Theoretical Economics Letters, 2012, 2, 351-354
http://dx.doi.org/10.4236/tel.2012.24064 Published Online October 2012 (http://www.SciRP.org/journal/tel)
On the Closed-Form Solution to the Endogenous Growth
Model with Habit Formation
Ryoji Hiraguchi
Faculty of Economics, Ritsumeikan University, Kyoto, Japan
Email: rhira@fc.ritsumei.ac.jp
Received June 25, 2012; revised July 23, 2012; accepted August 20, 2012
ABSTRACT
We study the AK growth model with external habit formation. We show that there exists a unique solution path expressed
in terms of the Gauss hypergeometric function. Using the closed-form solution, we also show that the opti mal path con-
verges to a balanced growth path.
Keywords: Endogenous Growth; Closed-Form Solution; Habit Formation
1. Introduction
The Gauss hypergeometric functions are typically used in
mathematical physics, but are not so common in eco-
nomics. As far as we know, Boucekkine and Ruiz-
Tamarit [1] are the first to find that the functions are also
useful in dynamic macroeconomics. They obtain an ex-
plicit solution path to Lucas-Uzawa two-sector endoge-
nous growth model by using the hypergeometric function.
They express the optimal path as a system of four dif-
ferential equations and two transversality conditions and
then use the hypergeometric functions for solving the
system of equations.
Several authors get analytical solution paths to the
exogenous growth models. Pérez-Barahona [2] investi-
gates the model with non-renewable energy resources
and find that the optimal path has a closed form solution
path by using the hypergeometric function. Hiraguchi [3]
finds that a solution path to the neoclassical growth
model with endogenous labor is also represented by the
special function.
Boucekkine and Ruiz-Tamarit [1] argue (see page 34)
that the hypergeometric functions will also be useful in
the investigation of the endogenous growth models.
They guess that the transition dynamics of the models
will be easier to understand if we can use the special
functions. However, there is only a few literature that
applies the special functions to the endogenous growth
models other than the Lucas-Uzawa model. One ex-
ample is Guerrini [4] who uses the special functions and
obtains a closedform solution path to the AK model
with logistic population growth. Broad applicability of
the hypergeometric functions to the endogenous growth
models is uncertain at this point, and more investiga-
tions are needed.
In this paper, we study the AK endogenous model with
external habit formation. The model has been investi-
gated by many authors including Carroll et al. [5] and
Gómez [6]. The utility function of the agent depends on
both the absolute level of consumption and the ratio be-
tween consumption and habit stock. Here the habit for-
mation is external and the level of the habit stock is ex-
ogenous to each agent. We show that there exists a unique
solution path and it is represen ted by the hypergeometric
function.
Habit formation in consumption is now popular in
modern macroeconomics. Authors have explained some
empirical facts by incorporating habits into the dynamic
macroeconomic models. Abel [7] and Gal [8] show that
habit formation can solve the equity premium puzzle in
asset pricing models and Carroll et al. [9] provide an
explanation of strong correlations between saving and
growth. Some authors characterize the properties of the
optimal paths in these models. Alvarez-Cuadrado et al.
[10], Alonso-Carrera et al. [11] and Gómez [6] in-
vestigate the transitio nal dyn amics and the stability of the
optimal paths in endogenous growth models with habits,
both analytically and numerically.
The problem of the previous papers is that they assume
the existence and the uniqueness of the optimal path
without proof. These properties are not at all obvious
here, because there exists no general theorem on the
existence of a solution path in an infinite horizons opti-
mization problem with externalities. Here we utilize the
special functions to show that their assumptions are in
fact correct.
The note is organized as follows. Section 2 describes
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