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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The framework used in EPI theory to derive a Lagrangian is similar to the framework used when a Lagrangian is constructed by maximizing the uncertainty or Shannon entropy (Frieden 2004). The two approaches are compared by Hawkins and Frieden (2004), calculating density functions in bond and option prices.
- 2.
The solution to the differential equation [2.15] is straightforward after the change of variable x → exp(x) and q(x) → x. Applying this change of variable, Eq. [2.15] becomes the typical second-order differential equation \( a\,\tilde{q}\prime\prime\,(\tilde{x}) + b\,\tilde{q}\prime(\tilde{x}) + c\,\tilde{q}(\tilde{x}) = 0 \), with \( a = 4 \), \( b = \frac{\kappa }{{t{{(\tilde{x})}^2}}} - 4 \) and \( c = - \beta \).
- 3.
The transcendental equation is calculated and used to deduce \( {k_1} \), but it is not presented here because it is too long and its form does not contribute to the discussion.
- 4.
The routines to calculate parameters of the Pareto distribution using an MLE method was programmed using Mathematica™ based on Clauset et al. (2009).
References
Anderson C (2006) The Long Tail: How Endless Choice Is Creating Unlimited Demand. Random House Business Books
Andriani P, McKelvey B (2007) Beyond gaussian averages: Redirecting international business and management research toward extreme events and power laws. J Int Bus Stud 38(7):1212–1230
Andriani P, McKelvey B (2009) Perspective–from gaussian to paretian thinking: Causes and implications of power laws in organizations. Organ Sci 20(6):1053–1071
Arfken GB, Weber HJ (1995) Mathematical Methods for Physicists, 4th edn. Academic Press
Babel M, Gupta A, Pradhan P (2007) A multivariate econometric approach for domestic water demand modeling: An application to kathmandu, nepal. Water Resour Manag 21:573–589
Barabási AL, Jeong H, Neda Z, Ravasz E, Schubert A, Vicsek T (2002) Evolution of the social network of scientific collaborations. Physica A 311(3–4):590–614
Barnett RR, Levaggi R, Smith P (1992) Local authority expenditure decisions: A maximum likelihood analysis of budget setting in the face of piecewise linear budget constraints. Oxford Econ Pap 44(1):113–134
Bettencourt L, Lobo J, Helbing D, Kühnert C, West G (2007) Growth, innovation, scaling, and the pace of life in cities. Proceedings of the National Academy of Sciences 104(17):7301–7306
Beuthe M, Eeckhoudt L, Scannella G (2000) A practical multicriteria methodology for assessing risky public investments. Socio Econ Plan Sci 34(2):121–139
Blomquist S, Newey W (2002) Nonparametric estimation with nonlinear budget sets. Econometrica 70(6):2455–2480
Bonilla R, Zarama R, Valdivia JA (2011) Theoretical model to deduce a pdf with a power law tail using extreme physical information. arXiv:11025313v2 [physicsdata-an]
Cheng TC (2005) Robust regression diagnostics with data transformations. Comput Stat Data An 49(3):875–891
Clauset A, Shalizi C, Newman M (2009) Power-law distributions in empirical data. SIAM Review 51(4):661–703
Conley BC (1967) Price elasticity of the demand for water in southern california. Ann Regional Sci 1:180–189
Dekking F, Kraaikamp C, Lopuhaa H, Meester L (2005) A Modern Introduction to Probability and Statistics. Springer-Verlag, London
Durbin J (1973) Distribution theory for tests based on the sample distribution function. Society for Industrial and Applied Mathematics
Efron B (1998) R. a. fisher in the 21st century. invited paper presented at the 1996 r. a. fisher lecture. Statistical Science 13(2):95–122
Fisher RA (1925) Theory of statistical estimation. Mathematical Proceedings of the Cambridge Philosophical Society 22(05):700–725
Frieden B, Hawkins R (2010) Asymmetric information and economics. Physica A 389(2):287–295
Frieden BR (2004) Science from Fisher Information: A Unification. Cambridge University Press
Frieden BR, Cocke WJ (1996) Foundation for fisher-information-based derivations of physical laws. Phys Rev E 54(1):257–260
Frieden BR, Gatenby RA (2005) Power laws of complex systems from extreme physical information. Phys Rev E 72(3, Part 2)
Frieden BR, Soffer BH (1995) Lagrangians of physics and the game of fisher-information transfer. Phys Rev E 52(3):2274–2286
Fullerton D, Gan L (2004) A simulation-based welfare loss calculation for labor taxes with piecewise-linear budgets. J Public Econ 88(11):2339–2359
Gatenby RA, Frieden BR (2002) Application of information theory and extreme physical information to carcinogenesis. Cancer Res 62(13):3675–3684
Gurka MJ, Edwards LJ, Muller KE, Kupper LL (2006) Extending the box – cox transformation to the linear mixed model. J Roy Stat Soc A Sta 169(2):273–288
Hausman JA (1985) The econometrics of nonlinear budget sets. Econometrica 53(6):1255–1282
Hawkins RJ, Frieden BR (2004) Fisher information and equilibrium distributions in econophysics. Phys Lett A 322(1–2):126–130
Hawkins RJ, Aoki M, Frieden BR (2010) Asymmetric information and macroeconomic dynamics. Physica A 389(17):3565–3571
Johnson N (2007) Two’s company, three is complexity. Oxford, Oneworld Publications
Johnson N, Jefferies P, Hui P (2003) Financial market complexity. Oxford Finance Series, Oxford University Press
Kumar A (2010) Nonparametric estimation of the impact of taxes on female labor supply. J Appl Econom
Lise S, Paczuski M (2001) Self-organized criticality and universality in a nonconservative earthquake model. Phys Rev E 63(3 II):361,111–361,115
Mandelbrot B (1960) The pareto-lévy law and the distribution of income. Int Econ Rev 1(2):79–106
Mandelbrot B (2009) New methods of statistical economics, revisited: Short versus long tails and gaussian versus power-law distributions. Complexity 14:55–65
Marsaglia G, Tsang WW, Wang J (2003) Evaluating kolmogorov’s distribution. J Stat Softw 8(18):1–4
Mitzenmacher M (2003) A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1(2):226–251
Moffitt R (1984) The effects of grants-in-aid on state and local expenditures : The case of afdc. J Public Econ 23(3):279–305
Moffitt R (1986) The econometrics of piecewise-linear budget constraints: A survey and exposition of the maximum likelihood method. J Bus Econ Stat 4(3):317–328
Moffitt R (1990) The econometrics of kinked budget constraints. J Econ Perspect 4(2):119–39
Monteiro HPC (2010) Residential water demand in portugal: checking for efficiency-based justifications for increasing block tariffs. Working Papers ercwp0110, ISCTE, UNIDE, Economics Research Centre
Montroll EW, Shlesinger MF (1983) Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: A tale of tails. J Stat Phys 32:209–230
Newman MEJ (2005) Power laws, pareto distributions and zipf’s law. Contemp Phys 46(5):323–351
Olmstead SM, Hanemann WM, Stavins RN (2007) Water demand under alternative price structures. J Environ Econ Mang 54(2):181–198
Pareto V (1971) Manual of Political Economy, transl. by A Schwier. London: Macmillan
Reed WJ (2003) The pareto law of incomes–an explanation and an extension. Physica A 319:469–486
Sakia RM (1992) The box-cox transformation technique: a review. Statistician 41:169–178
Scannella G, Beuthe M (2003) Valuation of road projects with uncertain outcomes. Transport Rev 23(1):35–50
Seko M (2002) Nonlinear budget constraints and estimation: effects of subsidized home loans on floor space decisions in japan. J Hous Econ 11(3):280–299
Strogatz SH (2001) Exploring complex networks. Nature 410(6825):268–276
Strong A, Smith VK (2010) Reconsidering the economics of demand analysis with kinked budget constraints. Land Econ 86(1):173–190
Teugels JL, Vanroelen G (2004) Box-cox transformations and heavy-tailed distributions. J Appl Probab 41:213–227
Zaman A, Rousseeuw PJ, Orhan M (2001) Econometric applications of high-breakdown robust regression techniques. Econ Lett 71(1):1–8
Acknowledgments
We acknowledge EAAB for sharing micro data to perform the actual analysis. We acknowledge COLCIENCIAS’s financial support for Bonilla, and the financial support for Zarama from the Research Fund of the School of Engineering at Universidad de los Andes.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendices
Appendix 1
Definitions of functions, variables and parameters used in the document are presented in Tables 2, 3 and 4, respectively.
Appendix 2
This appendix shows the variable change to write the bounded information \( J \).
To minimize the information functional
the second term on the right,
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).
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
where \( {a_1} + {a_2} = 1 \).
Using equations Eq. 2.35 in Eq. 2.34, we get
An expression for \( h\prime(z) \) is calculated taking the derivative \( \tfrac{\partial }{{\partial z}} \) of Eq. 2.36,
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
where \( z ^{*} = {t_1}{x_0} \) and \( z ^{* *} = {t_2}{x_0} \), and \( h({t_i}x) = \tfrac{1}{{{t_i}}}\ f(x) \).
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bonilla, R., Zarama, R. (2012). An Information Approach to Deriving Domestic Water Demand: An Application to Bogotá, Colombia. In: Mejía, G., Velasco, N. (eds) Production Systems and Supply Chain Management in Emerging Countries: Best Practices. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-26004-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-26004-9_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-26003-2
Online ISBN: 978-3-642-26004-9
eBook Packages: EngineeringEngineering (R0)