*2.2. Environmental DEA*

Policy makers must consider environmental efficiency assessment at a country level in order to regulate to promote environmental protection and economic development. In this way, some studies have involved undesirable outputs in the definition of DEA. The treatment of these undesirable outputs within the DEA literature has been presented in three ways, according to Dyson et al. [25]: (1) inverting the anti-isotonic factor, (2) subtracting the value of the undesirable factor from a large number or (3) treating the undesirable output as an input. We have opted for the third strategy.

The DEA model for environmental assessment requires the incorporation of different production factors (desirable outputs, undesirable outputs and inputs), and this requires all variables to be greater than or equal to zero. Here, non-radial models satisfy this requirement; therefore, they can measure the efficiency of DMUs (decision-making units) that contain negative or zero values in any of their inputs or outputs.

Conventional energy efficiency measures that do not consider undesirable outputs are biased because firms can lose their productive efficiency due to a negative output [26]. Following Faere et al. [9], when evaluating the performance of producers, it makes sense to compensate for the supply of desirable outputs, as well as to penalize the provision of undesirable outputs. In other words, "positive" and "negative" factors should be treated asymmetrically when measuring a producer's performance. The performance measures outlined above, in fact, treat positive and negative factors asymmetrically, valuing the former and ignoring the latter. This extreme form of asymmetry characterizes much of the literature on measuring productivity and efficiency, so it is necessary to introduce concepts that allow the smoothing of this approach.

Unlike traditional DEA models, the model proposed by Faere et al. [9] assumes that the reduction of undesirable outputs is costly in terms of desirable outputs. To reduce undesirable outputs, part of the production of desirable outputs must be sacrificed. In the literature, this implies moving from the assumption that the technology of undesirable outputs is "freely (or strongly) disposable", where the variation of undesirable outputs does not represent any cost in terms of production, to the assumption of "weakly disposable" outputs, where such variation involves a cost, given the conceptual incorporation that implies that desirable and undesirable outputs are jointly produced. In this work, the desirable outputs - *yd*- *R*+ are distinguished from the undesirable outputs (*yu*- *R*+) and the inputs are denoted by *x* -*R*+.

According to Faere et al. [9], mathematically, the concept of strong disposability between desirable and undesirable outputs can be expressed as follows:

$$P\left(y^{\mu}, y^{d}\right) \in P(\mathbf{x}) \quad \to \ \left(y^{d} - \mathbf{s}\right) \in P(\mathbf{x}), \ \mathbf{s} \ge 0 \tag{1}$$

Given a vector of inputs (*x*) and a production possibility frontier *P*(*x*), if a level *y<sup>d</sup>* can be reached, then *y<sup>d</sup>* − *s* can also be produced for any *s* ≥ 0.

On the other hand, it is common that certain bad outputs cannot be separated from the corresponding good outputs; therefore, to reduce a bad output, it is necessary to reduce the good output [27]. Within the DEA literature, this is the concept of weak disposability, and it can be denoted as follows:

$$P(y^{\mu}, y^{d}) \in P(\mathbf{x}) \to \left(\partial y^{\mu}, \partial y^{d}\right) \in P(\mathbf{x}), \text{ with } 0 \le \theta \le 1. \tag{2}$$

Given a vector of inputs (*x*) and a production possibility frontier *P*(*x*), on the one hand, a total decrease of the undesirable output (*yu* = 0) is not possible unless the desirable output is also zero - *y<sup>d</sup>* = 0 ; on the other hand, it can only be decreased proportionally - *yu*, *yd* when 0 ≤ θ ≤ 1. In this case, *yu* and *yd* are called non-separable undesirable outputs and non-separable desirable outputs, respectively. We consider that the weak disposability assumption in the activity generation activity is necessary considering that it is not possible to generate electricity using fossil fuels without incurring CO2 emissions.
