Download An OLG-macroeconomic model in a non

AN OLG MACROECONOMIC
MODEL IN A NON-NEUTRALITY
MONEY CONTEXT: COMPLEX
DYNAMICS
FERNANDO BIGNAMI and ANNA AGLIARI
Catholic University of Piacenza
DOMENICO DELLI GATTI and
TIZIANA ASSENZA
Catholic University of Milano
FRAMEWORK OF THE MODEL
• A large empirical literature has shown that the inflation rate
could affect real activity leading to an increase in saving output
or the capital stock (i.e. Loayza et al. 2000, Kahn et al. 2001).
In the last decade the long run real effects of inflation have
been detected even in models with financial market
imperfections (i.e. Body and Smith, 1998, Cordoba and Ripoll,
2004, Ragot 2006.)
• The present paper can be classified in this framework. We
present a new chanel of non-neutrality of monetary policy in the
presence of financial frictions. When agents face a borrowing
constraint a redistribution of real assets can occur due to the
interaction between net worth and inflation. A change in the
growth rate of money supply can affect real output through the
impact of inflation on borrowers’ net worth.
• This model is an overlapping generation
version of a Kyotaki and Moore (1997)
economy, with money and bequest.
• In our model the novel feature is the role of
the money as a store of value and of the
bequest as a source of founds to be
invested in landholding.
• In this setting we explore the properties of
the two-dimensional model that represents the
law of evolution of the economy.
OLG-KM MODEL
In each period there are four classes of agents:
YOUNG FARMER
OLD FARMER
YOUNG GATHERER
OLD GATHERER
(YF)
(OF)
(YG)
(OG)
And two types of goods:
Output (y)
Non-reproducible asset (land, K) whose total supply
is fixed
The production functions of the YF and of the YG
are:
ytF   K tF1 ,   0


ytG  G K tG1 , G . is increasing,
strictly concave and satisfies the Inada conditions.
A bequest motive is rooted in the intergenerational altruism.
The generical utility function si:

U i  U cti,t 1 , ati1 , mti,t

i  F,G
where cti,t 1 is consumption of the agent of type i and generation t in t+1 (the old agent);
ati1 is bequest left by the same agent to his/her child;
m :
i
t ,t
M ti,t
Pt
are real money balances of the agent of type i and generation t in t (the young agent).
FARMER’S OPTIMIZATION
PROBLEM
The farmer are three constraints:
• flow-of-found (FF) constraint when young;
• FF constraint when old;
• financing constraint.

The YF is endowed at birth with bequest atF . He employs the bequest and the credit bt to invest in land qt KtF  KtF1
and hold money balances m :
F
t ,t
M tF,t
Pt

.


The flow-of found (FF ) constraint of the YF (in real terms) is: qt KtF  KtF1  mtF,t  bt  atF
where: qt :
where R :
q
Qt
is the real price of land. The YF is financialially coinstrained: bt  t 1 K tF
R
Pt
P
1  it
is the real gross interest rate and 1   t 1 : t 1 is the gross rate of inflation. R is given and costant.
Pt
1   t 1
The FF constraint of the old farmer (OF) is: ctF,t 1  atF1  Rbt   K tF  mtF,t 1
• The farmer maximizes own utility function subject to
three constraints:
max U
s.t
F

 ln ctF,t 1  1    ln atF1  
F
mtF,t
FF constraint of the YF
FF constraint of the OF
Financing constraint
0< <1,

F
 0
Money has two different and contrasting effects on the net worth:
• Given the bequest, the higher is money of the young, the lower
net worth and landholding.
• The higher is money of the old, the higher resources available to
him and the higher the bequest the hold leaves to the young
GATHERER’S OPTIMIZATION
PROBLEM
• Being unconstrained from the financial
point of view, the gatherer maximizes own
utility function subject to the FF constraint
of the YG and of the OG.
FF constraint of the YG:
mtG,t  bt  qt
K
G
t
 K tG1
a
G
t
FF constraint of the OG:

ctG,t 1  atG1  CMFt G  G K tG
  Rb
t
 mtG,t 1
where CMFt G   mtG,t 1 t 1 , 0< <1,  t 1 :
Pt
1

Pt 1
1   t 1
• Then, the optimization problem is
max U G   ln ctG,t 1  1    ln atG1   G mtG,t
s.t FF constraint of the YG
FF constraint of the OG
From F.O.C. we obtain the asset price
equation
qt 

g K tF
R


with g  G ' K  K tF

RESOURCE CONSTRAINTS AND
MONEY FLOWS
• Since the total amount of land is fixed, an
increase of landholding for the farmer can occur
only if there is a corresponding decrease of
landholding for the gatherer (aggregate resource
constraint).
The sum of aggregate output and real money balances of the old
agents is equal to the sum of aggregate consumption of the old agents
and real money balances of the young agents.
Moreover: the total amount of real money balances of the young
agents is equal to the total amount of real money balances of the old
agents.
• In order to describe the way in which money flows in
the economy, let’s assume that the OF consumes
less than the output he has produced, while the OG
consumes more than the output he has produced.
The OF sells units of output “savings” to the OG in
order to let him consume in excess of his output. The
OG pays this output by means of money. After the
transaction, the OF use this money to remburse
debit to the OG and leave the bequest to the YF. The
YF receives this bequest from OF and credit from
YG and employs these resources to invest

qt 1 K tF1  K tF

and holds money balances.
• From these considerations, we obtain an equation
that represent a sort of quantity theory of money in
this model.
mtF,t 
1


1
ytF  Yt G
1
where  :

mtG,t
mtF,t
(i.e. the ratio of money of the gatherer to money of farmer of the same generation is constant)
The dynamic of the economy is described by:
•The law of motion of the farmer’s land;
•The asset price equation;
•The quantity theory of money.
Since the dimensionality of the system can be reduced,
we obtain the following map T
THE MAP T
1

k

k

 t
4 R 2 qt2



T :

 AB   qt  kt 1  A k  kt 1
q


R
q

t
 t 1
1

k



4 R 2 qt2


where A=A'
and A ' 
1-
 1+ 
B
 1+ 1    
1+ 1      
1+ 1     






      1  
1+ 1    
 
1
 '
x

x


4R2 y 2





 y '  R  y   AB   y  x  A

1

x


2
2

4
R
y



x  x 




where: kt 1  x; k  x '; qt  y; qt 1  y '
FIXED POINTS
If B  0 then there exists an unique fixed point
2 

2
x
1

1

4
B
x

E  x , y   
;
 1  1  4B2 x

4
BRx




If B  0 then no fixed point exists
COEXISTENCE
q
q
C1*
E

k
E
k
INVARIANT CLOSED CURVE
q
q
C1*
C1*

E

E
k
k
FLIP BIFURCATION SEQUENCE
q
q
E
E
k
k
STRANGE ATTRACTORS
q
q
E
E

k
k
MAP WITH DENOMINATOR
• The map of this model is a plane map with
denominator. If the denominator vanish, then
the map is not defined in the whole plane and
some particular behaviors can be related to this
fact. In particular if one of the components of the
map (or of its inverse) assume the form 0/0 in a
point of plane, then some particular dynamic
properties of the map can be evidenced, related
to the presence of this points [see i.e. Bischi,
Gardini and Mira 1999].
DIFFERENT KINDS OF CONTACT
BIFURCATIONS
• For this map in which the denominator vanish (or
one of its inverse) can be occur different kinds of
contact bifurcations. This bifurcations are
explained by contacts between arc of phase
curve and some singularity of the map as:
SET OF NONDEFINITION;
PREFOCAL CURVE (FOCAL POINT).
These bifurcations cause the creation of
particular structures of the basin boundaries,
denoted as lobes and crescents
• The set of nondefinition coincides with the locus of
points in which at least one denominator vanishes.
For the map T, we obtain:


 s   x, y  


2


: y  0 y  

2R x 

1
• The focal points are defined which are simple roots
of the algebric system:
 N  x, y   0

 D  x, y   0
where N and D are numerator and denominator respectively of the map T
Focal points
E
q
q
Q1
E
k
k
enlargement
• The prefocal curve is a set of points for which at
least one inverse exist which focalizes the whole
set into a single point, called focal point [Mira,
1996]. For the map T the prefocal curve is located
on the y axis.
• The bifurcations due to tangential contacts
between arcs of phase curve and a prefocal curve
or a set nondefinition are denoted as
bifurcations of first class
• The bifurcations due to the merging of focal points
or to the merging of focal points and fixed points,
or to contacts between prefocal curves and critical
curves are denoted as bifurcations of second
calss
Disappearance of the closed
invariant curve
q
q
C1*
C1*

E
k
Map T5
k
REFERENCES
• G.I. Bischi, L. Gardini and C. Mira, 1999. Plane Maps with
Denomiantor. I. Some Geometric Properties, International
Journal of Bifurcation and Chaos, 9, 119-153.
• J. Body, B. Smith, 1998. Capital Market Imperfections in a
Monetary Growth Model, Economy Theory, 11, 241-273.
• M. Cordoba, J.C. Ripoll, 2004. Collateral Constraints in a
Monetary Economy, Journal of European Economic
Association, 2, n° 6, December
• N. Kiyotaky, J. Moore, 1997. Credit Cycle, Journal of Political
Economy, 105, 211-248.
• C. Mira, 1996. Some Properties of Two Dimensional Maps
not Defined in the Whole Plane, Proc. ECIT 96 Urbino, in
Grazer Mathematische Berichte.
• X. Ragot, 2006. A Theory of Law Inflation with Credit
Constraints, mimeo.