AN OLG MACROECONOMIC MODEL IN A NONNEUTRALITY MONEY CONTEXT: COMPLEX DYNAMICS

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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

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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.

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• 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.

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OLG-KM MODEL

• In each period there are four classes of agents

YOUNG FARMER (YF) OLD FARMER
(OF) YOUNG GATHERER (YG) OLD GATHERER (OG)
And two types of goods
Output (y) Non-reproducible asset (land, K) whose
total supply is fixed
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• The production functions of the YF and of the YG
  are

A bequest motive is rooted in the
intergenerational altruism. The generical utility
function si
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FARMERS OPTIMIZATION PROBLEM

• The farmer are three constraints
• flow-of-found (FF) constraint when young
• FF constraint when old
• financing constraint.

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• The farmer maximizes own utility function
  subject to three constraints

• 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

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GATHERERS 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.

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• Then, the optimization problem is

From F.O.C. we obtain the asset price equation
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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.
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• In order to describe the way in which money flows
  in the economy, lets 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
• and holds money balances.

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• From these considerations, we obtain an equation
  that represent a sort of quantity theory of money
  in this model.

• The dynamic of the economy is described by
• The law of motion of the farmers 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

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THE MAP T
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(No Transcript)
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FIXED POINTS
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COEXISTENCE
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INVARIANT CLOSED CURVE
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FLIP BIFURCATION SEQUENCE
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STRANGE ATTRACTORS
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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.

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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

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• The set of nondefinition coincides with the locus
  of points in which at least one denominator
  vanishes.
• For the map T, we obtain
• The focal points are defined which are simple
  roots of the algebric system
  
  

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Focal points
q
q
k
k
enlargement
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• 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

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Disappearance of the closed invariant curve
q
q
k
k
Map T5
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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.