P is the profit before tax of the i-th firm in the t-th period, it q represents the vector of output
quantities of the i-th firm in the t-th period; it p is vector of quantity of variable outputs of the
i-th firm in the t-th period. To develop a profit efficiency mode for this study we replaced
total cost (TC) in the cost efficiency model with pre-tax profit, while the inputs prices and
outputs remain the same.