Theoretical Economics Letters, 2012, 2, 351-354
http://dx.doi.org/10.4236/tel.[phone] 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
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
C
opyright © 2012 SciRes. TEL

R. HIRAGUCHI 352
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 Ap-
pendix.
2. 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 continuu m of agents
with unit measure. There is no population growth. The
instantaneous utility func tion of each agent is
 


11
111
,= =
11
ttt tt
tt
cch ch
uch






 .



Here is his own consumption, t is the habit
stock, t
c
>0 h

is the parameter on the utility curvature
and shows the importance of relative con-
sumption level

0,

 1

tt
ch

on the utility function. When
there =0

and the utility is time-separable, the pa-
rameter

coincides with the coefficient of the relative
risk aversion. The habit stock is exogenous to the con-
sumer and is accumulated by the following differential
equation:


=
ttt
hch  
. (1)
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:


0
,
:,d,s.t.=
max ttttt t
ck
tt
PeuchtkAk


.c
(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
k
0
h
>0A
k
t
x

ˆ=
ttt
x
xx
.
The current value Hamiltonian is




1
=11
tttt t
H
chAkc




 
where t

is the multiplier. The first order conditions (FOCs) and
the transversality condition (TC) are
FOC(k) :=,
t
t
A



 (3)
(1 )
FOC(c) :=,
tt t
ch


 (4)
TC:() =0.
lim t
tt
tke



 (5)
Here FOC(x) means the FOC on the variable
x
. In
equilibrium, the individual consumption is equal to
t
c
and the habit stock is accu-
mulated according to


=.
ttt
h  
 hc (6)
The path


,,
ttt
ckh
)-(6) for sois optimal if Eq and only if it satisfies uations (2me 0
t

.
The next lemma shows that when t
A
he productivity
is too high, the interior optimal path does not exist1.
Lemma 1. If




11>A

  , the optimal path
do Appendix. is too low and satisfies es not exists.
Proof. See the 
Moreover, if the productivity
>
A

, Equations (3) and (4) together imply that the
ed growth rate of consumption (and also habit
stock) is negative. Thus we impose the following re-
striction on the parameters to ensure that the optimal path
is interior and that the optimal growth rate is positive:
balanc

 
11<<.
A
A   (7)
3. Closed-Form Solution
, we first obtain a linear To characterize the optimal path
differential equation on the habit-consumption ratio
=
ttt
zhc. Note that ˆˆ
ˆ=
ttt
zhc. Substitution
ˆ
ˆ=.
tt
zh A  
of
Equation (4) into Equation (3) yields

 (8)
where


=1 >0   .
written as On the other hand, Equation (6) is


1
t
z. Thus Equation (8)
implies
ˆ=
t
h  1

 
1
t
zA

ˆ=1
t
z

 


he equation by t
z, we get. Multiplying both sides of t a linear differential
equation


=
tt
zzz 


ith w


=A  nd

>0   a=z
>0A
   .
(9)
Since
The solution is

0
=.
t
t
zze zz
  

>0

atio z con- , the habit consumption r
ve t rges to z

as t goes to .
Next wuse Euation (9)to e q obtain the equilibrium
consumption path. Since ˆ
ˆˆ=
tt t
czh, Equation (8) is re-
expressed as


ˆ=
t
z 
ˆt
c   A

. Thus



00
=A
tt
cce t zz




consump and the tion is
t
c

100
,zh
   (10) = gt
tt
cez
where the parameter
g
is defined as

==>
1
AA
g0.
   (11)
Note that by definition,
 
00
=chz
0
. Since
=<
tt
zz


lim

, the co growth rate co
we show that the growth rate of the
nsumptionn-
verges to g. Later
1It is not a simple task to prove Lemma 1 only by using the transversal-
ity condition (5). This is because we cannot easily obtain the asymp-
totic growth rate of physical capital by using the system of the differ-
ential Equations (2)-(6). the aver age consumptio n t c

R. HIRAGUCHI 353
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 re-
source constraint (2) can be written as

00
=d
t
At As
ts
kek ecs

. Using Equatio
 1
tAg
s
 

00 0
0
=d.
ts
hez s





Under the parametric restriction (7),
At
ke zk

1=1 1<AA

  and then

>=Ag A

 . Thus the term
 1
0d
te


constant as t goes
lemma simplifies the transversality condition (5).
Lemma 2. The transversality condition (5) holds
Ag ss
zs
 
to . The next
if
an
converges to a finite
d only if
 1
000
0
=d
Agss
kzhezs
 



. (12)
There exists a unique that satisfies the condition.
(H
the equilibrium capital
as
0
z
enere 0 k and 0 h are giv.)
Proo See the ppendix.  f. A
Using Lemma 2, we can re-write


1
00
=d
Ags
At
ts
t
kezhe zs




 .
We now express the equilibrium capital without using
the integral. As Hiraguchi (2012) shows, the hyper-
geometric function
 


21
F,,;=abcz =0 !
n
nnn n
abcz n


with a satisfies ()=()/ ()
n
aan 

3
a
y
12
12
3 111
12
2
21 3
1221
d
=F,,1;,
ay a
x
a ax ax
ebbe y
ebaab
ae
aaab







 (13)
where , , , and are
1>0a
t2. A sim2>0a
ilar equ3>0a, ion is a1>0b o prove2
b
ouconstan atlsd by Bcek-
kine and Ruiz-Tamarit [1] (see Proposition 1 in page 40).
Thus we can express the integral part of t k as



/1
d
Ags
ezs

 

/1
21
0
=F
1,,1;.
s
t
Ag t
t
ze
Ag
zz
Ag Ag e
z



 












Recall that

0
=t
t
zzezz



. Finally we get the
following proposition.
Proposition 1. The optimal path exists, is
unique and is expressed as

,,
ttt
chk




1
00
=,
gt t
t
ceze zzzh
 


 

0
(14)
00
=={( )}()
gt t
ttt
hzcezezzz h
  

0
,

(15)

 
1
gt
t
zh
Ag
k





0
0
0
t
zz Ag ez





(16)
The parameters are
21
=F1, ,1;.ezA
g

 






=1 >0   ,


=>0  ,

=>A   0  and zA  







=1gA

 

. The value of
termined by Equation (12).
It is known that for any a, b and c,
Thus the growth rate of the capital also c
Therefore the optimal path
to a balanced growth path
0
z is de-

21
F,,;0=1abc.
onverges to g.
always converges
h rate g.

,,
ttt
chk
with the growt
If =0

(no habit =formation), then


n Equa
and the
first te the hypergeetric function itio
is rm ofomn (16) 1=0



. Thus the olly: ptimal consumption path and
capital path grows exponentia

00
=,
g
t
t
cz he
  (17)

1
 
0.
0
=
g
t
zh

t
ke
zA
g

 (18)
It is well-kown that the basic AK growth model does
not have transitional dynamics and
 
n the optimal growth
rate is always constant.
4. Conclusion
In this paper, we obtain a closed-form so
the AK growth model with habit formation. As Boucek-
ki
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.
REFERENCES
[1] R. Boucekkine and R. Ruiz-Tamarit, “Special Functions
for the Study of Economic Dy namics: The Case of the Lu-
cas—Uzawa Model,” Journal of Mathematical Econom-
ics, Vol. 44, No. 1, 2008, pp. 33-54.
doi:10.1016/j.jmateco.2007.05.001
[2] A. Pérez-
2If we let , we can easily show that the integral is equal to
1
=ay ze



1
11
ax
bb



21 3
132
20d
a
e
aaa
abz z
. As many authors have already
shown, the integral can be obtained by using the hy-

d
c
b
az z

p
ergeometric function. Barahona, “Nonrenewable Energy Resources as

R. HIRAGUCHI 354
Input for Phytion: A New Ap- ing Up with the Joneses,” American Economic Review,
Vol. 80, No. 2, 1990, pp. 38-42.
[8] J. Gal, “Keeping up with the Jone
sical Capital Accumula
proach,” Macroeconomic Dynamics, Vol. 15, No. 1, 2011,
pp. 1-30. doi:10.1017/S1365100509090415
[3] R. Hiraguchi, “A Note on the Analytical Solution to the
Neoclassical Growth Model with Leisure,” Macroeco-
nomic Dynamics, 2012, pp. 1-7.
ses: Consumption Ex-
ternalities, Portfolio Choice, and Asset Prices,” Journal of
Money, Credit, and Banking, Vol. 26, No. 1, 1994, pp. 1-
8. doi:10.2307/2078030
doi:10.1017/S1365100512000442
[4] L. Guerrini, “Transitional Dynamics in the Ramsey Mod-
el with AK Technology and Logistic Population Change,”
Economics Letters, Vol. 109, No. 1, 2010, pp. 17-19.
[9] C. Carroll, J. Overland and D. Weil, “Saving and Growth
with Habit Formation,” American Economic Review, Vol.
90, No. 3, 2000, pp. 341-355. doi:10.1257/aer.90.3.341
[10] F. Alvarez- Cuad ra do, G. Mo nte ir o and S. T urn ov sky , “Ha bit
doi:10.1016/j.econlet.2010.07.002
[5] C. Carroll, J. Overland and D. Weil, “Comparison Utility
in a Growth Model,” Journal of Economic Growth, Vol. 2
No. 4, 1997, pp. 339-367.
Formation, Catching up with the Joneses, and Economic
Growth,” Journal of Economic Growth, Vol. 9, No. 1,
2004, pp. 47-80. ,
doi:10.1023/A:1009740920294 doi:10.1023/B:JOEG.0000023016.26449.eb
[11] J. Alonso-Carrera, J. Caballé and X. Raurich, “Growth,
Habit Formation, and Catching up with the Joneses,” Euro-
pean Economic Re
1691.
[6] M. A. Gómez, “Convergence Speed in the AK Endoge-
nous Growth Model with Habit Formation,” Economics
Letters, Vol. 100, No. 1, 2008, pp. 16-21.
doi:10.1016/j.econlet.2007.10.022 view, Vol. 49, No. 6, 2005, pp. 1665- doi:10.1016/j.euroecorev.2004.03.005
[7] A. Abel, “Asset Prices under Habit Formation and Catch-
Appendix
. Proof of Lemma 1
Let Consider a path s
th is feasible since the
as
1and then the term t
tt
ke



ality condition
can be expressed as.
the transversholds if and At
tt
ke 
only if Thus

=1 1>0A   .
e pa
ritten
uch
that

=/2 >0
tt
ck  . Th
resource constraint can be w

 1
00 0
00.
t

=d=
lim limAg
s
At ts
tt
ekkzh ezs
 


 





Under Equation (7) >
A
g and then


=2>0
t
kk
the path, we can easily
show
tA
 .
that Along
1d
Ags
ezs

 
 is finite. Thus the above equation
0s
equivalent to EqWe next show
exists a unique Equation (12). If we de-
fin
is
that thereuation (1
0 satisfying
2).
z
e a function t
G as

ˆˆ
ˆ==2
lim tt
tt
ckA=
lim lim
t
t
h


the instantaneous utility as
.
If we denote
=
t
U


11(1 )
ttt
ech









,
o. The
growth rate of converges to a positive constant be-
cause
1
=1
tt
t
Gzze zze


 

 
 ,
Equation (12) iwritten as then the intertemporal utility is equal t 0d
t
Ut


t
U

s


00 0
0
=d
Agt t
kheGz t

.

ˆ=11 2=2>0
limtt
UA 
  .
Thus 
Since 10
t
e
 

, the function




1
=1
tt
t
Gzezze




0d=
t
Ut
.
Then the optimal path does not exist.
oof of Lem 2
quivalent to Equa-
f the multiplier



1
tt
eez z


  g
fuof. Mor and
 is a strictly decreasin
nction eover,

0=
t
G



=0 zt G.
Thus there exts a uniqueng is 0
z satisfyi

00 0
0
=t G
2. Prmakh zdt

.
We first show that Equation (12) is e is
A


tion (5). The growth rate ot
