Stochastic Frontier Analysis

This group has 2%-points higher efficiency potentials on average compared to the average of all the multi-product sewage firms. The lowest average efficiency potentials are seen in the very small product companies or the very large product companies. However, the fewest multi-product sewage companies fall in to these two size categories. Thus, the tendency of economies of scope does not seem linked with size.

Clearly, as with water, North Jutland has around 32% of the multi-product firms and in the sewage multi-product industry this region has the lowest average efficiency potential compared to the remaining regions. The difference is at least 11%-points which we find noticeable and thus, the tendency of economies of scope may be affected by the regional factor or vice versa.

The noise of the model can also be calculated with:

SFA was performed in R under the benchmarking package as well as the DEA. The data is the same as used in DEA, but with FADO as output, and NVM, DC_NVM, AC_NVM as input⁴. We had to

determine the best functional form to describe the data, thus we did analyses using linear-linear, log-linear, log-log and the inverse.

The log-log for both the sewage and the water gave the best results in respect to the t- and p-values of the inputs. The log-log also showed applicable results in terms of the sign of lambda. We got negative lambdas for a few of the other functional forms, which should not be possible. Therefore, we chose the log-log function form for this analysis.

The water companies are presented only with municipal companies on the efficient frontier in the analysis. This was done since we were unable to get acceptable results otherwise, (i.e., lambdas were negative and/or p-values were extremely high). We tried running the SFA with the both the municipal companies and the private companies, as well as both groups separately. However, all analyses including the private firms did not give us any acceptable results. We believe the reason SFA cannot be performed with private firms is because there is a wide range of size within the private firms. Since SFA tries to make a frontier based on the chosen functional form, and the private firms are so different in sizes these two factors might not work together. Therefore, for the water companies, municipal firms only are presented.

4 This is the opposite order compared to DEA. However, it is due to the restriction of the SFA-function in the

The SFA results obtained from R for municipal water firms are presented below:

Table 15: R-Output of the SFA for the Water Companies

Parameters Std.err t-value Pr(>|t|) (Intercept) 2.3765 0.4594 5.1731 0.000

NVM -0.4879 0.2653 -1.8392 0.069

AC_NVM -0.1048 0.2395 -0.4377 0.662

DC_NVM 1.4517 0.2025 7.1671 0.000

lambda 1.2386 0.8180 1.5143 0.133

= 0.1067

= 0.04210429

= 0.06459501

Remark: NVM, AC_NVM and DC_NVM are variables used in the SFA as input. Lambda is the square root of the fraction: the variance due to inefficiency divided by the variance due to noise. The three sigmas describe the variation of the model.

.

The total variation within the model is: . The variation due to inefficiency is:

and the variation due to noise is: . The percentage of variation in the water model that is due to inefficiency is:

Therefore the percentage of the total variation of the model due to noise and lack of cost drivers is:

To us, this seems high, which may be an explanation for the general larger efficiency potential scores for the water firms compared to the sewage firms (see Figure 7).

For the sewage industry, the SFA was performed with all the companies, and we did not encounter the same problems (high p-values, negative lambda) as with the water companies.

The results obtained from R for sewage are presented below.

Table 16: R-Output of the SFA for the Sewage Companies

Parameters Std.err t-value Pr(>|t|) (Intercept) 2.0436 0.5349 3.820 0.000

NVM 0.3502 0.1891 1.851 0.067

AC_NVM 0.2414 0.1516 1.593 0.114

DC_NVM 0.2989 0.2042 1.464 0.146

lambda 2.5424 0.7519 3.382 0.001

= 0.14958

= 0.0200399

= 0.1295268

Remark: NVM, AC_NVM and DC_NVM are variables used in the SFA as input. Lambda is the square root of the fraction: the variance due to inefficiency divided by the variance due to noise. The three sigmas describe the variation of the model.

.

The total variation within the model is: . The variation due to inefficiency is:

and the variation due to noise is: . Hence, the percentage of variation in the sewage model that is due to inefficiency is:

Therefore the percentage of the total variation of the model due to noise and lack of cost drivers is:

The noise in data for the sewage companies is lower than that for the water companies. It should also be noted that the sewage companies have two more cost drivers (Figure 2 and 3) compared to the water companies in the calculation of the NVM.

The total variation is higher for the sewage companies than for the water companies, but that may be due to the lack of private water firms in the SFA. Hence, the total variation is for the whole

market (population) with the sewage firms, whereas the water firms’ total variation is only a sample of the market (only the municipal water firms).

SFA has demonstrated that there is noise within the model especially in regards to the water companies but also with the sewage companies. Therefore, we deem it acceptable that a correction of efficiency potentials is made to compensate for the noise or insufficient model.