2.1.4. Total Factor Productivity (TFP)

TFP is defined as the portion of output not explained by the number of inputs used in production [29]. TFP is also interpreted as a proxy for advancements in production technology [30]. TFP can be increased by reducing the input factors while maintaining the same amount of production or by increasing the production amount with the same input factors. Therefore, TFP reflects the overall production technology, which is a key factor in gaining market competitiveness.

This study measures TFP change by examining the relative productivity among the energy sectors of 42 countries using a directional distance function (DDF) model. The formula for calculating the distance function for country k can be computed using the following optimization problem:

$$\overrightarrow{\mathbf{D}} (\mathbf{x}\_{k'}^{l} \mathbf{y}\_{k'}^{m} \mathbf{g}\_{x'} \mathbf{g}\_{y^{m}}) = \text{Maximize} \mathfrak{P}\_{k} \tag{1}$$

$$\text{s.t. } \sum\_{i=1}^{N} \lambda\_i \mathbf{x}\_i^l \le \mathbf{x}\_k^l + \beta\_k \mathbf{g}\_{\mathbf{x}^l} \qquad l = 1, \dots, L \tag{2}$$

$$\sum\_{i=1}^{N} \lambda\_i y\_i^m \ge y\_k^m + \beta\_k \mathbf{g}\_{y^m} \qquad m = 1, \cdots, M \tag{3}$$

$$
\lambda\_i \ge 0, \qquad \left( i = 1, \dots, N \right) \tag{4}
$$

where β*k* is the production ine fficiency score of country *k*, and *i* is the country name. λ*i* is the weight variable used to identify the reference point on the production frontier line. *l* and *m* are the input and output variable names, respectively; *x* is the production input factor in the *L* × *N* input factor matrix; and *y* is the output in the *M* × *N* output factor matrix. In addition, g*x* is the directional vector of the input factor, and g*y* is the directional vector of the output factors. To estimate the production ine fficiency score of all countries, a model calculation must be applied independently *N* times for each country.

To estimate the productivity change indicators, this study sets the directional vector = g*x<sup>l</sup>* , g*y<sup>m</sup>* = - *xl k* , *y<sup>m</sup> k* . This type of directional vector assumes that an ine fficient firm can decrease its productive ine fficiency while increasing its desirable outputs and that it can decrease its inputs in proportion to the initial combination of actual outputs. Under this directional vector setting and the selection of data variables in Figure 2, the following equation can be obtained:

$$\overrightarrow{\text{D}}(\mathbf{x}\_{k'}^{l}\,\mathbf{y}\_{k'}^{m}\,\mathbf{g}\_{x'}\,\mathbf{g}\_{y''}) = \text{Maximize}\boldsymbol{\beta}\_{k}\tag{5}$$

$$\sum\_{i=1}^{N} \lambda\_i \mathbf{x}\_i^l \le (1 - \beta\_k) \mathbf{x}\_k^l \qquad l = \text{Labor, Capital stock, material} \tag{6}$$

$$\sum\_{i=1}^{N} \lambda\_i y\_i^m \ge (1 + \beta\_k) y\_k^m \qquad m = \text{gross output} \tag{7}$$

$$
\lambda\_i \ge 0 \qquad i = 1, \dots, N \tag{8}
$$

This study employs the Luenberger productivity indicator (LPI) as a TFP measure because the LPI is believed to be more robust than the widely used Malmquist indicator [31]. The LPI is computed with the results of the DDF model and is derived as follows [31,32]:

$$\text{TFP}^{t+1}\_{\text{t}} = \frac{1}{2} \Big\{ \overset{\cdot}{\text{D}}^{t+1}(\text{x}\_{\text{t}}, y\_{\text{t}}) - \overset{\cdot}{\text{D}}^{t+1}(\text{x}\_{\text{t}+1}, y\_{\text{t}+1}) + \overset{\cdot}{\text{D}}^{t}(\text{x}\_{\text{t}}, y\_{\text{t}}) - \overset{\cdot}{\text{D}}^{t}(\text{x}\_{\text{t}+1}, y\_{\text{t}+1}) \Big\} \tag{9}$$

where *x*t is the input for year t, *<sup>x</sup>*t+1 is the input for year t + 1, *y*t is the desired output for year t, and *y*t+<sup>1</sup> is the desired output for year t + 1. → D t (*<sup>x</sup>*t, *y*t) is the ine fficiency score of year t based on the frontier curve in year t. Similarly, → D t+1 (*<sup>x</sup>*t, *y*t) is the ine fficiency score of year t + 1 based on the frontier curve in year t + 1.
