Technical efficiency in the Italian performing arts companies

Abstract

This paper evaluates the determinants of firms’ technical efficiency in the Italian Performing Arts (PA) sector, by estimating a stochastic production frontier for an unbalanced panel of 107 firms over the period 2005–2012. The panel data setting allows us to control for both unobserved and observed heterogeneity of PA firms, reaching several interesting and robust findings. Firstly, it finds that the Italian PA firms are scale inefficient as they generally operate in either an increasing or decreasing returns to scale landscape. This result is reinforced by the fact that the smaller firms (10–49 employees) are the most technically efficient firm size class. Secondly, this research proves that the efficiency score is on average 66%, demonstrating that PA firms’ output could be substantially increased without the use of new inputs. Thirdly, it confirms that the quality matters and competences increase the efficiency in the sector. Finally, the environmental factors (especially the quality of institutions) have a strong impact on technical efficiency of PA firms, supporting that regional differences also exist in this sector.

Appendix

1.1 Modelling TE using pooled, TRE, TREM and LCM models

The technical efficiency (TE) can be modelled as either output-oriented or an input-oriented technical efficiency. Given our data set, we estimate an output-oriented technical efficiency for the production frontier model. Following Aigner et al. (1977), the simple pooled stochastic production frontier (PM) for panel data can be written in log-linear form:

$$ \ln {y}_{\mathrm{it}}=f\left({x}_{\mathrm{it}};\beta \right)+{v}_{\mathrm{it}}-{u}_{\mathrm{it}} $$

(5)

where lny _it is the log of observed output (revenue) for firm i in year t, and x _it is the vector of inputs (in logs); β is a J × 1 vector of the corresponding production function parameters, including the constant that is common to all firms, v _it is the statistical noise term with zero mean and constant variance; u _it ≥ 0, is a non-negative one-sided inefficiency term which follows a half-normal distribution and ${u}_{\mathrm{it}}\sim {N}^{+}\left(0,{\sigma}_u^2\right)$. The parameters of the model given in Eq. (5) are estimated by maximum likelihood (LM) and the inefficiency term is computed as the conditional mean of the inefficiency using the technique of Jondrow et al. (1982) so that E[−u _it| v _it − u _it]. The variance parameters for this half-normal model in terms are given by: ${\lambda}^2={\sigma}_u^2/{\sigma}_v^2$ and ${\lambda}^2={\sigma}_u^2/{\sigma}_v^2\ge 0$. If λ = 0, then there are no technical inefficiency effects and all deviations from the production frontier are due to noise.

The term u _it is the log-difference between the maximum output $\left(\ln {y}_{\mathrm{it}}^{\ast }=f\left({x}_{\mathrm{it}};\beta \right)+{v}_{\mathrm{it}}\right)$ and the actual output (lny _it), so that the technical inefficiency equals the percentage by which the actual output could be increased without increasing the inputs of production. Consequently, the technical efficiency index (TE_it) for firm i in year t, is obtained as the ratio of the observed production over the maximum technical output obtainable for a firm, defined by the frontier production frontier:

$$ {\mathrm{TE}}_{\mathrm{it}}==\frac{f\left({x}_{\mathrm{it}};\beta \right)\exp \left({v}_{\mathrm{it}}-{u}_{\mathrm{it}}\right)}{f\left({x}_{\mathrm{it}};\beta \right)\exp \left({v}_{\mathrm{it}}\right)}=\exp \left(-{u}_{\mathrm{it}}\right) $$

(6)

In order to integrate both the unobserved heterogeneity and the time-varying inefficiency into the stochastic production frontier model, Greene (2004, 2005) proposed a TRE model which extends the model given in Eq. (6) by adding a firm-specific stochastic term, w _i , which is an i.i.d. random component:

$$ \ln {y}_{\mathrm{it}}={w}_i+f\left({x}_{\mathrm{it}};\beta \right)+{v}_{\mathrm{it}}-{u}_{\mathrm{it}} $$

(7)

The TRE model given in Eq. (7) has a two-part composite error ε _it = v _it − u _it which has asymmetric distribution as in the previous specification. The model parameters are estimated by applying simulated maximum likelihood procedure proposed by Greene (2005). The inefficiency term, u _it, is obtained by the conditional mean of the inefficiency term so that E[−u _it | w _i + ε _it] and the TE_it score is obtained in line with Eq. (6) as before.

We also extend the TRE model and use a Mundlak’s (1978) formulation to reduce the possible bias that results from correlations between the unobserved heterogeneity term (w _i ) and the explanatory variables (inputs). The extended model, called here the TREM model, is defined by introducing the within-group means of inputs into the TRE model in Eq. (7) and it is given by Eq. (8):

$$ {w}_i={\lambda}^{\hbox{'}}\cdot \overline{X_i}+{\eta}_i $$

(8)

where $\overline{X_i}=\left(1/{T}_i\right){\sum}_{t=1}^T{X}_{it}$ are firm-specific means, T _i is the number of time periods for firm i, λ’ is the corresponding vector of coefficients to be estimated, and ${\eta}_i\sim N\left(0,{\sigma}_{\eta}^2\right)$. Eq. (8) divides the firm-specific stochastic term into two components: the first term explains the relationship between the exogenous variables and the firm-specific effect (with the auxiliary coefficients λ _i ) and the second component, η _i , is orthogonal to the explanatory variables. In this way, we control for any correlation between the exogenous variables and the heterogeneity component eliminating any bias.

Furthermore, the TFE model of Greene (2005) allows that the time-invariant heterogeneity (firms’ effects) is correlated with the inputs as w _i is assumed a firm-fixed effect (i.e. a firm’s specific dummy). An important drawback of this specification is, however, that the vector of firm-specific dummies creates an incidental parameter problem when the number of time periods (T) is small as it is the case with our panel. Hence, with small T, the w _i are inconsistent and subject to small sample bias which may affect the TE scores (Kumbhakar et al. 2014). Therefore, this model is not applied in our setting.

When heterogeneous technologies are present for different groups of firms, the LCM permits us to exploit the information contained in the data more efficiently as it can incorporate technological heterogeneity. Following Greene (2005), we obtain the LCM by writing Eq. (5) as follows:

$$ {\left.\operatorname{}\ln\;{y}_{\mathrm{it}}=f\left({x}_{\mathrm{it}};\beta \right)\right|}_j+{v}_{\mathrm{it}}{\left|{}_j-{u}_{\mathrm{it}}\right|}_j $$

(9)

where subscript i denotes each performing arts firm, t indicates time as before and j represents the different classes (groups). The vertical bar means that there is a different model for each class j. Assuming as before that v _it is normally distributed and that u _it follows a half-normal distribution as before, the likelihood function (LF) for each firm i at time t for group j is presented in Kumbhakar and Lovell (2000), Greene (2005, p. 291, Eq. 34), and Alvarez and del Carrol (2010, p. 3). The LF, for each firm i in group j, is obtained as the product of the LFs in each time period t:

$$ {\mathrm{LF}}_{\mathrm{ij}}=\prod_{t=1}^T{\mathrm{LF}}_{\mathrm{ij}\mathrm{t}} $$

(10)

The LF for each firm is then obtained as a weighted average of its LF for each group j, using prior probabilities (P _ij) of class j membership as weights:

$$ {\mathrm{LF}}_i=\sum_{j=1}^J{P}_{\mathrm{ij}}{\mathrm{LF}}_{\mathrm{ij}} $$

(11)

The prior probabilities must lie between 0 and 1 and the sum of these probabilities must be 1. The firm i resides permanently in a specific class but this is unknown and hence P _ij reflects the analyst’s uncertainty. The overall log-likelihood function is the sum of the individual log LFs:

$$ \log \mathrm{LF}=\sum_{i=1}^N\log {\mathrm{LF}}_i $$

(12)

The log LF is maximised with respect to the parameter set θ _j  = (β _j , σ _j , λ _j ), using the conventional methods. The posterior probabilities of class membership are estimated using the Bayes theorem:

$$ P\left(\mathrm{jt}/i\right)=\frac{P_{\mathrm{ij}}{\mathrm{LF}}_{\mathrm{ij}\mathrm{t}}}{\sum_{i=1}^J{P}_{\mathrm{ij}}{\mathrm{LF}}_{\mathrm{ij}\mathrm{t}}} $$

(13)

In the final stage, comparing the posterior probabilities for all j, the most probable latent class j* and inefficiencies u _it|j* are estimated. Thus, predictions of inefficiency are computed as before using Jondrow et al. (1982) for the group with the largest posterior probability.

Table 9 Information on the collected data sample

Table 10 Description of variables used

Table 11 Frequency distribution of TE scores

Table 12 Correlations coefficients of TE scores for the basic SFA model

Table 13 SFA model with efficiency determinants, using employees as size variable

Table 14 SFA model with efficiency determinants, using lnPrice

Fig. 6

Scatter diagram matrix of TE scores obtained for the basic SFA models