Existence, uniqueness and stability of equilibria
in an OLG model with persistent inherited
habits
Giorgia Mariniy
University of York
February 7, 2007
Abstract
This paper presents su¢ cient conditions for existence and uniqueness of a
steady-state equilibria in an OLG model with persistent inherited habits
and analyses the implications of persistence of inherited habits for the
local stability of the steady-state equilibria. The conditions for a stable
solution are derived under the assumption that habits are transmitted
both across and within generations. Under this assumption monotonic
convergence to the steady state is not always assured. Both competitive
and optimal equilibria may display explosive dynamics.
keywords: Habits, Existence, Uniqueness, Stability, Overlapping generations
jel classification: D50, D91, E13, E32, O41
Earlier versions were presented at the DeFiMS Wednesday Research Seminars in 2002,
2003, and 2004, at the University of Rome La Sapienza Seminar Series in 2005 and at the
11th Spring Meeting of Young Economists held in Seville, Spain, May 2006. I am indebt to
Pasquale Scaramozzino for his invaluable supervision and to Giacomo Rodano for his constant
encouragement and support. All errors are mine.
y Centre for Health Economics, University of York, Heslington, YORK, YO10 5DD, UK.
Phone: +44 1904 321421, Fax: +44 1904 321454, E-mail: [email protected]
1
1
Introduction
This paper derives su¢ cient conditions for existence and uniqueness of a steadystate equilibria in an overlapping generations model with persistent inherited
habits and analyses the implications of persistence of inherited habits for the
local stability of the steady-state equilibria.
The main assumption of this paper is that habits are transmitted both across
and within generations: that is the situation in which habits are transmitted
from one generation to the next one (inheritance) and individuals carry these
habits into old age (persistence). This assumption is modelled in the paper with
non-separable preferences both across and within generations.1
In the literature on stability of the equilibrium in OLG models with habits,
Diamond’s [2] OLG model represents the benchmark model with separable preferences both across and within generations. Galor and Ryder [5] develop suf…cient conditions that guarantee the existence of a unique and globally stable
nontrivial steady-state equilibrium in Diamond’s [2] model, and de la Croix and
Michel [1] prove that the optimal solution in Diamond’s [2] model is always
characterised by monotonic convergence to the steady state.
Michel and Venditti [7] provide su¢ cient conditions for stability of the equilibrium in an OLG model with separable preferences across generations only
and prove that the optimal solution may be oscillating and optimal cycles may
exist.
de la Croix and Michel [1] provide su¢ cient conditions for existence and
uniqueness of the equilibrium in an OLG model with separable preferences
within generations only. They prove that the optimal solution may display
damped oscillations even when the social planner does not discount the utility
of future generations.
The contribution of this paper is to provide su¢ cient conditions for existence, uniqueness and stability of the equilibrium in an OLG model in which
preferences are non-separable both across and within generations.
Since the intergenerational transmission of habits generates an intergenerational externality, both the competitive equilibrium and the social planner problems are analysed. The paper derives the conditions under which the unique
competitive equilibrium converges to or diverges from the steady state and shows
that the competitive equilibrium may display either ‡uctuations or explosive behaviour. Then it studies the conditions under which the unique optimal solution
is stable. Under the assumption of persistence of inherited habits, convergence
of the optimal solution to the steady state is not always assured. The optimal
solution may display either locally explosive dynamics or damped oscillations.
This paper shows that the assumption of persistence of inherited habits is
not innocuous: introducing the assumption of persistence in the contest of an
1 The assumption of persistent inherited habits in the contest of an OLG with production
of a single good (consumption good such as food) is supported by plenty of empirical evidence
on eating habits. This empirical evidence suggests that the risk of becoming obese is greater
among children with obese parents (Dietz [3]) (inheritance) and that obesity beginning in
childhood may persist through the life span (Law [6]) (persistence).
2
OLG model in which habits are inherited, dynamics of the model and stability
of the equilibrium are considerably a¤ected. Therefore researchers should be
more cautious in neglecting persistence of inherited habits.
2
The model
The economy is populated by three generations, each one living three periods.
The rate of growth of population is zero. The young generation passively inherits habits from their parents: et = cat 1 ; 8t > 0. et is the stock of habits inherited
from the parents and cat 1 is the level of consumption of the parents. Since inherited habits are out of children’s control, they generate an intergenerational
externality. The adult generation derives utility from consumption of the quantity cat , given the stock of inherited habits et . The old generation derives utility
not only from consumption of the quantity cot+1 but also from consumption in
t + 1 relative to consumption in t: ht+1 = cat ; 8t > 0. ht+1 represents the stock
of habits that individuals form during their adulthood and that persist into old
age and cat the level of consumption of the parents. Since habits persistence is
under individual control, the stock h is internalised in the utility maximisation
process and it does not generate any intragenerational externality.
The intertemporal utility function of each individual is
U cat ; cot+1 ; et ; ht+1 = u (cat
et ) + v cot+1
ht+1
(1)
in which 2 (0; 1) is the parameter measuring the intensity of parental habits
and 2 (0; 1) is the parameter measuring the persistence of adult habits into old
age. Note that the rate of depreciation of habits inherited from the parents is
so high that parental habits do not directly a¤ect their children’s consumption
when their children are old: the function v is independent of the stock e. On
the other hand, parental habits indirectly predispose children to a persistent
behaviour from adulthood into old age: the function v depends on the stock h.
Thus, intergenerational and intragenerational interactions work separately on
the instantaneous utility functions u and v.
Assumption 1 (i) u : R2+ ! R is twice di¤ erentiable over R2++ ; (ii) u is
strictly quasi concave over R2++ , so that uca > 0; uca ca < 0; (iii) ue < 0; uee < 0
and uca e > 0.
Assumption 2 (i) v : R2+ ! R is twice di¤ erentiable over R2++ ; (ii) v is
strictly quasi concave over R2++ , so that vco > 0; vco co < 0; (iii) vh < 0; vhh < 0
and vco h > 0.
Conditions (iii) de…ne reaction to the stocks e and h. The patterns inherited
from the parents have a negative impact on the individual utility function (ue <
0) and an increase in the stock e of past experiences has a positive e¤ect on the
desire for consumption (uca e > 0). Similar comments hold for the assumptions
vh < 0 and vco h > 0.
3
Assumption 3 (i) ue
2
uca ; uee
uca ca ; (ii) vh
vco ; vhh
2
vco co .
Assumption 4 (Starvation is Avoided in Both Periods) limcat !0 uca +vh
= limcot+1 !0 vco = 1; 8 cat ; cot+1 2 R2++ .
Assumption 5 (Second Period Consumption is a Normal Good) (uca +
vh ) vco h > vco (uca ca + vhh ).
Assumption 6 (E¤ective Consumption is Always Non-Negative) cat
et > 0 and cot+1
ht+1 > 0.
Lemma 1 Under Assumption 6, the indi¤ erence curve associated to the intertemporal utility function (1) is downward sloping if < uca =vco .
Proof. The slope of the indi¤erence curve is
dcot+1
=
dcat
t+1
uca + vh @h
@ca
t
vco
=
uca
vco
vco
(2)
under Assumption 3 and @ht+1 =@cat = 1. If = 0 (benchmark Diamond’s [2]
model with separable preferences), downward slope is always ensured by the
standard set of assumptions on marginal utility, i.e. uca ; vco > 0. If > 0,
the slope is upward sloping if > uca =vh , ‡at if = uca =vh and negative
if < uca =vh . Therefore, the indi¤erence curve is downward sloping only if
2 (0; uca =vh ].
No restriction is necessary for the domain of .
In the second period of their life (adulthood) each household supplies inelastic labour to the labour market for which they receive a wage. Individual …rst
and second period budget constraints are
cat + st
cot+1
= wt
= Rt+1 st
(3)
(4)
in which st are the savings, Rt+1 is the interest factor upon which consumers
plan lifetime consumption and saving and wt is the real wage. The former
constraint (3) states that individuals can decide how to allocate income between
current consumption and saving; the latter constraint (4) states that during
retirement agents spend their savings completely. Agents optimally choose the
consumption plan cat ; cot+1 , solving the following problem
max u (cat
o
ca
t ;ct+1
et ) + v cot+1
ht+1
(5)
subject to the individual constraints (3) and (4), to the stock of persistent
habits ht+1 = cat and given the external stock of inherited habits et , wages wt
and correctly foreseen interest rates Rt+1 .
4
First-order conditions lead to the following Euler equation2
uca
vco = Rt+1 vco
(6)
With respect to the standard Diamond’s [2] model in which = 0, marginal
utility of the young is lower, as vco > 0: in order to achieve the same level of
satisfaction when old, adults need to correct their satisfaction by the (negative)
habit e¤ect.
The saving function associated with the maximisation problem is de…ned as
st = s (wt ; Rt+1 ; et ): it is independent of ht+1 since the constraint ht+1 = cat is
under the agent’s control, but it depends on et since et = cat 1 is an externality.
Given (wt ; Rt+1 ), Assumptions 1 and 2 imply that st exists and is unique.
Assumption 5 implies that it is an increasing function of the real wage wt
sw =
uca ca + (Rt+1 + ) vco co
2
uca ca + (Rt+1 + ) vco co
>0
(7)
By Assumptions 1 and 2 it may be an increasing or decreasing function of real
return to capital
h
i
vco + vco co cot+1 1 + Rt+1
sr =
(8)
2
uca ca + (Rt+1 + ) vco co
while by Assumption 1 it is a decreasing function in the stock of externality et
se =
uca ca
2
uca ca + (Rt+1 + ) vco co
<0
(9)
The e¤ect of the real wage is positive since an increase in real wage increases
both current consumption and saving for retirement. The e¤ect of the interest
rate is ambiguous since it depends on the intertemporal elasticity of substitution.
The e¤ect of the consumption externality is negative: an increase in e implies an
increase in the desire for consumption when an adult and therefore a reduction
in saving, ceteris paribus.
In each period a single consumption good is produced. Production of the
single good is realised with a constant returns to scale technology. Per capita
output yt is a function of capital intensity kt :
yt = f (kt )
(10)
Assumption 7 (i) f : R+ ! R+ is twice continuously di¤ erentiable; (ii)
fk > 0 and fkk < 0 8k > 0; (iii) f (0) = 0; limkt !1 fk = 0; limkt !0 fk = 1.
Firm maximises pro…ts and inputs are paid their marginal products
Rt = fk (kt )
(11)
2 Second-order conditions are satis…ed under Assumptions 1 and 2. Strictly quasi concavity
guarantees that the sequence of minors of the Hessian matrix is f ; +g and therefore that the
matrix is de…nite negative.
5
and
wt = f (kt )
kt fk (kt )
(12)
in which Rt is the interest factor on capital and wt is the real wage paid to
workers.
Under the assumption that capital completely depreciates over time and
that the economy is endowed with k0 and e0 = ca 1 at time t = 0, the resource
constraint of the economy is
yt = cat + cot + kt+1
(13)
The consumption good can be either consumed by the adult and the old generations, or accumulated as capital for future production.
3
The competitive economy
Considering the market clearing conditions (11), (12) and the resource constraint
(13), if follows that the capital accumulation is given by
kt+1 = st (wt (kt ) ; Rt+1 (kt ) ; et )
(14)
De…nition 1 Under Assumptions 1, 2 and 7, the intertemporal equilibrium
1
of the competitive economy is a sequence fkt ; et gt=1 following from individual
constraints (3) and (4), equilibrium of production factors (11) and (12), saving
function and market clearing (14), in which k0 , e0 and the condition et = cat 1
are exogenously given:
kt+1
et+1
= s (f (kt ) kt fk (kt ) ; fk (kt ) ; et )
= f (kt ) kt fk (kt ) s (f (kt ) kt fk (kt ) ; fk (kt ) ; et )
(15)
(16)
De…nition 2 Under Assumptions 1, 2 and 7, the steady-state equilibrium of
the intertemporal equilibrium (15)-(16) is a combination of (k; e) 2 R2++ such
that
k = s (f (k) kfk (k) ; fk (k) ; e)
e = f (k) kfk (k) s (f (k) kfk (k) ; fk (k) ; e)
3.1
(17)
(18)
Existence, uniqueness and stability of the steady-state
equilibrium (17)-(18)
Proposition 1 The steady-state equilibrium (17)-(18) exists and is unique if
and only if det I JCE 6= 0, where JCE is the Jacobian matrix associated
to the intertemporal equilibrium (15)-(16) and evaluated at steady state (k; e)
(Galor [4]).
Proof. See Appendix A.
6
Stability of the steady state (17)-(18) depends on parameters and . Since for
some values of these parameters the hyperbolic condition may not be satis…ed,
it is necessary to look for the critical value of and , ^ and ^, at which the
change in trajectory takes place (bifurcation) and the …xed point becomes non^; ^ = 1 if the roots
hyperbolic. Non-hyperbolicity may arise when mod
of the characteristic polynomial associated with the Jacobian matrix are real
or when r = 1 if these roots are complex. If one of these conditions is met,
linear approximation cannot be used to determine the stability of the system.
Otherwise, local stability properties of the linear approximation carry over to
the non-linear system.
Proposition 2 Suppose Proposition 1 holds and suppose that se = (1 kfsrkkfkk ) ,
h
i
1
kk j
sr > fkk
and 1 jssrefj kk < 1+ 1 s1w kjf
sr fkk . Then the determinant of the Jacobian
matrix JCE is positive and equal to 1, the trace TCE
is positive and smaller than
J
2, and the discriminant
is negative.
Proof. See Appendix B.
Proposition 3 Suppose Proposition 2 holds. Then the eigenvalues of JCE are
^; ^ <
complex and conjugate and the steady state is stable if and only if mod
1 and unstable otherwise.
Proof. If mod
^; ^ < 1, the system is characterised by a spiral convergence
^; ^ = 1, the system exhibits a period
to the steady-state equilibrium; if mod
^; ^ > 1, the system exhibits a spiral divergence from the
orbit; and if mod
steady-state equilibrium (Galor [4]). The system is therefore characterised by a
bifurcation identi…ed at ^; ^ which are the roots of det JCE = 1.
The competitive equilibrium is thus characterised by spillovers from one generation to the next and from adulthood to old age. The main components of the
intergenerational spillovers are: the savings by old people …nance the capital
stock required for production in the next period directly, and wages of adults
indirectly; past consumption of the adults feeds the inherited habits of the next
generation. While the process transforming the savings by the old into income
for the adult displays decreasing returns to scale due to the characteristics of the
production function, the process transforming past consumption of the adults
into consumption of the next generation displays constant returns to scale due
to the characteristics of the utility function. The intragenerational spillovers
is only given by the individual past consumption that feeds individual’s habits
from adulthood to old age. This process displays constant returns to scale, again
due to the characteristics of the utility function. Thus, even though the intergenerational bequest in terms of higher wages will not be su¢ cient to cover the
intergenerational bequest in terms of higher inherited habits, the intragenerational spillover leaves a bequest in terms of higher persistence. The combination
7
of the positive bequests in terms of higher wages and higher persistence is suf…cient to o¤set the negative bequest in terms of the higher externality. This
leads to an increase in saving to maintain future standards of consumption that
induces an expansion. When the enrichment is strong enough, the externality
has already reverted to higher levels, allowing a fall in savings and the start of a
recession. As the e¤ect due to persistence is stronger than the e¤ect due to the
externality, the model is characterised by converging cycles. Thus, the competitive equilibrium still displays ‡uctuations, but the bifurcation corresponds to
di¤erent critical values of and . Depending on the parameters and , the
economy may converge to or diverge from the steady state.
4
The optimal solution
As the inheritance from the parents introduces an externality in the model, the
decentralised equilibrium is implicitly sub-optimal compared to the equilibrium
that would maximise the planner’s utility. Thus, this paper studies the social
planner’s optimisation problem.
The social planner’s goal is to choose the optimal allocation that maximises
social welfare. Given the discount factor , the social planner chooses fcat ; cot g
and fkt ; et g such that
max
a o
ct ;ct ;kt ;et
1
X
t
u (cat
et ) +
1
v (cot
ht )
subject to the resources constraint (13), et = cat 1 , ht = cat
k0 .
First order conditions are:
uca (cat
et ) + ue cat+1
et+1
vco (cot
ht )
1
and given e0 and
=
=
1
(19)
t=0
(20)
1
vco (cot
= vco cot+1
ht )
vh cot+1
ht+1 fk (kt+1 )
ht+1
(21)
Equation (20) represents the optimal intergenerational allocation of consumption between the middle aged and old, in which the social planner internalises the
externality associated with e. Marginal utility of the adult, corrected for the externality through the additional term ue , is equal to the marginal utility of the
old. Note that this social planner’s …rst-order condition does not respect equation (6). Moreover, with respect to the standard Diamond [2] model in which =
= 0, marginal utility of the adult, uca (cat
et ) + ue cat+1
et+1 , is lower,
1 o
ht ) vh cot+1
ht+1 ,
as ue < 0, while marginal utility of the old, vc (cot
is higher, as vh < 0. Equation (21) sets the optimal intertemporal allocation.
De…nition 3 Under Assumptions 1, 2 and 7, the intertemporal equilibrium of
1
the optimal solution is a sequence fcat ; cot ; kt ; et gt=1 following from …rst order
8
conditions (20)-(21) and the resource constraint (13), under the condition et =
ht = cat 1 :
uca (cat
et ) + ue cat+1
et+1
=
=
1
vco (cot
ht )
et
kt+1
(22)
1
vco (cot
= vco cot+1
= ht = cat 1
= f (kt ) cat
ht )
vh cot+1
ht+1
ht+1 fk (kt+1 )
(23)
(24)
(25)
cot
De…nition 4 Under Assumptions 1, 2 and 7, the steady-state equilibrium of
the intertemporal equilibrium (22)-(25) is a combination (ca ; co ; k; e) 2 R2++
such that
uca (ca
e) + ue (ca
e)
1
=
1
vco (co
h)
= fk (kt+1 )
e = h = ca
k = f (k) ca
vh (co
h)
(26)
(27)
o
c
(28)
(29)
Equation (26) shows that in an economy with persistent e¤ects of parental
habits, the marginal utility of the adult is lower than the corresponding marginal utility in Diamond’s [2] model: the inheritance represents a benchmark
from which individuals want to depart. Even the marginal utility of the old
is higher than the corresponding marginal utility in Diamond’s [2] model: the
same interpretation carries on. Once the externality associated with parents’
habits is internalised, persistence a¤ects marginal utility in the same way as the
externality: they both induce consumers to save. Equation (27) is the modi…ed
Golden Rule. Equations (28) and (29) have been already discussed in the paper.
4.1
Existence, uniqueness and stability of the steady-state
equilibrium (26)-(29)
Proposition 4 The steady-state equilibrium (26)-(29) exists and is unique if
and only if det I JSO 6= 0, where JSO is the Jacobian matrix associated to
the intertemporal equilibrium (22)-(25) and evaluated at steady state (ca ; co ; k; e)
(Galor [4]).
Proof. See Appendix C.
Proposition 5 Assume that capital stock and the externality are state variables
and that adult and old consumption are jump variables. Locally-explosive dynamics is possible, depending on the sign of the trace TSO
and of the element Z
J
9
of the Jacobian matrix JSO . If
0, the eigenvalues are real and local dynamics is either explosive or monotonic. If < 0, the eigenvalues are complex and
conjugate and local dynamics displays either explosive or damped oscillation.
Proof. See Appendix D.
The above proposition states all the possible patterns of dynamics of the
optimal steady-state equilibrium.
Under the assumption that the trace TSO
and the element Z are both posJ
itive, locally explosive dynamics is identi…ed by an unstable node if the eigenvalues are real and by an unstable focus if the eigenvalues are complex and
conjugate. Under the assumption that TSO
and Z are both negative, the opJ
timal solution is a stable saddle point, only if the constraints on the elements
of the Jacobian matrix JSO respect the condition on negativity of the trace.
Under the assumption that TSO
J and Z have opposite sign, the optimal solution
may be either stable or unstable. In the …rst case, dynamics displays damping
oscillation to the steady state. In the second, locally explosive dynamics occurs
when the constraints on the elements of the matrix JSO do not respect the
condition on negativity of the trace.
The stability of the stationary point depends on the assumption that habits
are transmitted both across and within generations, which a¤ects the sign of the
trace TSO
and of element Z. Apart from the benchmark model (Diamond [2]),
J
monotonic convergence to the steady state is ensured only under the assumption that the stock of habits that individuals form during their adulthood does
not persist into their old age, i.e. only if = 0. Contrary to the competitive
equilibrium, the optimal solution is characterised only by a positive intergenerational spillover: the savings by old people that directly …nance the capital stock
required for production in the next period, and that indirectly sustain wages
of adults. The intergenerational and the intragenerational spillovers associated to consumption decisions are a priori internalised by the social planner in
the maximisation problem (19). As in the competitive equilibrium, the process
transforming the savings by the old into income for the adult displays decreasing
returns to scale, due to the characteristics of the production function. However,
the intergenerational bequest in terms of higher wages does not interact with
any other spillover. The intergenerational bequest in terms of higher wages will
lead to a constant increase in saving that induces a permanent expansion. The
model is thus characterised by a diverging explosive dynamics.
5
Numerical example
Assume that the utility function is logarithmic, ln (cat
et )+ ln cot+1
ht+1 ,
and that the production function is Cobb-Douglas, yt = kt . The steady-state
10
= 0:25; = 0:43
= =0
(Diamond’s [2] model)
0:609
0:457
1:109
0:137
y
w
R
k
e
ca
co
= 0:25; = 0:43
= 0:65; = 0:3
0:628
0:471
1:011
0:155
0:316
0:316
0:157
0:319
0:152
Table 1: Steady-state values in the competitive economy
equilibrium (17)-(18) becomes (see Appendix E.1)
k
1
=
( + ) 1
"
#
1 2
( + )1
( + )+
e =
Once steady-state values of capital and stock of externality are found, use market clearing conditions (11) and (12), resource constraint (13) and individual
constraints (3) and (4) to …nd steady-state values of output, real wages, interest
factor, consumption of the adult and old
y
=
( + )
w
R
= (1
)( + )
=
( + )
1
ca
=
1
co
=
(1
1
)( + )
( + )
( + )
1
1
1
Assigning values to the relevant parameters, it is possible to make a comparison with Diamond’s [2] model (see Table 1). Note that consumption of the
adult is much higher than consumption of old people because individuals do not
internalise the intergenerational externality e a¤ecting their own consumption
(ca
e) and therefore they tend to consume more than what is optimal.
11
Also note that the choice of the above parameters veri…es Assumptions 1-7,
Lemma 1, conditions (7)-(9), and Propositions 1 and 2 as
JCE
det I
1
=
0:449 6= 0
sr
=
0:057 >
jse j
sr fkk
=
0:270 < 1:517 = 1 + 1
TCE
J
=
0:517 < 2
=
0:5172
0:205 =
4=
1
fkk
sw k jfkk j
1 sr fkk
3:733 < 0 if se =
1
kfkk
sr fkk
As
< 0, the eigenvalues of JCE are complex and conjugate (Proposition 3),
0:379 0:250i, and the bifurcation is identi…ed at ^ = 0:510; ^ = 0:001 which
are the roots of
)
(
(1
)
(1
)
(
+
)
=1
det JCE =
( + )
2 +
1+
[ ( + )+ ]
under = 0:25 and = 0:43. Therefore the system is characterised by a spiral
convergence to the steady-state equilibrium if mod (0:510; 0:001) < 1; the system exhibits a period orbit if mod (0:510; 0:001) = 1; and the system exhibits
a spiral divergence from the steady-state equilibrium if mod (0:510; 0:001) > 1.
Analogously, when the utility function is logarithmic and that the production
function is Cobb-Douglas, the steady-state equilibrium (26)-(29) becomes (see
Appendix E.2)
co
o
c
=
=
a
c(
1
1
(1
)
(1
1
)
(1
)
)
1
1
a
e = h = c = (1
)
(1
)
1
(1
)
1
1
k
=
(1
)
Assigning values to the relevant parameters, it is possible to make comparisons
between Golden Rule ( = 1) and Modi…ed Golden Rule ( = 0:5; 0:99) (see
Table 2). Note that consumption of the adult is still higher than consumption
of old people but intertemporal consumption plan is now smoother as the social
planner internalises the intergenerational externality e.
12
y
k
e = h = ca
co
= 0:25; = 0:43
= 0:65; = 0:3
= 0:5
0:762
0:337
0:234
0:190
= 0:25; = 0:43
= 0:65; = 0:3
= 0:99
0:770
0:352
0:226
0:193
= 0:25; = 0:43
= 0:65; = 0:3
=1
0:771
0:354
0:224
0:193
Table 2: Steady-state values in the optimal solution
Modi…ed Golden Rule ( = 0:5)
Modi…ed Golden Rule ( = 0:99)
Golden Rule ( = 1)
0:685
0:536
0:535
0:107i
0:179i
0:178i
2:851
1:695
1:682
0:446i
0:568i
0:561i
Table 3: Eigenvalues in the optimal solution
Also note that the choice of the above parameters veri…es Propositions 4 and
5 as
det I
TSO
J
Z
JSO = 0:200 6= 0
= 0:5
4:483
7:228
= 0:99
4:462
7:148
=1
4:434
7:061
As the eigenvalues of JSO are complex and conjugate (see Table 3) and as TSO
J
and Z are both positive, the dynamics of the social optimal is identi…ed by a
locally unstable focus. This result holds even when the social planner does not
discount the utility of future generations, i.e. when = 1 (the Golden Rule).
6
Concluding remarks
This paper derives su¢ cient conditions for existence and uniqueness of a steadystate equilibria in an overlapping generations model with persistent inherited
habits and analyses the implications of persistence of inherited habits for the
local stability of the steady-state equilibria.
It derives the conditions for the steady-state stability of competitive equilibrium and shows that the competitive equilibrium may display either ‡uctuations
or explosive behaviour. Then it studies the conditions for the steady-state stability of optimal solution. It is proved that the optimal solution may display
damped oscillations or locally explosive dynamics. This result depends on the
assumption that habits are transmitted from one generation to the next one
(inheritance) and individuals carry these habits into old age (persistence).
This paper shows that the assumption of persistence of inherited habits
is not innocuous: when we introduce persistence in the contest of an OLG
model in which habits are inherited, dynamics of the model and stability of the
13
equilibrium are dramatically a¤ected. For this reason, researchers should be
more cautious in neglecting this assumption.
The results presented in this paper are therefore fundamental to understanding the mechanisms underneath models with habit formation and habit persistence, as habits seem to play a signi…cant role in many aspects of macroeconomic
theory.
14
A
Proof of Proposition 1
First, linearise the non-linear dynamic system (15)-(16) around the steady state
(17)-(18)
dkt+1
det+1
= sw [ kfkk ] dkt + sr fkk dkt+1 + se det
=
kfkk dkt sw [ kfkk ] dkt sr fkk dkt+1
(30)
(31)
se det
Then solve the …rst equation by dkt+1
dkt+1 =
1
[ sw kfkk dkt + se det ]
sr fkk
1
and substitute the solution into the second equation of the system (30)
det+1 =
1
[(sw
sr fkk
1
1 + sr fkk ) kfkk dkt
se det ]
The linearised system (15)-(16) around the steady state (17)-(18) is therefore
dkt+1
det+1
=
1
sr fkk
1
in which
J
CE
=
"
sw kfkk
1 + sr fkk ) kfkk
(sw
(sw
sw kfkk
1 sr fkk
1+sr fkk )kfkk
1 sr fkk
se
1 sr fkk
se
1 sr fkk
se
se
dkt
det
#
(32)
is the Jacobian matrix evaluated at steady state (k; e). It is immediate to show
that the matrix I JCE is equal to
#
"
se
kk
1 + 1swskf
1 sr fkk
CE
r fkk
I J
=
(sw 1+sr fkk )kfkk
1 + 1 ssrefkk
1 sr fkk
and that its determinant is
det I
JCE
=
=
1+
1
sw kfkk
+
1 sr fkk
1
se
sw kfkk
+
sr fkk
(sw
(1
1 + sr fkk ) kfkk
2
sr fkk )
sr fkk + se + (sw + se ) kfkk
6= 0
1 sr fkk
(33)
under Assumptions 1-3 and 7, partial derivatives (7)-(9), equilibrium of production factors (11) and (12) and De…nition 2.
B
Proof of Proposition 2
The determinant of the matrix JCE is equal to
det JCE =
se sw kfkk
(1
se (sw
2
sr fkk )
1 + sr fkk ) kfkk
(1
15
se =
2
sr fkk )
the trace is equal to
TCE
=
J
sw kfkk
1 sr fkk
se
sr fkk
1
and the discriminant is de…ned as
2
= TCE
J
4 det JCE
Therefore, under the following conditions
1
se
=
sr
>
jse j
sr fkk
(1
sr fkk )
kfkk
1
fkk
< 1+ 1
sw k jfkk j
1 sr fkk
it is immediate to prove that the determinant is equal to 1
det JCE =
1
se kfkk
=1
sr fkk
and that the trace becomes in absolute values smaller than 2
TCE
=
J
sw k jfkk j
jse j
+
<2
1 sr fkk
1 sr fkk
Finally, since
TCE
< 2 ) TCE
J
J
2
<4
the discriminant is negative
=
C
jse j
sw k jfkk j
+
1 sr fkk
1 sr fkk
2
4<0
Proof of Proposition 4
First, linearise the non-linear dynamic system (22)-(25) around the steady state
(26)-(29):
uca ca dcat + uca e det + uca e dcat+1 + uee det+1
=
1
vco co dcot +
vco h dcot+1
1
vco co dcot +
1
vco h dht
vco co fk (kt+1 ) dcot+1
det+1
dkt+1
16
1
vco h dht +
vhh dht+1
= vco h fk (kt+1 ) dht+1 +
+vco fkk (kt+1 ) dkt+1
= dht+1 = dcat
= fk (kt ) dkt dcat dcot
Then solve the …rst equation by dcat+1 and the second by dcot+1 , using the third
and the fourth equations, under the assumption that det =dht :
dcat+1
dcot+1
det+1
dkt+1
vco h
1
vco fkk (kt+1 )
vco co
+
dcot +
2u a
uca e
fk (kt+1 ) vco co fk (kt+1 )
c e
1
vco h
vc2o h
+
+ 2
det +
2
uca e
uca e vco co fk (kt+1 )
uca ca
uee
vhh
vco h
vco fkk (kt+1 )
+
+
uca e
uca e
uca e
uca e vco co fk (kt+1 )
vco h vco fkk (kt+1 )
vco h
dcat +
fk (kt ) dkt
vco co
uca e vco co fk (kt+1 )
1
vco h
vco fkk (kt+1 )
+
det +
dcot +
=
vco co fk (kt+1 )
fk (kt+1 )
vco co fk (kt+1 )
vco fkk (kt+1 )
vco h
vco fkk (kt+1 )
+
fk (kt ) dkt
dcat
vco co fk (kt+1 ) vco co
vco co fk (kt+1 )
= dht+1 = dcat
= fk (kt ) dkt dcat dcot
=
The linearised system (22)-(25) around the steady state (26)-(29) is therefore
32
2
3 2
0
1
0
0
det
det+1
[B(1
E) 1]
6 dcat+1 7 6
A B (D E) DB C B (1 + D) 7
dcat
76
6
6
7=6
74
1
4 dkt+1 5 6
0
1
1
5 dkt
4
D
dcot
dcot+1
1+D
E
D E
3
7
7
5
in which
A
B
C
D
E
uca ca + uee + vhh
>0
uca e
vco h
>0
uca e
vco co
<0
2u a
c e
vco fkk (kt+1 )
>0
vco co
vco h
<0
vco co
under the assumption that in steady state fk (kt+1 ) = fk (kt ) = 1 . As
2
3
0
1
0
0
6 [B(1 E) 1] A B (D E) DB C B (1 + D) 7
6
7
JSO = 6
7
1
0
1
1
4
5
D
E
D E
1+D
17
(34)
is the Jacobian matrix evaluated at steady state (ca ; co ; k; e), it is immediate to
show that the matrix I JSO is equal to
2
3
1
1
0
0
6 [B(1 E) 1] 1 A + B (D E)
DB
C + B (1 + D) 7
6
7
I JSO = 6
7
1
0
1
1
4
5
D
E
D+E
1 D
and that its determinant is
JSO
det I
=
1
=
=
A
B (D
A + B (D
1
E) +
1
E)
+ (1 + D) +
D+
1
1
B + CE
2
B + CE
2
uca ca + uee + vhh
1
1
1
uca e
vco fkk (kt+1 ) vco h
vco h
+
uca e
vco co
=
=
+
vco fkk (kt+1 )
6= 0
vco co
under Assumptions 1-3 and 7, conditions (28) and (29) and De…nition 4.
D
Proof of Proposition 5
Study the characteristic polynomial P in the eigenvalues
Jacobian matrix (34)
4
P( )=
TSO
J
3
+Z
2
1
associated to the
TSO
+ det JSO = 0
J
in which
det JSO
1
=
TSO
J
=
Z
=
B + CE
2
2
=
>0
1 + 1 + A B (D E) + D ? 0
if 1 + 1 + A + D ? B (D E)
2 1 + (A + BE) 1 + 1 + (A C) D ? 0
1
if 2
C) D ?
+ (A
(A + BE) 1 +
Then, factorise the polynomial P into
P( )=(
1
1)
(
1
2)
1
which is equivalent to
2
1
+
1
2
2
18
=0
2
+
1
=0
1
Now analyse all possible scenarios, due to the sign’s ambiguity of the trace TSO
J
and of the element Z.
Proof. First, assume that TSO
and Z are both positive and analyse the two
J
possible cases:
2
i
1.
1
4
> 0; i = 1; 2. The four eigenvalues are real and they can
be:
(a) four negative roots. This case implies that 1 + 2 < 0 and 1 2 > 0.
This case is excluded as it violates TSO
J > 0.
(b) two negative and two positive roots. This case implies that 1 + 2 ?
0 and 1 2 < 0. This case is excluded as it violates Z > 0.
(c) four positive roots. This case implies that 1 + 2 > 0 and 1 2 > 0.
This case is accepted as it respects both conditions on TSO
and Z.
J
2.
2
i
1
4
< 0; i = 1; 2. Look at the real parts only. Since the real
part a =
i 6= 0; i = 1; 2, the eigenvalues are complex and conjugate
and they can be:
1
2
(a) four negative roots. This case implies that 1 + 2 < 0 and 1 2 > 0.
This case is excluded as it violates TSO
J > 0.
(b) two negative and two positive roots. This case implies that 1 + 2 ?
0 and 1 2 < 0. This case is excluded as it violates Z > 0.
(c) four positive roots. This case implies that 1 + 2 > 0 and 1 2 > 0.
This case is accepted as it respects both conditions on TSO
and Z.
J
Under the assumption that TSO
and Z are both positive, the only admissible
J
case is c). It identi…es an unstable node if the eigenvalues are real and an
unstable focus if the eigenvalues are complex and conjugate. Locally explosive
dynamics is highly likely.
Then, assume that TSO
and Z are both negative and analyse the two possible
J
cases:
2
i
1.
1
4
> 0; i = 1; 2. The four eigenvalues are real and they can
be:
(a) four positive roots. This case implies that 1 + 2 > 0 and 1 2 > 0.
This case is excluded as it violates both conditions on TSO
and Z.
J
(b) two negative and two positive roots. This case implies that 1 + 2 ?
0 and 1 2 < 0. This case is admissible only if 1 + 2 < 0 as
TSO
J < 0.
(c) four negative roots. This case implies that 1 + 2 < 0 and 1 2 > 0.
This case is excluded as it violates condition on Z > 0.
2.
2
i
1
4
< 0; i = 1; 2. Look at the real parts only. Since the real
part a =
i 6= 0; i = 1; 2, the eigenvalues are complex and conjugate
and they can be:
1
2
19
(a) four positive roots. This case implies that 1 + 2 > 0 and 1 2 > 0.
This case is excluded as it violates both conditions on TSO
and Z.
J
(b) two negative and two positive roots. This case implies that
0 and 1 2 < 0. This case is admissible only if 1 +
TSO
J < 0.
2
(c) four negative roots. This case implies that 1 + 2 < 0 and
This case is excluded as it violates condition on Z > 0.
1
?
< 0 as
1+ 2
2
> 0.
Under the assumption that TSO
and Z are both negative, the only admissible
J
case is b), but only if 1 + 2 < 0. It identi…es a stable saddle point that ensures
monotonic local convergence.
Finally, assume that TSO
J and Z have opposite sign and distinguish two possible
cases:
2
i
1.
1
4
> 0; i = 1; 2. The four eigenvalues are real and they can
be:
(a) four positive roots. This case implies that 1 + 2 > 0 and 1 2 > 0.
This case is excluded as it violates both conditions on TSO
and Z.
J
(b) two negative and two positive roots. This case implies that 1 + 2 ?
0 and 1 2 < 0. This case is admissible only if 1 + 2 > 0 as is
ensures TSO
J > 0.
(c) four negative roots. This case implies that 1 + 2 < 0 and
This case is admissible only if TSO
J < 0 and Z > 0.
2.
2
i
1
2
> 0.
1
4
< 0; i = 1; 2. Look at the real parts only. Since the real
=
6
0; i = 1; 2, the eigenvalues are complex and conjugate
part a =
i
and they can be:
1
2
(a) four positive roots. This case implies that 1 + 2 > 0 and 1 2 > 0.
This case is excluded as it violates both conditions on TSO
and Z.
J
(b) two negative and two positive roots. This case implies that 1 + 2 ?
0 and 1 2 < 0. This case is admissible only if 1 + 2 > 0 as it
ensures TSO
J > 0.
(c) four negative roots. This case implies that 1 + 2 < 0 and
This case is admissible only if TSO
J < 0 and Z > 0.
1
2
> 0.
Under the assumption that the trace and the element Z have opposite sign, case
b) identi…es an unstable solution as TSO
> 0. Locally-explosive dynamics is
J
highly likely. Case c) identi…es a stable node for real eigenvalues and a stable
focus for complex and conjugate eigenvalues, and therefore it ensures damped
convergence to the steady state.
20
E
Closed-form solutions
E.1
Intertemporal equilibrium (15)-(16)
Assume that the utility function is logarithmic, ln (cat
et )+ ln cot+1
ht+1 ,
and that the production function is Cobb-Douglas, yt = kt , and …nd the steadystate values of k and e. The intertemporal equilibrium (15)-(16) becomes
kt+1
=
et+1
=
[(1
1+
(1
kt
) kt
(1
et ] +
1
kt
) kt2 1
+
1
+ (1 + ) 1 +
) kt
(1 + )
+
(35)
et
(36)
and the associated system (17)-(18) becomes
k
=
e =
[(1
1+
(1
( k
)k
e] +
(1
1
( k
) k2 1
+
1 + ) (1 + )
1+
e
k2
k
0
)k
+ ) (1 + )
(37)
(38)
Find e from equation (38)
e=
(1
[1 +
)
(1
)]
1
1
=
+
k2
k
1
1
(39)
+
in which 0
(1
) = [1 + (1
)], and substitute e into the equation (37)
and get the following second order equation in k 1
[
Since
0
(1
0
[
(1
0
)] k 2(
1)
) = [1 +
(1
)]
+ (1 + ) [
(1
=
=
)
1
+ (1 + ) = 0
)],
and the second order equation in k
) (1 + ) (1
1 + (1
)
) ]k
(1
)
(1
) =
1 + (1
)
(1
)
(1
) (1 +
1 + (1
)
(1
)
[
1
+ ]=
1 + (1
)
(1
) (1 + ) (1
)
1 + (1
)
=
=
(1
(1
k 2(
(1
[1 +
1)
1
)
=
becomes
+ (1 + ) [
) (1
) 2(
k
(1
)]
21
(1
1)
+
) ]k
[
(1
1
+ (1 + )
) ]
k
1
+1
=
0
=
0
[
(1
) ]
(1
) (1 )
, c 1 and k 1
x, it is possible
Setting a
[1+ (1 )] , b
1
to rewrite the above second order equation in k
as a quadratic equation
ax2 + bx + c = 0
(40)
Equation (40) has the following set of possible solutions
rn
o2
[
(1
) ]
[
(1
) ]
p
+4
2
b
b
4ac
x1;2 =
=
2a
2 (1 ) (1 )
[1+ (1
(1
) (1 )
[1+ (1 )]
)]
Since capital cannot be an imaginary number, the discriminant
[
(1
) ]
2
(1
[1 +
+4
) (1
)
(1
)]
must be non-negative. Since , , , (1
), (1
) and [1 + (1
)] are all
positive, the discriminant is strictly positive. Now study the sign of the sequence
fa; b; cg. By Descartes’ theorem, equation (40) has a positive and a negative
solution, whatever the sign of the coe¢ cient b. Since physical capital cannot
be negative, the negative solution is excluded a priori. Therefore, the unique
solution for equation (40) is
rn
o
[
(1
) ]
[
+
x =
=
2
1
[
2
+
Using k
k
=
1
1
[
2
1
+
2
1
2
(1
) ]
2
+4
(1
) (1 )
[1+ (1 )]
(1
) (1 )
[1+ (1 )]
=
1 + (1
)
+
(1
) (1
)
s
1 + (1
)
(1
) (1
)
2
[
(1
) ] +4
(1
) (1
)
[1 + (1
)]
(1
) ]
x, the steady-state value of k is
(1
1 + (1
)
+
(1
) (1
)
s
) ]
1 + (1
)
(1
) (1
)
[
(41)
(1
) (1
)
) ] +4
[1 + (1
)]
(1
2
)
1
1
Therefore, the solution of the steady-state system (17)-(18) is given by equations
(41) and (39)
k
=
e =
1
( + ) 1
"
#
1 2
( + )1
( + )+
22
(42)
(43)
in which
1
[
2
(1
1
2
1 + (1
)
(1
) (1
)
(1
1+
E.2
1 + (1
)
(1
) (1
)
s
) ]
[
(1
2
) ] +4
(1
) (1
)
[1 + (1
)]
)
(1
)
Intertemporal equilibrium (22)-(25)
Assume that the utility function is logarithmic and that the production function
is Cobb-Douglas, the intertemporal equilibrium (22)-(25) becomes
1
cat
1
cat+1
1
et
cot
1
1
=
et+1
1
ht
et
kt+1
cot
ht
+
1
cot+1
ht+1
1
k 1
cot+1
ht+1 t+1
= ht = cat 1
= kt
cat cot
=
and the steady-state equilibrium (26)-(29) becomes
1
1
ca
ca
e
e
1
1
=
(co
=
h)
+
1
k
1
co
h
(44)
(45)
e = h = ca
k
= ca + co + k
(46)
(47)
Then solve equation (44) by co and get
co =
in which
(1
)k
( 1 + )(1
ca
(48)
)
+ . Using the fact that in equilibrium ca =
k, substitute (48) into equation (47) and get
1
k = (1 + ) [(1
)k
k] + k
which can be solved by k
1
1
k=
(1
)
23
(49)
Now use (49) into (46) and (48) and …nd the steady-state values of e and co :
1
1
a
e = h = c = (1
(
o
c
=
(1
)
)
(1
1
)
(1
)
1
1
(1
)
1
(1
24
)
)
References
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Journal of Economic Dynamics and Control 23, 519-537.
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[3] Dietz, W.H., 1983. Childhood obesity: Susceptibility, cause, and management. Journal of Paediatrics, 103, 676-686.
[4] Galor, O., 2003. Introduction to stability analysis of discrete dynamical systems, mimeo, Brown University
[5] Galor, O., Ryder, H., 1989. Existence, uniqueness and stability of equilibrium in an overlapping-generations model with productive capital. Journal
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25