Firm efficiency, industry performance and the economy: three-way decomposition with an application to Andalusia


An economy may perform better because the firms become more efficient, the industries are better organized, or the allocation between industries is improved. In this paper we extend the literature on the measurement of industry efficiency (a decomposition in firm contributions and an organizational effect) to a third level, namely that of the economy. The huge task of interrelating the performance of an economy to industrial firm data is accomplished for Andalusia.

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  1. 1.

    Details and complete DEA descriptions may be found in specific books such as Charnes et al. (1995) or Cooper et al (2000). We included constant e T y ik (which is total production of firm i of industry k, Tis transposition) in the objective function; this monotonic transformation will prove useful for the price normalization.

  2. 2.

    In national accounts statistics, firms are classified within industries following national or international classifications (NAICS in US, Canada and Mexico; NACE in the EU; ISIC in the UN) according to its primary output. However, firms usually produce also other commodities than its primary output (secondary ones) taking advantage of idle resources, i.e. a milk producing farm, whose primary output is producing milk for a diary products firms can also make home-made cheese to sell in a small shop and even provide rural tourism services with idle spaces and labor-force. Formally \(\hat{y}_{jk}\) is the kth component of vector y and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y}_{jk}\) is the vector comprising all other components. Industry and commodity classifications usually match at the most common aggregation levels. i.e. NACE (Nomenclature of Activities of the European Union) matches CPA (Classification of Products by Activity) up to two digits level (breakdown in 89 industries/products): NACE: 01 Crop and animal production, hunting and related service activities CPA: 01 products of agriculture, hunting and related services. Then, the primary product of industry k is the product k. The difference among outputs made in (3) is conceptually different from that made by Lozano and Villa (2004) in their ‘hybrid’ centrally planned DEA models. In the present paper, the difference is made on the basis of the consideration of specific industrial output. On the other hand, the difference in Lozano and Villa (2004) among inputs in output-oriented models is based whether on being centrally planned or not. At first, one could think that both approaches are some how related, since we ‘plan’ to expand only the specific industrial output. However, it is to be highlighted that we include the difference among primary and secondary outputs on the oriented side of the model, this is, the output side, while these authors differentiate between both types of inputs (not outputs) on output oriented models and vice versa.

  3. 3.

    In the averaging procedure described in (5), the weighted harmonic mean is used because it is the most suitable procedure for averaging productivities or performances, Casas Sánchez and Santos Peñas (1996, pp. 78–81). Indeed, according to Proposition 1 of ten Raa (2011) the harmonic mean of disaggregated efficiencies is an upper bound for aggregated efficiency.

  4. 4.

    Note that 22 of them are industries with a single observation (see # k in Table 1), which are efficient by definition, and consequently such industries are also efficient. See “Appendix 2” for further details.

  5. 5.

    Exceltur (2005).


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Firstly, we thank Jose M. Rueda-Cantuche (IPTS-Joint Research Center EU and Pablo de Olavide University) for his essential help in building the database. We also thank the feedback from: Mikuláš Luptáčik (WU Vienna University of Economics and Business) during the 24th EURO Conference held in Lisbon, July 12th, 2010; Jan Oosterhaven, Erik Dietzenbacher and Bart Los (Groningen University), Geoffrey Hewings (REAL-University of Illinois) and Tobias Kronenberg (Forschungszentrum Jülich) during the 4th Spanish Input–Output Conference held in Madrid, September 2011. Additionally, the first author thanks Thanh Le Phuoc (University of Maastricht and MERIT) and very specially to Michael R. Bussiek (GAMS Corp.) for their precious cooperation in constructing the computational model; Mònica Serrano (University of Barcelona) and Michael C. Ferris (University of Wisconsin at Madison) for his help in early calculations. He also thanks CentER for hospitality and the financial support of Junta de Andalucía (Regional Government of Andalusia, Spain) and Pablo de Olavide University. We are grateful to two anonymous referees and an associate editor for critical comments that prompted many changes.

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Correspondence to Antonio F. Amores.

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Appendix 1: Proof

We demonstrate that the feasible set of Eq. (3) is larger or equal than the feasible set of the industry model presented in ten Raa (2011). The latter is characterized by the pair of constraints \(\sum\nolimits_{{j \in I_{k} }} {\lambda_{jk} x_{jk} } \le \sum\nolimits_{{i \in I_{k} }} {x_{ik} } ,\sum\nolimits_{{j \in I_{k} }} {\lambda_{jk} y_{jk} } \ge \sum\nolimits_{{i \in I_{k} }} {y_{ik} /\varepsilon_{k} }\). In Eq. (3) the first constraint is copied, as is the k-th component of the second constraint. The other components, \({{\sum\nolimits_{{j \in I_{k} }} {\lambda_{jk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y}_{jk} } \ge \sum\nolimits_{{i \in I_{k} }} {\lambda_{ik} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y}_{ik} } } \mathord{\left/ {\vphantom {{\sum\nolimits_{{j \in I_{k} }} {\lambda_{jk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y}_{jk} } \ge \sum\nolimits_{{i \in I_{k} }} {\lambda_{ik} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y}_{ik} } } {\varepsilon_{k} }}} \right. \kern-0pt} {\varepsilon_{k} }}\) are replaced by \(\sum\nolimits_{{j \in I_{k} }} {\lambda_{jk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y}_{jk} } \ge \sum\nolimits_{{i \in I_{k} }} {\lambda_{ik} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{y}_{ik} }\). Because ε k is an efficiency score between zero and one, this replacement is a relaxation.

Appendix 2: Data and computation details

The IEA (Instituto de Estadística de Andalucía—Regional Statistical Office of Andalusia) provided the cross-section inputs and outputs establishment data. These data were used for the elaboration of the Input–Output Andalusian Framework 2000—MIOAN00 (IEA 2006), which is the input–output table for Andalusia, based on the European System of Accounts (ESA-95) published by EUROSTAT (1996). IEA publishes two use tables, which differ by valuation. One is valued at purchasers’ prices and the other at basic prices, which is the same as the former but excluding trade and transport margins and net commodity taxes (see Viet 1994, p. 28). Trade and transport margins needs simply be reallocated from the commodities where they are included, at purchasers’ values, to the use matrix rows of trade and transport services. The make table is published exclusively at basic prices. The United Nations System of National Accounts (SNA) recommends basic values; production costs of good and services are measured before they are conveyed to the market for consumption so that the effects of tax and subsidy policies as well as of differences in types of economic transactions are isolated. Valuations are in basic prices. ten Raa and Rueda-Cantuche (2007) detail the procedure, including the assumed equality of margins and net commodity taxes between establishments in a given industry, consuming a given commodity.

There is a single capital type and a single labour type. Data for each establishment is obtained from capital consumption and total equivalent employees figures in the IO dataset. The capital endowment and the total labour force are the sum across establishments of their capital consumption and total equivalent employees figures.

Sales and purchases were classified into 86 commodities, then 88 inputs (86 commodities and 2 factor inputs: capital and labour) and 86 outputs were considered for 39,272 observations: 39,258 obtained by IEA from specific surveys done to build MIOAN00 while the other 14 observations represent data of sectors which data are obtained by IEA from different statistical sources when building MIOAN00, instead of by specifically surveying establishments: The list of this latter group of sectors is presented in Table 5.

Table 5 Industries not surveyed for MIOAN00

We do not claim that the data were measured without error. Particularly, basic prices building follow some usual assumptions. For a sensitivity analysis we refer to ten Raa (2005).

The results have been computed using a specifically designed and optimized GAMS v21.6 code that uses Cplex Solver. It has taken 10 h to run it on a laptop with a processor Pentium Centrino Duo 1.66 Ghz with 32-bits architecture and 2 Gb RAM.

Appendix 3: Industry/commodity classification

See Table 6.

Table 6 Industry classification in MIOAN00

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Amores, A.F., ten Raa, T. Firm efficiency, industry performance and the economy: three-way decomposition with an application to Andalusia. J Prod Anal 42, 25–34 (2014).

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  • Input–output
  • Industrial organization
  • Comparative advantage
  • Allocative efficiency
  • Efficiency decomposition

JEL Classification

  • L10
  • D24
  • O47