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An evolutive financial market model with animal spirits: imitation and endogenous beliefs

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Abstract

We propose a financial market model with optimistic and pessimistic fundamentalists who, respectively, overestimate and underestimate the true fundamental value due to ambiguity in the stock market. We assume that agents form their beliefs about the fundamental value through an imitative process, considering the relative ability shown by optimists and pessimists in guessing the realized stock price. We also introduce an endogenous switching mechanism, allowing agents to switch to the other group of speculators if they performed better in terms of relative profits. Moreover, the stock price is determined by a nonlinear mechanism. We study, via analytical and numerical tools, the stability of the unique steady state, its bifurcations and the emergence of complex behaviors, with possible multistability phenomena. To quantify the global propensity to optimism/pessimism of the market, we introduce an index, depending on pessimists’ and optimists’ beliefs and shares, thanks to which we are able to show that the occurrence of the waves of optimism and pessimism are due to the joint effect of imitation and switching mechanism. Finally, we perform a statistical analysis of a stochastically perturbed version of the model, which high lights fat tails and excess volatility in the returns distributions, as well as bubbles and crashes for stock prices, in agreement with the empirical literature.

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Notes

  1. 1 We stress that such a notion of ambiguity differs from the one employed in general equilibrium theory, where agents, in making their choices in stochastic frameworks, are assumed to take into account different probability laws describing the distribution of relevant random variables. See e.g. Dow and Werlang (1992) and Ellsberg (1961).

  2. 2 Starting with the crucial paper by De Bondt and Thaler (1985), a well-grounded empirical literature has arisen to show the presence of overreaction phenomena in financial markets. We recall that, as said in France et al. (1994), page 19, overreaction is “defined as a movement in price that overshoots the equilibrium value and then subsequently returns to its true value”. Subsequently, some authors have given further foundations to overreaction events through the formulation and analysis of mathematical models. See, for instance, the works by Barberis et al. (1998), by Hong and Stein (1999) and by Veronesi (1999).

  3. We only provide sufficient conditions for the occurrence of each scenario. We remark thatwe may indeed obtain the same scenario for other parameter configurations. Moreover, wenotice that some of the stability intervals we shall derive below may in principle be empty forany parameter configuration. We will provide numerical examples in Section 4 to show thatthere exist parameter settings for which such intervals are nonempty and the correspondingscenarios do actually occur.

  4. Actually, if \(\mu _{1}=\mu _{2}\in (\max \left \{\mu _{\ell },0\right \},\mu _{r})\) we find two disjoint adjacent stability intervals. Such limit case can however be encompassed in the mixed scenario.

References

  • Anderson S, de Palma A, Thisse J (1992) Discrete choice theory of product differentiation. The MIT Press, Cambridge

    Google Scholar 

  • Akerlof GA, Shiller RJ (2009) Animal spirits: how human psychology drives the economy and why it matters for global capitalism. Princeton University Press, Princeton

    Google Scholar 

  • Barberis N, Shleifer A, Vishny R (1998) A model of investor sentiment. J Financ Econ 49:307–343

    Article  Google Scholar 

  • Barberis N, Thaler R (2003) A survey of behavioral finance. In: Constantinides G M, Harris M, Stulz R (eds) Handbook of the economics of finance, 1st edn. Elsevier, Amsterdam, pp 1053–1128

    Google Scholar 

  • Brock WA, Hommes CH (1997) A rational route to randomness. Econometrica 65:1059–1095

    Article  Google Scholar 

  • Brock WA, Hommes CH (1998) Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J Econ Dyn Control 22:1235–1274

    Article  Google Scholar 

  • Chiarella C, Dieci R, He X-Z (2009) Heterogeneity, market mechanisms, and asset price dynamics. In: Hens T, Schenk-Hoppé K R (eds) Handbook of financial markets: dynamics and evolution. Elsevier, Amsterdam, pp 277–344

    Chapter  Google Scholar 

  • Chiarella C, He X-Z, Hommes CH (2006) A dynamic analysis of moving average rules. J Econ Dyn Control 30:1729–1753

    Article  Google Scholar 

  • De Bondt W, Thaler R (1985) Does the stock market overreact? J Financ 40:793–805

    Article  Google Scholar 

  • De Grauwe P (2012) Lectures on behavioral macroeconomics. Princeton University Press, Princeton

    Google Scholar 

  • De Grauwe P, Macchiarelli C (2015) Animal spirits and credit cycles. J Econ Dyn Control 59:95–117

    Article  Google Scholar 

  • De Grauwe P, Rovira Kaltwasser P (2012) Animal spirits in the foreign exchange market. J Econ Dyn Control 36:1176–1192

    Article  Google Scholar 

  • Dieci R, He X-Z, Hommes C (eds) (2014) Nonlinear economic dynamics and financial modelling. Essays in honour of Carl Chiarella. Springer, Berlin

  • Dow J, Werlang S (1992) Uncertainty aversion, risk aversion, and the optimal choice of portfolio. Econometrica 60:197–204

    Article  Google Scholar 

  • Ellsberg D (1961) Risk, ambiguity, and the savage axioms. Q J Econ 75:643–669

    Article  Google Scholar 

  • Epstein S (2003) Cognitive-experiential self-theory of personality. In: Millon T, Lerner M J (eds) Comprehensive handbook of psychology, vol 5. Wiley, Hoboken, New Jersey, pp 159–184

    Google Scholar 

  • Farebrother RW (1973) Simplified Samuelson conditions for cubic and quartic equations. Manch Sch 41:396–400

    Article  Google Scholar 

  • France V, Kodres L, Moser J (1994) A review of regulatory mechanisms to control the volatility of prices. Federal Reserve Bank of Chicago. Econ Perspect 18:15–26

    Google Scholar 

  • Gilbert D (2002) Inferential correction. In: Gilovich T, Griffin D, Kahneman D (eds) Heuristics and biases: the psychology of intuitive thought. Cambridge University Press, Cambridge, pp 167–184

    Chapter  Google Scholar 

  • Harris L (1998) Circuit breakers and program trading limits: what have we learned? In: Litan R, Santomero A (eds) Brookings-wharton papers on financial services. Brookings Institutions Press, Washington, pp 17–63

    Google Scholar 

  • Hommes CH (1991) Adaptive learning and roads to chaos: the case of the cobweb. Econ Lett 36:127–132

    Article  Google Scholar 

  • Hommes CH (1994) Dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand. J Econ Behav Organ 24:315–335

    Article  Google Scholar 

  • Hommes CH (2006) Heterogeneous agent models in economics and finance. In: Tesfatsion L, Judd KL (eds) Hand-book of computational economics, vol. 2: agent-based computational economics. Elsevier Science B.V., Amsterdam, pp 1109–1186

    Chapter  Google Scholar 

  • Hommes CH (2013) Behavioral rationality and heterogeneous expectations in complex economic systems. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Hong H, Stein JC (1999) A unified theory of underreaction, momentum trading and overreaction in asset markets. J Financ 54:2143–2184

    Article  Google Scholar 

  • Keynes JM (1936) The general theory of employment interest and money. Macmillan, London

    Google Scholar 

  • Kindleberger CP, Aliber RZ (2005) Manias, panics, and crashes: a history of financial crises. Wiley, Hoboken, New Jersey

    Book  Google Scholar 

  • Kyle A (1988) Trading halts and price limits. Rev Futures Mark 7:426–434

    Google Scholar 

  • Naimzada A, Pireddu M (2015a) Real and financial interacting markets: a behavioral macro-model. Chaos Solit Fract 77:111–131

    Article  Google Scholar 

  • Naimzada A, Pireddu M (2015b) A financial market model with endogenous fundamental values through imitative behavior. Chaos 25:073110. doi:10.1063/1.4926326

    Article  Google Scholar 

  • Naimzada A, Pireddu M (2015c) Introducing a price variation limiter mechanism into a behavioral financial market model. Chaos 25:083112. doi:10.1063/1.4927831

    Article  Google Scholar 

  • Naimzada A, Ricchiuti G (2008) Heterogeneous fundamentalists and imitative processes. Appl Math Comput 199:171–180

    Google Scholar 

  • Naimzada A, Ricchiuti G (2009) The dynamic effect of increasing heterogeneity in financial markets. Chaos Solit Fract 41:1764–1772

    Article  Google Scholar 

  • Naimzada A, Ricchiuti G (2014) Complexity with heterogeneous fundamentalists and a multiplicative price mechanism. Econ Notes 43:233–247

    Article  Google Scholar 

  • Schlag K (1998) Why imitate, and if so, how?: a boundedly rational approach to multi-armed bandits. J Econ Theory 78:130–156

    Article  Google Scholar 

  • Schmitt N, Westerhoff F (2014) Speculative behavior and the dynamics of interacting stock markets. J Econ Dyn Control 45:262–288

    Article  Google Scholar 

  • Tuinstra J (2002) Nonlinear dynamics and the stability of competitive equilibria. In: Hommes C H, Ramer R, Withagen C H (eds) Equilibrium, markets and dynamics. Essays in honour of claus weddepohl. Springer, Berlin, pp 329–343

    Chapter  Google Scholar 

  • Tversky A, Kahneman D (1974) Judgement under uncertainty: heuristics and biases. Science 185:1124–1131

    Article  Google Scholar 

  • Veronesi P (1999) Stock market overreactions to bad news in good times: a rational expectations equilibrium model. Rev Financ Stud 12:975–1007

    Article  Google Scholar 

  • Weddepohl C (1995) A cautious price adjustment mechanism: chaotic behavior. J Econ Behav Organ 27:293–300

    Article  Google Scholar 

  • Westerhoff F (2009) Exchange rate dynamics: a nonlinear survey. In: Rosser Jr J B (ed) Handbook of research on complexity. Edward Elgar, Cheltenham, pp 287–325

    Google Scholar 

  • Wilson T (2002) Strangers to ourselves: discovering the adaptive unconscious. Harvard University Press, Cambridge

    Google Scholar 

  • Zhu M, Chiarella C, He X-Z, Wang D (2009) Does the market maker stabilize the market? Physica A 388:3164–3180

    Article  Google Scholar 

Download references

Acknowledgments

The authors thank Professor He and Professor Hommes for their valuable comments about the interpretation of the model and the consistency with real data, and all the participants to NED 2015 for helpful discussions.

The authors wish also to thank the anonymous Reviewers and Professor Dieci, Guest Editor of the Special Issue on “Nonlinear Economic Dynamics”, for the useful suggestions.

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Appendix

Appendix

Proof of Proposition 1

The expression of the steady state for the population shares follows by noticing that, in equilibrium, P(t + 1) = P(t) and thus, by Eq. 3, we have π X = π Y = 0 at the steady state, so that \(\omega ^{\ast }=\frac {1}{2}\). Moreover, from the stock price equation we find \(P^{\ast }=\omega ^{\ast } X^{\ast }+(1-\omega ^{\ast })Y^{\ast }=\frac {X^{\ast }+Y^{\ast }}{2}\), so that \(X^{\ast }=F-\frac {\Delta }{2}\) and \(Y^{\ast }=F+\frac {\Delta }{2}\). Inserting such expressions in P , we get P = F, as desired. □

Proof of Proposition 2

Since from Eq. 7 it follows that

$$\begin{array}{@{}rcl@{}} F&=&X(t+1)+\frac{\Delta}{1+e^{\mu((X(t)-P(t))^{2}-(Y(t)-P(t))^{2})}}\\ &=&Y(t+1)-\frac{\Delta}{1+e^{-\mu((X(t)-P(t))^{2}-(Y(t)-P(t))^{2}))}}, \end{array} $$

then we find

$$\begin{array}{@{}rcl@{}} Y(t+1)&=& X(t+1)+{\Delta}\left( \frac{1}{1+e^{\mu((X(t)-P(t))^{2}-(Y(t)-P(t))^{2})}}+ \frac{1}{1+e^{-\mu((X(t)-P(t))^{2}-(Y(t)-P(t))^{2}))}}\right)\\ &=& X(t+1)+{\Delta}, \end{array} $$

as desired. □

Proof of Proposition 3

To prove (i′) − (iii′), we use the conditions in Farebrother (1973).

To such aim, we need to compute the Jacobian matrix for G in correspondence to (X , P , ω ),which reads as

$$ \left[\begin{array}{ccc} \frac{-\mu\Delta^{2}}{2} & \frac{\mu\Delta^{2}}{2} & 0\\ \widetilde{\gamma} & 1-\widetilde{\gamma} & -\widetilde{\gamma}{\Delta}\\ \frac{-\beta\widetilde{\gamma}\Delta}{4} & \frac{\beta\widetilde{\gamma}\Delta}{4} & \quad \frac{\beta\widetilde{\gamma}{\Delta}^{2}}{4} \end{array}\right]. $$
(13)

The Farebrother conditions are the following:

  1. (i)

    1 − C 1 + C 2C 3 > 0;

  2. (ii)

    1 − C 2 + C 1 C 3 − (C 3)2 > 0;

  3. (iii)

    3 − C 2 > 0;

  4. (iv)

    1 + C 1 + C 2 + C 3 > 0,

where C i , i ∈ {1, 2, 3}, are the coefficients of the characteristic polynomial

$$\lambda^{3}+C_{1} \lambda^{2}+C_{2} \lambda+C_{3}=0.$$

In our framework, we have:

$$C_{1}=\frac{\mu{\Delta}^{2}}{2}+\widetilde{\gamma}-1-\frac{\beta\widetilde{\gamma}{\Delta}^{2}}{4}, \quad C_{2}=\frac{-\mu{\Delta}^{2}}{2}\left( 1+\frac{\beta\widetilde{\gamma}{\Delta}^{2}}{4}\right)+ \frac{\beta\widetilde{\gamma}{\Delta}^{2}}{4},\quad C_{3}=\frac{\mu\beta\widetilde{\gamma}{\Delta}^{4}}{8}, $$

and thus simple computations allow to notice that conditions (i) − (i i i) above read,respectively, as (i′) − (iii′), while (i v) reduces to \(\widetilde {\gamma }>0\), which is indeed true. □

Proof of Corollary 1

Firstly, we keep Δ ≠ 0, μ and \(\widetilde {\gamma }\) fixed and we solve conditions \((i^{\prime })-(iii^{\prime })\) with respect to β.

If 2 − μΔ2 ≤ 0, we have that condition (i′) is not satisfied by any β > 0, which means that we are in the unconditionally unstable scenario.

Let us now consider 2 − μΔ2 > 0.Condition (i′) is then equivalent to

$$ \beta>\beta_{\ell}=\frac{4}{\widetilde{\gamma}{\Delta}^{2}}\left( \frac{\widetilde{\gamma}} {2-\mu{\Delta}^{2}}-1\right). $$
(14)

From (iii′) we have

$$ \beta<\beta_{r_{2}}=\frac{4}{\widetilde{\gamma}{\Delta}^{2}}\cdot\frac{6+\mu{\Delta}^{2}}{2-\mu{\Delta}^{2}}. $$
(15)

Finally, we notice that we can rearrange condition (ii′) as k 1 β 2 + k 2 β + k 3 > 0 with \(k_{1}=- {\Delta }^{8}\widetilde {\gamma }^{2}\mu ^{2} - 2{\Delta }^{6}\widetilde {\gamma }^{2}\mu ,\, k_{2}=4{\Delta }^{6} \widetilde {\gamma } \mu ^{2} + 8{\Delta }^{4}\widetilde {\gamma }^{2}\mu - 16{\Delta }^{2}\widetilde {\gamma }\) and \(k_{3}=32\mu {\Delta }^{2} + 64\).

If μ > 0, then k 1 < 0 < k 3, and thus condition (ii′) is fulfilled by

$$ \beta_{r_{0}}<\beta<\beta_{r_{1}}, $$
(16)

with \(\beta _{r_{0}}<0<\beta _{r_{1}}\) given by

$$\beta_{r_{0}}=\frac{2}{\mu\widetilde{\gamma}{\Delta}^{4}}\cdot\frac{\mu^{2}{\Delta}^{4}+2\mu\widetilde{\gamma} {\Delta}^{2}-4-\sqrt{\xi}}{2+\mu{\Delta}^{2}},\; \beta_{r_{1}}=\frac{2}{\mu\widetilde{\gamma}{\Delta}^{4}}\cdot \frac{\mu^{2}{\Delta}^{4}+2\mu\widetilde{\gamma} {\Delta}^{2}-4+\sqrt{\xi}}{2+\mu{\Delta}^{2}}, $$

where

$$\xi=\mu^{4}{\Delta}^{8} +4\mu^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}+16+4\mu^{3}\widetilde{\gamma}{\Delta}^{6}-16\mu \widetilde{\gamma}{\Delta}^{2}+8\mu^{3}{\Delta}^{6}+24\mu^{2}{\Delta}^{4}+32\mu{\Delta}^{2}. $$

Combining Eqs. 1416 and recalling that β > 0,we find that system (i′) − (iii′) is equivalent to

$$ \max\{\beta_{\ell},0\}<\beta<\min\{\beta_{r_{1}},\beta_{r_{2}}\}. $$
(17)

In particular, if β ≤ 0,i.e., if \(\widetilde {\gamma }\le 2-\mu {\Delta }^{2}\), and\(\min \{\beta _{r_{1}},\beta _{r_{2}}\}>0\), we have the destabilizing scenario, while when \(0<\beta _{\ell }<\min \{\beta _{r_{1}},\beta _{r_{2}}\}\)the mixed scenario occurs; in all other cases we have the unconditionally unstable scenario.

If, instead, μ = 0, then k 1 = 0 and it is easy tosee that conditions (i′) − (iii′) are equivalent to \(\max \{\beta _{\ell },0\}<\beta <\beta _{r_{1}}\),with \(\beta _{\ell }=\frac {4}{\widetilde {\gamma }{\Delta }^{2}} \left (\frac {\widetilde {\gamma }}{2}-1\right )\)and\(\beta _{r_{1}}=\frac {4}{\widetilde {\gamma }{\Delta }^{2}}\). Then, in this case we havea destabilizing scenario (\(\beta _{\ell }\le 0<\beta _{r_{1}}\))if and only if \(\widetilde {\gamma }\in (0,2]\), while a mixed scenario (\(0<\beta _{\ell }<\beta _{r_{1}}\))occurs if and only if \(\widetilde {\gamma }\in (2,4)\).

Now we keep Δ ≠ 0, β and \(\widetilde {\gamma }\) fixed and we solve conditions (i′) − (iii′) with respect to μ. Solving (i′) we find

$$ \mu<\mu_{r}=\frac{2}{{\Delta}^{2}}\cdot\frac{\beta\widetilde{\gamma}{\Delta}^{2}+4-2\widetilde{\gamma}}{\beta \widetilde{\gamma}{\Delta}^{2}+4}, $$
(18)

while (iii′) provides

$$ \mu> \mu_{\ell}=\frac{2}{{\Delta}^{2}}\cdot\frac{\beta\widetilde{\gamma}{\Delta}^{2}-12}{\beta \widetilde{\gamma}{\Delta}^{2}+4}. $$
(19)

Condition (ii′) can be rewritten as

$$ q_{1}\mu^{2}+q_{2}\mu+q_{3}>0, $$
(20)

where \(q_{1}={\Delta }^{6}\beta \widetilde {\gamma }(4-\beta \widetilde {\gamma }{\Delta }^{2}), \, q_{2}=-2{\Delta }^{2}({\Delta }^{4}\beta ^{2}\widetilde {\gamma }^{2} - 4{\Delta }^{2}\beta \widetilde {\gamma }^{2} - 16)\)and \(q_{3}= 16(4-\beta \widetilde {\gamma }{\Delta }^{2})\).

First we notice that, if q 1 = 0, i.e., if \(\beta =4/(\widetilde {\gamma }{\Delta }^{2})\), then also q 3 = 0and thus (20) is satisfied for all μ > 0 if and only if \({\Delta }^{4}\beta ^{2}\widetilde {\gamma }^{2}={\Delta }^{4} \left (4/(\widetilde {\gamma } {\Delta }^{2})\right )^{2}\widetilde {\gamma }^{2}=16< 4{\Delta }^{2}\beta \widetilde {\gamma }^{2}+ 16=32\), which is indeed true. We stress that if μ = 0 = q 1, then condition (ii′) is again never fulfilled and thus we do not have stability for μ = 0 if q 1 = 0. Hence, when q 1 = 0 conditions (i′) − (iii′) simply reduce to\(\mu \in (\mu _{\ell },\mu _{r})\cap (0,+\infty )\), and this provides the destabilizing scenario for μ < 0 < μ r , the mixed scenariofor 0 < μ < μ r , and the unconditionally unstable scenario for \(\mu _{r}\le \max \{\mu _{\ell },0\}\).

Let us now assume that q 1 ≠ 0. If \(\chi ={q_{2}^{2}}-4q_{1}q_{3}>0\), we can introduce the real numbers

$$\mu_{1}=\frac{\beta^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}-16-4\beta{\widetilde{\gamma}}^{2}{\Delta}^{2}- \sqrt{\chi}}{\beta\widetilde{\gamma}{\Delta}^{4}(4-\beta\widetilde{\gamma}{\Delta}^{2})}, $$

and

$$\mu_{2}=\frac{\beta^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}-16-4\beta{\widetilde{\gamma}}^{2}{\Delta}^{2}+\sqrt{\chi}} {\beta\widetilde{\gamma}{\Delta}^{4}(4-\beta\widetilde{\gamma}{\Delta}^{2})}, $$

where

$$\begin{array}{@{}rcl@{}} \chi&=&256+16\beta^{2}{\widetilde{\gamma}}^{4}{\Delta}^{4}+\beta^{4}{\widetilde{\gamma}}^{4}{\Delta}^{8}+128 \beta{\widetilde{\gamma}}^{2}{\Delta}^{2}+96\beta^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}\\ &&-8\beta^{3}{\widetilde{\gamma}}^{4}{\Delta}^{6}-16\beta^{3}{\widetilde{\gamma}}^{3}{\Delta}^{6}-256 \beta{\widetilde{\gamma}}{\Delta}^{2}. \end{array} $$

We then have that Eq. 20 is solved by \(\mu <\mu _{1}\,\cup \,\mu >\mu _{2}\) if q 1 > 0 and by μ 1 < μ < μ 2if q 1 < 0. Let us examine theformer case. Since we have q 1 > 0 and q 3 > 0, the sign of μ 1/2 is the same of − q 2. Since from q 1 > 0 we have \(\beta <4/(\widetilde {\gamma }{\Delta }^{2})\), we obtain

$$\beta^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}-16-4\beta{\widetilde{\gamma}}^{2}{\Delta}^{2}<\left( \frac{4}{{\Delta}^{2} \widetilde{\gamma}}\right)^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}-16-4\beta{\widetilde{\gamma}}^{2}{\Delta}^{2}=-4 \beta{\widetilde{\gamma}}^{2}{\Delta}^{2}<0, $$

so that, in such case, we have q 2 > 0 and thus μ 1 < μ 2 < 0and Eq. 20 is fulfilled by any μ ≥ 0. Combining this with Eqs. 18 and 19 we obtain \(\mu \in (\mu _{\ell },\mu _{r})\cap [0,+\infty )\), which provides the destabilizing scenario if μ < 0 < μ r , the mixed scenario if 0 < μ < μ r and the unconditionally unstable scenario if \(\mu _{r}\le \max \{\mu _{\ell },0\}\).

Conversely, if q 1 < 0, combining μ 1 < μ < μ 2 with Eqs. 18 and 19 we have \(\mu \in (\max \left \{\mu _{1},\mu _{\ell }\right \},\min \left \{\mu _{2},\mu _{r}\right \})\cap [0,+\infty )\), which can again give rise to either a destabilizing, a mixed or an unconditionally unstable scenario.

If \({q_{2}^{2}}-4q_{1}q_{3}<0\), we have that Eq. 20 is always fulfilled if \(4-\beta \widetilde {\gamma }{\Delta }^{2}>0\) and never fulfilled when \(4-\beta \widetilde {\gamma }{\Delta }^{2}<0\). In the former case, recalling Eqs. 18 and 19, we obtain \(\mu \in (\mu _{\ell },\mu _{r})\cap [0,+\infty )\), and thus we can have destabilizing, mixed and unconditionally unstable scenarios. If instead \(4-\beta \widetilde {\gamma }{\Delta }^{2}<0\), conditions (i′) − (iii′) can not be satisfied and we just find the unconditionally unstable scenario.

It can be easily seen that the remaining situation \({q_{2}^{2}}-4q_{1}q_{3}=0\) can only provide the previous scenarios.Footnote 4

Finally, the stability conditions with respect to Δ when μ = 0 and β ≠ 0 read as

$$\sqrt{\frac{2\widetilde{\gamma}-4}{\widetilde{\gamma}\beta}}<{\Delta}<\frac{2}{\sqrt{\widetilde{\gamma}\beta}}\,, $$

when \(\widetilde {\gamma }\ge 2\), and simply as \(0\le {\Delta }<\frac {2}{\sqrt {\widetilde {\gamma }\beta }}\,\), when \(\widetilde {\gamma }< 2\). For \(\widetilde {\gamma }\ge 2\) we then find the mixed scenario and for \(\widetilde {\gamma }<2\) the destabilizing scenario. The proof is complete. □

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Cavalli, F., Naimzada, A. & Pireddu, M. An evolutive financial market model with animal spirits: imitation and endogenous beliefs. J Evol Econ 27, 1007–1040 (2017). https://doi.org/10.1007/s00191-017-0506-8

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