Financial fragility and global dynamics:

1
Financial fragility and global dynamics
Two routes to complexity
Alessandro Vercelli University of Siena (Italy),
E-mail vercelli_at_unisi.it
(Structural Change and Economic Dynamics, 2000)
Serena Sordi University of Siena (Italy), E-mail
sordi_at_unisi.it
(Journal of Economic Behavior and Organization,
2004, forthcoming)
Roberto Dieci University of Bologna (Italy),
E-mail rdieci_at_rimini.unibo.it
(Chaos, Solitons Fractals, 2005, submitted)
2
This paper deals with a simple model of financial
fluctuations, where a crucial role is played by
the dynamic interaction between the aggregate
current financial ratio and the aggregate
intertemporal financial ratio
The model results in a 4D discrete-time dynamical
system, capable of generating complex dynamics,
which is analyzed by means of both analytical
tools, such as local stability analysis and
bifurcation theory, and numerical simulation.
The behavior of the model is studied for
different regimes of parameters. We show that the
dynamics is very sensitive to the parameters
which represent (1) the speed with which the
aggregate current financial ratio reacts to the
deviation of the intertemporal financial ratio
from its target level and (2) the intensity with
which aggregate current financial decisions
affect future financial constraints.
In particular, different regimes of parameters
are identified, which determine two different
routes to complexity, one leading to chaotic
dynamics, the other to coexistence of attractors
and path-dependence.
3
Purpose to propose a simple prototype model that
describes the complex dynamics of a sophisticated
monetary economy
It is a widely recognized fact that the financial
constraints and objectives of economic agents
have assumed a crucial role in shaping their
behaviour
The analysis of the financial determinants of
economic behaviour is therefore becoming a
general issue that affects the entire economy. We
refer specifically in what follows to a
sophisticated monetary economy, i.e., to one that
has fully developed financial infrastructures and
interrelations
In this case it is reasonable to model all
decision makers as financial units, focusing on
the interaction between their current and
intertemporal financial constraints
In this paper we focus on the aggregate outcomes
4
The model
Each financial unit operating in the economy is
characterized in each period t by

• cash outflows (purchases of goods and services,
  as well as payment of interest and repayment of
  principal on outstanding debt)
• cash inflows (sales of goods and services and new
  loans)
• ? current financial ratio cash outflows/inflows
• Intertemporal financial ratio (the ratio between
  the sum of discounted expected future cash
  outflows and the sum of discounted expected
  future inflows )

By aggregating the outflows and inflows of all
the private financial units we obtain
? aggregate outflows of the private (public)
sector
? aggregate inflows of the private sector
? current financial ratio for the private
sector of the entire economy
(? ? desired current financial ratio)

? aggregate intertemporal financial ratio
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Derivation of the dynamical system
Financial units, at the end of period t ? 1,
decide their cash outflows in period t on the
basis of their desired financial ratio for that
period, with a lag of one period between realized
inflows and outflows
The intertemporal financial ratio varies over
time as the (common) expectations of future cash
inflows and outflows vary. In turn, expectations
about future cash inflows/outflows are affected
by new information contained in recent
realizations if realized outflows, or inflows,
in period t are higher or lower than expected for
that period, expectations are revised according
to the sign of the deviation. The simplest way to
incorporate this idea into the model is to assume
that the intertemporal financial ratio varies
through an adaptive mechanism
0 lt ? lt 1
6
While financial units may be willing to expand
their activity (and thus to increase their
financial exposure) in the short-run when their
expectations are optimistic, we assume that they
are reluctant to go beyond a long-run threshold
of financial fragility, which we denote by 1 ? ?
(0 lt ? lt 1). It is thus reasonable to assume
that financial units decision to vary their
desired current financial position depends on the
distance between the intertemporal financial
ratio and its target level
The model is closed by assuming a lag of one
period between aggregate cash outflows and
production (or aggregate inflows)
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4-dimensional nonlinear dynamical system

Steady states
?
8
Local stability conditions
The Jacobian matrix of the system evaluated at
the equilibrium point is the following
Characteristic equation
where
9
Necessary and sufficient conditions for the
characteristic equation to only have roots of
absolute value less than one are the following
(see Farebrother, 1973, p. 399)
In terms of the parameters of the model
10
(Local) stability region
(the stability region shows very little
sensitivity to changes in the parameter ? )
? 0.25
Neimark-Sacker boundary G(a,bm)0
?
FIG. 1
Neimark-Sacker boundary G(a,bm)0
0
0.5
1.0
1.5
2.0
2.5
?
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Global dynamics
The aim of the section is twofold 1) to explore
the dynamics of the model for those ranges of the
parameters for which the steady state is unstable
(parameter combinations outside the stability
region of Fig. 1) 2) to discuss the global
stability of the steady state for parameter
ranges inside the stability region of Fig. 1
We limit our analysis to the case 0 lt ? lt 1
Within this case, moreover, we distinguish
further between the normal case, where 0 lt ? lt 1,
and the case of overreaction, where ? 1
(? 0.25, Epu 50 ?
)
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Endogenous fluctuations without bankruptcy
We start from sufficiently low levels of the
parameters ? and ? and analyze the dynamic
effects of increasing the parameter ? (with ?
0.1)
? the local stability conditions imply that the
steady state is locally asymptotically stable (an
attracting focus) for low levels of ?
FIG. 2
13
When ? is increased beyond a certain threshold,
a N-S bifurcation occurs to the steady state,
which becomes a repelling focus
Numerical simulation provides evidence that the
N-S bifurcation is of supercritical type (at
least when ? is sufficiently low)
FIG. 2
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The effect of further increasing ? is that the
size of the limit cycle, and therefore the
amplitude of the fluctuations, increases
FIG. 2
Remark when a stable limit cycle exists for low
values of ?, the range of the fluctuations in the
current financial ratio is in general wider than
that of the fluctuations in the intertemporal
financial ratio.
The intertemporal financial ratio adjusts slowly
to changes in the current financial exposure
A characteristic feature of this regime is that
financial units never go bankrupt on average
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A first route to complexity bankruptcy and
chaotic dynamics
We start with ? and ? both high (with ? 0.95)
and analyze the dynamic effects of increasing the
parameter ?
? Unlike the case with a low ?, the steady
state proves now to be locally stable even for
very high values of the reaction coefficient ?
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The N-S bifurcation brings about an oscillatory
regime where the desired current financial ratio
and the intertemporal financial ratio have
fluctuations of similar amplitude
FIG. 3
With a high ?, as in the present case, changes in
the current financial exposure have heavy effects
in the long-run and financial units are likely to
go bankrupt (Fig. 3c)
The ceiling to forces the dynamics and
produces chaotic fluctuations (see Figs. 3d,e,f)
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FIG. 3
18
Chaos Plot
2.5
?
FIG. 4
0
0
2.5
?
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A second route to complexity Coexistence of
attractors and path-dependence
We finally explore a third dynamic scenario we
start from a sufficiently low level ? ( 0.2) and
analyze the dynamic effects of increasing the
parameter ? from 1.278 to 1.45, i.e., in a range
such that the pair (?, ?) is within the stability
region, but close to the N-S boundary
?
?
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The steady state is stable only locally, but not
globally , because another attractor exists in
the phase-space
In most of our numerical simulations, this
attractor is a stable closed orbit of great
amplitude, characterized by periodic or
quasi-periodic motion
Fig. 5 follows the sequence of qualitative
changes that occur to the attractors and to their
basins of attraction
FIG. 5
Stable steady state (the only bounded attractor)
? 1.278
Basin of infinity
BA of the steady state
(a)
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For a higher ?, a competing stable closed orbit
exists in the phase space, whose appearance is
due to a global bifurcation
The new attractor appears abruptly, and the BA of
the stable closed orbit becomes wider and wider
as ? increases, while the BA of the steady state
is gradually reduced to a small neighborhood
(Figs. 5b,c,d,e)
FIG. 5
? 1.279
? 1.3
BA of the closed orbit
(c)
(b)
? 1.35
? 1.44
(d)
(e)
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Stable periodic orbit (the only
bounded attractor)
FIG. 5
? 1.15 ? 0.16
(f)
Remark apart from the two extreme cases, changes
of the parameter ? determine a drastic modificatio
n of the structure of the basins of attractions,
without any important modification of the
structure of the attractors
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Other cases of coexistence of attractors (FIG. 6)
? 1.15 ? 0.16
Unstable steady state
Two stable closed orbits
(a)
? 2.3 ? 0.5
Strange attractor
(b)
Stable steady state
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Path dependence in the stable regime (FIG. 7)
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Conclusions

The model of financial fluctuations developed in
the present paper builds on a stylized
representation of the feedback between current
and intertemporal financial conditions of the
economy
Such an interaction is governed a) by the firms'
speed of adjustment towards desired financial
conditions (captured by the parameter ? of our
model) and b) by the intensity with which
tomorrows financial conditions are affected by
current financial decisions (expressed by the
parameter ?)
Despite its simplicity, the model is capable of
producing a wide range of dynamic scenarios
according to the values of the parameters

• convergence to a (globally) stable steady state
• regular fluctuations on an attracting closed
  orbit
• chaotic fluctuations
• path-dependence, when a locally stable steady
  state coexists with a competing attractor
  characterized by regular, or chaotic, oscillatory
  motion

The first and second scenario are typically
detected in the case of weak firms reaction (low
?), the third and fourth scenario are possible
under firms over-reaction (? gt 1), provided that
the level of intertemporal financial ratio is
sensitive enough to current aggregate financial
decisions (sufficiently high ?)