Responses to Changes in Domestic Water Tariff Structures: A Latent Class Analysis on Household-Level Data from Granada, Spain

Abstract

A problem that affects the estimation of water demand functions is the presence of unobserved individual heterogeneity, which means that a common demand function is unlikely to represent the behavior of all users. We implement Latent Class Models to estimate water demand functions for four groups of users who are classified according to their unobservable preferences. This more flexible approach makes it possible to distinguish four different response patterns to changes in the drivers of water use, including different price elasticities. These results should be of particular interest to regulators who would like to tailor water demand management policy to heterogeneous users. Our analysis exploits household-level panel data on residential water demand and consumers’ characteristics obtained by combining information from a survey of 1,465 domestic users in the city of Granada and bimonthly price and consumption data supplied by this city’s water supplier from the period 2009–2011.

Appendices

Appendix 1

As explained in Sect. 5, the estimation using LCMs is nonlinear, therefore, two-stage least squares models are likely to be inconsistent (Howard and Roe 2013). Therefore, we used a control function approach to correct for price endogeneity.

Consider the model:

$$\begin{aligned} WD=\delta z_{1}+ \alpha AvgP+u_{1} \end{aligned}$$

(9)

where WD is residential water demand, AvgP the average price (i.e., the endogenous explanatory variable) and $z_{1}$ a vector of exogenous explanatory variables.

This methodology uses the same first stage that would be used in 2SLS, that is, the endogenous explanatory variable is regressed on the exogenous explanatory variables and the set of instruments. Let $z_{2}$ denote a vector of instruments.

$$\begin{aligned}&AvgP=\pi _{1}z_{1}+\pi _{2}z_{2}+u_{2}\nonumber \\&E(z_{1}^{\prime }u_{2})=0 \quad E(z_{2}^{\prime }u_{2})=0 \end{aligned}$$

(10)

The average price would be endogenous if and only if $u_{1}$ is correlated with $u_{2}$, being $\gamma =E(u_{2}u_{1})/E(u_{2}^{2})$.

$$\begin{aligned} u_{1}=\gamma u_{2}+\epsilon \end{aligned}$$

(11)

Since $u_{1}$ and $u_{2}$ are uncorrelated with $z_{1}$ and $z_{2},\,E(u_{2}\epsilon )=0$ and $E(z_{2}\epsilon )=0$. Substituting Eq. (11) into Eq. (9):

$$\begin{aligned} WD=\delta z_{1}+\alpha AvgP+\gamma u_{2}+\epsilon \end{aligned}$$

(12)

In this equation $u_{2}$ is included as an explanatory variable. As explained above, $\epsilon$ is uncorrelated with $u_{2},\,z_{1}$ and $z_{2}$. Moreover, AvgP is defined as a linear function of the explanatory variables, the instruments and the residual $u_{2}$, so AvgP is uncorrelated with $\epsilon$ and therefore, $\delta$ and $\alpha$ can be consistently estimated with Eq. (12).

Results from the first stage estimation as defined in Eq. (10) are presented in Table 12. Following Hewitt and Hanemann (1995) and Olmstead (2009b), we used the full set of marginal prices in each block as instruments. The Hansen test indicated that all the instruments are exogenous. The residuals were obtained from this estimation and then, the second stage estimation included the average price and the control function as defined in Eq. (11), that is, the residuals from the first stage and a standard normal random variable (Howard and Roe 2013).

Table 12 Control function estimation

Appendix 2

1.1 Selected questions from the survey used to construct a water habits index

P.17. In general, do you have any of the following water conservation habits in the household?

• (a) Do you recycle water, for example, making use of the water while you wait for the shower to get hot?

• (b) Do you store drinking water in the refrigerator rather than letting the tap run every time you want a cool glass of water?

• (c) Do you defrost food in advance in order to avoid using running hot water to thaw meat or other frozen foods?

• (d) Do you fill the sink with water when washing dishes by hand?

• (e) Do you operate automatic dishwashers and washing machines only when they are fully loaded?

• (f) Do you slightly turn off the backflow valve to reduce the tap flow?

• (g) Do you use a rubbish bin in the toilet rather than flushing the toilet unnecessarily?

• (h) Do you avoid letting water run while brushing your teeth?

• (i) Do you take shorter showers?

• (j) Do you avoid washing the cars with drinking water?