# Endogenous Instability in Credit-Constrained Emerging Economies with Leontief Technology

Authors: Cristiana Mammana; Elisabetta Michetti
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

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.


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## Body

## 1. Introduction

The 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 [1].
Furthermore Corsetti et al. [2], 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. [3–5], Caballero and
Krishnamurthy [6], Cespedes et al. [7] and Krugman
[8].) 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 [9].
(The relevance of asymmetric information problems
in such crises has been largely recognized by Furman and Stiglitz [10] and
Mishkin [11]. The fact that the level cash flow of the firm plays an
important role in the investment is widely recognized in Hubbard [12] and in
Bernanke et al. [13].)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 [14]
and Kiyotaki and Moore [15], the authors studied a closed economy and they
showed that credit constraints can lead to oscillations. Differently,
considering open economies, in Aghion et al. [4], 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. [16], where they proved
that the existence of nominal price rigidities can lead to multiple equilibria.While the models in Aghion et al. [4, 16]
focused on the monetary sector, we study a real economy model of the kind
considered in Aghion et al. [17], in Aghion et al. [18], and in Caballé et al. [19].In Aghion et al. [17], 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. [18],
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 [20].)Finally, in Caballé et al. [19], 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. [19] 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, fixed point; 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. [19] 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 [21], Petrei and Tybout
[22], and De Melo et al. [23].)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.  (See
World Bank [24] 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 Section2 we
present the model. In Section 3 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 4 we present numerical simulations that enforce the results we
proved in Section 3. We give our conclusions in Section 5.


## 2. The model

We consider a small open economy that produces a
single tradeable good using capitalK and a country-specific input Z (like land or real estate) whose price p is expressed in terms of produced goods. We
assume that the supply of Z 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 rater=1+R>1 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 t,
produced in the economy is given by using the following Leontief technology
which prevents any kind of substitution among the different
inputs: (2.1)yt=min⁡{Kta,Z}, where 1/a>r 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 [9],
entrepreneurs can borrow at most a proportional amount μ≥0 of their own wealth at time t,Wt, that is μWt.
Let L be the amount that entrepreneurs can borrow
then, in time t, Lt≤μWt.
The parameter μ represents the level of financial development
reached by the economy. As a limit case, when μ=0,
entrepreneurs can invest only their own wealth, while the bigger μ is,  as the  more
possibility they have to borrow from the capital market, and so the financial
system is well developed. (Holmstrom and Tirole
[25] 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 It in each time is upper bounded
by (2.2)It=(1+μ)Wt.At each period, entrepreneurs maximize their profits
and this program determines their optimal demandzt of the input Z.
Given the production function (2.1), the profit maximization implies that the
entrepreneurs' optimal demand of the country-specific input is zt=Kt/a,
where (2.3)Kt=It−ptzt, that is, the difference between
the total amount invested and the cost of the production factor demanded.In an equilibrium situation, it must be(It−ptzt)=azt so we reach the following
equation: (2.4)It−ptzt=azt.Since Z is constant, each of the following cases can
be verified.(Z>Kt/a). There  is   an   excess   in   the   supply   ofZ,
.  sub‐stituting (2.3) in (2.1), placingpt=0,
and  considering  that  the  credit  con‐straint  (2.2).  (2.5)yt=1a(1+μ)Wt.(Z≤Kt/a).  Z  yt=Z.
,    pt Z relation (2.4) and  (2.2)  zt=Z.
(2.6)pt=(1+μ)Wt−aZZ. Relation (2.6)
states a positive relationship between pt and Wt.
It depends on the existence of the credit constraint (2.2) in the sense that
greater wealth implies greater investment via credit constraint and so greater
demand of Z and, consequently, its price increases.Now we can derive the dynamic model describing the
economy. Considering that the price of the country-specific production factorpt,
the investment, It,
and the production, yt,
are all expressed in terms of entrepreneurs wealth Wt,
the dynamic system is the net wealth produced by entrepreneurs and saved by
consumers, available for the next period, that is given by (2.7)Wt+1=(1−α)(e+yt−rμWt), where α∈(0,1) is the consumption rate, so (1−α) is the constant fraction the consumers save of
their own wealth, and e≥0 is an exogenous income. This relation can be
understood considering that at time t entrepreneurs borrow, invest, produce, and pay
their debt rμWt 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(2.8)(yt−rμWt)=rWt. In this case, there is no pure
profit, and substituting (2.8) in the dynamic equation (2.7), we obtain the
following increasing function of the entrepreneurs' wealth:(2.9)Wt+1=(1−α)(e+rWt)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 factorZ so its relative price is zero. Greater current
wealth implies greater investment, and therefore greater production and because
of p=0,  greater profits and future wealth. In this case,
the dynamic system is given by substituting (2.5) in the wealth dynamic equation
(2.7) so we obtain the following increasing function: (2.10)Wt+1=(1−α)[e+(1+μa−rμ)Wt]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 factorZ and, according to the Leontief production
function, the production yt=Z.
Substituting such equation in the dynamic relation (2.7), we have the following
decreasing function: (2.11)Wt+1=(1−α)[e+Z−rμWt] that holds for intermediate
financial developed economies. Equations (2.10),
(2.11), and (2.9) 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 mapWt+1=f(Wt) that is given by (2.10) for Wt∈[0,WM),
by (2.11) for Wt∈[WM,Wm),
and by (2.9) for Wt∈[Wm,+∞),
where the turning points WM and Wm are given by (2.12)WM=Za1+μ,(2.13)Wm=Zr(1+μ). 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:(2.14)f(Wt)={f1(Wt)=(1−α)[e+(1+μa−rμ)Wt],0≤Wt<WM;f2(Wt)=(1−α)[e+Z−rμWt],WM≤Wt<Wm;f3(Wt)=(1−α)(e+rWt),Wm≤Wt, where α∈(0,1),μ≥0,1/a>r>1,e≥0,
and Z>0 are the economically plausible definition
domains of the parameters.


## 3. Qualitative dynamics

In this section,
we study the qualitative dynamics of the continuous bimodal piecewise linear
map given by (2.14) when varying its parameters. In particular we consider the
case of e>0 and (1−α)r<1.  (The
study of the special case e=0 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 (1−α)r<1 is economically plausible considering that α≫0 while r=1+ϵ,ϵ is sufficiently
low.) It must be remembered that f is increasing on D1=[0,WM) and on D3=[Wm,+∞) while it is decreasing on D2=[WM,Wm). (Note
that if μ=0,f2 is constant on D2.)
Furthermore, its constant slopes in each of such pieces are, respectively,
given by (3.1)f1′(Wt)=(1−α)(1+μa−rμ),(3.2)f3′(Wt)=(1−α)r,(3.3)f2'(Wt)=−(1−α)rμ. The following proposition gives
sufficient conditions on the parameter μ with respect to the other parameters of the
model such that the fixed point lies on each of the three linear pieces of map
(2.14).Proposition 3.1.
Letf  be given by (2.14)
and let e>0 and (1−α)r<1.
Then:
(a)
for allμ∈[0,μm),f has a unique positive fixed point W1⋆∈D1;(b)
for allμ∈[μm,μM),f has a unique positive fixed point W2⋆∈D2;(c)
for allμ∈[μM,+∞),f has a unique positive fixed point W3⋆∈D3; where μm=(Za−(1−α)(Z+e))/(1−α)(e+Z+Zar),μM=(Z(1−(1−α)r))/(1−α)er−1,
and μM>μm>0.Proof.
Letg(Wt)=f(Wt)−Wt.
To prove part (a) we consider thatg(0)=(1−α)e>0,
while g(WM)=(μ(1−α)(e+Z−rZ)+[(1−α)(e+Z)−Za])/(1+μ),
where g(WM)<0, for all μ<μm.
So f has at least one fixed point in D1.
The uniqueness in D1 follows because f is linear in D1 with slope other than one because we are
assuming (1−α)r<1.
In this case, the fixed point is given by (3.4)W1⋆=(1−α)e1−(1−α)[((1+μ)/α)−rμ]. Furthermore, it is the unique
fixed point of f in ℝ+ because f is continuous, f2 is a decreasing function and f3 is an increasing function with slope less than
one for the hypothesis (1−α)r<1.
To prove part (b) we first consider that ifμ=μm then g(WM)=0 so the existence of the fixed point in D2 is proven. Otherwise, if μ∈(μm,μM), then the same arguments we use to prove part (a) show that g(WM)>0 while g(Wm)=(μ[re(1−α)]+[r(1−α)(e+Z)−Z])/(1+μ),
where g(Wm)<0 for all μ∈(μm,μM).
So f has at least one fixed point in D2.
The uniqueness in D2 follows because f is decreasing in D2.
In this case, the fixed point is given by (3.5)W2⋆=(1−α)(e+Z)1+(1−α)rμ. Furthermore, it is the unique
fixed point of f in ℝ+ because f is continuous, f1 is an increasing function with positive
intercept for the hypothesis e>0, and f3 is an increasing function with slope less than
one for the hypothesis (1−α)r<1.
To prove part (c) we first consider that ifμ=μM,
then g(Wm)=0 so the existence of the fixed point in D3 is proven. Otherwise, if μ∈(μM,+∞),
then the same arguments we use to prove part (b) show that g(Wm)>0 while it does have
a value of Wt,
for instance, W¯=(1−α)e/(1−(1−α)r),
where W¯>Wm because μ>μM,
such that g(W¯)<0.
So f has at least one fixed point in D3.
The uniqueness in D3 follows because f is linear in D3 with slope lesser than one for the hypothesis (1−α)r<1.
In such a case, the fixed point is given by (3.6)W3⋆=(1−α)e1−(1−α)r. Furthermore, it is the unique
fixed point of f in in ℝ+ because f is continuous and piecewise linear and f1 is an increasing function with positive
intercept for the hypothesis e>0.
Cases (a), (b), and (c) are depicted in Figure1.Scheme off as defined in (2.14) in the three cases of
localization of the fixed point: in (a), μ=1.5,
in (b), μ=6,
and in (c), μ=60.
(a)(b)(c)The following proposition states the global stability
of economies at high- or low-financial development levels.Proposition 3.2.
Letf be given by (2.14) and let e>0 and (1−α)r<1.
Then
(a)
for allμ∈[0,μm),W1⋆ is a globally stable fixed point;(b)
for allμ∈[μM,+∞),W3⋆ is a globally stable fixed point.Proof.
To prove statement (a) we first consider that for allμ∈[0,μm),
the map f has a unique fixed point W1⋆ such that W1⋆∈D1,
as proved in Proposition 3.1 part (a).
Furthermore f is a linear increasing function in D1 and f(0)>0 (because of e>0) while f(WM)<WM,
so its multiplier  is given by (3.1), f1′∈(0,1), for all Wt∈D1.
Then the fixed point W1⋆∈D1 is  asintotically stable in D1.
Considering that this case set D1 is globally attractive and positively
invariant, then we conclude that W1⋆ is globally stable.
To prove statement (b) we first consider that for allμ∈[μM,+∞),
the map f has a unique fixed point W3⋆∈D3 as proved in Proposition 3.1, part (c).
Now we have to consider two different cases. First, if μ∈(μM,+∞),
then W3⋆∈D3′,
where D3′=D3∖{Wm}.
Furthermore, its multiplier is given by (3.2), f3′∈(0,1) for the hypothesis (1−α)r<1,
so the fixed point W3⋆ is  asintotically
stable in the set D3′.
The global stability is trivially proved. Secondly, if μ=μM,
then W3⋆=Wm that is a no-differentiable point, so we
cannot calculate its eigenvalue. However, Wm is asintotically
stable from the right in the sense that for all W0∈D3′,
the sequence of the iteratives converges to Wm.
Furthermore, D3′ is positively invariant and globally
attractive. So we conclude that W3⋆∈D3 is globally stable for all μ∈[μM,+∞).
About such cases see, Figures2 and 3.Koenigs Lemerary staircase diagram for two different
initial conditions: in(a),W0=15∈(W1⋆,WM) while in (b),W0=1<W1⋆.
In both cases μ=1.5.
(a)(b)Koenigs Lemerary staircase diagram for two different
initial conditions: in(a),W0=15>W3⋆ while in (b),W0=1<W3⋆.
In both cases μ=60.
(a)(b)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 ofμ∈[μm,μM),
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 W2⋆ belongs to the decreasing piece of f,
that is, the set D2,
so every point at the right of W2⋆ 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 whenμ is small enough (see Figure 4 panel (a).Repelling period-2 orbits forμ≃4.9.
In (a),
we have a first cycle two for the initial condition W0=20 that is different from the one in (c) for W0=5.
In (b),
the fixed invariant interval of f2 is illustrated.
(a)(b)(c)Proposition 3.3.
Letf be given by (2.14) and let e>0 and (1−α)r<1.
Then the fixed point W2⋆ is globally stable for all μ∈[μm,μ⋆),
where μ⋆=1/(1−α)r∈[μm,μM).Proof.
As we proved in Proposition3.1,
part (b),
if μ∈[μm,μM),
then the unique fixed point W2⋆ belongs to D2 so its multiplier is given by (3.3) that
belongs to (−1,0)⇔μ<μ⋆.
It is straightforward to conclude that the fixed point is also globally stable.Now we have to study the case ofμ∈[μ⋆,μM).
A previous consideration is that if μ=μ⋆,
then the unique fixed point W2⋆ is not hyperbolic: its multiplier, given by
(3.3) is in fact f2′(W2⋆)|μ=μ⋆=−1.
So, when μ=μ⋆,
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 mapf 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-two stable orbit; 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.
Letf be given by (2.14) and let e>0 and (1−α)r<1.
Let also μ=μ⋆.
Then there exists a period-2 orbit, say O2,
that involves the maximum point, given by (2.12), that is, O2={WM,f(WM)}.Proof.
The proof is straightforward as it
is only based on the computation thatf2(WM)|μ=μ⋆=WM|μ=μ⋆.Once known thatf has a cycle-2 involving the maximum point for
the bifurcation value μ=μ⋆,
we are interested in knowing if such invariant set is stable. However, since
the map is not differentiable in WM,
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−1,
so an attractor could die without a double-period one being born. Furthermore,
as proved in Nusse and Yorke [20], border collision bifurcations are possible
so that the map could pass from a stable fixed point to a variety of attractors
like a period-m attractor (m≥2), or a 2m-piece chaotic attractor, or a m-piece chaotic attractor or finally a one-piece chaotic attractor.In order to study the stability of the cycle-2,O2,
we found out for μ=μ⋆,
we have to consider that the point WM that belongs to O2 has no derivative (it has two one-sided
derivatives). However, the following proposition proves that the bifurcation
parameter value μ⋆ opens a chaotic region via a border collision
bifurcation.Proposition 3.5.
Letf be given by (2.14) and let e>0 and (1−α)r<1.
Let also μ=μ⋆.
Then WM is a Misiurewicz point. (A preperiodic point is usually called a
Misiurewicz point.)Proof.
For allWt∈[WM,f(WM)],
we have that f2(Wt)=Wt (it can be verified by simple calculations,
that is, f2(Wt)=f2∘f2|μ=μ⋆=Wt) as a consequence of the fact that f′(W2⋆)=−1.
So each point in the set I2=[WM,f(WM)] is a fixed point for the second iterate, f2.
Then each point in I2,
except W2⋆,
is involved by a cycle-2 and each of such periodic orbits must be unstable, so
also O2 is an unstable period-two orbit. Because O2 involves the maximum WM,
as we proved in Proposition 3.4, 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 atμ⋆,
the map is 2-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 Section4 we use the numerical analysis to support the results we derived in
this section and we present numerical simulations fixing all the parameters but μ 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 dynamics

In this
section, we provide some numerical simulations by fixing the values of all the
parameters of the model butμ.
In such a way, we are able to prove quantitatively the qualitative results
reached in Section 3 and also to pursue other results that cannot be proved
rigorously. Let α=0.8,r=1.02,a=0.5 so 1/a=2,e=10,
and Z=100.
(As it can be proved, the chosen value of Z 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 μ.
We choose these parameter values according to what is considered in Aghion et al. [3].)In Figure1, we present the scheme of f for different values of μ.
As we proved in Proposition 3.1, the fixed point can belong to each of the
pieces of f depending on μ,
where μm≃2.37 while μM≃39.02. (We
are approximating an error less than 10−2.)As we proved in Proposition3.2 the economy is
globally stable for 0≤μ<μm and μ≥μM and the generic orbit definitively converges
monotonically to W1⋆ or W3⋆,
as determined in (3.4) and (3.6), that are points in which f is increasing. In fact, the Koenigs Lemerary
staircase diagram in Figures 2 and 3 shows the converging trajectory for an
arbitrary initial condition.As we proved in Proposition3.1, the fixed point W2⋆ belongs to the decreasing piece when μm≤μ<μM that is the case of economies going through a
phase of financial development. Furthermore, because of the bimodality of f,
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
3.3, if μ<μ⋆,
where μ⋆≃4.9,
the fixed point is still globally stable even if the generic orbit is
asymptotically oscillating.In caseμ=μ⋆ the map exhibits a bifurcation and it gives
rise to an infinite number of repelling period-2 cycles. In fact, Propositions
3.4 and 3.5 show such evidence. Note that for such value of μ,
the fixed point is not hyperbolic while each point Wt∈[WM,f(WM)] is fixed for the second iterate of f,
as it is clear when looking at Figure 4(b).
So all the period-2 orbits are unstable. Furthermore, the set [WM,f(WM)] 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 kindO2={W2⋆+γ,W2⋆−γ},
for all γ∈(0,(f(WM)−WM)/2],
where W2⋆ is given by (3.5). Two of such orbits are, for
example, the ones in Figure 4(a) and 4(c).
The bifurcation occurring at μ⋆ is not canonical: one of the repelling
cycle-2 involves the maximum point WM 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 5 shows the trajectory for an
initial condition once the bifurcation happens; the trajectory is also represented versus time.(a) The generic aperiodic orbit covers two
disjoint invariant sets. (b) The trajectory with respect to time. In both
casesμ=5.5 and W0=5.
(a)(b)As we said, after the bifurcation atμ⋆,
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 [26].)Figure6 shows the bifurcation diagram of the map for
different values of μ.
The black intervals are those in which the dynamics is chaotic or periodic with
very high period. Furthermore, we observe both large intervals of μ where the asymptotic behavior is a cycle-2
(if μ∈(μ1,μ2) with μ1≃8.39 and μ2≃23.5) or a cycle-3 (if μ∈(μ1′,μ2′) with μ1′≃29.1 and μ2′≃39.02).  (In such a
case, the chaotic properties could also be proven by the well-known Li and
Yorke Theorem, see Li and Yorke [27].) In this
case, the dynamics are still predictable even in the long run.Figure 6
Bifurcation diagram with respect toμ.However, inside the two chaotic regions, that are
visible in the following Figures7 and 8, 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 μ that are periodic windows inside the chaotic
region are visible in such figures.Figure 7
Bifurcation diagram forμ∈(4,9).Figure 8
Bifurcation diagram forμ∈(22,32).


## 5. Conclusions

In 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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*Source: 196494-2008-09-09.xml*