A DEA-based Evaluation of Latin America and the Caribbean Agricultural Environmental Performance under the Assumption of Natural and Managerial Efficiency

Revista de economía mundial 47, 2017, 157-178 ISSN: 1576-0162 A DEA-bAsED EvAluAtion of lAtin AmEricA AnD thE cAribbEAn AgriculturAl EnvironmEntAl PErformAncE unDEr thE AssumPtion of nAturAl AnD mAnAgEriAl EfficiEncy Evaluación dEl dEsEmpEño mEdioambiEntal dEl sEctor agrícola dE américa latina y El caribE bajo El supuEsto dE EficiEncia natural y gErEncial, utilizando dEa Juan-Javier Moreno-Moreno Doctoral Fellowship Programme of the IFHARU and SENACYT Government of Panama [email protected] Teresa Sanz-Diaz Universidad de Sevilla [email protected] Francisco Velasco-Morente Universidad de Sevilla [email protected] Carlos Ludena Senior Independent Consultant (PNUD, BID and FAO) [email protected] Recibido: marzo de 2017; aceptado: mayo de 2017. AbstrAct This study uses the Data Envelopment Analysis to measure the agricultural environmental performance under the assumption of natural and managerial disposability, but also evaluates the Returns to Scale and Damages to Scale in Latin America and the Caribbean during 2012. Seven agricultural variables are analyzed: animal feed, fertilizers, capital stock, labor, land, value of the gross agricultural production and agricultural emissions (CO2eq). The results found that the Bahamas, Brazil, Chile, Costa Rica and Mexico attained the maximum efficiency level in all efficiency measurements. These countries present the best performances compared with the other countries evaluated. Keywords: Agricultural; Environmental; Efficiency; DEA. rEsumEn Este estudio utiliza el Análisis de Envolventes de Datos para medir el desempeño medioambiental agrícola de América Latina y el Caribe; con suposición de natural y managerial disposability, y rendimientos a escala y daños a escala; durante 2012. Se utilizan siete variables: alimentación animal, fertilizantes, reserva de capital, mano de obra, tierra, valor de la producción agrícola y emisiones agrícolas (CO2eq). Los resultados muestran que las Bahamas, Brasil, Chile, Costa Rica y México alcanzaron el máximo nivel de eficiencia en todas las mediciones de eficiencia. Estos países presentan el mejor desempeño en comparación con el resto de países evaluados. Palabras clave: Agricultura; Medioambiente; Eficiencia; DEA. Clasificación JEL: C6, Q5, R5. Revista de economía mundial 47, 2017, 157-178 1. introDuction Climate change is a global issue that threatens the Latin America and the Caribbean (LAC) countries to a greater or lesser intensity and causes negative social, environmental and economic consequences (CAF, 2013). This phenomenon is brought about by global warming, due to the increase in greenhouse gas (GHG) emissions (Sánchez and Reyes, 2015). The new regulations present the need to adapt and mitigate climate change, as a major international challenge to reduce GHG emissions (Vergara et al., 2013; Espinosa and Mezouar, 2016). Most LACs have signed and ratified the Paris agreement on climate change (COP 21) and they are among the most ambitious seeking to increasingly reduce GHG emissions to create a green economy (UNFCCC, 2016). However, LACs have been implementing the UN Framework Convention on Climate Change (UNFCCC) guidelines for a number of years with different sectoral initiatives for climate change mitigation and adaptation, which are essential for governments, private sectors and the general population to create sustainable development (CAF, 2013). In LAC, one of the economic sectors most vulnerable to climate change is agriculture (CAF, 2013). So, LAC share a set of common environmental challenges to reduce their environmental footprint caused by unsustainable production practices (ECLAC, 2016; IDB, 2016). This new scenario raises the role of agriculture of the LAC countries particularly in sustainable development, to attain comprehensive cross-environmental policies to reconcile its agricultural productivity and its environmental performance (UNEP, 2010). A number of empirical studies have analyzed the agricultural productivity in LAC in recent periods. These studies have been mainly based on variables such as labor, land, fertilizers, irrigation, capital stock, livestock, crop and value of the gross agriculture production. Likewise, they have used diverse methods, such as Data Envelopment Analysis (DEA), the Malmquist Index, the stochastic frontier, and the Cobb-Douglas and Translog functions. Authors such as Pfeiffer (2003), who studied the agricultural productivity in the Andean Community between 1972 and 2000 must be highlighted. Also, Bharati and Fulginiti (2007) studied the agricultural productivity growth in ten Mercosur countries between 1972 and 2002. A feature of both studies is that they concluded that technical progress was the main contributor to agricultural 160 Juan-Javier Moreno-Moreno, María Teresa sanz-Díaz, Francisco velasco-MorenTe, carlos e. luDena productivity growth. They also concluded that farmers benefited from research and the introduction of new technological innovation, especially in countries such as Brazil, but the gap between countries in the use of better agricultural practices was widening. On the other hand, Hutchinson (2007) found that in many Caribbean countries the level of total agricultural productivity fell between 1961-2000, which meant that the Caribbean countries were becoming more dependent on food imports and exposed to global shocks in the food system. It is worth noting in this respect the important work by Solís et al. (2009) which focused on 639 farms in El Salvador and Honduras, and concluded that improvements in technical efficiency are financially beneficial to farm households while also contributing to environmental sustainability. Analysis by Ludena (2010) has found that agricultural productivity has grown in LAC between 1961-2007. Yet, the total factor has increased mostly because of technological changes rather than changes in efficiency. Additionally, Zúniga (2011) found gains in agricultural productivity in Central America between 1994 and 2010 due to technical changes but also because of efficiency changes. A prior significant study by Ebata (2011) analyzed the agricultural productivity in Central America and the Caribbean between 1976 and 2006. In particular, this study has included the CO2 emissions variable in its analysis. It found that over this period agricultural productivity increased when not taking into account own environmental variables such as emissions. Conversely, a lower growth of agricultural productivity was recorded between 1992 and 2006 when CO2 emissions are added to the analysis. Martín-Retortillo et al. (2014) found that the increases in efficiency made a rather modest contribution to enhancement of the production growth in LAC countries between 1950 and 2010. Above all, the use of capital explained the production growth. Further, Nin-Pratt et al. (2015) highlighted that the agricultural performance in LAC between 1980 and 2012 was marked by the differences in the use of inputs per worker, the quality of natural resources and the availability and use of new technologies. In fact, they concluded that when countries increase the inputs per worker, then the productivity improves more rapidly than in the countries with a limited access to capital and land. Trindade and Fulginiti (2015) have estimated agricultural productivity between 1969 and 2009 in South America. They concluded that the increase in agricultural productivity was brought about by new innovations in the sector. Also, they showed that spending on education and R&D in agriculture are important to understanding agricultural performance among the countries of this region. So, in order to support the analysis, Lachaud et al. (2015) has analyzed agricultural productivity growth in LACs, covering the period 1961-2012. This research has included climate variables in the study for a better understanding of agricultural performance in the region. The results showed that agricul- 161 Revista de economía mundial 47, 2017, 157-178 a dea-based evaluation of latin ameRica and the caRibbean agRicultuRal enviRonmental PeRfoRmance undeR the assumPtion of natuRal and manageRial efficiency tural production in LACs is very sensitive to labor, animal stock, and land. In addition, an extreme temperature has a negative and significant impact on agricultural production, while rainfall has a small but positive effect. Moreover, the combined effect of all climate variables considered in the study have, on average, an increasingly negative impact. In view of the above, Moreno-Moreno et al. (2017) studied the operational efficiency and environmental efficiency in the agriculture sector of eighteen LAC countries. They have proved that 6 countries fulfill the efficiency measures: Argentina, Belize, Bolivia, Brazil, Costa Rica and Mexico present the best agricultural performance. Lastly, Lachaud et al. (2015) and Moreno-Moreno et al. (2017) have concluded that new technology in agriculture will increase the ability to reduce pollution in agriculture but also achieve better agricultural performance. Thereupon, this empirical comparison has the intention of identifying the progress of each country toward accomplishing the important challenge: to reconcile its agricultural productivity and its environmental performance. This study applies DEA for environmental assessment through the natural disposability, managerial disposability and joint performance –which is the integration of natural and managerial disposability– of the agricultural sector of 25 LACs in 2012. As well, the types of Returns to Scale (RTS) and Damages to Scale (DTS) will be provided. Therefore, animal feed, fertilizers, capital stock, labor and land are considered as input variables and value of the gross agricultural production and agricultural emissions (CO2eq) as output variables. Based on the results, relevant knowledge will be provided to the governments so that they can work on strengthening environmental strategies, in such way as to adapt new technology to reduce pollution in order to create sustainable agriculture. The rest of this article is organized as follows. Section 2 explains the DEA methodology based on natural disposability, managerial disposability and joint performance. Section 3 defines the descriptive statistics. Section 4 analyzes the results obtained, comparing the situation of the LACs by indicating three types of efficiency, as well as the RTS and DTS. Lastly, the conclusions are in Section 5. 2. mEthoDology DEA is a mathematical technique that studies the relations between the inputs and outputs used in a production process in order to determine the levels of efficiency (Farrell, 1957). It is a nonparametric method first described by Charnes et al. (1978) to measure the relative efficiency of a set of Decision Making Units (DMU) and the subsequent formalization described by Banker et al. (1984). DEA has been applied in various fields of management science and operations research (Campos et al., 2016; Expósito et al., 2017 and Fernández-Serrano et al., 2017). In addition, it has been used to assess the environmental performance by several authors such as Scheel (2001); Färe et al. (2004); Zhou et al. (2008); 162 Juan-Javier Moreno-Moreno, María Teresa sanz-Díaz, Francisco velasco-MorenTe, carlos e. luDena Sueyoshi and Goto (2012; 2014); Halkos and Tzeremes (2014); Sanz-Díaz et al. (2017); Vlontzos and Pardalos (2017), who emphasize the benefit of DEA as an application for environmental management, which is a problem of undoubted relevance in the policies of any state. An important feature of DEA for environmental assessment is that it includes desirable outputs and undesirable outputs and its own production variables, which cannot be separated into an environmental analysis model of these features (Dios-Palomares et al., 2014). In this way, Sueyoshi and Goto (2012) have developed a non-radial and radial model of DEA for environmental assessments. This approach classifies the outputs into desirable (good) and undesirable (bad) outputs and establishes two concepts: natural disposability and managerial disposability. So, this study uses the DEA radial approach for the environmental assessment proposed by Sueyoshi and Goto (2014). An important feature of this approach is that it employs the DEA-RAM (Range-Adjusted Measure), first proposed by (Cooper et al., 1999) to incorporate the natural disposability and managerial disposability measurements into a unified treatment. Natural disposability: refers to a DMU seeking to achieve its efficiency by decreasing its directional vector of inputs in order to decrease the directional vector of undesirable outputs, as well as to increase the vector of desirable outputs as much as possible. Managerial disposability: a DMU’s search for reaching its efficiency by increasing the directional vector of inputs to decrease the directional vector of undesirable outputs, as well as to increase the vector of desirable outputs as much as possible. 2.1. nAturAl DisPosAbility In model (1), each j-th DMU j=1,…n, uses inputs Xj=(x_1j,…,xmj)T to produce desirable outputs Gj=(g_1j,…,g_sj)T and undesirable outputs Bj=(b1j,…,bhj)T. Furthermore, di x,i=1,…m, dr g,r=1,…,s and df b,f=1,…,h are all slack variables related to inputs, desirable outputs and undesirable outputs, respectively. λ=(λ1,…,λn)T are structural or intensity variables, which are unknown and are used for connecting the input and output vectors by a convex combination. R is the range resolute through the upper and lower bounds of inputs, desirable outputs and undesirable outputs, expressed by: The natural efficiency of the kth country is evaluated by the following radial model:   In model (1), each j-th DMU 𝑗𝑗 = 1, … 𝑛𝑛, uses inputs 𝑋𝑋(= (𝑥𝑥+(,…,𝑥𝑥,(). to produce desirable outputs𝐺𝐺(= (𝑔𝑔+(,…,𝑔𝑔2(). and undesirable outputs 𝐵𝐵(= (𝑏𝑏+(,…,𝑏𝑏5().. Furthermore,d7 8, 𝑖𝑖 = 1, … 𝑚𝑚,d; <, 𝑟𝑟 = 1, … , 𝑠𝑠 and d? @, 𝑓𝑓 = 1, … , ℎ are all slack variables related to inputs, desirable outputs and undesirable outputs, respectively. 𝜆𝜆 = (𝜆𝜆+,…,𝜆𝜆D). are structural or intensity variables, which are unknown and are used for connecting the input and output vectors by a convex combination. R is the range resolute through the upper and lower bounds of inputs, desirable outputs and undesirable outputs, expressed by: Ri x=(m+s+h)-1 max xij j=1,…,n - min xij j=1,…,n -1 ,  Rr g = (m + s + h)-1 max grj j=1, …, n - min grj j=1, …, n -1 and Rf b=(m+s+h)-1 max bfj j=1,…,n - min bfj j=1,…,n -1 . The natural efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj+di x n j=1 =xik i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s ,df b≥0 f=1,…,h and ξ:unrestricted (1) The natural efficiency is measured by: θ*=1- ξ*+ε Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 .   All slack variables resolved in the optimality of model (1) indicate the level of inefficiency. The model (1) has the following dual formulation: Min vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ 𝑠𝑠. 𝑡𝑡. vixij m i=1 - urgrj s r=1 + wfbfj h f=1 +σ≥0 j=1,…,n , ur  grk s r=1 + wfbfk = 1 h f=1 ,  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. (2) Where 𝜈𝜈7, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢;, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑛𝑛𝑎𝑎𝑤𝑤?, 𝑓𝑓 = 1, … , ℎ are all positive dual variables associated with the first three constraints from the model (1). The dual variable σ is obtained from the fourth equation of model (1). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.2. MEASUREMENT OF THE RTS BY USING SCSCS. (1) 163 Revista de economía mundial 47, 2017, 157-178 a dea-based evaluation of latin ameRica and the caRibbean agRicultuRal enviRonmental PeRfoRmance undeR the assumPtion of natuRal and manageRial efficiency The natural efficiency is measured by: All slack variables resolved in the optimality of model (1) indicate the level of inefficiency. The model (1) has the following dual formulation: Where νi,i=1,…,m;ur,r=1,…,s and wf,f=1,…,h are all positive dual variables associated with the first three constraints from the model (1). The dual variable is obtained from the fourth equation of model (1). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.2. mEAsurEmEnt of thE rts by using scscs According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the RTS unambiguously. Therefore, model (3) calculates the upper bound (τ *) and lower bound (τ *) of the dual variable (σ) and the influence ζk g from undesirable outputs as follows:The type of RTS is determined as follows:   In model (1), each j-th DMU 𝑗𝑗 = 1, … 𝑛𝑛, uses inputs 𝑋𝑋(= (𝑥𝑥+(,…,𝑥𝑥,(). to produce desirable outputs𝐺𝐺(= (𝑔𝑔+(,…,𝑔𝑔2(). and undesirable outputs 𝐵𝐵(= (𝑏𝑏+(,…,𝑏𝑏5().. Furthermore,d7 8, 𝑖𝑖 = 1, … 𝑚𝑚,d; <, 𝑟𝑟 = 1, … , 𝑠𝑠 and d? @, 𝑓𝑓 = 1, … , ℎ are all slack variables related to inputs, desirable outputs and undesirable outputs, respectively. 𝜆𝜆 = (𝜆𝜆+,…,𝜆𝜆D). are structural or intensity variables, which are unknown and are used for connecting the input and output vectors by a convex combination. R is the range resolute through the upper and lower bounds of inputs, desirable outputs and undesirable outputs, expressed by: Ri x=(m+s+h)-1 max xij j=1,…,n - min xij j=1,…,n -1 ,  Rr g = (m + s + h)-1 max grj j=1, …, n - min grj j=1, …, n -1 and Rf b=(m+s+h)-1 max bfj j=1,…,n - min bfj j=1,…,n -1 . The natural efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. x ij λ j +d i x n j=1 =x ik  i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g ≥0 r=1,…,s ,df b ≥0 f=1,…,h and ξ:unrestricted (1) The natural efficiency is measured by: θ*=1- ξ*+ε Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 .   All slack variables resolved in the optimality of model (1) indicate the level of inefficiency. The model (1) has the following dual formulation: Min vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ 𝑠𝑠. 𝑡𝑡. vixij m i=1 - urgrj s r=1 + wfbfj h f=1 +σ≥0 j=1,…,n , ur  grk s r=1 + wfbfk = 1 h f=1 ,  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. (2) Where 𝜈𝜈7, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢;, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑛𝑛𝑎𝑎𝑤𝑤?, 𝑓𝑓 = 1, … , ℎ are all positive dual variables associated with the first three constraints from the model (1). The dual variable σ is obtained from the fourth equation of model (1). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.2. MEASUREMENT OF THE RTS BY USING SCSCS.   In model (1), each j-th DMU 𝑗𝑗 = 1, … 𝑛𝑛, uses inputs 𝑋𝑋(= (𝑥𝑥+(,…,𝑥𝑥,(). to produce desirable outputs𝐺𝐺(= (𝑔𝑔+(,…,𝑔𝑔2(). and undesirable outputs 𝐵𝐵(= (𝑏𝑏+(,…,𝑏𝑏5().. Furthermore,d7 8, 𝑖𝑖 = 1, … 𝑚𝑚,d; <, 𝑟𝑟 = 1, … , 𝑠𝑠 and d? @, 𝑓𝑓 = 1, … , ℎ are all slack variables related to inputs, desirable outputs and undesirable outputs, respectively. 𝜆𝜆 = (𝜆𝜆+,…,𝜆𝜆D). are structural or intensity variables, which are unknown and are used for connecting the input and output vectors by a convex combination. R is the range resolute through the upper and lower bounds of inputs, desirable outputs and undesirable outputs, expressed by: Ri x=(m+s+h)-1 max xij j=1,…,n - min xij j=1,…,n -1 ,  Rr g = (m + s + h)-1 max grj j=1, …, n - min grj j=1, …, n -1 and Rf b=(m+s+h)-1 max bfj j=1,…,n - min bfj j=1,…,n -1 . The natural efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj+di x n j=1 =xik i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s ,df b≥0 f=1,…,h and ξ:unrestricted (1) The natural efficiency is measured by: θ*=1- ξ*+ε Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 .   All slack variables resolved in the optimality of model (1) indicate the level of inefficiency. The model (1) has the following dual formulation: Min vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ 𝑠𝑠. 𝑡𝑡. vixij m i=1 - urgrj s r=1 + wfbfj h f=1 +σ≥0 j=1,…,n , ur  grk s r=1 + wfbfk = 1 h f=1 ,  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. (2) Where 𝜈𝜈7, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢;, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑛𝑛𝑎𝑎𝑤𝑤?, 𝑓𝑓 = 1, … , ℎ are all positive dual variables associated with the first three constraints from the model (1). The dual variable σ is obtained from the fourth equation of model (1). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.2. MEASUREMENT OF THE RTS BY USING SCSCS. (2)   In model (1), each j-th DMU 𝑗𝑗 = 1, … 𝑛𝑛, uses inputs 𝑋𝑋(= (𝑥𝑥+(,…,𝑥𝑥,(). to produce desirable outputs𝐺𝐺(= (𝑔𝑔+(,…,𝑔𝑔2(). and undesirable outputs 𝐵𝐵(= (𝑏𝑏+(,…,𝑏𝑏5().. Furthermore,d7 8, 𝑖𝑖 = 1, … 𝑚𝑚,d; <, 𝑟𝑟 = 1, … , 𝑠𝑠 and d? @, 𝑓𝑓 = 1, … , ℎ are all slack variables related to inputs, desirable outputs and undesirable outputs, respectively. 𝜆𝜆 = (𝜆𝜆+,…,𝜆𝜆D). are structural or intensity variables, which are unknown and are used for connecting the input and output vectors by a convex combination. R is the range resolute through the upper and lower bounds of inputs, desirable outputs and undesirable outputs, expressed by: Ri x=(m+s+h)-1 max xij j=1,…,n - min xij j=1,…,n -1 ,  Rr g = (m + s + h)-1 max grj j=1, …, n - min grj j=1, …, n -1 and Rf b=(m+s+h)-1 max bfj j=1,…,n - min bfj j=1,…,n -1 . The natural efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj+di x n j=1 =xik i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s ,df b≥0 f=1,…,h and ξ:unrestricted (1) The natural efficiency is measured by: θ*=1- ξ*+ε Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 .   All slack variables resolved in the optimality of model (1) indicate the level of inefficiency. The model (1) has the following dual formulation: Min vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ 𝑠𝑠. 𝑡𝑡. v i x ij m i=1 - u r g rj  s r=1 + w f b fj h f=1 +σ≥0 j=1,…,n , ur  grk s r=1 + wfbfk = 1 h f=1 ,  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. (2) Where 𝜈𝜈7, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢;, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑛𝑛𝑎𝑎𝑤𝑤?, 𝑓𝑓 = 1, … , ℎ are all positive dual variables associated with the first three constraints from the model (1). The dual variable σ is obtained from the fourth equation of model (1). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.2. MEASUREMENT OF THE RTS BY USING SCSCS.   According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the RTS unambiguously. Therefore, model (3) calculates the upper bound 𝜏𝜏∗ and lower bound τ* of the dual variable (σ) and the influence ζ$ % from undesirable outputs as follows: Max/Minσ+ζk b  s.t.allconstraintsinboth(1)and(2)  ζk b  - wfbfk h f=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁$ L≥ 0.  (3) The type of RTS is determined as follows: a) Increasing RTS ↔ 𝜏𝜏*≥ τ* > 0: The proportion of desirable outputs is more than the increase in all factors. b) Constant RTS ↔ 𝜏𝜏 ≥ 0 ≥ τ*: The proportion of desirable outputs is the same proportion as the increase in all factors. c) Decreasing RTS ↔0> 𝜏𝜏*≥ τ*: The proportion of desirable outputs is less than the increase in all factors. 2.3. MANAGERIAL DISPOSABILITY. The managerial efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj  -di x n j=1 =xik  i=1,…,m , grjλj n j=1 -dr g  -ξgrk=grk  r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk  f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s anddf b≥0 f=1,…,h and ξ:unrestricted. (4) The explanation of all the variables and the criterion in model (4) are the same as for the model (1), with the sole change that the model (4) considers only single-sided input deviations −dT U. The managerial efficiency is measured by: θ*=1- 𝜉𝜉∗ + 𝜀𝜀 Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 . All slack variables resolved in the optimality of model (4) indicate the level of inefficiency. The dual formulation of the model (4) is as follows: Min- vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ (5) (3) 164 Juan-Javier Moreno-Moreno, María Teresa sanz-Díaz, Francisco velasco-MorenTe, carlos e. luDena a) Increasing : The proportion of desirable outputs is more than the increase in all factors. b) Constant : The proportion of desirable outputs is the same proportion as the increase in all factors. c) Decreasing : The proportion of desirable outputs is less than the increase in all factors. 2.3. mAnAgEriAl DisPosAbility The managerial efficiency of the kth country is evaluated by the following radial model: The explanation of all the variables and the criterion in model (4) are the same as for the model (1), with the sole change that the model (4) considers only single-sided input deviations -di x. The managerial efficiency is measured by: All slack variables resolved in the optimality of model (4) indicate the level of inefficiency. The dual formulation of the model (4) is as follows: Where νi,i=1,…,m; ur,r=1,…,s and wf,f=1,…,h are all dual variables associated with the first three constraints from the model (4). The dual variable σ is obtained from the fourth equation of model (4). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor.   According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the RTS unambiguously. Therefore, model (3) calculates the upper bound 𝜏𝜏∗ and lower bound τ* of the dual variable (σ) and the influence ζ$ % from undesirable outputs as follows: Max/Minσ+ζk b  s.t.allconstraintsinboth(1)and(2)  ζk b  - wfbfk h f=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁$ L≥ 0.  (3) The type of RTS is determined as follows: a) Increasing RTS ↔ 𝜏𝜏*≥ τ* > 0: The proportion of desirable outputs is more than the increase in all factors. b) Constant RTS ↔ 𝜏𝜏 ≥ 0 ≥ τ*: The proportion of desirable outputs is the same proportion as the increase in all factors. c) Decreasing RTS ↔0> 𝜏𝜏*≥ τ*: The proportion of desirable outputs is less than the increase in all factors. 2.3. MANAGERIAL DISPOSABILITY. The managerial efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj  -di x n j=1 =xik  i=1,…,m , grjλj n j=1 -dr g  -ξgrk=grk  r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk  f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s anddf b≥0 f=1,…,h and ξ:unrestricted. (4) The explanation of all the variables and the criterion in model (4) are the same as for the model (1), with the sole change that the model (4) considers only single-sided input deviations −dT U. The managerial efficiency is measured by: θ*=1- 𝜉𝜉∗ + 𝜀𝜀 Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 . All slack variables resolved in the optimality of model (4) indicate the level of inefficiency. The dual formulation of the model (4) is as follows: Min- vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ (5)   According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the RTS unambiguously. Therefore, model (3) calculates the upper bound 𝜏𝜏∗ and lower bound τ* of the dual variable (σ) and the influence ζ$ % from undesirable outputs as follows: Max/Minσ+ζk b  s.t.allconstraintsinboth(1)and(2)  ζk b  - wfbfk h f=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁$ L≥ 0.  (3) The type of RTS is determined as follows: a) Increasing RTS ↔ 𝜏𝜏*≥ τ* > 0: The proportion of desirable outputs is more than the increase in all factors. b) Constant RTS ↔ 𝜏𝜏 ≥ 0 ≥ τ*: The proportion of desirable outputs is the same proportion as the increase in all factors. c) Decreasing RTS ↔0> 𝜏𝜏*≥ τ*: The proportion of desirable outputs is less than the increase in all factors. 2.3. MANAGERIAL DISPOSABILITY. The managerial efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj  -di x n j=1 =xik  i=1,…,m , grjλj n j=1 -dr g  -ξgrk=grk  r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk  f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s anddf b≥0 f=1,…,h and ξ:unrestricted. (4) The explanation of all the variables and the criterion in model (4) are the same as for the model (1), with the sole change that the model (4) considers only single-sided input deviations −dT U. The managerial efficiency is measured by: θ*=1- 𝜉𝜉∗ + 𝜀𝜀 Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 . All slack variables resolved in the optimality of model (4) indicate the level of inefficiency. The dual formulation of the model (4) is as follows: Min- vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ (5)   According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the RTS unambiguously. Therefore, model (3) calculates the upper bound 𝜏𝜏∗ and lower bound τ* of the dual variable (σ) and the influence ζ$ % from undesirable outputs as follows: Max/Minσ+ζk b  s.t.allconstraintsinboth(1)and(2)  ζk b  - wfbfk h f=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁$ L≥ 0.  (3) The type of RTS is determined as follows: a) Increasing RTS ↔ 𝜏𝜏*≥ τ* > 0: The proportion of desirable outputs is more than the increase in all factors. b) Constant RTS ↔ 𝜏𝜏 ≥ 0 ≥ τ*: The proportion of desirable outputs is the same proportion as the increase in all factors. c) Decreasing RTS ↔0> 𝜏𝜏*≥ τ * : The proportion of desirable outputs is less than the increase in all factors. 2.3. MANAGERIAL DISPOSABILITY. The managerial efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj  -di x n j=1 =xik  i=1,…,m , grjλj n j=1 -dr g  -ξgrk=grk  r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk  f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s anddf b≥0 f=1,…,h and ξ:unrestricted. (4) The explanation of all the variables and the criterion in model (4) are the same as for the model (1), with the sole change that the model (4) considers only single-sided input deviations −dT U. The managerial efficiency is measured by: θ*=1- 𝜉𝜉∗ + 𝜀𝜀 Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 . All slack variables resolved in the optimality of model (4) indicate the level of inefficiency. The dual formulation of the model (4) is as follows: Min- vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ (5)   According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the RTS unambiguously. Therefore, model (3) calculates the upper bound 𝜏𝜏∗ and lower bound τ* of the dual variable (σ) and the influence ζ$ % from undesirable outputs as follows: Max/Minσ+ζk b  s.t.allconstraintsinboth(1)and(2)  ζk b  - wfbfk h f=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁$ L≥ 0.  (3) The type of RTS is determined as follows: a) Increasing RTS ↔ 𝜏𝜏*≥ τ* > 0: The proportion of desirable outputs is more than the increase in all factors. b) Constant RTS ↔ 𝜏𝜏 ≥ 0 ≥ τ*: The proportion of desirable outputs is the same proportion as the increase in all factors. c) Decreasing RTS ↔0> 𝜏𝜏*≥ τ*: The proportion of desirable outputs is less than the increase in all factors. 2.3. MANAGERIAL DISPOSABILITY. The managerial efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. x ij λ j  -d i x n j=1 =x ik  i=1,…,m , grjλj n j=1 -dr g  -ξgrk=grk  r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk  f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g ≥0 r=1,…,s anddf b ≥0 f=1,…,h and ξ:unrestricted. (4) The explanation of all the variables and the criterion in model (4) are the same as for the model (1), with the sole change that the model (4) considers only single-sided input deviations −dT U. The managerial efficiency is measured by: θ*=1- 𝜉𝜉∗ + 𝜀𝜀 Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 . All slack variables resolved in the optimality of model (4) indicate the level of inefficiency. The dual formulation of the model (4) is as follows: Min- vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ (5)   According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the RTS unambiguously. Therefore, model (3) calculates the upper bound 𝜏𝜏∗ and lower bound τ* of the dual variable (σ) and the influence ζ$ % from undesirable outputs as follows: Max/Minσ+ζk b  s.t.allconstraintsinboth(1)and(2)  ζk b  - wfbfk h f=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁$ L≥ 0.  (3) The type of RTS is determined as follows: a) Increasing RTS ↔ 𝜏𝜏*≥ τ* > 0: The proportion of desirable outputs is more than the increase in all factors. b) Constant RTS ↔ 𝜏𝜏 ≥ 0 ≥ τ*: The proportion of desirable outputs is the same proportion as the increase in all factors. c) Decreasing RTS ↔0> 𝜏𝜏*≥ τ*: The proportion of desirable outputs is less than the increase in all factors. 2.3. MANAGERIAL DISPOSABILITY. The managerial efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj  -di x n j=1 =xik  i=1,…,m , grjλj n j=1 -dr g  -ξgrk=grk  r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk  f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s anddf b≥0 f=1,…,h and ξ:unrestricted. (4) The explanation of all the variables and the criterion in model (4) are the same as for the model (1), with the sole change that the model (4) considers only single-sided input deviations −dT U. The managerial efficiency is measured by: θ*=1- 𝜉𝜉∗ + 𝜀𝜀 Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 . All slack variables resolved in the optimality of model (4) indicate the level of inefficiency. The dual formulation of the model (4) is as follows: Min- vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ (5) (4)   According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the RTS unambiguously. Therefore, model (3) calculates the upper bound 𝜏𝜏∗ and lower bound τ* of the dual variable (σ) and the influence ζ$ % from undesirable outputs as follows: Max/Minσ+ζk b  s.t.allconstraintsinboth(1)and(2)  ζk b  - wfbfk h f=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁$ L≥ 0.  (3) The type of RTS is determined as follows: a) Increasing RTS ↔ 𝜏𝜏*≥ τ* > 0: The proportion of desirable outputs is more than the increase in all factors. b) Constant RTS ↔ 𝜏𝜏 ≥ 0 ≥ τ*: The proportion of desirable outputs is the same proportion as the increase in all factors. c) Decreasing RTS ↔0> 𝜏𝜏*≥ τ*: The proportion of desirable outputs is less than the increase in all factors. 2.3. MANAGERIAL DISPOSABILITY. The managerial efficiency of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj  -di x n j=1 =xik  i=1,…,m , grjλj n j=1 -dr g  -ξgrk=grk  r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk  f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n ,di x≥0 i=1,…,m , dr g≥0 r=1,…,s anddf b≥0 f=1,…,h and ξ:unrestricted. (4) The explanation of all the variables and the criterion in model (4) are the same as for the model (1), with the sole change that the model (4) considers only single-sided input deviations −dT U. The managerial efficiency is measured by: θ*=1- 𝜉𝜉∗ + 𝜀𝜀 Ri xdi x* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 . All slack variables resolved in the optimality of model (4) indicate the level of inefficiency. The dual formulation of the model (4) is as follows: Min- vi  xik m i=1 - ur  grk s r=1 + wfbfk h f=1 +σ (5) (5)   𝑠𝑠. 𝑡𝑡. - vixij m i=1 - urgrj s r=1 + wfbfj h f=1 +σ≥0 j=1,…,n , u r  g rk s r=1 + w f b fk h f=1 = 1  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. Where 𝜈𝜈E, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢J, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑤𝑤P, 𝑓𝑓 = 1, … , ℎare all dual variables associated with the first three constraints from the model (4). The dual variable 𝜎𝜎 is obtained from the fourth equation of model (4). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.4. MEASUREMENT OF THE DTS BY USING SCSCS. According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the DTS unambiguously. Therefore, model (6) can get the upper bound 𝜋𝜋∗ and lower bound 𝜋𝜋∗ of the dual variable (σ) and the influence ζk g from desirable outputs as follows: Max/Minσ-ζk Z  s.t.allconstraintsinboth(4)and(5) ζk g  - urgrk s r=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = - vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁c Z≥ 0.   (6) The type of DTS is determined as follows: a) Increasing DTS ↔ 𝜋𝜋*≥ π* > 0: The proportion of undesirable outputs is more than the increase in all factors. b) Constant DTS ↔𝜋𝜋*≥ 0 ≥ π*: The proportion of undesirable outputs is the same proportion as the increase in all factors. c) Decreasing DTS ↔0> 𝜋𝜋*≥ π*: The proportion of undesirable outputs is less than the increase in all factors. 2.5. JOINT PERFORMANCE: INTEGRATION OF NATURAL AND MANAGERIAL DISPOSABILITY. The joint performance of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x+ +di xm i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj-di x+ n j=1 +di x- =xik  i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n , (7) 165 Revista de economía mundial 47, 2017, 157-178 a dea-based evaluation of latin ameRica and the caRibbean agRicultuRal enviRonmental PeRfoRmance undeR the assumPtion of natuRal and manageRial efficiency 2.4. mEAsurEmEnt of thE Dts by using scscs According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the DTS unambiguously. Therefore, model (6) can get the upper bound (π*) and lower bound (π*) of the dual variable (σ) and the influence ζ k g from desirable outputs as follows: The type of DTS is determined as follows: a) Increasing DTS The proportion of undesirable outputs is more than the increase in all factors. b) Constant DTS The proportion of undesirable outputs is the same proportion as the increase in all factors. c) Decreasing DTS The proportion of undesirable outputs is less than the increase in all factors. 2.5. Joint PErformAncE: intEgrAtion of nAturAl AnD mAnAgEriAl DisPosAbility. The joint performance of the kth country is evaluated by the following radial model: The difference of model (7) compared with model (1) and model (4) is that the slack variable related with the inputs is divided into two types: di x+ and di x- and which symbolize the slacks of inputs under managerial disposability and natural disposability, respectively. The joint performance is measured by:   𝑠𝑠. 𝑡𝑡. - vixij m i=1 - urgrj s r=1 + wfbfj h f=1 +σ≥0 j=1,…,n , ur  grk s r=1 + wfbfk h f=1 = 1  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. Where 𝜈𝜈E, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢J, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑤𝑤P, 𝑓𝑓 = 1, … , ℎare all dual variables associated with the first three constraints from the model (4). The dual variable 𝜎𝜎 is obtained from the fourth equation of model (4). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.4. MEASUREMENT OF THE DTS BY USING SCSCS. According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the DTS unambiguously. Therefore, model (6) can get the upper bound 𝜋𝜋∗ and lower bound 𝜋𝜋∗ of the dual variable (σ) and the influence ζk g from desirable outputs as follows: Max/Minσ-ζk Z  s.t.allconstraintsinboth(4)and(5) ζk g  - urgrk s r=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = - v i x ik m i=1 - u r g rk s r=1 + w f b fk h f=1 +σ, 𝜁𝜁c Z≥ 0.   (6) The type of DTS is determined as follows: a) Increasing DTS ↔ 𝜋𝜋*≥ π* > 0: The proportion of undesirable outputs is more than the increase in all factors. b) Constant DTS ↔𝜋𝜋*≥ 0 ≥ π*: The proportion of undesirable outputs is the same proportion as the increase in all factors. c) Decreasing DTS ↔0> 𝜋𝜋*≥ π*: The proportion of undesirable outputs is less than the increase in all factors. 2.5. JOINT PERFORMANCE: INTEGRATION OF NATURAL AND MANAGERIAL DISPOSABILITY. The joint performance of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x+ +di xm i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj-di x+ n j=1 +di x- =xik  i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n , (7) (6)   𝑠𝑠. 𝑡𝑡. - vixij m i=1 - urgrj s r=1 + wfbfj h f=1 +σ≥0 j=1,…,n , ur  grk s r=1 + wfbfk h f=1 = 1  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. Where 𝜈𝜈E, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢J, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑤𝑤P, 𝑓𝑓 = 1, … , ℎare all dual variables associated with the first three constraints from the model (4). The dual variable 𝜎𝜎 is obtained from the fourth equation of model (4). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.4. MEASUREMENT OF THE DTS BY USING SCSCS. According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the DTS unambiguously. Therefore, model (6) can get the upper bound 𝜋𝜋∗ and lower bound 𝜋𝜋∗ of the dual variable (σ) and the influence ζk g from desirable outputs as follows: Max/Minσ-ζk Z  s.t.allconstraintsinboth(4)and(5) ζk g  - urgrk s r=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = - vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁c Z≥ 0.   (6) The type of DTS is determined as follows: a) Increasing DTS ↔ 𝜋𝜋* ≥ π * > 0 : The proportion of undesirable outputs is more than the increase in all factors. b) Constant DTS ↔𝜋𝜋*≥ 0 ≥ π*: The proportion of undesirable outputs is the same proportion as the increase in all factors. c) Decreasing DTS ↔0> 𝜋𝜋*≥ π*: The proportion of undesirable outputs is less than the increase in all factors. 2.5. JOINT PERFORMANCE: INTEGRATION OF NATURAL AND MANAGERIAL DISPOSABILITY. The joint performance of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x+ +di xm i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj-di x+ n j=1 +di x- =xik  i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n , (7)   𝑠𝑠. 𝑡𝑡. - vixij m i=1 - urgrj s r=1 + wfbfj h f=1 +σ≥0 j=1,…,n , ur  grk s r=1 + wfbfk h f=1 = 1  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. Where 𝜈𝜈E, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢J, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑤𝑤P, 𝑓𝑓 = 1, … , ℎare all dual variables associated with the first three constraints from the model (4). The dual variable 𝜎𝜎 is obtained from the fourth equation of model (4). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.4. MEASUREMENT OF THE DTS BY USING SCSCS. According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the DTS unambiguously. Therefore, model (6) can get the upper bound 𝜋𝜋∗ and lower bound 𝜋𝜋∗ of the dual variable (σ) and the influence ζk g from desirable outputs as follows: Max/Minσ-ζk Z  s.t.allconstraintsinboth(4)and(5) ζk g  - urgrk s r=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = - vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁c Z≥ 0.   (6) The type of DTS is determined as follows: a) Increasing DTS ↔ 𝜋𝜋*≥ π* > 0: The proportion of undesirable outputs is more than the increase in all factors. b) Constant DTS ↔𝜋𝜋*≥ 0 ≥ π * : The proportion of undesirable outputs is the same proportion as the increase in all factors. c) Decreasing DTS ↔0> 𝜋𝜋*≥ π*: The proportion of undesirable outputs is less than the increase in all factors. 2.5. JOINT PERFORMANCE: INTEGRATION OF NATURAL AND MANAGERIAL DISPOSABILITY. The joint performance of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x+ +di xm i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj-di x+ n j=1 +di x- =xik  i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n , (7)   𝑠𝑠. 𝑡𝑡. - vixij m i=1 - urgrj s r=1 + wfbfj h f=1 +σ≥0 j=1,…,n , ur  grk s r=1 + wfbfk h f=1 = 1  vi≥εRi x  i=1,…,m , ur≥εRr g  r=1,…,s , wf≥εRf b  f=1,…,h , σ:unrestricted. Where 𝜈𝜈E, 𝑖𝑖 = 1, … , 𝑚𝑚; 𝑢𝑢J, 𝑟𝑟 = 1, … , 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑤𝑤P, 𝑓𝑓 = 1, … , ℎare all dual variables associated with the first three constraints from the model (4). The dual variable 𝜎𝜎 is obtained from the fourth equation of model (4). The dual variables show the level of increase in inefficiency due to a unit increase in each production factor. 2.4. MEASUREMENT OF THE DTS BY USING SCSCS. According to Charnes et al. (1989) and Sueyoshi and Goto (2014), the Strong Complementary Slackness Conditions (SCSCs) have been incorporated to obtain the DTS unambiguously. Therefore, model (6) can get the upper bound 𝜋𝜋∗ and lower bound 𝜋𝜋∗ of the dual variable (σ) and the influence ζk g from desirable outputs as follows: Max/Minσ-ζk Z  s.t.allconstraintsinboth(4)and(5) ζk g  - urgrk s r=1 =0, ξ+ε Ri xdi x m i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  = - vixik m i=1 - urgrk s r=1 + wfbfk h f=1 +σ, 𝜁𝜁c Z≥ 0.   (6) The type of DTS is determined as follows: a) Increasing DTS ↔ 𝜋𝜋*≥ π* > 0: The proportion of undesirable outputs is more than the increase in all factors. b) Constant DTS ↔𝜋𝜋*≥ 0 ≥ π*: The proportion of undesirable outputs is the same proportion as the increase in all factors. c) Decreasing DTS ↔ 0 > 𝜋𝜋* ≥ π *: The proportion of undesirable outputs is less than the increase in all factors. 2.5. JOINT PERFORMANCE: INTEGRATION OF NATURAL AND MANAGERIAL DISPOSABILITY. The joint performance of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x+ +di xm i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj-di x+ n j=1 +di x- =xik  i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n , (7)   Maxξ+ε Ri xdi x+ +di xm i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. xijλj-di x+ n j=1 +di x- =xik  i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n , di x+ ≥0 i=1,…,m , di x- ≥0 i=1,…,m , dr g≥0 r=1,…,s , df b≥0 f=1,…,h and 𝜉𝜉: 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢. (7) The difference of model (7) compared with model (1) and model (4) is that the slack variable di x related with the inputs is divided into two types: di x+ and di xwhich symbolize the slacks of inputs under managerial disposability and natural disposability, respectively. The joint performance is measured by: θ*=1- ξ*+ε Ri xdi x+* +di x-* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 .  Where, all slack variables are the inefficiency levels and are resolute for the optimality of model (7). 3. DATA. The variables on the agriculture sector were obtained from the study of Nin-Pratt et al. (2015) and FAOSTAT (2016) for the year 2012. The analysis includes 25 LAC countries, divided by sub-regions (see Table 1). As mentioned above this study uses five conventional agricultural input variables: Animal feed commodities, expressed in metric tons of maize equivalents. Fertilizers refers to the quantity of nitrogen (N), phosphate (P205) and potash (K20), expressed in metric tons. Capital stock that includes (1) crop capital and (2) livestock capital, expressed at constant prices of 2005 as base year. Labor is the total economically active population in agriculture in thousands of people. Land involves land under temporary crops, temporary meadows for mowing or pasture, land under market and kitchen gardens, land that is temporarily fallow, land cultivated with permanent crops, and pasture land in thousands of hectares. Table 1: Agricultural variables for 2012. Country Animal feed Fertilizers Capital Stock Labor Land Agri. production Agri. emissions Andean Community: Bolivia 2091 47807 10807 2056 37596 3732400 23342 Colombia 14051 755150 112632 3484 42618 14014336 52263 Ecuador 7872 249324 21184 1219 7507 7335057 13734 Peru 4363 330161 24533 3728 24332 9613351 23450 Venezuela 4071 391819 30593 698 21600 7234275 35661   The joint performance of the kth country is evaluated by the following radial model: Maxξ+ε Ri xdi x+ +di xm i=1 + Rr gdr g s r=1 + Rf bdf b h f=1  s.t. x ij λ j -d i x+ n j=1 +d i x- =x ik  i=1,…,m , grjλj n j=1 -dr g-ξgrk=grk r=1,…,s , bfjλj n j=1 +df b+ξbfk=bfk f=1,…,h , λj n j=1 =1, λj≥0j=1,…,n , di x+ ≥0 i=1,…,m , di x- ≥0 i=1,…,m , dr g≥0 r=1,…,s , df b ≥0 f=1,…,h and 𝜉𝜉: 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢. (7) The difference of model (7) compared with model (1) and model (4) is that the slack variable di x related with the inputs is divided into two types: di x+ and di xwhich symbolize the slacks of inputs under managerial disposability and natural disposability, respectively. The joint performance is measured by: θ*=1- ξ*+ε Ri xdi x+* +di x-* m i=1 + Rr gdr g* + Rf bdf b* h f=1 s r=1 .  Where, all slack variables are the inefficiency levels and are resolute for the optimality of model (7). 3. DATA. The variables on the agriculture sector were obtained from the study of Nin-Pratt et al. (2015) and FAOSTAT (2016) for the year 2012. The analysis includes 25 LAC countries, divided by sub-regions (see Table 1). As mentioned above this study uses five conventional agricultural input variables: Animal feed commodities, expressed in metric tons of maize equivalents. Fertilizers refers to the quantity of nitrogen (N), phosphate (P205) and potash (K20), expressed in metric tons. Capital stock that includes (1) crop capital and (2) livestock capital, expressed at constant prices of 2005 as base year. Labor is the total economically active population in agriculture in thousands of people. Land involves land under temporary crops, temporary meadows for mowing or pasture, land under market and kitchen gardens, land that is temporarily fallow, land cultivated with permanent crops, and pasture land in thousands of hectares. Table 1: Agricultural variables for 2012. Country Animal feed Fertilizers Capital Stock Labor Land Agri. production Agri. emissions Andean Community: Bolivia 2091 47807 10807 2056 37596 3732400 23342 Colombia 14051 755150 112632 3484 42618 14014336 52263 Ecuador 7872 249324 21184 1219 7507 7335057 13734 Peru 4363 330161 24533 3728 24332 9613351 23450 172 Juan-Javier Moreno-Moreno, María Teresa sanz-Díaz, Francisco velasco-MorenTe, carlos e. luDena sub-regions need to continue their efforts to enhance their efficiency scores by using technological innovation. 4.1. mEAsurEmEnt of rts AnD Dts It is important to emphasize that the implications of RTS and DTS are completely opposite, however the existence of DTS depends on RTS. That is, the combination of RTS and DTS provides a strategic direction on how to enhance efficiency scores by controlling the agricultural activity size, and indicates if a country needs to introduce technological innovation to reduce pollution. So, Table 5 shows that Argentina, Bahamas, Barbados, Belize, Bolivia, Brazil, Costa Rica, Jamaica, Nicaragua, Paraguay, Suriname, Trinidad and Tobago and Uruguay belong to constant RTS. Conceptually, they may maintain their agricultural activity’s current size to keep their operational performance. Meanwhile, Chile, Colombia, the Dominican Republic, Ecuador, El Salvador, Guatemala, Guyana, Honduras, Mexico, Panama, Peru and Venezuela exhibit decreasing RTS. Conceptually, a large size of the agricultural sector is not recommended for these countries, because their proportion of the value of the gross agricultural production is less than the increase in all factors. It is important to note that since 2010 LACs have suffered six years of slowdown, including two of recession, and have halted social development (World Bank, 2017). Besides, LACs’ GHG emissions contribution has been estimated at 10% of global emissions in 2012 (Sánchez and Reyes, 2015). Thus, from the economic growth perspective, in LACs it is not practical to maintain or to reduce their agricultural activity’s current size. LACs need to keep themselves up in order to register a positive growth to increase the total wage bill (Moreno-Moreno et al., 2017). Thus, it is better for LACs to use technological innovation that will be the key to enhancing their operational and environmental performance. tAblE 4: tyPEs of rts AnD Dts. Country RTS DTS Argentina Constant Constant Bahamas Constant Constant Barbados Constant Increasing Belize Constant Decreasing Bolivia Constant Constant Brazil Constant Constant Chile Decreasing Constant Colombia Decreasing Constant 173 Revista de economía mundial 47, 2017, 157-178 a dea-based evaluation of latin ameRica and the caRibbean agRicultuRal enviRonmental PeRfoRmance undeR the assumPtion of natuRal and manageRial efficiency Costa Rica Constant Decreasing Dominican Republic Decreasing Decreasing Ecuador Decreasing Decreasing El Salvador Decreasing Increasing Guatemala Decreasing Constant Guyana Decreasing Decreasing Honduras Decreasing Decreasing Jamaica Constant Constant Mexico Decreasing Constant Nicaragua Constant Decreasing Panama Decreasing Decreasing Paraguay Constant Increasing Peru Decreasing Constant Suriname Constant Decreasing Trinidad and Tobago Constant Decreasing Uruguay Constant Increasing Venezuela Decreasing Decreasing Source: Own elaboration. So, from the DTS point of view Table 5 shows that Argentina, Bahamas, Bolivia, Brazil, Jamaica, Chile, Colombia, Guatemala, Mexico and Peru belong to constant DTS, which implies that they should adopt technological innovation as an alternative to enhance their environmental performance. Meanwhile, Barbados, El Salvador, Paraguay and Uruguay exhibit increasing DTS, which indicates that it is strongly recommended for them to adopt technological innovation to reduce pollution. Besides, Belize, Costa Rica, Nicaragua, Suriname, Trinidad and Tobago, the Dominican Republic, Ecuador, Guyana, Honduras, Panama and Venezuela belong to decreasing DTS. They have an advantage compared with the other LACs: they do not immediately need technological innovation because their marginal environmental cost is decreasing. However, if their agricultural activity size varies without considering technological innovation, this is not recommended in the long term. 174 Juan-Javier Moreno-Moreno, María Teresa sanz-Díaz, Francisco velasco-MorenTe, carlos e. luDena 5. conclusions This empirical study demonstrated that in LACs during 2012 the efficiency under natural disposability appears comparatively higher (0.9408) than the efficiency under managerial disposability (0.8137) and joint performance (0.6877) (See Table 3). In general terms, LACs have negatively adopted the regulation for environmental protection. That is, they are more oriented toward reaching their agricultural production targets and secondly use technological innovation to reduce their pollutants. However, between the 25 LACs evaluated during 2012, there are 18 countries which have reached the maximum efficiency under natural disposability (1.000). Meanwhile, 12 countries have attained the maximum efficiency under managerial disposability and only 5 countries have achieved the maximum efficiency score under joint performance. Here, Argentina, Bahamas, Brazil, Chile, Mexico and Costa Rica presented the best agricultural performance during 2012. These countries have gained widespread international recognition for their increasingly important role in agricultural productivity and for using technology innovation in agriculture to confront the important challenge of reducing their environmental footprint (ECLAC et al., 2013; BEST, 2001; Bouroncle et al., 2015). It can be concluded that a smaller number of LAC countries direct their efforts toward creating a sustainable and environmentally-friendly agriculture. Thus, those countries that had inefficiency levels should compare themselves to the efficient countries so that they adopt the best practices to enhance their efficiency scores. Comparing sub-regions, the best performance in terms of natural disposability is in the Southern Cone. This is followed by the Andean Community with 0.9402. Next is Central America with 0.9180, and the Caribbean comes last with 0.9050. Nonetheless, the Kruskal-Wallis H test concluded that these subregions did not have major efficiency differences between them under natural disposability. Meanwhile, from efficiency under the managerial perspective, all the subregions failed to attain the maximum efficiency score. Yet, the Andean Community shows a comparatively high efficiency score of 0.8696, followed by the Southern Cone with 0.8147. Then come Central America and the Caribbean with 0.7927 and 0.7781, respectively. Still, the Kruskal-Wallis H test concluded that these sub-regions did not have major efficiency differences between them under managerial disposability. In terms of joint performance, i.e., when the natural disposability and managerial disposability coexist, all the sub-regions failed to attain the maximum efficiency score under the assumption of joint performance. Yet, the Southern Cone had a comparatively high efficiency score of 0.7360, followed by Central America with 0.7086. Then, come the Caribbean and the Andean Community with 0.6787 and 0.6274, respectively. It can be observed that the joint performance is always below efficiency under natural and managerial disposability. The Kruskal-Wallis H test concluded that there were no major technological 175 Revista de economía mundial 47, 2017, 157-178 a dea-based evaluation of latin ameRica and the caRibbean agRicultuRal enviRonmental PeRfoRmance undeR the assumPtion of natuRal and manageRial efficiency innovation progress differences between the sub-regions to reduce agricultural emissions (CO2eq.). Finally, under the natural disposability assumption, 13 countries exhibit constant RTS and 12 countries decreasing RTS. On the other hand, under the managerial disposability assumption, 11 countries show decreasing DTS, 10 constant DTS and 4 increasing DTS. The results suggest that for LAC countries, it is not simply a matter of varying the current size of their agricultural activity to improve their operational performance. Instead, they should direct their efforts toward creating a sustainable agriculture by implementing new technologies and developing their management to modernize agriculture to reduce pollution. AcknowlEDgmEnts This research was mainly supported by the IFHARU and SENACYT of Panama. Besides, the second author is grateful for the financial support received from the Departamento de Análisis Económico y Economía Política (Universidad de Sevilla) and the Andalusian Regional Government (project SEJ-132), and the Energy and Environmental Economics Chair at the University of Seville (ref. 1394/0103). The third. author wishes to acknowledge the funding provided by the Project of the Spanish Ministry of Economy and Competitiveness, HERMES (TIN2013-46801-C4-1-r), and Simon (P11-TIC-8052). We are also thankful to Nin-Pratt et al. (2015) of the Inter-American Development Bank for the collaboration with part of the data used in this study. rEfErEncEs Banker, R.D., Charnes, A. and Cooper, W.W. (1984): “Some Models for the Estimation of Technical and Scale Efficiencies in Data Envelopment Analysis”, Management Science, 30 (9), 1078–1092. 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