# Measure the operational and quality-adjusted efficiency of Chilean water companies

### Effectiveness evaluation

This section describes the methodology used to assess the OE, QAE and O & QAE of the Chilean water industry. Under the EO, water utilities are supposed to contract inputs and increase outputs at the same time^{24}. Thus, this measurement does not include undesirable results. Under QAE, water utilities are assumed to contract unwanted outputs^{25}. O & QAE incorporates both operational and quality of service variables and therefore water services are assumed to increase desirable outlets while at the same time unwanted outlets are contracted. Suppose there is *m* water services and *j*water service ( left ({j = 1, .., m} right) ) uses a vector of *m* contributions (X_j = left ({x_ {j1}, .., x_ {jn}} right) ) produce a vector of *q* desirable outings (Y_j = left ({y_ {j1}, .., y_ {jq}} right) ) and a vector of *p* unwanted exits (B_j = left ({b_ {j1}, .., b_ {jp}} right) ). The individual’s OE *k*The water service is derived using the following RAM-DEA model: [Eqs. 1-6]

$$ max frac {{ left ({ mathop { sum} nolimits_ {i = 1} ^ n { frac {{d_i ^ x}} {{R_i ^ x}}} + mathop { sum} nolimits_ {s = 1} ^ q { frac {{d_s ^ y}} {{R_s ^ y}}}} right)}} {{ left ({n + q} right)}} $$

(1)

*s*.*t*

$$ mathop { sum} limits_ {j = 1} ^ m {x_ {ij} lambda _ {ij} + d_i ^ x = x_ {ik} , (i = 1, ldots, n)} $$

$$ mathop { sum} limits_ {j = 1} ^ m {y_ {sj} – d_s ^ y = y_ {sk} , (s = 1, ldots, q)} $$

$$ mathop { sum} limits_ {j = 1} ^ m { lambda _j = 1, lambda _j ge 0 , left ({j = 1, ldots, n} right)} $ $

$$ d_i ^ x ge 0 quad (i = 1, ldots, n) $$

$$ d_s ^ y ge 0 , left ({s = 1, ldots, q} right) $$

or (d_i ^ x , and , d_s ^ y ) present the deviations for desirable inputs and outputs, respectively. The term *??* are intensity variables which are used to construct the efficient frontier^{5}. What’s more, (R_i ^ x )and (R_s ^ y ) are desirable input and output ranges, respectively, which are calculated based on the upper and lower limits of desirable inputs and outputs. These ranges take the following form:

$$ R_i ^ x = overline {x_i} – underline {x_i} , and , R_s ^ y = overline {y_s} – underline {y_s} $$

(2)

or ( overline {x_i} = max _j left {{x_ {ij}} right } , and , underline {x_i} = min _j left {{x_ {ij}} to the right}) are the upper and lower limits of the inputs, respectively, and ( overline {y_p} = max _j left {{y_ {pj}} right } , and , underline {y_p} = min _j left {{y_ {pj}} to the right}) are the upper and lower limits of desirable outputs, respectively^{15}. Then we calculate the OE as follows:

$$ OE = 1 – frac {{ left ({ mathop { sum} nolimits_ {i = 1} ^ n { frac {{d_i ^ {x ast}}} {{R_i ^ x}} } + mathop { sum} nolimits_ {s = 1} ^ q { frac {{d_s ^ {y ast}}} {{R_s ^ y}}}} right)}} {{ left ( {n + q} right)}} $$

(3)

where the clue^{presents the optimal values obtained from the model (1)}15 *. Unlike the EO, the assessment of the AQI requires the inclusion of undesirable outputs. Consequently, the AQI of some*k

The water service is derived by solving the following RAM-DEA model:

$$ max frac {{ left ({ mathop { sum} nolimits_ {i = 1} ^ n { frac {{d_i ^ x}} {{R_i ^ x}}} + mathop { sum} nolimits_ {r = 1} ^ p { frac {{d_r ^ b}} {{R_r ^ b}}}} right)}} {{ left ({n + p} right)}} $$

*(4)*st

.

$$ mathop { sum} limits_ {j = 1} ^ m {x_ {ij} lambda _ {ij} – d_i ^ x = x_ {ik} , (i = 1, ldots, n)} $$

$$ mathop { sum} limits_ {j = 1} ^ m {b_ {rj} + d_r ^ b = b_ {rk} , (r = 1, ldots, p)} $$

$$ mathop { sum} limits_ {j = 1} ^ m { lambda _j = 1, , lambda _j ge 0 , left ({j = 1, ldots, m} right) } $$

$$ d_i ^ x ge 0 , (i = 1, ldots, n) $$

$$ d_r ^ b ge 0 , left ({r = 1, ldots, p} right) $$ or (d_r ^ b ) show deviations for unwanted exits. In addition, (R_r ^ b )

denotes the range for unwanted outputs which is derived based on the lower and upper limits of unwanted outputs:

$$ R_r ^ b = overline {b_r} – underline {b_r} $$

(5) or ( overline {b_r} = max _j left {{b_ {rj}} right } , and , underline {b_r} = min _j left {{b_ {rj}} to the right})^{are the upper and lower limits of the unwanted outputs, respectively}26

. The QAE is calculated as follows:

$$ QAE = 1 – frac {{ left ({ mathop { sum} nolimits_ {i = 1} ^ n { frac {{d_i ^ {x ast}}} {{R_i ^ x}} } + mathop { sum} nolimits_ {r = 1} ^ p { frac {{d_r ^ {b ast}}} {{R_r ^ b}}}} right)}} {{ left ( {n + p} right)}} $$

(6)^{Finally, Sueyoshi et al.} 15^{and Sueyoshi and Goto} 24 *developed the following RAM-DEA model to estimate the unified efficiency (operational and quality-adjusted efficiency) of the*k

water service. This model takes into account the negative and positive parts of the input gap variables, the desirable and undesirable output gap variables. It is as follows:

$$ max frac {{ left ({ mathop { sum} nolimits_ {i = 1} ^ n { frac {{(d_i ^ {x +} + d_i ^ {x -})}} { {R_i ^ x}}} + mathop { sum} nolimits_ {s = 1} ^ q { frac {{d_s ^ y}} {{R_s ^ y}}} + mathop { sum} nolimits_ {r = 1} ^ p { frac {{d_r ^ b}} {{R_r ^ b}}}} right)}} {{ left ({n + q + p} right)}} $$

*(seven)*st

.

( mathop { sum} limits_ {j = 1} ^ m {x_ {ij} lambda _ {ij} – d_i ^ {x +} + d_i ^ {x -} = x_ {ik} , ( i = 1, ldots, n)} )

( mathop { sum} limits_ {j = 1} ^ m {y_ {sj} – d_s ^ y = y_ {sk} , (s = 1, ldots, q)} )

( mathop { sum} limits_ {j = 1} ^ m {b_ {rj} + d_r ^ b = b_ {rk} , (r = 1, ldots, p)} )

( mathop { sum} limits_ {j = 1} ^ m { lambda _j = 1, , lambda _j ge 0 , left ({j = 1, ldots, m} right) } )

(d_i ^ {x +} ge 0 , , (i = 1, ldots, n) )

(d_i ^ {x -} ge 0 , (i = 1, ldots, n) )

(d_s ^ y ge 0 , left ({s = 1, ldots, q} right) )

(d_r ^ b ge 0 , left ({r = 1, ldots, p} right) )

The unified O&QAE is defined as follows:

$$ {{{ mathrm {O}}}} , & , {{{ mathrm {QAE}}}} = {{{ mathrm {1}}}} – frac {{ left ( { mathop { sum} nolimits_ {i = 1} ^ n { frac {{(d_i ^ {x + ast} + d_i ^ {x – ast})}} {{R_i ^ x}} + } mathop { sum} nolimits_ {s = 1} ^ q { frac {{d_s ^ {y ast}}} {{R_s ^ y}} +} mathop { sum} nolimits_ {r = 1} ^ p { frac {{d_r ^ {b ast}}} {{R_r ^ b}}}} right)}} {{ left ({n + q + p} right)}} $ $

### (8)

Influence of environmental variables on efficiency^{In this study, we are also interested in identifying factors that could have an impact on the O & QAE of water utilities. So after getting the O & QAE scores from the equations. (7 and 8), we regressed them against a set of environmental variables (see the next section for more details). Since the O & QAE score takes a value between zero and one, we use the Tobit regression}7.37.38.39 [Eq. 9]

. Thus, the model takes the following form:

$$ beta _ {j, t} = gamma _0 + gamma _j zeta _ {j, t} ^ prime + delta _jt + eta _j + upsilon _ {j, t} $$

(9) or ( beta _ {j, t} ) designates the O & QAE of each water service (j ) at any time (t ) , ( gamma _0 ) is the intercept term (constant), ( zeta _ {jt} ^ prime ) *is the vector of environmental variables and* t capture time. Moreover, in the regression model above, ( eta _j ) capture company-specific mannequins and ( upsilon _ {l, t} )^{is the error term (noise), which follows the standard normal distribution. Several studies in the past have used the Tobit regression model to assess the impact of several environmental variables on the efficiency of public services.}40.41.42.43^{. However, this approach is not exempt from limitations since it requires the restrictive condition of separability between the input-output space and the space of exogenous variables.}36^{. Alternatively, partial border methods can be used to determine efficiency scores taking into account the influence of exogenous variables. The smoothed nonparametric regression between the ratio of conditional and unconditional efficiencies makes it possible to analyze the influence of exogenous variables on the production process by avoiding the problem of endogeneity.}44

### .

Selection of data and samples^{Our case study focuses on 21 water utilities in Chile during the years 2007-18, which provide water and sanitation services to around 93% of the Chilean urban population.}31^{. The sample consists of 11 fully-fledged private water services, 9 concessionary water services and 1 public water service. Being natural monopolies, an economic regulator has been put in place to control financial performance and quality of service and set prices.}45^{. The data used to estimate OE, QAE and O & QAE comes from the website of the economic regulator, Superintendencia de Servicios Sanitarios (SISS)}11

.^{The inputs, desirable and undesirable outputs were chosen on the basis of previous studies in the water industry and available statistical information.}12.46.47.48

. We used two inputs in our analysis. The first input was operating expenses (costs) measured in thousands of Chilean pesos per year (CLP per year). It includes all operating costs borne by the water companies with the exception of labor costs. Therefore, the second entry was the number of employees per year. We used two desirable outputs. The first production was represented by the volume of water delivered in cubic meters per year. The second desirable outcome was the annual number of clients receiving wastewater treatment services. The quality of service variables, i.e. unwanted outputs, were defined as the volume of water leakage measured in cubic meters per year and the number of unplanned water supply interruptions measured in hours per year.^{Previous studies have demonstrated the importance of including environmental variables in the analysis of efficiency as they have an impact on the input needs of water companies and, therefore, on inefficiency.}49.50

Table 3 Descriptive statistics of the variables used.

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