Financial models with interacting heterogeneous agents: modeling assumptions and mathematical tools from discrete dynamical system theory. Minicourse for the PhD Program in Methods and Models for Economic Decisions, Insubria University Marina Pireddu University of Milano-Bicocca Dept. 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
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Financial models with interacting heterogeneous agents ...pireddu/Beamer-Insubria-2D.pdf · Outline 1 2D discrete dynamical systems 2 Other Heterogeneous Agents Models Marina Pireddu
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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.
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.
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.
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.
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.
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.
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.
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.
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.
(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.
(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.
(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.
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
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
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
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
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
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
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.
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.
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.
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.
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.
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.
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.
– 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
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).
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).
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).
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.
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.
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.
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.
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.
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.
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.
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 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 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 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):
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):
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):
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.
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.
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.
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.
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,
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 = 5
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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)
eµπX (t+1) + 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.
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)
eµπX (t+1) + 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.
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)
eµπX (t+1) + 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.
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)
eµπX (t+1) + 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.
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)
eµπX (t+1) + 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.
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)
eµπX (t+1) + 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.
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)
eµπX (t+1) + 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.
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)
eµπX (t+1) + 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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
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.
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.
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.
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.
The belief about the fundamental of pessimistic agents is given by:
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
The belief about the fundamental of optimistic agents is given by:
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
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 ;
(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.
The belief about the fundamental of pessimistic agents is given by:
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
The belief about the fundamental of optimistic agents is given by:
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
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 ;
(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.
The belief about the fundamental of pessimistic agents is given by:
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
The belief about the fundamental of optimistic agents is given by:
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
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 ;
(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.
The belief about the fundamental of pessimistic agents is given by:
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
The belief about the fundamental of optimistic agents is given by:
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
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 ;
(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.
The belief about the fundamental of pessimistic agents is given by:
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
The belief about the fundamental of optimistic agents is given by:
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
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 ;
(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.
The belief about the fundamental of pessimistic agents is given by:
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
The belief about the fundamental of optimistic agents is given by:
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
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 ;
(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.
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 .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
where D(t) is the excess demand, γ is a positive parameter influencingthe price variation reactivity and a1, a2 are positive parameters limitingprice variation.
where D(t) is the excess demand, γ is a positive parameter influencingthe price variation reactivity and a1, a2 are positive parameters limitingprice variation.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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.
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.
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.
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.
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.
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.
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.
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)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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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