First, regarding the question why the Lerner index calculation is based on our Equation (7) (see Figure 2
), instead of having been based on the cost function (our Equation (3)), the answer is that the direct estimation of our Equation (3) could not provide the residual term u
(see our Equation (7)), which captures the mark-up. We have clearly stated that the methodology we have followed is that of Kumbhakar et al.
). The calculation of the residual term u
is a very essential task in the implementation of their stochastic frontier methodology.
Second, regarding the equation Θ = u/ECQ (mentioned in the above MT’s remark), we confirm that we have indeed used the “return-to-dollar” specification in our Equation (7) in order to calculate the term u, through which we first calculated Θ and then L (please note that the equation Θ = u/ECQ, mentioned in the above remark, is exactly the same as our Equation 8, although using different notation).
To summarize, we followed the steps below:
We defined the translog cost function (our Equation (3)).
We “fitted” its first derivative (our Equation (6)) into our Equation (5) in order to finally arrive at our Equation (7).
We estimated our Equation (7) to derive the value of the residual u, which captures the mark-up.
We used the value of u to calculate Θ and L.
All the above steps have been performed following the Kumbhakar et al.
) stochastic frontier methodology.
The alternative procedure proposed in the criticism (as we understand it) follows the steps below:
It estimates the translog cost function in order to calculate Ecq, where Ecq = (∂lnTC)/(∂lnQ).
Based on the estimated value of Ecq, derived in step (a), it estimates either Equation (6) (in the criticism) in order to derive directly the value of L, or, alternatively, it estimates Equation (7) (in the criticism) in order to first derive Θ and then L.
First, it is clear that the alternative approach proposed by MT is the usual approach followed in the empirical literature.
Second, in response to the MT’s comment regarding the statistical noise, we clarify that in the methodology of Kumbhakar et al.
) the term u
is uniquely related to the mark-up. For further details about the term u
, the reader can refer to Kumbhakar et al.
(2012, p. 115
). Regarding the statistical noise, it is captured by v
, which is a symmetric two-sided noise term (Kumbhakar et al.
(2012, p. 114
)). We also take the opportunity here to explain that although the term u
is calculated in a way that resembles the estimation of cost inefficiency in a cost frontier model, in Kumbhakar et al.
) the term u
is uniquely related to the mark-up, leading Kumbhakar et al.
) to consider their approach as a non-standard application of stochastic frontier models. More precisely, their approach is not a cost frontier model, but a revenue share to total cost (TR/TC) frontier model. The bigger the distance between the observed TR/TC value from the minimum level it can reach (the frontier), the bigger the mark-up (and the market power).