An Information Approach to Deriving Domestic Water Demand: An Application to Bogotá, Colombia

Abstract

This work presents a new model for studying water demand or use under an alternative pricing structure. The model takes into account the inequality shown by the distribution function of water usage: a large amount of water used by a small number of households. Rather than discard these extreme events (outliers), the model permits the description of the whole distribution function. The model builds on information by using extreme physical information theory, which provides a methodology to construct a dynamic equation describing water-use behavior. The solution to the dynamic equation gives the distribution function of water use, allowing calculations to be carried out in different scenarios for indicators such as the price elasticity of demand. The model is used to examine a case study of water use in Bogotá, Colombia.

Appendices

Appendix 1

Definitions of functions, variables and parameters used in the document are presented in Tables 2, 3 and 4, respectively.

Table 2 Functions

Table 3 Variables

Table 4 Parameters

Appendix 2

This appendix shows the variable change to write the bounded information $J$.

To minimize the information functional

$$ F = \int {\frac{{f\prime{{(x)}^2}}}{{f(x)}}{\hbox{d}}x} - \kappa \int {\frac{{h\prime{{(z)}^2}}}{{h(z)}}\,{\hbox{d}}z}, $$

(2.32)

the second term on the right,

$$ J = \int {\frac{{h\prime{{(z)}^2}}}{{h(z)}}\,{\hbox{d}}z,} $$

(2.33)

must be written in terms of $f\prime(x)$, $f(x)$ and $x$.

The variable change is done using Eqs. 2.6 and 2.7, and the conservation of the probability. Supposing that $x$ and $t(x)$ are random variables, the PDF $h(z)$ can be written as the product of two random variables, by definition (Dekking et al. 2005).

$$ h(z = t(x)\,x) = \int {\frac{1}{{|x|}}f(x)p(\frac{z}{x}|x)\,{\hbox{d}}x,} $$

(2.34)

where the function $p(t|x)$ is the probability of $t = {t_1}$ or $t = {t_2}$, given $x$. This probability function, $p(t|x)$, can be written in term of the delta function $\delta (x - {x_0})$ and the Heaviside step function $H(x - {x_0})$ (Arfken and Weber 1995) as

$$ p(t|x) = {a_1}\delta (t - {t_1})(1 - H(x - {x_0}) + {a_2}\delta (t - {t_2})H(x - {x_0}), $$

(2.35)

where ${a_1} + {a_2} = 1$.

Using equations Eq. 2.35 in Eq. 2.34, we get

$$ \begin{array}{llllll} h(z) = \int \limits_1^{\infty } \,\frac{1}{{|x|}}f(x)\left\{ {{a_1}\delta (\frac{z}{x} - {t_1})(1 - H(x - {x_0}) + {a_2}\delta (\frac{z}{x} - {t_2})H(x - {x_0})} \right\}\,{\hbox{d}}x \hfill \cr = \int \limits_1^{{{x_0}}} \,\frac{1}{{|x|}}f(x){a_1}\delta (\frac{z}{x} - {t_1})\,{\hbox{d}}x + \int \limits_{{{x_0}}}^{\infty } \,\frac{1}{{|x|}}f(x){a_2}\delta (\frac{z}{x} - {t_2})\,{\hbox{d}}x. \end{array} $$

(2.36)

An expression for $h\prime(z)$ is calculated taking the derivative $\tfrac{\partial }{{\partial z}}$ of Eq. 2.36,

$$ \begin{array}{llll} \frac{{\partial h(z)}}{{\partial z}} = \int \limits_0^{{{x_0}}} \,\frac{1}{{|x|}}f(x){a_1}\,\frac{\partial }{{\partial z}}\delta (\frac{z}{x} - {t_1})\,{\hbox{d}}x + \int \limits_{{{x_0}}}^{\infty } \,\frac{1}{{|x|}}f(x){a_2}\,\frac{\partial }{{\partial z}}\delta (\frac{z}{x} - {t_2})\,{\hbox{d}}x \hfill \cr = - \int \limits_0^{{{x_0}}} \,\frac{1}{{|x|}}f(x){a_1}\,\frac{{\left| y \right|}}{x}\delta (z - {t_1}x)\,{\hbox{d}}x - \int \limits_{{{x_0}}}^{\infty } \,\frac{1}{{|x|}}f(x){a_2}\,\frac{{\left| y \right|}}{x}\delta (z - {t_2}x)\,{\hbox{d}}x \hfill \cr = - \frac{1}{z}h(z). \end{array} $$

(2.37)

This last development takes into account that $f(\tfrac{z}{{{t_i}}}) = {t_i}h(z)$ and properties of the delta special function.

Finally, using Eqs. 2.37 and 2.6, we write $J = \int {\tfrac{{h\prime{{(z)}^2}}}{{h(z)}}\,{\hbox{d}}z}$ as

$$ \begin{array}{llll} J = \int \frac{{h(z)}}{{{z^2}}}\,{\hbox{d}}z \hfill \\ = \int \limits_1^{{{z^{ * }}}} \,\frac{{h(z)}}{{{z^2}}}\,{\hbox{d}}z + \int \limits_{{z^{ * *} }}^{\infty } \,\frac{{h(z)}}{{{z^2}}}\,{\hbox{d}}z \hfill \\ = \int \limits_1^{{{y_0}}} \,{t_1}\frac{{h({t_1}x)}}{{t_1^2{x^2}}}\,{\hbox{d}}x + \int \limits_{{{y_0}}}^{\infty } \,{t_2}\frac{{h({t_2}x)}}{{t_2^2{x^2}}}\,{\hbox{d}}x \hfill \cr = \int {\frac{1}{{t{{(x)}^2}}}\frac{{f(x)}}{{{x^2}}}\,{\hbox{d}}x,}\end{array} $$

(2.38)

where $z ^{*} = {t_1}{x_0}$ and $z ^{* *} = {t_2}{x_0}$, and $h({t_i}x) = \tfrac{1}{{{t_i}}}\ f(x)$.