### Technical efficiency and the Stochastic Frontier Analysis (SFA)

According to Farrell’s well-known model (Farrell 1957), technical efficiency is defined as the measure of the ability of a firm to obtain the best production from a given set of inputs (*output-increasing oriented)*, or as a measure of the ability to use the minimum feasible amount of inputs, given a determined level of output (*input-saving oriented*) (Greene, 1980; Atkinson and Cornwell, 1994) ^{[b]} .

This sub-section briefly illustrates how technical efficiency output-oriented measures can be obtained from Stochastic Frontier Analysis (SFA) models. SFA was originally and independently proposed by Aigner *et al.* (1977) and Meeusen and van der Broeck (1977). A stochastic frontier production function for panel data can be written as:

{y}_{\mathit{it}}=f\left({x}_{\mathit{it}},t;\beta \right)\mathit{\text{\xb7exp}}\phantom{\rule{0.25em}{0ex}}\left(e\right);

(1a)

e=\left({v}_{\mathit{it}}-{u}_{\mathit{it}}\right)i=1,2,\dots .Nt=1,2,\dots .T\phantom{\rule{0.75em}{0ex}}

(1b)

where *y*_{
it
} denotes the level of output for the *i*-th observation at year *t*; *x*_{
it
} is the row vector of inputs; *t* is the time index, *ß* is the vector of parameters to be estimated; *f* (·) is a suitable functional form for the frontier (generally Translog or Cobb-Douglas); *v*_{
it
} is a symmetric random error assumed to account for measurement error and other factors not under the control of the firm; and *u*_{
it
} is an asymmetric non-negative error term assumed to account for technical inefficiency (Kumbhakar and Lovell, 2000).

The *v*_{
i
}’s are usually assumed to be independent and identically distributed N\phantom{\rule{0.5em}{0ex}}\left(0,\phantom{\rule{0.5em}{0ex}}{\sigma}_{v}^{2}\right) random errors, independent of the *u*_{
it
}’s, which are assumed to be independent and identically distributed and with truncation (at zero) of the normal distribution \left|N\phantom{\rule{0.5em}{0ex}}\left(0,\phantom{\rule{0.5em}{0ex}}{\sigma}_{u}^{2}\right)\right|. The Maximum Likelihood Estimation (MLE) of (1) allows us to estimate the vector *ß* and the variance parameters {\sigma}^{2}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\sigma}_{u}^{2}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\sigma}_{v}^{2} and \gamma \phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\sigma}_{u}^{2}\phantom{\rule{0.5em}{0ex}}/\phantom{\rule{0.5em}{0ex}}{\sigma}_{u}^{2}+{\sigma}_{v}^{2}; where 0 ≤ γ ≤ 1 (Coelli, 1996) ^{[c]}. The technical efficiency (TE_{
i
}) measure is obtained from the ratio of *y*_{
it
} to the maximum achievable level of output (y*) that lies on the frontier (*u*_{
it
} = 0) and it is given by:

\mathit{TE}=\phantom{\rule{0.5em}{0ex}}\frac{\mathit{\text{yit}}}{y*}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\frac{f\phantom{\rule{0.25em}{0ex}}\left(\mathit{\text{xit}},t,\beta \right)\xb7\text{exp}\left(\epsilon \right)}{f\phantom{\rule{0.25em}{0ex}}\left(\mathit{\text{xit}},t,\beta \right)\xb7\text{exp}\left(\mathit{vi}\right)}\phantom{\rule{0.5em}{0ex}}=\mathit{exp}\phantom{\rule{0.25em}{0ex}}\left(-{u}_{i}\right)

(2)

It means that technical efficiency depends only on the inefficiency component under the control of the firm (*u*_{
it
}). Assuming a semi-normal distribution for *u*_{
it
} and according to Jondrow et al. (1982), we are able to estimate the degree of technical efficiency of each firm.

Some authors have proposed SFA models in which the inefficiency effects (u_{i}) are expressed as a function of a vector of observable explanatory variables and all parameters – frontier production and inefficiency effects – are estimated simultaneously (Kumbhakar *et al.*, 1991; Reisfschneider and Stevenson, 1991; Huang and Liu, 1994). These approaches were adapted by Battese and Coelli (1995) to take into account panel data. They proposed a model where the functional relationship between inefficiency effects and the firm-specific factors was directly incorporated into the MLE. The inefficiency term *u*_{
it
} has a truncated (at zero) normal distribution with mean *m*_{
it
}:

{u}_{\mathit{it}}={m}_{\mathit{it}}+{W}_{\mathit{it}}\phantom{\rule{0.5em}{0ex}}

(3a)

where *W*_{
it
} is a random error term which is assumed to be independently distributed, with a truncated (at -*m*_{
it
}) normal distribution with mean zero and variance *σ*^{2} (*i.e*. *W*_{
it
} ≥ - *z*_{
it
} such that *u*_{
it
} is non-negative). The mean *m*_{
it
} is defined as:

{m}_{\mathit{it}}=Z\phantom{\rule{0.25em}{0ex}}\left({z}_{\mathit{it}},d\right)\phantom{\rule{3.25em}{0ex}}i=1,2,\dots .N\phantom{\rule{3.5em}{0ex}}t=1,2,\dots .T\phantom{\rule{4.25em}{0ex}}

(3b)

where Z is the vector (Mx1) of the *z*_{
it
} firm-specific inefficiency variables of inefficiency; and *δ* is the (1xM) vector of unknown coefficients associated with *z*_{
it
}. This allows us to estimate the inefficiency effects arising from the *z*_{
it
} explanatory variables.

### Data and empirical model

Data were collected on a balanced panel data from 36 sheep dairy processors in Sardinia. The sample consisted of 18 private firms and 18 cooperatives ^{[d]}.

The panel data covered a six-year period from 2004 to 2009 for a total of 216 observations. Information came from the Official Register of Accounts, specifically from the income statement, that firms have to submit to the “*Register of the Companies*” of the Italian Chambers of Commerce, Industry, Handicrafts and Agriculture (C.C.I.A.A.) ^{[e]}. In other words, we used economic and financial data reported in the available official balance sheets to describe the sheep dairy production process in Sardinia. We assumed a Translog functional form as the frontier technology specification for the sheep dairy firms. This model is similar to the Battese and Coelli (1995) model, with a non-neutral specification for the production frontier function. Basically, following Huang and Liu (1994), the model assumes that technical efficiency depends on both the method of application of inputs and the intensity of input use (Karagiannis and Tzouvelekas, 2005).

Thus the inefficiency term *u*_{
it
} explained by (3) is equal to:

{u}_{\mathit{it}}={d}_{0}+{d}_{\mathit{it}}{z}_{i}+{d}_{m}\mathit{ln}\phantom{\rule{0.25em}{0ex}}{x}_{\mathit{\text{mit}}}+{W}_{\mathit{it}}\phantom{\rule{4.25em}{0ex}}i=1,2,\dots .N\phantom{\rule{1.5em}{0ex}}t=1,2,\dots .T

(4)

which allows us to evaluate the role of inputs in conditioning inefficiency.

The Translog stochastic function production model is specified as follows:

\mathit{ln}{Y}_{\mathit{it}}={\beta}_{0}+{\displaystyle \sum _{j=1}^{5}{\beta}_{j}\phantom{\rule{0.25em}{0ex}}\mathrm{ln}\phantom{\rule{0.25em}{0ex}}x\mathit{\text{jit}}}+\frac{1}{2}{\displaystyle \sum _{j\le}^{4}{\displaystyle \sum _{k=1}^{4}{\beta}_{\mathit{jk}}\phantom{\rule{0.25em}{0ex}}\mathrm{ln}\phantom{\rule{0.25em}{0ex}}x\mathit{\text{jit}}\xb7\phantom{\rule{0.25em}{0ex}}\mathrm{ln}\phantom{\rule{0.25em}{0ex}}x\mathit{\text{kit}}}\phantom{\rule{0.5em}{0ex}}+\left(\mathit{\text{vit}}-\mathit{\text{uit}}\right)\phantom{\rule{0.25em}{0ex}}}

(5a)

where four explanatory variables are used to describe the production frontier (a fifth dummy variable is added in order to identify possible technological differences between private firms and cooperatives).

The dairy output was aggregated into one category (Y), which represents the *value of sheep cheeses* (and by-products) produced by each firm in a certain year (measured in Euros).

The aggregate inputs that were included as variables of the production function are as follows:

X_{1}*Intermediate inputs* used in the production process: the cost of intermediate inputs spent by firms (measured in Euros);

X_{2}*Labour* used in each firm in terms of total wages paid to workers (measured in Euros);

X_{3}*Capital:* measured in terms of annual depreciation rate so as to have a measure of the average annual use of the capital stock (measured in Euros);

X_{
t
}*Time* *:* the year of observation, so that the technological progress component can be captured (2004 = 1; 2005 = 2; ….2009 = 6).

As mentioned above, a further variable was inserted in the model, a *dummy* variable X_{p} that describes the organisational form of the firms, *i.e*. private firms (X_{p} = 1) or cooperatives (X_{p} = 0). This variable was inserted in order to evaluate if there were technological differences between the two types of firm and so to analyse whether they are part of a single or two different production frontiers.

Taking into account formula (4), the inefficiency model (5a) has the following form:

{u}_{\mathit{it}}={\delta}_{0}+{\delta}_{1}{Z}_{1\mathit{it}}++{W}_{\mathit{it}}

(5b)

Explanatory firm-specific variables of the inefficiency effects were represented by Z_{1.} This reflects the *Age* of the firms and - according to the non-neutral model proposed by Huang and Liu (1994) – the same pool of variables (included time) used to describe the production frontier function (x_{
it
}). Thus the (5b) function is expressed by:

\begin{array}{l}{u}_{\mathit{it}}={\delta}_{0}+{\delta}_{1}\mathit{Ag}{e}_{\mathit{it}}+{\delta}_{2}\mathit{\text{lnIntermediate}}\phantom{\rule{0.25em}{0ex}}\mathit{\text{input}}{s}_{\mathit{it}}+{\delta}_{3}\mathit{\text{lnLabou}}{r}_{\mathit{it}}+{\delta}_{4}\mathit{\text{lnCapita}}{l}_{\mathit{it}}+\\ \phantom{\rule{2.7em}{0ex}}{\delta}_{t}\mathit{\text{Tim}}{e}_{\mathit{it}}+{W}_{\mathit{it}}\end{array}

(6)

The variable *Age* was selected in order to evaluate the role of experience in technical efficiency. We assume that older firms are more efficient than newer ones because they have accumulated experience and knowledge (*learning by doing*) that allow them to improve their technical and economic performance (Nelson and Winter, 1982).