The Measurement of Environmental Economic Inefficiency with Pollution-generating Technologies

This study introduces the measurement of environmental inefficiency from an economic perspective that integrates, in addition to marketed good outputs, the negative environmental externalities associated with bad outputs. We develop our proposal using the latest by- production models that consider two separate and parallel technologies: a standard technology generating good outputs, and a polluting technology for the by-production of bad outputs (Murty et al., 2012). While research into environmental inefficiency incorporating undesirable or bad outputs from a technological perspective is well established, no attempts have been made to extend it to the economic sphere. Our model defines an economic inefficiency measure that accounts for suboptimal behavior in the form of foregone private revenue and social cost excess (environmental damage). We show that economic inefficiency can be consistently decomposed according to technical and allocative criteria, considering the two separate technologies and market prices, respectively. We illustrate the empirical implementation of our approach on a set of established and complementary models using a dataset on agriculture at the level of US states.


Introduction
Measuring the environmental inefficiency of production units is an increasingly important topic of recent economic research. Environmental inefficiency assessment integrates marketed (desirable, intended, or good) outputs with negative environmental externalities into inefficiency modeling (the production of so-called undesirable, unintended, detrimental, or bad outputs). Such analysis is important from the perspective of sustainable production because it provides valuable insights for firms on how to adopt environmentally friendly strategies, and for policy makers to improve the design of pollutant-abatement instruments, accounting for environmental challenges.
Since the seminal work of Pittman (1983), the literature on modeling production technologies that account for bad outputs has developed into two main frameworks: one involving parametric methods (such as stochastic frontier analysis, SFA; Aigner et al., 1977), and one based on nonparametric methods (such as data envelopment analysis, DEA;Charnes et al., 1978;Banker et al., 1984). The present study relies on data envelopment techniques because they are flexible and do not impose restrictive assumptions on the parametric specification of the technology, nor on the distribution of environmental inefficiency. 1 Using these alternative frameworks, many different approaches have been proposed to assess environmental efficiency of production units. Lauwers (2009) classified these approaches into three groups. The first group concerns environmentally adjusted production efficiency models, in which undesirable outputs are incorporated into the production technology. In general, two main branches of studies within this group can be distinguished: (i) treating bad outputs as strong (free) disposable inputs (Haynes et al., 1993;Hailu and Veeman, 2001) 2 or (ii) treating bad outputs as weekly disposable outputs and assuming the null-jointness of both bad and good outputs (Färe et al., 1986;Färe et al., 1989). 3 The second group of studies consists of frontier eco-efficiency models (Korhonen and Luptacik, 2004;Kuosmanen and Kortelainen, 2005), which do not follow axiomatic production efficiency frameworks, but relate aggregate ecological outcomes with economic outcomes only. In other words, eco-efficiency is measured either through minimization of environmental outcomes given economic outcomes (for 3 example, value added) or the alternative maximization of economic outcomes given the environmental outcomes. The third group of studies is based on the introduction of the materials balance principle into production models (Lauwers and Van Huylenbroeck, 2003;Coelli et al., 2007). The materials balance principle states that flows into and out of the environment are equal, linking the raw materials used in the production system to outputs, both intended and residual ones.
While these three groups of approaches are currently in use, their principles have been heavily debated. The branches of studies assuming bad outputs as free disposable inputs or weakly disposable outputs have confronted each other (see, for example, the discussion between Hailu and Veeman (2001), Färe and Grosskopf (2003) and Hailu (2003)). Further, the main criticisms of these studies are inconsistency with physical laws or violating the materials balance principle (Coelli et al., 2007;Murty et al., 2012). Eco-efficiency models have been criticized mainly for their incomplete characterization of the production process . Finally, critics of the materials balance approach have noted that it does not specify how bad outputs are generated, focuses mainly on material inputs, and requires all variables to be measured in the same measurement unit (Førsund, 2009;Hoang and Rao, 2010;Murty et al., 2012). As a result, many subsequent extensions, as well as empirical applications, have followed one of these three diverging approaches (see, for example, Reinhard et al., 2000;Mahlberg and Sahoo, 2001 for the first approach; Pérez Urdiales et al., 2016;Picazo-Tadeo et al., 2011 for the second approach; and Welch and Barnum, 2009;and Hampf and Rødseth, 2015 for the third approach). Dakpo et al.'s (2016) recent survey of environmental efficiency studies extended the Lauwers (2009) classification into the fourth, most recent, category of by-production models, which are based on the idea of defining two subtechnologies in parallel: one that generates good outputs and a second that generates bad outputs. This approach was introduced by Murty et al. (2012) and, as a consistent and relatively new approach, its empirical applications are flourishing (e.g., Dakpo et al., 2017;Arjomandi et al., 2018;Ray et al., 2018) as are its extensions (e.g., Serra et al., 2014;Lozano, 2015;Dakpo, 2016;Førsund, 2018).
Regardless the modeling approach under the four listed categories, a common feature of all previous studies is that they are only capable of measuring technical efficiency by focusing on the technological side of the production process, while neglecting the measurement of environmental efficiency from an economic perspective. The determination of economic efficiency is important from a managerial standpoint focused on market-oriented performance.
Managers are interested in increasing performance not only in physical terms by taking Electronic copy available at: https://ssrn.com/abstract=3383443 advantage of the best technology available, but also by realizing the economic gains associated with allocative efficiency improvements; that is, the choice of optimal output and input mixes, leading to either maximum profit, revenue, or minimum cost. In the current framework including undesirable outputs, economic efficiency not only relates to the private objectives listed above, but must be extended to the social cost associated to the by-production of undesirable outputs. Indeed, the economic damage associated with their production, represented by a social cost function, shows how their production is detrimental to the economy. Yet, the existing models fail to take this step forward and internalize the negative economic effects associated to their by-production. In other words, they only consider the technological side, while it still remains an externality from an economic perspective.
This study enhances these models by introducing a measure of environmental economic inefficiency that includes undesirable outputs and implements them from theoretical and empirical perspectives. To fill in the gap in the literature we postulate a comprehensive framework that is consistent with the economic behavior of organizations in their attempt to maximize revenue, but also accounts for the environmental inefficiency that results from the failure to minimize the economic cost associated to environmental damage. This results in the definition of an "environmental profit function" that maximizes the difference between private (market) revenue less social (environmental) cost, using the prices of good and bad outputs. 4 Hence, we develop a framework that is capable of balancing private gains (revenue) and social losses (cost) into a measure of economic inefficiency that can be decomposed according to technical and allocative criteria. Furthermore, within our framework we show how to decompose overall profit inefficiency into desirable (marketed output) inefficiency, and ecodamage inefficiency.
In this regard, we define the DEA programs that allow the empirical implementation of our novel approach. 5,6 Our point of departure is the by-production model introduced by Murty 4 The model can be easily enhanced to include the minimization of inputs cost, but instead we keep the definition of "environmental profit inefficiency" as a trade-off between private revenue and social cost. 5 Brännlund et al. (1995) measured profit inefficiency under a quota system and the production of undesirable outputs by DEA models. However, they did not use prices for weighting the negative externalities and do not decompose profit inefficiency into its drivers, something that we will do in this paper. Additionally, we note that Pham and Zelenyuk (2018) defined revenue inefficiency in the banking industry accounting for nonperforming loans (NPLs), which are modeled as undesirable outputs under the approach of weak disposability. However, the model is internal to the firm (that is, private revenue), as it does not include environmental indicators, while they do not implement it empirically. 6 Also, Coelli et al. (2007) used the materials balance approach to estimate both environmental efficiency and cost efficiency separately, but they did not relate them to estimate an overall measure of cost efficiency incorporating environmental factors. The studies of Welch andBarnum (2009), Nguyen et al. (2012) and Hoang and Alauddin (2012) are similar to that of Coelli et al. (2007). Although other studies invoke the concept of revenue inefficiency in the context of production with undesirable outputs, they do so for the purpose of estimating shadow prices 5 et al. (2012), as it represents the most recent extension of previous approaches and can arguably be seen as a generalization that, by considering two independent technologies for desirable and undesirable outputs, avoids some of their inconsistencies (namely, the multiplicity of optimal combinations of desirable and undesirable outputs for a given level of inputs, and erroneously signed marginal rates of transformation − shadow prices − between outputs and inputs). Nevertheless, our model could be easily particularized for previous approaches. 7 We also consider recent qualifications of the original by-production model by Dakpo (2016) and Førsund (2018). 8 We demonstrate the practical usefulness of our newly developed methodology through an application to state-level data of the United States agricultural sector. Agriculture involves the production of not only good outputs such as primary food commodities, but also of bad outputs related with, for example, the need for fuel, the usage of pesticides, fertilizers and other agriculture chemicals, or the management of manure (Skinner et al., 1997;Reinhard et al., 1999Reinhard et al., , 2000. Examples of bad outputs associated to these polluting inputs in agriculture are greenhouse gas emissions, pesticide and nitrogen leaching and runoff, risk to human health and fish from exposure to pesticides and fertilizers, etc. (see Ball et al., 2001;Kellog et al., 2002;Dakpo et al., 2017). In the empirical application we are capable of considering two of these bad outputs: CO2 emissions and pesticide exposures.
The remainder of this paper is structured as follows. The next section reviews the byproduction models of technical inefficiency and introduces their mathematical underpinnings.
The subsequent section develops our extension allowing the measurement of economic (profit) inefficiency. We then discuss our empirical application, briefly commenting the dataset and presenting the results. Conclusions are drawn in the final section.

The by-production models
Pitmann (1983) and Färe et al. (1986) initiated the asymmetric modeling of outputs when measuring efficiency depending on their nature, increasing those that are market-oriented between good and bad outputs. Examples of such studies are Färe et al. (2005Färe et al. ( , 2006. In addition, productivity change analyses accounting for bad outputs have been also undertaken, but without considering the economic side (Ball et al., 2005). 7 Details on the characteristics of the by-production approach are presented in the next section. 8 Although we are aware of other methodological developments that rely on the by-production model, such as Serra et al. (2014) or Lozano (2015), we have not considered them since their general idea is to mix the byproduction approach with other efficiency frameworks, and not the modification of the model per se. Hence, if applied, their results would not be comparable to those of the original by-production model. 6 while reducing those that are detrimental to the environment. A key question is how to axiomatically model the production technology when calculating technical efficiency through distance functions. Most particularly, as commented in the introduction to this paper, should the axioms underlying the production technology reflect their strong or weak disposability, and eventually, be modeled as outputs or as if they were inputs? Among the existing approaches for dealing with undesirable outputs and efficiency, the by-production model introduced by Murty and Russell (2002) and Murty et al. (2012) is currently considered a preferred option (for applications in agriculture see, for example, Serra et al., 2014, andDakpo et al., 2017).
The by-production approach posits that complex production systems are made up of several independent processes (Frisch, 1965). In this model, the technology can be separated into sets of sub-technologies; one for the production of good outputs and one for the generation of bad outputs. The "global" technology implies interactions between several separate subtechnologies. Førsund (2018) and Murty and Russell (2018) recently classified the byproduction approach among the multi-equation modeling approaches and argued that an important advantage of this approach is that it represents pollution-generating technologies by accounting for the Material Balance Principle, thereby satisfying the laws of thermodynamics.
Additionally, as Murty et al. (2012) remarked, the by-production model avoids two inconsistencies of previous approaches. In particular, several technical efficiency combinations of good and bad outputs, with varying levels of bad output, could be possible when holding (polluting and non-polluting) input quantities fixed. However, in the absence of abatement activities implemented by the firm, this type of combination is contrary to the phenomenon of by-production, since by-production implies that, at fixed levels of inputs, there is only one level of pollution at the frontier of the production possibility set. Moreover, it is possible to observe a negative trade-off between the inputs associated with pollution, like fuel, and their associated bad output, such as CO2, which represents a clear inconsistency (more fuel but less CO2). These are the reasons why the by-production approach is utilized in the current study to introduce the concept of environmental economic inefficiency taking market prices into account.
In order to briefly review the standard by-production approach, let us formally define  n x R , with 1 2   n n n . 9 The first set could comprise land, labor, and so on, while the second set, in the context of our empirical application on agriculture, consists of inputs like fuel, fertilizers, and pesticides, which produce certain pollutants as byproducts, such as CO2 emissions and pesticide exposures. In this way, the 'global' technology, denoted by T , is the intersection of two sub-technologies, 1 T and 2 T . Whereas 1 T is the standard production technology with only good outputs, 2 T represents the production of bad outputs. In the model by Murty et al. (2012), both technologies are linked through the level of the polluting inputs.
In the non-parametric framework of DEA, the two sub-technologies may be expressed mathematically under variable returns to scale (VRS) as: Note that the sub-technologies are defined with two different intensity variables:  and  . Additionally, as Murty et al. (2012) highlighted, 1 T satisfies the standard free-disposability property of inputs (pollutant and non-pollutant) and the good output. On the pollution side, the bad outputs satisfy the assumption of costly disposability, which implies the possibility of observing inefficiency in the generation of pollution.
Regarding the measurement of technical efficiency, Murty et al. (2012) showed that some conventional approaches, like the hyperbolic and directional distance function defined on 1 2   T T T , are inadequate in the context of by-production. We use the term "outputoriented" in this context because these distance functions measure efficiency with respect to both good and bad outputs simultaneously. In this way, the weakness is due to the fact that the two aforementioned measures use the same coefficient (decision variable) for determining efficiency both in 1 T for the good outputs and 2 T for the bad outputs. This implies that it is possible to reach the efficiency frontier for some of the sub-technologies, but the observation can fall short of achieving the frontier of the other one. For consistency, efficiency in the by-8 production approach requires models that project the assessed observations onto both the efficient frontier of 1 T and the efficient frontier of 2 T .
The abovementioned drawbacks of standard approaches motivated Murty et al. (2012) to propose a different measure for dealing with good and bad outputs under by-production. For DMU0, this measure is good-output-specific and bad-output-specific, and is based on the index previously defined by Färe et al. (1985): The optimal value of (3) coincides with the mean of the standard good-output-oriented efficiency and the environmental bad-output-oriented efficiency. Note also that the above model is separable. In this case, this means that the optimal value can be determined as the mean of a model that minimizes To introduce our economic inefficiency model we extend the state-of-the-art of byproduction approach (Murty et al. 2012, Dakpo et al. 2017and Førsund, 2018 by incorporating information on market prices. To do that, we resort to duality theory following Chambers et al. (1998), and, more recently, Aparicio et al. (2015), Aparicio et al. (2016a), and Aparicio et al. (2016b). In particular, we recall relevant duality results concerning the directional distance function. Consequently, we start out by defining this type of measure from an output-oriented perspective in the context of by-production. Under the viewpoint introduced by Murty et al.
(2012), we need a measure that allows us to project the assessed observations onto the efficient frontiers of 1 T and 2 T simultaneously. In this way, the "by-production" directional outputoriented distance function for the Murty et al. (2012)   , , , 0 (5.8) The exogenous coefficients 1 0   and 2 0   , 1 2 1     , are weights that are prefixed by the corresponding decision maker (manager, politician, regulator, etc.) to reflect the relative importance of the standard (traditional) way of producing versus the new and clean paradigm for generating goods and services. Additionally, its linear dual is: , , , , 0, (6.5) , free (6.6) Finally, to complete this opening section, we recall the first additive measure and decomposition of economic inefficiency proposed in the literature. We refer to the Nerlovian profit inefficiency measure, which can be decomposed into technical inefficiency (the directional distance function) and a residual term interpreted as allocative inefficiency (Chambers et al., 1998).
In the standard production context, considering private revenue and cost only, and given a vector of input and output prices   ,    m s w p R and technology T , the profit function  is defined as DMU0 is defined as optimal profit (that is, the value of the profit function at market prices) minus observed profit, both normalized by the value of a reference vector Additionally, Chambers et al. (1998) 3. Measuring economic inefficiency with by-production models in DEA

Economic inefficiency model considering Murty et al.'s (2012) technology
We will first introduce some notation and definitions. Given a fixed level of input the maximum feasible revenue given the output price vector Next, we explicitly show how the value of   The dual program of (10) is (11) To evaluate economic loss due to revenue inefficiency, in the context of the directional output distance functions, Färe and Primont (2006) proved that a normalized measure of revenue inefficiency, in particular the ratio   14 Likewise, we can introduce cost efficiency following the same rationale, and based on the cost function. However, in our context we are interested in "social/environmental" cost functions rather than private costs, representing a measure of the (monetary) minimal damage caused by the production of undesirable outputs. The cost function represents a "monetized metric" of the ecological footprint such as the social cost of carbon (SCC); for example, the damage per ton of CO2 (see Pearce et al., 1996). Correspondingly, an observation is economically inefficient in environmental terms if, given the amount of undesirable outputs produced, it causes larger damage than that represented by the minimum "social/environmental" cost function (either as a result of technical or allocative inefficiencies).
Let us assume that it is possible to observe or estimate prices for the undesirable outputs: Murty et al.'s (2012) approach, the eco-damage function will be non-parametrically determined directly from 2 T as follows.

 
The dual program of (12) is (13) , We now derive, by duality, a normalized measure of economic inefficiency and show how it can be decomposed into (desirable) revenue inefficiency and eco-damage inefficiency.
In order to do that, we first prove the following technical proposition.
be an optimal solution of (11) and let   be an optimal solution of (13) when 0 (acting as w ) are taken as arguments. We will prove that is a feasible solution of (6). Constraints (6.5) and (6.6) are trivially satisfied. Regarding (6.1),  As for (6.2), In the same way, it is possible to prove that (6.3) and (6.4) are also satisfied. In particular, constraint (6.3) holds by (13.1) and (13.2). Consequently,   is a feasible solution of (6). Regarding the objective function of (6) evaluated at this point, (10) and (11) have the same optimal value and models (12) and (13) also have the same optimal value. In this way, is any optimal solution of (11) when  q t , which is the inequality that we were seeking. ■ be market prices for good and bad outputs, respectively. Then, Consequently, applying Proposition 1, we get .
Finally, given that   min , Note that the left-hand side of (15) may be interpreted as a (normalized) measure of economic environmental inefficiency. Additionally, following Farrell's tradition, the righthand side can be interpreted as (environmental) technical inefficiency and the residual term associated with closing the inequality could be interpreted as allocative inefficiency. Moreover, it is possible to decompose the left-hand side of (15) into However, note that the normalization term used in (15) and (16) − that is, − depends on two different terms, in contrast to what happens with respect to the Nerlovian profit inefficiency measure in (7). By analogy with the standard approach based on the directional distance function, we suggest resorting to an endogenous value for 1  T and, therefore, also for 2 . It is easy to check that this value is 1 *

Economic inefficiency model considering Dakpo et al.'s (2012) approach
We now turn to Dakpo et al.'s (2017) approach. In this case, the projection points in the two subtechnologies for the input dimensions must coincide. The "by-production        .
Proof. Following the same steps than in Proposition 1, we get the desired result. ■ Applying Proposition 2, with market prices   , q w , we get the following inequality.
The left-hand side in (21) may be interpreted as a measure of economic environmental inefficiency, which could be decomposed into technical inefficiency (the right-hand side in (21)) and a residual term, interpreted as allocative inefficiency. Førsund's (2018) proposal , , , 0, (22.8)

Economic inefficiency model considering
min , The left-hand side may be interpreted as a measure of economic environmental inefficiency. In particular, it is possible to decompose it into  min , Inequalities (24) and (28)

Dataset and variables
The empirical illustration relies on state-level data in the United States that comes from multiple agencies. The main source of data is the US Department of Agriculture's (USDA) Economic Research Service (ERS), which compiled the data necessary to calculate agricultural productivity in the US, and, in particular, the price indices and implicit quantities of farm outputs and inputs for each of the 48 continental states for 1960−2004. The dataset has been validated and used extensively in previous research (for example, in Ball et al., 1999;Zofío and Lovell, 2001;Huffman and Evenson, 2006;Sabasi and Shumway, 2018). A critical review of the data in light of recent developments can be found in Shumway et al. (2015;2016). To illustrate our models, we consider the most recent year available in the dataset (2004) and assume that the production process is characterized by the following three non-polluting inputs (capital services excluding land, land service flows, and labor services), two polluting inputs (energy and pesticides), and two good outputs (livestock and crops). 12 All these variables are calculated as implicit (real) quantity indices, expressed in thousands of dollars, at constant prices of 1996, using the first state (Alabama) as reference benchmark. An index of relative real output (alternatively, real input) is obtained by dividing the nominal output (input) value ratio for the two states by the corresponding output (input) price index. The details on the method of construction of all variables are contained in the following webpage of USDA-ERS: https://www.ers.usda.gov/data-products/agricultural-productivity-in-the-us/methods/.
As for the undesirable output production generated by energy consumption, we consider carbon dioxide (CO2) emissions from the agricultural sector associated with fuel combustion, also for 2004 (expressed in tons of CO2 equivalents), obtained from the US Environmental Protection Agency (EPA). Since these data are given in overall terms for the whole country, we further disaggregate it by state, using for that purpose the share that each state has in farm production expenses for gasoline, fuels, and oils, as reported by the US Department of The ERS dataset also contains data on one more output: other farm-related output. However, we do not take it into account since it is a residual to capture additional farm income and therefore usually consists of a very marginal fraction of the total farm output (rarely above 5 percent of the total farm output) and is therefore negligible. In addition, the ERS dataset provides the information on inputs of agricultural chemicals, fertilizers, and other intermediate inputs.
Since we cannot relate these inputs with their associated bad outputs for which data are unavailable, we drop these inputs in our analysis. 13 In particular, we use the data on expenses in gasoline, fuels and oils from the Census of Agriculture as reported by the US Department of Agriculture, expressed in thousands of dollars. Since the census was not conducted in 2004, we take the average of the values reported for 2002 and 2007 censuses, which consists of the closest approximation of the 2004 data. 14 On the contrary, a general US market for sulfur dioxide (SO2) has existed for many years (see https://www.epa.gov/airmarkets/so2-allowance-auctions). 15 California's GHG emissions program is the second largest in the world after the European Union's Emissions Trading System. From its beginning in 2012 it covered the power and industrial facilities, and it expanded to natural gas and transportation fuels in 2015, allowing it to cover approximately 85 percent of California's GHG emissions. Another program exists in the US, called the Regional Greenhouse Gas Initiative (RGGI), in which nine states participate. However, it is much narrower than the Californian program since it applies only to some power plants and only to CO2 emissions. Because of this, we decided to exclusively use the data from Californian program in our research. Also, mixing data on prices from both Californian and RGGI initiatives would not be appropriate since the two sets of values are incompatible.

26
The measure of bad output related to pesticides is the number of pesticide exposures

Technical, allocative and profit frontiers.
When solving our four reference economic models -that is, Murty et al. (2012) and Dakpo et al. (2017), each complemented with Førsund's (2018) proposal − it is relevant to determine, from a technological perspective, the number of observations that are efficient, thereby defining the frontier of the global by-production technology T, consisting of both the intended production 1 T technology, (1) (hereafter, conventional or standard technology) and the pollution-generating technology 2 T , (2) (hereafter, polluting technology). Table 3 shows that the number of observations defining the production frontier is greater in the conventional technology 1 T than in the polluting technology 2 T , except in the case of the Murty et al. (2012) model incorporating Førsund's proposal. Nevertheless, the number of observation jointly defining the by-production technology by being efficient in both 1 T and 2 T is greatly reduced.
Interestingly, all states that are efficient according to Murty et al.'s model are also efficient in the other three models. These are California, Delaware, Iowa, Illinois, Rhode Island, and Vermont, whose different production scales indicate that they represent alternative most productive scale sizes, serving as benchmark for the remaining states.

Technical inefficiency: results within and between models
Departing from this general portrait of inefficiency frequencies at the technical, allocative, and overall profit inefficiency levels, we now focus on the technological side, with iii) Technical inefficiency in the residual (polluting) technology 2 T differs across the four models. However, while there are not significant differences between Murty et al.'s (5) and Dakpo (17) and (28). 16 16 We acknowledge that this practically implies a "business as usual" evaluation of technical inefficiency in the by-production model. However, our proposed model is general enough to accommodate other weights, as would be the case if other (subjective) weights were chosen, or in other applications where the difference in economic value between private revenue and social cost would not be that large (as could be seen if average private revenue and social cost were calculated using the mean quantity and price values presented in Table 1). One should keep in mind that our proposal to choose delta simply reflects the empirical balance between the former economic values (that is, private benefit and social cost), since a weight equal to 0.5 would simply imply that both monetary valuations are equal. The correlations between the technical efficiency scores using Spearman's definition show that the conventional and environmental efficiency performance are weakly or even negatively correlated in most cases (see Table 4). This should come as no surprise given that observations do have market incentives to perform better in the conventional side of the production process 1 T (that is, to maximize output revenue), but these incentives are weak or absent in the case of environmental cost minimization. Since the production of undesirables outputs (CO2 emissions and pesticide exposures) is not normally internalized by the economic system, productive efficiency in 2 T is not tightly pursued, which means that a negative correlation between both rankings is a likely outcome. This can be seen clearly in the unmodified Murty et al. and Dakpo et al. models,where 2 T inefficiencies are greater on average. Nevertheless, we note that the variability in the rankings is so high that none of these correlations are significant at the standard confidence levels. It is also worth remarking that the correlations between models' rankings for the polluting technology 2 T are generally smaller than for the standard technology 1 T (except for the Murty et al. and Dakpo et al. models enhanced with Førsund's proposal, whose correlation is    Besides focusing on absolute values, we can gain information on the weight that each technology ( 1 T and 2 T ) has on the by-production technology T. The right-hand panel of Figure   1 shows the average percentage weight that each one of them has on the aggregate result. On average, for US agriculture, both 1 T and 2 T inefficiencies account for a significant share of aggregate by-production inefficiency, their values being driven by the large number of efficient observations in either technology, resulting in a null contribution to aggregate inefficiency, and explaining the relative balanced average percentage values for the four models, regardless of 1  T . This is particularly the case for the Dakpo et al. approach, where 2 T technical inefficiency is rather large in absolute values. Nevertheless, since mean values provide only a rough first approximation to inefficiency results, we now study their different distributions.
A visual comparison of the values of the technical inefficiency scores for 1 T and 2 T is presented in Figure 2, where box-plots of the different distributions make it possible to identify extreme values. The different boxes (grouped in pairs by models) represent the intervals between the first and third quartiles of the ranking distribution (that is, the interquantile range (IQR) between Q1 and Q3), with its median represented by the horizontal line within it (the median can then be compared to the mean values presented in Figure 1). The dispersion in the rankings within this interquartile range is relatively low, particularly for the polluting Still focusing on the box-plots, it is relevant to test whether the distributions of the conventional and polluting technologies, 1 T and 2 T , are equal within each one of the four models. For this purpose, we have performed the test proposed by Simar and Zelenyuk (2006).
Their method adapts the nonparametric test for the equality of two densities developed by Li (1996). For this test we use algorithm II with 1000 replications, which computes the Li statistic Alternatively, it is also interesting to test if the 1 T and 2 T inefficiency distributions are different between models. Figure 3 depicts their kernel distributions in each one of the four models (left and right panels for 1 T and 2 T , respectively). When plotting these distributions we follow the procedure proposed by Simar and Zelenyuk (2006), which in short: (i) uses Gaussian 17 Simar and Zelenyuk (2006) developed their algorithm for radial distance functions, in which the efficiency values equal to one are smoothed. We adapted their algorithm to our additive context by smoothing the inefficiency scores equal to zero. 18 The level of significance changes to 1 percent when the hypothesis is tested for Dakpo et al.'s model enhanced with Førsund's assumption. kernels, (ii) employs the reflection method to overcome the issue of a zero-bounded support of the inefficiency scores (Silverman, 1986), and (iii) determines the bandwidths using Sheather and Jones's (1991) method. As commented, 1 T distributions are the same across pairwise models; that is, Murty et al.'s model (5), and Dakpo et al.'s model (17), are equal to their respective Førsund's extensions,(22) and (26)   34 rejected at the 1 percent level. Detailed results on these bilateral tests for both 1 T and 2 T are presented in Appendix A.1. Therefore, as for technical inefficiencies, we can conclude that 1 T and 2 T results are statistically different within models, but in many cases not between models.

Profit inefficiency: technical and allocative inefficiencies between models
We now discuss the economic efficiency dimension of the agricultural sector at the state level. The left-hand panel of Figure (25) and (28), the difference in results between the former and the latter models is marginal.
Consequently, while the difference within each type of model is minimal (that is, M PI vs. M&F PI , and D PI vs. D&F PI ), the differences between models remain the same at the 50 percent level. Regarding the difference between technical and allocative inefficiencies in absolute terms, the former doubles the latter in absolute terms on average. We have also calculated Spearman's correlations between the different pairs of models. Similarly, the alternative allocative inefficiency rankings (AIs) present rather comparable results, both between and within models.
Finally, as presented in the lower central panel of   We conclude the empirical section by checking whether these distributions are statistically different. As before, we first test whether the technical and allocative efficiencies components of profit inefficiency are different from each other within the same model (M TI vs. M AI , D TI vs. D AI , etc.) following the method proposed by Simar and Zelenyuk (2006). In all models except Dakpo et al.'s, the null hypothesis testing the equality of the densities cannot be rejected, even at the 10 percent level of significance.
Hence, technical and allocative inefficiencies are not statistically different in the three models. This can be visually corroborated by comparing the TI and AI distributions in the box-plots corresponding to each model in Figure 5, or comparing the technical and allocative distributions presented in the left and central panels of Figure 6.
As for the differences in profit, technical and allocative inefficiencies between models (M vs. D, M vs. M&F, etc.), the test of Simar and Zelenyuk (2006) returns varied results on the (in)existence of statistical differences between these distributions, which can be visually anticipated in Figure 6.

Conclusions
This paper introduces the theory and practice of environmental economic inefficiency measurement. Environmental economic inefficiency represents the ability of firms to maximize the difference between private revenue and "social/environmental" cost given the production technology and market prices. This objective can be likened to a "profit" function that economically weighs private gains and social losses by internalizing the damage associated with the production of undesirable outputs. Resorting to duality theory enabled us to demonstrate how this (supporting) economic function relates to a technical counterpart represented by the directional distance function, effectively extending the analytical framework of Chamber et al. (1998) to the field of environmental economics. Since the directional distance function can be regarded a measure of technical efficiency, the gap between technical and optimal economic performance can be attributed to allocative inefficiencies. Hence, profit inefficiency can be consistently decomposed into its technical and allocative sources.
The new model departs from one of the most recent proposals characterizing the production technology in the presence of undesirable outputs; the so-called by-production model put forward by Murty et al. (2012). This analytical framework differentiates between two separate sub-technologies, one corresponding to the conventional (privately oriented) approach and one characterizing the production of pollutants only. Our model makes it possible to assign different weights to each technology in order to account for the modeler or stakeholder preferences (managerial, political, legal/regulatory, etc.).
Although the new economic model could be developed adopting other technological characterizations, the by-production approach overcomes prior limitations and is becoming increasingly popular among practitioners. Moreover, it is subject to continuous qualifications such as those recently introduced by Dakpo et al. (2017) andFørsund (2018).
We develop our new model within the data envelopment analysis framework, which allows us to illustrate its empirical viability using a real-life data set on US agriculture for 2004. The production technology is characterized by five inputs − capital, land, labor, energy, and pesticide (the latter two of which are polluting inputs) − and two outputs: livestock and crops. We implement four models corresponding to the original proposal by Murty et al. (2012), a modified version corresponding to Dakpo et al.'s (2017) qualification that ensures that the projections points for input dimensions are the same in the conventional and polluting technologies, and their corresponding modifications that incorporate Førsund's (2018) proposal of bringing non-contaminating inputs into the polluting technology, so as to allow for substitution effects. The main empirical findings are the following: This simply reflects the fact that farmers do not have market incentives to perform better in the environmental side of the production process (that is, reducing social costs), as opposed to the conventional side, where falling short from the production frontier results in lower (private) revenue. In passing, we note that these empirical results comparing technical efficiencies for alternative models are novel, since they had not been confronted until now.
 As for the new economic inefficiency framework, statistical differences can be found across the alternative models. Profit inefficiency is generally larger in Murty et al.'s model than in Dakpo et al.'s. This result extends to their technical and allocative components. Our results show that technical inefficiency is generally larger than allocative inefficiency, suggesting that there is more room for economic improvements by taking advantage of the existing technology than by reallocating the relative demand for inputs and outputs given their market prices (that is, the relative specialization in input usage and output production). As for the extension of these two models with Førsund's proposal, no statistically significant differences emerge.
We conclude from these results that, as expected, the analytical approach chosen to evaluate environmental economic efficiency is highly dependent on the technological model upon which it is based. Choosing alternative models leads to significant differences in the magnitude of technical and allocative inefficiency, which may question the credibility of results given their lack of robustness, and lead to contradictions and faulty managerial and policy decision making. Therefore, caution should be exerted when implementing the new analytical framework, which nevertheless opens the door to a 43 whole new range of models capable of internalizing the social cost of environmental damage when assessing economic performance. This is a key extension in the measurement of environmental efficiency that was not available until now. Appendix A.1. Results of Simar and Zelenyuk (2006) Dakpo et al. & Førsund: (26). *** Denotes statistically significant differences between models at the critical 1 percent level. ** Denotes statistically significant differences between models at the critical 5 percent level. * Denotes statistically significant differences between models at the critical 10 percent level.
Appendix A.2. Results of Simar and Zelenyuk (2006) Dakpo et al. & Førsund: (26). ** Denotes statistically significant differences between models at the critical 5 percent level. * Denotes statistically significant differences between models at the critical 10 percent level.