6 Analyses
6.1 Returns to Scale Analyses
6.1.3 Efficiency Potential Analyses
It can be seen in the CRS graphs that the frontiers are straight lines, whereas the VRS frontiers have kinks. In the frontier graphs of the water companies a cluster is seen close to the origin (southwest corner). Furthermore, it is noted that the water companies’ distance to the frontier is greater under CRS than VRS. A graphical representation of the increase in frontier companies for VRS compared to CRS can be seen.
For the sewage companies the cluster is not as extreme as for the water companies. Nonetheless, the distance to the frontier is also greater under CRS than VRS, especially for the companies situated to the east, which benefit from a shift of CRS to VRS, since they would seem the most inefficient under the CRS-assumption (furthest away from the frontier). Likewise, more companies constitute the frontier under VRS compared to CRS.
These graphs visually enhance the statement that CRS is considered harsher than VRS.
Below are the results for water firms:
Figure 5: Distribution of Efficiency Potentials for the Water Firms
For the water firms, the efficiency potentials, when using CRS first band, show a cluster between 40-80%. A shift in the efficiency potentials towards the middle is apparent when changing from CRS (F1) to CRS (F2). It can also be seen that percentage of firms above the 65% efficiency potential has decreased.
0 2 4 6 8 10 12 14 16 18 20
0-5 11-15 21-25 31-35 41-45 51-55 61-65 71-75 81-85 91-95
Percentage of Companies
Intervals of Efficiency Potentials in Percentage
CRS (F1)
0 2 4 6 8 10 12 14 16 18 20
0-5 11-15 21-25 31-35 41-45 51-55 61-65 71-75 81-85 91-95
Percentage of Companies
Intervals of Efficiency Potentials in Percentage
CRS (F2)
0 2 4 6 8 10 12 14 16 18 20
0-5 11-15 21-25 31-35 41-45 51-55 61-65 71-75 81-85 91-95
Percentage of Companies
Intervals of Efficiency Potentials in Percentage
VRS (F1)
0 2 4 6 8 10 12 14 16 18 20
0-5 11-15 21-25 31-35 41-45 51-55 61-65 71-75 81-85 91-95
Percentage of Companies
Intervals of Efficiency Potentials in Percentage
VRS (F2)
When using VRS (F1) there is also a large percentage of firms that have an efficiency potential between 35-70% and a larger percentage of firms have lower efficiency potentials (between 0-40%) compared to the CRS (F1) approach.
When using VRS (F2) the efficiency potentials have shifted further to the west. However, the VRS assumption shows more even distribution and lower efficiency potentials in general, as well as a lower percentage of firms present in each category compared to the CRS approach. This could be linked to the number of companies constituting the frontier, and that increases the likelihood of a company having a smaller relative distance from the frontier (See Table 3).
The same analysis has been done for the sewage companies.
Figure 6: Distribution of Efficiency Potentials for the Sewage Firms
For the sewage firms, we do not see as clear of a pattern as with water in regard to using CRS (F1) or CRS (F2) as the distribution of efficiency potentials seem somewhat similar. There is a cluster of companies with an efficiency potential between 20-50% in both cases (F1 and F2).
0 2 4 6 8 10 12 14 16 18 20
0-5 11-15 21-25 31-35 41-45 51-55 61-65 71-75 81-85 91-95
Percentage of Companies
Intervals of Efficiency Potentials in Percentage
CRS (F1)
0 2 4 6 8 10 12 14 16 18 20
0-5 11-15 21-25 31-35 41-45 51-55 61-65 71-75 81-85 91-95
Percentage of Companies
Intervals of Efficiency Potentials in Percentage
CRS (F2)
0 2 4 6 8 10 12 14 16 18 20
0-5 11-15 21-25 31-35 41-45 51-55 61-65 71-75 81-85 91-95
Percentage of Companies
Intervals of Efficiency Potentials in Percentage
VRS (F1)
0 2 4 6 8 10 12 14 16 18 20
0-5 11-15 21-25 31-35 41-45 51-55 61-65 71-75 81-85 91-95
Percentage of Companies
Intervals of Efficiency Potentials in Percentage
VRS (F2)
When going from CRS (F1) to VRS (F1), there is a very small shift in the companies’ efficiency potentials. There are more companies that have a lower efficiency potential and the cluster is now between 20-45%, which is not very different from CRS (F1).
Once more, a bigger shift occurs when using VRS (F2) compared to CRS (F2). The largest proportion of companies lies within the intervals of 0-5%, whereas the majority of sewage firms would have efficiency potential between 41-45% in the VRS (F1) approach; as with the water companies, the second frontier under VRS consists of more frontier companies than the first frontier.
Hence, a summary of results is presented to give a concise view. Below are two graphs, water (blue) and sewage (red), that represent the 95% confidence intervals of the efficiency potentials under the CRS and VRS assumption using the first and second band.
Figure 7: Efficiency Potentials of Water and Sewage Companies under Different Scale and Frontier Assumptions
Remarks: The graph to the left is for water companies and the graph to the right is for sewage companies.
Note that the second axes have different ranges.
This means that with 95% confidence the average efficiency potential of a water company using the first band as the efficient frontier under the CRS assumption is 56% and that the efficiency potential lies between 54-58%.
For water firms, there is a large shift (approximately 13%-points) from VRS (F1) to VRS (F2). There is also a 10%-point decrease in average efficiency potentials when comparing CRS (F1) and VRS (F1).
58
50 49
36 56
47 46
33 54
45 43
30 25
30 35 40 45 50 55 60
CRS (F1) CRS (F2) VRS (F1) VRS (F2)
Efficiency Potentials in Percentage
Returns to Scale Assumption Upper Bound Mean Lower Bound
37
34 36
30 34
31 32
27 31
28 29
23 20
25 30 35 40
CRS (F1) CRS (F2) VRS (F1) VRS (F2)
Efficiency Potentials in Percentage
Returns to Scale Assumptions Upper Bound Mean Lower Bound
This can be interpreted that the size of the companies in water differs substantially, and as the theory says, this has an impact depending on which RTS is used (see section 5.1.1 Returns to Scale in Data Envelopment Analysis). Additionally, the difference in using the first or second band under CRS affects the average efficiency potentials by 9%-points.
In regard to the sewage companies, the results do not have as great of difference. The average potential of CRS (F1) and CRS (F2) differ by 3%-points, and there is little difference between using CRS (F1) and VRS (F1). Thus, the size of the sewage companies must be more similar than with the water companies. There is a bigger shift going from VRS (F1) to VRS (F2) but more companies are also present on the second frontier, VRS (F2).
Clearly, the average efficiency potentials are in general higher with the water companies compared to the sewage companies. Comparing the average efficiency potentials for water to sewage
companies there is a large difference especially when using CRS (approximately 20%-points). This could indicate several things: the water companies are more inefficient in general, the variation in company size affects the results or the model is not as accurate as the model for sewage companies (the model for sewage companies does contain more cost drivers). However, it is important to remember that the size of the efficiency potentials of the water companies might also be affected by the fact that private companies are allowed on the efficient frontier.
The differences can also be seen when looking at the total market savings. The total savings in Danish kroner (DKK) can be calculated by multiplying the efficiency potential by the DOiPL.
When we sum up each individual savings, the total for the Danish market can be found.
Table 6: Potential Market Savings within the Water and Sewage Companies
In DKK CRS (F1) CRS (F2) VRS (F1) VRS (F2) Water 867,989,164 749,517,123 486,194,862 354,408,769 Sewage 1,163,955,051 1,050,422,360 1,065,631,102 819,536,276 Total 2,031,944,215 1,799,939,483 1,551,825,964 1,173,945,045
Remark: CRS is Constant Returns to Scale, VRS is Variable Returns to Scale, F1 is first frontier, F2 is second frontier.
From this table, it is clear to see that the largest shift in market savings lie within the shift from CRS to VRS. By using CRS (F1) for water over VRS (F1), the potential savings are 381,794,302 DKK increase. Sewage does not have a big difference; by using CRS (F1) over VRS (F1), the more potential savings is 98,323,949 DKK.
This table also displays the impact between using first and second band. The Water Department uses CRS (F2) over CRS (F1), and that choice has a total (water and sewage) impact of 232,004,732 DKK potential market savings.
There is a difference between using first or second band, but the reason behind the usage for the second band is subjective. We will base the remaining analyses on the use of the first frontier only, as this offers the most savings and the first frontier was found to be representative of the market, thus, we found it unnecessary to use the second frontier.
In regard to using CRS to VRS, sewage companies do not show a wide variety of results between the two models. This cannot be said for the water firms. The efficiency potentials differ depending on the assumption chosen. As mentioned, CRS is often used in regulatory environments since it is harsher, and is more representative of a competitive market (Munksgaard et al. 2005); therefore, we believe that arguments can be made for using VRS within water, but since regulation is to protect the consumers, using CRS is an acceptable assumption to us. This will be discussed later on when a better understanding of the whole picture is obtained.