On the Closed-Form Solution to the Endogenous Growth Model with Habit Formation

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 optimal path converges to a balanced growth path.


Introduction
The Gauss hypergeometric functions are typically used in mathematical physics, but are not so common in economics.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 explicit solution path to Lucas-Uzawa two-sector endogenous growth model by using the hypergeometric function.They express the optimal path as a system of four differential 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] investigates 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 example 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 investigated 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 between consumption and habit stock.Here the habit formation is external and the level of the habit stock is exogenous to each agent.We show that there exists a unique solution path and it is represented 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] investigate the transitional dynamics 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 optimization 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 the model and obtains the first order conditions.Section 3 obtains the closed-from solution path.The conclusions are in Section 4. Proofs of the propositions are in Appendix.

Set-Up
In this section, we construct the one-sector endogenous growth model with external habit formation and obtains the first order conditions.There is a continuum of agents with unit measure.There is no population growth.The instantaneous utility function of each agent is Here is his own consumption, t is the habit stock,


and the utility is time-separable, the parameter  coincides with the coefficient of the relative risk aversion.The habit stock is exogenous to the consumer and is accumulated by the following differential equation: Here t c is the average level of consumption and > 0


is a parameter.The parameter  is high, the habit stock responds to the recent consumption quickly.The consumer solves the following problem: (2) here > 0


is the discount factor, Equation ( 2) is the resource constraint, t is physical capital and is the technology parameter.We assume that there is no capital depreciation.The initial capital stock 0 and the initial habit stock are given.In what follows, we denote the growth rate of a variable as is the multiplier.The first order conditions (FOCs) and the transversality condition (TC) are FOC(c) : = , The next lemma shows that when t A he productivity is too high, the interior optimal path does not exist 1 .3) and ( 4) together imply that the ed growth rate of consumption (and also habit stock) is negative.Thus we impose the following restriction on the parameters to ensure that the optimal path is interior and that the optimal growth rate is positive: balanc

Closed-Form Solution
, we first obtain a linear To characterize the optimal path differential equation on the habit-consumption ratio = where

Since
The solution is  atio z con-, the habit consumption r ve t rges to z  as t goes to  .Next w use E uation (9) to e q obtain the equilibrium consumption path.Since = where the parameter g is defined as Note that by definition,  , the co growth rate co we show that the growth rate of the nsumption nverges to g.Later equilibrium capital also converges to the same value g.
In Equation ( 10), 0 z is unknown.To fix its value, we ha n (10), we get ve to use the transversality condition (5).Here the resource constraint (2) can be written as Under the parametric restriction (7), There exists a unique that satisfies the condition.(H the equilibrium capital as 0 z en ere 0 k and 0 h are giv .)Proo See the ppendix. f.A Using Lemma 2, we can re-write We now express the equilibrium capital without using the integral.As Hiraguchi (2012) shows, the hypergeometric function .
. Finally we get the following proposition.
Proposition 1.The optimal path exists, is unique and is expressed as The parameters are   It is well-k own that the basic AK growth model does not have transitional dynamics and the optimal growth rate is always constant.

Conclusion
In this paper, we obtain a closed-form so the AK growth model with habit formation.As Boucekki s are applicable to many kinds of economic models.As a future study, igate the different kinds of the endo-lution path to ne and Ruiz-Tamarit [1] claim, the hypergeometric functions are actually very useful in the investigation of the endogenous growth models.We guess that the Gauss hypergeometric function the dynamic macro we hope to invest genous growth models, especially the growth models with R & D.  Then the optimal path does not exist.

oof of Lem 2
quivalent to Equaf the multiplier c h  on the utility function.When there = 0 cekkine and Ruiz-Tamarit[1] (see Proposition 1 in page 40).Thus we can express the integral part of t k as

.
As many authors have already shown, the integral can be obtained by using the hy-

.
We first show that Equation (12) is e is A   tion(5).The growth rate o t  Thus the term