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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.

Page 3

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.

13

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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

Page 4

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

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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

Page 6

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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

Page 7

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

Page 8

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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

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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

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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

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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

Page 12

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

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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

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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

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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,

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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,

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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,

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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

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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

Page 20

18 A. Naimzada, M. Pireddu / Chaos, Solitons and Fractals xxx (2015) xxx–xxx

ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]

t70

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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

Page 21

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

1041

1042

1043

1044

1045

1046

1047

1048

1049

1050

1051

1052

1053

1054

1055

1056

1057

1058

1059

1060

1061

1062

1063

1064

1065

1066

1067

1068

1069

1070

1071

1072

1073

1074

1075

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

Page 22

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

1106

1107

1108

1109

1110

1111

1112

1113

1114

1115

1116

1117

1118

1119

1120

1121

1122

1123

1124

1125

1126

1127

1128

1129

1130

1131

1132

1133

1134

1135

1136

1137

1138

1139

1140

1141

1142

1143

1144

1145

1146

1147

1148

1149

1150

1151

1152

1153

1154

1155

1156

1157

1158

1159

1160

1161

1162

1163

1164

1165

1166

1167

1168

1169

1170

1171

1172

1173

1174

1175

1176

1177

1178

1179

1180

1181

1182

1183

1184

1185

1186

1187

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

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ARTICLE IN PRESSJID: CHAOS [m3Gdc;May 21, 2015;15:42]

1188

1189

1190

1191

1192

1193

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1200

1201

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1207

1208

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1220

1221

1222

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1227

1228

1229

1230

1231

1232

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1234

1235

1236

1237

1238

1239

1240

1241

1242

1243

1244

1245

1246

1247

1248

1249

1250

1251

1252

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1256

1257

1258

1259

1260

1261

1262

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1264

1265

12661267126812691270127112721273127412751276127712781279128012811282128312841285128612871288128912901291129212931294129512961297129812991300130113021303130413051306130713081309131013111312131313141315131613171318

“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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