Our reference: CHAOS7667

(1)

Journal: CHAOS

Article Number: 7667

Please e-mail your responses and any corrections to:

E-mail:

correctionsaptara@elsevier.com

Dear Author,

Please check your proof carefully and mark all corrections at the appropriate place in the proof (e.g., by using on-screen

annotation in the PDF file) or compile them in a separate list. Note: if you opt to annotate the file with software other

than Adobe Reader then please also highlight the appropriate place in the PDF file. To ensure fast publication of your

paper please return your corrections within 48 hours.

Your article is registered as a regular item and is being processed for inclusion in a regular issue of the journal. If

this is NOT correct and your article belongs to a Special Issue/Collection please contact

s.alagarsamy@elsevier.com

immediately prior to returning your corrections.

For correction or revision of any artwork, please consult

http://www.elsevier.com/artworkinstructions

Any queries or remarks that have arisen during the processing of your manuscript are listed below and highlighted by

flags in the proof. Click on the ‘Q’ link to go to the location in the proof.

Location

Query / Remark:

click on the Q link to go

in article

Please insert your reply or correction at the corresponding line in the proof

Q1

AU: Please confirm that given names and surnames have been identified correctly.

Q2

AU: Please verify the presentation of both the affiliations and corresponding author.

Q3

AU: Please provide better quality artwork for all figures.

Q4

AU: Figs. [3, 6, 10, 17, 18 and 20] have been submitted as color images; however, the captions

have been reworded to ensure that they are meaningful when your article is reproduced both in

color and in black and white. Please check and correct if necessary.

Please check this box or indicate your approval if

you have no corrections to make to the PDF file

(2)

ARTICLE IN PRESS

JID: 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. •

(3)

Contents lists available atScienceDirect

Chaos, Solitons and Fractals

Nonlinear 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_Dep_ar_t_ment_{of Economics, Management and Statistics, University of Milano-Bicocca, U6 Building, Piazza dell’Ateneo Nuovo 1,} 20126 Milan, Italy

b_Dep_ar_t_ment_{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.

1. Introduction 1

Both empirical and theoretical arguments show that

in-Q2 2

stabilities are a common feature of all markets: the prod-3

uct markets, the labor market, and the ﬁnancial markets. 4

As recalled in[1], over the last twenty years many stock 5

market models have been proposed in order to study the 6

∗ _{Corresponding author. Tel.: +39 026 4485 7}₆₇_{; fax: +39 026 4485 705.} E-mail addresses: ahmad.naimzada@unimib.it (A. Naimzada), marina.pireddu@unimib.it,marina.pireddu@gmail.com(M. Pireddu).

1_{Tel.: +39 026 4485 813; fax: +39 026 4483 085.}

dynamics of ﬁnancial markets (see[2,3]). According to such 7 models, even in the absence of stochastic shocks, the inter- 8 action between boundedly rational, heterogeneous specula- 9 tors accounts for the dynamics of ﬁnancial markets. Those 10 models, when endowed with stochastic shocks, are able to 11 replicate some important phenomena, such as bubbles and 12 crashes, excess volatility and volatility clustering (see, for in- 13 stance,[4–12]).2_{Indeed, differently from DSGE models, be-} ₁₄

2_{In the past decades, ﬁnancial markets have been analyzed also via}

ap-proaches based on Heston model, e.g. in[13–17]. http://dx.doi.org/10.1016/j.chaos.2015.05.007

(4)

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

ARTICLE IN PRESS

havioral models are able to better reproduce, for instance, the 15

high kurtosis, the presence of fat tails, the strong autocorrela-16

tion of real data. However, in this kind of models authors have 17

restricted their attention to the representation and the dy-18

namics of ﬁnancial markets only and the existing feedbacks 19

between the real and ﬁnancial sides of the economy are often 20

completely neglected. 21

The ﬁnancial crisis of 2008 created new research issues 22

for economists. Recently, a growing literature has investi-23

gated how speculative phenomena in ﬁnancial markets gets 24

transmitted to the real economy and whether or not real 25

market developments feed back on the ﬁnancial sector. One 26

simple way to answer such questions consists in integrating 27

the standard New Keynesian Macroeconomic (NKM) model 28

with the tools of the Agent-Based Computational (ACE) ﬁ-29

nance literature. We stress that, in the present context, 30

we will use the expression “Agent-Based” in a loose sense, 31

meaning frameworks with hetereogeneous and/or bound-32

edly rational agents. 33

For instance, Scheffknecht and Geiger[18]present a ﬁ-34

nancial market model with leverage-constrained heteroge-35

neous agents integrated with a New Keynesian standard 36

model; all agents are assumed to be boundedly rational. 37

Those authors show that a systematic reaction by the central 38

bank to ﬁnancial market developments dampens macroeco-39

nomic volatility. Moreover, Lengnick and Wohltmann[19], 40

Kontonikas and Ioannidis[20], Kontonikas and Montagnoli 41

[21]and Bask[22]consider New Keynesian models intercon-42

nected with ﬁnancial markets models. The results are en-43

dogenous developments of business cycles and stock price 44

bubbles. 45

Contributions in the macroeconomic literature on the in-46

teraction between the real and the ﬁnancial sides that do 47

not build upon NKM for the description of the real sector 48

have been proposed by Charpe et al.[23], Westerhoff[24]

49

andNaimzada and Pireddu[1]. In particular, the latter two

50

works employ a classical Keynesian demand function only to 51

represent the real sector. The advantage of this approach is 52

simplicity. Models are typically of small scale, so that analyt-53

ical solutions are tractable. 54

More precisely, Charpe et al.[23]propose an integrated 55

macro-model, using a Tobin-like portfolio approach, and con-56

sider the interaction among heterogeneous agents in the ﬁ-57

nancial market. They ﬁnd that unorthodox ﬁscal and mone-58

tary policies are able to stabilize unstable macroeconomies. 59

Westerhoff[24]describes the real economy via a Keynesian 60

good market approach, while the set-up for the stock mar-61

ket includes heterogeneous speculators, i.e., fundamentalists 62

and chartists. In[24]it is shown that interactions between 63

the real sector and the stock market appear to be destabiliz-64

ing, giving rise to chaotic dynamics through bifurcations. Fi-65

nally, Naimzada and Pireddu[1]consider a framework sim-66

ilar to that in[24]but, in order to analyze the interactions 67

between product and ﬁnancial markets, a parameter repre-68

senting the degree of interaction is introduced. With the aid 69

of analytical and numerical tools it is shown that, under the 70

assumption of equilibrium for the stock market, an increas-71

ing degree of interaction between markets tends to locally 72

stabilize the system. 73

One important aspect to be considered when integrating 74

the real and ﬁnancial sectors is the identiﬁcation of the chan-75

nels through which the two sectors influence each other. Sev- 76 eral channels have been proposed, but the literature has not 77 yet agreed upon which channels are the most crucial ones. 78 Possible assumptions for describing the channels through 79 which the financial market influences the real sector are the 80 wealth effects[1,21,22,24], a collateral-based cost effect[19] 81 or a balance-sheet based leverage targeting effect[18]. Ex- 82 amples for channels going in the opposite direction, from the 83 real sector towards the financial market, are a misperception 84 effect[1,19,24], or a mixture of a misperception effect with 85 negative dependence on the real interest rate[20–22]. 86 Our paper belongs both to the strand of literature on the 87 interactions between real and financial markets, as well as to 88 the literature on heterogeneous fundamentalists (see, for in- 89 stance,[25–33]). In fact, we here propose a model in which 90 the real economy, described via a Keynesian good market 91 approach, and the stock market, with heterogeneous funda- 92 mentalists, interact. The two papers that bear a stronger re- 93 semblance to our framework are[24,26]. More precisely,sim- 94

ilarto[26], we assume that the financial side of the econ- 95 omy is represented by a market where traders behave in two 96 different ways: optimism and pessimism. Optimists (pes- 97 simists) systematically overestimate (underestimate) the ref- 98 erence value due to a belief bias. Moreover, like in[26]agents 99 may switch between optimism and pessimism according to 100 which behavior is more profitable. On the other hand, in[26] 101 only the financial sector is considered and the connection 102 with the good market is missing. When comparing our set- 103 ting to the one in[24], we stress that,similarto what done 104 in that paper, we assume the real economy to be represented 105 by an income-expenditure model in which expenditures de- 106 pend also on the dynamics of the stock market price. On the 107 other hand, in[24]the real market subsystem is described 108 by a stable linear relation, while the financial sector is repre- 109 sented by a cubic functional relation, that is, by a nonlinear 110 relation. In that way, the oscillating behavior is generated by 111 the financial subsystem only. In our paper we present instead 112 a model in which the oscillating behavior is generated also by 113 the real subsystem. Indeed, our stock price adjustment mech- 114 anism is linear, but not always stable, while the real subsys- 115 tem is described by a nonlinear relation. To be more precise, 116 the nonlinearity of the real subsystem is due to the nonlin- 117 earity of the adjustment mechanism of the good market with 118 respect to the excess demand. In particular, the sigmoidal 119 nonlinearity we deal with has been recently considered in 120

[34]. Another difference with respect to[24]is the way we 121 represent and analyze the interaction between the two mar- 122 kets. We assume in fact that economic agents operating in 123 the ﬁnancial sector base their decisions on a weighted av- 124 erage between an exogenous fundamental value and an en- 125 dogenous fundamental value depending on the current real- 126 ization of income, while in the real market we assume that 127 private expenditures depend also, with a given weight, on 128 the stock market price. In particular, in our model the pa- 129 rameter describing that weight represents also the degree 130 of interaction between the two markets. The extreme values 131 of the weighting parameter correspond to the two cases an- 132 alyzed in[24], i.e., the isolated market framework and the 133 interacting market scenario. Finally, we remark that in[24] 134 no heterogeneous fundamentalists and no switching mecha- 135

nism are considered. 136

(5)

Hence, summarizing, our main difference with respect to 137

[26]is that we also consider the real sector of the economy, 138

while with respect to[24]is that we introduce the interaction 139

degree parameter and the switching mechanism, in order to 140

describe the changes in the share of agents in the ﬁnancial 141

market. 142

We stress that, to the best of our knowledge, the present 143

paper is the ﬁrst contribution considering a model with both 144

real and ﬁnancial interacting markets and an evolutionary 145

selection process for the population for which an analytical 146

study is performed. Indeed, in the existing literature, just few 147

papers[18,19,35]deal both with interacting sectors and an 148

evolutionary approach, and all of them propose just a numer-149

ical analysis of the framework under consideration. 150

The present work represents the third step of a line of re-151

search started with[1]and[34]. Indeed, in[34]we analyzed 152

the effects of the introduction of a nonlinearity in the adjust-153

ment mechanism of income with respect to the excess de-154

mand in the standard Keynesian income-expenditure model, 155

showing that it was able to generate complex dynamics and 156

multistability phenomena. However, in[34]only the real sec-157

tor was considered and the relation with the stock market 158

was missing. In[1]we then dealt with the same Keyne-159

sian model in[34]to which we added the connection, mod-160

eled via the interaction degree parameter aforementioned, 161

with the ﬁnancial subsystem, represented by an equilibrium 162

stock market with heterogeneous speculators, i.e., chartists 163

and fundamentalists. The equilibrium assumption, equiva-164

lent to the hypothesis that the stock market speed of adjust-165

ment tends to inﬁnity, was motivated by the functioning of ﬁ-166

nancial markets and allowed to reduce our two-dimensional 167

model to a one-dimensional system. We here add some fur-168

ther elements of interest to our achievements in[1]and[34]. 169

Indeed, starting from the Keynesian model in[34],similarto 170

what done in[1], we still consider a framework with real and 171

ﬁnancial sectors connected via the interaction degree param-172

eter, to which we add three new ingredients: the stock mar-173

ket is no more assumed to be always in equilibrium and thus 174

we need to analyze one more equation, describing the stock 175

price dynamics; instead of dealing with chartists and funda-176

mentalists, as already stressed, we consider the case of het-177

erogeneous (optimistic and pessimistic) fundamentalists; ﬁ-178

nally, we here allow agents to switch between optimism and 179

pessimism, according to which behavior is more proﬁtable, 180

and this leads us to add a further equation to our model, 181

describing the evolutionary dynamics of the population of 182

traders. 183

We notice that our model displays several behavioral fea-184

tures: indeed, agents are not optimizing and follow instead 185

adaptive rules, based on the difference between the realized 186

price and the perceived fundamental value, which depends 187

on the trend of the economy; also the switching mechanism 188

is adaptive in nature, as realized relative proﬁts are taken into 189

account. 190

The main contribution of this paper to the existing litera-191

ture lies in focusing on the role of real and ﬁnancial feedback 192

mechanisms, not only in relation to the dynamics and stabil-193

ity of a single market, but also for those of the economy as 194

a whole. Analytical and numerical tools are used to investi-195

gate the role of the parameter describing the degree of inter-196

action, in order to detect the mechanisms and the channels 197

through which the stability of the isolated real and financial 198 sectors leads to instability for the two interacting markets. 199 More precisely, we start by introducing the “interaction de- 200 gree approach”, which allows us to study high-dimensional 201 systems with many parameters by decomposing them into 202 subsystems easier to analyze, that are then interconnected 203 through the “interaction parameter”. Next, we introduce our 204 model and we derive the stability conditions both for the iso- 205 lated markets and for the whole system with interacting mar- 206 kets. In particular, we find that in the stability conditions it 207 is possible to isolate the parameter describing the degree of 208 interaction between the two markets and that the stability 209 conditions are fulfilled if that parameter belongs to a range 210 characterized by two lower bounds and two upper bounds. 211 Finally, we show how to apply the “interaction degree ap- 212 proach” to our model. To this aim, we first classify the possi- 213 ble scenarios according to the stability/instability of the iso- 214 lated financial and real markets: in such way we are led to an- 215 alyze four frameworks. For each of those we consider differ- 216 ent possible parameter configurations and we show, both an- 217 alytically and numerically, which are the effects of increasing 218 the degree of interaction between the two markets. The con- 219 clusions we get are not univocal: indeed, depending on the 220 value of the other parameters, an increase in the interaction 221 parameter may either have a stabilizing or a destabilizing ef- 222 fect, but also other phenomena are possible. Namely, accord- 223 ing to the mutual position of the before-mentioned lower 224 and upper bounds for the stability range, it may also happen 225 that the stabilization of the system occurs just for interme- 226 diate values of the interaction parameter, neither too small, 227 nor too large, or it may as well happen that one of the upper 228 bounds is always negative or smaller than one of the lower 229 bounds and thus the system never achieves a complete stabi- 230 lization, even if its complexity may decrease and we observe 231 some periodicity windows interrupting the chaotic band. 232 We conclude our theoretical analysis by showing which 233 are the effects of an increasing belief bias for optimists and 234 pessimists. Although its effect is clearly destabilizing when 235 markets are isolated, its role becomes more ambiguous when 236 the markets are interconnected. Indeed, increasing the bias 237 may have either a stabilizing or a destabilizing role, according 238 to the value of the other parameters. However, our numerical 239 simulations suggest that increasing the bias has generally a 240 destabilizing effect, as usually we do not reach a complete 241 stabilization, or we achieve it just in small intervals for the 242

bias. 243

(6)

ARTICLE IN PRESS

produce the alternation of booms and busts, respectively 259

characterized by income growth and contraction in the real 260

sector and by the waves of optimism and pessimism in the 261

ﬁnancial market. 262

Even for the empirical veriﬁcation of our model we con-263

sider the destabilizing scenario, in which the two separately 264

stable sectors become unstable when coupled. In particular, 265

in order to describe the accidental ﬂuctuations of the com-266

position of the many individual digressions from the sim-267

ple rules they are supposed to follow, according to[36]we 268

add a noise term to the demand functions of optimists and 269

pessimists. In such framework we show that our model is 270

able to reproduce three crucial stylized facts about cyclical 271

movements of output and empirical foundation of the con-272

cept of animal spirit: high autocorrelation for output caus-273

ing strong ﬂuctuations, a non-normal distribution for out-274

put, characterized by a high kurtosis and fat tails, and ﬁnally 275

a strong correlation between output and a long-period opti-276

mism index, which describes the waves of optimism and pes-277

simism. For the empirical veriﬁcation of our model we follow 278

the approach by De Grauwe[2,37–39]where, starting from 279

macroeconomic models with heterogeneous and boundedly 280

rational agents, but without ﬁnancial sector, shows that the 281

obtained dynamics are in agreement with (some or all of) the 282

three stylized facts we consider, too. 283

The remainder of the paper is organized as follows. In 284

Section 2we present the stylized facts we wish to replicate 285

with our model. InSection 3we illustrate the interaction 286

degree approach we use to analyze the two linked subsys-287

tems. InSection 4we introduce the model, composed by the 288

real and ﬁnancial sectors. InSection 5we derive the condi-289

tions for the steady state stability, both in the case of iso-290

lated and interacting markets. InSection 6we classify and 291

investigate, analytically and numerically, the possible scenar-292

ios determined by the stability/instability of the real and ﬁ-293

nancial markets when isolated, and we ﬁnally show which 294

are the effects of an increasing bias. InSection 7we discuss 295

our model and give an economic interpretation of our main 296

results. InSection 8we add stochastic shocks to the deter-297

ministic framework considered so far and we show that the 298

model reproduces the stylized facts presented inSection 2. 299

InSection 9we draw some conclusions and outline possible 300

future research directions. 301

2. Stylized facts 302

Economic systems are characterized by periods of strong 303

growth in output followed by periods of decline in economic 304

growth, that is, by booms and busts. Every macro-economic 305

model should be able to explain and reproduce such booms 306

and busts in economic activity. 307

Before proposing our model, it is then useful to present 308

some stylized facts about cyclical movements of output and 309

empirical foundation of the concept of animal spirit, we wish 310

our framework to be able to replicate. 311

Fig. 1shows the strong cyclical movements of the output 312

gap in the US since 1978, that is, the difference between the 313

output and the potential output, the latter being determined 314

in the short run by the capital stock. 315

These cyclical movements are caused by a strong auto-316

correlation in the output gap numbers, i.e., if in period t the 317

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 output gap assumes a certain value, it is likely that its value 318 in period t+ 1 will not vary too much. In particular, if in a cer- 319 tain period the output gap is positive (negative), it is likely to 320

remain positive (negative) for some periods. 321

A second stylized fact about the movements in the output 322 gap, clearly visible inFig. 2for the US, is that such movements 323 are not normally distributed in two aspects. The ﬁrst is that 324 there is a relatively high kurtosis (kurtosis_{= 3.62) and thus,} 325 with respect to the normal distribution, there is too much 326 concentration of observations around the mean. The second 327 aspect are fat tails, i.e., there are larger movements in the out- 328 put gap than is compatible with the normal distribution. In 329 particular, booms and busts are more likely to happen. 330 This also means that, basing the forecasts on the normal 331 distribution, the probability that in some period a large in- 332 crease or decrease in the output gap can occur would be un- 333

derestimated. 334

InFigs. 1and2we report the plots for the US, but similar 335 autocorrelation coeﬃcients are found in other countries (see 336

[40,41]). 337

A third stylized fact we take into account is the high cor- 338 relation between the consumer sentiment index and the out- 339 put gap. The best-known among the sentiment indicators is 340 the Michigan consumer conﬁdence indicator. Such sentiment 341 indicators are developed on the basis of how the individuals 342 perceive the present and the future economic conditions. 343 InFig. 3the Michigan consumer conﬁdence indicator is 344 plotted together with the US output gap in period 1978–2008 345 on the basis of quarterly data, showing a high correlation be- 346

tween the two variables. 347

We will return on the above presented stylized facts in 348

Sections 7and8, where we will show how our model, both 349 in its original formulation and when endowed with stochas- 350 tic errors, is able to replicate such kind of behaviors. In par- 351 ticular, in regard to the third stylized fact, a feature of the 352 correlation of our theoretical framework is that the causal- 353 ity goes both ways, i.e., animal spirits affect output and vice 354

versa. 355

(7)

Fig. 2. The frequency distribution of US output gap from 1960 to 2009, having kurtosis= 3.61 and Jarque–Bera= 7.17 withp-value= 0.027. Source: US Department of Commerce and Congressional Budget Oﬃce.

agents, but without ﬁnancial sector, the author shows that 361

the obtained dynamics are in agreement with (some or all 362

of) the three stylized facts we consider, too. 363

3. The interaction degree approach 364

As we shall see inSection 4, when considering both ﬁ-365

nancial and real markets, we are led to analyze a nonlinear 366

high-dimensional system with many parameters. Such fea-367

tures do not allow to easily handle that system from an ana-368

lytical viewpoint when all the parameters vary, even if we are 369

able to analytically determine its steady state and the corre-370

sponding stability conditions. For this reason, we propose an 371

approach that consists in studying, as a ﬁrst step, the frame-372

work with isolated markets, which are described by lower-373

dimensional subsystems that are simpler to investigate and 374

whose different behaviors can be quickly classiﬁed. Then we 375

make the parameter representing the interaction degree in-376

crease, keeping the other parameters ﬁxed. In such way we 377

are able to analytically ﬁnd (if it exists) the set of values of 378

the interaction parameter that implies stability. Moreover, 379

the use of numerical tools allows us to understand what hap-380

pens also in the unstable regime. 381

This is the strategy we are going to employ inSection 6to 382

classify and investigate the various scenarios for our system. 383

However, in order to make the exposition more ﬂuent, we 384

startSection 5with the analysis of the stability conditions 385

for the case of interacting markets and we derive next the 386

stability conditions for the framework with isolated markets. 387

In symbols, if we denote our integrated system by 388

S_ω

(

x1, . . . , xN

)

, where

ω

∈ [0, 1] is the interaction

de-389

gree parameter and x1, . . . , xNare the endogenous variables

390

governing the system, when setting

ω

= 0 we are led to 391

study two (or, in general, more) isolated subsystems, we de- 392 note by S1

0

(

x1, . . . , xm

)

and S₀2

(

xm+1, . . . , xN

)

, for some m ∈ 393

{

1_{, . . . , N − 1}

}

_{. When instead}

ω

_{= 1 the subsystems are fully} 394 integrated. The case with

ω

∈ (0, 1) represents a partial inter- 395

action between the subsystems. 396

In our framework, when

ω

_{= 0 we find two isolated sub-} 397 systems, corresponding to the financial and real markets. 398 The former is described by two variables, i.e., the stock price 399 and the difference between the shares of optimistic and pes- 400 simistic agents, while the latter is described by a unique vari- 401 able, that is, the level of income. We stress that the influence 402 of the real market on the financial market is due to the fact 403 that the reference value used in the decisional mechanism 404 by the agents in the financial market is determined by the 405 level of income; on the other hand, the investments depend 406 also on the price of the financial asset. Such a double inter- 407 action is described by the parameter

ω

. Notice that we chose 408 a unique parameter,

ω

, to represent both the influence of the 409 real market on the financial sector and vice versa of the finan- 410 cial sector on the real market, not only to limit the already 411 large number of parameters in our model, but also in view 412 of applying the interaction degree approach precisely in the 413 formulation described above. Of course the framework with 414 two different parameters describing the mutual relationships 415 between the two markets could be analyzed in a similar 416

manner. 417

4. The model 418

4.1. The stock market 419

(8)

6 A .N aimzada, M .P ir eddu /C haos, Solit ons a nd Fr act a ls xxx (20 15) xxx–xxx

ARTICLE

IN

PRESS

JID: CHA O S [m3Gdc;Ma y 21, 2015;15:42]

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

(9)

suppose instead that they form believes about the funda-422

mental and, on the basis of this belief, they operate in the 423

stock market. In particular, we consider the trading behav-424

ior of two types of speculators: optimists and pessimists. The 425

label optimist (pessimist) refers to traders that systemati-426

cally overestimate (underestimate) the reference value used 427

in their decisional mechanism[26]. Both types of agents be-428

long to the class of fundamentalists as, believing that stock 429

prices will return to their fundamental value, they buy stocks 430

in undervalued markets and sell stocks in overvalued mar-431

kets. To be more precise, we should say that we model agents 432

as fundamentalists, but their effective behavior depends on 433

the relative position of the stock price with respect to the per-434

ceived reference values (see(4.3)and(4.4)). Optimists and 435

pessimists behave in a similar manner, a part from the fact 436

that the beliefs they have about the reference value differ. 437

The perceived reference values, we denote by F_toptand F_tpes_, 438

are a weighted average between an exogenous value (F∗+ a 439

and F∗− a, respectively, with a > 0) and a term depending 440

on the income value. For simplicity, according to[1]and[24], 441

we assume for the latter term a direct relationship with the 442

economic activity value, both for optimists and pessimists.3

443

In particular, in our model the endogenous term of the funda-444

mental value perceived by optimists and pessimists is given 445

by kYt+ a and kYt− a, respectively, where Ytis the national

446

income and k is a positive parameter capturing the above de-447

scribed direct relationship. Hence, we assume that 448 F_topt=

(

1−

ω

)(

F∗+ a

)

+

ω

(

kYt+ a

)

=

(

1−

ω

)

F∗+

ω

kYt+ a (4.1) and 449 F_tpes=

(

1−

ω

)(

F∗− a

)

+

ω

(

kYt− a

)

=

(

1−

ω

)

F∗+

ω

kYt− a, (4.2)

where a_{> 0 is the belief bias and F}∗is the true unobserved 450

fundamental, both exogenously determined. The constant

ω

451

∈ [0, 1] represents the weighting average parameter. In par-452

ticular, when

ω

= 0 the reference value is completely exoge-453

nous and coincides with the reference value of an isolated 454

stock market like in[26]. When instead

ω

= 1 the reference 455

value is endogenous. 456

Optimists’ demand is given by 457

dopt_t =

α

F_topt− Pt

, (4.3)

and, similarly, pessimists’ demand is given by 458

dpes_t =

α

F_tpes− Pt

, (4.4)

where Pt is the stock price and

α

> 0 is the reactivity

459

parameter. 460

The market maker determines excess demand and adjusts 461

the stock price for the next period. In particular, we denote 462

by ni

t, i ∈

{

opt,pes

}

, the fraction of traders of type i in the

463

market at time t and we assume the market maker behavior 464

to be described by the linear price adjustment mechanism 465

Pt+1= Pt+

μ

nopt_t dopt_t + npes t d

pes t

, (4.5)

where

μ

> 0 is the market maker price adjustment

param-466

eter. For simplicity, we normalize the population size to 1. 467

3_{For an economic justiﬁcation of such hypothesis, see}_{[24, p. 4]}_.

According to(4.5), the market maker increases (decreases) 468 the stock price if excess demand nopt_t d_topt+ npes

t d pes t is posi- 469 tive (negative). 470 We set xt= ntopt− n pes

t , in order to express the fraction of 471

optimistic (pessimistic) traders as n_topt₌ 1+xt

2

(

n

pes t =

1−xt

2

)

, 472

so that we can rewrite(4.5)as 473

Pt₊₁= Pt+

αμ

2

F_topt− Pt

(

1+ xt

)

+

F_tpes− Pt

(

1− xt

)

. (4.6) Recalling the deﬁnition of F_toptand F_tpesfrom(4.1)and(4.2), 474

respectively, we rewrite(4.6)as 475

Pt₊₁= Pt+

αμ{

[

(

1−

ω

)

F∗+

ω

kYt]− Pt+ axt

}

. (4.7)

We observe that the evolution of the stock price is 476 determined by two factors. The ﬁrst one is the devia- 477 tion of the unbiased reference value from the stock price 478

(

[

(

1−

ω

)

F∗+

ω

kYt]− Pt

)

: when the price in period t is be- 479

low (above) the unbiased reference value, the price will 480 increase (decrease) in the next period. The second factor 481 involves the fraction of optimists and pessimists in the mar- 482 ket. If xtis positive (negative) there are more (less) optimists 483

than pessimists, so that the price will increase (decrease) in 484 the next period. The strength of such effect is inﬂuenced by 485 the belief bias a. Finally, we notice that with a completely ex- 486 ogenous reference value, i.e., when

ω

= 0, and without bias,4 ₄₈₇

(4.7)has a unique steady state given by P∗= F∗_. ₄₈₈

Defining now the dynamics of the population of traders, 489 we assume that they will start trying the optimistic or pes- 490 simistic behavior and, if it turns out to be the most profitable, 491 they will stick to it; otherwise they will switch to the other 492 behavior in the next period. Such an evolutionary process is 493 governed by the profits that traders make in each period. Let 494 us define the profits

π

i

trealized by type i, i ∈

{

opt,pes

}

, as 495

π

i

t = dti−1

(

Pt− Pt−1

)

. (4.8)

Following[4,42], we assume that the fraction ni

tof traders of 496

type i is given by the discrete choice model 497

ni t=

exp

(

βπ

i

t

)

exp

(

βπ

_topt

)

+ exp

(

βπ

pes

t

)

, (4.9)

where

β

≥ 0 is the parameter representing the intensity of 498 choice. In particular, if

β

= 0 the difference between prof- 499 its is not considered and the behavior choice is purely ran- 500 dom, so that nopt_t = npes

t = 12. At the other extreme, when 501

β

→ +∞, the switches are fully governed by the rational 502 component and all traders are of the optimistic type (xt→ 503 1) if

π

_topt>

π

pes

t , while all traders are of the pessimistic type 504

(xt→ −1) if

π

_topt<

π

_tpes; ﬁnally, if

π

_topt=

π

_tpes, we ﬁnd again 505

nopt_t = npes

t =12 and thus xt= 0. We observe that right hand 506

side in(4.9)may be seen as a representation of the relative 507

proﬁts of the traders of type i. 508

From(4.1)–(4.4),(4.7)and(4.8)it follows that 509

π

opt t −

π

pes t =

dopt_t₋₁− dpes t−1

(

Pt− Pt−1

)

= 2a

μα

2

_{

_[

₍

₁₋

_ω

₎

_F∗₊

_ω

_kYt −1] −Pt−1+ axt−1

}

, (4.10)

(10)

ARTICLE IN PRESS

and thus, from the deﬁnition of xtand(4.9), we have

510 xt = tanh

β

π

opt t −

π

pes t

2

= tanh

(

a

μβα

2_[

₍

₁₋

_ω

₎

_F∗ +

ω

kYt₋₁− Pt−1+ axt−1]

)

. (4.11)

4.2. The real market

511

Similarto[1]and[24], we consider a Keynesian good

mar-512

ket interacting with the stock market, in a closed economy 513

with public intervention. It is assumed that both private and 514

government expenditures depend on national income and 515

that private expenditure depends also on the performance 516

in the stock market. The dynamic behavior in the real econ-517

omy is described by an adjustment mechanism depending 518

on the excess demand: if aggregate excess demand is posi-519

tive (negative), production will increase (decrease). Indeed, 520

income Yt+1in period t+ 1 is deﬁned in the following way:

521

Yt+1= Yt+

γ

g

(

Zt− Yt

)

, (4.12)

where g is an increasing function with g

(

0

)

= 0, Ztis the

ag-522

gregate demand in a closed economy, deﬁned as 523

Zt= Ct+ It+ Gt,

where Ct, It and Gtstand for consumption, investment and

524

government expenditure, respectively, and

γ

> 0 is the real

525

market speed of adjustment between demand and supply. In 526

order to conduct our analysis, denoting by Et= Zt− Ytthe

ex-527

cess demand, we specify the function g as 528 g

(

Et

)

= a2

_a 1+ a2 a1e−Et+ a2 − 1

, (4.13)

with a1, a2positive parameters.

529

With such a choice, g is increasing and g

(

0

)

= 0. More-530

over, g is bounded from below by−a2and from above by a1.

531

The presence of the two horizontal asymptotes prevents too 532

large variations in income and prevents the real market from 533

diverging, creating a real oscillator. We stress that this kind 534

of nonlinearity has been recently considered in[34]. 535

The rationale for introducing in(4.12)a nonlinear map 536

g, rather than a linear one, is that in the latter framework

537

the income variation

t+1= Yt+1− Yt may grow

unbound-538

edly and, in particular, when Etlimits to±∞, the same does

539

t+1. However, this is an unrealistic assumption because of

540

the material constraints in the production side of an econ-541

omy. Indeed, when excess demand increases, capacity con-542

straints will surely lead to lower increases in income, due to 543

the limited expansion from time to time of capital and la-544

bor stock; when excess demand decreases, capital cannot be 545

destroyed proportionally to excess demand as the only fac-546

tors that may reduce productivity are attrition of machines 547

from wear, time, and innovations. Moreover, also the labor 548

factor imposes constraints: indeed, due to the presence of 549

trade unions, it is diﬃcult, or impossible, to reduce employ-550

ment below a certain threshold level. 551

Like commonly assumed, private and government expen-552

ditures are partly exogenous and partly increase with na-553

tional income. Moreover, as in[1]and[24], we suppose that 554

the ﬁnancial situation of households and ﬁrms depends on 555

the stock market performance, too. If the stock price in-556

creases, the same does private expenditure. On the basis of 557

these considerations, we can write the relation between pri- 558 vate and government expenditures and national income and 559

stock price as 560

Zt= Ct+ It+ Gt= A + bYt+

ω

cPt, (4.14)

where A> 0 deﬁnes autonomous expenditure, b ∈ [0, 1] is 561 the marginal propensity to consume and invest from current 562 income, c∈ [0, 1] is the marginal propensity to consume and 563 invest from current stock market wealth, and

ω

∈ [0, 1] rep- 564 resents the degree of interaction between the real and the 565 stock markets. In particular, when

ω

= 0 the real market is 566 completely isolated from the ﬁnancial market; when

ω

_{= 1} 567 the two markets are fully interconnected; for

ω

∈ (0, 1) we 568

have a partial interaction. 569

We stress that it would also be possible to assume that 570 aggregate demand Ztdepends, rather than on the stock price 571

Ptas in(4.14), on the price variation Pt− Pt−1. Notice however 572 that this would increase the dimensionality of our system. 573 We will deal with such a new formulation in a future paper. 574 Inserting Ztfrom(4.14)into(4.12)and recalling the deﬁ- 575 nition of g in(4.13), we obtain the dynamic equation of the 576

real market 577 Yt₊₁= Yt+

γ

a2

_a 1+ a2 a1e−(A+bYt+ωcPt−Yt)+ a2 − 1

.

Summarizing, when taking into account both the ﬁnan- 578 cial and the real markets, we are led to study the following 579

system describing the whole economy: 580

⎧

⎪

⎨

⎪

⎩

Pt+1= Pt+

αμ{

[

(

1−

ω

)

F∗+

ω

kYt]− Pt+ axt

}

xt₊₁= tanh

(

a

μβα

2_[

(

₁₋

ω

)

_F∗₊

_ω

_kYt_{− P}_t_{+ ax}_t_]

₎

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- 581

ates of the three-dimensional map 582

G=

(

G1, G2, G3

)

:

(

0, +∞

)

×

(

−1, 1

)

× [0, +∞

)

→ R3,

583

(

P, x,Y

)

→

(

G1

(

P, x,Y

)

, G2

(

P, x,Y

)

, G3

(

P, x,Y

))

,

deﬁned as: 584

⎧

⎨

⎩

G1

(

P, x,Y

)

= P +

αμ

((

1−

ω

)

F∗+

ω

kY− P + ax

)

G2

(

P, x,Y

)

= tanh

(

μ

a

α

2

β

[

(

1−

ω

)

F∗+

ω

kY− 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 585

In order to classify inSection 6the various scenarios oc- 586 curring for

ω

_{= 0 and investigate their local stability when}

ω

587 increases, hereinafter we derive the sufficient conditions for 588 stability both in the case of interacting and isolated markets. 589 In fact, the classification we will adopt in the next section re- 590 lies on the stability/instability features of the real and finan- 591 cial subsystems when they are isolated. Then, for any such 592 scenario, we will study what happens when the degree of in- 593

terconnection between the two markets increases. 594

(11)

Sections 5.1and5.2and the numerical results inSection 6. 597

In fact, for the reader’s convenience, in correspondence to 598

the scenarios inSections 6.1and6.2we will check what the 599

corresponding stability conditions say and we will compare 600

those conditions with the numerical simulations performed 601

therein. The trivial veriﬁcations for the remaining scenarios 602

considered in the next section can be performed analogously. 603

5.1. Interacting markets

604

The map in(4.16)has a unique ﬁxed point given by 605

(

P∗, x∗_,Y∗

₎

=

ω

Ak+

(

1−

ω

)

F∗

(

1− b

)

1− b −

ω

2_ck , 0, A+

ω

c

(

1−

ω

)

F∗ 1− b −

ω

2_ck

.

The Jacobian matrix for G computed in correspondence to it 606 reads as 607 JG

(

P∗, x∗_,Y∗

₎

₌

⎡

⎣

1−

αμ

a

αμω

k −

μ

a

α

2

_{β α}

2

_μ

_a2

_{β α}

2

_μ

_a

_βω

_k γa1a2ωc a1+a2 0 1− γa1a2(1−b) a1+a2

⎤

⎦

. (5.1) In order to check the stability of the steady state in the 608

various scenarios considered inSection 6, we are going to use 609

the following conditions (see[43]): 610 (i) 1+ C1+ C2+ C3> 0; 611 (ii) 1_{− C}1+ C2− C3> 0; 612 (iii) 1− C2+ C1C3−

(

C3

)

2> 0; 613 (iv) 3− C2> 0, 614

where Ci, i∈ {1, 2, 3}, are the coeﬃcients of the

character-615

istic polynomial 616

λ

3_{+ C}

1

λ

2+ C2

λ

+ C3= 0.

In our framework, we have 617 C1=

γ

a1a2

(

1− b

)

a1+ a2 − 2 +

αμ

−

μ

a2

_α

2

_β

_; C2= 2

μ

a2

α

2

β

+ 1 −

αμ

−

γ

a1a2

ω

2ck

αμ

a1+ a2 −

γ

a1a2

(

1− b

)

a1+ a2

(

1−

αμ

+

μ

a2

_α

2

_β

₎

_; C3=

μ

a2

α

2

β

γ

a1a2

(

1− b

)

a1+ a2 − 1

.

Notice that, making

ω

explicit, it is possible to rewrite Con-618

ditions (i)–(iv) above respectively as follows: 619 (i)

ω

2_<

₍

₁_{+ C} 1+ C+ C3

)

_γ_aa1+a2 1a2ckαμ:= B1; 620 (ii)

ω

2_<

₍

₁_{− C} 1+ C− C3

)

_γ_aa1+a2 1a2ckαμ:= B2; 621 (iii)

ω

2_>

₍

_{−1 +}_C_{− C} 1C3+ C32

)

_γaa₁1a+a₂ck2αμ:= B3; 622 (iv)

ω

2_>

₍

_C_{− 3}

₎

a1+a2 γa₁a₂ckαμ:= B4, 623

where we have set 624 C= 2

μ

a2

_α

2

_β

_{+ 1 −}

_αμ

₋

γ

a1a2

(

1− b

)

a1+ a2 ×

(

1−

αμ

+

μ

a2

_α

2

_β

₎

_. Hence, if 625 min

{

1+ C1+ C+ C3, 1 − C1+ C− C3

}

> 0 and max

{

−1 + C− C1C3+ C32, C− 3

}

< 1,

the integrated system is locally asymptotically stable at the 626

steady state if 627

max

{

B3, B4

}

<

ω

2< min

{

B1, B2

}

,

ω

∈ [0, 1]. (5.2)

If instead 628

min

{

1+ C1+ C+ C3, 1 − C1+ C− C3

}

≤ 0 or

max

{

−1 + C− C1C3+ C32, C− 3

}

≥ 1,

it is not possible to have local stability at the steady state, for 629

any

ω

∈ [0, 1]. 630

5.2. Isolated markets 631

In the special case in which

ω

= 0, System(4.15)can be 632

rewritten as 633

⎧

⎪

⎨

⎪

⎩

Pt₊₁= Pt+

αμ

(

F∗− Pt+ axt

)

xt+1= tanh

(

μ

a

α

2

β

[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 634

(

P∗, x∗_,Y∗

₎

₌

F∗, 0, A 1− b

. (5.4)

Since in such framework the ﬁrst two equations in(5.3)de- 635 pend just on Ptand xt, and the last one just on Yt, imply- 636 ing that the real and stock markets are completely discon- 637 nected, as explained inSection 3, instead of considering the 638 three-dimensional system in(5.3), we will rather deal with 639 the two-dimensional subsystem related to the stock market 640

Pt+1= Pt+

αμ

(

F∗− Pt+ axt

)

xt₊₁= tanh

(

μ

a

α

2

_β

_[F∗_{− P}_t_{+ ax}_t_]

₎

and with the one-dimensional subsystem related to the real 641

market 642 Yt₊₁= Yt+

γ

a2

_a 1+ a2 a1e−(A+(b−1)Yt)+ a2 − 1

.

In this way, in agreement with the ﬁndings in[26], the steady 643

state in(5.4)should be split as 644

(

P∗, x∗

₎

₌

₍

_F∗_{, 0}

₎

_{, Y}∗₌ A

1− b

and, similarly, the Jacobian matrix in(5.1)computed in cor- 645 respondence to the steady state when

ω

= 0 should be split 646

as 647 J1

₍

_P∗_{, x}∗

₎

₌

1−

αμ

a −

μ

a

α

2

β α

2

μ

_a2

β

, J2

₍

_Y∗

₎

_{= 1 −}

γ

a1a2

(

1− b

)

a1+ a2 .

(12)

ARTICLE IN PRESS

Fig. 4. The bifurcation diagram with respect toω∈ [0, 1] for P, forμ= 5 and the initial conditions P0= 5, x0= 0.8 and Y0= 25.

subsystem read then as 649

det J1₌

_μα

2_a2

_β

_{< 1,}

1+ tr J1_{+ det J}1_{= 2 −}

_μα

_{+ 2}

_μα

2_a2

_β

_{> 0,}

1− tr J1_{+ det J}1₌

_μα

_{> 0.}

Notice that the third condition is always fulﬁlled, while the 650

ﬁrst two can be rewritten, making

β

explicit, as 651

αμ

− 2

2

μα

2_a2 <

β

<

1

μα

2_a2. (5.5)

From(5.5) we easily infer the destabilizing role of the be-652

lief bias on the ﬁnancial side of the economy when the two 653

markets are isolated: indeed, the stability interval for

β

gets 654

reduced when a increases. 655

On the other hand, the real subsystem is locally asymp-656

totically stable at the steady state if−1 < 1 −γa₁a₂(1−b) a1+a2 < 1. 657

The right inequality is always fulﬁlled (except for b= 1, but 658

we will always deal with the case 0_{< b < 1), while the left} 659

inequality holds if and only if 660

γ

< 2

(

a1+ a2

)

a1a2

(

1− b

)

.

Hence, when

ω

_{= 0 both subsystems are stable if} 661

αμ

− 2 2

μα

2_a2 <

β

< 1

μα

2_a2 and

γ

< 2

(

a1+ a2

)

a1a2

(

1− b

)

. (5.6) 6. Possible scenarios 662

Starting from the various stability/instability scenarios for 663

the ﬁnancial and real subsystems when isolated, in the next 664

pages we shall investigate what happens in each framework 665

when the degree of interaction between the two markets in-666

creases, in order to show that modifying the parameter

ω

667

may produce very different effects depending on the value 668

of the other parameters and on the speciﬁc framework con- 669

sidered. 670

We will conclude the section by analyzing the effects of 671 an increasing belief bias on the stability of the whole system. 672

6.1. Stable ﬁnancial and real subsystems 673 In this framework, when isolated, both markets are sta- 674 ble. As

ω

increases, the steady state can either remain sta- 675 ble until

ω

= 1 or can undergo a ﬂip bifurcation, followed 676 by a secondary double Neimark–Sacker bifurcation, accord- 677 ing to the considered value of the other parameters. In 678 particular, the parameter

μ

seems to play a crucial role 679 in this respect. In fact, in Figs. 4–6 below we have ﬁxed 680 the parameters as follows: F∗_{= 5, k = 0.25,}

α

_{= 0.08,}

β

₌ 681 1, c = 1, a = 2,

γ

= 3.5, a1= 2, a2= 4, A = 5, b = 0.7, and 682

μ

= 5 inFig. 4, while

μ

= 28 inFigs. 5and6. InFig. 4the 683 steady state remains stable until

ω

_{= 1, while in}Figs. 5and 684

6a destabilization occurs for

ω

ࣃ0.515. More precisely, in 685

Figs. 4and5(A) we show the bifurcation diagram for P with 686 respect to

ω

_{∈ [0, 1], while in}Fig. 5(B) we draw the bifur- 687 cation diagram for Y with respect to

ω

∈ [0, 1]; inFig. 5(C) 688 we show the Lyapunov exponent when

ω

varies in [0, 1]. In 689

Fig. 6(A) and (B) we depict, in the phase plane, the ﬁxed point 690 when

ω

= 0.25 and the period-two cycle when

ω

= 0.70, 691 respectively; ﬁnally, inFig. 6(C) we show the time series 692 for P (in red) and Y (in blue) when

ω

= 0.95, which high- 693 light a quasiperiodic behavior characterized by long mono- 694 tonic increasing motions, followed by oscillatory decreasing 695

motions. 696

Let us now check whether the theoretical results in 697

Section 5are in agreement with the numerical achievements 698

above. 699

(13)

Fig. 5. The bifurcation diagrams with respect toω∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, forμ= 28 and the initial conditions P0= 5, x0= 0.8 and Y0= 25.

Fig. 6. The (P, Y)-phase portraits forμ= 28, andω= 0.25 in (A) andω= 0.70 in (B), respectively; in (C) the time series for P in red (below) and Y in blue (above)

whenμ= 28 andω= 0.95.(For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

the case, as the ﬁrst chain of inequalities reads as−6.25 < 704

1< 7.812 and the last inequality is 3.5 < 5 when

μ

= 5, 705

while the ﬁrst chain of inequalities reads as 0.167_{< 1 < 1.395} 706

and the last inequality is again 3.5< 5 when

μ

= 28. As it 707

is immediate to see from such calculations, although for the 708

considered parameter values both isolated markets are sta-709

ble, when

μ

increases the stability region of the ﬁnancial 710

subsystem decreases: this may explain the detected grow-711

ing destabilization of the whole system in correspondence to 712

larger values of

μ

. 713

As concerns the stability conditions for

ω

varying in 714

[0, 1], when

μ

_{= 5 we have B}1= 1.2, B2= 2.386, B3=

715

−2.4509, B4= −6.778, and thus(5.2) reads as

ω

∈ [0, 1],

716

that is, the system is stable for any

ω

, in agreement with 717

Fig. 4; when instead

μ

= 28 we have B1= 1.2, B2=

718

0.274, B3= −0.098, B4= −0.793, and thus (5.2) reads as

719

ω

∈ [0,

B2

)

= [0, 0.523

)

, that is, the system is stable just for

720

small values of

ω

, in agreement withFig. 5. 721

What we can then conclude in this scenario is that in-722

creasing

μ

has a destabilizing effect. In fact, ﬁxing all the 723

other parameters as above and letting just

μ

vary, we ﬁnd 724

that ∂B1

∂μ = 0,∂∂μB2 < 0, ∂∂μB3 > 0 and ∂∂μB4 > 0. Hence, the sta-725

bility region decreases when

μ

increases (as B1 does not

726

vary with

μ

and the upper bound B2 decreases, while the

727

lower bounds B3and B4increase), conﬁrming the highlighted

728

destabilizing role of the parameter

μ

. 729

Summarizing, for the above parameter configurations, the 730 interaction between the financial and real markets either 731 maintains the stability of the system, or it has a destabilizing 732 effect, through a flip bifurcation. We stress that such bifurca- 733 tion differs from the flip bifurcation detected in[26]in two 734 aspects. The first one is that, as already stressed in the Intro- 735 duction, those authors deal just with the isolated financial 736 market, while our flip bifurcation concerns the interaction 737 between the real and financial markets. The second aspect 738 is that after the flip bifurcation in[26]some numerical sim- 739 ulations we performed suggest that the system diverges and 740 thus such bifurcation would not lead to complex behaviors: 741 in our framework, the flip bifurcation is followed instead by 742 a stable period-two cycle which, increasing further the inter- 743 action parameter, undergoes a secondary double Neimark– 744 Sacker bifurcation, giving rise to quasiperiodic motions. 745

(14)

ARTICLE IN PRESS

Fig. 7. The bifurcation diagrams with respect toω∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, forγ= 5 and the initial

conditions P0= 12, x0= −0.3 and Y0= 61.

Fig. 8. The bifurcation diagrams with respect toω∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, forγ= 8 and the initial

Fig. 9. The bifurcation diagrams with respect toω∈ [0, 1] for P in (A) and Y in (B), and the Lyapunov exponent in (C), respectively, forγ= 8.8 and the initial

bifurcation and then a secondary double Neimark–Sacker bi-754

furcation. For even larger values of

γ

, we just obtain a reduc-755

tion of the complexity of the system for suitable intermediate 756

values of

ω

, but the system is never stabilized. 757

More precisely, inFigs. 7–9below,

ω

varies in [0, 1] and 758

the other parameters are: F∗= 2, k = 0.1,

α

= 0.08,

β

= 759

1_{, c = 1, a = 2.4,}

μ

_{= 28, a}1= 3, a2= 1, A = 12, b = 0.7,

760

and

γ

= 5 inFig. 7,

γ

= 8 inFig. 8, and

γ

= 8.8 inFig. 9. In 761

Fig. 7the ﬁxed point becomes stable for

ω

ࣃ 0.2 and remains 762

stable until

ω

_{= 1. In}Fig. 8, instead of remaining stable, it 763

undergoes a ﬂip bifurcation for

ω

ࣃ 0.5 and then a secondary 764

double Neimark–Sacker bifurcation for

ω

ࣃ 0.96. InFig. 9the 765

ﬁxed point is never stable: we just observe a reduction of the 766 complexity of the system for

ω

∈ (0.2, 0.8), where we have 767 a stable period-two cycle. In more detail, inFigs. 7(A)–9(A) 768 we show the bifurcation diagrams with respect to

ω

∈ [0, 769 1] for P, while inFigs. 7(B)–9(B) we draw the bifurcation 770 diagrams for Y; inFigs. 7(C)–9(C) we depict the Lyapunov 771

exponents when

ω

varies in [0, 1]. 772

Let us now check whether the theoretical results in 773

Section 5are in agreement with the numerical achievements 774

above. 775