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ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Highlights
• We propose a model in which the real sector and the stock market interact. • In the stock market there are optimistic and
pessimistic fundamentalists. • We detect the mechanisms through which instabilities get transmitted between markets. •
In order to perform such analysis, we introduce the “interaction degree approach”. • We show the effects of increasing the
interaction degree between the two markets.
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Chaos, Solitons and Fractals xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Chaos, Solitons and FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier.com/locate/chaos
Real and financial interacting markets: A behavioral
macro-model
Ahmad Naimzada a,1, Marina Pireddu b,∗Q1
a Department of Economics, Management and Statistics, University of Milano-Bicocca, U6 Building, Piazza dell’Ateneo Nuovo 1,
20126 Milan, Italyb Department of Mathematics and Applications, University of Milano-Bicocca, U5 Building, Via Cozzi 55, 20125, Milan, Italy
a r t i c l e i n f o
Article history:
Received 30 September 2014
Accepted 9 May 2015
Available online xxx
a b s t r a c t
In the present paper we propose a model in which the real side of the economy, described via
a Keynesian good market approach, interacts with the stock market with heterogeneous spec-
ulators, i.e., optimistic and pessimistic fundamentalists, that respectively overestimate and
underestimate the reference value due to a belief bias. Agents may switch between optimism
and pessimism according to which behavior is more profitable. To the best of our knowledge,
this is the first contribution considering both real and financial interacting markets and an
evolutionary selection process for which an analytical study is performed. Indeed, employing
analytical and numerical tools, we detect the mechanisms and the channels through which
the stability of the isolated real and financial sectors leads to instability for the two interacting
markets. In order to perform such analysis, we introduce the “interaction degree approach”,
which allows us to study the complete three-dimensional system by decomposing it into two
subsystems, i.e., the isolated financial and real markets, easier to analyze, that are then linked
through a parameter describing the interaction degree between the two markets. Next, we
derive the stability conditions both for the isolated markets and for the whole system with
interacting markets. Finally, we show how to apply the interaction degree approach to our
model. Among the various scenarios we are led to analyze, the most interesting one is that in
which the isolated markets are stable, but their interaction is destabilizing. We choose such
setting to give an economic interpretation of the model and to explain the rationale for the
emergence of boom and bust cycles. Finally, we add stochastic noises to the optimists and
pessimists demands and show how the model is able to reproduce the stylized facts for the
real output data in the US.
© 2015 Published by Elsevier Ltd.
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1. Introduction
Both empirical and theoretical arguments show that in-
stabilities are a common feature of all markets: the prod-
uct markets, the labor market, and the financial markets.
As recalled in [1], over the last twenty years many stock
market models have been proposed in order to study the
∗ Corresponding author. Tel.: +39 026 4485 767; fax: +39 026 4485 705.
E-mail addresses: [email protected] (A. Naimzada),
[email protected], [email protected] (M. Pireddu).1 Tel.: +39 026 4485 813; fax: +39 026 4483 085.
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http://dx.doi.org/10.1016/j.chaos.2015.05.007
0960-0779/© 2015 Published by Elsevier Ltd.
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
dynamics of financial markets (see [2,3]). According to such
models, even in the absence of stochastic shocks, the inter-
action between boundedly rational, heterogeneous specula-
tors accounts for the dynamics of financial markets. Those
models, when endowed with stochastic shocks, are able to
replicate some important phenomena, such as bubbles and
crashes, excess volatility and volatility clustering (see, for in-
stance, [4–12]).2 Indeed, differently from DSGE models, be-
2 In the past decades, financial markets have been analyzed also via ap-
proaches based on Heston model, e.g. in [13–17].
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
2 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
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havioral models are able to better reproduce, for instance, the
high kurtosis, the presence of fat tails, the strong autocorrela-
tion of real data. However, in this kind of models authors have
restricted their attention to the representation and the dy-
namics of financial markets only and the existing feedbacks
between the real and financial sides of the economy are often
completely neglected.
The financial crisis of 2008 created new research issues
for economists. Recently, a growing literature has investi-
gated how speculative phenomena in financial markets gets
transmitted to the real economy and whether or not real
market developments feed back on the financial sector. One
simple way to answer such questions consists in integrating
the standard New Keynesian Macroeconomic (NKM) model
with the tools of the Agent-Based Computational (ACE) fi-
nance literature. We stress that, in the present context,
we will use the expression “Agent-Based” in a loose sense,
meaning frameworks with hetereogeneous and/or bound-
edly rational agents.
For instance, Scheffknecht and Geiger [18] present a fi-
nancial market model with leverage-constrained heteroge-
neous agents integrated with a New Keynesian standard
model; all agents are assumed to be boundedly rational.
Those authors show that a systematic reaction by the central
bank to financial market developments dampens macroeco-
nomic volatility. Moreover, Lengnick and Wohltmann [19],
Kontonikas and Ioannidis [20], Kontonikas and Montagnoli
[21] and Bask [22] consider New Keynesian models intercon-
nected with financial markets models. The results are en-
dogenous developments of business cycles and stock price
bubbles.
Contributions in the macroeconomic literature on the in-
teraction between the real and the financial sides that do
not build upon NKM for the description of the real sector
have been proposed by Charpe et al. [23], Westerhoff [24]
and Naimzada and Pireddu [1]. In particular, the latter two
works employ a classical Keynesian demand function only to
represent the real sector. The advantage of this approach is
simplicity. Models are typically of small scale, so that analyt-
ical solutions are tractable.
More precisely, Charpe et al. [23] propose an integrated
macro-model, using a Tobin-like portfolio approach, and con-
sider the interaction among heterogeneous agents in the fi-
nancial market. They find that unorthodox fiscal and mone-
tary policies are able to stabilize unstable macroeconomies.
Westerhoff [24] describes the real economy via a Keynesian
good market approach, while the set-up for the stock mar-
ket includes heterogeneous speculators, i.e., fundamentalists
and chartists. In [24] it is shown that interactions between
the real sector and the stock market appear to be destabiliz-
ing, giving rise to chaotic dynamics through bifurcations. Fi-
nally, Naimzada and Pireddu [1] consider a framework sim-
ilar to that in [24] but, in order to analyze the interactions
between product and financial markets, a parameter repre-
senting the degree of interaction is introduced. With the aid
of analytical and numerical tools it is shown that, under the
assumption of equilibrium for the stock market, an increas-
ing degree of interaction between markets tends to locally
stabilize the system.
One important aspect to be considered when integrating
the real and financial sectors is the identification of the chan-
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
nels through which the two sectors influence each other. Sev-
eral channels have been proposed, but the literature has not
yet agreed upon which channels are the most crucial ones.
Possible assumptions for describing the channels through
which the financial market influences the real sector are the
wealth effects [1,21,22,24], a collateral-based cost effect [19]
or a balance-sheet based leverage targeting effect [18]. Ex-
amples for channels going in the opposite direction, from the
real sector towards the financial market, are a misperception
effect [1,19,24], or a mixture of a misperception effect with
negative dependence on the real interest rate [20–22].
Our paper belongs both to the strand of literature on the
interactions between real and financial markets, as well as to
the literature on heterogeneous fundamentalists (see, for in-
stance, [25–33]). In fact, we here propose a model in which
the real economy, described via a Keynesian good market
approach, and the stock market, with heterogeneous funda-
mentalists, interact. The two papers that bear a stronger re-
semblance to our framework are [24,26]. More precisely, sim-
ilar to [26], we assume that the financial side of the econ-
omy is represented by a market where traders behave in two
different ways: optimism and pessimism. Optimists (pes-
simists) systematically overestimate (underestimate) the ref-
erence value due to a belief bias. Moreover, like in [26] agents
may switch between optimism and pessimism according to
which behavior is more profitable. On the other hand, in [26]
only the financial sector is considered and the connection
with the good market is missing. When comparing our set-
ting to the one in [24], we stress that, similar to what done
in that paper, we assume the real economy to be represented
by an income-expenditure model in which expenditures de-
pend also on the dynamics of the stock market price. On the
other hand, in [24] the real market subsystem is described
by a stable linear relation, while the financial sector is repre-
sented by a cubic functional relation, that is, by a nonlinear
relation. In that way, the oscillating behavior is generated by
the financial subsystem only. In our paper we present instead
a model in which the oscillating behavior is generated also by
the real subsystem. Indeed, our stock price adjustment mech-
anism is linear, but not always stable, while the real subsys-
tem is described by a nonlinear relation. To be more precise,
the nonlinearity of the real subsystem is due to the nonlin-
earity of the adjustment mechanism of the good market with
respect to the excess demand. In particular, the sigmoidal
nonlinearity we deal with has been recently considered in
[34]. Another difference with respect to [24] is the way we
represent and analyze the interaction between the two mar-
kets. We assume in fact that economic agents operating in
the financial sector base their decisions on a weighted av-
erage between an exogenous fundamental value and an en-
dogenous fundamental value depending on the current real-
ization of income, while in the real market we assume that
private expenditures depend also, with a given weight, on
the stock market price. In particular, in our model the pa-
rameter describing that weight represents also the degree
of interaction between the two markets. The extreme values
of the weighting parameter correspond to the two cases an-
alyzed in [24], i.e., the isolated market framework and the
interacting market scenario. Finally, we remark that in [24]
no heterogeneous fundamentalists and no switching mecha-
nism are considered.
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 3
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
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Hence, summarizing, our main difference with respect to
[26] is that we also consider the real sector of the economy,
while with respect to [24] is that we introduce the interaction
degree parameter and the switching mechanism, in order to
describe the changes in the share of agents in the financial
market.
We stress that, to the best of our knowledge, the present
paper is the first contribution considering a model with both
real and financial interacting markets and an evolutionary
selection process for the population for which an analytical
study is performed. Indeed, in the existing literature, just few
papers [18,19,35] deal both with interacting sectors and an
evolutionary approach, and all of them propose just a numer-
ical analysis of the framework under consideration.
The present work represents the third step of a line of re-
search started with [1] and [34]. Indeed, in [34] we analyzed
the effects of the introduction of a nonlinearity in the adjust-
ment mechanism of income with respect to the excess de-
mand in the standard Keynesian income-expenditure model,
showing that it was able to generate complex dynamics and
multistability phenomena. However, in [34] only the real sec-
tor was considered and the relation with the stock market
was missing. In [1] we then dealt with the same Keyne-
sian model in [34] to which we added the connection, mod-
eled via the interaction degree parameter aforementioned,
with the financial subsystem, represented by an equilibrium
stock market with heterogeneous speculators, i.e., chartists
and fundamentalists. The equilibrium assumption, equiva-
lent to the hypothesis that the stock market speed of adjust-
ment tends to infinity, was motivated by the functioning of fi-
nancial markets and allowed to reduce our two-dimensional
model to a one-dimensional system. We here add some fur-
ther elements of interest to our achievements in [1] and [34].
Indeed, starting from the Keynesian model in [34], similar to
what done in [1], we still consider a framework with real and
financial sectors connected via the interaction degree param-
eter, to which we add three new ingredients: the stock mar-
ket is no more assumed to be always in equilibrium and thus
we need to analyze one more equation, describing the stock
price dynamics; instead of dealing with chartists and funda-
mentalists, as already stressed, we consider the case of het-
erogeneous (optimistic and pessimistic) fundamentalists; fi-
nally, we here allow agents to switch between optimism and
pessimism, according to which behavior is more profitable,
and this leads us to add a further equation to our model,
describing the evolutionary dynamics of the population of
traders.
We notice that our model displays several behavioral fea-
tures: indeed, agents are not optimizing and follow instead
adaptive rules, based on the difference between the realized
price and the perceived fundamental value, which depends
on the trend of the economy; also the switching mechanism
is adaptive in nature, as realized relative profits are taken into
account.
The main contribution of this paper to the existing litera-
ture lies in focusing on the role of real and financial feedback
mechanisms, not only in relation to the dynamics and stabil-
ity of a single market, but also for those of the economy as
a whole. Analytical and numerical tools are used to investi-
gate the role of the parameter describing the degree of inter-
action, in order to detect the mechanisms and the channels
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
through which the stability of the isolated real and financial
sectors leads to instability for the two interacting markets.
More precisely, we start by introducing the “interaction de-
gree approach”, which allows us to study high-dimensional
systems with many parameters by decomposing them into
subsystems easier to analyze, that are then interconnected
through the “interaction parameter”. Next, we introduce our
model and we derive the stability conditions both for the iso-
lated markets and for the whole system with interacting mar-
kets. In particular, we find that in the stability conditions it
is possible to isolate the parameter describing the degree of
interaction between the two markets and that the stability
conditions are fulfilled if that parameter belongs to a range
characterized by two lower bounds and two upper bounds.
Finally, we show how to apply the “interaction degree ap-
proach” to our model. To this aim, we first classify the possi-
ble scenarios according to the stability/instability of the iso-
lated financial and real markets: in such way we are led to an-
alyze four frameworks. For each of those we consider differ-
ent possible parameter configurations and we show, both an-
alytically and numerically, which are the effects of increasing
the degree of interaction between the two markets. The con-
clusions we get are not univocal: indeed, depending on the
value of the other parameters, an increase in the interaction
parameter may either have a stabilizing or a destabilizing ef-
fect, but also other phenomena are possible. Namely, accord-
ing to the mutual position of the before-mentioned lower
and upper bounds for the stability range, it may also happen
that the stabilization of the system occurs just for interme-
diate values of the interaction parameter, neither too small,
nor too large, or it may as well happen that one of the upper
bounds is always negative or smaller than one of the lower
bounds and thus the system never achieves a complete stabi-
lization, even if its complexity may decrease and we observe
some periodicity windows interrupting the chaotic band.
We conclude our theoretical analysis by showing which
are the effects of an increasing belief bias for optimists and
pessimists. Although its effect is clearly destabilizing when
markets are isolated, its role becomes more ambiguous when
the markets are interconnected. Indeed, increasing the bias
may have either a stabilizing or a destabilizing role, according
to the value of the other parameters. However, our numerical
simulations suggest that increasing the bias has generally a
destabilizing effect, as usually we do not reach a complete
stabilization, or we achieve it just in small intervals for the
bias.
We analyze all the possible scenarios in order to show
the variety of dynamics generated by the model, compar-
ing the numerical simulations and the bifurcation diagrams
with the analytical results. Nonetheless, in order to discuss
our model, we choose to illustrate and economically inter-
pret the stronger result we obtain, i.e., that increasing the
interaction degree, which measures how much the specula-
tors in the financial market are influenced by the behavior
of the real market and vice versa, may be destabilizing even
when the two isolated sectors are stable. More precisely, we
explain and show how an increasing value for the interac-
tion parameter, together with the switching mechanism for
agents in the financial market and the presence of upper and
lower bounds imposed by the sigmoidal nonlinearity in our
real sector, describing the output capacity constraints, can
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
4 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
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Fig. 1. The output gap in the US on the basis of quarterly data from 1960 to
2009. Source: US Department of Commerce and Congressional Budget Office. Q3
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produce the alternation of booms and busts, respectively
characterized by income growth and contraction in the real
sector and by the waves of optimism and pessimism in the
financial market.
Even for the empirical verification of our model we con-
sider the destabilizing scenario, in which the two separately
stable sectors become unstable when coupled. In particular,
in order to describe the accidental fluctuations of the com-
position of the many individual digressions from the sim-
ple rules they are supposed to follow, according to [36] we
add a noise term to the demand functions of optimists and
pessimists. In such framework we show that our model is
able to reproduce three crucial stylized facts about cyclical
movements of output and empirical foundation of the con-
cept of animal spirit: high autocorrelation for output caus-
ing strong fluctuations, a non-normal distribution for out-
put, characterized by a high kurtosis and fat tails, and finally
a strong correlation between output and a long-period opti-
mism index, which describes the waves of optimism and pes-
simism. For the empirical verification of our model we follow
the approach by De Grauwe [2,37–39] where, starting from
macroeconomic models with heterogeneous and boundedly
rational agents, but without financial sector, shows that the
obtained dynamics are in agreement with (some or all of) the
three stylized facts we consider, too.
The remainder of the paper is organized as follows. In
Section 2 we present the stylized facts we wish to replicate
with our model. In Section 3 we illustrate the interaction
degree approach we use to analyze the two linked subsys-
tems. In Section 4 we introduce the model, composed by the
real and financial sectors. In Section 5 we derive the condi-
tions for the steady state stability, both in the case of iso-
lated and interacting markets. In Section 6 we classify and
investigate, analytically and numerically, the possible scenar-
ios determined by the stability/instability of the real and fi-
nancial markets when isolated, and we finally show which
are the effects of an increasing bias. In Section 7 we discuss
our model and give an economic interpretation of our main
results. In Section 8 we add stochastic shocks to the deter-
ministic framework considered so far and we show that the
model reproduces the stylized facts presented in Section 2.
In Section 9 we draw some conclusions and outline possible
future research directions.
2. Stylized facts
Economic systems are characterized by periods of strong
growth in output followed by periods of decline in economic
growth, that is, by booms and busts. Every macro-economic
model should be able to explain and reproduce such booms
and busts in economic activity.
Before proposing our model, it is then useful to present
some stylized facts about cyclical movements of output and
empirical foundation of the concept of animal spirit, we wish
our framework to be able to replicate.
Fig. 1 shows the strong cyclical movements of the output
gap in the US since 1978, that is, the difference between the
output and the potential output, the latter being determined
in the short run by the capital stock.
These cyclical movements are caused by a strong auto-
correlation in the output gap numbers, i.e., if in period t the
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
output gap assumes a certain value, it is likely that its value
in period t + 1 will not vary too much. In particular, if in a cer-
tain period the output gap is positive (negative), it is likely to
remain positive (negative) for some periods.
A second stylized fact about the movements in the output
gap, clearly visible in Fig. 2 for the US, is that such movements
are not normally distributed in two aspects. The first is that
there is a relatively high kurtosis (kurtosis = 3.62) and thus,
with respect to the normal distribution, there is too much
concentration of observations around the mean. The second
aspect are fat tails, i.e., there are larger movements in the out-
put gap than is compatible with the normal distribution. In
particular, booms and busts are more likely to happen.
This also means that, basing the forecasts on the normal
distribution, the probability that in some period a large in-
crease or decrease in the output gap can occur would be un-
derestimated.
In Figs. 1 and 2 we report the plots for the US, but similar
autocorrelation coefficients are found in other countries (see
[40,41]).
A third stylized fact we take into account is the high cor-
relation between the consumer sentiment index and the out-
put gap. The best-known among the sentiment indicators is
the Michigan consumer confidence indicator. Such sentiment
indicators are developed on the basis of how the individuals
perceive the present and the future economic conditions.
In Fig. 3 the Michigan consumer confidence indicator is
plotted together with the US output gap in period 1978–2008
on the basis of quarterly data, showing a high correlation be-
tween the two variables.
We will return on the above presented stylized facts in
Sections 7 and 8, where we will show how our model, both
in its original formulation and when endowed with stochas-
tic errors, is able to replicate such kind of behaviors. In par-
ticular, in regard to the third stylized fact, a feature of the
correlation of our theoretical framework is that the causal-
ity goes both ways, i.e., animal spirits affect output and vice
versa.
See also [2] for the discussion of other behavioral mod-
els reproducing the same stylized facts. Indeed, the approach
we followed for the empirical verification of our model is the
one by De Grauwe [2,37–39] where, starting from macroe-
conomic models with heterogeneous and boundedly rational
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 5
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 2. The frequency distribution of US output gap from 1960 to 2009, having kurtosis = 3.61 and Jarque–Bera = 7.17 with p-value = 0.027. Source: US Department
of Commerce and Congressional Budget Office.
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agents, but without financial sector, the author shows that
the obtained dynamics are in agreement with (some or all
of) the three stylized facts we consider, too.
3. The interaction degree approach
As we shall see in Section 4, when considering both fi-
nancial and real markets, we are led to analyze a nonlinear
high-dimensional system with many parameters. Such fea-
tures do not allow to easily handle that system from an ana-
lytical viewpoint when all the parameters vary, even if we are
able to analytically determine its steady state and the corre-
sponding stability conditions. For this reason, we propose an
approach that consists in studying, as a first step, the frame-
work with isolated markets, which are described by lower-
dimensional subsystems that are simpler to investigate and
whose different behaviors can be quickly classified. Then we
make the parameter representing the interaction degree in-
crease, keeping the other parameters fixed. In such way we
are able to analytically find (if it exists) the set of values of
the interaction parameter that implies stability. Moreover,
the use of numerical tools allows us to understand what hap-
pens also in the unstable regime.
This is the strategy we are going to employ in Section 6 to
classify and investigate the various scenarios for our system.
However, in order to make the exposition more fluent, we
start Section 5 with the analysis of the stability conditions
for the case of interacting markets and we derive next the
stability conditions for the framework with isolated markets.
In symbols, if we denote our integrated system by
Sω(x1, . . . , xN), where ω ∈ [0, 1] is the interaction de-
gree parameter and x1, . . . , xN are the endogenous variables
governing the system, when setting ω = 0 we are led to
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
study two (or, in general, more) isolated subsystems, we de-
note by S10(x1, . . . , xm) and S2
0(xm+1, . . . , xN), for some m ∈{1, . . . , N − 1}. When instead ω = 1 the subsystems are fully
integrated. The case with ω ∈ (0, 1) represents a partial inter-
action between the subsystems.
In our framework, when ω = 0 we find two isolated sub-
systems, corresponding to the financial and real markets.
The former is described by two variables, i.e., the stock price
and the difference between the shares of optimistic and pes-
simistic agents, while the latter is described by a unique vari-
able, that is, the level of income. We stress that the influence
of the real market on the financial market is due to the fact
that the reference value used in the decisional mechanism
by the agents in the financial market is determined by the
level of income; on the other hand, the investments depend
also on the price of the financial asset. Such a double inter-
action is described by the parameter ω. Notice that we chose
a unique parameter, ω, to represent both the influence of the
real market on the financial sector and vice versa of the finan-
cial sector on the real market, not only to limit the already
large number of parameters in our model, but also in view
of applying the interaction degree approach precisely in the
formulation described above. Of course the framework with
two different parameters describing the mutual relationships
between the two markets could be analyzed in a similar
manner.
4. The model
4.1. The stock market
With respect to the stock market, we assume that agents
are not able to observe the true underlying fundamental. We
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
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Fig. 3. US output gap (in purple-red, with corresponding scaling on the right vertical axis) and Michigan sentiment index (in blue, with corresponding scaling on the left vertical axis) in period 1978–2008. Source: US
Department of Commerce, Bureau of Economic Analysis, and University of Michigan: Consumer Sentiment Index. (For interpretation of the references to color in this figure legend, the reader is referred to the web version
of this article.) Q4
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suppose instead that they form believes about the funda-
mental and, on the basis of this belief, they operate in the
stock market. In particular, we consider the trading behav-
ior of two types of speculators: optimists and pessimists. The
label optimist (pessimist) refers to traders that systemati-
cally overestimate (underestimate) the reference value used
in their decisional mechanism [26]. Both types of agents be-
long to the class of fundamentalists as, believing that stock
prices will return to their fundamental value, they buy stocks
in undervalued markets and sell stocks in overvalued mar-
kets. To be more precise, we should say that we model agents
as fundamentalists, but their effective behavior depends on
the relative position of the stock price with respect to the per-
ceived reference values (see (4.3) and (4.4)). Optimists and
pessimists behave in a similar manner, a part from the fact
that the beliefs they have about the reference value differ.
The perceived reference values, we denote by Fopt
t and Fpes
t ,
are a weighted average between an exogenous value (F∗ + a
and F∗ − a, respectively, with a > 0) and a term depending
on the income value. For simplicity, according to [1] and [24],
we assume for the latter term a direct relationship with the
economic activity value, both for optimists and pessimists.3
In particular, in our model the endogenous term of the funda-
mental value perceived by optimists and pessimists is given
by kYt + a and kYt − a, respectively, where Yt is the national
income and k is a positive parameter capturing the above de-
scribed direct relationship. Hence, we assume that
F optt = (1 − ω)(F ∗ + a) + ω(kYt + a)
= (1 − ω)F∗ + ωkYt + a (4.1)
and
F pest = (1 − ω)(F∗ − a) + ω(kYt − a)
= (1 − ω)F∗ + ωkYt − a, (4.2)
where a > 0 is the belief bias and F∗ is the true unobserved
fundamental, both exogenously determined. The constant ω∈ [0, 1] represents the weighting average parameter. In par-
ticular, when ω = 0 the reference value is completely exoge-
nous and coincides with the reference value of an isolated
stock market like in [26]. When instead ω = 1 the reference
value is endogenous.
Optimists’ demand is given by
doptt = α
(F opt
t − Pt
), (4.3)
and, similarly, pessimists’ demand is given by
dpest = α
(F pes
t − Pt
), (4.4)
where Pt is the stock price and α > 0 is the reactivity
parameter.
The market maker determines excess demand and adjusts
the stock price for the next period. In particular, we denote
by nit , i ∈ {opt, pes}, the fraction of traders of type i in the
market at time t and we assume the market maker behavior
to be described by the linear price adjustment mechanism
Pt+1 = Pt + μ(nopt
t doptt + npes
t dpest
), (4.5)
where μ > 0 is the market maker price adjustment param-
eter. For simplicity, we normalize the population size to 1.
3 For an economic justification of such hypothesis, see [24, p. 4].
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
According to (4.5), the market maker increases (decreases)
the stock price if excess demand noptt d
optt + n
pest d
pest is posi-
tive (negative).
We set xt = noptt − n
pest , in order to express the fraction of
optimistic (pessimistic) traders as noptt = 1+xt
2 (npest = 1−xt
2 ),
so that we can rewrite (4.5) as
Pt+1 = Pt + αμ
2
[(F opt
t − Pt
)(1 + xt ) +
(F pes
t − Pt
)(1 − xt )
].
(4.6)
Recalling the definition of Fopt
t and Fpes
t from (4.1) and (4.2),
respectively, we rewrite (4.6) as
Pt+1 = Pt + αμ{[(1 − ω)F∗ + ωkYt ] − Pt + axt}. (4.7)
We observe that the evolution of the stock price is
determined by two factors. The first one is the devia-
tion of the unbiased reference value from the stock price
([(1 − ω)F∗ + ωkYt ] − Pt ) : when the price in period t is be-
low (above) the unbiased reference value, the price will
increase (decrease) in the next period. The second factor
involves the fraction of optimists and pessimists in the mar-
ket. If xt is positive (negative) there are more (less) optimists
than pessimists, so that the price will increase (decrease) in
the next period. The strength of such effect is influenced by
the belief bias a. Finally, we notice that with a completely ex-
ogenous reference value, i.e., when ω = 0, and without bias,4
(4.7) has a unique steady state given by P∗ = F∗.Defining now the dynamics of the population of traders,
we assume that they will start trying the optimistic or pes-
simistic behavior and, if it turns out to be the most profitable,
they will stick to it; otherwise they will switch to the other
behavior in the next period. Such an evolutionary process is
governed by the profits that traders make in each period. Let
us define the profits π it realized by type i, i ∈ {opt, pes}, as
π it = di
t−1(Pt − Pt−1). (4.8)
Following [4,42], we assume that the fraction nit of traders of
type i is given by the discrete choice model
nit = exp(βπ i
t )
exp(βπ optt ) + exp(βπpes
t ), (4.9)
where β ≥ 0 is the parameter representing the intensity of
choice. In particular, if β = 0 the difference between prof-
its is not considered and the behavior choice is purely ran-
dom, so that noptt = n
pest = 1
2 . At the other extreme, when
β → +∞, the switches are fully governed by the rational
component and all traders are of the optimistic type (xt →1) if πopt
t > πpest , while all traders are of the pessimistic type
(xt → −1) if πoptt < πpes
t ; finally, if πoptt = πpes
t , we find again
noptt = n
pest = 1
2 and thus xt = 0. We observe that right hand
side in (4.9) may be seen as a representation of the relative
profits of the traders of type i.
From (4.1)–(4.4), (4.7) and (4.8) it follows that
π optt − πpes
t =(dopt
t−1− dpes
t−1
)(Pt − Pt−1)
= 2aμα2{[(1 − ω)F∗ + ωkYt−1]
−Pt−1 + axt−1}, (4.10)
4 Else, the steady state is still unique and its expression reads as P∗ = F ∗ +ax∗.
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
8 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
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and thus, from the definition of xt and (4.9), we have
xt = tanh
(β(π opt
t − πpest
)2
)= tanh(aμβα2[(1 − ω)F∗
+ωkYt−1 − Pt−1 + axt−1]). (4.11)
4.2. The real market
Similar to [1] and [24], we consider a Keynesian good mar-
ket interacting with the stock market, in a closed economy
with public intervention. It is assumed that both private and
government expenditures depend on national income and
that private expenditure depends also on the performance
in the stock market. The dynamic behavior in the real econ-
omy is described by an adjustment mechanism depending
on the excess demand: if aggregate excess demand is posi-
tive (negative), production will increase (decrease). Indeed,
income Yt+1 in period t + 1 is defined in the following way:
t+1 = Yt + γ g(Zt − Yt ), (4.12)
where g is an increasing function with g(0) = 0, Zt is the ag-
gregate demand in a closed economy, defined as
Zt = Ct + It + Gt ,
where Ct, It and Gt stand for consumption, investment and
government expenditure, respectively, and γ > 0 is the real
market speed of adjustment between demand and supply. In
order to conduct our analysis, denoting by Et = Zt − Yt the ex-
cess demand, we specify the function g as
g(Et ) = a2
(a1 + a2
a1e−Et + a2
− 1
), (4.13)
with a1, a2 positive parameters.
With such a choice, g is increasing and g(0) = 0. More-
over, g is bounded from below by −a2 and from above by a1.
The presence of the two horizontal asymptotes prevents too
large variations in income and prevents the real market from
diverging, creating a real oscillator. We stress that this kind
of nonlinearity has been recently considered in [34].
The rationale for introducing in (4.12) a nonlinear map
g, rather than a linear one, is that in the latter framework
the income variation �t+1 = Yt+1 − Yt may grow unbound-
edly and, in particular, when Et limits to ±∞, the same does
�t+1. However, this is an unrealistic assumption because of
the material constraints in the production side of an econ-
omy. Indeed, when excess demand increases, capacity con-
straints will surely lead to lower increases in income, due to
the limited expansion from time to time of capital and la-
bor stock; when excess demand decreases, capital cannot be
destroyed proportionally to excess demand as the only fac-
tors that may reduce productivity are attrition of machines
from wear, time, and innovations. Moreover, also the labor
factor imposes constraints: indeed, due to the presence of
trade unions, it is difficult, or impossible, to reduce employ-
ment below a certain threshold level.
Like commonly assumed, private and government expen-
ditures are partly exogenous and partly increase with na-
tional income. Moreover, as in [1] and [24], we suppose that
the financial situation of households and firms depends on
the stock market performance, too. If the stock price in-
creases, the same does private expenditure. On the basis of
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
these considerations, we can write the relation between pri-
vate and government expenditures and national income and
stock price as
Zt = Ct + It + Gt = A + bYt + ωcPt , (4.14)
where A > 0 defines autonomous expenditure, b ∈ [0, 1] is
the marginal propensity to consume and invest from current
income, c ∈ [0, 1] is the marginal propensity to consume and
invest from current stock market wealth, and ω ∈ [0, 1] rep-
resents the degree of interaction between the real and the
stock markets. In particular, when ω = 0 the real market is
completely isolated from the financial market; when ω = 1
the two markets are fully interconnected; for ω ∈ (0, 1) we
have a partial interaction.
We stress that it would also be possible to assume that
aggregate demand Zt depends, rather than on the stock price
Pt as in (4.14), on the price variation Pt − Pt−1. Notice however
that this would increase the dimensionality of our system.
We will deal with such a new formulation in a future paper.
Inserting Zt from (4.14) into (4.12) and recalling the defi-
nition of g in (4.13), we obtain the dynamic equation of the
real market
Yt+1 = Yt + γ a2
(a1 + a2
a1e−(A+bYt +ωcPt −Yt ) + a2
− 1
).
Summarizing, when taking into account both the finan-
cial and the real markets, we are led to study the following
system describing the whole economy:⎧⎪⎨⎪⎩Pt+1 = Pt + αμ{[(1 − ω)F∗ + ωkYt ] − Pt + axt}xt+1 = tanh(aμβα2[(1 − ω)F ∗ + ωkYt − Pt + axt ])
Yt+1 = Yt + γ a2
(a1+a2
a1e−(A+bYt +ωcPt −Yt )+a2− 1
).
(4.15)
The associated dynamical system is generated by the iter-
ates of the three-dimensional map
G = (G1, G2, G3) : (0,+∞) × (−1, 1) × [0,+∞) → R3,
(P, x,Y ) �→ (G1(P, x,Y ), G2(P, x,Y ), G3(P, x,Y )),
defined as:⎧⎨⎩G1(P, x,Y ) = P + αμ((1 − ω)F∗ + ωkY − P + ax)
G2(P, x,Y ) = tanh(μaα2β[(1 − ω)F∗ + ωkY − P + ax])
G3(P, x,Y ) = Y + γ a2
(a1+a2
a1e−[A+bY+ωcP−Y ]+a2− 1
).
(4.16)
5. Some local stability results
In order to classify in Section 6 the various scenarios oc-
curring for ω = 0 and investigate their local stability when ωincreases, hereinafter we derive the sufficient conditions for
stability both in the case of interacting and isolated markets.
In fact, the classification we will adopt in the next section re-
lies on the stability/instability features of the real and finan-
cial subsystems when they are isolated. Then, for any such
scenario, we will study what happens when the degree of in-
terconnection between the two markets increases.
Straightforward computations show that there is a perfect
correspondence between the analytical conditions derived in
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 9
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
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Sections 5.1 and 5.2 and the numerical results in Section 6.
In fact, for the reader’s convenience, in correspondence to
the scenarios in Sections 6.1 and 6.2 we will check what the
corresponding stability conditions say and we will compare
those conditions with the numerical simulations performed
therein. The trivial verifications for the remaining scenarios
considered in the next section can be performed analogously.
5.1. Interacting markets
The map in (4.16) has a unique fixed point given by
(P∗, x∗,Y ∗)
=(
ωAk + (1 − ω)F ∗(1 − b)
1 − b − ω2ck, 0,
A + ωc(1 − ω)F∗
1 − b − ω2ck
).
The Jacobian matrix for G computed in correspondence to it
reads as
JG(P∗, x∗,Y ∗) =
⎡⎣1 − αμ αμa αμωk
−μaα2β α2μa2β α2μaβωkγ a1a2ωc
a1+a20 1 − γ a1a2(1−b)
a1+a2
⎤⎦.
(5.1)
In order to check the stability of the steady state in the
various scenarios considered in Section 6, we are going to use
the following conditions (see [43]):
(i) 1 + C1 + C2 + C3 > 0;(ii) 1 − C1 + C2 − C3 > 0;
(iii) 1 − C2 + C1C3 − (C3)2 > 0;(iv) 3 − C2 > 0,
where Ci, i ∈ {1, 2, 3}, are the coefficients of the character-
istic polynomial
λ3 + C1λ2 + C2λ + C3 = 0.
In our framework, we have
1 = γ a1a2(1 − b)
a1 + a2
− 2 + αμ − μa2α2β;
2 = 2μa2α2β + 1 − αμ − γ a1a2ω2ckαμ
a1 + a2
− γ a1a2(1 − b)
a1 + a2
(1 − αμ + μa2α2β);
3 = μa2α2β
(γ a1a2(1 − b)
a1 + a2
− 1
).
Notice that, making ω explicit, it is possible to rewrite Con-
ditions (i)–(iv) above respectively as follows:
(i′) ω2 < (1 + C1 + C̃ + C3)a1+a2
γ a1a2ckαμ:= B1;
(ii′) ω2 < (1 − C1 + C̃ − C3)a1+a2
γ a1a2ckαμ:= B2;
(iii′) ω2 > (−1 + C̃ − C1C3 + C32)
a1+a2γ a1a2ckαμ
:= B3;(iv′) ω2 > (C̃ − 3)
a1+a2γ a1a2ckαμ
:= B4,
where we have set
˜ = 2μa2α2β + 1 − αμ − γ a1a2(1 − b)
a1 + a2
×(1 − αμ + μa2α2β).
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
Hence, if
min{1 + C1 + C̃ + C3, 1 − C1 + C̃ − C3} > 0 and
max{−1 + C̃ − C1C3 + C32, C̃ − 3} < 1,
the integrated system is locally asymptotically stable at the
steady state if
max{B3, B4} < ω2 < min{B1, B2}, ω ∈ [0, 1]. (5.2)
If instead
min{1 + C1 + C̃ + C3, 1 − C1 + C̃ − C3} ≤ 0 or
max{−1 + C̃ − C1C3 + C32, C̃ − 3} ≥ 1,
it is not possible to have local stability at the steady state, for
any ω ∈ [0, 1].
5.2. Isolated markets
In the special case in which ω = 0, System (4.15) can be
rewritten as⎧⎪⎨⎪⎩Pt+1 = Pt + αμ(F ∗ − Pt + axt )
xt+1 = tanh(μaα2β[F ∗ − Pt + axt ])
Yt+1 = Yt + γ a2
(a1+a2
a1e−[A+(b−1)Yt ]+a2− 1
) (5.3)
and its steady state reads as
(P∗, x∗,Y ∗) =(
F ∗, 0,A
1 − b
). (5.4)
Since in such framework the first two equations in (5.3) de-
pend just on Pt and xt, and the last one just on Yt, imply-
ing that the real and stock markets are completely discon-
nected, as explained in Section 3, instead of considering the
three-dimensional system in (5.3), we will rather deal with
the two-dimensional subsystem related to the stock market{Pt+1 = Pt + αμ(F ∗ − Pt + axt )
xt+1 = tanh(μaα2β[F ∗ − Pt + axt ])
and with the one-dimensional subsystem related to the real
market
t+1 = Yt + γ a2
(a1 + a2
a1e−(A+(b−1)Yt ) + a2
− 1
).
In this way, in agreement with the findings in [26], the steady
state in (5.4) should be split as
(P∗, x∗) = (F ∗, 0), Y ∗ = A
1 − b
and, similarly, the Jacobian matrix in (5.1) computed in cor-
respondence to the steady state when ω = 0 should be split
as
J1(P∗, x∗) =[
1 − αμ αμa
−μaα2β α2μa2β
],
J2(Y ∗) = 1 − γ a1a2(1 − b)
a1 + a2
.
The Jury conditions (see [44]) for the stability of the financial
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
10 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 4. The bifurcation diagram with respect to ω ∈ [0, 1] for P, for μ = 5 and the initial conditions P0 = 5, x0 = 0.8 and Y0 = 25.
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subsystem read then as
det J1 = μα2a2β < 1,
1 + tr J1 + det J1 = 2 − μα + 2μα2a2β > 0,
1 − tr J1 + det J1 = μα > 0.
Notice that the third condition is always fulfilled, while the
first two can be rewritten, making β explicit, as
αμ − 2
2μα2a2< β <
1
μα2a2. (5.5)
From (5.5) we easily infer the destabilizing role of the be-
lief bias on the financial side of the economy when the two
markets are isolated: indeed, the stability interval for β gets
reduced when a increases.
On the other hand, the real subsystem is locally asymp-
totically stable at the steady state if −1 < 1 − γ a1a2(1−b)a1+a2
< 1.
The right inequality is always fulfilled (except for b = 1, but
we will always deal with the case 0 < b < 1), while the left
inequality holds if and only if
γ <2(a1 + a2)
a1a2(1 − b).
Hence, when ω = 0 both subsystems are stable if
αμ − 2
2μα2a2< β <
1
μα2a2and γ <
2(a1 + a2)
a1a2(1 − b).
(5.6)
6. Possible scenarios
Starting from the various stability/instability scenarios for
the financial and real subsystems when isolated, in the next
pages we shall investigate what happens in each framework
when the degree of interaction between the two markets in-
creases, in order to show that modifying the parameter ωmay produce very different effects depending on the value
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
of the other parameters and on the specific framework con-
sidered.
We will conclude the section by analyzing the effects of
an increasing belief bias on the stability of the whole system.
6.1. Stable financial and real subsystems
In this framework, when isolated, both markets are sta-
ble. As ω increases, the steady state can either remain sta-
ble until ω = 1 or can undergo a flip bifurcation, followed
by a secondary double Neimark–Sacker bifurcation, accord-
ing to the considered value of the other parameters. In
particular, the parameter μ seems to play a crucial role
in this respect. In fact, in Figs. 4–6 below we have fixed
the parameters as follows: F∗ = 5, k = 0.25, α = 0.08, β =1, c = 1, a = 2, γ = 3.5, a1 = 2, a2 = 4, A = 5, b = 0.7, and
μ = 5 in Fig. 4, while μ = 28 in Figs. 5 and 6. In Fig. 4 the
steady state remains stable until ω = 1, while in Figs. 5 and
6 a destabilization occurs for ω�0.515. More precisely, in
Figs. 4 and 5 (A) we show the bifurcation diagram for P with
respect to ω ∈ [0, 1], while in Fig. 5 (B) we draw the bifur-
cation diagram for Y with respect to ω ∈ [0, 1]; in Fig. 5 (C)
we show the Lyapunov exponent when ω varies in [0, 1]. In
Fig. 6 (A) and (B) we depict, in the phase plane, the fixed point
when ω = 0.25 and the period-two cycle when ω = 0.70,
respectively; finally, in Fig. 6 (C) we show the time series
for P (in red) and Y (in blue) when ω = 0.95, which high-
light a quasiperiodic behavior characterized by long mono-
tonic increasing motions, followed by oscillatory decreasing
motions.
Let us now check whether the theoretical results in
Section 5 are in agreement with the numerical achievements
above.
First of all let us verify that, for the chosen parameter
sets, when ω = 0 both the financial and real subsystems are
stable, i.e., let us check that all the inequalities in (5.6) are
fulfilled. A straightforward computation shows that this is
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 11
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 5. The bifurcation diagrams with respect to ω ∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, for μ = 28 and the initial
conditions P0 = 5, x0 = 0.8 and Y0 = 25.
Fig. 6. The (P, Y)-phase portraits for μ = 28, and ω = 0.25 in (A) and ω = 0.70 in (B), respectively; in (C) the time series for P in red (below) and Y in blue (above)
when μ = 28 and ω = 0.95. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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the case, as the first chain of inequalities reads as −6.25 <
1 < 7.812 and the last inequality is 3.5 < 5 when μ = 5,
while the first chain of inequalities reads as 0.167 < 1 < 1.395
and the last inequality is again 3.5 < 5 when μ = 28. As it
is immediate to see from such calculations, although for the
considered parameter values both isolated markets are sta-
ble, when μ increases the stability region of the financial
subsystem decreases: this may explain the detected grow-
ing destabilization of the whole system in correspondence to
larger values of μ.
As concerns the stability conditions for ω varying in
[0, 1], when μ = 5 we have B1 = 1.2, B2 = 2.386, B3 =−2.4509, B4 = −6.778, and thus (5.2) reads as ω ∈ [0, 1],
that is, the system is stable for any ω, in agreement with
Fig. 4; when instead μ = 28 we have B1 = 1.2, B2 =0.274, B3 = −0.098, B4 = −0.793, and thus (5.2) reads as
ω ∈ [0,√
B2) = [0, 0.523), that is, the system is stable just for
small values of ω, in agreement with Fig. 5.
What we can then conclude in this scenario is that in-
creasing μ has a destabilizing effect. In fact, fixing all the
other parameters as above and letting just μ vary, we find
that∂B1∂μ
= 0,∂B2∂μ
< 0,∂B3∂μ
> 0 and∂B4∂μ
> 0. Hence, the sta-
bility region decreases when μ increases (as B1 does not
vary with μ and the upper bound B2 decreases, while the
lower bounds B3 and B4 increase), confirming the highlighted
destabilizing role of the parameter μ.
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
Summarizing, for the above parameter configurations, the
interaction between the financial and real markets either
maintains the stability of the system, or it has a destabilizing
effect, through a flip bifurcation. We stress that such bifurca-
tion differs from the flip bifurcation detected in [26] in two
aspects. The first one is that, as already stressed in the Intro-
duction, those authors deal just with the isolated financial
market, while our flip bifurcation concerns the interaction
between the real and financial markets. The second aspect
is that after the flip bifurcation in [26] some numerical sim-
ulations we performed suggest that the system diverges and
thus such bifurcation would not lead to complex behaviors:
in our framework, the flip bifurcation is followed instead by
a stable period-two cycle which, increasing further the inter-
action parameter, undergoes a secondary double Neimark–
Sacker bifurcation, giving rise to quasiperiodic motions.
6.2. Unstable financial subsystem – stable real subsystem
In the framework we are going to consider, when iso-
lated, the financial subsystem is unstable and characterized
by quasiperiodic motions, while the real subsystem is sta-
ble. For not too large values of the parameter γ , when ω in-
creases, the fixed point becomes stable through a reversed
Neimark–Sacker bifurcation. According to the value of γ , that
fixed point can either persist until ω = 1 or can undergo a flip
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
12 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 7. The bifurcation diagrams with respect to ω ∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, for γ = 5 and the initial
conditions P0 = 12, x0 = −0.3 and Y0 = 61.
Fig. 8. The bifurcation diagrams with respect to ω ∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, for γ = 8 and the initial
conditions P0 = 12, x0 = −0.3 and Y0 = 61.
Fig. 9. The bifurcation diagrams with respect to ω ∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, for γ = 8.8 and the initial
conditions P0 = 12, x0 = −0.3 and Y0 = 61.
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bifurcation and then a secondary double Neimark–Sacker bi-
furcation. For even larger values of γ , we just obtain a reduc-
tion of the complexity of the system for suitable intermediate
values of ω, but the system is never stabilized.
More precisely, in Figs. 7–9 below, ω varies in [0, 1] and
the other parameters are: F∗ = 2, k = 0.1, α = 0.08, β =1, c = 1, a = 2.4, μ = 28, a1 = 3, a2 = 1, A = 12, b = 0.7,
and γ = 5 in Fig. 7, γ = 8 in Fig. 8, and γ = 8.8 in Fig. 9. In
Fig. 7 the fixed point becomes stable for ω � 0.2 and remains
stable until ω = 1. In Fig. 8, instead of remaining stable, it
undergoes a flip bifurcation for ω � 0.5 and then a secondary
double Neimark–Sacker bifurcation for ω � 0.96. In Fig. 9 the
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
fixed point is never stable: we just observe a reduction of the
complexity of the system for ω ∈ (0.2, 0.8), where we have
a stable period-two cycle. In more detail, in Figs. 7 (A)–9 (A)
we show the bifurcation diagrams with respect to ω ∈ [0,
1] for P, while in Figs. 7 (B)–9 (B) we draw the bifurcation
diagrams for Y; in Figs. 7 (C)–9 (C) we depict the Lyapunov
exponents when ω varies in [0, 1].
Let us now check whether the theoretical results in
Section 5 are in agreement with the numerical achievements
above.
First of all let us verify that, for the chosen parameter
sets, when ω = 0 just the real subsystem is stable, i.e., let
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 13
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 10. The bifurcation diagram with respect to ω ∈ [0, 1] for Y in (A), for μ = 20 and the initial conditions P0 = 8, x0 = 0.3 and Y0 = 75 for the blue (darker)
points, and P0 = 4, x0 = 0.25 and Y0 = 60 for the green (lighter) points; the (P, Y)-phase portrait showing the coexistence of the period-six cycle (in blue) with a
period-two cycle (in green) when ω = 0.7948 in (B), for μ = 20 and the initial conditions P0 = 5, x0 = 0.3 and Y0 = 65 for the period-two cycle and P0 = 4, x0 =0.3 and Y0 = 65 for the period-six cycle; the (P, Y)-phase portrait showing the coexistence of the period-six cycle (in blue) with two closed invariant curves (in
green) when ω = 0.826 in (C), for μ = 20 and the initial conditions P0 = 4, x0 = 0.3 and Y0 = 65 for the period-six cycle and P0 = 4, x0 = 0.1 and Y0 = 64 for the
invariant curves. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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us check that the last inequality in (5.6) is fulfilled, but not
the first chain of inequalities therein. A straightforward com-
putation shows that this is the case, as the first chain of in-
equalities reads as 0.116 < 1 < 0.968 and the last inequality
is 5 < 8.888 when γ = 5; the first chain of inequalities reads
again as 0.116 < 1 < 0.968 and the last inequality becomes
8 < 8.888 when γ = 8; finally, the first chain of inequalities
reads once again as 0.116 < 1 < 0.968 and the last inequal-
ity becomes 8.8 < 8.888 when γ = 8.8. As it is immediate
to see from such calculations, although for the considered
parameter values the isolated real market is stable, when γincreases it progressively approaches the instability region:
this may explain the detected growing destabilization of the
whole system when ω increases, in correspondence to larger
values of γ .
As concerns the stability conditions for ω varying in
[0, 1], when γ = 5 we have B1 = 3, B2 = 1.9004, B3 =0.0379, B4 = −2.3117, and thus (5.2) reads as ω ∈ (
√B3, 1] =
(0.194, 1], that is, the system is stable for large values of ω,
in agreement with Fig. 7; when γ = 8 we have B1 = 3, B2 =0.271, B3 = 0.035, B4 = −1.34, and thus (5.2) reads as ω ∈(√
B3,√
B2) = (0.189, 0.521), that is, the system is stable
just for intermediate values of ω, neither too small, nor too
large, in agreement with Fig. 8; finally, when γ = 8.8 we
have B1 = 3, B2 = 0.024, B3 = 0.038, B4 = −1.193, and thus,
since B2 < B3, (5.2) implies that there exists no ω for which
the system is stable, in agreement with Fig. 9.
What we can then conclude is that increasing γ has a
destabilizing effect. In fact, fixing all the other parameters as
above and letting just γ vary, we find that∂B1∂γ
= 0,∂B2∂γ
< 0,
∂B3∂γ
changes sign (in particular, it is negative for γ = 5 and
positive for γ = 8 and γ = 8.8), but B3 always lies in (0, 0.1)
for γ ∈ [5, 8.8] and thus it does not restrict the stability region
too much, and finally∂B4∂γ
> 0. Hence, the stability region de-
creases when γ increases (as B1 does not vary with γ and the
upper bound B2 decreases, while B3 is small and it does not
vary a lot, and the lower bound B4 increases), confirming the
highlighted destabilizing role of γ .
Summarizing, for the above parameter configurations, for
small values of ω, the instability of the financial market gets
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
transmitted to the real market. However, increasing ω de-
creases the complexity of the whole system. This effect may
either persist until ω = 1 or it may vanish for larger values of
ω, where we can find instead quasiperiodic motions, follow-
ing a secondary double Neimark–Sacker bifurcation of the
period-two cycle. Pursuing further the comparison with [26],
we stress that when the markets are isolated we are led to
consider instability regimes characterized by quasiperiodic
motions, like those in [26]. On the other hand, our analysis
additionally shows that increasing the interaction degree it is
possible to obtain a stabilization of the whole system, maybe
interrupted by a flip bifurcation.
We remark that in the existing literature on nonlinearities
it is possible to find some examples of stabilization phenom-
ena for interacting oscillating subsystems via an increase of
the interaction degree between the subsystems (see, for in-
stance, [45]). However, differently from our model, in those
examples the subsystems are symmetric and described by
the same functional relation: moreover, to the best of our
knowledge, they do not belong to the economic literature,
but they rather concern biological or physical systems.
6.3. Stable financial subsystem – unstable real subsystem
In this framework, when isolated, the financial sub-
system is stable, while the real subsystem is unsta-
ble. When ω increases, different possible behaviors
may arise. We depict some of them in Figs. 10–13 be-
low, where we have fixed the parameters as follows:
F∗ = 2, k = 0.1, α = 0.08, β = 0.5, c = 1, a = 2.4, γ =11.5, a1 = 3, a2 = 1, A = 12, b = 0.7, while we have μ = 20
in Fig. 10, μ = 28 in Fig. 11 and μ = 31 in Figs. 12 and 13.
More precisely, in Fig. 10 (A) we show the bifurcation di-
agram with respect to ω ∈ [0, 1] for Y, when μ = 20. For
such parameter configuration, when ω ∈ [0, 0.68] there is
just a stable period-two cycle. For ω � 0.68 a period-six cy-
cle emerges, which coexists with the period-two cycle un-
til ω � 0.823 (see the (P, Y)-phase portrait in Fig. 10 (B)
for ω = 0.7948). For ω � 0.823, through a double Neimark–
Sacker bifurcation of the period-two cycle, quasiperiodic mo-
tions emerge, which coexist with the period-six cycle (see
the (P, Y)-phase portrait in Fig. 10 (C) for ω = 0.826). For ω
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
14 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 11. The bifurcation diagrams with respect to ω ∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, for μ = 28 and the initial
conditions P0 = 10, x0 = 0.5 and Y0 = 61.
Fig. 12. The bifurcation diagrams with respect to ω ∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, for μ = 31 and the initial
conditions P0 = 10, x0 = 0.5 and Y0 = 61.
Fig. 13. The (P, Y)-phase portrait for μ = 31 and ω = 0.1 in (A), ω = 0.4 in (B), ω = 0.7 in (C) and ω = 0.9 in (D), respectively.
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� 0.830, the quasiperiodic motions end and only the period-
six cycle survives until ω � 0.88, where new quasiperiodic
motions emerge, lasting until ω = 1.
We recall that the coexistence of different kinds of attrac-
tors is also known as multistability. This feature may be con-
sidered as a source of richness for the framework under anal-
ysis because, other parameters being equal, i.e., under the
same institutional and economic conditions, it allows to ex-
plain different trajectories and evolutionary paths. The ini-
tial conditions, leading to the various attractors, represent
indeed a summary of the past history, which in the presence
of multistability phenomena does matter in determining the
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
evolution of the system. Such property, in the literature on
complex systems, is also called “path-dependence” [46].
In Figs. 11 (A) and 12 (A), we show the bifurcation dia-
grams with respect to ω ∈ [0, 1] for P, while in Figs. 11 (B) and
12 (B), we show the bifurcation diagrams with respect to ω∈ [0, 1] for Y; in Figs. 11 (C) and 12 (C), we represent the Lya-
punov exponents when ω varies in [0, 1]. Finally, in regard to
the parameter configuration considered in Fig. 12, we show
in Fig. 13 (A) the (P, Y)-phase portrait for ω = 0.1, where we
have a period-two cycle, in Fig. 13 (B) the (P, Y)-phase portrait
for ω = 0.4, where we have a period-eight cycle, in Fig. 13 (C)
the (P, Y)-phase portrait for ω = 0.7, where we have again a
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 15
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 14. The bifurcation diagrams with respect to ω ∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, for the initial conditions
P0 = 6, x0 = 0.25 and Y0 = 0.63.
Fig. 15. The (P, Y)-phase portrait for ω = 0 in (A), ω = 0.5 in (B), ω = 0.8 in (C) and ω = 1 in (D), respectively.
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period-two cycle, and in Fig. 13 (D) the (P, Y)-phase portrait
for ω = 0.9, where we have a chaotic attractor.
Straightforward computations, analogous to those per-
formed in Sections 6.1 and 6.2, show that the theoretical
results in Section 5 are in agreement with the numerical
achievements above.
Hence, for the selected parameter values, the system is
never stable. From the pictures we also notice that increas-
ing μ has a further destabilizing effect, as the complexity of
the system grows when μ moves from 20 to 31. In particular,
in Figs. 11 and 12 we highlight the presence of the so-called
“bubbles” (see [47,48]) and, as shown in Figs. 12 and 13, for
μ = 31 we have even chaotic dynamics for values of ω close
to 1.
Summarizing, in the case of isolated stable financial mar-
ket and unstable real market, for the above parameter config-
urations we have been not able to find stabilizing values for
ω, even if intermediate values of ω may lead to a reduction
of the complexity of the system. In this sense we could argue
that the instability of the real market seems to have stronger
destabilizing effects than the instability of the financial mar-
ket: in fact, the former gets transmitted and possibly ampli-
fied by the connection with the financial market, while, as
we saw in Section 6.2, the latter gets dampened and possibly
eliminated by the connection with the real market.
6.4. Unstable financial and real subsystems
In this last framework, when isolated, the financial
and the real subsystems are unstable. In particular, when
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
ω = 0 both of them may be chaotic. When ω increases,
we do not find a complete stabilization of the system,
but we observe some periodicity windows in Figs. 14
and 15 below, where we have fixed the parameters as fol-
lows: F∗ = 2, k = 0.1, α = 0.08, β = 1, c = 1, a = 2.4, μ =28, γ = 20, a1 = 3, a2 = 1, A = 12, b = 0.7. Notice that
these are the same parameter values considered in the
second scenario, except for a larger value of γ . We already
observed in Section 6.2 that increasing γ has a destabilizing
effect for the above parameter configuration and this is
confirmed by Figs. 14 and 15. More precisely, in Fig. 14 (A)
and (B) we show the bifurcation diagrams with respect to
ω ∈ [0, 1] for P and Y, respectively; in Fig. 14 (C) we show
the Lyapunov exponent when ω varies in [0, 1]. In Fig. 15
(A)–(D) we depict, in the (P, Y)-phase plane, the chaotic
regime when ω = 0, a period-eleven cycle when ω = 0.5, a
chaotic attractor when ω = 0.8 and a period-fourteen cycle
when ω = 1, respectively.
Again, it is trivial to check that the theoretical results in
Section 5 are in agreement with the numerical achievements
above, so that, for the above parameter configuration (as well
as for many other ones we investigated), when both the iso-
lated financial and real markets are unstable, we may have a
reduction of the complexity of the whole system until peri-
odic motions, but not a complete stabilization.
6.5. The role of an increasing belief bias
We already stressed at the end of Section 5 the desta-
bilizing role of the belief bias for the financial side of the
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
16 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 16. The bifurcation diagrams for P with respect to a ∈ [2, 7] with μ = 5 in (A) and with respect to a ∈ [2, 3.9] with μ = 28 in (B), respectively, both obtained
for the initial conditions P0 = 10, x0 = 0.25 and Y0 = 50.
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economy when the two markets are isolated (see (5.5)
and the subsequent comments). We now investigate which
are the effects of an increasing bias on the stability of
the whole system when the markets are interconnected.
Since the results we got are uniform enough across the
various scenarios considered so far, we illustrate our find-
ings just for the scenario in Section 6.1, in which, when
isolated, both the financial and the real subsystems are
stable. Indeed, in Fig. 16 below we have fixed the pa-
rameters as follows: F∗ = 5, k = 0.25, α = 0.08, β = 1, c =1, γ = 3.5, a1 = 2, a2 = 4, A = 5, b = 0.7, ω = 0.9, and μ =5 in Fig. 16 (A), where we show the bifurcation diagram for P
with respect to a ∈ [2, 7], while μ = 28 in Fig. 16 (B), where
we depict the bifurcation diagram for P with respect to a ∈[2, 3.9]. In Fig. 16 (A) the steady state gets destabilized for a �5.9 through a Neimark–Sacker bifurcation; in Fig. 16 (B) the
steady state is instead unstable for values of a close to the ex-
treme values of the considered interval, while it is stable for
intermediate values of a.
Hence, even if the destabilizing role of the bias is clear
when markets are isolated, Fig. 16 suggests that its role
becomes more ambiguous when the markets are intercon-
nected. Indeed, increasing a may have either a destabilizing
or a stabilizing role, according to the value of the other pa-
rameters. However, taking into account also the conclusions
we got for the scenarios in Sections 6.2–6.4 (we do not report
here for the sake of brevity), it seems that increasing a has
generally a destabilizing effect, as usually we do not reach a
complete stabilization of the system, or we achieve it just in
small intervals for the belief bias.
7. Interpretation of the results
It is commonly recognized that agents in making their
choices in the real market are influenced also by the condi-
tions in the asset market, and vice versa. Hence, we wish to
analyze the role of an increase in the intensity of the interac-
tion between the two sectors, starting from a framework in
which both isolated markets are stable, in order to show the
arising economic fluctuations, i.e., booms and busts. To such
aim, we will fix the parameters like in the destabilizing sce-
nario considered in Section 6.1, with μ = 28.5 and ω = 0.99,
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
and we will interpret the dynamics of the time series for the
main economic variables of our dynamical system, for t ∈[1030, 1050] in Fig. 17 and for t ∈ [1045, 1075] in Fig. 18. The
variables we will consider are output Y, stock price P and the
profit differential �π = πopt − πpes from (4.10). Moreover,
in analogy with the Michigan consumer confidence indica-
tor considered in Section 2, we define the long-period opti-
mism index as X10,t+1 = xt +xt−1+···+xt−910 . Such variable mea-
sures the average share of optimistic agents in the last ten
periods and its behavior is meant to indicate the dynamics of
animal spirits in the medium run, avoiding considering sud-
den and transient fluctuations, which may make the overall
dynamics more difficult to interpret.
Let us start our analysis, by considering a boom, i.e., a
phase of persistent economic growth. In particular, we focus
our attention on the time series in Fig. 17, where we put in ev-
idence t ′ = 1036. Starting from t = t ′ we observe a predom-
inance of long-period optimism among the agents in the fi-
nancial market, as X10 is positive, and increasing output in the
real sector. Since the excess demand in the stock market, i.e.,
α{[(1 − ω)F∗ + ωkYt ] − Pt + axt} is positive, recalling (4.7), it
follows that the stock price grows. By (4.14) this implies that
output increases due to the wealth effect, as the output dif-
ferential is proportional to aggregate demand. This in turns
makes the stock price increase because, recalling (4.1) and
(4.2), the perceived reference values depend on income and
the same holds for the demands of optimists and pessimists
in (4.3) and (4.4). By (4.5) this makes the stock price increase
further and consequently also the relative profits of optimists
grow by (4.10). The latter phenomenon maintains the over-
all optimism level, so that the share of optimists increases,
together with the long-period optimism index. Due to the
presence of our sigmoidal function in the real sector, such
a virtuous cycle persists until an upper turning point, that in
Figs. 17 and 18 corresponds to t ′′ = 1050, after which a bust
phase starts. Such turning point exists because, due to physi-
cal physical and material constraints, the system is not able to
maintain further positive output variations in the short run,
as capital and labor forces are fixed in the short run. See also
[49], where the authors identify the combination of bound-
edly rational managerial behavior with rigidities and delays
in capacity adjustments as crucial for the occurrence and the
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 17
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
Fig. 17. The time series for Y (in blue), P (in red), �π (in green) and X10 (in black) for the boom phase, corresponding to the time periods t ∈ [1030, 1050], in
which we put in evidence t ′ = 1036. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 18. The time series for Y (in blue), P (in red), �π (in green) and X10 (in black) for the bust phase, corresponding to the time periods t ∈ [1045, 1075], in which
we put in evidence t ′′ = 1050. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
nature of boom and bust cycles, and [50] where ample ev-
idence from experiments and case studies is provided that
boundedly rational behavior are quite persistent in boom and
bust cycles. We recall that the role of capacity constraints
is also discussed in the literature on Hicksian business cycle
model [51].
Hence, after the turning point income increases less and
less, and the same holds for the perceived reference values,
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
so that the optimists’ profits are reduced and their share in
the population decreases. Thus, also the demand in the finan-
cial market decreases and the stock price starts falling. Due
to the wealth effect expressed in (4.14), this makes the out-
put differential become negative because aggregate demand
decreases and this in turns drives down the perceived refer-
ence values, and consequently, also the excess demand in the
financial market diminishes, making the stock price go down.
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
18 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
t70
075
080
085
090
095
010
00
Yt 20253035404550556065
t70
075
080
085
090
095
010
00
Pt
24681012141618
t70
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080
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095
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00
xt
-1
-0.8
-0.6
-0.4
-0.20
0.2
0.4
0.6
0.81
t70
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080
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090
095
010
00
X10,t
-0.2
-0.1
5
-0.1
-0.0
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0.050.
1
0.150.
2
0.25
Y20
25
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35
40
45
50
55
60
65
05
10
15
20
25
30
P2
46
810
12
14
16
18
05
10
15
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x-1
-0.8
-0.6
-0.4
-0.2
00.2
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0.6
0.8
105
10
15
20
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X10
-0.2
-0.1
5-0
.1-0
.05
00
.05
0.1
0.1
50
.20
.25
02468
10
12
14
16
18
20
Lag
02
46
810
1214
1618
20
AutocorrelationofY -0.20
0.2
0.4
0.6
0.81
Lag
02
46
810
1214
1618
20
AutocorrelationofP -0.20
0.2
0.4
0.6
0.81
Lag
02
46
810
1214
1618
20
Autocorrelationofx -0.6
-0.4
-0.20
0.2
0.4
0.6
0.81
Lag
02
46
810
1214
1618
20
AutocorrelationofX10
-0.20
0.2
0.4
0.6
0.81
Sta
ndar
dN
orm
alQ
uant
iles
-3-2
-10
12
3
QuantilesofY 2025303540455055606570
Sta
ndar
dN
orm
alQ
uant
iles
-3-2
-10
12
3
QuantilesofP
2468101214161820
Sta
ndar
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orm
alQ
uant
iles
-3-2
-10
12
3
Quantilesofx -1.5-1
-0.50
0.51
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Sta
ndar
dN
orm
alQ
uant
iles
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-10
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-0.2
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-0.2
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0.050.
1
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2
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Fig
.19
.T
he
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ese
rie
s(i
nth
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rst
row
),th
eh
isto
gra
ms
(in
the
seco
nd
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),th
ea
uto
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ela
tio
ns
(in
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dro
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an
dth
eQ
–Q
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plo
ts(i
nth
efo
urt
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for
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10
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ly,c
orr
esp
on
din
gto
the
tim
ep
eri
od
s
t∈
[70
1,1
00
0].
Please cite this article as: A. Naimzada, M. Pireddu, Real and financial interacting markets: A behavioral macro-model, Chaos,
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 19
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
t910 920 930 940 950 960 970 980 990 1000
X10
,t
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Yt
25
30
35
40
45
50
55
60
65
Fig. 20. The time series for output Y (in purple-red, with corresponding scaling on the right vertical axis) and the long-period optimism index X10 (in blue, with
corresponding scaling on the left vertical axis) for the last nearly 100 periods considered in Fig. 19. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
1036
1037
1038
1039
1040
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1042
1043
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1045
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1076
1077
1078
1079
1080
t t t t
Hence, by (4.10) the relative pessimists’ profits increase and
thus also their share, so that the long-period optimism in-
dex decreases. Moreover, by the wealth effect, falling stock
prices make the output differential become more and more
negative. Such a vicious cycle corresponds to a bust phase
which, again due to the presence of our sigmoidal function
in the real sector, persists until a lower turning point, that in
Fig. 18 corresponds to t ′′′ = 1070 (non-explicitly indicated in
that picture) after which a new growth phase starts. Such
turning point exists because, due to physical and material
constraints, the system is not able to maintain further neg-
ative output variations in the short run, as capital and la-
bor forces are fixed in the short run. Hence, after the turn-
ing point income decreases less and less, and the same holds
for the perceived reference values, so that the pessimists’
profits are reduced and their share in the population de-
creases. Thus, the demand in the financial market increases
and the stock price starts raising. Due to the wealth effect,
this makes the output differential become positive because
aggregate demand increases. This in turns raises the per-
ceived reference values, and consequently, also the excess
demand in the financial market grows, making the stock
price goes up. Hence, by (4.10) the relative optimists’ prof-
its increase and thus also their share, so that the long-period
optimism index increases. In such way a new boom gets
triggered.
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
We stress that the whole mechanism works for suffi-
ciently large values of the interaction parameter ω. Indeed,
for ω = 0 no relationships between the real and the financial
sectors exist: namely, on the one hand, by (4.14) no wealth
effect is present and, on the other hand, by (4.1) and (4.2) the
reference values are exogenous and do not depend on the real
market performance.
8. Introducing stochastic shocks
Following the approach in [36], we add a noise term to
each of the demand components in (4.3) and (4.4). Such
terms are meant to reflect a certain within-group hetero-
geneity, describing the accidental fluctuations of the com-
position of the many individual digressions from the simple
rules they are supposed to follow. The heterogeneity is rep-
resented by two independent and normally distributed ran-
dom variables εoptt and εpes
t , for the optimists and pessimists,
respectively. Combining the deterministic and stochastic ele-
ments, the optimists’ demand reads as
doptt = α(F opt
t − Pt ) + εoptt , εopt
t ∼ N(0, (σ opt)2) (8.1)
and, similarly, pessimists’ demand is given by
dpes = α(F pes − Pt ) + εpes, εpes ∼ N(0, (σ pes)2) (8.2)
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
20 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
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1107
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1109
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1111
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where σopt and σpes are two positive parameters rep-
resenting the standard deviations of the normal random
variables.
Going through the same steps in Section 4, we find that
the price equation in (4.7) becomes
Pt+1 = Pt + αμ{[(1 − ω)F∗ + ωkYt ] − Pt + axt} + εt , (8.3)
where εt ∼ N(0, σ 2t ), with σ 2
t = μ2((noptt σ opt)2 +
(npest σ pes)2), and consequently a stochastic term will enter
also the dynamic equation (4.11) governing the switching
mechanism, due to the presence of the profits.
In the next simulations, reported in Fig. 19, we con-
sider the following parameter set: F∗ = 5, k = 0.25, α =0.08, β = 1, c = 1, a = 2, γ = 3.5, a1 = 2, a2 = 4, A =5, b = 0.7,μ = 21, ω = 0.9 and σopt = 0.1 = σpes, where we
denote by σopt and σpes the standard deviation for optimists
and pessimists, respectively. For simplicity, we assumed that
σopt and σpes coincide. Hence, we add the stochastic noises
and observe which are the generated dynamics for variables
Y, P, x, X10 in the most interesting of the previously consid-
ered scenarios, i.e., that in Section 6.1. More precisely, for
Y, P, x, X10 in Fig. 19 we plot the time series (in the first row),
the histograms (in the second row), the autocorrelations (in
the third row) and the Q–Q test plots (in the fourth row). The
initial conditions are Y0 = 0.2, P0 = 0.3, x0 = 0.25, x−1 =0.1, x−2 = · · · = x−9 = 0, and we report 300 values after a
transient of 700 iterations.
The values corresponding to Fig. 19 for the mean, stan-
dard deviation (SD), skewness, kurtosis and Jarque–Bera test
for normality (abbreviated in J–B), with its precise value,
can be found in Table 8.4. We stress that J–B = 0 means
normality, while J–B = 1 means non-normality. Hence, we
find that Y (as well as P) is not normally distributed. Such
conclusion is in agreement with the histograms in Fig. 19,
showing that Y (and P, too) has an higher kurtosis than the
normal distribution (which is equal to 3) and fatter tails. A
further confirmation for those findings is given by Q–Q test
plots in Fig. 19, which shows that Y (and P) are not nor-
mally distributed and that their distributions are fat-tailed.
We recall that the Q–Q plots (Quantile–Quantile plots) plot
the quantiles of one distribution against those of the nor-
mal, contrasting the two cumulative distribution functions.
If the variable under analysis is normally distributed, then
its plot lies on the 45-degree line, which corresponds to
the normal distribution. Moreover, if the considered vari-
able is not normally distributed and its tails lay below
(above) the 45-degree line, then its distribution is fat-tailed
(thin-tailed).
The non-normality of the distribution, and in particular
the presence of fat tails, implies that in our model there are
larger movements in output than is compatible with the nor-
mal distribution, and thus, as desired, booms and busts are
more likely to happen.
Moreover, looking at the time series in Fig. 19, we observe
strong cyclical movements for Y and P, implying that Y and
P are highly autocorrelated, as confirmed by the autocorrela-
tion plots reported in the same figure. We stress that in our
model output Y is the variable to be compared with the out-
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
put gap considered in the stylized facts in Section 2.
Mean SD Skewness Kurtosis J − B J − Bvalue
Y 48.1370 7.3197 −0.7195 3.8505 1 34.9293
P 11.3703 2.5983 −0.8192 4.5586 1 63.9219
x 0.0059 0.4208 −0.0061 2.6425 0 1.5990
X10 0.0039 0.0683 −0.0295 2.7397 0 0.8904
(8.4)
Hence, in analogy with the first two stylized facts in
Section 2, we find a strong autocorrelation and a non-normal
distribution for output.
As regards the third stylized fact described in Section 2,
for the same parameter configuration used for Fig. 19, we
show in Fig. 20 the high correlation between the long-period
optimism index X10 (introduced in Section 7) and the move-
ments of output Y. Since X10 plays the role of the Michigan
sentiment index in Fig. 3 and Y that of the output gap therein,
we then find that our model is able to reproduce the third
stylized fact, too.
We stress that we performed the empirical verification
of our model in its stochastic version as adding shocks ac-
counts for taking into account all those aspects that our sim-
ple deterministic model cannot explicitly consider. Of course,
it would be interesting to analyze the consequences of adding
further stochastic noises to the stock price and to the real side
equations, as well as to the switching mechanism. Nonethe-
less we showed that, even adding stochastic terms just to the
demand functions, we have been able to replicate the desired
stylized facts.
9. Conclusion and future directions
In this paper we proposed a model belonging both to the
strand of literature on the interactions between real and fi-
nancial markets, e.g. [24], as well as to the literature about
heterogeneous fundamentalists, as for instance [26]. In fact,
in the model we presented the real economy, described via a
Keynesian good market approach, interacts with a stock mar-
ket with heterogeneous fundamentalists. Agents may switch
between optimism and pessimism according to which be-
havior is more profitable.
More precisely, our main difference with respect to [26] is
that we also considered the real sector of the economy, while
with respect to [24] is that we introduced the interaction de-
gree parameter and the switching mechanism, in order to
describe the changes in the share of agents in the financial
market. To the best of our knowledge, this was the first con-
tribution considering both real and financial interacting mar-
kets and an evolutionary selection process for the popula-
tion for which an analytical study is performed. Indeed, we
employed analytical and numerical tools to investigate the
role of the parameter describing the degree of interaction, in
order to detect the mechanisms and the channels through
which the stability of the isolated real and financial sectors
leads to instability for the two interacting markets. The main
contribution of the present paper to the existing literature
lies in fact in focusing on the role of real and financial feed-
back mechanisms, not only in relation to the dynamics and
stability of a single market, but for those of the economy as
a whole. In order to perform our analysis, we introduced the
cial interacting markets: A behavioral macro-model, Chaos,
.05.007
A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx 21
ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]
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“interaction degree approach”, which allowed us to study the
complete three-dimensional system by decomposing it into
two subsystems, i.e., the isolated financial and real markets,
easier to analyze, that are then interconnected through the
“interaction parameter”. We classified the possible scenarios
according to the stability/instability of the isolated financial
and real markets: in this way we have been led to analyze
four frameworks. For each of those we considered different
possible parameter configurations and we showed, both ana-
lytically and numerically, which are the effects of increasing
the degree of interaction between the two markets. The con-
clusions we got are not univocal: indeed, depending on the
value of the other parameters, an increase in the interaction
parameter may either have a stabilizing or a destabilizing ef-
fect, but also other phenomena are possible.
We concluded our theoretical analysis by investigating
which are the effects of an increasing belief bias. Although it
is clearly destabilizing when markets are isolated, we found
that its role becomes more ambiguous when the markets are
interconnected. Indeed, increasing the bias may have either
a stabilizing or a destabilizing effect, according to the value
of the other parameters. However, our numerical simulations
suggested that increasing the bias has generally a destabiliz-
ing effect, as usually we did not reach a complete stabiliza-
tion of the system, or we achieved it just in small intervals
for the bias.
We analyzed all the possible scenarios in order to show
the variety of dynamics generated by the model, compar-
ing the numerical simulations and the bifurcation diagrams
with the analytical results. Nonetheless, in order to discuss
our model, we chose to illustrate and economically interpret
the stronger result we obtained, i.e., that increasing the in-
teraction degree, which measures how much the speculators
in the financial market are influenced by the behavior of the
real market and vice versa, may be destabilizing even when
the two isolated sectors are stable. More precisely, we ex-
plained and showed how an increasing value for the interac-
tion parameter, together with the switching mechanism for
agents in the financial market and the presence of upper and
lower bounds imposed by the sigmoidal nonlinearity in our
real sector, describing the output capacity constraints, can
produce the alternation of growth periods (booms) and re-
cessions (busts), characterized by income contraction in the
real sector and by the predominance of waves of pessimism
in the financial market.
Even for the empirical verification of our model we con-
sidered the destabilizing scenario, in which the two sepa-
rately stable sectors become unstable when coupled. In par-
ticular, in order to describe the accidental fluctuations of
the composition of the many individual digressions from the
simple rules they are supposed to follow, according to [36]
we added a noise term to the demand functions of optimists
and pessimists. In such framework, following the approach
in [2,37–39], we showed that our model is able to reproduce
three crucial stylized facts about cyclical movements of out-
put and empirical foundation of the concept of animal spirit:
high autocorrelation for output causing strong cyclical move-
ments, a non-normal distribution for output, characterized
by a high kurtosis and fat tails, and finally a strong correlation
between output and a long-period optimism index, which
describes the waves of optimism and pessimism.
Please cite this article as: A. Naimzada, M. Pireddu, Real and finan
Solitons and Fractals (2015), http://dx.doi.org/10.1016/j.chaos.2015
Future research should focus, for instance, on extending
the model so as to include money and other financial assets.
A different possible extension could concern the introduc-
tion in our model of chartists, in order to check whether they
have a destabilizing effect also in the present framework, or
of unbiased fundamentalists, as already done in [26], in view
of comparing the results in the different contexts. Finally,
we are going to investigate the consequences of some pol-
icy rules, such as the target and the countercyclical adjusting
policies (see for instance [52]), analyzing how their introduc-
tion affects the dynamics of our model.
Acknowledgments
The authors thank Dr. Fausto Cavalli for his software assis-
tance during the preparation of the statistic section and the
anonymous Reviewers for their helpful and valuable com-
ments.
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