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Unformatted text preview: Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model R. Boucekkine J. R. Ruiz-Tamarit December, 2004 Abstract The special functions are intensively used in mathematical physics to solve differ- ential systems. We argue that they should be most useful in economic dynamics, notably in the assessment of the transition dynamics of endogenous growth models. We illustrate our argument on the Lucas-Uzawa model, which we solve by the means of Gaussian hypergeometric functions. We show how the use of Gaussian hyperge- ometric functions allows for an explicit representation of the equilibrium dynamics of the variables in level . In contrast to the preexisting approaches, our method is global and does not rely on dimension reduction. Keywords : Special functions, hypergeometric functions, optimal control, Lucas- Uzawa model, economic dynamics JEL classification: C61, C62, O41. We are indebted to Vadim Kuznetsov, Alphonse Magnus, Nico Temme, Benteng Zou and David de la Croix for useful suggestions and comments. R. Boucekkine acknowledges the support of the Belgian research programmes PAI P4/01 and ARC 03/08-302. J. R. Ruiz-Tamarit acknowledges the financial sup- port from the Spanish CICYT, Project SEC2000-0260, and the Grant PR2003-0107 from the Secretar a de Estado de Educacion y Universidades, Spanish MECD. Corresponding author. Department of Economics and CORE, Universit e Catholique de Louvain, Place Montesquieu 3, B-1348 Louvain-la-Neuve (Belgium). E-mail: email@example.com Department of Economics, Universitat de Val` encia (Spain) and Universit e catholique de Louvain. E-mail: firstname.lastname@example.org 1 Introduction Special functions refer to some specific functions with special characteristics, typically used in mathematical physics and computational mathematics. Prominent examples in- clude the Gamma function, the Beta function, the Bessel function and the more general, hypergeometric functions. There exist some excellent reviews on the properties and uses of such functions, among them Luke (1969), Abramowitz and Stegun (1972), and more recently Temme (1996) and Andrews, Askey and Roy (1999). The area of special func- tions is, by no way, a new research area: it traces back at least to Euler and Gauss. Among their multiple uses, the resolution of differential equations and systems is not the least important nor the least historically recognized. This is the case of Gauss hyperge- ometric functions, which allowed to solve nicely the so-called hypergeometric differential equation (Kummer, 1836, or Goursat, 1881). We shall also use this class of functions in our study. Indeed, we argue in this paper that researchers in economic dynamics should use much more intensively these tools, which can be decisive if one aims at getting be- yond the typical computational and/or local approaches. More specifically, we argue that such functions might be most useful in the assessment of the transition dynamics and...
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