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Discrete Dynamics in Nature and SocietyVolume ), Article ID
Research Article
Endogenous Instability in Credit-Constrained Emerging Economies with
Leontief Technology
Department of Economic and Financial Institutions, University of Macerata, 62100 Macerata, Italy
Received 20 March 2008; Accepted 16 June 2008
Academic Editor: Masahiro Yabuta
Copyright © 2008 Cristiana Mammana and Elisabetta Michetti. This is an open access article distributed under the , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
AbstractThis work provides a framework to analyze the role of financial development
as a source of endogenous instability in emerging economies
subject to moral hazard problems. We propose and study a dynamic
model describing a small open economy with a tradeable good produced
by internationally mobile capital and a country specific input, using Leontief
technology. We demonstrate that emerging markets could be endogenously
unstable since large capital inflows increase risk and exacerbate
asymmetric information problems, according to empirical evidences. Using
bifurcation and stability analysis, we describe the properties of the
system attractors, we assess the plausibility for complex dynamics and, we
find out that border collision bifurcations can emerge due to the fact that
the state space is piecewise smooth. As a consequence, when a fixed or
periodic point loses its stability, the final dynamics may become suddenly
chaotic. This fact may explain how financial crises occurred in emerging
economies.
Discrete Dynamics in Nature and SocietyVolume ), Article ID
pagesdoi:10.6494Research ArticleEndogenous Instability in Credit-Constrained Emerging Economies with Leontief Technology and Department of Economic and Financial Institutions, University of Macerata, 62100 Macerata,
ItalyReceived 20 March 2008; Accepted 16 June 2008Academic Editor: Masahiro Yabuta Copyright (C) 2008 Cristiana Mammana and Elisabetta Michetti. This is an open access article distributed under the , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This work provides a framework to analyze the role of financial development
as a source of endogenous instability in emerging economies
subject to moral hazard problems. We propose and study a dynamic
model describing a small open economy with a tradeable good produced
by internationally mobile capital and a country specific input, using Leontief
technology. We demonstrate that emerging markets could be endogenously
unstable since large capital inflows increase risk and exacerbate
asymmetric information problems, according to empirical evidences. Using
bifurcation and stability analysis, we describe the properties of the
system attractors, we assess the plausibility for complex dynamics and, we
find out that border collision bifurcations can emerge due to the fact that
the state space is piecewise smooth. As a consequence, when a fixed or
periodic point loses its stability, the final dynamics may become suddenly
chaotic. This fact may explain how financial crises occurred in emerging
economies.1. IntroductionThe facts leading to the financial crisis in the
emerging markets of South-East Asia in summer 1997 have shown how a crisis can
emerge after a boom in the fundamentals, therefore they open new theoretical
approaches to financial crises and a need for new explanations. In the case of
emerging markets, we witness a new phenomenon because, differently from past
crises (like Mexico 1994 or European Monetary System 1992), such crisis was
characterized by a large capital inflows with borrowing excess in a financial
liberalizationcontext, a fast economic growth driven by fundamentals with
poverty reduction (Asian miracle) and an increase in the financial risk assumed
without a prudential regulation and a financial supervision
system. (The fact that macroeconomic factors, especially
a boom in lending, played a key role in the vulnerability of emerging markets
to financial crises, has been discussed in World Economic Outlook [].
Furthermore Corsetti et al. [], state that in Thailand the boom in lending
caused problems in its financial sector. In Hong Kong and Singapore, the
development has been accompanied by strong risk supervision and control, so the
financial crisis was prevented.)According to such considerations, a model that can
explain such financial crisis must prove that an inversion in the real
aggregates with a fall in investment and save, is not only possible but can
also appear in an unpredictable and sudden way when the economy goes through
financial development.In this work, we present a framework that provides an
explanation to these peculiar events according to the balance-sheet view to
crises. (Contributions to this line of research are
in Aghion et al. [–], Caballero and
Krishnamurthy [], Cespedes et al. [] and Krugman
[].) Using the discrete dynamic system theory,
we prove that economies are endogenously unstable when going through a phase of
financial development if the entrepreneurs face credit constraint due to moral
hazard problems, as proved by Bernanke and Gertler [].
(The relevance of asymmetric information problems
in such crises has been largely recognized by Furman and Stiglitz [] and
Mishkin []. The fact that the level cash flow of the firm plays an
important role in the investment is widely recognized in Hubbard [] and in
Bernanke et al. [].)Many authors considered that financial constraints on
firms due to asymmetric information considerations can play a role in the
propagation of the business cycle. For instance, in Azariadis and Smith []
and Kiyotaki and Moore [], the authors studied a closed economy and they
showed that credit constraints can lead to oscillations. Differently,
considering open economies, in Aghion et al. [], the authors studied a
credit-constrained model where firms have debt both
in domestic and foreign currency, and they prove that the economy can easily
suffer a financial crisis. A revised version of this monetary model is offered
in a later work of the same authors, Aghion et al. [], where they proved
that the existence of nominal price rigidities can lead to multiple equilibria.While the models in Aghion et al. [, ]
focused on the monetary sector, we study a real economy model of the kind
considered in Aghion et al. [], in Aghion et al. [], and in Caballé et al. [].In Aghion et al. [], the authors
developed a simple macroeconomic model where the combination between moral
hazard problems in capital markets and unequal access to investment
opportunities across individuals generates endogenous and permanent
fluctuations in aggregate GDP, investment, and interest rates. In their work,
the endogenous cycles are the product of two separate forces: high investment
begets high profits and high investment but, at the same time, high investment
pushes up interest rates and reduces future profits and investment. We will consider
a similar mechanism in which high investment pushes up the price of the
constant country-specific input and reduces future profits and investment.Similarly, in Aghion et al. [],
the authors developed a dynamic, open, credit-constrained, real economic model
to conclude that financial underdeveloped or very developed economies are
stable while at an intermediate level of financial development, economies may
be unstable. Anyway in that work the authors only provide numerical simulations
showing the existence of a stable 2-period cycle. However, the existence of either cycles of
higher period or chaotic dynamics
is not proved from their
work.In this paper, we prove instead the existence of a
noncanonical route to chaos due to border collision bifurcations. Border
collision bifurcations occur in piecewise smooth maps when a fixed point
collides with a borderline separating two smooth regions. The discontinuous
change in the Jacobian elements results in many atypical bifurcation phenomena,
like a periodic orbit turning directly into a chaotic orbit, or multiple
attractors coming into existence or going out of existence as the parameter is
varied across some critical value, and so forth. (About such
kinds of noncanonical route to chaos, see Nusse and Yorke [].)Finally, in Caballé et al. [], the
authors studied the dynamics exhibited by an economic model based on the real
side describing a small open economy subject to credit constraint due to moral
hazard problems using a Cobb-Douglas production function. They proved that
complex dynamics are exhibited at intermediate level of financial development
to conclude that economies experiencing a process of financial development are
more unstable than both very underdeveloped and very developed economies. We
present a similar model to that proposed in Caballé et al. [] while
assuming Leontief technology, and consequently the macroeconomic model here
studied is a piecewise-linear dynamic system.The main results of the study herewith conducted are
that economies with very developed or very undeveloped financial markets have a
unique globally structurally stable, while emerging markets could
be endogenously unstable. In fact, we prove that an intermediate level of
financial development does exist such that the system exhibits a border
collision bifurcation that opens a two-piece chaotic region. When entering in
the aperiodic region, the chaotic properties of the attractor make the
evolution of the system sensitively depending on the initial condition, the
dynamics are unpredictable and structurally unstable, so perturbations on the
parameters (exogenous shocks) produce large and persistent effects. In the
chaotic region, we observe also periodic windows so that the dynamics are
predictable even though the period of the periodic orbit could be so high as to
make impossible the distinction between such a cycle and a proper aperiodic
orbit.The properties we demonstrate allow us to argue that
when going through a phase of financial development, the dynamics shown by the
system could drastically change and pass from a
stable fixed point to chaotic, aperiodic,
unpredictable behavior. A similar result has been reached by Caballé et al. [] only for economies characterized by a Cobb-Douglas
technology, while we show that their results still hold also in the case of
Leontief technology, that is, if there is no substitutability between
production factors.The basic mechanism we describe is a combination of
two opposite forces deriving from an increase in the investment level. Firstly,
a greater investment leads to greater output and profits. Higher profits
improve credit worthiness and fuel borrowing thus leading to greater
investment. Simultaneously, this boom increases the demand for
country-specific input and rises its relative price. This rise in input prices
leads to lower profits and reduces credit worthiness, borrowing and investment
with a subsequent fall in aggregate output. So we will be able to conclude that
financial development may destabilize economies that start from an intermediate
level of financial development according to the experience documented in a
number of countries.
(E.g., in the years
leading up to the crisis of the early 1980's in Southern Cone countries, there
is evidence that profits in the tradeable sector sharply deteriorated due to a
rise in domestic input prices. See Galvez and Tybout [], Petrei and Tybout
[], and De Melo et al. [].)In fact, the endogenous explanation we pursue in this
work is consistent with the experience of several emerging markets where the
liberalization process has taken place (like South-East Asia) where, as a
result of a rapid financial liberalization process, capital
inflowed in large quantities allowing rapid growth
in lending and a boom in investment. When large capital inflows are associated
with growing imbalances, the crisis came, and most of these forces got
reversed: capital flowed out, currency collapsed, real-estate prices dropped,
lending stopped, and investment collapsed.
World Bank [] for a description of the link between capital-flow reversal
and currency crises.) It is however important to
emphasize that the aim of this paper is not to explain exactly what happened in
some specific country but rather to propose and study a unified, dynamic,
macroeconomic model that awards a central role to financial constraints and
financial development.The paper is organized as follows. In Section
present the model. In Section
we study the qualitative dynamics of the model:
we prove the global stability of economies with low- or
high-financial development and we assess the plausibility
for instability of economies at an intermediate level of financial development.
In Section
we present numerical simulations that enforce the results we
proved in Section . We give our conclusions in Section .2. The modelWe consider a small open economy that produces a
single tradeable good using capital
  and a country-specific input
  (like land or real estate) whose price
  is expressed in terms of produced goods. We
assume that the supply of
  is constant.In such an economy, there are two categories of
individuals: first the lenders who can lend their wealth to the entrepreneurs
or invest in the international capital market given the international
equilibrium interest rate
  but they cannot invest directly in the
production, second the borrowers that are the entrepreneurs investing in the
production or in the international capital market. The total output, in time
produced in the economy is given by using the following Leontief technology
which prevents any kind of substitution among the different
  is the capital
productivity. (This hypothesis is necessary because
otherwise the entrepreneurs have no incentive to
invest in the production.) The
tradeable good can be consumed or accumulated as productive capital for the
production in the next period. We assume that capital fully depreciates after
one period.Asymmetric information considerations generate moral
hazard so, according to the results reached by Bernanke and Gertler [],
entrepreneurs can borrow at most a proportional amount
  of their own wealth at time
  that is
  be the amount that entrepreneurs can borrow
then, in time
The parameter
  represents the level of financial development
reached by the economy. As a limit case, when
entrepreneurs can invest only their own wealth, while the bigger
  is,
possibility they have to borrow from the capital market, and so the financial
system is well developed. (Holmstrom and Tirole
[] also consider the direct relationship between capital market rules and
the level of the financial development due to moral hazard considerations.)
Since the maximum amount entrepreneurs can borrow from capital markets is
fixed, the total investment
  in each time is upper bounded
At each period, entrepreneurs maximize their profits
and this program determines their optimal demand
  of the input
Given the production function (), the profit maximization implies that the
entrepreneurs' optimal demand of the country-specific input is
that is, the difference between
the total amount invested and the cost of the production factor demanded.In an equilibrium situation, it must be
  so we reach the following
  is constant, each of the following cases can
be verified.
  .  
  (),
   ().  
 .  
   
 ,    
  
  
   ()  
Relation ()
states a positive relationship between
  and
It depends on the existence of the credit constraint () in the sense that
greater wealth implies greater investment via credit constraint and so greater
  and, consequently, its price increases.Now we can derive the dynamic model describing the
economy. Considering that the price of the country-specific production factor
the investment,
and the production,
are all expressed in terms of entrepreneurs wealth
the dynamic system is the net wealth produced by entrepreneurs and saved by
consumers, available for the next period, that is given by
  is the consumption rate, so
  is the constant fraction the consumers save of
their own wealth, and
  is an exogenous income. This relation can be
understood considering that at time
  entrepreneurs borrow, invest, produce, and pay
their debt
  while consumers save.Now we have to consider the role played by the credit
constraint. To do this, we need to study three different cases.(1)If the financial system is well developed,
entrepreneurs invest in the production only up to the point in which the
productive investment return is equal to the capital market return so
In this case, there is no pure
profit, and substituting () in the dynamic equation (), we obtain the
following increasing function of the entrepreneurs' wealth:
that holds for well-developed
economies.(2)If the financial system is underdeveloped, the
investment—which is constrained—does not absorb the total supply of the
country specific factor
  so its relative price is zero. Greater current
wealth implies greater investment, and therefore greater production and because
greater profits and future wealth. In this case,
the dynamic system is given by substituting () in the wealth dynamic equation
() so we obtain the following increasing function:
that holds for less-developed
economies.(3)Finally, if the financial system is at an
intermediate level of development, the investment absorbs the supply of the
country-specific production factor
  and, according to the Leontief production
function, the production
Substituting such equation in the dynamic relation (), we have the following
decreasing function:
that holds for intermediate
financial developed economies. Equations (),
(), and () describe the dynamic system for low, intermediate, and high level
of financial development of the economy respectively and, similarly, low, intermediate,
and high level of entrepreneurs' wealth.From the previous considerations, we derive the map
  that is given by () for
and by () for
where the turning points
  and
  are given by
The dynamic model we want to
study is given by the following continuous first-order piecewise linear map,
whose iterates describe the dynamics of entrepreneur wealth we
investigate:
  are the economically plausible definition
domains of the parameters.3. Qualitative dynamicsIn this section,
we study the qualitative dynamics of the continuous bimodal piecewise linear
map given by () when varying its parameters. In particular we consider the
  and
study of the special case
  needs a partially different approach because
the system would also have a fixed point at the origin even if it is always
unstable. The hypothesis
  is economically plausible considering that
  while
  is sufficiently
low.) It must be remembered that
  is increasing on
  and on
  while it is decreasing on
 . (Note
  is constant on
Furthermore, its constant slopes in each of such pieces are, respectively,
The following proposition gives
sufficient conditions on the parameter
  with respect to the other parameters of the
model such that the fixed point lies on each of the three linear pieces of map
().Proposition 3.1.
be given by ()
  and
Then:(a)for all
  has a unique positive fixed point
 ;(b)for all
  has a unique positive fixed point
 ;(c)for all
  has a unique positive fixed point
 ; where
 .Proof. Let
 .To prove part (a) we consider that
  for all
  has at least one fixed point in
The uniqueness in
  follows because
  is linear in
  with slope other than one because we are
In this case, the fixed point is given by
Furthermore, it is the unique
fixed point of
  because
  is continuous,
  is a decreasing function and
  is an increasing function with slope less than
one for the hypothesis
 .To prove part (b) we first consider that if
  then
  so the existence of the fixed point in
  is proven. Otherwise, if
  then the same arguments we use to prove part (a) show that
  while
  for all
  has at least one fixed point in
The uniqueness in
  follows because
  is decreasing in
In this case, the fixed point is given by
Furthermore, it is the unique
fixed point of
  because
  is continuous,
  is an increasing function with positive
intercept for the hypothesis
  and
  is an increasing function with slope less than
one for the hypothesis
 .To prove part (c) we first consider that if
  so the existence of the fixed point in
  is proven. Otherwise, if
then the same arguments we use to prove part (b) show that
  while it does have
a value of
for instance,
  because
  has at least one fixed point in
The uniqueness in
  follows because
  is linear in
  with slope lesser than one for the hypothesis
In such a case, the fixed point is given by
Furthermore, it is the unique
fixed point of
  in in
  because
  is continuous and piecewise linear and
  is an increasing function with positive
intercept for the hypothesis
 .Cases (a), (b), and (c) are depicted in Figure .Figure 1: Scheme of
  as defined in () in the three cases of
localization of the fixed point: in (a),
and in (c),
 .The following proposition states the global stability
of economies at high- or low-financial development levels.Proposition 3.2.
  be given by () and let
  and
Then(a)for all
  is a globall(b)for all
  is a globally stable fixed point.Proof. To prove statement (a) we first consider that for all
  has a unique fixed point
  such that
as proved in Proposition
Furthermore
  is a linear increasing function in
  and
  (because of
 ) while
so its multiplier
is given by (),
  for all
Then the fixed point
asintotically stable in
Considering that this case set
  is globally attractive and positively
invariant, then we conclude that
  is globally stable.To prove statement (b) we first consider that for all
  has a unique fixed point
  as proved in Proposition , part (c).
Now we have to consider two different cases. First, if
Furthermore, its multiplier is given by (),
  for the hypothesis
so the fixed point
asintotically
stable in the set
The global stability is trivially proved. Secondly, if
  that is a no-differentiable point, so we
cannot calculate its eigenvalue. However,
  is asintotically
stable from the right in the sense that for all
the sequence of the iteratives converges to
Furthermore,
  is positively invariant and globally
attractive. So we conclude that
  is globally stable for all
 .About such cases see, Figures
and .Figure 2: Koenigs Lemerary staircase diagram for two different
initial conditions: in
  while in
In both cases
 .Figure 3: Koenigs Lemerary staircase diagram for two different
initial conditions: in
  while in
In both cases
 .As we proved, the dynamics exhibited by economies at high or
low levels of financial development are tame: the
generic orbit converging to the unique positive fixed point is definitively
monotone. Furthermore, the economy is structurally stable, because of the
hyperbolicity of the fixed point, so its behavior is predictable.Now we have to consider the case of
that is, the case of economies at an intermediate level of financial
development. First, we consider that in such a case the generic orbit that
eventually converges to the fixed point (or to another attractor) is not
monotone. In fact, the only fixed point
  belongs to the decreasing piece of
that is, the set
so every point at the right of
  is mapped to its left and vice versa. So, even
though the fixed point is stable, the dynamics of the trajectory is
definitively oscillating and economic fluctuations are observed.The following proposition proves the stability of
economies at an intermediate level of financial development when
  is small enough (see Figure
panel (a).Figure 4: Repelling period-2 orbits for
we have a first cycle two for the initial condition
  that is different from the one in (c) for
the fixed invariant interval of
  is illustrated.Proposition 3.3.
  be given by () and let
  and
Then the fixed point
  is globally stable for all
 .Proof. As we proved in Proposition ,
then the unique fixed point
  belongs to
  so its multiplier is given by () that
belongs to
It is straightforward to conclude that the fixed point is also globally stable.Now we have to study the case of
A previous consideration is that if
then the unique fixed point
  is not hyperbolic: its multiplier, given by
() is in fact
the map exhibits a bifurcation: its fixed point becomes unstable and we have to
identify the new attractor that is eventually born.Before studying this case, we consider that the map
  is piecewise linear so it is only piecewise
smooth and it can exhibit noncanonical bifurcation phenomena. While in the
well-known period-doubling route to chaos when a fixed point becomes
unstable, we observe a period- in such a case this does not
happen even if we find out a cycle-2 owned by the map as proved in the
following proposition.Proposition 3.4.
  be given by () and let
  and
Then there exists a period-2 orbit, say
that involves the maximum point, given by (), that is,
 .Proof. The proof is straightforward as it
is only based on the computation that
 .Once known that
  has a cycle-2 involving the maximum point for
the bifurcation value
we are interested in knowing if such invariant set is stable. However, since
the map is not differentiable in
we cannot compute its multiplier so we cannot study its stability in the
typical way.The discontinuity in the first derivative of the map
implies that it can jump without crossing the bifurcation value
so an attractor could die without a double-period one being born. Furthermore,
as proved in Nusse and Yorke [], border collision bifurcations are possible
so that the map could pass from a stable fixed point to a variety of attractors
like a period-
  attractor (
 ), or a
 -piece chaotic attractor, or a
 -piece chaotic attractor or finally a
 -piece chaotic attractor.In order to study the stability of the cycle-2,
we found out for
we have to consider that the point
  that belongs to
  has no derivative (it has two one-sided
derivatives). However, the following proposition proves that the bifurcation
parameter value
  opens a chaotic region via a border collision
bifurcation.Proposition 3.5.
  be given by () and let
  and
  is a Misiurewicz point. (A preperiodic point is usually called a
Misiurewicz point.)Proof. For all
we have that
  (it can be verified by simple calculations,
 ) as a consequence of the fact that
So each point in the set
  is a fixed point for the second iterate,
Then each point in
is involved by a cycle-2 and each of such periodic orbits must be unstable, so
  is an unstable period-two orbit. Because
  involves the maximum
as we proved in Proposition , then the critical point is attracted by an
unstable orbit so it is a Misiurewicz point.Since the topological entropy at the Misiurewicz point
is greater than 1, it reveals that we have
entered into a (aperiodic) chaotic region. (At the
preperiodic point, we have no attracting cycles since they cannot capture the
critical point, which is preperiodic.) In
particular, after the bifurcation occurred at
the map is
 -piece chaotic.Here we cannot prove other results with respect to all
the parameters of the system however, since other qualitative dynamics that
could eventually emerge strictly depend on the fixed values of the parameters,
in Section
we use the numerical analysis to support the results we derived in
this section and we present numerical simulations fixing all the parameters but
  at economically plausible values. In such a
way, we pursue numerical results about the dynamics exhibited by the model. The
quantitative analysis allow us to show the dynamic evolution of the system and
to conclude about its properties.4. Quantitative dynamicsIn this
section, we provide some numerical simulations by fixing the values of all the
parameters of the model but
In such a way, we are able to prove quantitatively the qualitative results
reached in Section
and also to pursue other results that cannot be proved
rigorously. Let
(As it can be proved, the chosen value of
  only affects the quantitative dynamics, that
is, the width of the invariant interval where the dynamics are exhibited, but
not the qualitative dynamics, that is, the bifurcation sequence occurs at the
same parameter values of
We choose these parameter values according to what is considered in Aghion et al. [].)In Figure , we present the scheme of
  for different values of
As we proved in Proposition , the fixed point can belong to each of the
  depending on
  while
 . (We
are approximating an error less than
 .)As we proved in Proposition
the economy is
globally stable for
  and
  and the generic orbit definitively converges
monotonically to
as determined in () and (), that are points in which
  is increasing. In fact, the Koenigs Lemerary
staircase diagram in Figures
shows the converging trajectory for an
arbitrary initial condition.As we proved in Proposition , the fixed point
  belongs to the decreasing piece when
  that is the case of economies going through a
phase of financial development. Furthermore, because of the bimodality of
there is a stretching and folding action that could generate complex dynamics
like cycles of every period and chaos. However, as we stated in Proposition
the fixed point is still globally stable even if the generic orbit is
asymptotically oscillating.In case
  the map exhibits a bifurcation and it gives
rise to an infinite number of repelling period-2 cycles. In fact, Propositions
show such evidence. Note that for such value of
the fixed point is not hyperbolic while each point
  is fixed for the second iterate of
as it is clear when looking at Figure .
So all the period-2 orbits are unstable. Furthermore, the set
  is positively invariant, so every initial
condition will converge to one of such repelling periodic orbit.Numerical computations also show that all these
cycles-2 are of the kind  ,
for all  ,
where   is given by (). Two of such orbits are, for
example, the ones in Figure
The bifurcation occurring at   is not canonical: one of the repelling
cycle-2 involves the maximum point   that is a Misiurewicz point. Therefore, such
border-collision bifurcation opens a chaotic region in which the generic orbit
covers two disjoint invariant sets. Figure
shows the trajectory for an
initial condition once the the trajectory is also represented versus time.Figure 5: (a) The generic aperiodic orbit covers two
disjoint invariant sets. (b) The trajectory with respect to time. In both
cases   and  .As we said, after the bifurcation at  ,
the dynamics are suddenly chaotic so the map has the properties of density of
the periodic orbits, topological transitivity, and sensitively dependence on
the initial condition. (We are referring to the
definition of chaotic set given in Devaney [].)Figure
shows the bifurcation diagram of the map for
different values of  .
The black intervals are those in which the dynamics is chaotic or periodic with
very high period. Furthermore, we observe both large intervals of   where the asymptotic behavior is a cycle-2
(if   with   and  ) or a cycle-3 (if   with   and  ).
(In such a
case, the chaotic properties could also be proven by the well-known Li and
Yorke Theorem, see Li and Yorke [].) In this
case, the dynamics are still predictable even in the long run.Figure 6: Bifurcation diagram with respect to  .However, inside the two chaotic regions, that are
visible in the following Figures
and , the dynamics are both chaotic and
periodic with eventually a very high period. So, while in the first case the
asymptotic behavior is not predictable, in the second case it is still
predictable even though the attractor could be nonhyperbolic, and so the system
would be structurally unstable. Some small intervals of   that are periodic windows inside the chaotic
region are visible in such figures.Figure 7: Bifurcation diagram for  .Figure 8: Bifurcation diagram for  .5. ConclusionsIn this work,
we studied a piecewise linear dynamic system describing a small open economy
where the reached level of financial development plays a central role as a
source of endogenous instability.By analyzing the qualitative dynamics, we proved
rigorously the global stability of economies at a low or high level of
financial development. On the contrary, the economies at an intermediate level
of financial development could not converge to the steady state. Consequently,
we assess the existence of chaotic behavior in the patterns. In this case, we
have been able to prove by qualitative and also quantitative study the
following results.(i)Economies at an intermediate level of
financial development eventually converge to the fixed point by oscillations or
they fluctuate indefinitely.(ii)They can be unstable but predictable if the
attractor is a stable periodic orbit, even with high period, that can also
belong to a window in the chaotic region.(iii)They can be unstable and unpredictable if we
are in a proper chaotic region because of the sensitivity to the initial
conditions.(iv)Economies can be structurally unstable when
going trough regions governed by different asymptotic dynamics because of the
lack of hyperbolicity.(v)The bifurcation phenomenon is atypical because
of the presence of no differentiable points.The instability of economies that are financially
developing can be understood according to the hypothesis of the model studied.
In fact, during a boom, the investment expands and so does the demand for the
country-specific factor. It increases its price and pushes down future
profits. Less profits lead to less creditworthiness because of the presence of
the credit constraint and consequently less investments. Finally, the
country-specific factor will not be completely exhausted so its prices will
fall down with high future profits and a new possible economic boom.
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