María Molinos-Senante

Simon Porcher

Alexandros Maziotis

IMPACT OF REGULATION ON ENGLISH AND WELSH WATER-ONLY COMPANIES: AN INPUT-DISTANCE FUNCTION APPROACH

Keywords: productivity growth, service quality, efficiency, regulation, water industry, technical change

The assessment of productivity change over time and its drivers is of great significance for water companies and regulators when setting urban water tariffs. This issue is even more relevant in privatized water industries, such as those in England and Wales, where the price-cap regulation is adopted. In this paper, an input-distance function is used to estimate productivity change and its determinants for the English and Welsh water-only companies (WoCs) over the period 1993-2009. The impacts of several exogenous variables on companies' efficiencies are also explored. From a policy perspective, this study describes how regulators can use this type of modeling and results to calculate illustrative X factors for the WoCs. The results indicate that the 1994 and 1999 price reviews stimulated technical change and there were small efficiency gains. However, the 2004 price review did not accelerate efficiency change or improve technical change. The results also indicated that during the whole period of study, the excessive scale of the WoCs contributed negatively to productivity growth. On average, WoCs reported relatively high efficiency levels, which suggests that they had already been investing in technologies that reduce long-term input requirements with respect to exogenous and service-quality variables.

Finally, an average WoC needs to improve its productivity toward that of the best company 2 by 1.58%. The methodology and results of this study are of great interest to both regulators and water-company managers for evaluating the effectiveness of regulation and making informed decisions.

INTRODUCTION

In many countries, the water and sewerage industry is regulated because water services are provided through a monopoly regime. In this context, the assessment of the total factor productivity (TFP) has been evidenced as a useful tool for both water companies and regulators. The importance of evaluating productivity growth is especially marked in countries where the performance evaluation of water companies is part of the process of setting water tariffs [Maziotis et al., 2016a]. While there are several countries whose process to update water tariffs is based on price-cap regulation [START_REF] Marques | Regulation of Water and Wastewater Services: an international comparison[END_REF], the English and Welsh water industry has become a paradigmatic case [Molinos-Senante et al., 2016].

After the privatization of the water companies in England and Wales in 1989, the regulator (Ofwat) adopted the RPI-K price-cap methodology to regulate the water industry [START_REF] Saal | Estimating OPEX Productivity Growth in English and Welsh Water and Sewerage Companies[END_REF] (see more details in section 2). Moreover, since the privatization, there have been two types of water companies operating in England and Wales: Water and Sewerage Companies (WaSCs) and Water Only Companies (WoCs) [Molinos-Senante et al., 2015a].

Given the importance of the productivity-growth estimation in the framework of the English and Welsh water-industry regulation, several researchers have assessed the TFPs of the water companies in England and Wales. To this end, from a methodological point of view, three main approaches have been used. Most of the previous studies employed stochastic-frontier techniques to provide TFP growth estimates [START_REF] Saal | The impact of privatization and regulation on the water and sewerage industry in England and Wales: a translog cost function model[END_REF][START_REF] Saal | Productivity and price performance in the privatized water and sewerage companies in England and Wales[END_REF]2006;Botasso and Conti, 2009a;2009b;[START_REF] Saal | Determining the contribution of technical efficiency, and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF]. A second methodological approach based on index numbers was applied by [START_REF] Maziotis | Profit, productivity and price performance changes in the water and sewerage industry: An empirical application for England and Wales[END_REF]2016a]. In addition, alternative productivity indices (Malmquist, Luenberger, Färe-Primont) based on nonparametric techniques were also employed to evaluate TFP changes of water companies in England and Wales [START_REF] Portela | Productivity change in the water industry in England and Wales: Application of the meta-malmquist index[END_REF][START_REF] Molinos-Senante | The Luenberger productivity indicator in the water industry: an empirical analysis for England and Wales[END_REF]2016;[START_REF] Maziotis | Assessing the impact of quality of service on the productivity of water industry: A Malmquist-Luenberger productivity index approach for England and Wales[END_REF].

In spite of the methodological differences among these previous studies, all of them have in common that they evaluated the productivity growth of WaSCs or WaSCs and WoCs.

However, with the exception of Bottasso and Conti [2009a], none of the previous papers focused exclusively on English and Welsh WoCs. In this context, it should be highlighted that [START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF] and Molinos-Senante et al. [2015a] evidenced that English and Welsh WaCs and WoCs operate under different technologies and therefore cannot be modelled together.

In the framework of benchmarking studies applied to the water industry, the importance of accounting for the operational environment in which the water companies are working has been illustrated [START_REF] Carvalho | The influence of the operational environment on the efficiency of water utilities[END_REF]. Thus, some exogenous factors, along with the quality of the service provided by water companies, may influence their operational costs [START_REF] Saal | Determining the contribution of technical efficiency, and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF][START_REF] De Witte | Influential observations in frontier models, a robust non-oriented approach to the water sector[END_REF]. Hence, it is essential to integrate these variables into productivity-growth assessment. This issue has not been ignored by previous studies on this topic since [START_REF] Saal | Determining the contribution of technical efficiency, and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF] pioneered by integrating quality adjusted outputs in TFP estimates for English and Welsh WaSCs. However, they focused only on analyzing the effect of exogenous variables on the productivity growth of the English and Welsh water companies. In other words, none of the previous studies assessing the TFP change based on stochastic methods have integrated quality-of-service to customers as operating variables that might influence input requirements.

Against this background, the objectives of this paper are threefold. The first is to analyze the impact of regulation on the productivity growth and its drivers, namely, efficiency, technical and scale changes of the English and Welsh WoCs from 1993 to 2009. It should be highlighted that regulation is essential for leading water companies towards long-term sustainability [Molinos-Senante and Sala-Garrido, 2016]. The second objective is to explore explanatory variables (external factors and quality of service) affecting the productivity changes of WoCs. Finally, an X factor for each WoC is proposed to be applied in CPI-X price regulation.

To the best of our knowledge, only [START_REF] Bottasso | Scale economies, technology and technical change: evidence from the English water only sector[END_REF] focused on the performance assessment of English and Welsh WoCs instead of WaSCs or both WaSCs and WoCs.

Moreover, their study only involved 144 observations since the period observed was 1995/96-2004/05. In this context, our study extends significantly the period of time analyzed. The main objective of [START_REF] Bottasso | Scale economies, technology and technical change: evidence from the English water only sector[END_REF] was to evaluate the presence of scale economies in the UK WoCs. By contrast, our study goes much further because it also evaluates the contributions of technical and efficiency changes to productivity growth of WoCs. Our paper also innovates with respect to the explanatory variables of productivity change of English and Welsh water companies since quality-of-service variables are also integrated into the model.

The methodology and results of this research are of great interest from the policy point of view. First, the methodology applied allows estimating an X factor for each WoC, which is essential to promote innovation and competition in the CPI-X price-regulation framework.

Second, the integration of quality of service into the assessment provides essential information to the water regulator to develop policies (incentives or new standards) to improve the sustainability of the urban water cycle.

THE ENGLISH AND WELSH WATER INDUSTRY

The water and sewerage industry in England and Wales was privatized in 1989 and before privatization there were 10 Regional Water Authorities responsible for the water and sewerage supply in England and Wales and 20 Statutory Water companies, which were already privatized companies that were only responsible for the supply of water. After 1989, the Regional Water Authorities were privatized and formed the WaSCs and the Statutory Water Companies became WoCs. While in 1993 the total number of companies was 30, they were 22 in 2008 due to mergers and acquisitions. The 10 WaSCs are responsible for the supply of water in areas that are not supplied by the WoCs, and the collection, treatment and disposal of sewerage in all areas. Due to mergers and takeovers between WaSCs/WoCs and WoCs/WoCs occurred during the period 1993-2010, the number of WoCs fell since privatization. In 1993 there were 20 WoCs and subsequent mergers in 1997 and 1998 led to a reduction in the number of WoCs, from 20 to 16. Further mergers between WaSCs/WoCs and WoCs/WoCs in 2010, reduced the number of WoCs to 12. There are three regulatory bodies in the UK water and sewerage industry. The Office of Water Services (Ofwat), which is the economic regulator and sets the price limits for each company every five years, the Environment Agency (EA), which is responsible for pollution control, licensing and regulation of water abstraction, and the Drinking Water Inspectorate (DWI), which is responsible for controlling and monitoring drinking water quality.

In England and Wales, water industry is regulated based on the RPI-K price-cap methodology [START_REF] Saal | Estimating OPEX Productivity Growth in English and Welsh Water and Sewerage Companies[END_REF]. K can be decomposed into two factors, namely, Q and X, where Q is the price increase necessary to finance the required environmental and quality improvements and X is the productivity offset [Bottasso and Conti, 2009a]. At privatization in 1989, price limits were set by the Secretary of State for a period of ten years and were, on average, RPI +5.2 per annum for the industry, RPI+5 per annum for WaSCs and RPI+6.1 per annum for WoCs. The K factor was set at a high level in order to make up for years of underinvestment before privatization and to ensure that the shares of the public companies would be attractive to potential investors. However, as documented in past studies [see for instance [START_REF] Maziotis | Profit, productivity and price performance changes in the water and sewerage industry: An empirical application for England and Wales[END_REF], the first price caps were relatively lax and as a result, Ofwat exercised its right to reset price caps in 1994. Thus, the average K factor after the 1994 review was RPI+0.9 for the industry, RPI+1.0 for the WaSCs and RPI-0.4 for the WoCs, representing a considerable tightening of price caps [Ofwat, 1994]. This continued in the price review of 1999 with an average K factor of RPI-2.1 for the industry, RPI-2.0 for the WaSCs and RPI-2.8 for the WoCs [Ofwat, 1999]. In the 2004 price review the K factor increased again to an average of 4.2% per annum [Ofwat, 2004b], whereas in 2009 Ofwat published its final price determinations suggesting an average K factor of RPI+0.5 per annum for WaSCs ,and RPI+0.3 per annum for WoCs for the next five years [Ofwat, 2009].

The determination of X-factors in the UK water and sewerage industry, and therefore of price limits, is carried out through benchmarking techniques, which provide information about the relative performance of companies. As there are companies that are regulated under the same framework, the regulator can compare the performance of each company against the performance of the others in the industry. When establishing price limits, Ofwat uses cross section econometric techniques and unit cost models to assess operating expenditure (OPEX) and capital expenditure (CAPEX) relative efficiency separately, for water and sewerage services. For instance, operating expenditure relative efficiency for water services is derived by the use of four econometric models: water distribution, resources and treatment, power and business activities. OPEX efficiency of sewerage services is derived using two econometric models, namely network including power and large sewerage treatment works and three unit cost models, namely small sewerage treatment works, sludge treatment and disposal and business activities [Ofwat, 2005a].

Once the most efficient company has been determined by Ofwat from these models, then the regulator tests how much each company needs to reduce its actual costs to appear as efficient as the most efficient company. Finally, Ofwat also takes into consideration factors that are not included in the efficiency models, called "special factors", for instance the high levels of deprivation in a region supplied by a water company which might result in high levels of bad debt.

METHODOLOGY

To explore the input-output relationship affecting the performance of the water industry, we adopt an input-distance function estimated using a translog function following [START_REF] Orea | A parametric decomposition of generalized Malmquist productivity index[END_REF] and [START_REF] Coelli | A primer on efficiency measurement for utilities and transport regulations[END_REF]. The input-distance function provides an input-oriented measure of technical efficiency by finding the maximum possible radial contraction of an input that can produce a given quantity of output. The nature of production and regulation in the water industry in England and Wales justifies the use of an input-distance function rather than an output-distance function. In this context, water companies do not seek to increase outputs given an exogenous input allocation but rather try to reduce input use for a given exogenous output level, such as the water produced or the number of customers. The relationship can be stated formally as

𝑑 𝑖𝐼 (𝑌 𝑖𝑡 , 𝑋 𝑖𝑡 ) = max {Ф: ( 𝑋 𝑖𝑡 Ф ) ∈ 𝐿(𝑋 𝑖𝑡 , 𝑌 𝑖𝑡 ), Ф > 0 }, ( 1
)
where Ф is the distance by which the input vector can be deflated, and the input set 𝐿(𝑋, 𝑌) represents the set of all input vectors that can produce the output vector. We follow the axioms of [START_REF] Färe | Multi-output production duality: theory and applications[END_REF], according to which the input-distance function has the following properties: (i) 𝑑 𝑖𝑛𝑝𝑢𝑡 (𝑋, 𝑌) is non-decreasing, positively linearly homogenous and concave in 𝑋, and non-increasing and quasi-concave in 𝑌; (ii) 𝑋 belongs to the input set of 𝑌 if and only if 𝑑 𝑖𝑛𝑝𝑢𝑡 (𝑋, 𝑌) > 1. When a company operates on the frontier of the period 𝑡 technology, isoquant 𝐿(𝑋, 𝑌), then 𝑑 𝑖𝑛𝑝𝑢𝑡 (𝑋, 𝑌) = 1.

The translog function can be used as a second-order approximation to a true but unknown technology. This characteristic is of particular interest because it reduces the estimation bias resulting from an improper assumption concerning functional form [START_REF] Saal | Determining the contribution of technical efficiency, and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF].

For 𝑁 (𝑖: 1, … , 𝑁) companies observed in 𝑇 (𝑡: 1, … , 𝑇) time periods, 𝑀 (𝑚: 1, … , 𝑀) outputs and 𝐾 (𝑘: 1, … , 𝐾) inputs, the translog input-distance function is specified as follows:

ln 𝑑 𝐼 (𝑋 𝑖𝑡 , 𝑌 𝑖𝑡 , 𝑡) = 𝑇𝐿(𝑌 𝑖𝑡 , 𝑋 𝑖𝑡 , 𝑡) = 𝛼 0 + ∑ 𝜋 𝑚 ln 𝑥 𝑚𝑖𝑡 + 1 2 𝑀 𝑚=1 ∑ ∑ 𝛾 𝑚𝑛 𝑀 𝑛=1 𝑙𝑛𝑥 𝑚𝑖𝑡 ln 𝑥 𝑛𝑖𝑡 + 𝑀 𝑚=1 ∑  𝑘 ln 𝑦 𝑘𝑖𝑡 + 1 2 ∑ ∑ 𝛽 𝑘𝑙 𝐿 𝑙=1 𝑙𝑛𝑦 𝑘𝑖𝑡 ln 𝑦 𝑙𝑖𝑡 + ∑ ∑  𝑚𝑘 𝐾 𝑘=1 ln 𝑥 𝑚𝑖𝑡 𝑙𝑛𝑦 𝑘𝑖𝑡 + 𝑀 𝑚=1 𝐾 𝑘=1 𝐾 𝑘=1 ∑  𝑘 𝐾 𝑘=1 ln 𝑦 𝑘𝑖𝑡 𝑡 + ∑  ln 𝑥 𝑚𝑖𝑡 𝑡 +  1 𝑡 + 1 2  2 𝑡 2 + 𝜀 𝑖𝑡 𝑀 𝑚=1 , (2)
where 𝑦 𝑚𝑖𝑡 is the m-th output quantity and 𝑥 𝑘𝑖𝑡 is the k-th input quantity for company 𝑖 at time 𝑡, and 𝜋, 𝛾, , 𝛽, , , 𝜓 and  are the parameters to be estimated. Equation ( 1) satisfies the properties of the distance function.

Technical efficiency is given in our case by the ratio of the minimal inputs required to the actual inputs used. This ratio tells the regulator or the company manager the amount by which all inputs could be reduced for a given level of output. After the estimation of the parameters, the value of the input-oriented distance function is predicted as 𝑇𝐸 𝑖𝑡 = 𝑒 -𝑢 𝑖𝑡 , which varies between 0 and 1. A value of 1 means that the unit evaluated (WoC in this study) is efficient, and 𝑢 𝑖𝑡 is an inefficiency index. Consequently, ln 𝑑 𝐼 = 𝑢 𝑖𝑡 .

As ln 𝑑 𝐼 (𝑋 𝑖𝑡 , 𝑌 𝑖𝑡 , 𝑡) is not observable, we can select one of the outputs and take advantage of the distance function's homogeneity properties to write an estimable distance-function specification. Indeed, homogeneity of degree 1 implies:

∑ 𝜋 𝑚 𝑀 𝑚=1 + ∑ ∑ 𝛾 𝑚𝑛 𝑀 𝑛=1 𝑙𝑛𝑥 𝑛𝑖𝑡 + 𝑀 𝑚=1 ∑ ∑ 𝜑 𝑚𝑘 𝐾 𝑘=1 𝑙𝑛𝑦 𝑘𝑖𝑡 + ∑ 𝜂 𝑚 t = 1 𝑀 𝑚=1 , 𝑀 𝑚=1
(3)

which will be satisfied if ∑ 𝜋 𝑚 𝑀 𝑚=1 = 1 and ∑ 𝛾 𝑚𝑛 = 𝑀 𝑚=1 ∑ 𝜑 𝑚𝑘 = 𝑀 𝑚=1 ∑ 𝜂 𝑚 = 𝑀 𝑚=1 0.
Homogeneity of degree 1 in input means also that 𝑑(𝜏𝑋 𝑖𝑡 , 𝑌 𝑖𝑡 , 𝑡) = 𝜏𝑑(𝑋 𝑖𝑡 , 𝑌 𝑖𝑡 , 𝑡) ∀ 𝜏 > 0;

normalizing the input-distance function by one of input is equivalent to setting 𝜏 = 1/𝑥 𝑖𝑡 𝑘 and then 𝑑(𝑋 𝑖𝑡 /𝑥 𝑖𝑡 𝑘 , 𝑌 𝑖𝑡 ) = 𝑑(𝑋 𝑖𝑡 ,𝑌 𝑖𝑡 ,𝑡) 𝑥 𝑖𝑡 𝑘 [START_REF] Mellah | Performance of the Tunisian Water Utility: An inputdistance function approach[END_REF]. Accounting for these restrictions in Eq. ( 2) yields the following estimating form:

-𝑙𝑛𝑥 𝑀𝑖𝑡 = 𝛼 𝑖 + ∑ 𝜋 𝑚 𝑀-1 𝑚=1 ln 𝑥 ̃ 𝑚𝑖𝑡 + 1 2 ∑ ∑ 𝛾 𝑚𝑛 𝑀-1 𝑛=1 ln 𝑥 ̃𝑚𝑖𝑡 ln 𝑥 ̃𝑛𝑖𝑡 + ∑  𝑘 𝑙𝑛𝑦 𝑘𝑖𝑡 + 𝐾-1 𝑘=1 𝑀-1 𝑚=1 1 2 ∑ ∑ 𝛽 𝑘𝑙 𝐾-1 𝑙=1 ln 𝑦 𝑘𝑖𝑡 ln 𝑦 𝑙𝑖𝑡 + ∑ ∑  𝑚𝑘 𝑀-1 𝑘=1 𝑀-1 𝑚=1 ln 𝑥 ̃𝑚𝑖𝑡 𝑙𝑛𝑦 𝑘𝑖𝑡 + 𝐾-1 𝑘=1 ∑  𝑚 𝑀-1 𝑚=1 𝑙𝑛𝑥 ̃𝑚𝑖𝑡 𝑡 + ∑  𝑘 𝐾 𝑘=1 ln 𝑦 𝑘𝑖𝑡 𝑡 +  1 𝑡 + 1 2  2 𝑡 2 + ∑  𝑝 𝑃 𝑝=1 ʓ 𝑝𝑖𝑡 + 𝑣 𝑖𝑡 -𝑢 𝑖𝑡 , (4)
where 𝑥 ̃𝑘𝑖𝑡 = 𝑥 𝑘𝑖𝑡 𝑥 𝐾𝑖𝑡 and 𝜀 𝑖𝑡 = 𝑣 𝑖𝑡 -𝑙𝑛𝑑 𝑖𝑡 = 𝑣 𝑖𝑡 -𝑢 𝑖𝑡 . In this specification, the error term has two components: the symmetric error term 𝑣 𝑖𝑡 , which represents random error and is assumed to be independent and identically distributed, 𝜈 𝑖𝑡 ̴ 𝑁(0, 𝜎 𝜈 2 ); and the asymmetric error term 𝑢 𝑖𝑡 , which captures the inefficiency effects and is drawn from an exponential distribution, 𝑢 𝑖𝑡 ̴ exp (𝜃) [START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF].

Following the approach of [START_REF] Saal | Determining the contribution of technical efficiency, and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF], we add 𝑝 exogenous operating characteristics, whose impact on the input requirements is captured in the term ∑

 𝑝 𝑃 𝑝=1
ʓ 𝑝𝑖𝑡 . We also replace the single intercept parameter with firm-specific dummies 𝛼 𝑖 Di to control for unobserved heterogeneity that has not been specifically controlled for in the model [START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF], Kunbahakar et al., 2015]. The inclusion of firm-specific dummies is appropriate in our case as 13 years of data are observed, covering several regulatory changes, such as the imposition of new environmental and drinking-water quality regimes, and different price reviews from the regulator. Because of the many years considered in this study and the changes in the companies' operating and management conditions, the firmspecific dummies capture unobserved heterogeneity in the companies' operating characteristics, which would otherwise not be controlled for in the model.

As our approach is characterized by multiple inputs and outputs, TFP is measured as the ratio of the aggregate output produced relative to the aggregate input used. In our case study, a useful TFP change measure provides insight into potential productivity improvements. The widely used generalized Malmquist Index is the measure of TFP change from various sources of productivity change. In this paper, we follow the approach of [START_REF] Orea | A parametric decomposition of generalized Malmquist productivity index[END_REF], [START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF] and [START_REF] Mellah | Performance of the Tunisian Water Utility: An inputdistance function approach[END_REF] for the parametric estimation of a translog input-oriented distance function. The parametric generalized Malmquist Index is written as follows: [START_REF] O'donnell | A Bayesian approach to imposing curvature on distance functions[END_REF][START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF][START_REF] Saal | Determining the contribution of technical efficiency, and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF][START_REF] Mellah | Performance of the Tunisian Water Utility: An inputdistance function approach[END_REF].

VARIABLE SPECIFICATION AND DATA

The selection of variables was guided by a review of the literature [START_REF] Worthington | A review of frontier approaches to efficiency and productivity measurement in urban water utilities[END_REF][START_REF] See | Exploring and analysing sources of technical efficiency in water supply services: Some evidence from Southeast Asian public water utilities[END_REF]Mellan and Ben Amor, 2016], the specific cost drivers of the English and Welsh water industry such as the value of assets, the length of mains, number of employees [START_REF] Saal | Determining the contribution of technical efficiency, and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF][START_REF] Molinos-Senante | The Luenberger productivity indicator in the water industry: an empirical analysis for England and Wales[END_REF]CEPA, 2014;Maziotis et al., 2016a] and the available data. It should be noted that productivity assessment integrates physical parameters such as the volume of water supplied, economic parameters such as the operational, capital and labor costs [START_REF] Jabran | Economic assessment of different mulches in conventional and water-saving rice production systems[END_REF] and exogenous factors such as the proportion of water that is abstracted from rivers, reservoirs or boreholes, average pumping head, population density, proportion of metered properties [CEPA, 2014].

The volume of drinking water distributed (𝑦 1 ), expressed in megaliters per day, and the number of connected properties (𝑦 2 ) were selected as outputs. Following Portela et al.

[2011], the water distributed was considered as a proxy for the volume of water abstracted from several water sources and therefore does not include water leakage. Three variables were used as inputs: (i) capital stock (𝑥 1 ); operating expenses (OPEX) as variable input usage (𝑥 2 ); and (iii) labor (𝑥 3 ). In general, the measurement of capital stock is a challenge since these data are not readily available [START_REF] Mellah | Performance of the Tunisian Water Utility: An inputdistance function approach[END_REF]. However, for the UK water industry, previous studies [START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF][START_REF] Maziotis | Profit, productivity and price performance changes in the water and sewerage industry: An empirical application for England and Wales[END_REF] measured this variable based on the inflation-adjusted modern-equivalent asset estimates of the replacement cost of net tangible fixed assets. Following the approach of [START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF], operating costs were deflated using the Office of National Statistics' producer price index for material and fuels purchased in the collection, purification and treatment by the water industry. Operating expenses involve business costs, energy costs, and resource and treatment costs. In other words, they are total operating costs less labor costs. Labor was proxied by full-time equivalent (FTE) employee numbers, which are available for each WoC.

A literature review [e.g. [START_REF] Marques | Nonparametric benchmarking of Japanese water utilities: Institutional and environmental factors affecting efficiency[END_REF][START_REF] Ananda | Evaluating the performance of urban water utilities: Robust nonparametric approach[END_REF][START_REF] Molinos-Senante | The role of environmental variables on the efficiency of water and sewerage companies: a case study of Chile[END_REF] illustrated that a large variety of variables might be used as explanatory variables of water companies´ performances. Actually, there is no formal theory as to what should be the determinants of the performances of water companies. Based on extant literature [START_REF] Saal | Economies of integration in the English and Welsh water only companies and the assessment of alternative unbundling policies[END_REF][START_REF] Marques | Nonparametric benchmarking of Japanese water utilities: Institutional and environmental factors affecting efficiency[END_REF]CEPA, 2014;[START_REF] Pinto | Water services performance: do operational environmental and quality factors account? Urban Water Journal[END_REF][START_REF] Brea-Solis | Regulatory incentives to water losses reduction: the case of England and Wales[END_REF] six exogenous variables were integrated in the assessment: (i) customer density; (ii) average pumping head; (iii) percentage of water taken from reservoirs; (iv) percentage of drinkingwater losses; (v) percentage of billing contacts (e.g., complaint and payment handling, meter reading, meter installation) not responded to within 5 working days, with respect to the total number of billing contacts; and (vi) percentage of bursts in mains per connected property. As [START_REF] Marques | Nonparametric benchmarking of Japanese water utilities: Institutional and environmental factors affecting efficiency[END_REF] pointed out, some factors are not completely exogenous for long run management of water companies, but in the short run the ability of the managers is limited and therefore they should be considered as potential explanatory factors.

Our empirical application focused on the English and Welsh WoCs for the period 1993-2009. The data used for estimation were retrieved from the "July Returns for the Water and Sewerage Industries in England and Wales" published by Ofwat each year. The panel is unbalanced due to the consolidation process that occurred in the UK water industry. Thus, the number of WoCs declined over the sample period due to both WoC/WoC and WaSC/WoC mergers [START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF]. When mergers occurred between firms of similar size, we considered the merged entity as a new firm entering the panel, whereas if a WoC was acquired by a WaSC, we simply dropped the company from the sample [START_REF] Bottasso | The appropriateness of the poolability assumption for multiproduct technologies: Evidence from the English water and sewerage utilities[END_REF]. In 2009, Ofwat published new accounting requirements as part of the wider process of further developing competition in the water industry. These new reporting requirements are seen by Ofwat as a necessary first step to improve data quality, before such accounting separation could be finalized in updated regulatory accounting guidelines (RAGs) a [Ofwat, 2009a]. Hence, these changes in the UK water and sewerage regulatory framework did not allow us to collect consistent data to accurately assess the productivity growth of water companies in subsequent years [START_REF] Maziotis | Assessing the impact of quality of service on the productivity of water industry: A Malmquist-Luenberger productivity index approach for England and Wales[END_REF]. Therefore, our study involves 243 observations. The descriptive statistics for the variables are shown in Table 1.

EMPIRICAL RESULTS

The estimated parameters of the distance functions are reported in Table 2. We also check if the input-distance function satisfies monotonicity and curvature conditions, i.e., nondecreasing and concave in inputs, and non-increasing and quasi-concave in outputs. The results confirm that the estimated function is non-decreasing in inputs and non-increasing in outputs, as shown by the first-order coefficients of inputs and outputs, respectively. a Interested readers can consult the Regulatory accounting guidelines 2016-17 published by Ofwat at its webpage: www.ofwat.gov.uk/wp-content/uploads/2016/10/prs_in1609RAG1617.pdf Moreover, the Hessian matrix of the translog input-distance function with the second-order coefficients and the interaction term between inputs as elements is negative semi-definite [START_REF] Mellah | Performance of the Tunisian Water Utility: An inputdistance function approach[END_REF]. Every principal minor of odd order is non-positive and every principal minor of even order is non-negative [START_REF] Simon | Mathematics for Economists[END_REF]. The condition of quasi-concavity in output is only partially satisfied at the sample mean b . This does not imply the absence of an underlying cost-minimization process but may reflect the inability of the translog input-distance function to approximate the true input distance over the range of the data [START_REF] Wales | On the flexibility of functional forms: an empirical approach[END_REF]. As [START_REF] Färe | On functional form representation of multi-output production technologies[END_REF] and [START_REF] Wolf | Imposing curvature and monotonicity on flexible functional forms: an efficient regional approach[END_REF] noted, the translog function may lose flexibility when subjected to curvature restrictions. Overall, the estimated translog input-distance function is acceptable. The firm-specific dummies are all statistically different from zero. Note that a negative coefficient suggests an increase (decrease) in technical efficiency (inefficiency), and the reverse is also true. Since all variables have been normalized around their means, the firstorder coefficients of outputs and inputs can be, respectively, interpreted as distancefunction output and input elasticities for the average WoC of the sample.

The output elasticities are statistically significant and suggest that creating more waterconnected properties requires more input requirements than providing additional volumes of water. This also evident from the second-order coefficient of water volumes and waterconnected properties. The scale elasticity, the sum of the inverses of the output elasticities, is 1.04 at the sample mean, which implies that a 1% increase in outputs will require an increase in input of 0.96%. This finding is consistent with other studies that reported economies of scale for English and Welsh WoCs [START_REF] Bottasso | The appropriateness of the poolability assumption for multiproduct technologies: Evidence from the English water and sewerage utilities[END_REF]. As far as the input elasticities are concerned, they are all positive and statistically different from zero, implying that the distance function is increasing with respect to inputs. The input elasticities of capital, operating expenses and labor are 0.445, 0.436 and 0.118, respectively. Labor input was used as the normalized variable in the distance function, so its elasticity is recovered from the sum of capital and operating-expense (OPEX) elasticities. This finding suggests that both capital and operating expenses are the main drivers of increased input requirements to abstract, treat and distribute water to downstream users. The second-order coefficient of water distribution and the water-connected property is positive and insignificant, which suggests that these outputs are not complementary. This finding is consistent with the results [START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF] and [START_REF] Bottasso | Scale economies, technology and technical change: evidence from the English water only sector[END_REF]. The second-order coefficient of inputs with respect to capital and operating expenses suggest that input requirements increase more when additional capital investment increases than when operating expenses increase. This is also evident when looking at the interaction term between time and inputs. Capital elasticity over time is positive and statistically significant, which means that WoCs have carried out substantial capital-investment programs since privatization. The estimated coefficient of the time factor is negative and statistically significant, suggesting that the sample-average firm has undergone technological regression from 1994 to 2009 at a small rate of 0.28%. The time-squared coefficient is relatively small and insignificant, suggesting that the estimated rate of technical change has been further declining, by 0.02% per year. The coefficients of time related to each of the input variables are statistically significant, suggesting that technical change has resulted in the use of capital and the reduction of operating expenses. The statistically significant parameter of the interaction term between time and outputs suggests that technical change increases the relative magnitude of connection elasticity, whereas it decreases the relative magnitude of the elasticity of the amount of water delivered.

Looking at the estimated parameters of the exogenous factors, it is concluded that the elasticity of input requirements with respect to density is positive and statistically significant. This implies that increased density reduces input requirements, which may be explained by the fact that a company with a relatively high customer to area-size ratio might use shorter pipes and hence have lower distribution costs [Torres and Maurison, 2006]. Increased billing contacts not dealt with within five working days reduces input requirements, which implies that companies had already achieved high levels of service to customers [Ofwat, 2004a;[START_REF]Levels of service for the water industry in England & Wales 2004-2005 report[END_REF]. The elasticity of input requirements with respect to reservoirs is positive and significant. This implies that water taken from reservoirs does not require higher inputs than water taken from rivers or groundwater resources, which may be more energy intensive than reservoirs and therefore might require higher inputs. Finally, the estimated parameter of bursts per connected property is positive and significant, whereas water losses are still positive but insignificant. This implies that increased investments in reducing bursts or water losses may reduce long-term input requirements because of inputsaving effects resulting from the investment in technologies that improve prediction of bursts in mains and manage water leakage. It should be noted that customer research carried out by Ofwat and other stakeholders suggested that customers in general believe that water companies in England and Wales have achieved a broadly satisfactory level of service in relation to costs [Ofwat, 2004a;[START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF]2009b]. Several conclusions can be drawn from the analysis of the trend in the efficiency levels of WoCs. First, the tightening of the regulatory regime till 2001/02 had a positive impact on the efficiency levels of WoCs, but did not have a positive effect on companies' efficiency levels in the subsequent years. Moreover, since the WoCs' average and maximum efficiency levels were relatively high at privatization, it was difficult to achieve substantial efficiency gains after privatization. Finally, the minimum efficiency levels have improved considerably since privatization, which means that price-cap regulation had a positive impact on those companies that were less inefficient when privatized. The year-by-year average for TFP change (TFPC) and its decomposition into EC, TC and SEC are depicted in Figure 1. TFPC increased till 1994/95 by 2.52%, with technical change being the major determinant of productivity growth. The tight 1994 price review led to a declining trend for TFPC, however it remained at a positive level. In the year ending 1998, average technical change remained at 0.40% per year, suggesting that the potential for productivity improvements through technical change still remained in the industry. The further tightening of the regulatory regime created more volatility in TFPC. In the year ending 2001/02, TFPC remained at 0.56% per year with efficiency change increasing by 1.26% per year. Technical change and scale change had consistent negative impacts on productivity growth. However, the efficiency gains were not sustained the following years and in 2004/05, TFPC dropped by -1.44%, with TC and TEC being the major determinants of deterioration in productivity. This finding is consistent with [START_REF] Portela | Productivity change in the water industry in England and Wales: Application of the meta-malmquist index[END_REF] and [START_REF] Molinos-Senante | The Luenberger productivity indicator in the water industry: an empirical analysis for England and Wales[END_REF][START_REF] Maziotis | Assessing the impact of quality of service on the productivity of water industry: A Malmquist-Luenberger productivity index approach for England and Wales[END_REF] where the authors concluded that less efficient companies improved their performance towards the frontier company whereas the frontier company did not continue to improve its performance. This conclusion reinforces Ofwat's Figure 2 shows cumulative TEC, TC, SEC and TFP growth indices for an average WoC.

The 1994 tight price review had a positive impact on TFPC, which increased by 5.4% in 1998 relative to 1993, mainly due to an improvement in TC. This finding is consistent with [START_REF] Bottasso | Scale economies, technology and technical change: evidence from the English water only sector[END_REF] where the authors reported small but positive rates of technical change for WoCs. Using both WaSCs and WoCs, [START_REF] Ashton | Capital Utilisation and Scale in the English and Welsh Water Industry[END_REF] reported a rate of technical change 1.8% during the period 1991-1996 and [START_REF] Portela | Productivity change in the water industry in England and Wales: Application of the meta-malmquist index[END_REF] reported an increasing trend for technical change for the years 1993-1997. A declining trend was then observed for TFPC for the subsequent years, which was attributed to decreases in SEC and TEC. In 2000, TFPC increased by 4.8% relative to 1993 due to a substantial increase in technical change of 5.36% and small efficiency gains. The 1999 price review helped less efficient companies to improve their efficiency, whereas the best performing companies continued to improve their performance but at a lower rate. This finding is consistent with [START_REF] Bottasso | Scale economies, technology and technical change: evidence from the English water only sector[END_REF] and [START_REF] Portela | Productivity change in the water industry in England and Wales: Application of the meta-malmquist index[END_REF] where the industry showed an increasing trend in technical change till 2000 but in the second half of the regulatory period 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Linking the TFPC results with the regulatory cycle, it is concluded that the 1994 price review had a positive impact on productivity growth, which was attributed to TC, i.e., the shift to the efficient frontier. This finding suggests that during the years 1994-2000, the frontier company improved its managerial practices significantly. The scale effect became negative during the years 1997-2000, which implies that the acquisition / mergers of WoCs from WaSCs did not have a positive impact on WoCs' productivities. Moreover, the additional tight price review in 1999 had a positive effect on WoCs' performances. During the years 2001 -2005, less efficient companies moved closer to the frontier company, the productivity of which was following a downward trend. This finding is consistent with [START_REF] Portela | Productivity change in the water industry in England and Wales: Application of the meta-malmquist index[END_REF] and Ofwat's statement that several companies were at or near the frontier company in terms of both operating and capital-maintenance efficiencies [Ofwat, 2004b;[START_REF] Portela | Productivity change in the water industry in England and Wales: Application of the meta-malmquist index[END_REF]. TC also improved but followed a downward trend, whereas the scale effect remained constantly negative. Overall, the 2004 price review did not further accelerate efficiency changes or stimulate technical changes. There were deteriorations in productivity for both best firms and other firms, meaning that less efficient companies did not move closer to the frontier, nor did the benchmark company continue to improve productivity. This finding is consistent with the results of Molinos-Senante et al. [2014],

which showed that during the years 2004 -2008, the UK water industry experienced a decline in productivity growth mainly due to technical regression.

From a policy and managerial perspective, this study allowed the identification of the primary drivers of productivity change and the assessment of the impact of regulation on as that of England and Wales, where water companies consolidate and make up a technologically mature industry, some exogenous factors become significant in the assessment of the performances of water companies. This issue is of great importance for the English and Welsh water industry which is regulated based on RPI-K price-cap methodology. Given the importance of this topic, several studies have aimed to assess TFP of water companies in England and Wales. However, they focused on assessing the productivity performances of WaSCs or both WaSCs and WoCs, whereas the only study that specifically addressed the assessment of WoCs´ performances aimed to evaluate the presence of scale economies. Hence, this study adds to the debate over the impact of regulation on WoCs' productivity, technology and scale changes. The primary findings of our empirical research can be summarized as follows. First, the 1994 and 1999 price reviews stimulated technical change, and there were small efficiency gains. During the years 1993/94 -2001/02, the best company improved its productivity over time and several companies were at or near the frontier company in terms of both operating and capital-maintenance efficiencies. Mergers between WoCs or WaSCs and WoCs did not have positive effects on WoC performance, as the scale change was negative.

The 2004 price review did not speed up TEC or stimulate TC, whereas SEC remained negative. Second, the average and maximum efficiency levels of WoCs had been relatively high at privatization, so it was difficult to achieve substantial efficiency gains after privatization. Moreover, the minimum efficiency levels have been improved considerably since privatization, which means that price-cap regulation had a positive impact on those companies that were less inefficient when privatized. Furthermore, the inclusion of several exogenous factors did not broadly increase long-term input requirements. On average, companies reported relatively high efficiency levels during the period of study, which suggests that they had already been investing in cost-efficient technologies or using inputs with respect to exogenous factors in a cost-efficient way. Finally, an average WoC needs to improve its productivity or reduce its costs relative to the benchmarking company by 1.58%.

From a policy and managerial perspective, our empirical study is of great significance for researchers and policy makers for the following reasons. First, our methodology allows the identification of the factors that affect productivity change over time, which could aid regulators and managers to define measures that can be employed to improve performance in a regulated industry. Moreover, calculating appropriate X factors in price-cap regulation is essential to create incentives to improve the efficiency and productivity of water companies. Finally, it provides a better understanding of the relative importance of various productivity components, which is essential to water regulators and water companies for the sustainable and efficient management of the urban water cycle.

  terms on the right side represent technical efficiency change (TEC), technological change (TC), and scale efficiency change (SEC), respectively. TEC represents the catching-up effect of the company. TC occurs as a feasible input-output combination set, which either expands or contracts. SEC refers to the movement of firm production along the boundary set, making use of its curvatures. Our model thus allows one to better evaluate the impact of SEC on productivity growth even if technology is characterized by globally decreasing or increasing returns to scale. Finally, following standard practice, we normalize the inputs, outputs and time variables around their mean values, and monotonicity and curvature conditions on the distance function's partial derivatives with respect to inputs and outputs are imposed before estimation [O' Donnell

Figure 1 .

 1 Figure 1. Evolution of total factor productivity change (TFPC), technical efficiency change (TEC), technical change (TC) and scale efficiency change (SEC) of the English and Welsh water only companies.

  that 'the improvement in relative efficiency since 1999 is striking. We now see companies clustering around the industry frontier for operating costs and capital expenditure, with several companies showing at or near best in class performance in both operating and capital maintenance efficiencies'[Ofwat, 2004b]. Moreover, managing a stepped change in price limits to remove excess returns from past price reviews and continuing high capital investment to improve overall performance was challenging for the frontier companies[Molinos-Senante et al. 2016a]. The introduction of new prices in 2005 led to small efficiency gains in the year ending 2006, which were offset by substantial reductions in TC and scale effect. Overall, the imposition of the price-cap regulation regime was meant to stimulate efficiency improvements and TC; in the longer term, it was not particularly effective in maintaining efficiency and productivity growth. Evidence of technical regress for WoCs was also found in[START_REF] Saal | Economies of integration in the English and Welsh water only companies and the assessment of alternative unbundling policies[END_REF] where the authors estimated a quadratic total cost function for the period 1993-2009 without controlling for exogenous factors. In contrast, other authors such as[START_REF] Bottasso | Scale economies, technology and technical change: evidence from the English water only sector[END_REF] reported positive technical change for WoCs by estimating a variable cost function for the period 1995-2005.Breas-Solis et al.[2017] also found technical progress by estimating an input distance using both WaSCs and WoCs. Therefore, the model specification, the choice of outputs/inputs, the sample period, the inclusion or not exogenous factors and the type of companies can lead to different conclusions with regard to the existence of technical progress. We also note that there might be other factors beyond the regulatory cycle that have impacted companies' productivity growth. For instance, water companies use substantial amounts of electricity to treat water and in 2005/06 electricity prices increased more than the producer price index for material and fuels purchased in the collection, purification and treatment (the index used to deflate operational costs). Moreover,Molinos-Senante et al. (2016a) noted that exogenous climate characteristics might explain the decline in companies' performance. For instance, during the extended dry period which started in the winter of 2004-05 water companies successfully minimized the risk of serious supply problems by using restrictions and undertaking additional water efficiency programs, increasing therefore their costs[Ofwat, 2007].

Figure 2 .

 2 Figure 2. Cumulative measures of total factor productivity change (TFPC), technical efficiency change (TEC), technical change (TC) and scale efficiency change (SEC) of the English and Welsh water only companies.

  This paper reports our empirical research dealing with the estimation of technology, assessment of efficiency and identification of determinants of productivity change in WoCs in England and Wales during the years 1993 -2009. To achieve this, an input-distance function is used to estimate and decompose the productivity change into TEC, TC and SEC using stochastic frontier techniques. We also explore the impacts of several exogenous and service-quality variables on companies' efficiencies. Finally, we show how the utility's regulator can use this type of modeling and the results to calculate illustrative X factors for the WoCs.

  ln (𝑇𝐹𝑃 𝑖,𝑡+1 /𝑇𝐹𝑃 𝑖,𝑡 ) = ln (𝑇𝐸 𝑖,𝑡+1 /𝑇𝐸 𝑖,𝑡 ) +

	1 2 (𝜕𝑙𝑛𝐷 𝑖,𝑡+1 /𝜕𝑡 + 𝜕ln𝐷 𝑖,𝑡 /𝜕𝑡) +	1 2	∑ 𝑘=𝐾 𝑘=1	[(1 +
	𝜕𝑙𝑛𝐷 𝑖,𝑡+1 /𝜕𝑙𝑛𝑦) + (1 + 𝜕ln𝐷 𝑖,𝑡 /𝜕𝑙𝑛𝑦)] (𝑙𝑛𝑦 𝑖,𝑡+1 -𝑙𝑛𝑦 𝑖,𝑡 ) ,

Table 3

 3 illustrates the trends in the average, maximum and minimum efficiency levels of WoCs over the period from 1993/94 to 2008/09.

	Year	Average	Maximum	Minimum
	1993-94	0.927	0.989	0.678
	1994-95	0.933	0.992	0.737
	1995-96	0.926	0.991	0.707
	1996-97	0.936	0.992	0.722
	1997-98	0.942	0.990	0.824
	1998-99	0.929	0.991	0.796
	1999-00	0.938	0.991	0.820
	2000-01	0.957	0.978	0.900
	2001-02	0.963	0.987	0.910
	2002-03	0.956	0.989	0.897
	2003-04	0.941	0.991	0.838
	2004-05	0.939	0.993	0.736
	2005-06	0.945	0.985	0.852
	2006-07	0.936	0.984	0.805
	2007-08	0.933	0.988	0.781
	2008-09	0.949	0.988	0.843

Table 3 .

 3 Average, maximum and minimum efficiency scores of English and Welsh water only companies.

  In 2004/05, the average TFPC index fell below its level at the first years of privatization. Deterioration of TFPC is attributed to a decrease in TC. TEC and SEC also decreased, but at lower rates. The introduction of new prices in 2005 did not

	1.00 1.05 1.10 technical change falls. improve the average company's productivity growth, which declined substantially due to
	0.95 substantial reductions in TC and SEC. Overall, relative to 1993, TFPC dropped by almost
	14%, mainly due to the reduction in TC by 10%, whereas SEC dropped by 3% and TEC by
	0.90 1%. This finding implies that it will be wiser for water companies to not base productivity
	0.85 plans only on technical efficiency but to focus further on scale efficiency and technological
	0.80 progress.	2009
	TFPC 1.000 1.001 1.026 1.038 1.052 1.054 1.046 1.048 1.054 1.051 1.030 1.004 0.992 0.977 0.939 0.904 0.866
	TEC 1.000 0.993 0.999 0.988 1.000 0.999 0.992 1.001 1.014 1.018 1.010 0.996 0.994 0.999 0.991 0.988 0.992
	TC 1.000 1.012 1.023 1.041 1.054 1.059 1.058 1.054 1.049 1.042 1.033 1.022 1.009 0.992 0.968 0.939 0.903
	SEC 1.000 0.996 1.004 1.009 0.998 0.997 0.996 0.993 0.991 0.991 0.987 0.986 0.989 0.986 0.980 0.977 0.971

companies' performances. This step is of great importance, as it will aid regulators and water companies in defining measures that can be employed to improve performance in a regulated industry. Moreover, the utility's regulators can use this type of modeling and the results to propose X factors for the WoCs over a five-year period. This was illustrated by using results from the year 2008 as an example. The potential (anticipated) productivity improvement of WoC1 obtained by catching up to company WoC10 over time is 0.988/0.859 = 1.149. Assuming that WoC1 achieves 50% catch-up for total costs over a 5-year period, then its productivity should improve by Analogously, illustrative X factors are computed for the other WoCs and for the average company.

Company

CONCLUSIONS

The assessment of the productivity change over time and its determinants is of great importance for regulators when setting urban water tariffs. In a regulatory framework, such


María Molinos-Senante

Simon Porcher

Alexandros Maziotis

Productivity change and its drivers for the Chilean water companies: A comparison of full private and concessionary companies

Keywords: environmental factors, performance, privatization, productivity growth, quality of service, water and sanitation industry

The privatization of the water industry has aroused interest in comparing the performance of public vs. private water companies. However, little research has been conducted to compare the performances of full private (FPWCs) and concessionary water companies (CWCs). This study estimates and compares the productivity growth and its drivers (efficiency, technical and scale change) for a sample of Chilean FPWCs and CWCs over the 2007-2015 period using the input distance function. Both types of water companies showed deteriorations in productivity growth, with CWCs exhibiting higher rates of negative productivity growth than FPWCs. For FPWCs, any gains in efficiency and scale were outstripped by negative technical change. CWCs did not improve their performance in any of the three components of productivity change. The comparison of productivity change between FPWCs and CWCs is essential to support decision-making therefore, this study is of great interest for policymakers worldwide who are developing policies aimed at privatizing water companies.

Introduction

Efficient management of water companies (WCs) is essential to ensure sustainable urban water activities. Performance assessments of WCs are relevant and have strong implications for regulators, utilities, customers, and stakeholders [START_REF] Pinto | Water services performance: do operational environment and quality factors count?[END_REF][START_REF] Dong | Benchmarking sustainability of urban water infrastructure systems in China[END_REF].

Since the 1980s, many studies have used different benchmarking methods to evaluate and compare the performances of water utility companies around the globe [START_REF] Haider | Intra-utility performance management model (In-UPM) for the sustainability of small to medium sized water utilities: Conceptualization to development[END_REF]. Some research has focused on assessing the efficiency of water utility companies through static comparisons of their performance levels [START_REF] Tutusaus | The ambiguity of innovation drivers: The adoption of information and communication technologies by public water utilities[END_REF]. This approach provides information on WCs assessed at a given moment in time. Alternative studies have evaluated the change in productivity of WCs. Unlike efficiency studies, productivity change studies integrate the temporal component in the assessment [START_REF] Portela | Productivity change in the water industry in England and Wales: Application of the meta-malmquist index[END_REF] and focus on how the performance levels of WCs change over time [START_REF] O'donnell | Measuring and analysing productivity change in a metafrontier framework[END_REF].

After the pioneering privatization of the English and Welsh water industry in 1989, several countries privatized some or all of their WCs. Since then, there has been controversial debate, politically and scientifically, about the appropriateness of privatizing utilities [START_REF] Cheung | Risk Factors of Public-Private Partnership Projects in China: Comparison between the Water, Power, and Transportation Sectors[END_REF]. Case studies analyzed in the literature [START_REF] Megginson | From state to market: A survey of empirical studies on privatization[END_REF] evidenced that private sector participation in the water industry is likely to result in improved managerial practices and higher operating efficiency. However, published literature also provides case studies where privatization of water utilities involved negative impacts on the society mainly due to increases in the water tariffs which led to water affordability problems (Al-Madfaei, 2017) 1 . Two major issues that have been investigated over the past 20 years include: i) the performance implications of public vs. private ownership of WCs [START_REF] Berg | Quantitative studies of water and sanitation utilities: A benchmarking literature survey[END_REF][START_REF] Suárez-Varela | Ownership and Performance in Water Services Revisited: Does Private Management Really Outperform Public?[END_REF], and ii) the change in productivity of WCs before and after their privatization [START_REF] Saal | Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF]. Most previous studies focusing on the second issue have referred to private WCs without taking into account that private WCs can be differentiated into two types, full private (FPWCs) and concessionary water companies (CWCs), depending on their approach towards privatization.

For FPWCs, ownership, including infrastructure, is privatized, usually by selling strategic participations and shares to private consortia. For CWCs, water and sewerage services are privatized for a certain period of time, such that water regulators enter into long-term contracts with private entities [START_REF] Petrova | At the Frontiers of the Rush for Blue Gold: Water Privatization and the Human Right to Water[END_REF].

For urban planning and from a policy perspective, it is essential to compare not only the productivity change of public vs. private WCs but also of FPWCs vs. CWCs. This comparison will provide relevant information to policymakers regarding the proper approach for privatizing urban water industries. Despite the usefulness of assessing the productivity change of FPWCs and CWCs, to the best of the author's knowledge, only Molinos-Senante and Sala-Garrido (2015) have conducted an empirical case study on this topic. They evaluated the productivity change of 18 Chilean WCs, including FPWCs and CWCs, from 1997 to 2013 by estimating the Luenberger productivity indicator. Although these authors pioneered the estimation of productivity change for FPWCs and CWCs, their paper had several limitations.

To compute changes in the productivity of WCs, parametric or nonparametric methods can be used [START_REF] Zhang | Evaluating the method of total factor productivity growth and analysis of its influencing factors during the economic transitional period in China[END_REF]. The first limitation of the study by Molinos-Senante and Sala-Garrido (2015) regards the use of the nonparametric Luenberger productivity indicator. This approach does not allow for the control of exogenous factors and quality of service. Several papers (see for instance [START_REF] Carvalho | The influence of the operational environment on the efficiency of water utilities[END_REF][START_REF] Tanner | The water Utility Adoption Model (wUAM): Understanding influences of organisational and procedural innovation in a UK water utility[END_REF] evidenced the importance of external variables in the performance assessment of WCs. The second limitation is that the authors decomposed the total factor productivity change (TFPC) into two drivers only: efficiency change (EC) and technical change (TC). This approach assumes that a change in the scale of a WC does not influence the change in productivity. A third limitation regards the presentation and interpretation of results. Molinos-Senante and Sala-Garrido ( 2015) presented results at the company level without any reference to the type of WC (i.e., FPWC or CWC). Moreover, no analysis was conducted to check the statistical significance of differences in productivity change between the two types of WCs.

Over the last 30 years, the Chilean water industry has implemented several major regulatory and institutional reforms to improve water and sewerage services [START_REF] Hearne | Water institutional reforms in Chile[END_REF]. One of the most relevant reforms was the privatization of WCs, which began in 1998. Unlike in England and Wales, where all WCs were privatized at the same time following a common approach, the Chilean government privatized WCs in two stages following two different approaches (SISS, 2015). In the first stage (1998)(1999)(2000), the five main Chilean WCs were privatized as FPWCs. Public WCs sold strategic participations to private consortia, with privatization of the public urban water infrastructure. In the second stage (2001)(2002)(2003)(2004), public WCs transferred rights for the exploitation of water and sewerage services for 30 years, leading to CWCs. In 1998, 92.6% of urban customers were supplied by public WCs. In 2015, FPWCs and CWCs combined provided water and sewerage services to 95.8% of customers.

The main objective of this paper was to evaluate if there are any differences in the performance between FPWCs and CWCs. In doing so, we estimated and compared the TFPC values (and its drivers: EC, TC, and SC) of a sample of FPWCs and CWCs. An empirical application was carried out, which focused on the 22 main Chilean WCs (12 FPWCs and 10 CWCs) over the 2007-2015 period which provide water and sewerage services to 98% of the urban population.

To the best of the knowledge of the authors, no study estimate and compared so far the TFPC of FPWCs and CWCs using a robust parametric method that allows for the control of exogenous factors and inclusion of quality of service issues that might influence productivity growth. Another innovation of the paper is the assessment of the impact of SC in the TFPC of FPWCs and CWCs.

Urban planners (local government) and water regulators are responsible for ensuring that the new urban developments have high quality drinking water and sewerage services (Gabrielsoon et al., 2018). Hence, the findings of this study contribute to better understand the dynamic of WCs by comparing the productivity change of FPWCs and CWCs. This information is essential to develop sound policies. Results about the comparison of TFPC and its drivers between FPWCs and CWCs are essential to support the decision-making process in selecting the approach to the privatization of WCs. In the year 2000 alone, 93 countries had municipalities that carried out some form of WC privatization [START_REF] Petrova | At the Frontiers of the Rush for Blue Gold: Water Privatization and the Human Right to Water[END_REF].

Hence, it is essential that policymakers make informed decisions to promote the long-term technical and economic sustainability of WCs.

Methodology

To evaluate the TFPC of the analyzed WCs and the influence of the quality of services on TFPC, this research follows the methodological approach proposed by [START_REF] Saal | Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF] (Figure 1).

Step 1

• Select the water companies (FPWCs and CWCs) to be evaluated

Step 2

• Definition of the inputs, outputs and quality and environmental variables to be integrated in the performance assessment model

Step 3

• Collect data for each water company evaluated over the time period analysed

Step 4

• Estimate the input distance function (Eq. 1)

Step 5

• Compute the Generalized Parametric Production Index (GPPI) (Eq. 3)

Step 6

• Decompose the total factor productivity change (TFPC) into efficiency change (EF), technical change (TC) and scale change (SC) for each water company (Eq.5)

Figure 1. Main steps to evaluate and compare the TFPC and its driver for FPWCs and CWCs Some previous papers (e.g., [START_REF] Antonioli | The use of a variable cost function in the regulation of the Italian water industry[END_REF][START_REF] Bottasso | Scale economies, technology and technical change: evidence from the English water only sector[END_REF] estimated cost functions to evaluate both the productivity growth and profit change of WCs.

By contrast, the present study applies the distance function approach, because there are no available data about all input prices to build a cost frontier and therefore, this study focused on assessing the TFPC of WCs. Following [START_REF] Orea | Parametric decomposition of a generalized Malmquist productivity index[END_REF], [START_REF] Mugisha | Performance assessment and monitoring of water infrastructure: An empirical case study of benchmarking in Uganda[END_REF], and [START_REF] Mellah | Performance of the Tunisian Water Utility: An inputdistance function approach[END_REF], among others, this study uses the input distance function to characterize a production technology having multiple inputs and outputs.

The input distance function yields the maximum deflation factor that should be applied to an input set to project it onto the efficient frontier of input set . For output 𝑥 (𝐼 𝑡 (𝑦))

vector at time , the input distance function is defined as: 𝑦 𝑡

(1)

𝐷 𝐼 (𝑦,𝑥,𝑡) = 𝑚𝑎𝑥 { 𝛿 : 𝑥 𝛿 ∈ 𝐼 𝑡 (𝑦), 𝛿 > 0 }
This function enables computation of the technical efficiency of the WCs because 𝐷 𝐼 (𝑥,𝑦,𝑡)

, where is a Farrell measure of the input technical efficiency (Ferro and = ( 1

𝑇𝐸 𝑖𝑡 𝐼 ) ≥ 1 𝑇𝐸 𝑖𝑡 𝐼
Mercadier, 2016). In accordance with previous studies, this paper uses an input-oriented approach to assess the performance of WCs [START_REF] Worthington | A review of frontier approaches to efficiency and productivity measurement in urban water utilities[END_REF]. This approach involves that performance improves by reducing the use of inputs for a given level of outputs. In the water industry, as in other network industries, the demand of outputs (i.e., drinking water and wastewater treatment services) is outside the control of managers, who mainly act on minimizing the use of inputs for water and sewerage services [START_REF] Pinto | Water services performance: do operational environment and quality factors count?[END_REF]. [START_REF] Orea | Parametric decomposition of a generalized Malmquist productivity index[END_REF] defined the Malmquist parametric productivity index (MPPI) as the weighted index of output change minus the weighted index of input change, employing input distance elasticities to estimate the weights of inputs and outputs. The MPPI is defined as follows:

(2)

ln 𝑀𝑃𝑃𝐼 𝐼 =- 1 2 ∑ 𝐾 𝑘 = 1 (𝜀 𝑘𝑡 + 1 + 𝜀 𝑘𝑡 ) ( ln ( 𝑦 𝑘𝑡 + 1 𝑦 𝑘𝑡 )) -1 2 ∑ 𝑀 𝑚 = 1 (𝜀 𝑚𝑡 + 1 + 𝜀 𝑚𝑡 ) ( ln ( 𝑥 𝑚𝑡 + 1 𝑥 𝑚𝑡 ))
where and . 𝜀 𝑘𝑡 = ∂𝑙𝑛𝐷 𝐼 (𝑡)/∂𝑙𝑛𝑦 𝑘 𝜀 𝑚𝑡 = ∂𝑙𝑛𝐷 𝐼 (𝑡)/∂𝑙𝑛𝑥 𝑚 is homogeneous of degree one in inputs; therefore, the weights of the input change 𝐷 𝐼 (𝑥,𝑦,𝑡) must sum to unity. By contrast, the output weights do not sum to unity [START_REF] Saal | Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF] due to the effects of non-constant returns to scale. Increasing returns to scale involve economies of scale because an expansion of output can be achieved with a less-than-proportionate increase in all inputs. If outputs increase by less than the proportional change in inputs, then there are decreasing returns to scale [START_REF] Carvalho | Estimating size and scope economies in the Portuguese water sector using the most appropriate functional form[END_REF]. [START_REF] Saal | Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF] modified the MPPI so that the output weights were non-negative and summed to unity by definition, and the input weights were non-negative and summed to unity because of the homogeneity of degree one of the input distance function. Hence, it integrates in the assessment economies of scale. The generalized parametric productivity index (GPPI) which is defined as follows was applied in this study:

ln 𝐺𝑃𝑃𝐼 𝐼 = ln 𝑀𝑃𝑃𝐼 𝐼 + 1 2 ∑ 𝐾 𝑘 = 1 ( ( 𝜀 𝑘𝑡 + 1 ∑ 𝑗 = 𝐾 𝑗 = 1 𝜀 𝑗𝑡 + 1 ) + ( 𝜀 𝑘𝑡 ∑ 𝑗 = 𝐾 𝑗 = 1 𝜀 𝑘𝑡 )) ( ln ( 𝑦 𝑘𝑡 + 1 𝑦 𝑘𝑡 )) -1 2 (3) ∑ 𝑚 = 𝑀 𝑚 = 1 (𝜀 𝑚𝑡 + 1 + 𝜀 𝑚𝑡 )(ln (𝑥 𝑚𝑡 + 1 + 𝜀 𝑚𝑡 ))
Following [START_REF] Caves | The economic theory of index numbers and the measurement of input, output, and productivity[END_REF], to implement the GPPI, the quadratic identity lemma is applied to the input distance function, as follows:

-ln ( 𝐷 𝐼 (𝑦,𝑥,𝑡 + 1)

𝐷 𝐼 (𝑦,𝑥,𝑡) ≡- 1 2 ∑ 𝑚 (𝜀 𝑚𝑡 + 1 + 𝜀 𝑚𝑡 ) ( ln ( 𝑥 𝑡 + 1 𝑚 𝑥 𝑡 𝑚 ) ) -1 2 ∑ 𝑘 (𝜀 𝑘𝑡 + 1 + 𝜀 𝑘𝑡 ) ( ln ( 𝑦 𝑡 + 1 𝑘 𝑦 𝑡 𝑘 ) ) -1 2 (4) [ ∂𝑙𝑛𝐷 𝐼 (𝑦,𝑥,𝑡 + 1) ∂𝑡 + ∂𝑙𝑛𝐷 𝐼 (𝑦,𝑥,𝑡) ∂𝑡 ]
Given that the input distance function is the inverse of the Farrell technical efficiency, then , and the TFPC is decomposed into three components: -𝑙𝑛𝐷 𝐼 (𝑡) = 𝑙𝑛𝑇𝐸 𝐼 (𝑡)

(5) 𝑇𝐹𝑃𝐶 = 𝐸𝐶 + 𝑇𝐶 + 𝑆𝐶 Following [START_REF] Saal | Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF], this paper uses the translog approach to estimate the input distance function, where and are units (WCs) and time indices, respectively. There 𝑖 𝑡 are inputs , and outputs , , and abbreviating :

𝑀 𝑥 𝑚 𝑚 = 1…𝑀, 𝐾 𝑦 𝑘 𝑘 = 1…𝐾 𝑥 𝑚 ≡ ( 𝑥 𝑚 𝑥 𝑀 ) -𝑙𝑛𝑥 𝑀,𝑖,𝑡 = 𝛼 𝑖 + ∑ 𝑀 -1 𝑚 𝜃 𝑚 𝑙𝑛𝑥 𝑚,𝑖,𝑡 + 1 2 ∑ 𝑀 -1 𝑚 ∑ 𝑀 -1 𝑛 𝛾 𝑚,𝑛 𝑙𝑛𝑥 𝑚,𝑖,𝑡 𝑙𝑛𝑥 𝑛,𝑖,𝑡 + ∑ 𝑘 𝜋 𝑘 𝑙𝑛𝑦 𝑘,𝑖,𝑡 + 1 2 ∑ 𝑘 ∑ 𝑙 𝛽 𝑘,𝑙 𝑙𝑛𝑦 𝑘,𝑖,𝑡 𝑙𝑛𝑦 𝑙,𝑖,𝑡 + ∑ 𝑀 -1 𝑚 ∑ 𝑘 𝜙 𝑚,𝑘 𝑙𝑛 (6)
are stochastic errors (assumed to measure inefficiency), drawn from an independent half-𝑢 𝑖,𝑡 normal distribution that is truncated at zero. The term captures the impact of ∑ 𝑝 𝜉 𝑝 𝑧 𝑝,𝑖,𝑡 𝑝 environmental and quality variables on input requirements, allowing the estimated input distance function to capture better the true relationship between inputs and outputs [START_REF] Saal | Determining the contribution of technical change, efficiency change and scale change to productivity growth in the privatized English and Welsh water and sewerage industry: 1985-2000[END_REF].

Intercept parameter accounts for heterogeneity of the analyzed WCs. This 𝛼 𝑖 parameter is obtained through the true fixed effect method, proposed by [START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF], and it allows for the control of other factors influencing input requirements that have not been specifically controlled in the model. Following [START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF], maximum likelihood techniques are used to allow for firm-specific fixed effects and time-varying inefficiency specification. The unknown parameters to be estimated are as follows: , , , , , 𝛼 𝑖 𝜃 𝑚 𝛾 𝑚,𝑛 𝜆 𝑘 𝛽 𝑘,𝑙 , , , , , and . 𝜙 𝑚,𝑘 𝜓 1 𝜓 2 𝜂 𝑚 𝜅 𝑘 𝜉 𝑝

Sample and Data Description

Statistical information from the WCs evaluated was extracted from the management reports of water and sewerage services published by the national urban water regulator (Superintendencia de Servicios Sanitarios, SISS) from 2007 to 2015.

Chilean WCs are multi-output producers, providing water supply and wastewater collection and treatment services. Following past evidence (Molinos-Senante et al., 2016a;[START_REF] Pinto | Water services performance: do operational environment and quality factors count?[END_REF][START_REF] Li | The influence of the regulatory environment on Chinese urban water utilities[END_REF], this study considered two outputs: i) the volume of water distributed, expressed in thousands of cubic meters of water produced annually, and ii)

the number of customers with access to wastewater treatment services. Three inputs were assessed: i) the main length, defined as the sum of the water and sewerage networks (in km), which was used as a proxy for capital costs (Ananda, 2014, Ferro and[START_REF] Ferro | Technical efficiency in Chile´s water and sanitation provides[END_REF]; ii) operating costs (Chilean pesos/year), defined as the total operating costs of the water and sewerage industries, deflated by the consumer price index taken from national statistics; and

iii) number of employees, which represents labor input expressed as the number, not cost, to impose the homogeneity assumption. Environmental and quality of service variables that are expected to influence TFPC were: i) customer density, defined as the number of customers per length pipe (customers/km); ii) nonrevenue water, defined as the percentage of water that was produced and not charged due to real and apparent losses [START_REF] Neamtu | The use of water balance in determining the water loss strategy[END_REF]

Results

Estimated results from the input distance function are reported in Table 2. Firm-specific effects i.e., the intercept parameter ranged from 3.299 to 3.663 for FPWCs and CWCs, 𝛼 𝑖 , respectively. The range of 0.364 suggested that the time-invariant heterogeneity of operating characteristics not otherwise controlled in the model accounted for small but important differences in the input requirements of the WCs.

Regarding monotonicity and curvature conditions, the estimated results confirmed that the input distance function was nondecreasing in inputs and nonincreasing in outputs, as shown by their first-order coefficients. The estimated input distance function was concave because the Hessian matrix of the translog input distance function with the second-order coefficients and the interaction term between inputs as elements was negative and semidefinite [START_REF] Simon | Mathematics for Economists[END_REF]. There were some violations of the quasi-concavity assumption with respect to outputs. These violations did not imply the absence of an underlying cost-minimization process but may have reflected the inability of the translog input distance function to approximate the true input distance over the range of data [START_REF] Wales | On the flexibility of functional forms: an empirical approach[END_REF]. In particular, 79% (or 21%) of the observations satisfied (or violated) the quasiconcavity assumption in outputs. As [START_REF] Färe | On functional form representation of multi-output production technologies[END_REF] and [START_REF] Wolf | Imposing curvature and monotonicity on flexible functional forms: an efficient regional approach[END_REF] noted, the translog function may lose flexibility when subjected to curvature restrictions.

Overall, the estimated translog input distance function was acceptable, and all variables were normalized around their means. Thus, the first-order coefficients of the outputs and inputs can be interpreted as the distance function output and input elasticities, respectively, for the average water company of the sample. Estimated input elasticities were all positive and statistically different from zero, implying that the distance function was increasing with respect to inputs. Input elasticities of network length, operational costs, and labor were 0.767, 0.178, and 0.055, respectively.

Number of employees was used as the normalized variable in the distance function. Its elasticity was recovered from the sum of the elasticities of the network length and operational costs. This finding suggested that network length and operating costs were the main drivers of increased input requirements to supply water and treat wastewater. Moreover, the high network length elasticity implied that the Chilean water and sewerage industry was capitalintensive. Output elasticities were statistically significant. Thus, providing wastewater treatment to more customers required more input than providing additional volumes of water. This is because increase in the volume of water supplied will take advantage of economies of scale whereas provide wastewater service to more customers will increase operational costs and number of employees (inputs) since the network will be bigger. The scale elasticity (i.e., sum of the inverses of the output elasticities) was 1.027 at the sample mean. In other words, a 1% increase in outputs would require an increase in inputs of 0.973%. Consistent with this result, [START_REF] Ferro | Technical efficiency in Chile´s water and sanitation provides[END_REF] reported increasing returns to scale in the Chilean urban water industry from 2005 to 2013. The second-order coefficient of customers provided with drinking water and wastewater treatment services was positive and significant, suggesting that these outputs were not complementary. This finding was consistent with the results of previous studies carried out in various settings [START_REF] Saal | Assessing the performance of water operations in the English and Welsh water industry: a lesson in the implications of inappropriately assuming a common frontier[END_REF].

Elasticities of network length and operational costs were negative and statistically significant over time. Thus, water companies increased capital-investment programs to improve the network, and the operational costs increased the input requirements over time.

The estimated coefficient of the time factor was negative and statistically significant, suggesting that the average firm in the sample underwent technological regression at a small rate of 1%. Costs increased annually in part because of technical regress. However, the timesquared coefficient was relatively small and positive, suggesting that the estimated rate of technical change increased at 0.03% per year. Coefficients of time related to each of the input variables were negative and statistically significant, suggesting that water companies experienced technical regress resulting in increasing input requirements with respect to network length and operational costs. The statistically significant parameter of the interaction term between time and outputs suggested that technical change increased the relative magnitude of the number of customers of wastewater treatment services, whereas it decreased the relative magnitude of the elasticity of the amount of water supplied.

Density and density squared showed negative first-order and positive second-order terms. This result suggested that as population density increased, the input requirements increased. However, this effect would eventually be exhausted at sufficiently high levels of population density. Therefore, WCs operating in low-density areas might be less efficient than companies operating in high-density areas. In more densely populated areas, the input requirements may decrease because a company with a relatively high customer-to-network length ratio might use shorter pipes and, hence, have lower distribution costs [START_REF] Torres | Driving forces for consolidation or fragmentation in the US water utility industry: A cost function approach with endogenous outputs[END_REF][START_REF] Bottasso | Scale economies, technology and technical change: evidence from the English water only sector[END_REF]. This finding seemed to confirm the existence of economies of density in the Chilean water industry, consistent with past research in various settings [START_REF] Kirkpatrick | An Empirical Analysis of State and Private-Sector Provision of Water Services in Africa[END_REF][START_REF] Picazo-Tadeo | The Role of Environmental Factors in Water Utilities' Technical Efficiency. Empirical Evidence from Spanish Companies[END_REF][START_REF] Mellah | Performance of the Tunisian Water Utility: An inputdistance function approach[END_REF].

Elasticities of input requirements with respect to the drinking water quality and wastewater treatment quality were negative and statistically significant. This finding suggested that investments in improving the qualities of drinking water and wastewater treatment led to higher input requirements. Finally, increased nonrevenue water resulted in higher input requirements, which may be attributed to increased investments associated with the detection, repair, and control of water loss.

Average TFPC values for the Chilean WCs (Table 3) illustrated notable reductions in values for FPWCs and CWCs from 2007 to 2015. Although TFPC was negative in both cases, it was markedly larger for CWCs than for FPWCs. This finding was consistent with the results of Molinos-Senante and Sala-Garrido ( 2015), who suggested that Chilean FPWCs exhibited better performance across years than CWCs. For FPWCs, the worsening of productivity was due to the negative shift of the efficient frontier because the average EC and SC values were positive. By contrast, for CWCs, the three drivers of TFPC (i.e., efficiency change, technical change, and scale change) were negative. This result suggested that CWCs did not improve their performance in any of the three components of the TFPC. To test whether the TFPC and its drivers differed significantly between FPWCs and CWCs, nonparametric Mann-Whitney and Kolmorov-Smirnov Z tests were carried out. The null hypothesis was that TFPC, EC, TC, and SC would not be significantly different between FPWCs and CWCs. The -values for these tests (Table 4) illustrated that the null hypothesis 𝑝 could be rejected for TC and SC, but not for EC and TFPC. Distributions of the TC and SC values among CWCs and FPWCs were statistically significant. On the other hand, although large, the difference in the average TFPC values between FPWCs and CWCs was not statistically significant. Thus, it cannot be concluded that FPWCs generally presented better performance across years than CWCs. This finding revealed the importance of verifying results from a statistical perspective, to avoid obtaining biased conclusions. Next, this paper sought to identify trends in the evolution of the productivity change across years. Figure 2 shows the average values of TFPC for the FPWCs, CWCs, and total sample of WCs evaluated year by year. Detailed results at the company level are reported as supplemental material. The TFPC of the Chilean water companies was positive for only one year, 2011/12. Thus, there was significant reduction of productivity during the period assessed. The increase in the use of inputs was not balanced by growth in the provision of water and sewerage services.

Two patterns can be differentiated for the productivity change of FPWCs and CWCs.

From 2007 to 2011, although both types of water companies showed worsening productivity, CWCs exhibited worse performance (lower TFPC values) than FPWCs. In particular, from 2007 to 2011, TFPC declined by 12.5% and 6.5% for CWCs and FPWCs, respectively. By contrast, from 2011 to 2015, there was no clear pattern in the productivity change of the WCs.

From 2012 to 2014, FPWCs and CWCs presented an opposite behavior. In the 2012/13 period, the TFPC of the FPWCs increased, whereas CWCs showed a reduction in productivity. During the next year, the opposite behavior was observed; the TFPC of the FPWCs declined, whereas the TFPC of the CWCs showed a positive behaviour. Total factor productivity change (%)

Figure 2. Average values of total factor productivity change for the total sample analyzed, for full private and concessionary water companies.

To understand better the drivers of productivity change of Chilean WCs, average values of EC, TC, and SC of the FPWCs and CWCs were plotted (Figures 3 and4). The efficiency change, or catching-up index, reflected the capacity of water companies to be managed on the efficient frontier. Positive values of EC can be attributed mainly to managerial improvements [START_REF] Simoes | On the economic performance of the waste sector. A literature review[END_REF]. For FPWCs, remarkable improvement of the EC was observed in the 2009/10 period (2.98%). This improvement compensated for the small decrease of this driver in subsequent years, leading a positive average value for the 2007-2015 period. EC showed greater variability for CWCs, with alternating positive and negative values across years. From a managerial perspective, this finding means that in average terms, FPWCs made more efforts to adopt better management practices than CWCs. Nevertheless, neither FPWCs nor CWCs exhibited positive values for EC in all of the years of the analyzed period. This finding suggested that Chilean WCs did not implement specific plans for improving management issues in the long term. -Garrido, 2015). Average TC values of FPWCs and CWCs were negative for all 9 years evaluated (Figures 3 and4). From 2007 to 2015, there was a steady regression of the efficient frontier. In the last years, the deterioration of TC was smaller than at the beginning of the period for both FPWCs and

CWCs, but TC remained negative. Regulatory reform is one of the main driving forces to improve TC, making these findings very relevant for urban water regulators. Despite efforts made at the national level by regulators, reforms adopted in recent years were not sufficient to strengthen the Chilean water industry. This deficiency may be due to several reasons, with the small number of WCs providing water and sewerage services being among the most important. Twenty-two WCs supply water to 98% of the urban population. The eight largest Chilean WCs belong to two economic groups that provide water and sewerage services to 79% of urban customers. Under this ownership, it is difficult for regulators to introduce reforms to promote competitiveness, innovation, and, therefore, productivity among WCs.

The scale change reflects the type of returns to scale presented by the WCs. Positive SC values imply economies of scale, while negative SCs imply diseconomies of scale [START_REF] Maziotis | Profit change and its drivers in the English and Welsh water industry: Is output quality important?[END_REF]. Neither FPWCs nor CWCs presented a clear tendency regarding the presence of economies of scale (Figures 3 and4). For both types of WCs, the SC contributed positively to TFPC in some years but negatively in others. FPWCs had larger variability in SC values than CWCs, as evidenced by the maximum and minimum average SC values of +3.7% and -2.9%, respectively. Such variability in SC is unusual worldwide. Nevertheless, our findings were consistent with previous studies focused on the Chilean water industry, which were inconclusive about the presence of economies of scale (SCL Econometrics 2009;Molinos-Senante et al., 2015;[START_REF] Ferro | Technical efficiency in Chile´s water and sanitation provides[END_REF]. In this context, [START_REF] Ferro | Technical efficiency in Chile´s water and sanitation provides[END_REF] concluded that the different results regarding economies of scale between previous studies were due to differences in methodology. However, the present study suggested that there was inconsistency in the presence of economies and diseconomies of scale for the Chilean WCs depending on the year. In some years, Chilean WCs presented increasing returns to scale, whereas they presented decreasing returns to scale in other years.

This issue greatly complicates long-term planning by water regulators because there is no clear sign that favors or disfavors the horizontal integration of WCs.

Implications

Results (Figure 2) illustrate that the performance of the Chilean WCs changes across years, which may be due to several factors. In the framework of water governance, Berg (2016) identified seven elements affecting water sector performance: ideas, institutions, interests, information, incentives, ideals, and individuals. In the case of the Chilean water industry, FPWCs and CWCs share some of these factors, including institutions, incentives, interests, and ideals. Thus, according to Berg's (2016) methodological framework, the three main factors explaining performance differences between Chilean WCs are ideas, information, and individuals. Ideas are understood as the different conceptual frameworks to support decision-making processes. Data collection, verification, and analysis are essential to identify best practices and establish realistic targets. Finally, leadership is a relevant factor to improve water sector performance (Berg, 2016). Given that the Chilean WCs evaluated are private they do not share a common framework to support decision-making. Moreover, quality of service and efficiency targets are different among WCs. Both issues contributed unequivocally to the different performance between companies.

Based on the empirical application carried out in this study for the Chilean water industry, the following policy recommendations are proposed. Firstly, from this study it can be concluded that Chilean WCs (both FPWCs and CWCs) present notable economies of density. It indicates that significant cost savings can be achieved if WCs provide water and sewerage services in compact cities. This means that water companies to better understand the costs to deliver water and treat wastewater between urban and rural areas and develop strategies and make informed decisions to manage their assets more efficiently (e.g. more mains may need to be laid in rural than urban areas) so that they can achieve cost savings in those areas where significant costs exist. Secondly, the results of this study confirm that FPWCs present positive economies of scale which means that if WCs increase their size they can reduce their costs. The opposite occurs for CWCs. Hence, the water regulator should develop policies to encourage the merging of FPWCs forming larger WCs. However, at the same time the water regulator should promote innovation in order to increase the efficiency and quality of service of the WCs. Finally, CWCs have negative efficiency change across years whereas FPWCs presented positive values. This indicates that CWCs have not improved (or have done so to a lesser extent than FPWCs) their operational practices. Hence, the water regulator should introduce incentives for all water companies to adopt the water industry best practices.

Conclusions

This manuscript contributes to the current strand of literature in two main aspects. It evaluates and compares the productivity change for FPWCs and CWCs. Moreover, it assesses the impact of exogenous factors and quality of service on water companies´ efficiency. Finally, it evaluates the impact of efficiency change, technical change and economies of scale in the productivity performance of both types of WCs.

The empirical application conducted is this study to compare the performance of Chilean FPWCs and CWCs led to several interesting conclusions. First, FPWCs and CWCs showed reductions in their productivity growth, with CWCs exhibiting higher rates of negative TFP growth than FPWCs. Second, for FPWCs, any gains in EC and SC were outstripped by negative TC. Less efficient FPWCs improved their efficiency relative to the most frontier company, whereas the frontier company did not improve its performance over time. An average FPWC showed increasing returns to scale, suggesting that larger FPWCs can reduce their costs through scale effects (e.g., mergers among FPWCs). It evidenced that water regulators should target policies to encourage the merging of FPWCs forming larger WCs and promoting the adoption of best practices in the water industry. Moreover, CWCs did not improve their performance in any of the three components of productivity change.

The major determinants in the deterioration of their productivity were the negative scale effect and TC. Effective long-term strategic planning and timely capital investment are needed to improve the technical efficiency. Hence, the study shows that national-level reforms that have been adopted in recent years have not been sufficient to strengthen the Chilean water industry. In conclusion, despite expending more efforts to adopt better management practices, FPWCs did not perform better than CWCs.

From a policy perspective, the findings of this study can be of great importance for researchers, urban planners, and policymakers for several reasons. First, the methodology employed allows the identification of factors that affect productivity change over time, which could aid regulators and managers to define measures that can be employed to improve performance in a regulated industry. Second, the comparison of different types of privatization will allow urban planners and policymakers to make decisions regarding the privatization approach to be taken. Finally, this study will improve understanding on the relative importance of various productivity components, which are essential to policymakers to make informed decisions for the sustainable and efficient management of water companies.

As a limitation to our study, we acknowledge that our case study only integrates four environmental and quality of service variables. This is due to the limited number of Chilean

WCs. Future research should evaluate the TFPC on a larger sample of water companies which will allow to integrate additional environmental and quality of service variables that may affect the performance of both FPWCs and CWCs. Moreover, as future research we will extend our database by including information on prices for inputs so we can estimate and decompose productivity growth by using cost frontier approaches.

Figure 3 .Figure 4 .

 34 Figure 3. Drivers of the total factor productivity change (TFPC) for full private water companies

Table 1 .

 1 Average of the inputs, outputs and environmental variables of the 22 Chilean water companies evaluated.

	; iii) drinking

nonrevenue water problems. The last two variables were measured by the water regulator as quality indicators between 0 and 1, with a value of 1 indicating that the water company met all legal requirements regarding the quality of drinking water (e.g., concentrations of pollutants) and the quality of wastewater treatment (e.g., sampling issues). Table

1

shows average values of data used to evaluate TFPC values for FPWCs, CWCs, and the whole sample of water companies from 2007 to 2015.

2 Source: Own elaboration from Superintendencia de Servicios Sanitarios data.

Table 2 .

 2 Estimated parameters of the input distance function. Labour input is the dependent variable.

	Variables	Parameter	Coeff	St.Error	T-stat
	Network length	𝜃 1	0.767	0.018	41.736*
	Operational costs	𝜃 2	0.178	0.018	9.815*
	Water distributed	𝜋 1	-0.309	0.021	-14.537*
	Wastewater customers treated	𝜋 2	-0.664	0.022	-29.683*
	Time	𝜓 1	-0.010	0.002	-6.305*
	Network length 2	𝛾 1,1	-0.693	0.027	-25.803*
	Network Length * Operational costs	𝛾 1,2	-0.013	0.021	-0.594
	Operational costs 2	𝛾 2,2	0.378	0.049	7.768*
	Water distributed*Network length	𝜙 1,1	-0.124	0.067	-1.848**
	Wastewater treated * Network Length	𝜙 2,1	0.280	0.066	4.214*
	Water distributed*Operational costs	𝜙 1,2	0.445	0.055	8.067*
	Wastewater treated * Operational costs	𝜙 2,2	-0.528	0.050	-10.619*
	Water distributed 2	𝛽 1,1	-0.286	0.031	-9.150*
	Wastewater customers treated 2	𝛽 2,2	-0.241	0.029	-8.200*
	Water distributed*Wastewater treated	𝛽 1,2	0.254	0.028	8.972*
	Network Length * Time	𝜂 1	-0.008	0.002	-4.096*
	Operational costs * Time	𝜂 2	-0.013	0.004	-3.101*
	Water distributed*Time	𝜅 1	-0.017	0.004	-4.453*
	Wastewater customers treated * Time	𝜅 2	0.021	0.003	6.377*
	Time 2	𝜓 2	0.003	0.001	3.693*
	Customer density	𝜉 1	-2.487	0.027	-92.730*
	Customer density 2	𝜉 2	0.404	0.007	58.568*
	Wastewater treatment quality	𝜉 3	-0.479	0.089	-5.356*
	Drinking water quality	𝜉 4	-0.208	0.027	-7.620*
	Non-revenue water	𝜉 5	-0.036	0.009	-4.137*
	𝜎		0.144	0.002	48.979*
	𝜆		5.841	0.469	12.435*
	Log likelihood function	239.668
	Average technical efficiency	0.912

* Coefficients are significant from zero at the 5% level ** Coefficients are significant from zero at the 10% level.

Table 3 .

 3 Average values of efficiency change, technical change, scale change and total factor productivity change from 2007 to 2015 expressed in percentage.

		Efficiency	Technical	Scale	Total factor
		change	change	change	productivity
		(%)	(%)	(%)	change (%)
	Full private	0.75	-12.54	3.87	-7.93
	Concessionary	-2.14	-5.94	-5.79	-13.89
	Total sample	0.48	-9.47	-1.27	-10.26

Table 4 .

 4 -values of Mann-Whitney and Kolmorov-Smirnov tests 𝑝

		Efficiency change	Technical change	Scale change	Total factor productivity change
	Mann-Whitney	0.863	0.000	0.043	0.618

HIGHLIGHTS

The productivity of full private and concessionary water companies was compared over the 2007-2015 period.

Stochastic frontier techniques were used to compute productivity and its drivers Drinking water and wastewater treatment quality along with non-revenue water contributed significantly to the productivity regression Full private and concessionary water companies presented positive and negative economies of scale, respectively.