Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model

Abstract

The special functions are intensively used in mathematical physics to solve differential systems. We argue that they should be most useful in economic dynamics, notably in the assessment of the transition dynamics of endogenous economic growth models. We illustrate our argument on the famous Lucas-Uzawa model, which we solve by the means of Gaussian hypergeometric functions. We show how the use of Gaussian hypergeometric functions allows for an explicit representation of the equilibrium dynamics of all variables in level. The parameters of the involved hypergeometric functions are identified using the Pontryagin conditions arising from the underlying optimization problems. In contrast to the pre-existing approaches, our method is global and does not rely on dimension reduction.

Closed-form solutions for the Lucas–Uzawa model of economic growth via the partial Hamiltonian approach

“…We begin by taking two first integrals and show that our approach yields the same results as in the previous literature (see e.g. [13,14,16]). We then show that if we use the three first integrals, we obtain completely new solutions for all the variables in the Lucas-Uzawa model which in turn yield new solutions for the growth rates of these variables.…”

“…Our partial Hamiltonian methodology yielded two first integrals with no restriction on parameters and by utilizing these two first integrals we derived all the solutions which have been previously discussed in the literature (see e.g. [13,14,16] ).…”

“…[12][13][14]). Xie [12] provided explicit solutions and dynamical properties for the original variables under the parameter restriction that the inverse of the intertemporal elasticity of substitution equals to the elasticity of output with respect to physical capital.…”

“…Xie [12] provided explicit solutions and dynamical properties for the original variables under the parameter restriction that the inverse of the intertemporal elasticity of substitution equals to the elasticity of output with respect to physical capital. At the same time Boucekkine and Ruiz-Tamarit [13] and Hiraguchi [15] gave solutions in terms of hypergeometric functions without relying on parameter restrictions. Later on, Chilarescu [14] proposed a solution procedure using only classical mathematical tools and also compared his results with those derived by Boucekkine and Ruiz-Tamarit [13].…”

“…At the same time Boucekkine and Ruiz-Tamarit [13] and Hiraguchi [15] gave solutions in terms of hypergeometric functions without relying on parameter restrictions. Later on, Chilarescu [14] proposed a solution procedure using only classical mathematical tools and also compared his results with those derived by Boucekkine and Ruiz-Tamarit [13]. The approach here is distinct.…”