Ceris-Cnr, W.P. N° 2/2002

 VCft 
P 
P 
 = β 0 + β y lnY ft + β k ln K ft + ∑ βi ln i ft  + β SP ln SPft + ∑ βiy ln i ft  lnY ft +
ln
P 
P 
P 
i
i
 Fft 
 Fft 
 Fft 
 Pi ft 
P 
 ln K + ∑ β iSP ln i ft  ln SPft + β yk ln Y ft * ln K ft +
+ ∑ β ik ln
P 
P 
i
i
 Fft 
 Fft 
1
1
+ β ySP ln Y ft * ln SPft + β kSP ln K ft * ln SPft + β yy (lnY ft ) 2 + β kk (ln K ft ) 2 +
2
2
 Pi ft   Pj ft 
1
1
+ β τ +v +u
 ln
+ β SPSP (ln SP ft ) 2 + ∑ ∑ β ij ln
τ ft
ft
ft
P  P 
2 i j
2
 Fft   Fft 
i, j ∈ { L, MS },

[3]

where the normalization of the monetary variables, VC, PL and PMS, with respect to the
price of fuel, PF, is made to ensure the linear homogeneity of the cost function in input
prices15.
The x-inefficiency term, uft, reflects the inability of firm f at the observation t to
attain the potential minimum cost defined by the stochastic frontier. The specification
for this effect and the discussion of the estimation technique for the final stochastic
frontier model are given in the next two sections.
3.2. Modeling inefficiency effects
Several innovations concerning the estimation of inefficiency using the stochastic
production and cost frontier approach have been introduced since the pioneer
contributions of Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck
(1977)16.
Researchers have attempted to overcome the shortcomings present in the above
frontier models: by specifying distributional forms for the inefficiency effects more
general than the half-normal and exponential distributions (Stevenson, 1980; Greene,
1990)17; proposing functional forms alternative to the traditional Cobb-Douglas
15

16

17

Symmetry property (βij = βji for all i, j) is also imposed a priori, whereas the other regularity
conditions, viz., monotonicity of the cost function in input prices and output, and concavity in input
prices are checked ex-post.
A brief introduction to the literature on stochastic frontier modeling and efficiency measurement is
provided in Piacenza (2000a). For a recent and more detailed review see Kumbhakar and Lovell
(2000).
A common criticism of the stochastic frontier method is that there is no a priori justification for the
selection of any particular distributional form for the inefficiency effects, uft. The half-normal and the
exponential distributions are arbitrary selections. Since both of these distributions have a mode at
zero, it implies that there is the highest probability that the inefficiency effects are in the neighborhood
of zero. This, in turn, implies relatively high efficiency. In practice, it may be possible to have a few
very efficient firms, but a lot of quite inefficient firms.

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