Performance of Water Utilities Evaluated from Different Stakeholders Perspectives: An Application to the Ivorian Sector

Abstract

In this paper, we model a performance measure of water utilities, considering an undesirable output: the amount of water lost within the network during distribution. We improve three methodological approaches in traditional production technology modeling with an undesirable output. We first adapt the model to the specificities of water production and distribution network. We then extend the analysis to include the economic objectives of three different stakeholders - a regulator that seeks to preserve this natural resource, the operator who wants to maximize his profit, and the consumer who seeks maximum consumption - in the context of a developing country with unfulfilled water demand. Finally, using our model, we estimate the economic value of lost water by its shadow marginal cost, which enables us to evaluate the cost of loss on the water grid in each of the scenarios. We then propose an empirical application for water production in Côte d’Ivoire.

Appendix

A.1. Dealing with abatement factors in our specific frame.

In the Färe and Grosskopf (2003) we show below the abatement factor θ is equal to the unit.

We have:

$$\begin{array}{l}\left. {\begin{array}{*{20}{c}} {\theta V_{bill}^ \ast = V_{bill} + \delta \Leftrightarrow \theta V_{bill}^ \ast - V_{bill} = \delta } \\ {\theta V_{loss}^ \ast = V_{loss} - \delta \Leftrightarrow \theta V_{loss}^ \ast - V_{loss} = - \delta } \end{array}} \right\}\\ \Leftrightarrow \theta V_{bill}^ \ast - V_{bill} = V_{loss} - \theta V_{loss}^ \ast \\ \Leftrightarrow \theta \left( {V_{bill}^ \ast + V_{loss}^ \ast } \right) = V_{bill} + V_{loss}\end{array}$$

Given $V_{bill} + V_{loss} = V_{prod} = V_{bill}^ \ast + V_{loss}^ \ast$, hence θ = 1.

In the modified Kuosmanen (2005) and Kuosmanen and Podinovski (2009) model, we can show, equivalently, that DMU-specific abatement factors are also equal to the unit $\theta _i = 1,\,\forall i \in I,$. In NLP1 we can re-write the first constraint at the optimum as:

$$\mathop {\sum }\limits_{i = 1}^I \theta _i\lambda _iV_{bill_i} + \mathop {\sum }\limits_{i = 1}^I \theta _i\lambda _iV_{loss_i} = V_{bill_o} + \delta + V_{loss_o} - \delta = V_{prod_o}$$

$$\Leftrightarrow \mathop {\sum }\limits_{i = 1}^I \theta _i\lambda _i\left( {V_{bill_i} + V_{loss_i}} \right) = V_{prod_o}$$

$$\Leftrightarrow \mathop {\sum }\limits_{i = 1}^I \theta _i\lambda _iV_{prod_i} = V_{prod_o}$$

At the same time we also have that: $\mathop {\sum}\nolimits_{i = 1}^I {\lambda _iV_{prod_i}} = V_{prod_o} \Rightarrow \,\theta _i = 1,\,\forall i \in I.$

A.2. Equivalence between programs LP2 and LP3.

From LP2:

$$\begin{array}{l}\,\,\,\,\,\,\,\mathop {{\max }}\limits_{\lambda ,\delta } \,\delta \\ s.t.\quad \mathop {\sum}\limits_{j = 1}^J {\lambda _jV_{bill_j} = V_{bill_o} + \delta } \\ \quad \quad \mathop {\sum}\limits_{j = 1}^J {\lambda _jV_{loss_j} = V_{loss_o} - \delta } \\ \quad \quad \mathop {\sum}\limits_{j = 1}^J {\lambda _jVC_j \le VC_o} \\ \quad \quad \mathop {\sum}\limits_{j = 1}^J {\lambda _j = 1} \\ \quad \quad \lambda _j \ge 0\quad \forall j = 1,\,...,\,J\end{array}$$

From the second constraint of LP2, we have:

$$\delta = - \left( {\mathop {\sum}\limits_{j = 1}^j {\lambda _jV_{loss_j}} - V_{loss_o}} \right)$$

By reporting the result in the first constraint, we get:

$$\mathop {\sum}\limits_{j = 1}^J {\lambda _jV_{bill_j}} - V_{bill_o} = - \left( {\mathop {\sum}\limits_{j = 1}^J {\lambda _jV_{loss_j}} - V_{loss_o}} \right)$$

$$\mathop {\sum}\limits_{j = 1}^J {\lambda _j\left( {V_{bill_j} + V_{loss_j}} \right) = V_{bill_o} + V_{loss_o}}$$

Given: $V_{bill_j} + V_{loss_j} = V_{{\it{Pr}}o{\it{d}}_j},\,\forall \,j = 1, \ldots ,J$

Hence LP3:

$$\begin{array}{l}\,\,\,\,\,\,\,\mathop {{\max }}\limits_{\lambda ,\delta _1} \,\delta _1\\ s.t.\quad \mathop {\sum}\limits_{j = 1}^J {\lambda _jV_{loss_j} = V_{loss_o} - \delta _1} \\ \quad \quad \mathop {\sum}\limits_{j = 1}^J {\lambda _jV_{prod_j} = V_{prod_o}} \\ \quad \quad \mathop {\sum}\limits_{j = 1}^J {\lambda _jVC_j \le VC_o} \\ \quad \quad \mathop {\sum}\limits_{j = 1}^J {\lambda _j = 1} \\ \quad \quad \lambda _j \ge 0\quad \forall j = 1,\,...,\,J\end{array}$$

A.3. Descriptive statistics for DRs

Table 10b–k

Table 10 b: DR2 descriptive statistics. c: DR3 descriptive statistics. d: DR4 descriptive statistics. e: DR5 descriptive statistics. f: DR6 descriptive statistics. g: DR7 descriptive statistics. h: DR8 descriptive statistics. i: DR9 descriptive statistics. j: DR10 descriptive statistics. k: DR11 descriptive statistics