*3.1. The DEA Method*

DEA is a popular approach for analyzing eco-efficiency [6]. In order to account for non-radial adjustments in the inputs and outputs, a DEA slack-based model (SBM) is used [25]. The output variable is production-based CO<sup>2</sup> productivity as measured by the ratio of output to CO<sup>2</sup> emissions [5,26] and the four inputs are the capital to labor ratio, the output to labor ratio, the capital to energy ratio, and the share of non-fossil fuels in energy consumption. This choice of variables is based on related work that estimates ecological efficiency at the country level [4,5].

The basic set up of the model is as follows. The four inputs and output are represented by *x* ∈ *R <sup>m</sup>* and *<sup>y</sup>* <sup>∈</sup> *<sup>R</sup> s*1 , respectively. For a collection of n DMUs, define the following matrices: *X* = [*x*1, . . . , *xn*] ∈ *R m x n* and *<sup>Y</sup>* <sup>=</sup> [*y*1, . . . , *<sup>y</sup>n*] <sup>∈</sup> *<sup>R</sup> s x n*. Assume that X > 0 and Y > 0.

The production possibility set, P, is:

$$P = \{(x, y) | x \ge X\lambda, \ y \le Y\lambda, \lambda \ge 0\}.\tag{1}$$

In Equation (1), the intensity vector is λ, and P corresponds to constant returns to scale (CRS) technology. Variable returns to scale can be obtained by adding the constraint that the sum of the elements in λ equal unity. A DMU (*x*0, *y*0) is efficiency if there is no vector (*x*, *y* ) ∈ *P* such that *x*<sup>0</sup> ≥ *x* and *y*<sup>0</sup> ≤ *y* and there is at least one strict inequality. The SBM is:

$$\left[\text{SBM}\right]\,\varepsilon = \min \frac{1 - \frac{1}{m}\sum\_{i=1}^{m}\frac{s\_i^{-}}{x\_{i0}}}{1 + \frac{1}{s}\sum\_{i=1}^{S}\frac{s\_r^{+}}{y\_{i0}}}.\tag{2}$$

Subject to:

$$\mathbf{x}\_0 = \mathbf{X}\boldsymbol{\lambda} + \mathbf{s}^- \tag{3}$$

$$y\_0 = \mathcal{Y}\lambda - s^+\tag{4}$$

$$s^- \ge 0, \ s^+ \ge 0, \lambda \ge 0. \tag{5}$$

The vectors *s* <sup>−</sup> and s<sup>+</sup> refer to the excess in inputs and the shortage of output, respectively. The objective function in (2) satisfies 0 < ε ≤ 1. Eco-efficiency is represented by ε with higher values indicating a higher level of eco-efficiency.

Changes in eco-efficiency over time can be estimated using the Malmquist productivity index (MPI) [27,28]. The MPI is the product of a catch-up effect and a frontier-shift effect [7]. The catch-up effect refers to how much a DMU improves or worsens its efficiency over time and is sometimes referred to as the efficiency change component (EFFCH). The frontiershift effect is the change in the efficient frontier over time and is sometimes referred to as the technical change component (TECH).

$$MPI = (Catch - up)(Fonter - shift)\tag{6}$$

$$\text{Catch} - \upupupupup = \frac{\varepsilon \text{ of } \text{DMul}\_0^{t+1} \text{ wrt } \text{ period } t+1 \text{ frontier}}{\varepsilon \text{ of } \text{DMul}\_0^t \text{ wrt } \text{period } t \text{ frontier}} \tag{7}$$

$$= \frac{\text{Frotiter} - \text{shift} =}{\sqrt{\frac{\varepsilon \text{ of } \text{DMU}\_0^t \text{ wrt} \text{ period t} \text{ fromtier}}{\varepsilon \text{ of } \text{DMU}\_0^t \text{ wrt} \text{ period t} + 1 \text{ fromtier}}} \text{ } \frac{\varepsilon \text{ of } \text{DMU}\_0^{t+1} \text{ wrt period t} \text{ fromtier}}{\varepsilon \text{ of } \text{DMU}\_0^{t+1} \text{ wrt period t} + 1 \text{ fromtier}} \text{} \tag{8}$$

In Equations (7) and (8), the combination of letters wrt denotes "with respect to". A change in catch-up greater than unity means that the efficiency of a DMU in period t + 1 is greater than the efficiency in period t. Thus, there has been a relative improvement in efficiency. A change in frontier shift greater than unity means that the efficient frontier in period t + 1 is higher than in period t. This indicates technological innovation. Total productivity change is the product of catch-up and frontier-shift. The DEA estimations in this paper were done using the R programing language [29] and the DJL package [30].
