A Multioutput Cost Function for Port Terminals
Some Guidelines for Regulation
Beatriz Tovar
btovar@empresariales.ulpgc.es
Departamento de Análisis Económico Aplicado
Universidad de Las Palmas de Gran Canaria
Sergio Jara-Díaz
jaradiaz@cec.uchile.cl
Departamento Ingeniería de Transporte
Universidad de Chile
Lourdes Trujillo
lourdes@empresariales.ulpgc.es
Departamento de Análisis Económico Aplicado
Universidad de Las Palmas de Gran Canaria
World Bank Policy Research Working Paper 3151, October 2003
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange
of ideas about development issues. An objective of the series is to get the findings out quickly, even if the
presentations are less than fully polished. The papers carry the names of the authors and should be cited
accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors.
They do not necessarily represent the view of the World Bank, its Executive Directors, or the countries they
represent. Policy Research Working Papers are available online at http://econ.worldbank.org.
1
Abstract
Cargo handling in ports is a multioutput activity, as freight can arrive in many forms such as
containers, bulk, rolling stock, or non-containerised general cargo. In this paper the operation of
port terminals is analysed through the estimation of a multioutput cost model that uses monthly
data on three firms located at the Las Palmas port in Spain. This permits the calculation of
product specific marginal costs, economies of scale (general and by firm) and economies of
scope, which are key tools to help the regulators in their task.
Key words: multiproduct, economies of scale and scope, regulation, port terminals and cargo
handling.
JEL Classification system: L9
2
1. Introduction
Broadly defined, a port can be described as a group of facilities and movable equipment used to
provide different types of services which, in economic terms, are highly heterogeneous. Since
ports are a key component of the logistics chain, their deficient operation directly affects relevant
economic variables such as export competitiveness and import final prices, which can negatively
affecteconomic development. This explains governments' (concerns with setting adequate
competitive or regulatory conditions to enable the efficient operation of ports.
Generally, all the activities developed at each port are coordinated by an entity known as the Port
Authority. Although private port authorities exist, in most countries port authorities are typically
e public entities that act as the regulatory entity for all the companies operating at the port.
Port regulation is not an easy task considering the diversity of activities that occur at port
facilities.1 Among those activities, cargo handling is of special relevance since it generally
represents over 80% of the costs incurred by a ship loading or unloading goods at a port. In spite
of the importance of cargo handling for the regulation of the port sector, little is known in
practice about the economics of this service.
This paper is an effort to increase understanding of this issue by presenting an estimation of a
cost function for general cargo handling services at multi-purpose terminals. This estimation
provides some of the key concepts for the regulation of the sector, such as marginal costs and
economies of scale and scope.
This paper is structured as follows: Section 2 describes certain aspects of the organization and
regulation of the port sector in general and of cargo handling services in particular, from which it
can be inferred that cargo handling is a multioutput activity. Section 3 presents the main cost
concepts used by the multiproduct theory to describe an economic activity, which will be used for
the empirical application of this paper. Section 4 summarizes previous works estimating output or
cost functions in the port sector. Sections 5 and 6 present the information used to build the
database, as well as the findings of the analyses made. Lastly, section 7 presents the final
conclusions.
2. Production and Regulation in the Port Sector
Ports have traditionally been subject to some kind of governmental control, although the
applicable legal system and the degree of dependence and control may vary from country to
country. Even though there is not a uniform pattern for port organization, the landlord model is
the one most widely used in the world. Under this model, the public sector provides port
infrastructure in the strict sense (lighthouses, quays, loading and unloading areas, etc.) and
private companies supply the superstructure required to provide port services (office buildings,
machinery, etc.).
1For an updated summary, see Trujillo and Nombela (2002)
3
These services, which are generally provided by private companies, include cargo handling,
which encompasses all handling operations from placing cargo on the dock to loading it on the
ship and vice versa. The most common means of introducing private sector participation into
cargo handling services at port terminals is through the granting of concessions. In general,
international experience has shown that the substitution of public property by private property in
relation to certain port services produces a remarkable increase in productivity and a reduction in
the waiting time of vessels, thus improving the efficiency of such services (Estache, González
and Trujillo, 2002).
In the past several decades, new cargo handling and vessel design technologies have been
developed that allow the maximization of mechanization, a reduction in the need for labor and
consequently, an improvement in the productivity of the vessel by dramatically reducing her stay
time at the port. This new technology can be labeled as the "unitization" or "unit load" concept.
Cargo unitization implies packing several small cargo items into a standard unit which can be
handled with specifically designed equipment. The main standard units used are pallets,
containers, roll-on/roll-off trucks and trailers. The unitization process leads to ship, port and
terminal specialization and has placed emphasis on the fact that in regards to cargo handling, the
type of package used to unitize the cargo is more important than the nature of the cargo itself.
Multi-purpose terminals provide services to ships carrying cargo which are heterogeneous but
presents identical generic characteristics. As a result of the present global trend towards general
cargo containerization, many multi-purpose terminals will eventually become container
terminals.
General cargo handling operations vary depending on whether the cargo is in break-bulk or
unitized form, and if it is unitized, whether it is containerized or roll-on/roll-off cargo (cargo that
is driven on and off the vessel). These different handling processes imply that costs vary in each
case and therefore, it is logical to treat them as separate products and to acknowledge the
multioutput character of the activity under consideration.
3. Multiproduct Cost Concepts2
Generally, the main issue when discussing economic regulation is cost, and consequently cost is
the focus of this paper. Specifically, a multioutput cost function estimate provides regulators with
key concepts, such as marginal costs per company or product, which allow them to define tariff
caps, if this were the regulatory system to be applied. Also, it allows the calculation of global
and specific economies of scale, which are useful in determining the feasibility of a marginal cost
tariff structure and facilitating the development of an optimal tariff structure. On the other hand,
the economies of scope calculation show whether it is advisable to specialize the company. In the
case of multi-purpose port terminals, this information provides guidelines for adequate port
planning, since it objectifies the decision to make port terminals container-specialized or
diversified instead.
2This title is based on the seminal paper of Baumol et al (1982).
4
Unlike single-productive firms, whose cost-production structure can be described with relatively
few interrelated concepts, multiproductive firms' cost analysis requires the description of several
new concepts. This led to the development of a theory which, as could be expected considered
single-production as a particular case.
Empirical determination of all cost concepts for a certain industry can be achieved through the
econometric estimation of the corresponding cost function C(W,Y). The explanatory variables of
such function, after all variable factors have been assumed, are product vector Y and price vector
of productive factors W. The latter has been eliminated in the expressions below in order to
simplify the mathematical formula.
Thus, the marginal cost of product i can be obtained as a derivative of the cost function with
respect to such product.
C
(1)
yi = Cmi
On the other hand, the degree of global economies of scale is a technical property of the
productive process which is defined in transformation or production functions. However, dual
relations allow the calculation of the degree of the economies of scale directly through the cost
function (Panzar and Willig, 1977) as follows:
S = C(Y)
(2)
Y yC(Y)
The degree of global economies of scale represents the maximum growth rate that the product
vector can reach when the productive factors vector increases in a certain proportion. Therefore,
the presence of increasing returns of scale (S>1) implies that an increase of productive factors by
a certain proportion enables an increase of the set of products by a proportion greater than ,
showing that a production expansion enjoys advantages from the point of view of costs.
Another way in which the firm's operations can change is through the variation of a certain
output's production, considering the amount of the rest of the products is constant. In order to
study the cost of such variation in production, it is necessary to define the incremental cost of
product i.The incremental cost of product i is represented by the cost of adding ith product plus
the vector of products produced by the firm and can be expressed as:
CIi = C(y1, y2,...., yn ) - C(y1, y2,...., yi ,0, yi ,....yn )
-1 +1 (3)
Although the average cost is not defined in multiproduct because Y is a vector3, the average
incremental cost is defined and reads:
3In this case, it is possible to define a ray average cost C/ related to the product proportional expansion from a
bundle of products Y0.
5
CIMei = CIi(Y ) (4)
yi
Incremental cost and average incremental cost definitions are used to identify the specific returns
to scale of a given product yi :
Si (Y) = CIi (Y) CIMei (Y) CIMei (Y)
C(Y) = C(Y) = (5)
yi Ci (Y)
yi yi
where Ci (Y) is the marginal cost of product i. Then, the degree of scale economies specific to a
product yi are the quotient between the product's average incremental cost and marginal cost, and
they will be increasing, constant or decreasing depending on whether Si(Y) is larger than, equal
to, or smaller than one, respectively.
Incremental cost definition can be extended to a subset of products R and it is very useful since it
allows identification of the specific return to scale of a given subset of products. Accordingly, the
degree of economies of scale specific to subset R is defined as follows:
SR (Y) = CIR (Y) CIr (Y)
y C(Y) = (6)
y iCi (Y)
j jR
jR yi
so the economies of scale specific to subset of products R will be increasing, constant or
decreasing depending on whether SR(Y) is larger than, equal to, or smaller than one, respectively.
Consequently, if SR >1, the application of tariffs to the marginal cost would not cover incremental
costs. Note that equation (2) represents a particular case of equation (6) when R equals M.
Cost complementarity between two different products can be analyzed following the expression
below that when showing values smaller than or equal to zero, indicates a weak cost
complementarity:
Cij (Y') = 2C(Y') 0, i j, 0 Y' Y
yiyj (7)
On the other hand, the expansion of the output vector may mean the introduction of new products
in the production line giving rise to a new concept related to production diversification. This last
possibility leads to a specific concept of multiproduct called economies of scope.
The economies of scope concept is useful to analyze whether it is advisable or not to have the
firm diversified or specialized. Thus, economies of scope measure the relative cost increase that
would result from the division of the production of Y into two different production lines T and N-
T. Formally, if an orthogonal partition of product vector M into two subsets T and N-T is carried
6
out, the degree of economies of scope EDT of subset of products T with relation to its
complementary subset N-T will follow this expression:
EDT (Y ) = 1 [C(YT ) +C(YN ) -C(Y)] (8)
C(Y ) -T
in such a way that the partition of the production will increase, decrease or not alter total costs
depending on wether EDT (Y) is larger than, equal to, or smaller than zero, respectively.
Accordingly, if EDT(Y) >0, there are economies of scope and, therefore, it is cheaper to produce
product vector Y jointly than product vectors YT and YN-T separately. In other words, it is not
advisable to specialize but rather to diversify the production. It is easy to see that ED should be in
the interval (-1, 1).
Lastly, there is a relation between the degrees of economies of scale and scope represented by the
equation:
SN (Y ) =TST(Y) + (1-T)SN-T(Y) (9)
1- EDT (Y )
where
y C(Y)
j
T = jT yj (10)
y C(Y)
j
jN yj
This relation shows that in the absence of economies of scope (ED=0), S would be a weighted
average of the specific economies of scale of each subset. However, the existence of economies
of scope (ED>0) favours the presence of economies of scale.
4. Empirical Literature
In a systematic and detailed analysis of empirical works on the econometric estimation of
production and cost functions in the port sector, the first thing noticed was the limited literature
on this issue, particularly in connection with cargo handling activities. This may be because it is
difficult to gather the necessary data to produce such works. Table 1 summarizes the works on
the estimation of production and cost functions in a single-output environment in the port sector.
With respect to the three production function estimation papers, all of them analyze single-output
scenarios and use the same Cobb-Douglas functional form and measurement of economies of
scale. Furthermore, two out of the three papers (Reker et al., 1990 and Tongzon, 1993) assess the
same activity at the same port and during the same time period, but with a somewhat different
definition of variables. This allows a comparison of the results of the economies of scale
estimation but it is disappointing, because they are contradictory.
7
With regard to the single-output cost function estimation, the findings are more significant. In the
two studies shown in Table 1, it can be observed that they expressly acknowledge that the
activity under consideration is a multioutput activity. In the Kim and Sachis (1986) work, a
single-productive cost function is estimated since they only have a limited number of
observations. There are updated versions of the paper by Martinez Budría (1996) (i.e. Jara Díaz
et al., 1997 and Jara Díaz et al., 2002) which estimate a multioutput cost function. Although
these two studies analyze different activities, they both arrive at the conclusion that there are
increasing economies of scale at the approximation point.
The studies estimating cost functions in a multioutput environment, which are summarized in
Table 2, are also very limited in number. There are only three in total, out of which one is an
upgraded version of another (Jara Díaz et al., 1997 and Jara Díaz et al., 2002). As in the single-
output case, although the activities assessed differ, the estimated economies of scale are
increasing at the approximation point.
The comparison that yields the most interesting results is probably that between the studies by
Martinez Budría (1996) and Jara-Díaz et al., (1997, 2002) since they only differ in the approach
(the first one is single-output and the other two are multioutput) and the functional form used -
Cobb-Douglas and quadratic, respectively. Both papers lead to the conclusion that there are
increasing returns to scale in infrastructure provision services in Spanish ports; however, the
figure obtained in the single-output case was larger. The authors themselves point out that this is
due to the existence of economies of scope, which are present in the multioutput scenario but
which can not be revealed from an aggregate description of the product. This shows that it is not
irrelevant to disregard that this is a multioutput activity.
This study reveals that no previous paper analyzes the multioutput character of costs in the cargo
handling activity.
5. A Model as an Empiric Study on Port of La Luz and Las Palmas4
La Luz and Las Palmas Port, located in the Canary Islands, has been chosen to carry out a
practical application aimed at estimating a cost function based on the data coming from the three
container terminals operating within the port area. Although they are called as such, these
terminals are not container terminals in the strictest meaning of the term. Other types of goods
can be handled in the site facilities such as Ro-Ro goods and general break-bulk cargo, and
therefore they are considered to be pure multi-purpose terminals despite their denomination.
4Within the Spanish port system, it is the public authority that determines the conditions for the private sector to operate by fixing
prices, number and type of terminals, conditions for exploitation, concession terms and features, among others. In accordance
with the rules in force, in order for public authorities to decide on these conditions they should observe efficiency, economy,
productivity and security standards, which calls for a wide knowledge of service costs.
8
9
Table 1. Estimation of monoproductive functions of production and costs
Author Activity Functional
specification Economies of scale Other Measurements
Production Function
Chang Average Productivities
(1978) Infraestructure? Cobb-Douglas Constants Marginal Productivities
Reker el al.
(1990) Containers Manipulation Cobb-Douglas Diminishing None
Tongzon
(1993) Containers Manipulation Cobb-Douglas Increasing Berth efficiency
Cost functions
Kin y Sachis Minimal efficient scale
(1986) Infrastructure and services Translogarithmic Increasing in the point of approximation Factor demand price elasticity
Cross elasticities
Martínez Budría Strongly Increasing in the point of Cost factor elasticities
(1996) Infrastructure Cobb-Douglas approximation Individual specific effects of each port
Second stage analysis
Table 2. Estimation of multiproductive functions of cost
Author Activity Functional
specification Economies of Scale Other Measurements
Jara Díaz et al.
(1997,2002) Infrastructure Quadratic Moderate Increasing in the point of Marginal costs of the product i
approximation Economies of scope
Martínez Budría et al. Marginal costs of the product i
(1998) Activity of the Estiba and Elasticities costs - products
Desestiba public societies Translogarithmic Increasing in the point of approximation Total productivity of the factors for a subsample
(SEED) of 14 SEED
9
The Model
Generally, the method used for estimating a cost function implies a company's optimizing
behavior consisting in choosing, in each case, the optimal combination of productive factors
resulting in the product vector, as long as such prices are considered to be exogenous to the
company.
It is helpful for a cost analysis to distinguish between the long and short term. The difference
between them lies in whether or not it is possible to adjust productive factors. Thus, all
productive factors are adjustable in the long run, and this is why the model to be estimated
tries to explain the aggregated economic expenses using both the levels of production for each
of the products and the prices of the productive factors used in such production process as
explanatory variables. A short-run cost function implies the existence of non-adjustable fixed
factors, and therefore the model to be estimated tries to explain the economic expense using
the quantum of the fixed factors involved in addition to the vectors for products and prices for
variable productive factors.
The Data
The database used for this empirical work is an asymmetric pool of monthly data relating to
production, productive factors, and the expenses in relation with the three terminals operating
within the port area of La Luz and Las Palmas located in the Canary Islands--referred to
below as T.1, T.2, and T.3. More precisely, from 1992 through 1997 for T.1., from 1991
through 1999 for T.2., and from 1992 through 1998 for T.3.
The production of the three terminals may be aggregated into three products --general
break-bulk cargo ("general cargo") representing a mean value amounting to 9.9% of the
monthly moved tons, and general consolidated cargocontainers and Ro-Ro's, with respective
mean values for the entire sample of 87.4% and 2.7% of the monthly moved tons. Table 3
shows the monthly values obtained for the entire sample and for each of the three terminals,
both in terms of the three defined products as well as the total expense incurred during service
provision. It also includes a production-aggregated volume, which results from the addition of
the monthly moved tons concerning the three products in all.
The analysis of the information contained in Table 3 leads first to an approximation of the
size of companies. Thus, taking into consideration the aggregated product volume, the largest
company is T.3., followed by T.1 and by T.2. The average volume of movement in the entire
sample amounts to approximately 67,000 monthly tons.
It is interesting to observe the maximum and minimum values because they point to a
significant variability during the study period. In fact, maximum values reveal that monthly
figures have been five times the average value. Furthermore, nil as minimum values is
relevant because economies of scope calculations require products to reach those value levels.
10
Table 3. Total expense (million pesetas of December, 1999) and monthly average
production (thousands of tons) for the total sample
Terminals
Variable Sample T.1 T.2 T.3
Total monthly expense Mean 94,8 73,6 81,9 129,4
Containers Mean 59,2 53,1 33,5 97,4
max.-min. 310-15 74-32 62-15 310-49
General cargo Mean 5,6 0,6 9,9 4,4
max.-min. 29-0 3-0 29-0 14-0
Ro-Ro cargo Mean 2,1 1,0 0,8 4,7
max.-min. 11-0 3-0 4-0 11-0
Production-aggregated Mean 66,8 54,7 44,1 106,5
max.-min. 325-15 78-32 77-15 325-58
On the other hand, where the variable used as a size indicator is the total monthly production
expense (mean value), even though T. 3. is still found in the first place, the other two
companies, T.1 and T.2. interchange positions. The reason for this change in positions
between T.1 and T.2. is to be found in the combination of products produced by each. Indeed,
general cargo has a heavier bearing on T.2 than on the other two --1.9% as opposed to 0.9%
in T.1. and 4.7% in T.3. This would indicate that it is more expensive to move general cargo
than other products, which suggests that the marginal cost of general cargo will be higher.
Finally, as a first approach it is interesting to observe data as if it were the case of a single
product process. For that, a "pseudo-mean-cost" for the activity has been estimated based on
the aggregated production volume. The graphic representation generates a dotted area in the
shape of a curve for mean costs --as expected-- which suggests that the data retrieved are
sensible.
The variable to explain is the total monthly production expense for the terminals, which
results from the aggregation of expenses of all the productive factors defined below.
The productive factors used in production of the three referenced products have been
grouped into four categories: personnel, total area, capital and intermediate inputs. The
personnel working in port terminals may be classified in two categories of workers: the non-
port workers, who are those that carry out other tasks than the handling of cargo (
administrative, executive, control personnel, maintenance, among others); and stevedores or
port workers, who are charged with handling cargo. In addition, the port worker category
branches off into two: workers under Relación Laboral Común (RLC) (Ordinary Employment
Relationship) and workers under Relación Laboral Especial (RLE) (Special Employment
Relationship). The first category of port workers, i.e. RLC workers, are those which are on the
11
payroll, i.e., employed permanently by a company. Should their employment be terminated,
these types of workers will revert to their former situation as RLE workers --i.e. port workers
who are not on the payroll of a particular stevedore company and therefore are available to be
recruited on a provisional basis by any company to work 6-hour shifts, and who fall under the
management of the Sociedad Estatal de Estiba y Destiba (SEED) (State-owned Loading and
Unloading Company). The possibility of recruiting port workers on a per-task basis provides a
significant degree of flexibility to stevedore companies.
The information available regarding the amount of work used is expressed in number of men
per month for non-port workers, and in number of shifts per month for port workers. A shift is
a 6-hour work schedule. The price of each type of work is calculated as the quotient of the
cost of such type of work and the number of workers in the case of non-port workers, or the
number of worked hours in the case of port workers computed on a 6-hour shift basis.
With regard to area, the terminals under analysis may make use of an area that has been
granted under concession, which may be increased by provisionally renting --upon prior
request-- additional area from the port authority.5 The addition of both types of areas is called
total area and the area used is measured in monthly square meters. The price of the total area
is the quotient of the area-related expense divided by the total area square meters.
Capital encompasses all the components of tangible assets of the company --i.e. buildings,
machines, etc. The monthly cost results from the addition of the accounting depreciation for
the period plus the return on the active capital of the period and the shares of stock of the
SEED. 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.)
Lastly, the rest of the productive factors used by the company and that have not been included
in any of the three preceding categories, such as office supplies, water, electricity, and the
like, have been denominated under 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 a manner such that the
resulting monthly expense truly reflects consumption and not accountancy. The price of
electricity has been used as an indicator of the price of intermediate consumptions, as the
prices of the other components do not undergo variances.
The most important productive factor in terms of its share of total expense is personnel, with a
mean amounting to 53% of the monthly expense of the entire sample. The other referenced
5Note that this fact may turn area into a variable factor.
12
factors represent different shares of the mean total expense of the entire sample, with total
area representing 13%; capital, 8%; and intermediate consumption 26%.
Within personnel, non-port workers account for 21% of personnel expenses and port workers
account for a mean value amounting to the remaining 79% of the entire sample. Within this
latter group, the mean value of RLC and RLE workers for the entire sample amounts to 36%
and 43% respectively. The figures per company reveal similar patterns.
Because the database available for this estimation is made up of monthly observations, at first
it seems sensible to choose a short-run model as it does not appear to be an easy task for all
productive factors to be adjusted on a monthly basis. If any of the productive factors are not
adjusted, the amount used of such factor is considered instead of the price.
The factors eligible as fixed factors are non-port personnel, total area and equipment.6 This
last factor indicates the available machinery and movable equipment at the terminal. The
possibility for terminals to rent additional area and machinery and to recruit port personnel
under special labor relationships suggests certain adjustability in the short run.
The correlation matrix was analyzed in order to determine the type of model to estimate --
either short or long run. The correlation coefficients between production and the factors
prompt to be fixed factors show that non-port personnel, equipment and total area are not as
"fixed" as it could presumably been thought of at the outset. Such analysis leads to the
conclusion that terminals are somehow adapting these factors with production and a long-run
model is to be estimated. Even so, in an attempt to contrast the adequacy of a long-run model,
a short-run estimation will be carried out considering total area as the only fixed factor7, since
it appears as the most discrete candidate among all the others, and the rest of factors as
variables.
6. Model and Results Estimation
Both models estimate a system of equations made up by the total cost function and the
equation of expense in factors resulting from the application of Shephard's lemma.
In cost function estimations it is preferable to use flexible functional forms. The most popular
ones are translogarithmic and quadratic forms. The selection between them both depends on
6Unlike area and non-port personnel, which are measured in homogeneous units --sq. m and men/month respectively-- and
therefore they do not present any problems for aggregation purposes, equipment as a variable comprises such different
machinery as a postpanamax crane, a forklift truck, or a chassis. For aggregation purposes, two possible indicators were
considered: power and purchase value. The former was considered inadequate because it weighs very different machines on
an equal footing, such as a crane and a forklift truck because they have similar lifting power, and therefore purchase price of
equipment was chosen.
7As this is considered a fixed factor, it is used as an explanatory variable for the amount used of said factor, i.e. the total
monthly square meters.
13
the objective of the task. One of the advantages of quadratic function is its suitability to the
analysis of economies of scope and incremental costs, while it allows for the estimation of
marginal costs, which is the reason why it was chosen. For the long-run model, the
econometric specification of the long-run total cost function is as follows (9):
m n
CT = A0 + i ( yi - yi ) + i ( pi - pi ) +(T -T )
i=1 i=1
+ 1 m m
(yi - yi )(y j - y j ) + 1 n n
( pi - pi )( p j - p j )
2 ij ij (11)
i=1 j=1 2 i=1 j=1
m n m n
+ ij( yi - yi )( p j - p j ) + i ( yi - yi )(T -T ) + µi ( pi - pi )(T -T )
i=1 j=1 i=1 i=1
N
+(T -T)(T -T)+ iDi
i=1
Where: yi = Output i, pi = Input i price, m = Number of outputs, n = Number of inputs, T =
Temporal Trend, Di = Firms Dummy, N = Number of firms. All the variables marked with a
horizontal bar reflect the value of the entire sample mean.
The equations of expense in variable inputs consistent with the following equation:
Gi = pi xi = pi i + 2ii( pi - pi) + ij( pj - pj) + ij(yj - yj) +µi(T -T)
m n (12)
ji j=1
Where: Gi = Expense in factor i, pi= Price for the variable inputs i, xi = Demand derived by
input i. m = Number of outputs, n = Number of inputs, T = Temporal trend.
In addition, some company dummies have been included to capture specific effects, as well as
a temporal, linear and quadratic, cross trend with all the variables that reflect a possible
technical change.
In turn, the relevant short-run model is consistent with the following expression:
m n
CT = A0 + i(yi - yi) + i(pi - pi) +(stot-stot) +(T -T)
i=1 i=1
+ 1 m m
(yi - yi)(yj - yj)+ 1 n n
(pi - pi)(pj - pj) + (stot-stot)(stot-stot)
1
2 ij ij
i=1 j=1 2 i=1 j=1 2 (13)
m n m n
+ ij(yi - yi)(pj - pj) + i(yi - yi)(stot-stot)+ i(pi - pi)(stot-stot)
i=1 j=1 i=1 i=1
m n
+ i(yi - yi)(T -T)+ µi(pi - pi)(T -T)+(stot-stot)(T -T)
i=1 i=1
N
+(T -T)(T -T)+ iDi
i=1
14
Where: stot = used amount of fixed input: total area.
This only differs from the long-run total cost equation in the linear, quadratic and cross terms
that affect the only fixed factor selected -i.e., total area- and will also affect the variable inputs
in the respective expense equations.
Gi = pi xi = pi i +2ii(pi -pi) +jiij(pj -pj)+ (stot-stot)+ ij(yj -yj)+µi(T-T)
m n (14)
j=1
The method used for the simultaneous estimation of the equations system comprising the total
cost equation and the respective expense equations is the estimation of systems seemingly
unrelated equations. This is a recursive method by which estimated generalized least squares
are applied to a set of seemingly unrelated equations which are actually related through
positive or negative covariances between the error terms in the different equations at a given
moment in time. This method is actually a two-stage, consistent and asymptotically efficient
estimation procedure.
Both models are used to estimate deviations from the sample mean, both to avoid
multicollinearity issues and to facilitate the interpretation of parameters. The estimations
yielded highly similar results; these are summarized in table 4 as far as first-order parameters
are concerned. As regards the sign and statistic significance of the estimated parameters, the
previous table shows the good behavior in both models. All first-order coefficients carry the
expected sign. In addition, all of them are statistically significant but for short-run model
trend which, even though not significant, is nearly so. Nevertheless, the joint significance of
all first-order parameters in the short-run model was confirmed through the Wald Test.
Marginal Costs
The mean marginal costs estimated through both models for containers (C2), general cargo
(C3) and Ro/Ro cargo (C4) match the expected order and magnitudes as shown in table 4.
Indeed, containers were expected to rank first in a lower-to-higher marginal cost scale, closely
followed by Ro/Ro cargo and, lastly, general cargo further on behind. The main reason behind
this result lies with the different rates of return reached as regards the handling of these three
products.
Since there is no (product) price data available against which the marginal costs thus obtained
can be verified, maximum tariffs currently applied at the Port8, which were in force during the
study period, are resorted to instead. These tariffs were grouped based on the type of cargo
8Spanish ports apply maximum tariff for public services regarding stowing/unstowing, loading/unloading, and
reception/delivery.
15
involved. All estimates obtained have been verified to be below the aforementioned maximum
tariff, which indicates that the marginal costs arrived at are reasonable.
Table 4. Expense, marginal costs, demands for factors and trend.
Long-run Model Short-run Model
Parameter Estimation t statistic Estimation t statistic
Total cost (mean)-C1 96680 140,02 97394 138,92
Marginal cost of Containers -C2 745 28,48 684 20,72
Marginal cost of General cargo C3 1974 14,19 2056 14,96
Marginal cost of Ro-Ro cargo C4 1056 2,96 1139 3,051
Demand for RLC worker -C5 1,58 69,74 1,58 73,66
Demand for RLE workers-C6 2,34 45,86 2,33 49,13
Demand for intermediate consumption -C7 983 87,30 981 88,27
Demand for total area -C8 61593 106,85
Demand for capital -C9 583266 40,61 589240 44,83
Demand for not port workers -C10 0,02 76,67 0,02 78,28
Trend -C11 -67 -1,96 -64 -1,89
As could be expected, the similarities between the results arrived at through each model as far
as the marginal costs, inputs derivative demand levels, estimated expense and trends are
concerned, confirm the long-run equilibrium inferred from the data. The analysis is then
focused on drawing results and conclusions from the long-run model.
Table 5 shows the estimates from the final version of the long-run model, which does away
with a single second-order parameter (crossing between RLE workers and Ro/Ro cargo) not
significant and carrying a sign opposite to the expected one.
Table 5. Results of the estimation of the model of long term.
Parameter Variable (1) Estimate Standard error t Statistic
C1=A0 CONS 96680,2 690,475 140,02
C2=contt CONTT 744,568 26,1409 28,4829
C3=mg MG 1973,57 139,062 14,192
C4=rodt RODT 1055,81 356,65 2,96036
C5=plc PLC 1,57685 0,022611 69,7386
C6=ple PLE 2,33895 0,051002 45,8603
C7=pi PI 982,53 11,2547 87,2994
C8=pcanon PCANON 61592,9 576,436 106,851
C9=pk PK 583266 14363,4 40,6078
C10=pnph PNPH 0,021919 2,86E-04 76,6747
C11= T -67,0148 34,1904 -1,96005
16
Parameter Variable (1) Estimate Standard error t Statistic
C12=cdos CDOS -0,068971 0,049706 -1,38758
C13=cdos CMG 0,408093 0,557917 0,731457
C14=cdos CR 4,57755 1,54726 2,95849
C15=cplc CPLC 9,95E-03 7,10E-04 14,0028
C16=cple CPLE 0,02 1,45E-03 13,7777
C17=cpi CPI 5,818 0,511285 11,3792
C18=cpcanon CPCANON 180,843 21,0889 8,57526
C19=cpk CPK 7785,87 611,748 12,7272
C20=cpnph CPNPH 2,62E-04 1,34E-05 19,6209
C21=cte CTE 0,120936 0,150679 0,802607
C22=mg2 MG2 -0,518286 1,40314 -0,369375
C23=mgr MGR 1,15946 8,13486 0,14253
C24=mgplc MGPLC 0,037099 4,60E-03 8,06544
C25=mgple MGPLE 0,078918 0,01047 7,53744
C26=mgpi MGPI 8,45203 2,50495 3,37413
C27=mgpcanon MGPCANON -109,421 120,741 -0,906247
C28=mgpk MGPK 18048,7 2947,32 6,12378
C29=mgpnph MGPNPH 4,48E-04 5,61E-05 8,00074
C30=mgt MGT -0,469081 0,629254 -0,745456
C31=r2 R2 -40,9608 10,3971 -3,93965
C32=rplc RPLC 0,037754 0,011358 3,32395
C34=rpi RPI 20,5144 6,6753 3,07318
C35=rpcanon RPCANON -583,417 337,467 -1,72881
C36=rpk RPK -4811,55 7599,82 -0,633114
C37=rpnph RPNPH 7,67E-04 1,66E-04 4,62008
C38=rt RT -0,734985 1,88866 -0,389156
C39=plc2 PLC2 -7,20E-06 1,36E-06 -5,27695
C40=plcple PLCPLE -2,11E-05 7,46E-06 -2,83304
C41=plcpi PLCPI 9,25E-03 2,77E-03 3,33348
C42=plcpcanon PLCPCANON -0,307449 0,164174 -1,8727
C43=plcpk PLCPK 6,33105 2,33813 2,70774
C44=plcpnph PLCPNPH 1,90E-07 6,48E-08 2,93544
C45=µplct PLCT -0,015786 1,12E-03 -14,0831
C46=ple2 PLE2 -2,33E-05 9,31E-06 -2,50139
C47=plepi PLEPI 0,028954 7,71E-03 3,75454
C48=plepcanon PLEPCANON 0,788301 0,477103 1,65227
C49=plepk PLEPK -8,6808 5,87937 -1,47648
C50=plepnph PLEPNPH 1,69E-07 1,54E-07 1,09846
C51=µplet PLET 9,91E-03 2,68E-03 3,69417
C52=pi2 PI2 -15,0668 2,44814 -6,15439
C53=pipcanon PIPCANON 901,258 299,174 3,01249
17
Parameter Variable (1) Estimate Standard error t Statistic
C54=pipk PIPK 2696,68 2585,63 1,04295
C55=pipnph PIPNPH 4,67E-05 9,37E-05 0,497954
C56=µpit PIT 1,24028 0,688917 1,80033
C57=pcanon2 PCANON2 -10391,1 12917,9 -0,804397
C58=pcanonpk PCANONPK -209552 157560 -1,32998
C59=pcanonpnph PCANONPNPH -0,045851 7,98E-03 -5,74755
C60=µpcanont PCANONT 196,049 39,8325 4,92183
C61=pk2 PK2 -1,33E+06 1,36E+06 -0,977757
C62=pkpnph PKPNPH 0,231914 0,059244 3,91454
C63=µpkt PKT 125,511 696,051 0,180318
C64=pnph2 PNPH2 -5,59E-09 1,41E-09 -3,97384
C65=µpnpht PNPHT -9,87E-05 1,54E-05 -6,41019
C66= T2 0,142629 0,109505 1,30249
C67=T.1 T.1 -2460,71 220,639 -11,1526
C68=T.2 T.2 -2479,14 315,09 -7,86803
(1) For a more detailed description of the variables see Annex 1.
Dependent variable: Total Expenditure
Mean of dependent variable = 94783,3 Std. error of regression = 10802,5
Std. dev. of dependent var. = 34819,9 R-squared = 0,903733
Sum of squared residuals = 0,308070E+11 Durbin-Watson statistic =0,991747
Variance of residuals = 0,116693E+09 Corrected R-squared = 0,870825
Dependent variable: RLC Worker Expenditure
Mean of dependent variable = 17964,1 Std. error of regression = 3989,9
Std. dev. of dependent var. = 8563,97 R-squared = 0,84924
Sum of squared residuals = 0,420269E+10 Durbin-Watson statistic = 0,667191
Variance of residuals = 0,159193E+08 Corrected R-squared = 0,842659
Dependent variable: RLE Worker Expenditure
Mean of dependent variable = 21447,9 Std. error of regression = 7773,31
Std. dev. of dependent var. = 12515,1 R-squared = 0,613797
Sum of squared residuals = 0,159520E+11 Durbin-Watson statistic = 0,538230
Variance of residuals = 0,604243E+08 Corrected R-squared = 0,596938
Dependent variable: Non Port Worker Expenditure
Mean of dependent variable = 10410,9 Std. error of regression = 2223,66
Std. dev. of dependent var. = 4445,35 R-squared = 0,749944
Sum of squared residuals = 0,130539E+10 Durbin-Watson statistic = 0,773278
Variance of residuals = 0,494466E+07 Corrected R-squared = 0,739029
Dependent variable: Intermediate Consumption Expenditure
Mean of dependent variable = 24534,2 Std. error of regression = 4597,01
Std. dev. of dependent var. = 8445,03 R-squared = 0,702706
18
Sum of squared residuals = 0,557898E+10 Durbin-Watson statistic = 1,26826
Variance of residuals = 0,211325E+08 Corrected R-squared = 0,689728
Dependent variable: Total Area Expenditure
Mean of dependent variable = 7071,48 Std. error of regression = 1035,59
Std. dev. of dependent var. = 2897,86 R-squared = 0,871917
Sum of squared residuals = 0,283125E+09 Durbin-Watson statistic = 0,513589
Variance of residuals = 0,107244E+07 Corrected R-squared = 0,866326
Dependent variable: Capital Expenditure
Mean of dependent variable = 12985,4 Std. error of regression = 5212,91
Std. dev. of dependent var. = 7728,52 R-squared = 0,545660
Sum of squared residuals = 0,717404E+10 Durbin-Watson statistic = 0,391706
Variance of residuals = 0,271744E+08 Corrected R-squared = 0,525827
Certain cost-related concepts that play a relevant role in the analysis of multioutput activities
can be calculated based on the estimated parameters. Specifically, the following are
calculated.
Marginal costs by product and firm
Table 6 displays the results obtained and shows that all of them are statistically significant and
that the same order applicable to the sample mean higher to lower cost by products- is
maintained. Furthermore, no major variability exists between them.
The marginal costs for containers (in units) are 7595 pesetas/unit for T.1, 8435 pesetas/unit
for T.2 and 8425 pesetas/unit for T.3., all of which are reasonable figures considering that
maximum tariffs are set at 12145 pesetas/unit.
Table 6. Marginal costs for company at the mean (ptas / ton))
Estimate
Products Firms (Ptas/ton) t Statistic
T.1. 744 28,05
Containers T.2. 735 28,07
T.3. 757 28,97
T.1. 1994 13,99
General cargo T.2. 1915 13,96
T.3. 2032 14,38
T.1. 1117 3,089
Ro-Ro cargo T.2. 1017 2,853
T.3. 1053 2,91
19
Global and specific economies of scale
The global and specific economies of scale are calculated based on the estimated parameters;
the results are displayed in table 7.
Table 7. Global and product-specifics Economies of Scale estimated in the average
Economies of scale Estimate t Statistic
Global 1,64 33,18
Containers 1,01 254,80
General cargo 1,00 251,87
Ro-Ro cargo 1,08 32,30
As shown in table 7, all results are statistically significant. Global economies of scale above
one show that the average incremental cost for the mean decreases for proportional variations
in all products. Specific economies of scale are very close to one for all three products.
The results by company are displayed in table 8, where global economies of scale for
company T.3 are shown to be smaller than the other two which feature similar values. As
already explained, this is the largest terminal as well as the one with the greatest output levels;
this is thus a case of exhaustion of economies of scale.
Table 8. Global economies of scale and for terminal
Estimate T Statistic
Mean 1.64 33.18
T.1 2.26 32.61
T.2 2.13 24.63
T.3 1.07 37.17
Economies of scope
Based on the estimated parameters, all relevant orthogonal partitions of the product vector are
analyzed, i.e., the mean production cost of all products by a single company is compared to
the one that would apply if more companies were in charge of the production process:
* three companies: each one specializing in one product =ED;
* two companies: one specializing in containers and the other one offering the other two
products= EDC;
20
* two companies: one specializing in general cargo and the other offering the other two
products = EDMG;
* two companies: one specializing in Ro/Ro cargo and the other one offering the other
two products= EDR.
Table 9 summarizes the results obtained. As shown in the table, all ED estimates are within
the theoretically defined range (-1,1) and all of them are significant. The presence of different
types of economies of scope reinforces the existence of global returns of scale growth even
though the specific returns estimated by products remain almost constant, which is absolutely
feasible given the relation between S, Si and ED.
The results in table 9 show that specialization is not advisable since joint production always
carries savings as compared to specialized production. These savings are more noticeable
where one single company offering all products, as compared to three companies specializing
in one product each, where the joint production costs savings (at average values) would be as
high as 78%.
Table 9. Economies of scope
Estimate t Statistic
ED 0,782 21,20
EDC 0,387 20,95
EDMG 0,393 21,37
EDR 0,389 20,83
Also significant, even though not as much as the ones dealt with above, are the savings
obtained by comparing the situation of a single company in charge of all products against two
companies: one specializing in one product and the other manufacturing the remaining two,
where the savings through joint production (at average values) would range between 38.7%
and 39.3%, depending on each particular case.
Irrespective of the partition used, the savings obtained in these cases (EDC, EDMG, y EDR) are
highly similar. The reason behind this is that the only terms that depend upon the selected
partition are the second-order terms representing the products within T and N-T and these are
minor as compared to the first-order terms in the estimate arrived at.
7. Conclusions
There are three main reasons why a regulator will find it useful to know the cost structure.
First, estimating marginal costs as well as global and specific economies of scale is an
21
essential tool for tariff cap regulation since it provides the regulator with information on
whether the application of tariffs to marginal costs -considering the economies of scale- is
feasible.
Second, an economies of scope calculation provides the regulator with information on
whether it is advisable to have the terminals diversified or whether they should be specialized
instead, thus objectifying a decision which, when the right choice is made, inevitably carries
costs savings.
Lastly, the capacity to measure the degree of subadditivity provides the regulator with a
proper tool to decide the optimal port structure in terms of the adequate number of terminals.
Basically, all concept costs defined help add to the amount of information available to the
regulator and, accordingly, the regulator's knowledge of the reality to be regulated as well,
properly guiding the latter in exercising its powers and defining the framework for regulatory
action.9
As regards the first contribution made by cost models, this empirical study provides a first
estimate of the main indicators that are relevant to any regulator. Basically, the marginal cost
levels and their relative order confirm economic intuition. The lowest marginal costs are those
for containers, as could be expected, followed by Ro/Ro cargo, with a marginal cost 1.4 times
the marginal cost of the container and, lastly, further on behind, general cargo with a marginal
cost 2.65 times the cost of the container (evaluated at the mean). The marginal cost estimates
show the same behavior when estimated by company. In turn, the estimated global economies
of scale are above one, which shows that the average incremental cost at the sample mean
decreases on proportional variations for all products. Therefore, the application of tariffs at
the marginal cost would not cover total costs. Since the service is provided by privately
owned companies that are not subsidized, caps should be set above the marginal costs; a
contrary solution would amount to admitting that the companies are operating at a loss. A
comparison of these results to the caps in force during the sample period shows that the caps
defined are always above the marginal cost estimates, the result thus being in line with the
behavior of privately-owned companies10.
9For instance, a conclusion might be reached that the only feasible alternative is to have a single-terminal port.
This would call for stricter regulation as a result of the lack of competition in the port. Conversely, where more
than one terminal coexists on the market, there is less need for regulation, even though the regulator will still be
required to make sure that the different terminals operating at the port will not act in collusion. The Buenos Aires
port is clear evidence of the difficulties faced by regulators in designing the adequate port planning. Even though
originally divided into six terminals, only three are currently in operation at the Buenos Aires port. For a more
detailed explanation see Trujillo and Serebrisky (2003).
10The presence of competition, to a certain extent, among the three terminals seems to show that no excessive
profits are being obtained.
22
In regards to the second contribution, the estimates arrived at on economies of scope for all
the relevant orthogonal partitions of the product vector lead to the conclusion that, in this
particular case, specialization is not advisable. There are obvious savings as a result of joint
production when compared either to the extreme case of three companies specializing in one
product each or when compared against any partition into two companies (one of them a
specialized company).
Lastly, the presence of economies of scale and economies of scope is not sufficient to
conclude that the costs function analyzed is a subadditive one; this conclusion could not be
reached based on the information available when drafting this article. In any event, results of
scale and scope as a whole suggest that it is advisable to have the two smaller companies
grow, still handling all three products. More conclusive results would necessitate a larger
database so that all pertinent estimates for calculating this concept may be obtained. This
investigation will continue to develop towards that goal.
23
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25
ANNEX 1.
Glossary of variables
Cons = Constant
Contt = Monthly movement of containers
MG = = Monthly movement of general goods
RODT = Monthly movement of ro-ro cargo
PLC = RLC Worker Price
PLE = RLE Worker Price
PI = Intermediate Consumption Price
PCANON = Total Area Price
PK = Capital Price
PNPH = Non Port Worker Price
T = Temporal Trend
CDOS = Crossing container with itself
CMG = Crossing container with general cargo
CR = Crossing container with ro-ro cargo
CPLC= Crossing container with RLC Worker Price
CPLE= Crossing container with RLE Worker Price
CPI = Crossing container with Intermediate Consumption Price
CPCANON = Crossing container with Total Area Price
CPK = Crossing container with Capital Price
CPNPH = Crossing container with Non Port Worker Price
CTE = Crossing container with Temporal Trend
MG2 = Crossing of general cargo with itself
MGR = Crossing of general cargo with ro-ro cargo
MGPLC = Crossing of general cargo with RLC Worker Price
MGPLE = Crossing of general cargo with RLE Worker Price
MGPI = Crossing of general cargo with Intermediate Consumption Price
MGPCANON = Crossing of general cargo with Total Area Price
MGPK = Crossing of general cargo with Capital Price
MGPNPH = Crossing of general cargo with Non Port Worker Price
MGT = Crossing of general cargo with Temporal Trend
R2 = Crossing of ro-ro cargo with itself
RPLC = Crossing of ro-ro cargo with RLC Worker Price
RPI = Crossing of ro-ro cargo with Intermediate Consumption Price
RPCANON = Crossing of ro-ro cargo with Total Area Price
RPK = Crossing of ro-ro cargo with Capital Price
RPNPH = Crossing of ro-ro cargo with Non Port Worker Price
RT = Crossing of ro-ro cargo with Temporal Trend
PLC2 = Crossing RLC Worker Price with itself
PLCPLE = Crossing RLC Worker Price with RLE Worker Price
26
PLCPI = Crossing RLC Worker Price with Intermediate Consumption Price
PLCPCANON = Crossing RLC Worker Price with Total Area Price
PLCPK = Crossing RLC Worker Price with Capital Price
PLCPNPH = Crossing RLC Worker Price with Non Port Worker Price
PLCT = Crossing RLC Worker Price with Temporal Trend
PLE2 = Crossing RLE Worker Price with itself
PLEPI = Crossing RLE Worker Price Intermediate Consumption Price
PLEPCANON = Crossing RLE Worker Price with Total Area Price
PLEPK = Crossing RLE Worker Price with Capital Price
PLEPNPH = Crossing RLE Worker Price with Non Port Worker Price
PLET = Crossing RLE Worker Price with Temporal Trend
PI2 = Crossing Intermediate Consumption Price with itself
PIPCANON = Crossing Intermediate Consumption Price with Total Area Price
PIPK = Crossing Intermediate Consumption Price with Capital Price
PIPNPH = Crossing Intermediate Consumption Price with Non Port Worker Price
PIT = Crossing Intermediate Consumption Price with Temporal Trend
PCANON2 = Crossing Total Area Price with itself
PCANONPK = Crossing Total Area Price with Capital Price
PCANONPNPH = Crossing Total Area Price with Non Port Worker Price
PCANONT = Crossing Total Area Price with Temporal Trend
PK2 = Crossing of Capital Price with itself
PKPNPH = Crossing of Capital Price with Non Port Worker Price
PKT = Crossing of Capital Price with Temporal Trend
PNPH2 = Crossing Non Port Worker Price with itself
PNPHT = Crossing Non Port Worker Price with Temporal Trend
T2 = Crossing of the Temporal Trend with itself
T.1 = Dummy for firm T.1
T.2 = Dummy for the firm T.2
27