Measuring the Impact of Water Supply Interruptions on Household Welfare

Abstract

Water scarcity frequently leads to a need for rationing. The choice of an adequate rationing method should be based on the impact on consumer welfare that is produced by each rationing plan. Some rationing schemes, such as the frequently used supply interruption method, can be regarded as changes in the characteristics of the good (in this case, time availability) that do not modify the pre-set consumer budget. Under the standard theoretical restrictions on consumer behavior compensating or equivalent variations/surpluses cannot be used to identify the impacts of these methods on household welfare. In this paper, we propose a set of sufficient conditions with respect to the utility function that allows for the evaluation of the compensating or equivalent variations/surpluses associated with changes in goods’ quality, even if those goods are considered to be essential for consumers. We use these conditions to compare the welfare losses associated with the water supply cuts implemented in Seville (Spain) to those that would result if water was instead rationed using price changes.

Appendices

Appendix A

Given a monotonic twice differentiable transformation f, the function $f\left( U \right)$ only verifies P2$\left( {\frac{{\delta^{2} f\left( U \right)}}{{\delta x_{2} \delta c}} = 0} \right)$ if f is an affine transformation. The marginal utility of x₂ becomes:

$$ \frac{\delta f\left( U \right)}{{\delta x_{2} }} = \frac{\delta f\left( U \right)}{\delta U}\left( {\frac{\delta U}{{\delta U^{0} }}\frac{{\delta U^{0} }}{{\delta x_{2} }} + \frac{\delta U}{{\delta U^{1} }}\frac{{\delta U^{1} }}{{\delta x_{2} }}} \right) $$

(A1)

which, taking into account that $\frac{{\delta U^{0} }}{{\delta x_{2} }} = 0$ and $\frac{\delta U}{{\delta U^{1} }} = 1$ becomes:

$$ \frac{\delta f\left( U \right)}{{\delta x_{2} }} = \frac{\delta f\left( U \right)}{\delta U}\frac{{\delta U^{1} }}{{\delta x_{2} }} $$

(A2)

and, therefore:

$$ \frac{{\delta^{2} f\left( U \right)}}{{\delta x_{2} \delta c}} = \frac{{\delta^{2} f\left( U \right)}}{{\delta U^{2} }}\frac{\delta U}{\delta c}\frac{{\delta U^{1} }}{{\delta x_{2} }} + \frac{\delta f\left( U \right)}{\delta U}\frac{{\delta^{2} U^{1} }}{{\delta x_{2} \delta c}} $$

(A3)

Taking into account that $\frac{{\delta^{2} U^{1} }}{{\delta x_{2} \delta c}} = 0$, P2 is verified only if $\frac{{\delta^{2} f\left( U \right)}}{{\delta U^{2} }} = 0$ which implies that f is an affine transformation.

Appendix B

Table 8 Residential water demand functions based on a Stone-Geary utility function

Appendix C

Table 9 Fixed fee (€)

Table 10 Block dimension and price

Appendix D

The baseline linear demand model estimated is the following:

$$ \begin{aligned} x_{1} & = b_{1} + b_{NPER} NPER_{i} + b_{TEMP} TEMP_{t} + b_{RAIN} RAIN_{t} + b_{QUAL} QUAL_{t} + \mathop \sum \limits_{T = 1992}^{2000} b_{T} D_{T} \\ & \quad + b_{R} R + b_{c} c + b_{p} p \\ \end{aligned} $$

(D1)

Tables 11 and 12 show the main results of the estimation/simulation based on the previous linear demand function and the consumer surplus as a welfare method.

Table 11 Baseline linear demand model estimation^a

Table 12 Consumer surplus (€) under alternative rationing schemes: baseline model