Financial models with interacting heterogeneous agents ...pireddu/Beamer-Insubria-2D.pdf · Outline 1 2D discrete dynamical systems 2 Other Heterogeneous Agents Models Marina Pireddu

Financial models withinteracting heterogeneous agents:

modeling assumptions and mathematical toolsfrom discrete dynamical system theory.

Minicourse for the PhD Program in Methods and Modelsfor Economic Decisions, Insubria University

Marina Pireddu

University of Milano-BicoccaDept. of Mathematics and its Applications

[email protected]

Marina Pireddu (Univ. of Milano-Bicocca) Discrete-time heterogeneous agent models Insubria Univ., 20/03/2018 1 / 112

Outline

1 2D discrete dynamical systems

2 Other Heterogeneous Agents Models


2D discrete dynamical systems

Classification of 2D discrete dynamical systems

Consider the map F : R2 → R2, F = (f1, f2), withfi : R2 → R, (x1, x2)→ fi(x1, x2), i ∈ 1,2.

A first-order 2D discrete dynamical system is a sequence of vectorsXt = (x1,t , x2,t ), for t = 0,1,2, . . . , such that each vector after the first isrelated just to the previous vector by the relationship Xt+1 = F (Xt ),where F : R2 → R2.

If F is linear, i.e., F (Xt ) = AXt with A =

(a11 a12

a21 a22

), 2× 2 matrix,

the system is said to be linear;

if F is nonlinear, i.e., if F is not linear, then the system is said to benonlinear.







(a11 a12

a21 a22

), 2× 2 matrix,









(a11 a12

a21 a22

), 2× 2 matrix,









(a11 a12

a21 a22

), 2× 2 matrix,









(a11 a12

a21 a22

), 2× 2 matrix,





Equilibria and stability of 2D discrete dynamicalsystems

If Xt+1 = F (Xt ) is a 2D discrete dynamical system, then X ∗ = (x∗1 , x∗2 )

is a fixed point or equilibrium point of the system if F (X ∗) = X ∗, i.e.,fi(x∗1 , x

∗2 ) = x∗i , i ∈ 1,2.

For 2D linear systems, X ∗ = (0,0) is always an equilibrium.

Rather than the Euclidean distance, in R2 we use the norm ‖ · ‖1defined as ‖(x1, x2)‖1 = |x1|+ |x2|.

For ease of notation, we will denote ‖ · ‖1 simply by | · |.






∗2 ) = x∗i , i ∈ 1,2.









∗2 ) = x∗i , i ∈ 1,2.









∗2 ) = x∗i , i ∈ 1,2.









∗2 ) = x∗i , i ∈ 1,2.






Given Xt+1 = F (Xt ), with F : R2 → R2, the equilibrium point X ∗ ∈ R2 isstable if for all ε > 0 there exists δ > 0 such that for all X ∈ R2 with|X − X ∗| < δ it holds that |F t (X )− X ∗| < ε, for all t ∈ N \ 0.

If X ∗ is not stable then it is called unstable.

If X ∗ is stable and attracting, i.e., there exists η > 0 such that for allX ∈ R2 with |X − X ∗| < η it holds that limt→+∞ F t (X ) = X ∗, for t ∈ N,then X ∗ is called locally asymptotically stable.

If η = +∞, then X ∗ is called globally asymptotically stable.





















X ∗ = (0,0) is globally asymptotically stable



X ∗ = (0,0) is unstable

These are phase portraits in the (x1, x2)-plane.

They are useful to draw 2D orbits X0, F (X0), F 2(X0), . . . .













How do we check local stability?

First, we will deal with 2D linear dynamical systems.

Then, we will consider 2D nonlinear dynamical systems.













Local stability of 2D linear dynamical systems

We want to study the stability, at the equilibrium X ∗ = (0,0), of

F (Xt ) = AXt

with A =

(a11 a12

a21 a22

), 2× 2 matrix.

Given a 2× 2 matrix A, we define its spectral radius, and we denote itby ρ(A), as:

ρ(A) = max|λ1|, |λ2|, λ1 and λ2 eigenvalues of A.





F (Xt ) = AXt

with A =

(a11 a12

a21 a22

), 2× 2 matrix.







F (Xt ) = AXt

with A =

(a11 a12

a21 a22

), 2× 2 matrix.





TheoremFor F (Xt ) = AXt it holds that:

(i) if ρ(A) < 1, then X ∗ = (0,0) is globally asymptotically stable;(ii) if ρ(A) > 1, then X ∗ = (0,0) is unstable;(iii) if ρ(A) = 1, then X ∗ = (0,0) may be unstable or not.











Navigating the Trace-Determinant plane

We recall that, given A =

(a11 a12

a21 a22

),

the trace of A, denoted by tr(A), is defined as tr(A) = a11 + a22;the determinant of A, denoted by det(A), is defined asdet(A) = a11a22 − a12a21.

TheoremLet A be a 2× 2 matrix. Then ρ(A) < 1⇔ |tr(A)| − 1 < det(A) < 1.

Corollary (Jury conditions)Let A be a 2× 2 matrix. Thenρ(A) < 1⇔ 1 + tr(A) + det(A) > 0, 1− tr(A) + det(A) > 0 anddet(A) < 1.





(a11 a12

a21 a22

),








(a11 a12

a21 a22

),








(a11 a12

a21 a22

),








(a11 a12

a21 a22

),






Both eigenvalues are real when tr(A)2 − 4 det(A) ≥ 0.



Both eigenvalues are real when tr(A)2 − 4 det(A) ≥ 0.







Local stability of 2D nonlinear dynamical systems

If we are considering a nonlinear 2D system, i.e., Xt+1 = F (Xt ), forsome generic map F ∈ C1 having X ∗ as fixed point, then our matrix isJ = DF (X ∗), i.e., the Jacobian matrix of F computed at X ∗.

Indeed, if F : Ω→ R2, F ∈ C1(Ω), Ω ⊆ R2 open set and X ∗ ∈ Ω is afixed point of F , then we can linearize F in a neighborhood of X ∗ asfollows:

F (X )− X ∗ = DF (X ∗)(X − X ∗) + G(X − X ∗),

with G(X − X ∗) = o(|X − X ∗|) as X − X ∗ → 0.

Setting Y = X − X ∗ and H(Y ) = F (Y + X ∗)− X ∗, we obtain that0 = (0,0) is a fixed point of H and

H(Y ) = DH(0)Y + G(Y ),

with G(Y ) = o(|Y |) as Y → 0.






F (X )− X ∗ = DF (X ∗)(X − X ∗) + G(X − X ∗),

with G(X − X ∗) = o(|X − X ∗|) as X − X ∗ → 0.


H(Y ) = DH(0)Y + G(Y ),

with G(Y ) = o(|Y |) as Y → 0.






F (X )− X ∗ = DF (X ∗)(X − X ∗) + G(X − X ∗),

with G(X − X ∗) = o(|X − X ∗|) as X − X ∗ → 0.


H(Y ) = DH(0)Y + G(Y ),

with G(Y ) = o(|Y |) as Y → 0.






F (X )− X ∗ = DF (X ∗)(X − X ∗) + G(X − X ∗),

with G(X − X ∗) = o(|X − X ∗|) as X − X ∗ → 0.


H(Y ) = DH(0)Y + G(Y ),

with G(Y ) = o(|Y |) as Y → 0.






F (X )− X ∗ = DF (X ∗)(X − X ∗) + G(X − X ∗),

with G(X − X ∗) = o(|X − X ∗|) as X − X ∗ → 0.


H(Y ) = DH(0)Y + G(Y ),

with G(Y ) = o(|Y |) as Y → 0.






F (X )− X ∗ = DF (X ∗)(X − X ∗) + G(X − X ∗),

with G(X − X ∗) = o(|X − X ∗|) as X − X ∗ → 0.


H(Y ) = DH(0)Y + G(Y ),

with G(Y ) = o(|Y |) as Y → 0.



TheoremLet X ∗ be an equilibrium point of the 2D dynamical systemXt+1 = F (Xt ), with F : Ω→ R2, F ∈ C1(Ω), Ω ⊆ R2 open set.Denoting by J = DF (X ∗) the Jacobian matrix of F computed at X ∗, itholds that:

(i) if ρ(J) < 1, then X ∗ is locally asymptotically stable;

(ii) if ρ(J) > 1, then X ∗ is unstable;

(iii) if ρ(J) = 1, then X ∗ may be unstable or not.

Corollary (Jury conditions)Let X ∗ be an equilibrium point of the 2D dynamical systemXt+1 = F (Xt ), with F : Ω→ R2, F ∈ C1(Ω), Ω ⊆ R2 open set.Denoting by J = DF (X ∗) the Jacobian matrix of F computed at X ∗, itholds that, if 1 + tr(J) + det(J) > 0, 1− tr(J) + det(J) > 0 anddet(J) < 1, then X ∗ is locally asymptotically stable.






































Main 2D bifurcation phenomena

Let us consider the one-parameter family of 2D mapsF (X ;µ) : R2 × R→ R, with X = (x1, x2) ∈ R2, µ ∈ R and F ∈ Cr , for asuitable r (r ≥ 5).

If (X ∗, µ∗) is a fixed point of F , then we make a change of variables, sothat our fixed point is (0,0).

Let J = DX F (0,0).

Let T = tr(J) and D = det(J).






Let J = DX F (0,0).







Let J = DX F (0,0).







Let J = DX F (0,0).







Let J = DX F (0,0).




Then the following trace-determinant diagram illustrates the main 2Dbifurcation phenomena:



In addition to the bifurcations introduced for the 1D case, 2D maps canundergo Neimark-Sacker bifurcations, usually associated with theexistence of a (repelling or attracting) closed invariant curve.



We recall the table for the 1D bifurcations

Similar conditions characterize the 2D bifurcations, when replacing∂g∂x (x∗, µ∗) = ±1 with the existence of an eigenvalue= ±1 for J.

However, some of those conditions also involve the center manifold.













The Neimark-Sacker bifurcation is characterized by the presence of apair of complex conjugate eigenvalues of modulus 1.

A 1D analogue of the Neimark-Sacker bifurcation does not exist.



The Neimark-Sacker bifurcation is characterized by the presence of apair of complex conjugate eigenvalues of modulus 1.

A 1D analogue of the Neimark-Sacker bifurcation does not exist.



References on 2D discrete dynamical systems:

– Elaydi SN (2007) Discrete Chaos, Second Edition: With Applicationsin Science and Engineering. CRC Press, Taylor & Francis Group,Boca Raton, Florida. Chapters 4-5, Paragraphs 4.1, 4.8, 4.11, 5.2

– Jury EI (1964) Theory and Application of the z-transform Method.John Wiley and Sons, New York.

– Shone R (2002) Economic Dynamics. Phase Diagrams and TheirEconomic Application, second ed. Cambridge University Press,Cambridge. Chapter 5, Paragraphs 5.1, 5.3, 5.6, 5.9


Other Heterogeneous Agents Models

A 2D analysis of the model in Westerhoff (2012)

We recall the 2D framework in Westerhoff (2012) with (fully) interactingreal and financial markets:

Yt+1 = A + cYt + αPt ,

Pt+1 = Pt + η(Pt − dYt ) + σ(dYt − Pt )3.



A 2D analysis of the model in Westerhoff (2012)

We recall the 2D framework in Westerhoff (2012) with (fully) interactingreal and financial markets:

Yt+1 = A + cYt + αPt ,

Pt+1 = Pt + η(Pt − dYt ) + σ(dYt − Pt )3.



Proposition (interacting goods and stock markets)

The dynamics of the complete model is due to a two-dimensionalnonlinear map, given by Yt+1 = A + cYt + αPt andPt+1 = Pt + η(Pt − dYt ) + σ(dYt − Pt )

3. This map has three steadystates Y1 = A

1−c−dα , P1 = dY1 and Y 2,3 = Y1 ± α1−c−dα

√ησ ,

P2,3 = P1 ± 1−c1−c−dα

√ησ . All steady states of the model are positive if

c + dα < 1 and if A is sufficiently large. Given these requirements,steady state (Y1,P1) is unstable whereas steady states (Y 2,3,P2,3) arelocally asymptotically stable for η < (1 + c)/(1 + c + dα).







√ησ ,

P2,3 = P1 ± 1−c1−c−dα









√ησ ,

P2,3 = P1 ± 1−c1−c−dα





Indeed,

F : (0,+∞)2 → R2, F = (f1, f2), (Y ,P) 7→ fi(Y ,P), i ∈ 1,2,

withf1(Y ,P) = A + cY + αP,

f2(Y ,P) = P + η(P − dY ) + σ(dY − P)3.

Hence,∂f1∂Y

(Y ,P) = c

∂f1∂P

(Y ,P) = α

∂f2∂Y

(Y ,P) = −dη + 3dσ(dY − P)2 = d(3σ(dY − P)2 − η)

∂f2∂P

(Y ,P) = 1 + η − 3σ(dY − P)2



Indeed,

F : (0,+∞)2 → R2, F = (f1, f2), (Y ,P) 7→ fi(Y ,P), i ∈ 1,2,

withf1(Y ,P) = A + cY + αP,

f2(Y ,P) = P + η(P − dY ) + σ(dY − P)3.

Hence,∂f1∂Y

(Y ,P) = c

∂f1∂P

(Y ,P) = α

∂f2∂Y

(Y ,P) = −dη + 3dσ(dY − P)2 = d(3σ(dY − P)2 − η)

∂f2∂P

(Y ,P) = 1 + η − 3σ(dY − P)2



Thus,∂f1∂Y

(Y1,P1) =∂f1∂Y

(Y2,P2) =∂f1∂Y

(Y3,P3) = c

∂f1∂P

(Y1,P1) =∂f1∂P

(Y2,P2) =∂f1∂P

(Y3,P3) = α

∂f2∂Y

(Y1,P1) = −dη

∂f2∂P

(Y1,P1) = 1 + η

⇒ J(Y1,P1) =

(c α

−dη 1 + η

)

At (Y1,P1) we have tr(J) = c + 1 + η, det(J) = c + cη + dαη.



Thus,∂f1∂Y

(Y1,P1) =∂f1∂Y

(Y2,P2) =∂f1∂Y

(Y3,P3) = c

∂f1∂P

(Y1,P1) =∂f1∂P

(Y2,P2) =∂f1∂P

(Y3,P3) = α

∂f2∂Y

(Y1,P1) = −dη

∂f2∂P

(Y1,P1) = 1 + η

⇒ J(Y1,P1) =

(c α

−dη 1 + η

)




Thus,∂f1∂Y

(Y1,P1) =∂f1∂Y

(Y2,P2) =∂f1∂Y

(Y3,P3) = c

∂f1∂P

(Y1,P1) =∂f1∂P

(Y2,P2) =∂f1∂P

(Y3,P3) = α

∂f2∂Y

(Y1,P1) = −dη

∂f2∂P

(Y1,P1) = 1 + η

⇒ J(Y1,P1) =

(c α

−dη 1 + η

)




Recalling the Jury conditions

1 + tr(J) + det(J) > 0, 1− tr(J) + det(J) > 0, det(J) < 1,

at (Y1,P1) we have:

• 1 + tr(J) + det(J) = 2 + 2c + η(1 + c + dα) > 0 OK

• 1− tr(J) + det(J) = η(c + dα− 1) > 0 NO

• det(J) = c + cη + dαη < 1

so that (Y1,P1) is always unstable.





at (Y1,P1) we have:

• 1 + tr(J) + det(J) = 2 + 2c + η(1 + c + dα) > 0 OK

• 1− tr(J) + det(J) = η(c + dα− 1) > 0 NO

• det(J) = c + cη + dαη < 1






at (Y1,P1) we have:

• 1 + tr(J) + det(J) = 2 + 2c + η(1 + c + dα) > 0 OK

• 1− tr(J) + det(J) = η(c + dα− 1) > 0 NO

• det(J) = c + cη + dαη < 1






at (Y1,P1) we have:

• 1 + tr(J) + det(J) = 2 + 2c + η(1 + c + dα) > 0 OK

• 1− tr(J) + det(J) = η(c + dα− 1) > 0 NO

• det(J) = c + cη + dαη < 1






at (Y1,P1) we have:

• 1 + tr(J) + det(J) = 2 + 2c + η(1 + c + dα) > 0 OK

• 1− tr(J) + det(J) = η(c + dα− 1) > 0 NO

• det(J) = c + cη + dαη < 1




Since∂f2∂Y

(Y ,P) = d(3σ(dY − P)2 − η)

∂f2∂P

(Y ,P) = 1 + η − 3σ(dY − P)2

and dY 2,3 − P2,3 = ∓√

ησ , then:

∂f2∂Y

(Y2,P2) =∂f2∂Y

(Y3,P3) = d(

3ση

σ− η)

= 2dη

∂f2∂P

(Y2,P2) =∂f2∂P

(Y3,P3) = 1 + η − 3ση

σ= 1− 2η

⇒ J(Y2,P2) = J(Y3,P3) =

(c α

2dη 1− 2η

)At (Y2,P2) and (Y3,P3) we have:

tr(J) = c + 1− 2η, det(J) = c − 2cη − 2dαη.



Since∂f2∂Y

(Y ,P) = d(3σ(dY − P)2 − η)

∂f2∂P

(Y ,P) = 1 + η − 3σ(dY − P)2

and dY 2,3 − P2,3 = ∓√

ησ , then:

∂f2∂Y

(Y2,P2) =∂f2∂Y

(Y3,P3) = d(

3ση

σ− η)

= 2dη

∂f2∂P

(Y2,P2) =∂f2∂P

(Y3,P3) = 1 + η − 3ση

σ= 1− 2η

⇒ J(Y2,P2) = J(Y3,P3) =

(c α

2dη 1− 2η





Since∂f2∂Y

(Y ,P) = d(3σ(dY − P)2 − η)

∂f2∂P

(Y ,P) = 1 + η − 3σ(dY − P)2

and dY 2,3 − P2,3 = ∓√

ησ , then:

∂f2∂Y

(Y2,P2) =∂f2∂Y

(Y3,P3) = d(

3ση

σ− η)

= 2dη

∂f2∂P

(Y2,P2) =∂f2∂P

(Y3,P3) = 1 + η − 3ση

σ= 1− 2η

⇒ J(Y2,P2) = J(Y3,P3) =

(c α

2dη 1− 2η





Since∂f2∂Y

(Y ,P) = d(3σ(dY − P)2 − η)

∂f2∂P

(Y ,P) = 1 + η − 3σ(dY − P)2

and dY 2,3 − P2,3 = ∓√

ησ , then:

∂f2∂Y

(Y2,P2) =∂f2∂Y

(Y3,P3) = d(

3ση

σ− η)

= 2dη

∂f2∂P

(Y2,P2) =∂f2∂P

(Y3,P3) = 1 + η − 3ση

σ= 1− 2η

⇒ J(Y2,P2) = J(Y3,P3) =

(c α

2dη 1− 2η





Since∂f2∂Y

(Y ,P) = d(3σ(dY − P)2 − η)

∂f2∂P

(Y ,P) = 1 + η − 3σ(dY − P)2

and dY 2,3 − P2,3 = ∓√

ησ , then:

∂f2∂Y

(Y2,P2) =∂f2∂Y

(Y3,P3) = d(

3ση

σ− η)

= 2dη

∂f2∂P

(Y2,P2) =∂f2∂P

(Y3,P3) = 1 + η − 3ση

σ= 1− 2η

⇒ J(Y2,P2) = J(Y3,P3) =

(c α

2dη 1− 2η





Hence,

• 1 + tr(J) + det(J) = 2 + 2c − 2η(1 + c + dα) > 0

• 1− tr(J) + det(J) = 2η(1− c − dα) > 0 OK

• det(J) = c − 2cη − 2dαη < 1 OK (c < 1)

The first condition is satisfied for η < 1+c1+c+dα . This ensures the stability

of both (Y2,P2) and (Y3,P3).

Without the interaction degree approach, in order to compare thesystem stability when the the real and financial markets are isolated orinterconnected, Westerhoff (2012) compares the stability conditions atthe various equilibria.



Hence,

• 1 + tr(J) + det(J) = 2 + 2c − 2η(1 + c + dα) > 0

• 1− tr(J) + det(J) = 2η(1− c − dα) > 0 OK

• det(J) = c − 2cη − 2dαη < 1 OK (c < 1)






Hence,

• 1 + tr(J) + det(J) = 2 + 2c − 2η(1 + c + dα) > 0

• 1− tr(J) + det(J) = 2η(1− c − dα) > 0 OK

• det(J) = c − 2cη − 2dαη < 1 OK (c < 1)






We recall that for the isolated markets framework we had:

• Y ∗ = A+αP1−c is globally asymptotically stable;

• P∗1 = dY is unstable, P∗2,3 = P∗1 ±√

ησ are locally asymptotically

stable for η < η∗ = 1.

For the interacting markets framework it holds that:

• (Y1,P1) =(

A1−c−dα ,

dA1−c−dα

)is always unstable.

• (Y2,P2) and (Y3,P3) are locally asymptotically stable forη < η = 1+c

1+c+dα .

Since η < η∗, Westerhoff (2012) concludes that the interactionbetween markets impairs stability.









• (Y1,P1) =(

A1−c−dα ,

dA1−c−dα



1+c+dα .










• (Y1,P1) =(

A1−c−dα ,

dA1−c−dα



1+c+dα .










• (Y1,P1) =(

A1−c−dα ,

dA1−c−dα



1+c+dα .










• (Y1,P1) =(

A1−c−dα ,

dA1−c−dα



1+c+dα .




Moreover, since|P2 − P1| = |P3 − P1| = 1−c

1−c−dα

√ησ >

√ησ = |P2

∗ − P∗1 | = |P3∗ − P∗1 |,

Westerhoff (2012) concludes that the interaction between marketsmakes the model’s steady-state values more extreme.

We could study instead the stability of the steady states and considerthe bifurcation diagrams w.r.t. ω ∈ [0,1] of the map Fω associated to:

Yt+1 =A + cYt + α(ωPt + (1− ω)P)

Pt+1 =Pt + η(Pt − d(ωYt + (1− ω)Y )) + σ(d(ωYt + (1− ω)Y )− Pt )3



Moreover, since|P2 − P1| = |P3 − P1| = 1−c

1−c−dα

√ησ >

√ησ = |P2

∗ − P∗1 | = |P3∗ − P∗1 |,

Westerhoff (2012) concludes that the interaction between marketsmakes the model’s steady-state values more extreme.

We could study instead the stability of the steady states and considerthe bifurcation diagrams w.r.t. ω ∈ [0,1] of the map Fω associated to:

Yt+1 =A + cYt + α(ωPt + (1− ω)P)

Pt+1 =Pt + η(Pt − d(ωYt + (1− ω)Y )) + σ(d(ωYt + (1− ω)Y )− Pt )3



The bifurcation diagram for Y and P w.r.t. ω ∈ [0,1] of the map Fωwhen A = 3, c = 0.95, α = 0.02, d = 1, η = 1.63, σ = 0.3



A 3D framework: the model in Naimzada and Pireddu(2015b)

In addition to the real and the financial sectors, we now introduce ashare updating mechanism between optimistic and pessimisticfundamentalists, similar to De Grauwe and Rovira Kaltwasser (2012).

The real sector is described as a Keynesian good market.

Like in Westerhoff (2012) and in Naimzada and Pireddu (2014b), wesuppose that if the stock price increases, the same does privateexpenditure.





















Hence, aggregate demand is given by

Zt = Ct + It + Gt = A + bYt + ωcPt ,

where

A > 0 defines autonomous expenditure;

b ∈ [0,1] is the marginal propensity to consume and invest fromcurrent income;

c ∈ [0,1] is the marginal propensity to consume and invest fromcurrent stock market wealth;

ω ∈ [0,1] represents the degree of interaction between the realand the stock markets.

Assuming a sigmoidal income adjustment mechanism, we obtain

Yt+1 = Yt + γa2

(a1 + a2

a1e−(A+bYt +ωcPt−Yt ) + a2− 1).





where






Yt+1 = Yt + γa2

(a1 + a2






where






Yt+1 = Yt + γa2

(a1 + a2






where






Yt+1 = Yt + γa2

(a1 + a2






where






Yt+1 = Yt + γa2

(a1 + a2






where






Yt+1 = Yt + γa2

(a1 + a2




The financial sector is populated by optimistic and pessimisticfundamentalists.

Agents are not able to observe the true underlying fundamental.

Like in De Grauwe and Rovira Kaltwasser (2012), optimists(pessimists) systematically overestimate (underestimate) the referencevalue used in their decisional mechanism.

In De Grauwe and Rovira Kaltwasser (2012), the perceived referencevalues are exogenous, i.e., F opt = F ∗+a and F pes = F ∗−a, wherea > 0 is the belief bias and F ∗ is the true unobserved fundamental.





















The perceived reference values are for us a weighted averagebetween an exogenous value, like in De Grauwe and RoviraKaltwasser (2012), and a term depending on the income value,similarly to Westerhoff (2012) and Naimzada and Pireddu (2014b):

F optt = (1− ω)(F ∗+a) + ω(kYt +a) = (1− ω)F ∗ + ωkYt +a

and

F pest = (1− ω)(F ∗−a) + ω(kYt−a) = (1− ω)F ∗ + ωkYt−a,

where a > 0 is the belief bias and F ∗ is the true unobservedfundamental.

Moreover, k > 0 captures the direct relationship between theperceived reference values and income, while ω ∈ [0,1] is theweighting average parameter.





and








and






We assume the market maker behavior to be described by the linearprice adjustment mechanism

Pt+1 = Pt + µ(noptt dopt

t + npest dpes

t ),

where

µ > 0 is the market maker price adjustment parameter;

nit , i ∈ opt ,pes, is the fraction of traders of type i at time t ;

d it = α(F i

t − Pt ), i ∈ opt ,pes, is the demand of traders of type iand α > 0 is the reactivity parameter.

Normalizing the population to 1 and setting xt = noptt − npes

t , we obtain

Pt+1 = Pt + αµ [(1− ω)F ∗ + ωkYt ]− Pt + axt.





t + npest dpes

t ),

where



d it = α(F i



t , we obtain






t + npest dpes

t ),

where



d it = α(F i



t , we obtain






t + npest dpes

t ),

where



d it = α(F i



t , we obtain






t + npest dpes

t ),

where



d it = α(F i



t , we obtain




Following Anderson et al. (1992) and Brock and Hommes (1997), weassume that the fraction ni

t of traders of type i is given by the discretechoice model

nit =

exp(βπit )

exp(βπoptt ) + exp(βπpes

t ),

where β ≥ 0 is the parameter representing the intensity of choice andπi

t = d it−1(Pt − Pt−1) are the profits realized by type i , i ∈ opt ,pes.

In the limit β → 0 there is no switching and both the population sharescoincide with 1/2.

When instead β → +∞, the whole population moves towards optimismor pessimism, according to which option is more profitable.





nit =

exp(βπit )


t ),









nit =

exp(βπit )


t ),









nit =

exp(βπit )


t ),







Since

πoptt − πpes

t = (doptt−1 − dpes

t−1)(Pt − Pt−1)

= 2aµα2 [(1− ω)F ∗ + ωkYt−1]− Pt−1 + axt−1 ,

our model is described byPt+1 = Pt + αµ [(1− ω)F ∗ + ωkYt ]− Pt + axt

xt+1 =exp(2aβµα2[(1−ω)F∗+ωkYt−1]−Pt−1+axt−1)−1exp(2aβµα2[(1−ω)F∗+ωkYt−1]−Pt−1+axt−1)+1

Yt+1 = Yt + γa2

(a1+a2

a1e−(A+bYt +ωcPt−Yt )+a2− 1)

We stress that if x were exogenously fixed in (−1,1), the model wouldbecome 2D and, similarly to Westerhoff (2012), the real and thefinancial sectors would be described by one equation each.

However, in our case the nonlinearity would be present in the real,rather than in the financial, side of the economy.



Since

πoptt − πpes


t−1)(Pt − Pt−1)

= 2aµα2 [(1− ω)F ∗ + ωkYt−1]− Pt−1 + axt−1 ,



Yt+1 = Yt + γa2

(a1+a2






Since

πoptt − πpes


t−1)(Pt − Pt−1)

= 2aµα2 [(1− ω)F ∗ + ωkYt−1]− Pt−1 + axt−1 ,



Yt+1 = Yt + γa2

(a1+a2






Since

πoptt − πpes


t−1)(Pt − Pt−1)

= 2aµα2 [(1− ω)F ∗ + ωkYt−1]− Pt−1 + axt−1 ,



Yt+1 = Yt + γa2

(a1+a2






Isolated market framework (ω = 0)

The system above splits into the 2D subsystem related to the stockmarket Pt+1 = Pt + αµ (F ∗ − Pt + axt )

xt+1 =exp(2aβµα2F∗−Pt−1+axt−1)−1exp(2aβµα2F∗−Pt−1+axt−1)+1

and the 1D subsystem related to the real market

Yt+1 = Yt + γa2

(a1 + a2

a1e−(A−(1−b)Yt ) + a2− 1).

The unique steady state is given by

(P∗, x∗) = (F ∗,0), Y ∗ =A

1− b.







Yt+1 = Yt + γa2

(a1 + a2

a1e−(A−(1−b)Yt ) + a2− 1).


(P∗, x∗) = (F ∗,0), Y ∗ =A

1− b.







Yt+1 = Yt + γa2

(a1 + a2

a1e−(A−(1−b)Yt ) + a2− 1).


(P∗, x∗) = (F ∗,0), Y ∗ =A

1− b.







Yt+1 = Yt + γa2

(a1 + a2

a1e−(A−(1−b)Yt ) + a2− 1).


(P∗, x∗) = (F ∗,0), Y ∗ =A

1− b.



The Jacobian matrix computed in correspondence to the steady statesplits into

J1(P∗, x∗) =

[1− αµ αµa

−µaα2β α2µa2β

], J2(Y ∗) = 1− γa1a2(1− b)

a1 + a2.

The Jury conditions for the stability of the financial subsystem read as

1 + tr(J1) + det(J1) = 2− µα + 2µα2a2β > 0,

1− tr(J1) + det(J1) = µα > 0,

det(J1) = µα2a2β < 1.

The second condition is always fulfilled, while the other two can berewritten, making β explicit, as

αµ− 22µα2a2 < β <

1µα2a2 .




J1(P∗, x∗) =

[1− αµ αµa


], J2(Y ∗) = 1− γa1a2(1− b)

a1 + a2.


1 + tr(J1) + det(J1) = 2− µα + 2µα2a2β > 0,

1− tr(J1) + det(J1) = µα > 0,

det(J1) = µα2a2β < 1.


αµ− 22µα2a2 < β <

1µα2a2 .




J1(P∗, x∗) =

[1− αµ αµa


], J2(Y ∗) = 1− γa1a2(1− b)

a1 + a2.


1 + tr(J1) + det(J1) = 2− µα + 2µα2a2β > 0,

1− tr(J1) + det(J1) = µα > 0,

det(J1) = µα2a2β < 1.


αµ− 22µα2a2 < β <

1µα2a2 .



The real subsystem is locally asymptotically stable at the steady state if

−1 < 1− γa1a2(1− b)

a1 + a2< 1,

i.e., if

γ <2(a1 + a2)

a1a2(1− b).

Hence, when ω = 0 both subsystems are stable if

αµ− 22µα2a2 < β <

1µα2a2 and γ <

2(a1 + a2)

a1a2(1− b).

Thus, when isolated, both sectors can be stable, both can be unstable,and the mixed scenarios are possible, as well.




−1 < 1− γa1a2(1− b)

a1 + a2< 1,

i.e., if

γ <2(a1 + a2)

a1a2(1− b).


αµ− 22µα2a2 < β <

1µα2a2 and γ <

2(a1 + a2)

a1a2(1− b).





−1 < 1− γa1a2(1− b)

a1 + a2< 1,

i.e., if

γ <2(a1 + a2)

a1a2(1− b).


αµ− 22µα2a2 < β <

1µα2a2 and γ <

2(a1 + a2)

a1a2(1− b).





−1 < 1− γa1a2(1− b)

a1 + a2< 1,

i.e., if

γ <2(a1 + a2)

a1a2(1− b).


αµ− 22µα2a2 < β <

1µα2a2 and γ <

2(a1 + a2)

a1a2(1− b).




In order to investigate what happens to the system stability when thetwo markets are interconnected, we need to study the stability of the3D system for ω ∈ (0,1].

This can be done using the conditions in Farebrother (1973):

(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 characteristicpolynomial

λ3 + C1λ2 + C2λ+ C3 = 0

associated to the Jacobian matrix computed in correspondence to thesteady state.



In order to investigate what happens to the system stability when thetwo markets are interconnected, we need to study the stability of the3D system for ω ∈ (0,1].

This can be done using the conditions in Farebrother (1973):

(i) 1 + C1 + C2 + C3 > 0;

(ii) 1− C1 + C2 − C3 > 0;

(iii) 1− C2 + C1C3 − (C3)2 > 0;

(iv) 3− C2 > 0,


λ3 + C1λ2 + C2λ+ C3 = 0

associated to the Jacobian matrix computed in correspondence to thesteady state.



Calling G the map associated to the complete systemPt+1 = Pt + αµ [(1− ω)F ∗ + ωkYt ]− Pt + axt


Yt+1 = Yt + γa2

(a1+a2


it admits the unique steady state

(P∗, x∗,Y ∗) =

(ωAk + (1− ω)F ∗(1− b)

1− b − ω2ck,0,

A + ωc(1− ω)F ∗

1− b − ω2ck

).



Calling G the map associated to the complete systemPt+1 = Pt + αµ [(1− ω)F ∗ + ωkYt ]− Pt + axt


Yt+1 = Yt + γa2

(a1+a2


it admits the unique steady state

(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 (P∗, x∗,Y ∗)reads as

JG(P∗, x∗,Y ∗) =

1− αµ αµa αµωk

−µaα2β α2µa2β α2µaβωkγa1a2ωca1+a2

0 1− γa1a2(1−b)a1+a2

.Hence, in our framework we have

C1=γa1a2(1−b)a1+a2

− 2 + αµ− µa2α2β;

C2=2µa2α2β + 1− αµ− γa1a2ω2ckαµ

a1+a2− γa1a2(1−b)

a1+a2(1− αµ+ µa2α2β);

C3=µa2α2β(γa1a2(1−b)

a1+a2− 1).



The Jacobian matrix for G computed in correspondence to (P∗, x∗,Y ∗)reads as

JG(P∗, x∗,Y ∗) =

1− αµ αµa αµωk

−µaα2β α2µa2β α2µaβωkγa1a2ωca1+a2

0 1− γa1a2(1−b)a1+a2

.Hence, in our framework we have

C1=γa1a2(1−b)a1+a2

− 2 + αµ− µa2α2β;

C2=2µa2α2β + 1− αµ− γa1a2ω2ckαµ

a1+a2− γa1a2(1−b)

a1+a2(1− αµ+ µa2α2β);

C3=µa2α2β(γa1a2(1−b)

a1+a2− 1).



Making ω explicit, it is possible to rewrite Conditions (i)–(iv) as follows:

(i’) ω2 < (1 + C1 + C + C3) a1+a2γa1a2ckαµ := B1;

(ii’) ω2 < (1− C1 + C − C3) a1+a2γa1a2ckαµ := B2;

(iii’) ω2 > (−1 + C − C1C3 + C32) a1+a2γa1a2ckαµ := B3;

(iv’) ω2 > (C − 3) a1+a2γa1a2ckαµ := B4,

where we have set

C = 2µa2α2β + 1− αµ− γa1a2(1− b)

a1 + a2(1− αµ+ µa2α2β).

Hence, if minB1,B2 > 0 and maxB3,B4 < 1, the integrated systemis locally asymptotically stable at (P∗, x∗,Y ∗) for

maxB3,B4 < ω2 < minB1,B2, ω ∈ [0,1].

If instead minB1,B2 ≤ 0 or maxB3,B4 ≥ 1, the steady state isunstable for every ω ∈ [0,1].







(iv’) ω2 > (C − 3) a1+a2γa1a2ckαµ := B4,

where we have set

C = 2µa2α2β + 1− αµ− γa1a2(1− b)

a1 + a2(1− αµ+ µa2α2β).


maxB3,B4 < ω2 < minB1,B2, ω ∈ [0,1].








(iv’) ω2 > (C − 3) a1+a2γa1a2ckαµ := B4,

where we have set

C = 2µa2α2β + 1− αµ− γa1a2(1− b)

a1 + a2(1− αµ+ µa2α2β).


maxB3,B4 < ω2 < minB1,B2, ω ∈ [0,1].




Among the various detected scenarios, the most interesting one isprobably that in which the stable real and financial sectors becomeunstable when interconnected.

As ω increases, the steady state can either remain stable until ω = 1 orcan undergo a flip bifurcation, followed by a double Neimark-Sackerbifurcation.

The parameter µ plays a crucial role in this respect.

In the figures below we have fixed the other 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.





















The bifurcation diagram with respect to ω ∈ [0,1] for P when µ = 5



The bifurcation diagram with respect to ω ∈ [0,1] for P when µ = 28



Time series for P in red and for Y in blue when µ = 28 and ω = 0.95

The Neimark-Sacker bifurcation gives rise to a quasiperiodic behaviorcharacterized by the alternation of long monotonic increasing motionsand oscillatory decreasing motions.



Time series for P in red and for Y in blue when µ = 28 and ω = 0.95

The Neimark-Sacker bifurcation gives rise to a quasiperiodic behaviorcharacterized by the alternation of long monotonic increasing motionsand oscillatory decreasing motions.



In this scenario we can conclude that increasing µ has a destabilizingeffect.

In Naimzada and Pireddu (2015b) we give an economic interpretationof the model and we explain the rationale for the emergence of boomand bust cycles.

In the paper we also add stochastic noises to the optimists andpessimists demands, meant to reflect a certain within-groupheterogeneity, and we show how the model is able to reproduce thestylized facts for the real output data in the US.













Another interesting scenario is that in which there is a stabilization ofthe dynamics when interconnecting the unstable financial sector with astable real sector.

When isolated, the financial subsystem is unstable and characterizedby quasiperiodic motions, while the real subsystem is stable.

For not too large values of the parameter γ, when ω increases, thefixed point becomes stable through a reverse Neimark-Sackerbifurcation.

According to the value of γ, that fixed point can either persist untilω = 1 or can undergo a flip bifurcation and then a secondary doubleNeimark-Sacker bifurcation.

For even larger values of γ, we just obtain a reduction of thecomplexity of the system for suitable intermediate values of ω, but thesystem is never stabilized.































In the figures below we have fixed the other parameters as follows: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.

The bifurcation diagrams for P and Y w.r.t. ω ∈ [0,1] when γ = 5



In the figures below we have fixed the other parameters as follows: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.







The bifurcation diagrams for P and Y w.r.t. ω ∈ [0,1] when γ = 8.8



We can conclude that increasing γ has a destabilizing effect.

Summarizing, for the above parameter configurations, the instability ofthe stock market is transmitted to the real market for small values of ω.

However, increasing ω, the complexity of the whole system decreases.

This effect may either persist until ω = 1 or it may vanish for largervalues of ω, where we can find instead quasiperiodic motions.





















Other settings with heterogeneous fundamentalists

In Naimzada and Pireddu (2015c) and in Cavalli et al. (2017) weconsider just the financial sector.

The stock market is populated by optimistic and pessimisticfundamentalists.

Due to ambiguity in the stock market, generated by the uncertaintyabout the future stock price, agents do not rely on the truefundamental value in their speculations.

Agents form beliefs about the fundamental value, on the basis of animitative process.

Optimistic agents overestimate and pessimistic agents underestimatethe true fundamental value.

Differently from De Grauwe and Rovira Kaltwasser (2012) andNaimzada and Pireddu (2015b), the bias is no more exogenous.

























































Both in Naimzada and Pireddu (2015c) and in Cavalli et al. (2017) thestock price is determined by a nonlinear mechanism that preventsdivergence issues.

In Naimzada and Pireddu (2015c) population shares are exogenouslyfixed, while in Cavalli et al. (2017) shares evolve according to anupdating mechanism based on relative profits, similarly to Naimzadaand Pireddu (2015b).

Also the mechanisms governing the updating of the beliefs about thefundamental differ in the two papers.

Indeed, in Naimzada and Pireddu (2015c) agents update their beliefsproportionally to the relative profits realized by optimists andpessimists.

In Cavalli et al. (2017) agents consider instead the relative abilityshown by optimists and pessimists in guessing the realized stock price.































The setting in Naimzada and Pireddu (2015c)

The belief about the fundamental of pessimistic agents is given by:

X (t + 1) = feµπX (t+1)

eµπX (t+1) + eµπY (t+1)+ F

eµπY (t+1)

eµπX (t+1) + eµπY (t+1)

The belief about the fundamental of optimistic agents is given by:

Y (t + 1) = FeµπX (t+1)

eµπX (t+1) + eµπY (t+1)+ f

eµπY (t+1)


where

F is the true fundamental value;f is the lower bound for X . We assume 0 < f < F ;

f is the upper bound for Y . We assume f > F ;

πi(t + 1) = (P(t + 1)−P(t))σi(i(t)−P(t)) are the profits of agentsin group i ∈ X ,Y and σi > 0 their reactivity;µ ≥ 0 represents the intensity of the imitative process.





X (t + 1) = feµπX (t+1)


eµπY (t+1)



Y (t + 1) = FeµπX (t+1)


eµπY (t+1)


where








X (t + 1) = feµπX (t+1)


eµπY (t+1)



Y (t + 1) = FeµπX (t+1)


eµπY (t+1)


where








X (t + 1) = feµπX (t+1)


eµπY (t+1)



Y (t + 1) = FeµπX (t+1)


eµπY (t+1)


where








X (t + 1) = feµπX (t+1)


eµπY (t+1)



Y (t + 1) = FeµπX (t+1)


eµπY (t+1)


where








X (t + 1) = feµπX (t+1)


eµπY (t+1)



Y (t + 1) = FeµπX (t+1)


eµπY (t+1)


where








X (t + 1) = feµπX (t+1)


eµπY (t+1)



Y (t + 1) = FeµπX (t+1)


eµπY (t+1)


where








X (t + 1) = feµπX (t+1)


eµπY (t+1)



Y (t + 1) = FeµπX (t+1)


eµπY (t+1)


where






Agents, still remaining pessimists or optimists, proportionally imitatethose who obtain higher profits.

When µ = 0, X (t + 1) ≡ 12(f + F ) and Y (t + 1) ≡ 1

2(F + f ).

When µ→ +∞ :

– if πX > πY , then X (t + 1)→ f and Y (t + 1)→ F ;– if πX < πY , then X (t + 1)→ F and Y (t + 1)→ f .

The price adjustment mechanism is given by:

P(t + 1)− P(t) = γa2

(a1 + a2

a1 exp(−D(t)) + a2− 1),

where γ > 0 represents the market maker price adjustment reactivityand D(t) = ωσX (X (t)− P(t)) + (1− ω)σY (Y (t)− P(t)) is total excessdemand, with ω ∈ [0,1] the share of pessimists;a1 > 0 and a2 > 0 play the role, together with γ, of horizontalasymptotes.




When µ = 0, X (t + 1) ≡ 12(f + F ) and Y (t + 1) ≡ 1

2(F + f ).

When µ→ +∞ :



P(t + 1)− P(t) = γa2

(a1 + a2

a1 exp(−D(t)) + a2− 1),





When µ = 0, X (t + 1) ≡ 12(f + F ) and Y (t + 1) ≡ 1

2(F + f ).

When µ→ +∞ :



P(t + 1)− P(t) = γa2

(a1 + a2

a1 exp(−D(t)) + a2− 1),





When µ = 0, X (t + 1) ≡ 12(f + F ) and Y (t + 1) ≡ 1

2(F + f ).

When µ→ +∞ :



P(t + 1)− P(t) = γa2

(a1 + a2

a1 exp(−D(t)) + a2− 1),





When µ = 0, X (t + 1) ≡ 12(f + F ) and Y (t + 1) ≡ 1

2(F + f ).

When µ→ +∞ :



P(t + 1)− P(t) = γa2

(a1 + a2

a1 exp(−D(t)) + a2− 1),





When µ = 0, X (t + 1) ≡ 12(f + F ) and Y (t + 1) ≡ 1

2(F + f ).

When µ→ +∞ :



P(t + 1)− P(t) = γa2

(a1 + a2

a1 exp(−D(t)) + a2− 1),





When µ = 0, X (t + 1) ≡ 12(f + F ) and Y (t + 1) ≡ 1

2(F + f ).

When µ→ +∞ :



P(t + 1)− P(t) = γa2

(a1 + a2

a1 exp(−D(t)) + a2− 1),




In order to simplify our analysis, we assume that f = F −∆ andf = F + ∆, for some ∆ ≥ 0.

In this manner ∆ describes the maximum possible degree ofpessimism and optimism and it may be used as bifurcation parameter.

Indeed, we will use µ and ∆ as bifurcation parameters.

Our dynamical system reads as:X (t + 1) = F −∆

(1

1+e−µ(πX (t+1)−πY (t+1))

)Y (t + 1) = F + ∆

(1

1+eµ(πX (t+1)−πY (t+1))

)P(t + 1) = P(t) + γa2

(a1+a2

a1e−(ωσX (X(t)−P(t))+(1−ω)σY (Y (t)−P(t)))+a2− 1)

with πi(t + 1) = (P(t + 1)− P(t))σi(i(t)− P(t)), for i ∈ X ,Y.







(1

1+e−µ(πX (t+1)−πY (t+1))

)Y (t + 1) = F + ∆

(1

1+eµ(πX (t+1)−πY (t+1))

)P(t + 1) = P(t) + γa2

(a1+a2









(1

1+e−µ(πX (t+1)−πY (t+1))

)Y (t + 1) = F + ∆

(1

1+eµ(πX (t+1)−πY (t+1))

)P(t + 1) = P(t) + γa2

(a1+a2









(1

1+e−µ(πX (t+1)−πY (t+1))

)Y (t + 1) = F + ∆

(1

1+eµ(πX (t+1)−πY (t+1))

)P(t + 1) = P(t) + γa2

(a1+a2





PropositionOur system has a unique steady state in

(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )

).

Sketch of the proof:

– By the last equation in equilibrium it holds that

P∗ = ωσX X∗+(1−ω)σY Y∗

ωσX +(1−ω)σY.

– Moreover, in equilibrium πX = πY = 0, so that X ∗ = F − ∆2 and

Y ∗ = F + ∆2 .

– Inserting such expressions in P∗, we get the desired conclusion.




(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )

).




ωσX +(1−ω)σY.


Y ∗ = F + ∆2 .





(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )

).




ωσX +(1−ω)σY.


Y ∗ = F + ∆2 .





(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )

).




ωσX +(1−ω)σY.


Y ∗ = F + ∆2 .





(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )

).




ωσX +(1−ω)σY.


Y ∗ = F + ∆2 .




(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )

)The steady state values for X and Y are symmetric with respect to Fand lie at the middle points of the intervals in which they may vary.

When ∆ = 0 we find X ∗ = Y ∗ = P∗ = F , like in the classicalframework without belief biases and imitation.

It is possible to rewrite P∗ as

P∗ =ωσX (F − ∆

2 ) + (1− ω)σY (F + ∆2 )

ωσX + (1− ω)σY,

i.e., as a weighted average of X ∗ and Y ∗.



(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )




P∗ =ωσX (F − ∆

2 ) + (1− ω)σY (F + ∆2 )





(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )




P∗ =ωσX (F − ∆

2 ) + (1− ω)σY (F + ∆2 )





(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )




P∗ =ωσX (F − ∆

2 ) + (1− ω)σY (F + ∆2 )





Proposition

The variables X and Y in our system satisfy the following condition:Y (t) = X (t) + ∆, for all t ≥ 1.

For t ≥ 1, our dynamical system is then equivalent to that associatedto the 2D map

G = (G1,G2) : (f ,F )× (0,+∞)→ R2,

(X ,P) 7→ (G1(X ,P),G2(X ,P)),

defined as:G1(X ,P)=F−

∆

1+e−µ

(γa2

(a1+a2

a1e−(ωσX (X−P)+(1−ω)σY (X+∆−P))+a2−1

)(σX (X−P)−σY (X+∆−P))

)

G2(X ,P)=P + γa2

(a1+a2

a1e−(ωσX (X−P)+(1−ω)σY (X+∆−P))+a2− 1)

in the sense that the two systems generate the same trajectories.



Proposition

The variables X and Y in our system satisfy the following condition:Y (t) = X (t) + ∆, for all t ≥ 1.

For t ≥ 1, our dynamical system is then equivalent to that associatedto the 2D map

G = (G1,G2) : (f ,F )× (0,+∞)→ R2,

(X ,P) 7→ (G1(X ,P),G2(X ,P)),

defined as:G1(X ,P)=F−

∆

1+e−µ

(γa2

(a1+a2

a1e−(ωσX (X−P)+(1−ω)σY (X+∆−P))+a2−1

)(σX (X−P)−σY (X+∆−P))

)

G2(X ,P)=P + γa2

(a1+a2

a1e−(ωσX (X−P)+(1−ω)σY (X+∆−P))+a2− 1)

in the sense that the two systems generate the same trajectories.



For simplicity, we will deal with G to analytically derive the stabilityconditions for our model.

For the numerical simulations we will rely on the original 3Dformulation, in order to illustrate the behavior of all variables.



For simplicity, we will deal with G to analytically derive the stabilityconditions for our model.

For the numerical simulations we will rely on the original 3Dformulation, in order to illustrate the behavior of all variables.



The 2D system has a unique fixed point in

(X ∗,P∗) =

(F − ∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )

).

The Jacobian matrix for G in correspondence to (X ∗,P∗) reads as

JG(X ∗,P∗) =

[∆2µγσXσY

4 −∆2µγσXσY4

γ(ωσX + (1− ω)σY ) 1− γ(ωσX + (1− ω)σY )

],

where we set γ = γa1a2a1+a2

.



The 2D system has a unique fixed point in

(X ∗,P∗) =

(F − ∆

2,F − ∆(ωσX − (1− ω)σY )

2(ωσX + (1− ω)σY )

).

The Jacobian matrix for G in correspondence to (X ∗,P∗) reads as

JG(X ∗,P∗) =

[∆2µγσXσY

4 −∆2µγσXσY4

γ(ωσX + (1− ω)σY ) 1− γ(ωσX + (1− ω)σY )

],

where we set γ = γa1a2a1+a2

.



Denoting by det(J) and tr(J) the determinant and the trace of theabove Jacobian matrix, the Jury conditions read as follows

(i) 1 + tr(J) + det(J) > 0;

(ii) 1− tr(J) + det(J) > 0;

(iii) det(J) < 1.

In our framework, we have

det(J) =µ∆2γσXσY

4,

tr(J) =µ∆2γσXσY

4+ 1− γ(ωσX + (1− ω)σY ).

Making µ explicit, when ∆ 6= 0 (i)–(iii) are fulfilled if

2(γ(ωσX + (1− ω)σY )− 2)

γσXσY ∆2 < µ <4

γσXσY ∆2 .




(i) 1 + tr(J) + det(J) > 0;

(ii) 1− tr(J) + det(J) > 0;

(iii) det(J) < 1.



4,


4+ 1− γ(ωσX + (1− ω)σY ).


2(γ(ωσX + (1− ω)σY )− 2)


γσXσY ∆2 .




(i) 1 + tr(J) + det(J) > 0;

(ii) 1− tr(J) + det(J) > 0;

(iii) det(J) < 1.



4,


4+ 1− γ(ωσX + (1− ω)σY ).


2(γ(ωσX + (1− ω)σY )− 2)


γσXσY ∆2 .



Usually in the literature increasing the parameter, strongly related toour µ, describing the intensity of choice in the switching mechanismbetween different decisional rules has just a destabilizing effect, whilefor us it may also be stabilizing.

Indeed, when µ is positive but close to 0, through the imitative processthe instability of the financial market gets transmitted to the dynamicsof the fundamental values.

Increasing values for µ intensify the oscillations due to optimism andpessimism, but when µ is sufficiently large positive and negativeexcess demands for the two groups of agents balance out in theaggregate excess demand and this causes smaller price oscillations.

The profit differential decreases and this leads to smaller variations forthe fundamental values of optimists and pessimists.

When µ increases further, agents become however very reactive inupdating the fundamental values and this causes the emergence ofcomplex, quasiperiodic dynamics.































In (i)–(iii) it is also possible to make ∆ explicit, finding that when µ 6= 0the stability conditions read as

γ(ωσX + (1− ω)σY ) ≥ 2

and √2(γ(ωσX + (1− ω)σY )− 2)

γσXσYµ< ∆ <

2√γσXσYµ

,

or asγ(ωσX + (1− ω)σY ) < 2

and∆ <

2√γσXσYµ

.

When instead µ = 0 or ∆ = 0, the dynamics are generated by thefinancial market only, which is locally asymptotically stable at P∗ = F if

γ <2

ωσX + (1− ω)σY.




γ(ωσX + (1− ω)σY ) ≥ 2

and √2(γ(ωσX + (1− ω)σY )− 2)

γσXσYµ< ∆ <

2√γσXσYµ

,


and∆ <

2√γσXσYµ

.


γ <2





γ(ωσX + (1− ω)σY ) ≥ 2

and √2(γ(ωσX + (1− ω)σY )− 2)

γσXσYµ< ∆ <

2√γσXσYµ

,


and∆ <

2√γσXσYµ

.


γ <2




In the numerical simulations, for simplicity, we will focus on frameworkswith σX = σY = 1 and ω = 0.5.

The expression for the steady state then becomes

(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F)

and the financial market, when isolated, is stable if γ < 2.



In the numerical simulations, for simplicity, we will focus on frameworkswith σX = σY = 1 and ω = 0.5.

The expression for the steady state then becomes

(X ∗,Y ∗,P∗) =

(F − ∆

2,F +

∆

2,F)

and the financial market, when isolated, is stable if γ < 2.



Bifurcation analysis results

According to the value of γ, increasing values of µ may have:

– a destabilizing role

The bifurcation diagram with respect to µ ∈ [0,20] for X in blue, Y inred and P in green for γ = 1, F = 2, ∆ = 0.8, a1 = a2 = 1, and the

initial conditions X (0) = 1.25, Y (0) = 2.2 and P(0) = 3



Bifurcation analysis results

According to the value of γ, increasing values of µ may have:


The bifurcation diagram with respect to µ ∈ [0,20] for X in blue, Y inred and P in green for γ = 1, F = 2, ∆ = 0.8, a1 = a2 = 1, and the




– a mixed role

The bifurcation diagram with respect to µ ∈ [0,3.5] for X in blue, Y inred and P in green for γ = 5, F = 2, ∆ = 0.8, a1 = a2 = 1, and the




– a complexity decreasing role

The bifurcation diagram with respect to µ ∈ [0,12] for X in blue, Y inred and P in green, for γ = 4.8, F = 2, ∆ = 0.8, a1 = 2.6, a2 = 1, and

the initial conditions X (0) = 1.5, Y (0) = 2.5 and P(0) = 1



Similarly, according to the value of γ, increasing values of ∆ may have:


The bifurcation diagram with respect to ∆ ∈ [0,1] for X in blue, Y inred and P in green for γ = 1, F = 1.3, µ = 10, a1 = a2 = 1, and the




– a mixed role

The bifurcation diagram with respect to ∆ ∈ [0,0.5] for X in blue, Y inred and P in green for γ = 4.5, F = 1.3, µ = 10, a1 = a2 = 1, and the




– a complexity decreasing role

The bifurcation diagram with respect to ∆ ∈ [0,3] for X in blue, Y inred and P in green for γ = 5.4, F = 4, µ = 0.5, a1 = 3.3, a2 = 1, and

the initial conditions X (0) = 1.6, Y (0) = 4.5 and P(0) = 3



In Naimzada and Pireddu (2015c), starting from time series of themain variables, we explain the rules governing the dynamics of priceand of fundamental values.

The model is rich in multistability phenomena, characterized by thecoexistence of cyclic attractors of various periods with different chaoticattractors, in one or more pieces.



In Naimzada and Pireddu (2015c), starting from time series of themain variables, we explain the rules governing the dynamics of priceand of fundamental values.

The model is rich in multistability phenomena, characterized by thecoexistence of cyclic attractors of various periods with different chaoticattractors, in one or more pieces.



The bifurcation diagram with respect to µ ∈ [0,3] for P withγ = 5, F = 2.6, ∆ = 0.8, a1 = 2.6, a2 = 1, and the initial conditionsX (0) = 2.2, Y (0) = 3, and P(0) = 3 for the blue points, P(0) = 9 for

the red points and P(0) = 2.599 for the green points, respectively



The setting in Cavalli et al. (2017)

Two are the main differences with respect to Naimzada and Pireddu(2015c):

– optimistic and pessimistic shares are determined by an evolutionarymechanism based on relative profits;

– the beliefs about the fundamental are updated according to therelative ability shown by optimists and pessimists in guessing therealized stock price.






















X (t+1) = feµ(Y (t)−P(t))2

eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2 +Feµ(X(t)−P(t))2

eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2


Y (t+1) = Feµ(Y (t)−P(t))2

eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2 +feµ(X(t)−P(t))2

eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2

where



(i(t)− P(t))2 is the squared error between the fundamental valueperceived by agents of group i ∈ X ,Y and the stock price;µ ≥ 0 represents the intensity of the imitative process.




X (t+1) = feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2


Y (t+1) = Feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2

where







X (t+1) = feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2


Y (t+1) = Feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2

where







X (t+1) = feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2


Y (t+1) = Feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2

where







X (t+1) = feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2


Y (t+1) = Feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2

where







X (t+1) = feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2


Y (t+1) = Feµ(Y (t)−P(t))2


eµ(X(t)−P(t))2 + eµ(Y (t)−P(t))2

where






If |X (t)− P(t)| = |Y (t)− P(t)|, then X (t + 1) = (f + F )/2 andY (t + 1) = (F + f )/2⇒ X (t + 1) and Y (t + 1) lie at the middle point ofthe intervals in which they can vary.

If instead |X (t)− P(t)| < |Y (t)− P(t)|, then X (t + 1) will be closer to fthan to F and Y (t + 1) will be closer to F than to f .

The opposite conclusions hold in case |X (t)− P(t)| > |Y (t)− P(t)|.

When µ = 0, then X (t + 1) ≡ 12(f + F ) and Y (t + 1) ≡ 1

2(F + f )⇒there is no imitation.

When instead µ→ +∞ :

– if (X (t)− P(t))2 < (Y (t)− P(t))2, then X (t + 1)→ f andY (t + 1)→ F .– if (X (t)− P(t))2 > (Y (t)− P(t))2, then X (t + 1)→ F andY (t + 1)→ f .


































































The mechanism above only affects the value of the beliefs ofpessimists and optimists, but not the kind of decisional rule adopted byeach speculator.

The evolutionary competition between optimism and pessimism isdescribed by the switching rule used also in Naimzada and Pireddu(2015b):

ω(t + 1) =eβπX (t+1)

eβπX (t+1) + eβπY (t+1)=

11 + e−β(πX (t+1)−πY (t+1))

,

where ω(t) ∈ (0,1) represents the fraction of the population composedby pessimists at time t .

Since we assume a normalized population of size one, the fraction ofthe population composed by optimists at time t is 1− ω(t).

The profits are given by

πi(t + 1) = (P(t + 1)− P(t))(i(t)− P(t)), i ∈ X ,Y.





ω(t + 1) =eβπX (t+1)


11 + e−β(πX (t+1)−πY (t+1))

,




πi(t + 1) = (P(t + 1)− P(t))(i(t)− P(t)), i ∈ X ,Y.





ω(t + 1) =eβπX (t+1)


11 + e−β(πX (t+1)−πY (t+1))

,




πi(t + 1) = (P(t + 1)− P(t))(i(t)− P(t)), i ∈ X ,Y.





ω(t + 1) =eβπX (t+1)


11 + e−β(πX (t+1)−πY (t+1))

,




πi(t + 1) = (P(t + 1)− P(t))(i(t)− P(t)), i ∈ X ,Y.



The price variation mechanism is sigmoidal

P(t + 1)− P(t) = a2

(a1 + a2

a1e−γD(t) + a2− 1),

where D(t) is the excess demand, γ is a positive parameter influencingthe price variation reactivity and a1, a2 are positive parameters limitingprice variation.

Total excess demand reads as

D(t) = ω(t)(X (t)− P(t)) + (1− ω(t))(Y (t)− P(t))= ω(t)X (t) + (1− ω(t))Y (t)− P(t).



The price variation mechanism is sigmoidal

P(t + 1)− P(t) = a2

(a1 + a2

a1e−γD(t) + a2− 1),

where D(t) is the excess demand, γ is a positive parameter influencingthe price variation reactivity and a1, a2 are positive parameters limitingprice variation.

Total excess demand reads as

D(t) = ω(t)(X (t)− P(t)) + (1− ω(t))(Y (t)− P(t))= ω(t)X (t) + (1− ω(t))Y (t)− P(t).



Like in Naimzada and Pireddu (2015c) we assume that f and f lay atthe same distance ∆ from F , i.e., that f = F −∆ and f = F + ∆.

In this manner, ∆ ≥ 0 may be used as bifurcation parameter.

∆ is a measure of the heterogeneity degree among agents and thus ofthe bias in their beliefs.

∆ describes also the degree of ambiguity in the financial market, whichprevents agents from relying on the true fundamental value F in theirspeculations.





















Our model reads as

X (t + 1) = F −∆(

11+eµ((X(t)−P(t))2−(Y (t)−P(t))2)

)Y (t + 1) = F + ∆

(1

1+e−µ((X(t)−P(t))2−(Y (t)−P(t))2)

)P(t + 1) = P(t) + a2

(a1+a2

a1e−γ(ω(t)(X(t)−P(t))+(1−ω(t))(Y (t)−P(t)))+a2− 1)

ω(t + 1) = 11+e−β(πX (t+1)−πY (t+1))

When µ = 0, we enter the framework in De Grauwe and RoviraKaltwasser (2012) with bias a = ∆

2 , except for the presence of ournonlinear price adjustment mechanism.

If instead β = 0 we are in a particular case of Naimzada and Pireddu(2015c), with ω ≡ 0.5.

In the system above we can not disentangle the effects of the shareω(t) of pessimists from the role played by the value of beliefs.



Our model reads as

X (t + 1) = F −∆(

11+eµ((X(t)−P(t))2−(Y (t)−P(t))2)

)Y (t + 1) = F + ∆

(1

1+e−µ((X(t)−P(t))2−(Y (t)−P(t))2)

)P(t + 1) = P(t) + a2

(a1+a2


ω(t + 1) = 11+e−β(πX (t+1)−πY (t+1))







Our model reads as

X (t + 1) = F −∆(

11+eµ((X(t)−P(t))2−(Y (t)−P(t))2)

)Y (t + 1) = F + ∆

(1

1+e−µ((X(t)−P(t))2−(Y (t)−P(t))2)

)P(t + 1) = P(t) + a2

(a1+a2


ω(t + 1) = 11+e−β(πX (t+1)−πY (t+1))







Our model reads as

X (t + 1) = F −∆(

11+eµ((X(t)−P(t))2−(Y (t)−P(t))2)

)Y (t + 1) = F + ∆

(1

1+e−µ((X(t)−P(t))2−(Y (t)−P(t))2)

)P(t + 1) = P(t) + a2

(a1+a2


ω(t + 1) = 11+e−β(πX (t+1)−πY (t+1))







An optimistic behavior can be encompassed by values of X (t) andY (t) sufficiently close to F and F + ∆, respectively, even when ω(t) islarge, as well as by a large share of optimists even if beliefs are small.

To take into account such double nature of optimism/pessimism, wecould define a sentiment index as

I1(t) = ω(t)X (t) + (1− ω(t))Y (t),

i.e., an average of optimists and pessimists beliefs weighted by theircorresponding fractions.

On the other hand, in order to describe the temporal evolution of thewaves of optimism and pessimism, it is sometimes crucial to considerseveral consecutive periods.





I1(t) = ω(t)X (t) + (1− ω(t))Y (t),







I1(t) = ω(t)X (t) + (1− ω(t))Y (t),







I1(t) = ω(t)X (t) + (1− ω(t))Y (t),





To be able to neglect insignificant discording behaviors in isolatedperiods, we introduce the optimism/pessimism persistence index

IT (t) =t∑

j=t−T +1

ω(j)X (j) + (1− ω(j))Y (j)T

,

which is a moving average of I1(t) over the T ≥ 1 time stepspreceding t .

Since IT (t) ∈ (F −∆,F + ∆), we can say that optimism is realized overperiods of size T when IT (t) > F , and pessimism when IT (t) < F .




IT (t) =t∑

j=t−T +1

ω(j)X (j) + (1− ω(j))Y (j)T

,






IT (t) =t∑

j=t−T +1

ω(j)X (j) + (1− ω(j))Y (j)T

,





Proposition

Our system has a unique steady state in

(X ∗,Y ∗,P∗, ω∗) =

(F − ∆

2,F +

∆

2,F ,

12

).

At the equilibrium, index IT (t) coincides with the fundamental value Fand describes a neutral situation, where neither optimism norpessimism prevails.

Like in Naimzada and Pireddu (2015c) it holds that

Proposition

The variables X and Y satisfy the following condition:Y (t) = X (t) + ∆, for all t ≥ 1.



Proposition


(X ∗,Y ∗,P∗, ω∗) =

(F − ∆

2,F +

∆

2,F ,

12

).



Proposition




Proposition


(X ∗,Y ∗,P∗, ω∗) =

(F − ∆

2,F +

∆

2,F ,

12

).



Proposition




Hence, for t ≥ 1 our dynamical system is equivalent to that associatedto the 3D map

G = (G1,G2,G3) : (f ,F )× (0,+∞)× (0,1)→ R3,

(X (t),P(t), ω(t)) 7→ (G1(X (t),P(t), ω(t)),G2(X (t),P(t), ω(t)),G3(X (t),P(t), ω(t))),

defined as:

X (t + 1) = G1(X (t),P(t), ω(t))

= F −∆(

11+eµ((X(t)−P(t))2−(X(t)+∆−P(t))2)

)P(t + 1) = G2(X (t),P(t), ω(t))

= P(t) + a2

(a1+a2

a1e−γ(ω(t)(X(t)−P(t))+(1−ω(t))(X(t)+∆−P(t)))+a2− 1)

ω(t + 1) = G3(X (t),P(t), ω(t))

= 1

1+e−βa2∆

(a1+a2

a1e−γ(ω(t)(X(t)−P(t))+(1−ω(t))(X(t)+∆−P(t)))+a2−1

)

in the sense that the two systems generate the same trajectories forX (t), P(t) and ω(t).



We will deal with G both to derive the stability conditions for our modelusing the method in Farebrother (1973), and in the numericalsimulations, where we will specify the initial conditions for X (t), P(t)and ω(t) only, implicitly taking Y (0) = X (0) + ∆.

To study the stability of the 3D system at unique fixed point(X ∗,P∗, ω∗) =

(F − ∆

2 , F , 12

), we need to compute the Jacobian

matrix for G in correspondence to (X ∗,P∗, ω∗), which reads as−µ∆2

2µ∆2

2 0

γ 1− γ −γ∆

−βγ∆4

βγ∆4

βγ∆2

4

.



We will deal with G both to derive the stability conditions for our modelusing the method in Farebrother (1973), and in the numericalsimulations, where we will specify the initial conditions for X (t), P(t)and ω(t) only, implicitly taking Y (0) = X (0) + ∆.

To study the stability of the 3D system at unique fixed point(X ∗,P∗, ω∗) =

(F − ∆

2 , F , 12

), we need to compute the Jacobian

matrix for G in correspondence to (X ∗,P∗, ω∗), which reads as−µ∆2

2µ∆2

2 0

γ 1− γ −γ∆

−βγ∆4

βγ∆4

βγ∆2

4

.



The Farebrother conditions are the following:

(i) 1− C1 + C2 − C3 > 0;

(ii) 1− C2 + C1C3 − (C3)2 > 0;

(iii) 3− C2 > 0;

(iv) 1 + C1 + C2 + C3 > 0,


λ3 + C1λ2 + C2λ+ C3 = 0.

In our framework, we have: C1 =µ∆2

2+ γ − 1− βγ∆2

4,

C2 =−µ∆2

2

(1 +

βγ∆2

4

)+βγ∆2

4, C3 =

µβγ∆4

8.



Condition (iv) reduces to γ > 0, which is indeed true.

Conditions (i)− (iii) above can be respectively written as

(i ′)(

1 +βγ∆2

4

)(1− µ∆2

2

)>γ

2;

(ii ′)(

1− βγ∆2

4

)(1 +

βµ2γ∆6

16+µ∆2

2

(1 +

βγ∆2

4

))+βµγ2∆4

8> 0 ;

(iii ′) 6 + µ∆2 >βγ∆2

4(2− µ∆2) ,

where γ = γa1a2a1+a2

.

In the previous conditions, we may easily put in evidence β, µ and, forµ = 0, also ∆.





(i ′)(

1 +βγ∆2

4

)(1− µ∆2

2

)>γ

2;

(ii ′)(

1− βγ∆2

4

)(1 +

βµ2γ∆6

16+µ∆2

2

(1 +

βγ∆2

4

))+βµγ2∆4

8> 0 ;

(iii ′) 6 + µ∆2 >βγ∆2

4(2− µ∆2) ,


.






(i ′)(

1 +βγ∆2

4

)(1− µ∆2

2

)>γ

2;

(ii ′)(

1− βγ∆2

4

)(1 +

βµ2γ∆6

16+µ∆2

2

(1 +

βγ∆2

4

))+βµγ2∆4

8> 0 ;

(iii ′) 6 + µ∆2 >βγ∆2

4(2− µ∆2) ,


.




PropositionLet ∆ 6= 0, µ and γ be fixed. Then, on varying β, we can havedestabilizing, mixed and unconditionally unstable scenarios.Let ∆ 6= 0, β and γ be fixed. Then, on varying µ, we can havedestabilizing, mixed and unconditionally unstable scenarios.Let β 6= 0, µ = 0 and γ be fixed. Then, on varying ∆, we can haveeither a destabilizing or a mixed scenario.

We call a scenario destabilizing with respect to a parameter when thesteady state is stable below a certain threshold of that parameter andunstable above it.

We say that a scenario is mixed if the steady state is stable inside aninterval of parameter values and unstable outside it.

We say that a scenario is unconditionally unstable when the steadystate is unstable for all the values of the considered parameter.





















We may explain the presence of two thresholds for stability withrespect to µ and β as follows.

When γ is large enough, the isolated price adjustment mechanism isunstable and small positive values for µ and β allow the transmissionof such turbulence to the imitative process on beliefs, as well as to theswitching mechanism.

Further increasing values of β dampen large profits and this makes thepopulation shares stabilize on the mean value they may assume, i.e.,on their steady state values.

Similarly, intermediate values for µ dampen the role played by thedifference between the squared errors, making the beliefs for bothpessimists and optimists stabilize on their steady state values.

On the other hand, when β or µ are too large, they becomedestabilizing, because of a high degree of nervousness in the imitationand in the switching mechanisms.































(a) (b) (c)(a) : Stability region (in yellow) for F = 3, ∆ = 1, a1 = 10.2, a2 = 6and γ = 1. (b) : Bifurcation diagram on varying β for µ = 0.5. (c) :

Bifurcation diagram on varying µ for β = 2. Black (red) diagrams areobtained for X (0) = 2.6, P(0) = 3.0001(P(0) = 4), ω(0) = 0.5

The dashed (solid) red curve shows when a Neimark-Sacker(period-doubling) bifurcation occurs.



(a) (b) (c)(a) : Stability region (in yellow) for F = 3, ∆ = 1, a1 = 10.2, a2 = 6and γ = 1. (b) : Bifurcation diagram on varying β for µ = 0.5. (c) :

Bifurcation diagram on varying µ for β = 2. Black (red) diagrams areobtained for X (0) = 2.6, P(0) = 3.0001(P(0) = 4), ω(0) = 0.5

The dashed (solid) red curve shows when a Neimark-Sacker(period-doubling) bifurcation occurs.



(a) (b) (c)Stability regions (in yellow) for F = 3, β = 1, a1 = 5.1, a2 = 3 in (a)and a1 = 10.2, a2 = 6 in (b). (c) : Bifurcation diagram on varying ∆ forµ = 0.2, corresponding to the horizontal line plotted in the stability

diagram in (b). The black (red) diagram is obtained for initial conditionsX (0) = 2.6, P(0) = 3.0001 (P(0) = 4), ω(0) = 0.5



For null or moderate values of µ, an intermediate level of ambiguity inthe stock market may lead to a stabilization of the dynamics.

Indeed, if the level of ambiguity starts raising, agents no longer trustone another and they are discouraged from operating in the financialsector.

The reduced amount of speculations causes in turn a reduction in thestock price volatility, stabilizing the dynamics.

Such positive effect is destroyed both by an excessive imitationdegree, which makes agents too reactive to others’ choices, and by atoo high ambiguity level, which let orbits converge toward a periodic orchaotic attractor, rather than toward a fixed point.

Hence, we found that the three main model parameters (i.e., µ, ∆, β)have an ambiguous effect on the system equilibrium.































In Cavalli et al. (2017) we give an economic interpretation of theresults examining time series of beliefs, prices and shares ofoptimists/pessimists.

Moreover, we perform a statistical analysis of a stochasticallyperturbed version of the model, which highlights fat tails and excessvolatility in the returns distributions, as well as bubbles and crashes forstock prices, in agreement with the empirical literature.

Similarly to De Grauwe and Rovira Kaltwasser (2012), we assume thatthe true fundamental value follows a random walk.













A further extension

In Cavalli et al. (2018) we transformed the optimism/pessimismpersistence index into a variable on which agents base their decisions,in addition to considering price and profit dynamics.

The index plays no more a descriptive role, but it drives the dynamics.

The evolutionary selection depends on a weighted evaluation of theprofits realized by each group of fundamentalists and of a measure ofthe general sentiment perceived by the agents about the market.

The general feeling perceived by the agents about the market status isdescribed by the sentiment index

It = ωtXt + (1− ωt )Yt − F = Xt + (1− ωt )∆− F .

It measures the difference between the average belief about thefundamental value and the true fundamental value F .The sign of It gives information about the general degree of optimismor pessimism of the market.



A further extension









A further extension









A further extension









A further extension









A further extension









A further extension









The pessimists’ share evolves according to a convex combination ofthe general market sentiment and of the profits realized by the twokinds of speculators:

ωt+1 =eβ(σ(−It )+(1−σ)πX ,t+1)

eβ(σ(−It )+(1−σ)πX ,t+1) + eβ(σIt +(1−σ)πY ,t+1),

where:• β > 0 represents the intensity of choice of the switching mechanism;• σ ∈ [0,1] is the sentiment weight;• πj,t+1 = (Pt+1 − Pt )(jt − Pt ) are the profits realized by agents ofgroup j ∈ X ,Y.The opposite signs preceding It are a consequence of the differentattitude of optimists and pessimists toward positive or negative valuesof the sentiment index.When σ = 0, the above evolutionary mechanism is exactly the sameas in Cavalli et al. (2017).If σ = 1, the switching mechanism only depends on the sentimentindex.




ωt+1 =eβ(σ(−It )+(1−σ)πX ,t+1)






ωt+1 =eβ(σ(−It )+(1−σ)πX ,t+1)






ωt+1 =eβ(σ(−It )+(1−σ)πX ,t+1)






ωt+1 =eβ(σ(−It )+(1−σ)πX ,t+1)






ωt+1 =eβ(σ(−It )+(1−σ)πX ,t+1)






ωt+1 =eβ(σ(−It )+(1−σ)πX ,t+1)






ωt+1 =eβ(σ(−It )+(1−σ)πX ,t+1)





Like in Cavalli et al. (2017) for the beliefs it holds that Yt = Xt + ∆, forall t ≥ 1.

Moreover, the share of optimists is given by 1− ωt .



Like in Cavalli et al. (2017) for the beliefs it holds that Yt = Xt + ∆, forall t ≥ 1.

Moreover, the share of optimists is given by 1− ωt .



The model is described by the 3D map

G = (G1,G2,G3) : (F −∆,F )× (0,+∞)× (0,1)→ R3,

(Xt ,Pt , ωt ) 7→ (G1(Xt ,Pt , ωt ),G2(Xt ,Pt , ωt ),G3(Xt ,Pt , ωt )),

defined as:Xt+1 =G1(Xt ,Pt , ωt )=F − ∆

eµ((Xt−Pt )2−(Xt +∆−Pt )2) + 1Pt+1 =G2(Xt ,Pt , ωt )=Pt + f (γ(Xt − Pt + ∆(1− ωt )))

ωt+1 =G3(Xt ,Pt , ωt )=1

1 + eβ(2σ(Xt +(1−ωt )∆−F )+(1−σ)∆f (γ(Xt−Pt +(1−ωt )∆)))

where, in the last equation, we replaced It with its expression and weemployed the identity πY ,t+1−πX ,t+1 = (Pt+1−Pt )(Yt −Xt ) = f (γDt )∆.



Proposition

Our system hasa) a unique steady state S∗ = (X ∗,P∗, ω∗) = (F −∆/2,F ,1/2) if

σ ∈ [0,1] andσ ≤ 4

β∆(∆2µ+ 2);

b) three steady states S∗,So = (X o,Po, ωo) and Sp = (X p,Pp, ωp) if4

β∆(∆2µ+ 2)< σ ≤ 1.

In particular, So and Sp are symmetric w.r.t. S∗, withX p < X ∗ < X o, Pp < P∗ < Po and ωo < ω∗ < ωp.

The two new steady economic regimes that can be identified aspessimistic (Sp) and optimistic (So).Sp and So only exist if agents give a sufficiently large relevance to theperceived market mood.



Proposition


σ ∈ [0,1] andσ ≤ 4

β∆(∆2µ+ 2);


β∆(∆2µ+ 2)< σ ≤ 1.





Proposition


σ ∈ [0,1] andσ ≤ 4

β∆(∆2µ+ 2);


β∆(∆2µ+ 2)< σ ≤ 1.





Proposition


σ ∈ [0,1] andσ ≤ 4

β∆(∆2µ+ 2);


β∆(∆2µ+ 2)< σ ≤ 1.





Proposition


σ ∈ [0,1] andσ ≤ 4

β∆(∆2µ+ 2);


β∆(∆2µ+ 2)< σ ≤ 1.





The value of β, µ, ∆ and σ does not only influence the emergence ofthe steady states So and Sp, but affects their position, too.

Proposition

Let σ > 4/(β∆(∆2µ+ 2)), σ ∈ [0,1]. Then, on increasing σ, β,∆ andµ, we have that ωo decreases, while Po and Io increase, and that ωp

increases, while Pp and Ip decrease. Moreover, on increasing σ, β,∆and µ, the distance from X o and X p to X ∗ = F −∆/2 increases.



The value of β, µ, ∆ and σ does not only influence the emergence ofthe steady states So and Sp, but affects their position, too.

Proposition

Let σ > 4/(β∆(∆2µ+ 2)), σ ∈ [0,1]. Then, on increasing σ, β,∆ andµ, we have that ωo decreases, while Po and Io increase, and that ωp

increases, while Pp and Ip decrease. Moreover, on increasing σ, β,∆and µ, the distance from X o and X p to X ∗ = F −∆/2 increases.



Proposition

On varying σ ∈ [0,1], we have that the set on which the steady stateS∗ is locally asymptotically stable can be

a) connected, being an interval, in which case the sentiment weightcan have a destabilizing, stabilizing, mixed or neutral effect;

b) unconnected.



In the next bifurcation diagram, we set F = 10, a1 = 2, a2 = 1.

Beliefs are strongly polarized (∆ = 3), the price reactivity is high(γ = 4), there is no imitation (µ = 0) and the intensity of choice variesin the left plot and is moderate (β = 1) in the right plot.

The initial datum in the left plot, as well as for the black bifurcationdiagram in the right plot, is(X0,P0, ω0) = (X ∗ + 0.01,P∗ + 0.01, ω∗ + 0.01), while it is(X0,P0, ω0) = (X ∗ − 0.01,P∗ − 0.01, ω∗ − 0.01) in red plot.













2D bifurcation diagram in the left panel. In the right panel, thecorresponding bifurcation diagram for P when β = 1

The solid line refers to the pitchfork bifurcation, through which So andSp emerge when σ = 4/(β∆(∆2µ+ 2)).

The dashed and dash-dotted curves refer to the remaining stabilityconditions.













Considering β = 1, we have a mixed scenario, in which the polarizedprices undergo a period-doubling cascade of bifurcations.We observe an herding phenomenon as σ increases, which, accordingto the initial conditions, gives rise to price dynamics that endogenouslyfluctuate around large or small values.When σ = 0 the evolutionary selection only depends on profits.As σ increases, the switching mechanism is more affected by thesentiment index and less by the profits, which in this case are thesource of instabilities.Hence, endogenous oscillations decrease and disappear, so thatagents evenly distribute among beliefs and the stock price convergesto the fundamental value.Increasing σ further, we find a shares polarization, due to morerelevance given to the perceived market mood and thus to the utility ofbeing either pessimistic or optimistic.


















In Cavalli et al. (2018) we also show the basin of attraction of theoptimistic and pessimistic attractors for µ 6= 0 and σ = 1, from whichthe polarization of beliefs is clearly visible.

Indeed, a sufficiently high degree of optimism or pessimism,determined by both beliefs and shares values, uniquely determines theconvergence toward an attractor that reflects the same polarizedoptimism or pessimism.



In Cavalli et al. (2018) we also show the basin of attraction of theoptimistic and pessimistic attractors for µ 6= 0 and σ = 1, from whichthe polarization of beliefs is clearly visible.

Indeed, a sufficiently high degree of optimism or pessimism,determined by both beliefs and shares values, uniquely determines theconvergence toward an attractor that reflects the same polarizedoptimism or pessimism.



Future study directions:

Similarly to Brock and Hommes (1998) and De Grauwe andRovira Kaltwasser (2012), we could introduce also a group ofunbiased fundamentalists and a group of unbiased chartists.

The goal is to check whether, as in De Grauwe and RoviraKaltwasser (2012), the former group has a stabilizing role, i.e., itspresence makes the stability region become larger, while the lattergroup is destabilizing.

Another research direction will consist in deepening the study ofthe role of animal spirits as the drivers of economic decisions,extending the pursued approach to macroeconomic frameworksinvolving the real market side.





















References on the latest HAMs:

– Anderson SP, de Palma A, Thisse JF (1992) Discrete Choice Theoryof Product Differentiation. MIT Press, Cambridge.– Brock WA, Hommes CH (1997) A rational route to randomness.Econometrica 65, 1059–1095.– Brock WA, Hommes CH (1998) Heterogeneous beliefs and routes tochaos in a simple asset pricing model. Journal of Economic Dynamicsand Control 22, 1235–1274.– Cavalli F, Naimzada A, Pireddu M (2017) An evolutive financialmarket model with animal spirits: imitation and endogenous beliefs,Journal of Evolutionary Economics 27, 1007–1040.– Cavalli F, Naimzada A, Pecora N, Pireddu M (2018) Marketsentiment and heterogeneous fundamentalists in an evolutive financialmarket model, submitted.– De Grauwe P, Rovira Kaltwasser P (2012) Animal spirits in theforeign exchange market. Journal of Economic Dynamics and Control36, 1176–1192.



– Farebrother RW (1973) Simplified Samuelson conditions for cubicand quartic equations. The Manchester School 41, 396–400.– Naimzada A, Pireddu M (2015b) Real and financial interactingmarkets: A behavioral macro-model. Chaos Solitons Fractals 77,111–131.– Naimzada A, Pireddu M (2015c) A financial market model withendogenous fundamental values through imitative behavior, Chaos 25,073110. DOI: 10.1063/1.4926326


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