Under pressure: community water systems in the United States—a production model with water quality and organization type effects

Abstract

This paper examines the interaction of productive efficiency, water quality and organizational type. A water quality index is constructed employing various contaminants which are then treated together with variables including organizational type as endogenous in an input distance function model. The cost of drinking water quality and inefficiency are derived and hypothesis tests estimated concerning their variation by location, organization type and water quality. The key findings are that more technically efficient water utilities deliver higher quality drinking water and that private firms distribute higher quality water at a higher price and at a higher implicit cost than public utilities despite no overall inefficiency differences between the two organizational types.

Appendix

Following Caves et al. (1981) and Atkinson and Primont (2002), scale economies for a short run distance function is derived by assuming the existence of the short run transformation function:

$$ F(\ln Y_{1} , \ldots ,\ln Y_{m} ,\ln X_{1} , \ldots ,\ln X_{n - q} ,\ln Z_{n - q + 1} , \ldots ,\ln Z_{q} ,t) = 1 $$

(40)

where Y _i are outputs; X _i , inputs; Z _i , quasifixed inputs and t represents time. The total differential is:

$$ \sum\limits_{i = 1}^{m} {\frac{\partial F}{{\partial \ln Y_{i} }}d\ln Y_{i} } + \sum\limits_{i = 1}^{n - q} {\frac{\partial F}{{\partial \ln X_{i} }}d\ln X_{i} + \sum\limits_{i = n - q + 1}^{q} {\frac{\partial F}{{\partial \ln Z_{i} }}d\ln Z_{i} + } \frac{\partial F}{\partial t}} dt = 0. $$

(41)

For dt = 0, it follows that:

$$ \sum\limits_{i = 1}^{m} {\frac{\partial F}{{\partial \ln Y_{i} }}d\ln Y_{i} } + \sum\limits_{i = 1}^{n - q} {\frac{\partial F}{{\partial \ln X_{i} }}d\ln X_{i} + \sum\limits_{i = n - q + 1}^{q} {\frac{\partial F}{{\partial \ln Z_{i} }}d\ln Z_{i} } } = 0. $$

(42)

$$ \sum\limits_{i = 1}^{m} {\frac{\partial F}{{\partial \ln Y_{i} }}d\ln Y_{i} } = - \sum\limits_{i = 1}^{n - q} {\frac{\partial F}{{\partial \ln X_{i} }}d\ln X_{i} - \sum\limits_{i = n - q + 1}^{q} {\frac{\partial F}{{\partial \ln Z_{i} }}d\ln Z_{i} } } $$

(43)

Applying rule III of differentials in (Chiang 1984):

$$ \frac{{d\ln Y_{i} }}{{(d\ln X_{i} + d\ln Z_{i} )}} = - \frac{{\sum\nolimits_{i = 1}^{n - q} {\frac{\partial F}{{\partial \ln X_{i} }}} + \sum\nolimits_{i = n - q + 1}^{q} {\frac{\partial F}{{\partial \ln Z_{i} }}} }}{{\sum\limits_{i = 1}^{m} {\frac{\partial F}{{\partial \ln Y_{i} }}} }} = RTS^{R} $$

(44)

This expression for scale economies following Caves et al. (1981, pp. 995–996) is defined as the proportional increase in all outputs resulting from the proportional increase in all variable inputs holding time fixed and conditional on the levels of the quasifixed inputs.

Using the duality of the variable cost function, $VC = \sum\nolimits_{i = 1}^{n - q} {W_{i} X_{i} } ,$ with (44) by substituting Eqs. (8a), (8b) and (9) into equation (11) of Caves et al. (1981) in p. 996, Eq. (44) can also be derived [see also the scale elasticity expression, Eqs. (8a), (8b), where the Z variable represents fixed capital, in Garcia and Thomas (2001, p. 15):

$$ SCE = \left.\left( 1 - \sum\nolimits_{{i = n - q + 1}}^{q} \partial\ln VC / {\ln Z_{i} } \right)\right/ {\sum\nolimits_{{i = 1}}^{m}{{{\partial \ln VC}/{\ln Y_{i} }}} }=- \frac{ {\sum\nolimits_{{i =1}}^{{n - q}} {\frac{{\partial F}}{{\partial \ln X_{i} }}} }\left/{\sum\nolimits_{{i = 1}}^{{n - q}} {\frac{{\partial F}}{{\partial\ln X_{i} }}} } \right. + {\sum\nolimits_{{i = n - q + 1}}^{q}{\frac{{\partial F}}{{\partial \ln Z_{i} }}} } \left/{\sum\nolimits_{{i = 1}}^{{n - q}} {\frac{{\partial F}}{{\partial\ln X_{i} }}} } \right. } { {\sum\nolimits_{{i= 1}}^{m} {\frac{{\partial F}}{{\partial \ln Y_{i} }}} } \left/{\sum\nolimits_{{i = 1}}^{{n - q}} {\frac{{\partial F}}{{\partial\ln X_{i} }}} } \right.} $$

(45)

In terms of distance functions (Atkinson and Primont 2002) Eq. (45) becomes:

$$ RTS^{R} = - \frac{{1 + \sum\nolimits_{i = n - q + 1}^{q} {\frac{{\partial \ln D^{I} }}{{\partial \ln Z_{i} }}} }}{{\sum\nolimits_{i = 1}^{m} {\frac{{\partial \ln D^{I} }}{{\partial \ln Y_{i} }}} }} $$

(46)