression for input j we obtain ws i ws j ¼kij wi wj , (7)

where kij¼ki/kj. From (7) the degree to which

shadow price ratios differ from market price ratios is calculated. If kij¼1, there is allocative efﬁciency;

while if kij4(o)1, input i is under-utilized

(over-utilized) relative to input j.

If we have panel data, it is possible to obtain, for each pair of inputs, producer and time-speciﬁc allocative efﬁciency indices kij. Thus, with both

proposed approaches (the error component and the parametric approach) we can get time and ﬁrm varying indices of allocative efﬁciency. Moreover, it is important to distinguish between these two measures of allocative inefﬁciency (the parametric approach and the error components approach). In the error components approach, the aipts represent

the systematic allocative inefﬁciency for each input. In the parametric approach, kij indicates the

Moreover, in the error components approach, behind the aipt lies the assumption of an additive

relationship between wi and wis. In the parametric

approach a multiplicative relationship between wi

and wis is speciﬁed, yielding the coefﬁcients kij as

indices of allocative inefﬁciency.

However, it is also important to emphasize the existing relationship between the two approaches. When persistent inefﬁciency exists (that is, if the aipt

have mean values signiﬁcantly different from zero), their inclusion is necessary if the estimated para-meters, and consequently the estimated kij, are

unbiased. This implies that the inclusion of the parameter aiin the empirical model is indispensable,

in order to obtain unbiased estimates of the kij.

4. Econometric speciﬁcation

We now consider how to estimate systems (1)–(2). To do this, we have chosen a ﬂexible functional form, a translog short run multiproduct input distance function which is speciﬁed as

ln 1 ¼ B0þ
Xm
r¼1
arln yrptþ
1
2
Xm
r¼1
Xm
s¼1
ars ln yrptln yspt
þX
n
i¼1
biln xiptþ
1
2
Xn
i¼1
Xn
j¼1
bij ln xiptln xjpt
þxf ln Kptþxff
1
2ln K
2
ptþ
Xn
i¼1
xfiln Kpt ln xipt
þX
m
r¼1
xfrln Kpt ln yrptþ
Xn
i¼1
Xm
r¼1
r_{ri} ln y_{rpt}ln xipt
þX
T
T ¼1
gTDT þ uptþvpt, ð8Þ
xiptwipt
Cpt
¼biþ
Xn
j¼1
bij ln xjptþ
Xm
r¼1
rri ln yrpt
þxfi ln KptþAiptþvipt, ð9Þ
where y ¼ ðy1; . . . ; ymÞ is an output vector, x ¼
ðx1; . . . ; xnÞis a variable input vector, K is a
quasi-ﬁxed input, DT is a time dummy for year T, p
denotes port terminal; t denotes months and vpt, vipt,

uptand Aipt are disturbance terms which have been

already deﬁned. a, b, x, r, g are parameters related to outputs, inputs, quasi-ﬁxed input, cross term between outputs and inputs and time, respectively.

Homogeneity of degree +1 of the input distance function in variable inputs is enforced by imposing

the restrictions P n i¼1 bi¼1; Pn j¼1 bij¼0; Pn i¼1 rri¼ 0 ð8 r ¼ 1;. . . ; mÞ; P n i¼1

xfi ¼0: We also impose the symmetry conditions ars¼asr; bij¼bji; xfi ¼xif; xfr¼xrf; rri¼rir.

The estimation of this equation system provides the suitable empirical context to calculate the technical and allocative efﬁciency by means of the above proposed methodology.

5. Data and results 5.1. Data description

The sample consists of three multipurpose port terminals operating within the area of Las Palmas Port, located in the Canary Islands (Spain). The fact that all the port terminals considered are in the same port is a key advantage in the coherence and uniformity of the accounting data used. Thus, the regulation is the same, and environmental factors are equal or very similar for all of them.

We have monthly data from 1992 until 1997 for Terminal 1 (T.1), from 1991 until 1999 for Terminal 2 (T.2), and from 1992 till 1998 for Terminal 3 (T.3). The ﬁnal panel data set consists of 264 observations. Although these terminals mainly deal with con-tainers, they also operate roll-on/roll-off cargo (ro-ro) as well as general break-bulk cargo, so we distinguish three outputs measured in total tons: containers (CONT), ro-ro cargo (ROD) and general break-bulk cargo (MG).

The input variables are: port workers: ordinary (LC) and special6 (LE), capital (GK), intermediate consumption (GI). We also consider a quasi-ﬁxed input: total area (K).

The information available regarding the amount of workers7used is expressed in number of shifts per month. A shift is a 6 h work schedule. The total monthly labor expense for the terminals is calcu-lated as the sum of the cost of each type of worker. Capital covers all the components of tangible assets of the company —i.e. buildings, machines, etc. The monthly cost results from the addition of

6_{This can be recruited on a provisional basis by any company}

to work 6 h shifts.

7_{Information about port labor is very difﬁcult to obtain. Out of}

the previous papers dealing with port terminal efﬁciency only

the accounting depreciation for the period plus the return on the active capital of the period.8

With regard to area, the terminals under analysis can make use of a speciﬁc area that has been granted under concession, which may be increased by provisionally renting—upon prior request— additional area from the Port Authority. The addition of both types of areas is called total area, which is monthly measured in square meters.

Lastly, the rest of the productive factors used by the company that have not been included in any of the three preceding categories, such as ofﬁce supplies, water, electricity, and the like, have been denominated as intermediate consumption. The monthly expense results from the aggregation of the rest of the current expenses other than depreciation, personnel expenses and payment for area, after the pertinent corrections, in such a manner that, the resulting monthly expense truly reﬂects consumption and not accountancy.

The total monthly production expenses for the terminals result from the aggregation of expenses of all the productive factors deﬁned above. Table 1

shows the monthly values obtained for the entire sample and for each of the three terminals, both in terms of the deﬁned inputs and outputs, as well as the total expense incurred during service provision.9 It is worth stressing that data was gathered directly from the ﬁrms ﬁles and that all the details were discussed with executives when necessary, particu-larly for the monthly assignment of expenses. Data is described in detail inTovar (2002).

On the one hand, out of the three products, general break-bulk cargo (‘‘general cargo’’) repre-sents an average of 9.9% of the monthly total tons moved, containers represent a 87.4% and ro-ro 2.7%. On the other hand, labor costs account for an average of 53%10 of the monthly expenses for the entire sample. Total area represents 13%, capital amounts to 8% and intermediate consumption reaches 26%. Within port workers, ordinary ones account for 46% while special workers represent 54%. The ﬁgures reveal similar patterns per company.

Moreover, the analysis of the information con-tained in Table 1 leads to a ﬁrst approximation of the size of companies. Thus, taking into considera-tion the aggregated product volume, the largest company is T.3., followed by T.1. and by T.2. in last

Table 1

Monthly average input, output and expense values get for the entire sample and for each terminal

Variable Unit Terminals

Sample T.1 T.2 T.3

Mean S.E. Mean S.E. Mean S.E. Mean S.E.

Outputs

CONT 1000 Ton 59.2 41.57 53.1 9.72 33.5 7.45 97.4 54.36

MG 1000 Ton 5.6 6.35 0.6 0.78 9.9 7.39 4.4 3.12

ROD 1000 Ton 2.1 2.36 1.0 0.71 0.8 0.86 4.7 2.49

INPUTS

LC Number of shifts per month 336.4 206.13 344.0 140.28 251.0 49.90 439.8 306.94 LE Number of shifts per month 339.4 161.40 207.5 93.11 400.4 193.44 374.0 75.70 GI 1000 PTAS deﬂated 24,534.2 8445.04 21,961.4 5,485.22 20,573.2 3,556.72 31,832.1 10192.23 GK 1000 PTAS deﬂated 12,985.4 7728.52 6,063.2 429.87 11,043.0 3,939.85 21,416.1 7119.51 K M2 61,484.4 11758.16 63,971.8 7,892.55 57,530.6 2,597.86 64,435.8 18481.79 Expenditure

GLC 1000 PTAS deﬂated 17,964.1 8563.95 13,113.6 6,826.70 14,463.6 3,592.70 26,622.4 7979.02 GLE 1000 PTAS deﬂated 21,447.9 12515.12 18,759.9 6,911.54 20,738.9 9,453.66 24,663.6 17967.74

8_{This rate of return evidences the compensation earned by }

risk-free capital, which is made up of bank interest plus a risk premium. It has been considered that, for the period under analysis, the return for both concepts amounts to 8% per annum. The price of capital is the quotient of the cost of capital divided by the active capital of the period (net ﬁxed assets under exploitation for a given period t.)

9_{All monetary variables have been deﬂacted.}

10_{Note this is the same percentage that}_{Cullinane and Song}
(2003)report as a typical expenditure of a port.

position. It is to be noted that, where the variable used as the size indicator is the total monthly production expense (mean value), even though T.3. is still found to be in the ﬁrst place, the other two companies, T.1 and T.2. swap positions. This result is due to monthly expenses and do not vary monotonically with total production. This makes the different composition of outputs a likely explanation for cost differentials when factor prices are similar for the three companies. For example, the only explanation for the expenses of T2, being larger than those of T1 would be the difference in the trafﬁc mix, particularly, the larger volume of general cargo. This already suggests higher marginal costs for general cargo, which

reinforces the need for a multioutput analysis Jara-Dı´az et al. (2005).

5.2. Results

We have estimated systems (8)–(9) by means of iterative seemingly unrelated regressions (ITSUR), which is invariant to the omitted share equation.

In Tables 2–4 we present the estimated values from the input distance system. The variables have been divided by the geometric mean. Therefore, the ﬁrst-order coefﬁcients can be interpreted as elasti-cities. At the sample mean, the regularity conditions are satisﬁed: it is non-decreasing and quasi-concave in inputs and decreasing in outputs.

Table 2

Distance system estimated

Variable Coefﬁcient Standard error t-Statistic Variable Coefﬁcient Standard error t-Statistic L(CONT) 0.32822 0.3947 8.3160 L(ROD).L(GI) 0.00130 0.0009 1.4431 L(MG) 0.35813 0.0057 6.3068 L(ROD).L(K) 0.08451 0.0929 0.9100 L(ROD) 0.01642 0.0095 1.7225 L(ROD).L(GK) 0.00099 0.0005 2.1150 L(LC) 0.14139 0.0369 3.8291 L(LC).L(LC) 0.02116 0.0122 1.7274 L(LE) 0.25114 0.0279 8.9895 L(LC).L(LE) 0.05233 0.0114 4.6108 L(GI) 0.27117 0.0458 5.9193 L(LC).L(GI) 0.04950 0.0055 8.9437 L(K) 0.18176 0.1415 1.2847 L(LC).L(K) 0.01692 0.0340 0.4972 L(GK) 0.33629 0.0435 7.7245 L(LC).L(GK) 0.02399 0.0035 6.8374 L(CONT).L(CONT) 0.13236 0.0747 1.7714 L(LE).L(LE) 0.05444 0.0138 3.9455 L(CONT).L(MG) 0.02318 0.0086 2.6832 L(LE).L(GK) 0.03723 0.0030 12.4542 L(CONT).L(ROD) 0.01045 0.0111 0.9447 L(LE).L(GI) 0.06949 0.0054 12.7574 L(CONT).L(LC) 0.02655 0.0132 2.0089 L(LE).L(K) 0.10786 0.0311 3.4794 L(CONT).L(LE) 0.03048 0.0155 1.9698 L(GI).L(GI) 0.17636 0.0053 33.0481 L(CONT).L(GI) 0.00436 0.0074 0.5858 L(GI).L(GK) 0.05736 0.0031 18.4742 L(CONT).L(K) 0.33217 0.1905 1.7438 L(GI).L(K) 0.0433 0.0158 2.7515 L(CONT).L(GK) 0.00830 0.0046 1.8100 L(K).L(K) 0.31434 0.6175 0.5091 L(MG).L(MG) 0.00856 0.0016 5.6796 L(K).L(GK) 0.04762 0.0102 4.6748 L(MG).L(ROD) 0.00155 0.0014 1.1364 L(GK).L(GK) 0.11858 0.0029 41.0775 L(MG).L(LC) 0.00184 0.0017 1.1019 DT92 0.12548 0.0396 3.1696 L(MG).L(LE) 0.00071 0.0017 0.41710 DT93 0.09536 0.0790 1.2067 L(MG).L(GI) 0.00092 0.0008 1.0878 DT94 0.13487 0.0945 1.4279 L(MG).L(K) 0.03581 0.0317 1.1298 DT95 0.15500 0.1052 1.4727 L(MG).L(GK) 0.00021 0.0005 0.4520 DT96 0.02817 0.1173 0.2402 L(ROD).L(ROD) 0.00322 0.0027 1.2079 DT97 0.13772 0.1303 1.0566 L(ROD).L(LC) 0.00789 0.0022 3.6192 DT98 0.18434 0.1393 1.3232 L(ROD).L(LE) 0.00561 0.0022 2.5637 DT99 0.13160 0.1511 0.8710

Equation Mean R2 _{Std. error of regression}

Input distance function — — 0.079379

Ordinary worker share equation 0.236319 0.7541 0.044199

Special worker share equation 0.278165 0.8406 0.042865

Intermediate consumption share equation 0.323562 0.8505 0.020191

Capital share equation 0.161953 0.9602 0.012590

Statistically signiﬁcant at 10%. Statistically signiﬁcant at 5%.

InTable 2, the estimated parameters are shown. It can be seen that all ﬁrst-order parameters are statistically signiﬁcant and have the correct sign, except the quasi-ﬁxed input total area (K). As the quasi-ﬁxed input total area (K) is not statistically signiﬁcant this means that the marginal productivity of total area could be zero. This could be due to the fact that port terminal infrastructure and super-structure must be built with determined minimum

dimensions, so it is not possible to enlarge a terminal port in a continuous way. So, given that terminals keep growing, they are having more total area than previously needed.

5.2.1. Technical efficiency

InTable 3, and following Section 3.1 we present the estimated coefﬁcients from which it is possible to obtain the technical efﬁciency index (upt). These

indices have been represented in Fig. 1 (following Cornwell et al. (1990) approach). The temporal pattern of technical efﬁciency for the three terminals shows that T.3 is the most efﬁcient for almost the whole period, when its technical efﬁciency score begins getting worse. This occurs, more or less, when T.3 moves from a smaller to another bigger place inside the port area. This change of size requires a huge investment which probably drives this result. Remember that T3 is the largest company, so it seems to exhibit a relationship between size and efﬁciency (this relationship has also been reported in Cullinane et al. (2005). Terminal T.2 presents a declining technical efﬁ-ciency score in the ﬁrst years and an increasing one in the latter ones, being ﬁnally substituted by T.3 at the end of the period. Lastly, T.1 shows a general decline in technical efﬁciency.

Other technical efﬁciency scores are shown in Fig. 2. These are calculated considering the best observation as the reference technology.Fig. 2tells us a similar story. In terms of the trend in technical efﬁciency, T.3 shows a general decline (with the exception of the few ﬁrst periods), T.2 and T.1 present a similar, but softer, pattern than inFig. 1.

Table 3

bptComponents (from Eq. 3)

Parameters Coefﬁcients Standard error t-Statistic b1A 0.0425450 0.1105030 0.3850170 b1B 0.0000061 0.0057844 0.0010705 b1C 0.0000782 0.0000493 1.5836800 b2A 0.1322430 0.0458950 2.8814200** b2B 0.0049411 0.0035596 1.3880800 b2C 0.0000437 0.0000294 1.4861700 b3A 0.2295630 0.1167930 1.9655600** b3B 0.0251980 0.0061530 4.0951200** b3C 0.0002957 0.0000526 5.6142100** Table 4

AiptComponents (from Eq. 4)

Parameters Coefﬁcients Standard error t-Statistic
aLC1A 0.3719030 0.0456560 8.145760**
aLC1B 0.0091169 0.0020863 4.369740**
aLC1C 0.0000545 0.0000187 2.911360**
aLC2A 0.2436080 0.0470980 5.172390**
aLC2B 0.0061470 0.0007202 8.534300**
aLC2C 0.0000365 0.0000053 6.786440**
aLC3A 0.1496150 0.0527010 2.838960
aLC3B 0.0006403 0.0014934 0.428774
aLC3C 0.0000066 0.0000141 0.470838
aLE1A 0.3003560 0.0479150 6.268500**
aLE1B 0.0131560 0.0020900 6.294670**
aLE1C 0.0001023 0.1849460 5.534250**
aLE2A 0.0951980 0.0342090 2.782840
**
aLE2B 0.0056841 0.0008852 6.420770
aLE2C 0.0000380 0.0000068 5.569630
aLE3A 0.0117950 0.0440560 0.267731
aLE3B 0.0011804 0.0011371 1.038090
aLE3C 0.0000091 0.0000103 0.887793
aGK1A 0.1602610 0.0437710 3.661320**
aGK1B 0.014876 0.0004296 3.462580**
aGK1C 0.0000181 0.0000042 4.257970**
aGK2A 0.1904440 0.0437210 4.355870**
aGK2B 0.0000873 0.0002108 0.414233
aGK2C 0.0000018 0.0000017 1.057590
aKI3A 0.1849410 0.0456000 4.055760**
aGK3B 0.0006198 0.0005529 1.120990
aGK3C 0.0000031 0.0000051 0.622118
0
0.2
0.4
0.6
0.8
1
1.2
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
Time (CSS-1990)
**Technical Efficiency**
T.1 T.2 T.3

Fig. 1. Temporal pattern of technical efﬁciency for the three terminals.

5.2.2. Allocative efficiency: error components approach

In Table 4, following the error component approach, estimated coefﬁcients and asymptotic standard errors for AE parameters (Aipt

compo-nents) are shown. Remember that Aiptrepresent the

systematic allocative inefﬁciency for each input. The results of the estimation of the ﬁrm-speciﬁc temporal pattern of allocative inefﬁciencies follow-ing the error components approach are reported in Figs. 3 and 4for the three terminals with respect to ordinary and special port workers (LC and LE), respectively.

It is interesting to note that the two graphs represent the inverse image of each other. At the very beginning T.1 and T.2 over-utilized LC and underutilized LE, but as time passes these two terminals achieve allocative efﬁciency in LC but not in LE, which goes on being over-utilized. However, T.3 over-utilized LC and underutilized LE during the whole period. The joint analysis ofFigs. 3 and 4 shows that each terminal has a preferred adjustment factor. It is LC for T.1 and T.2, while it is LE for T.3. This would suggest that ﬁrms are operating with allocative efﬁciency with respect to one factor

and with inefﬁciency with respect to the other factor.

This behavior is best understood by considering the relevance of regulation. Indeed, it is useful to remember that regulation drives the level and composition of port labor (including the choice between LE and LC) for each terminal. Regulation impedes the labor adjustments for both types of workers jointly. This is why they end up trying to maintain one degree of freedom and may tend to focus on adjustments in one of the workers’ categories at a time. Figs. 3 and 4, indeed, clearly show that the employment policies of T.1 and T.2 differ from those followed by T.3.

Finally, as we can see fromFigs. 5 and 6the three terminals are allocatively inefﬁcient in the use of the two other factors considered: intermediate con-sumption (GI) is over-utilized and capital (GK) is under-utilized. The orders of magnitude are quite small in GI. In the case of capital this probably simply reﬂects the difﬁculty of adjusting capital perfectly in a situation of growth as was the case for these terminals during the period of analysis. The area comes ﬁrst and then, the rest of capital (buildings, cranes, and so on).

5.2.3. Allocative efficiency: parametric approach In the parametric approach we get ﬁrm and time-speciﬁc measures of AE for each pair of inputs. However, in order to conserve space we have only reported their values evaluated at the sample mean inTable 511which have been estimated from Eqs. (5) and (7).

Remember that kijo1 means that the ratio of the

shadow prices of input i to that input j is lower than
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1 9 17 25 33 41 49 57 65 73 81 89 97 105
**Time**
**Allocative Inefficiency**
LCT.1 LCT.2 LCT.3

Fig. 3. Firm-speciﬁc temporal pattern of allocative inefﬁciencies for LC (error components approach).

-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
1 10 19 28 37 46 55 64 73 82 91 100
**Time**
**Allocative Inefficiency**

LET.1 LET.2 LET.3

Fig. 4. Firm-speciﬁc temporal pattern of allocative inefﬁciencies for LE (error components approach).

0
0.2
0.4
0.6
0.8
1
1.2
1 9 17 25 33 41 49 57 65 73 81 89 97 105
**Time**
**Technical Efficiency**
T.1 T.2 T.3

Fig. 2. Technical efﬁciency score considering the best observa-tion as the reference technology.

11_{Firm and time-speciﬁc results are available from the authors}

the corresponding ratio of actual prices. The analysis of the average estimated values for the kij

reinforced the conclusions of the previous para-graphs and shows that LC is over-utilized relative to LE and GK. Moreover, GK is under-utilized relative to all the other production factors. Finally, the coefﬁcient kij which links GI with the other

productivity factor is always not statistically sig-niﬁcant, except in the case of GK.

With respect to labor, the result may reﬂect the labor-speciﬁc regulatory environment which im-pedes the adjustments needed by the operators. As for capital, it may be useful to point out that the levels of inefﬁciency are probably due to the impossibility of full adjustment as a result of indivisibility as previously mentioned; terminal or port infrastructure and superstructure must be built with determined minimum dimensions.12

The ﬁgures per company reveal similar patterns as we can see fromFig. 7.

In addition, it is possible to analyze the degree of substitutability between inputs (or the isoquant curve) by means of Morishima elasticities of substitution. Formally, Morishima elasticities in-form us about the variations of shadow prices given a variation in the input quantity. Positive values of the elasticities indicate complementarity between inputs and negative values imply substitutability. The Morishima elasticities have been divided into two components: Mij¼EijEii. The term Eijis the

cross-elasticity shadow price, and tells us whether the input pairs are net substitutes or net comple-mentaries, while Eii represents the elasticity price

characteristic of each input. Formally:

Mijðy; xÞ ¼ d ln ½ðDiðy; x=xiÞ=Djðy; x=xjÞ=d ln ½xi=xj

¼xiDijðy; xÞ=Djðy; xÞ xiDiiðy; xÞ=Diðy; xÞ

¼Eijðy; xÞ Eiiðy; xÞ. ð10Þ

The calculations made of the Morishima substitu-tion elasticities (Mij) and their two components (Eij

and Eii) are shown inTables 6–8. From the results,

we can deduce little scope for substitution possibi-lities among the different pairs of inputs. The estimation of the Eiishows the proper sign.

More-over, our estimation of the cross-elasticities shadow price Eij, behaves quite well. As expected, due to

regulation, LC and LE are net soft complemen-taries, and the degree of complementarity between

Table 5

Average values for coefﬁcients kij

Coefﬁcients Meana t-Statistic kOrdinary Worker, Special Worker 0.327803 1.726430

kLC,LE

KOrdinary Worker, Intermediate Consumption

0.287873 1.547900 KLC,GI

KOrdinary Worker, Capital 0.696075 6.500820

KLC,GK

kSpecial Worker, Intermediate Consumption

0.059402 0.504053 KLE,GI

KSpecial Worker, Capital 0.547863 6.949870

KLE,GK

KIntermediate Consumption, Capital 0.573215 10.37480

kGI,GK

Note: We have tested the signiﬁcance of the indices using the Wald test.

a_{Evaluated at the means of the data using parameter estimates}

of (8)–(9).

Statistically signiﬁcant different from one at 10% level. Statistically signiﬁcant different from one at 5% level.

0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
1 9 17 25 33 41 49 57 65 73 8 1 89 97 105
**Time**
**Allocative Inefficiency**

GIT.1 GIT.2 GIT.3

Fig. 5. Firm-speciﬁc temporal pattern of allocative inefﬁciencies for GI (error components approach).

-0.25
-0.2
-0.15
-0.1
-0.05
0
1 9 17 25 33 41 49 57 65 73 81 89 97 105
**Time**
**Allocative Inefficiency**
GKT.1 GKT.2 GKT.3

Fig. 6. Firm-speciﬁc temporal pattern of allocative inefﬁciencies for GK. (error components approach).

12_{It is not possible to enlarge a terminal port in a continuous}

LE and LC is higher than the one between LC and LE. This could be due to the fact that operators always have to include at least one LE port worker per shift. On the other hand, as we also expected, the cross-elasticities shadow price between port

labor (LC or LE) and capital show that these factors are net complementaries. It happens exactly the same way between capital and GI. Finally, our estimation of cross-elasticities shadow prices be-tween labor (LC or LE) and GI are not statistically signiﬁcant.

5.2.4. Technical change

The time year dummy coefﬁcients show the effect on the distance function of unobserved variables which, when evolving over time, affect all ﬁrms equally. We can check how these time effects inﬂuence the distance function from one year to another through the following expression:

TCTþ1;T¼gTþ1gT. (11) 0 0.2 0.4 0.6 0.8 1 1.2 1.4

KLCLE KLCGI KLCGK KLEGI KLEGK KGIGK

Sample T1 T2 T3

Fig. 7. Average ﬁrm allocative inefﬁciencies for each pair of inputs (parametric approach).

Table 6

Morishima substitution elasticities: Mij

Meana t-Statistic Meana t-Statistic Mlc,le 1,0587 10.2714** Mle, lc 1.1535 6.0648** Mlc, gi 0.6678 4.2113** Mgi, lc 0.004 0.0035 Mlc, gk 0.7790 10.4009** Mgk, lc 0.4777 10.019** Mle, gi 0.5271 6.7505** Mgi, le 0.0729 0.7081 Mgi, gk 0.1790 1.7666* Mgk, gi 0.4358 10.1179** Mle, gk 0.6726 9.9710 ** Mgk, le 0.4991 9.9215 ** Table 7

Cross and own elasticities: Eijand Eii

Meana t-Statistic Meana t-Statistic Elc, le 0.3497 7.5443** Ele, lc 0.6212 4.0677** Elc, gi 0.0411 0.9777 Egi, lc 0.0789 0.9052 Elc,gk 0.7000 2.3167** Egk, lc 0.1666 3.9478** Ele, gi 0.0051 0.1292 Egi, le 0.0055 0.1312 Egi, gk 0.1006 2.6778** Egk, gi 0.1247 3.1301** Ele, gk 0.1404 4.7940** Egk, le 0.1880 4.1426** Elc, lc 0.7098 8.3117** Ele, le 0.5312 9.7235** Egi, gi 0.0784 1.1577 Egk, gk 0.3110 33.7855**

Note: We have tested the signiﬁcance of the indices using the Wald test. * Statistically signiﬁcant at 10%. ** Statistically signiﬁcant at 5%. a

Evaluated at the means of the data using parameter estimates of (8)–(9). Table 8 Time effects Period TCa t-Statistic 1991–92 0.125484 3.169570** 1992–93 0.030120 0.605594 1993–94 0.039505 1.465350 1994–95 0.020130 0.880385 1995–96 0.183165 6.922290** 1996–97 0.109557 3.172850** 1997–98 0.046616 1.231000 1998–99 0.052731 1.272980

Note: We have tested the signiﬁcance of the indices using the Wald test.

*_{Statistically signiﬁcant at 10%.}
**_{Statistically signiﬁcant at 5%.}

a_{Evaluated at the means of the data using parameter estimates}

A positive (negative) value for TC indicates an upward (downward) shift in the distance function (see Fa¨re and Grosskopf, 1995). This measure is usually associated with technological change. The indices obtained from expression (10) are presented inTable 6.

There were only two periods where these indices were statistically signiﬁcant, and therefore, where time had an inﬂuence on ﬁrm activity. The ﬁrst one, from 1991 to 1992 where the indices evolved favorably, and the second one, from 1995 until 1997, where the indices had a negative sign, which indicates that time had a negative inﬂuence on ﬁrm activity. However, it can be observed that in the last period (1998–1999) the coefﬁcient returns to being positive, but not statistically signiﬁcant.

6. Conclusions

In this paper, we present an approach which allows us to estimate time-varying efﬁciency levels for individual ﬁrms without invoking strong dis-tributional assumptions for inefﬁciency or random noise. Using a Spanish ports panel, we have applied this methodology to a frontier input distance system. In this way, the operations of cargo handling ﬁrms in ports are analyzed by means of a multioutput input distance function estimation using monthly data on ﬁrms located at Las Palmas Port, Spain.

Both size and trafﬁc mix are ﬁrstly shown to be sufﬁciently diverse as to allow for a reliable estimation of a multioutput translog function which permitted us the calculation of ﬁrm and time-variant indices of technical and allocative inefﬁ-ciency (by using both the parametric and the error component approaches) which further add to the contribution of the paper to the literature.

Implementing our approach with typical medium size port terminals data we highlight two main conclusions. The ﬁrst one is that, it seems there is a relationship between ﬁrm size and technical efﬁ-ciency. The second one is that, our result with respect to allocative efﬁciency suggests that the port labor-speciﬁc regulatory environment impedes ad-justments that are needed by the operators. Finally, we have calculated Morishima elasticities.

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