A spatial efficiency index proposal: an empirical application to SMEs ...

Ann Reg Sci (2011) 47:353–371 DOI 10.1007/s00168-010-0382-8 SPECIAL ISSUE PAPER

A spatial efficiency index proposal: an empirical application to SMEs productivity Mari Luz Maté-Sánchez-Val · Antonia Madrid-Guijarro

Received: 19 January 2009 / Accepted: 1 February 2010 / Published online: 24 April 2010 © Springer-Verlag 2010

Abstract Our study introduces a Spatial Efficiency Index. Its definition is based on the traditional concepts of “technology set” and “distance function”, as well as on spatial econometric methods. This index measures productive efficiency in each productive unit in relation to its close environment, instead of comparing each unit with all others. This way both local efficiency patterns and the highest local efficiency inequalities can be identified. An empirical application was developed so as to analyse data from Small and Medium sized Spanish Industrial Enterprises, using nonparametric Data Envelopment Analysis techniques. Our proposal is a contribution to the development of indexes that explicitly include proximity relations—a field with countless research possibilities. JEL Classification

C43 · C61 · D24 · O18

1 Introduction Productivity is a key factor in economic development analysis due to its importance regarding regional growth and its role in business dynamism (Hulten 2001). Productivity Analysis has yielded a continuous stream of research lines and results (Taymaz 2005). Partial and global measures have to be considered in order to estimate productivity. Partial productivity measures can be very easily calculated from a single output generated by a single input, but productive units usually produce many outputs from

M. L. Maté-Sánchez-Val · A. Madrid-Guijarro (B) Accounting and Finance Department, Business Faculty, Technical University of Cartagena, Calle Real no 3, 30201 Cartagena (Murcia), Spain e-mail: [email protected] M. L. Maté-Sánchez-Val e-mail: [email protected]

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many inputs. Therefore, alternative global measures to calculate this variable needed to be developed. Among other authors, Caves et al. (1982) established the theoretical basis for the Malmquist Index, which measures productivity changes. One of the most important features of this index is the possibility to be broken down into its components (Färe et al. 1989, 1994). This kind of focussed analysis offers insights into the specific sources of productivity changes, including the efficiency change (catching-up) component. Efficiency indicates how well each productive unit is combining its resources to obtain the highest production level using a specific technology (Färe et al. 1994; Arestis et al. 2006). According to this, a productive unit is efficient if it cannot produce more of any particular output without decreasing some other output, or without consuming more input (Lin and Chen 2002). Some researchers have focussed on the analysis of this variable and its determinant factors. Thus, Caves and Barton’s study (1990) classified efficiency factors in two groups: firm specific and environmental factors. The first category is partially controlled by firms themselves, and includes factors such as size (Oczkowski and Sharma 2005), R&D intensity (Ornaghi 2006) and degree of outsourcing (Heshmati 2003). The second category—environmental factors—is not under direct control by the firm, and considers issues such as industry affiliation (Roudut 2006) or location (Li and Hu 2004). The study of location as an efficiency-explicative factor takes different variables into account. Along this line of research, Beeson and Husted (1989) detected that a great deal of efficiency variability can be explained by regional differences in labour and public infrastructure. Feldmann and Audretsch (1999) showed that agglomeration and interaction affect efficiency values. Cooke et al. (2004) determined that location may influence a firm’s innovation activities, with consequences on its production process and efficiency. In order to account for “location” as a factor, empirical applications have calculated the efficiency component of the Malmquist Productivity Index, comparing different units at the same point in time, as an alternative to introducing temporal differences (Berg et al. 1993). These studies are in accordance with the initial idea exposed by Caves et al. (1982) and suggest that all possible bilateral comparisons among units as well as the respective geometric means must be calculated in order to compare the efficiency components of two units during the same time period. Recent studies (Helfand and Levine 2004; Angeriz et al. 2006, among others) have applied this methodology and taken it one step further, by first finding the traditional Malmquist Productivity Index and its components, and then contrasting the existence of proximity effects through spatial econometric techniques. However, according to Schmidt and Moreira (2009) this double procedure does not take into account the uncertainty in the estimation of efficiencies. Taking this limitation into account, these authors propose an alternative way of introducing spatial effects in a stochastic frontier model. They include the spatial term as a latent component in the one-sided disturbance term. Schmidt and Moreira (2009) supersede previous literature integrating the analysed spatial pattern into the very model definition. Following this train of thought, the main aim in our study is to define an efficiency index which integrates the notion of space: The Spatial Efficiency Index. Such index determines a relative measure of productive efficiency for each productive unit in

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relation to its close environment, instead of comparing each unit with all others. A contiguity matrix is applied so as to take into account the neighbouring units of each productive unit. Our paper’s main contribution is the introduction of an index whose definition explicitly considers close proximity relations. The importance of such an index comes from its capacity to identify local efficiency patterns and determine the highest local efficiency inequalities. The detection of these local behaviour patterns can be used in the design, development and assessment of regional policies. In fact, as stated by Armstrong and Taylor (2000), local efficiency inequalities need to be eradicated in order to achieve regional parities. The identification of the best environment practices may provide rules and characteristics to guide the design of locally tailored regional policies, in order to get a positive interaction among close regions and efficiently assign scarce public resources. We present an empirical application of this spatial index, performing Data Envelopment Analysis (DEA) on a sample of Spanish industrial firms, and determining which provinces are characterised by efficiency advantages in relation to their close environment, as well as which provinces are in disadvantage. To develop our empirical analysis we distinguish among sub-sectors and perform a sensitivity analysis. The Spanish economic system is particularly suitable, due to the spatial dependence and the disadvantages in terms of productivity that characterize it, as compared to other European countries (OECD 2007). In section two, we give some definitions that are needed to develop this index and we specifically introduce the Spatial Efficiency Index. In section three, we present the results of the empirical application and end up with our main conclusions.

2 Spatial efficiency index proposal This section will be organised as follows: we firstly give the definition of the concepts of contiguity matrix, technological set and distance functions. Secondly, we propose the concepts of spatial technology set and spatial distance functions and prove that they confirm the characteristics of the previous non spatial definitions. Finally, we present a Spatial Efficiency Index and a non parametric DEA methodology adapted to solve it.

2.1 Preliminary concepts 2.1.1 Contiguity matrix Let C be a set of coordinates of cardinality R. We assume that for every c ∈ C there exists a productive unit. A first order contiguity matrix W (Moran 1948; Geary 1954) is a square and non stochastic matrix that connects each productive unit with its nearest neighbours. In this case, we considered, as neighbourhood criterion, the common border. Therefore, the wc f elements of this matrix, with c = f are unequal to zero if productive units have a common border and 0 otherwise. And wcc is equal to 0, ∀c, f ∈ C.

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2.1.2 Technology and distance functions Each productive unit c ∈ C transforms its corresponding input vector xc = (xc1 , p P into an output vector y = (y 1 , y 2 , . . . , y l ) ∈  L . Following Färe xc2 , . . . , xc ) ∈ + c c c c + et al. (1998), technology is represented by the production possibility set of feasible input–output combinations   p+l S = (x, y) ∈ + : x can produce y

(1)

S fulfills the standard assumptions of being non-empty, closed, comprehensive1 and convex. Furthermore, it is assumed that S satisfies constant returns to scale2 (Färe et al. 1998). For all y ∈ l+ , the input requirement sets are determined by   p L(y) = x ∈ + |(x, y) ∈ S

(2)

p

And, for all x ∈ + , the output possibility sets by   P(x) = y ∈ l+ |(x, y) ∈ S

(3)

According to Färe et al. 1998, L(y) is closed for all y and P(x) is closed for all x, and (x, y) ∈ S ⇔ x ∈ L(y) ⇔ y ∈ P(x)

(4)

Therefore, technology is completely characterised by the correspondence L : l+ → p p + or the correspondence,P : + → l+ , as well as by the set S.3 Based on these sets, an output distance function4 D : M → + is defined by: D(xc , yc ) = inf {θ > 0 : (xc , yc /θ ) ∈ S}

(5)

  p+l M = (xc , yc ) ∈ + |yc = 0 and (yc /θ ) ∈ P(xc ) for some θ > 0

(6)

θ

where

1 This assumption implies that technology satisfies the strong assumption of free disposability of outputs and inputs (see footnote 3). 2 Constant Returns to scale are defined as λS = S for all λ > 0. We consider this assumption due to the

problems in reconciling the concept of variable returns to scale with the components of Malmquist Index (Grifell-Tatjé and Lovell 1995; Angeriz et al. 2006). 3 Based on L(y) and P(x), technology verifies the assumption of free disposability of inputs if ∀x ∈ L(y) and x˜ ≥ x which implies that x˜ ∈ L(y), ∀y ∈ l+ and verifies the assumption of free disposability p of outputs if ∀y ∈ P(x) and y˜ ≤ y implies y˜ ∈ P(x), ∀x ∈ + . Notation: x˜ ≥ x if x˜i ≥ xi ∀i and y˜ ≤ y if y˜i ≥ yi ∀i. 4 A complementary analysis could be developed from the input perspective.

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To guarantee that the output distance function is well defined we needed to assume that p set P(x) is bounded for all x ∈ + . This assumption and the closeness assumption for S assure that the output possibility set is compact (Färe et al. 1998). Besides, p it is assumed that P(x) is a convex set for all x ∈ + and the correspondence p l P : + → + is continuous (Färe et al. 1994). 2.2 Spatial technology and spatial distance functions For each productive unit c ∈ C we define its spatial input and spatial output vectors applying a first order contiguity matrix W ⎛ W xc = ⎝





wcj x 1j ,



wcj x 2j , . . . ,

j∈NC

j∈NC

j∈NC







⎛ W yc = ⎝

j∈NC

wcj y 1j ,

wcj y 2j , . . . ,

j∈NC

⎞ p wcj x j ⎠

(7)

⎞ wcj y lj ⎠

(8)

j∈NC

where Nc denotes the set of nearest neighbours of each unit c, determined by W . Based on expressions (7) and (8), the spatial production possibility set is defined as a linear transformation of S where the elements of W are non-negative:   SS = (W x, W y) ∈  p+l |(x, y) ∈ S and W x can produce W y

(9)

Proposition 1 SS is non empty, closed, comprehensive, convex, and satisfies constant returns to scale Proof Let f : S → SS be the linear transformation defined by f (x, y) = (W x, W y), (i) Since S is non empty, there exist (x0 , y0 ) ∈ S therefore f (x0 , y0 ) = (W x0 , W y0 ) ∈ SS (ii) A linear transformation of a closed set is closed. (iii) Free disposability of inputs: Let x ∈ L(y), if x˜ ≥ x then implies W x˜ ≥ W x and the assumption of free disposability of inputs of S guarantees that W x ∗ ∈ SS. (iv) A linear transformation of a convex set is convex. (v) ∀λ > 0, λSS = λ f (S) = f (λS) = f (S) = SS. For all W y ∈ l+ , the spatial input requirement sets are defined by   p L(W y) = W x ∈ + |(x, y) ∈ S

(10)

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and, for all W x ∈ + , we determine the spatial output possibility sets by   P(W x) = W y ∈ l+ |(x, y) ∈ S

(11) 

Proposition 2 L(W y) is closed for all W y and P(W x) is closed for all W x, and (W x, W y) ∈ SS ⇔ W x ∈ L(W y) ⇔ W y ∈ P(W x)

(12)

Proof Immediately, considering that L(W y) and P(W x) are non negative linear transformations of L(y) and P(x) respectively. According to Proposition 2, and considering previous non spatial concepts, spatial p technology is completely characterised either by the correspondence L : l+ → + p or by correspondence P : + → l+ , as well as by set SS. Based on these sets, a spatial output distance function S D : H → + is defined by S D(W xc , W yc ) = inf {δ > 0 : (W xc , W yc /δ) ∈ SS} δ

(13)

where   p+l H = (W xc , W yc ) ∈ + /W yc = 0 and (W yc /θ ) ∈ P(W x) for some θ > 0 (14)  p

Proposition 3 P(W x)is bounded and convex for all W x ∈ + and the corresponp dence P : + → l+ is continuous. Proof Immediately, according to the definition of P(W x) and the previous assumptions of P(x) (Färe et al. 1994, 1998). The properties of P(W x) assure that spatial output distance function (13) is well defined in this case. 

2.3 Proposal of a spatial efficiency index Production theory is based on the assumption that the behaviour of production units is optimal. Under perfect competition conditions, a productive unit will reach the most efficient production point satisfying the profit maximization objective. However, this assumption is highly restrictive because not all production units succeed in working at an optimal level. The Malmquist Index analyzes the degree to which production units fail to be optimized. The index is consists of two components (Färe et al. 1994), one measuring the change in technical efficiency whereas the other measures the

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Fig. 1 Interpreting a spatial efficiency index above one

technology frontier shift. Our proposed index is based on the first efficiency component. This component allows us to determine the degree of efficiency for each productive unit under the assumption that there is no technological change. Traditionally, this index is built under the assumption that productive units are independent among them and, therefore, efficiency values for each of them are only determined by their particular characteristics. This is an unrealistic assumption. In order to overcome this, we present a specific Spatial Efficiency Index which integrates the spatial factor into its own definition. According to our proposal, the efficiency of each productive unit is going to be measured in relation to its closer environment. To build the index we assume constant returns to scale. For each productive unit c ∈ C, the Spatial Efficiency Index presents the following expression: S E P(xc , yc , W xc , W yc ) =

S D(W xc , W yc ) D(xc , yc )

(15)

Expression (15) compares the closeness of each unit c to its own production possibility set (S) to the closeness of the mean value of the neighbour units to the spatial production possibility set (SS). A value above one for this term means that, in average, neighbour units are more efficient than unit c, considering different technologies. A value under one indicates the opposite. To illustrate this concept we give a graphical example. Let’s consider any productive unit c which transforms its corresponding input vector xc ∈ 1+ into an output yc ∈ 1+ . Using a first order contiguity matrix, W , spatial input and the spatial output   are determined by W xc = ( j∈Nc wcj x 1j ) and W yc = ( j∈Nc wcj y 1j ), respectively, where Nc includes the nearest neighbours of c defined by W . Considering technology sets S and SS and both input–output combinations, a situation in which neighbour xc ,W yc ) > 1) is showed in Fig. 1 and a situnits are more efficient than unit c (i.e. S D(W D(xc ,yc ) uation in which neighbour units are less efficient than the unit c (i.e. is showed in Fig. 2.

S D(W xc ,W yc ) D(xc ,yc )

< 1)

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Fig. 2 Interpreting a spatial efficiency index below one

In Fig. 1, the spatial input–output combination (W xc , W yc ) is closer to the spatial technology SS than the input–output combination (xc , yc ) to S, keeping input level constant. Therefore, neighbour units are more efficient than unit c. In Fig. 2, the input–output combination (xc , yc ) is closer to the technology set S than the spatial input–output combination (W xc , W yc ) to SS, keeping input level constant. In this case, neighbour units are less efficient than unit c.

2.4 Computing distance functions through data envelopment analysis (DEA) To calculate the distance functions of expression (15) we solved several linear programming problems through DEA techniques. These procedures were used to compute the traditional Malmquist Index (Färe et al. 1998). Let c = 1, . . . , C be productive units in a given period, transforming an input vector p P into an output vector y = (y 1 , y 2 , . . . , y l ) ∈  L . xc = (xc1 , xc2 , . . . , xc ) ∈ + c c c c + A DEA technology may be defined as: S = (x, y) :

C 

λc ycl ≥ y l , l = 1, . . . , L

c=1 C 

p

λc x c ≤ x p ,

p = 1, . . . , P

c=1

λc ≥ 0; c = 1, . . . , C

and a spatial technology may be determined as:

123

(16)

A spatial efficiency index proposal

SS =

⎧ ⎨ ⎩

(W x, W y) :

361

C 

λc

c=1 C 

λi

c=1



wcj y lj

≥

wcj y lj , l = 1, . . . , L

c=1 j∈Nc

j∈Nc



C  

p

wcj x j ≤

C 



p

wcj x j ,

p = 1, . . . , P

c=1 j∈Nc

j∈Nc

λc ≥ 0; c = 1, . . . , C

⎫ ⎬

(17)

⎭

According to the corresponding technologies S and SS, the distance functions and the spatial distance functions are computed as: D(xc , yc ) = max θ C  λc ycl ≥ θ ycl , s.t. c=1 C 

p

p

λc x c ≤ x c ,

l = 1, . . . , L (18)

p = 1, . . . , P

c=1 λc ≥

0, c = 1, . . . , C S D(W xc , W yc ) = max δ C    λc wcj y lj ≥ δ wcj y lj , s.t. c=1 C 



c=1

j∈Nc

λc

j∈Nc

wcj x lj ≤

l = 1, . . . , L

j∈Nc



j∈Nc

p

wcj x j ,

p = 1, . . . , P

(19)

λc ≥ 0, c = 1, . . . , C

3 An empirical analysis of the Spanish productive system Spanish economy is characterised by low productivity. Between 1996 and 2006 Spanish productivity per employee decreased at an average rate of −0.44%, while EU15 countries experienced a rise in productivity at an average rate of 1.08% (Maroto and Cuadrado 2008). More specifically, in 2006 the manufacturing sector’s productivity was around 24.3 euros per employee, far from the level in such countries as Ireland, Belgium and Germany. The manufacturing sectors that have suffered the highest decrease in productivity since 1995 are the traditional ones: textile activity (−1.59%) and wood and cork activity (−0.44%). Spanish traditional sectors are labour-intensive, use low technology and do not require high qualified workers (Fonfría 2004). Another characteristic of Spanish Manufacturing productivity is the high regional disparity. According to previous descriptive studies (Zozaya 2007) high productivity regions are located in the north of Spain while low productivity regions are located in the south (Fig. 3). This regional distribution of productivity is a consequence of the industrial distribution of firms. In this sense, the high productivity regions are characterised by a larger number of high-tech firms. Consequently, the higher productivity in those

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Fig. 3 Regional distribution of industrial productivity in Spain, year 2006. Source: Own elaboration with GeoDa software

locations is caused by the role played by high-tech industry. High-tech industry can be found in just a few provinces, whereas low-tech industry is more homogenously distributed in the nation (Maté et al. 2009a). This spatial behaviour justifies that recent empirical studies (Dall’Erba 2005; Maté et al. 2009b) have focussed on the Spanish productive system in order to analyse the relation between spatial factors and productivity. These contributions conclude that heterogeneity and interaction among regions are significant in order to assess Spanish productivity. We performed our empirical analysis of the Spanish productive system taking those references into account. 3.1 Data and variables 3.1.1 Data In order to apply the Spatial Efficiency Index we computed the proposed index on a sample of 23,251 industrial Small and Medium Enterprises (SMEs). This sample was obtained from the SABI5 (Sistema de Análisis de Balances Ibéricos) database between 2004–2006. SMEs in the sample were firms that employed fewer than 250 employees and had annual sales of less than 50 million euros or total assets of less than 43 million euros (European Union 2003; Commission Recommendation 2003/361/EC). The choice of Small and Medium Enterprises as units of analysis is explained by their fundamental role in Spanish productive systems. According to the “Directorio Central de Empresas” (DIRCE 2007) (Central Directory of Companies from the National Institute of Statistics), 99.8% of Spanish companies are SMEs. This pattern is similar for all Spanish provinces as shown in Table 1. In order to achieve more homogeneous productive units we select just industrial SMEs. The consideration of the industrial sector is justified by its creativity and the tow-effect which this sector exerts on the rest of sectors (Hulten 2001; Trullén 2006). Moreover, previous literature has highlighted the advantage of taking into account only this sector in productivity studies (Timmer and Los 2005). We differentiate high and low technological subsamples within the industrial sector. 5 SABI database (Sistema Anual de Balances Ibéricos) contains accounting information for 850,000 Spanish firms. SABI covers 31% of firms with more than nine employees, and more than 50% of larger firms.

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Table 1 Percentage of SMEs over the total at a provincial level Province

2004

2005

2006

Province

2004

2005

2006

Álava

99.941

99.934

99.929

Logroño

99.996

99.991

99.995

Albacete

99.988

99.988

99.987

Lugo

99.979

99.983

99.973

Alicante

99.989

99.986

99.987

Madrid

99.861

99.865

99.865

Almería

99.963

99.964

99.964

Málaga

99.978

99.982

99.981

Ávila

99.982

99.972

99.961

Murcia

99.954

99.955

99.955

Badajoz

99.987

99.987

99.989

Navarra

99.934

99.931

99.927

Baleares

99.956

99.960

99.961

Orense

99.969

99.969

99.968

Barcelona

99.924

99.921

99.922

Asturias

99.954

99.957

99.954

Burgos

99.963

99.966

99.965

Palencia

99.981

99.981

99.981

Cáceres

99.980

99.980

99.976

Las Palmas

99.961

99.952

99.954

Cádiz

99.976

99.980

99.979

Pontevedra

99.973

99.968

99.968

Castellón

99.935

99.942

99.941

Salamanca

99.978

99.973

99.972

CiudadReal

99.990

99.993

99.990

Cantabria

99.968

99.970

99.969

Córdoba

99.979

99.985

99.986

Sta. Cruz de Tenerif

99.969

99.972

99.970 99.981

LaCoruña

99.957

99.955

99.955

Segovia

99.973

99.972

Cuenca

99.993

99.993

99.992

Sevilla

99.967

99.969

99.968

Gerona

99.968

99.970

99.971

Soria

99.983

99.983

99.982

Granada

99.981

99.982

99.982

Tarragona

99.973

99.975

99.972

Guadalajara

99.992

99.990

99.990

Teruel

99.967

99.977

99.954

Guipúzcoa

99.956

99.963

99.963

Toledo

99.988

99.993

99.992

Huelva

99.988

99.988

99.988

Valencia

99.954

99.948

99.955

Huesca

99.981

99.981

99.973

Valladolid

99.955

99.956

99.961

Jaén

99.977

99.980

99.982

Vizcaya

99.930

99.928

99.922

León

99.975

99.981

99.984

Zamora

99.992

99.992

99.992

Lérida

99.988

99.988

99.987

Zaragoza

99.943

99.942

99.934

Source: Own elaboration according to DIRCE database

3.1.2 Variables According to previous studies (Bergström 2000; Wang and Yao 2002; Helfand and Levine 2004; Angeriz et al. 2006), the production value of each SME is measured by the added value generated by each unit. This variable is defined as the difference between the value of goods and services produced and the value of external acquisitions. Considering this definition and the accounting information available for each firm in the SABI database, the added value is calculated by deducting “consumption of goods in stock” and “other operating expenses” from turnover (Domench et al. 2008).6 This definition does not include incomes or expenditures related to financial investments or non-operating activities. SMEs combine labour and capital resources in order to produce. Two possible measures of labour as a variable were the available: number 6 Labour expenses and depreciation expenses are not deducted from the turnover of the firm, therefore,

both inputs receive the same treatment in the calculation of the added value.

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Table 2 Sample descriptive statistics at province level Minimum

Maximum

Mean

SD

Panel A: low technology subsample for 2006 (50 observations) (number of firms: 19,869) Output variable Added Value (e)

96,237.46

510,120.26

272,699.37

100,690.37

2,410,156.08

9,662,596.61

5,058,693.91

1,394,941.82

435,690.26

1,332,320.64

821,038.34

219,516.50

20.89

45.11

30.74

5.04

Input Variables Capital Total Assets (e) Labour Personnel expenditures (e) Number of employees

Panel B: high technology subsample for 2006 (50 observations) (number of firms: 3,382) Output variable Added Value (e)

−150,571.50

1,645,982.57

361,179.95

368,715.92

967,344.00

21,563,714.50

6,805,969.95

5,064,500.31

258,581.00

2,489,477.00

957,898.67

479,599.02

15.00

66.68

33.29

11.16

Input Variables Capital Total Assets (e) Labour Personnel expenditures (e) Number of employees

of employees (measured as annual-average full-time staff) and personnel expenditure. In order to measure capital as a variable we considered total (non-current and current) assets of the firm as proposed by OCDE (2001). Next, we calculate a provincial (NUTS III7 ) representative value of each input–output variable. To obtain this aggregation we compute the mean value for the SMEs of each province. This statistic was chosen because it is robust in comparison with other alternatives. Table 2 displays a descriptive analysis of the variables distinguishing between firms that belong to medium-high and high technological industries (High Technology Subsample) and firms belonging to low and medium-low technological industries (Low Technology Subsample). As expected, low technological firms are characterised by lower added value and lower capital levels. These firms belong to traditional activity sectors that have historically become the basis of the Spanish productive system (Fonfría 2004). The standard deviations associated to the variables for low technological firms are acceptable, but those for high technological firms are very high. This fact evidences the high heterogeneity among high technological firms. In order to achieve more homogeneous units we only consider low technological firms. Furthermore, bearing in mind our research 7 The Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard for referencing the administrative division of countries for statistical purposes. The standard was developed by the European Union, and thus only covers the member states of the EU in detail. Eurostat also devised a hierarchy for the 10 countries which joined the EU in 2004, but these are subject to minor changes (Eurostat 1999). NUTS divisions do correspond to administrative divisions in Spain.

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A spatial efficiency index proposal Table 3 Different specifications of DEA models

365 Variables

Models 1

2

3

X

X

X

X

X

X

Output variable Added value Input variables Capital Total Assets Labour Number of employees

X

Personnel expenditure

X X

Weight matriz Binary Distances

X

X X

goal, the decision to take into account only low-tech firms is also supported by the fact that high-tech industry is not homogeneously distributed in the nation, while low-tech industry is—as pointed out by Maté et al. (2009a). We then had to define a weight matrix W to compute the Spatial Efficiency Index. This matrix establishes neighbourhood relationships among productive units. There are several criteria to define this matrix. We consider two alternative specifications which are independent of economic criteria.8 Firstly, we assume a binary first order contiguity matrix as weight matrix W1 (Moran 1948; Geary 1954). This is a square and non stochastic matrix whose wc f elements with c = f are equal to one if productive units have a common border, and 0 otherwise. The wcc elements are equal to 0, ∀c, f ∈ C. Secondly, we introduce a W2 weight matrix which is defined in terms of distances (in kilometres). In this case, each wc f element with c = f corresponds to the distance between the centroids of provinces c and f . The wcc elements are equal to 0. 3.2 DEA estimation and sensitivity analysis In this section we compute the Spatial Efficiency Index for Spanish low technological industrial Small and Medium Enterprises (SMEs) and support the robustness of our results when alternative model specifications are proposed. Taking into account the existence of different proxies to measure a variable, Table 3 shows alternative DEA specifications. The three DEA models of Table 3 use a similar output variable. The difference among them is based on input variable labour, for which we propose two alternative measures: number of employees and personnel expenditure. We also introduce two weight matrix definitions in the specifications of these models: a first order binary contiguity matrix and a distance matrix. 8 According to Manski (1993) a weight matrix defined as a function of economic criteria could generate

endogeneity in the results.

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3.2.1 DEA estimation In this section we comment Spatial Efficiency Index results for Model 1. DEA scores and the corresponding Quantile Map representation, which are shown in Table 4. Provinces with an index value below one are characterised by efficiency advantages in low technological firms, in comparison with their neighbour provinces. Provinces with an index value above one exemplify the opposite case. In order to get a better understanding of the results, we show the Spatial Efficiency Index regional distribution through a Quartile Map (Table 4, Panel B). In this graph, the provinces with a clear colour are characterised by higher efficiency values than the average of their corresponding neighbours. These provinces show high specialization in traditional low technological industries (some of them are Cáceres, Cádiz, Jaén and Orense) (Zozaya 2007). According to previous literature (Gandoy and González 2004), productive units based on traditional industries present efficiency advantages because of their specific characteristics. For instance, a business culture based on strict rules and budget constraints, promoting efficiency through employees who conform and adhere to such rules, avoiding the waste of time and of economic resources (Levitt 2002; Miron et al. 2004). The study of the conditions that have allowed these provinces to be in a relatively advantageous situation could help policymakers to design better tailored policies, leading to a better use of resources in close provinces. An alternative policy to achieve local parities in efficiency could be based on promoting partnership among units. In this sense, policymakers could implement measures to foster the mobility of resources as a way to transfer productive advantages from firms located in provinces with higher efficiency values to the others. Thus, there will be an improvement in the competitive position of more disadvantaged provinces.

3.2.2 Sensitivity analyses In a DEA analysis, it is possible to find alternative proxies to measure input or output variables but the results should be independent of which specific proxy is chosen (Hughes and Yaisawarng 2004). To corroborate this characteristic, in this section, we performed some sensitivity tests. We begin with the analysis of some descriptive statistics. Table 5 shows Spatial Efficiency Index results for the three previously defined models (see Table 3 for more detail about model specifications). The average efficiency values range from 0.95 to 0.97. This value is practically equal considering models 1 and 2, but there is a small difference when model 3 is observed. The lowest value corresponds to Model 3. Considering the number of efficient provinces, we noticed that there are more efficient provinces when a binary weight matrix is considered than when we include a distance weight matrix. Although these results suggest slight differences between models, statistical tools are necessary to corroborate the robustness of Spatial Efficient Index results when different specifications are applied. Table 6 shows Spearman correlation coefficients between alternative models. The results indicate extremely high correlation coefficients which are statistically different

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Table 4 Spatial efficiency index for Spanish Provinces in 2006 Province

Spatial efficiency

Province

Spatial efficiency

Panel A: DEA scores Álava

0.9981

Logroño

0.9693

Albacete

0.9942

Lugo

1.0049 0.9841

Alicante

0.9747

Madrid

Almería

0.9366

Málaga

1.0356

Ávila

1.0061

Murcia

0.9648

Badajoz

0.9677

Navarra

0.9721

Baleares

0.9754

Orense

0.9468

Barcelona

0.9632

Asturias

0.9705

Burgos

0.9818

Palencia

0.9488

Cáceres

0.9127

Las Palmas

0.9646

Cádiz

0.9140

Pontevedra

0.9741 0.9952

Castellón

0.9736

Salamanca

CiudadReal

0.9512

Cantabria

1.0037

Córdoba

0.9990

Sta. Cruz de Tenerife

0.9845 0.9821

LaCoruña

0.9904

Segovia

Cuenca

0.9685

Sevilla

0.9298

Gerona

0.9626

Soria

0.9881

Granada

0.9835

Tarragona

0.9754

Guadalajara

0.9660

Teruel

0.9807

Guipúzcoa

0.9840

Toledo

0.9898

Huelva

0.9715

Valencia

0.9808

Huesca

0.9662

Valladolid

1.0001

Jaén

0.9230

Vizcaya

0.9591

León

0.9784

Zamora

0.9547

Lérida

1.0088

Zaragoza

0.9614

Results obtained under a R routine Panel B: quantile map

Software used: Geoda Results obtained for the year 2004 and 2005 are practically analogous

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Table 5 Summary of spatial efficiency index results for the year 2006

Table 6 Spearman correlation coefficients

**Significant at 1% (bilateral)

Model 1

2

3

Average

0.9734

0.9755

0.9598

Minimum

0.9127

0.9277

0.8925

Number of efficient provinces

6

7

4

Model

1

2

3

1

1

–

–

2

0.905**

1

–

3

0.870**

0.874**

1

Fig. 4 Quartile Map of the spatial efficiency index for alternative specifications in the year 2006

from zero at level of significance of 1%. This result corroborates that Spatial Efficiency Index values are positively related and stable across specifications. Finally, we present a comparison of individual Spatial Efficiency Index scores across models. To achieve a better interpretation of this analysis, we displayed the spatial distribution for the Spatial Efficiency Index using a Quartile Map for each model (see Fig. 4). Figure 4 confirms the similarity among models. The above analysis confirms the robustness of our results when different DEA specifications are considered. Therefore, we confirm the usefulness of our results in order to adopt adequate policy measures (Hughes and Yaisawarng 2004). 4 Conclusions This paper proposes a new spatial efficiency index. In order to design and calculate it, we use the concepts of technology set and distance functions, previously developed by the Malmquist Index perspective. Based on the nature of these definitions, we develop a spatial technology set and spatial distance functions through the introduction of a contiguity matrix, making sure that they show the traditional properties that characterise previous non-spatial concepts. Once these properties are substantiated, we introduce a new index which combines spatial and non spatial expressions. Thus, we establish an efficiency measure in relation to each local environment and determine the highest local efficiency inequalities. We carried out an empirical application on a sample of low technological Spanish industrial SMEs in 2006. This kind of firms is suitable for the analysis due to its importance within the Spanish productive system (Fonfría 2004), its low productivity

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and its homogeneous distribution in the nation (Maté et al. 2009a). Labour and capital were taken as input variables, and the added value of each firm was used as output variable. Using accounting information available from the firms, representative variable values for each province and for its local environment were calculated. The results indicate the existence of a local behaviour which is characterised by the existence of efficiency advantaged provinces in comparison to their close environment. These provinces are specialised in traditional low technological subsectors (Zozaya 2007). Besides, we applied sensitivity tests to corroborate the adequacy of our results when alternative proxies were considered to represent each input–output variable. This analysis confirmed the robustness of Spatial Efficiency Index results when alternative DEA specifications are computed. Our study has clear implications for researchers and policymakers. Researchers may well consider our proposal as an important contribution in the search for indexes that explicitly include the spatial factor and consider proximity relations. Future studies could go into the analysis of such spatial productivity indexes and their alternative components in depth. Policymakers have a responsibility of locating provinces with efficiency advantages in relation to their close environment. Once these provinces are identified, policymakers can use the results from this methodology to design better tailored policy measures to mitigate local disparities among provinces. Appendix See Fig. 5.

Fig. 5 Spanish provincial map

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