Estimating Residential Water Demand Using the Stone-Geary Functional Form: The Case of Sri Lanka

Abstract

This paper formulates a demand model for residential water in Sri Lanka using the Stone-Geary functional form. This functional form considers water consumption to be composed of two parts—a fixed and a residual component. The presence of these two components means it is possible to estimate a threshold below which water consumption is non-responsive to price changes. In turn, this can provide policy makers with a better understanding of the degree to which price changes will affect water consumption and the extent to which price instruments can be utilised to raise additional revenues. These revenues could then be used to extend pipe-borne water infrastructure to a greater proportion of the population than is currently the case. The findings presented here show the portion of water use that is insensitive to price changes in Sri Lanka is between 0.64 and 1.06 m³ per capita per month. The results indicate that price elasticity ranges from -0.11 to -0.14 while income elasticity varies from 0.11 to 0.14. Combined, these findings suggest water authorities could raise revenue via price increases to fund critical infrastructure extension.

Appendices

Appendix 1: A Model to Estimate Residential Water Demand

A residential water demand model to estimate marginal price and the difference variable can be expressed as follows:

$$ {Q_w} = \left( {MP,D,W,Y} \right) $$

(1a)

$$ MP = f\left( {{Q_w},ratestructure} \right) $$

(1b)

$$ D = f\left( {{Q_w},ratestructure} \right) $$

(1c)

Where, Q _w is monthly water consumption for an average household; MP is marginal price for the last unit purchased; D is the difference between the actual water bill and what would be paid if all units were purchased at the MP; W is a vector of weather variables (temperature and rainfall) and; Y is a vector of socio-economic variables (household income and number of household members). The marginal price model includes three structural equations because quantity, marginal price, and the difference variable are simultaneously determined. Equation (1a) utilizes observed quantity values and is then solved for MP and D. The values of MP and D are then utilised in the demand equation.

Simultaneity arises because the quantity of water determines the price under the pricing schedule. With block-rate pricing, causality goes in both directions, from price to quantity and from quantity to price. Consumers choose the amount of water depending on some measure of price, and the price paid depends on the amount consumed. As a result, the OLS assumption that independence between the error term and the explanatory variables exists is violated. This will yield biased and inconsistent estimates (Hensen 1984).

To overcome this and produce more consistent parameters the IV approach has been used in studies similar to the one undertaken here (Billings 1982; Nieswiadomy and Molina 1989). The objective of the instruments is to provide a proxy for the endogenous price variables in water studies. The instruments should fulfil the requirement of being uncorrelated with the error structure while being correlated with the stochastic regressor (Deller et al. 1986).

The Stone-Geary model uses an artificial linearization of the tariff structure to derive instruments for the marginal price and difference variable. This assumes that, instead of taking the effort to learn how the tariff works and which block they are consuming in at each moment in time, households will roughly estimate the whole tariff as a ray line given by an intercept and a constant marginal price (Martinez-Espeneira 2003).

$$ TR = \alpha + \beta {Q_w} $$

(2)

The total revenue (TR) function is computed using the range of quantity (Q _w) values (10–33 m³) encountered in the entire data set. These values of the total revenue are then regressed against the corresponding quantity values. The first derivative of the estimated function gives

$$ \hat{\beta } = \frac{{\partial TR}}{{\partial {Q_w}}} = MP* $$

(3)

MP* is the instrumental marginal price variable to the consumer and marginal revenue to the water utility. Estimating the parameter α in Eq. (2) gives the difference between what consumers actually pay and what consumers would pay if all units were sold at the MP*, that is D*. This procedure results in the representation of each rate schedule by MP* and D*, which are constant for all observations under each specific rate structure.

Appendix 2: The Stone-Geary Functional Form

Let Q _w and Q _z be the demand for water and all other goods respectively; P _w and P _z are unit prices for water and other goods respectively; γ _w and γ _z are minimum amounts (subsistence level) for water and other goods; β _w and β _z are preference parameters (marginal budget shares) for water and other goods; U is the total utility and; I is income (Nauges and Martinez-Espineira 2004).

The Stone Geary utility function is expressed as follows:

$$ U = {\beta_w}1{\text{n}}\left( {{Q_w} - {\gamma_w}} \right) + {\beta_z}1{\text{n}}\left( {{Q_z} - {\gamma_z}} \right) $$

(4)

Where ${\beta_w} > 0,{\beta_z} > 0{\beta_w} + {\beta_z} = 1,\left( {{Q_z} - {\gamma_z}} \right) > 0\,and\,\left( {{Q_w} - {\gamma_w}} \right) > 0$

Normalizing the price of the aggregate goods to one result we use the following budget constraint:

$$ l = {Q_w}{P_w} + {Q_z} $$

(5)

Maximizing utility subject to the budget constraint yields the following demand function

$$ \matrix{{*{20}{c}} {L = {\beta_w}1{\text{n}}\left( {{Q_w} - {\gamma_w}} \right) + {\beta_z}1{\text{n}}\left( {{Q_z} - {\gamma_z}} \right) + \lambda \left( {I - {Q_w}{P_w} - {Q_z}} \right)} \hfill \\ {\frac{{\partial L}}{{\partial {Q_w}}} \to \frac{{{\beta_w}}}{{\left( {{Q_w} - {\gamma_w}} \right)}} = \lambda {P_w}} \hfill \\ {\frac{{\partial L}}{{\partial {Q_z}}} \to \frac{{{\beta_z}}}{{\left( {{Q_z} - {\gamma_z}} \right)}} = \lambda } \hfill \\ {\frac{{\partial L}}{{\partial \lambda }} \to I = {Q_w}{P_w} + {Q_z}} \hfill \\ {Assuming\,{\beta_w} + {\beta_z} = 1\quad {Q_z} = \frac{{{\beta_z}{P_w}\left( {{Q_w} - {\gamma_w}} \right) + {\gamma_z}{\beta_w}}}{{{\beta_w}}}} \hfill \\ {substituting\,in\,I = {Q_w}{P_w} + {Q_z}} \hfill \\ } $$

(6)

$$ {Q_w}\frac{{I{\beta_w} - {\gamma_z}{\beta_w} + {P_w}{\gamma_w}{\beta_w} - {P_w}{\gamma_w}}}{{{P_w}}}\,or\,{Q_w} = {\beta_w}\frac{{I - {P_w}{\gamma_w} - {\gamma_z}}}{{{P_w}}} + {\gamma_w} $$

(7)

The preference parameter for water can be given as;

$$ {\beta_w} = \frac{{{P_w}{Q_w} - {P_w}{\gamma_W}}}{{I - {P_w}{\gamma_w} - {\gamma_z}}} $$

(8)

It is assumed that γ_z = 0, so the demand function for water is given as;

$$ {Q_w} = \left( {1 - {\beta_w}} \right){\gamma_w} + {\beta_w}\frac{I}{{{P_w}}} $$

(9)

After assuming γ _z  = 0, γ _w can be renamed as the conditional water use threshold (Gaudin et al. 2001) and is a threshold below which consumption is not responsive to prices. The term conditional emphasizes that this threshold is dependent on the available technology, pricing structure, and price of durable goods during the period of estimation. The marginal budget share allocated to water is represented by β _w .

The income variable in this model is the virtual income that is, the difference between household income and the difference variable. Price and income elasticities can be derived from these estimates. In this particular case, the two elasticities have the same magnitude.

$$ {\xi_p} = - {\beta_w}\frac{{I - {\gamma_z}}}{{{P_w}{Q_w}}} = - {\xi_I} $$

(10)

In accordance with the approach used in the relevant literature (for example: McGuire (1979); Gaudin et al. (2001); Nauges and Martinez-Espineira (2004)) γ _z is abstracted from the equation and the price elasticity of water demand simplifies to:

$$ {\xi_P} = {\beta_w}\frac{I}{{P{Q_w}}} = - {\xi_I} $$

(11)

This simple specification is preferred because γ _z does not provide any relevant information to the study.