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.

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 [

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. [

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 [

While the models in Aghion et al. [

In Aghion et al. [

Similarly, in Aghion et al. [

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 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. [

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 [

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 [

The paper is organized as follows. In Section

We consider a small open economy that produces a
single tradeable good using capital

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

Asymmetric information considerations generate moral
hazard so, according to the results reached by Bernanke and Gertler [

At each period, entrepreneurs maximize their profits
and this program determines their optimal demand

In an equilibrium situation, it must be

Now we can derive the dynamic model describing the
economy. Considering that the price of the country-specific production factor

Now we have to consider the role played by the credit constraint. To do this, we need to study three different cases.

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

If the financial system is underdeveloped, the
investment—which is constrained—does not absorb the total supply of the
country specific factor

Finally, if the financial system is at an
intermediate level of development, the investment absorbs the supply of the
country-specific production factor

From the previous considerations, we derive the map

In this section,
we study the qualitative dynamics of the continuous bimodal piecewise linear
map given by (

Let

for all

for all

for all

Let

To prove part (a) we consider that

To prove part (b) we first consider that if

To prove part (c) we first consider that if

Cases (a), (b), and (c) are depicted in Figure

Scheme of

The following proposition states the global stability of economies at high- or low-financial development levels.

Let

for all

for all

To prove statement (a) we first consider that for all

To prove statement (b) we first consider that for all

About such cases see, Figures

Koenigs Lemerary staircase diagram for two different
initial conditions: in

Koenigs Lemerary staircase diagram for two different
initial conditions: in

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

The following proposition proves the stability of
economies at an intermediate level of financial development when

Repelling period-2 orbits for

Let

As we proved in Proposition

Now we have to study the case of

Before studying this case, we consider that the map

Let

The proof is straightforward as it
is only based on the computation that

Once known that

The discontinuity in the first derivative of the map
implies that it can jump without crossing the bifurcation value

In order to study the stability of the cycle-2,

Let

For all

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

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

In this
section, we provide some numerical simulations by fixing the values of all the
parameters of the model but

In Figure

As we proved in Proposition

As we proved in Proposition

In case

Numerical computations also show that all these
cycles-2 are of the kind

(a) The generic aperiodic orbit covers two
disjoint invariant sets. (b) The trajectory with respect to time. In both
cases

As we said, after the bifurcation at

Figure

Bifurcation diagram with respect to

However, inside the two chaotic regions, that are
visible in the following Figures

Bifurcation diagram for

Bifurcation diagram for

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.

Economies at an intermediate level of financial development eventually converge to the fixed point by oscillations or they fluctuate indefinitely.

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.

They can be unstable and unpredictable if we are in a proper chaotic region because of the sensitivity to the initial conditions.

Economies can be structurally unstable when going trough regions governed by different asymptotic dynamics because of the lack of hyperbolicity.

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.