*3.4. The Complete Model*

When all the components are considered the model takes a normal form **z***<sup>t</sup>* = *A***z***t*−<sup>1</sup> in *D* ∪ [*v* − *δ*, *v* + *δ*] where

$$A = \begin{pmatrix} a & b & 0 & \cdots & \cdots & 0 \\ -\varepsilon + \frac{\varepsilon}{S} - \frac{\varepsilon}{L} & d\_m & \cdots & 0 & -\frac{\varepsilon}{S} & \cdots & \frac{\varepsilon}{L} \\ & 1 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 1 & 0 & \cdots & 0 \\ & \vdots & 0 & 0 & 1 & 0 & \cdots & 0 \\ & \vdots & 0 & \vdots & 0 & \ddots & \cdots & 0 \\ & \vdots & 0 & 0 & \ddots & 1 & 0 \\ & \vdots & 0 & 0 & \ddots & 1 & 0 \\ 0 & \cdots & 0 & \cdots & 0 & 1 \end{pmatrix}.\tag{23}$$

Also, in this case, we can write the characteristic polynomial, however, the stability region study is not trivial, and it can be performed only partially. For this reason, in the following, we settle for numerical simulation to discuss the dynamics scenario and the economic implications.

#### **4. Numerical Simulation**

In the following, we study numerically all the presented cases. In detail, we start with the fundamental map; then we consider fundamental and market makers' demand together; as a third step we consider chartist demand firstly alone then jointly with fundamental demand and, finally, we combine all the demands. The market makers' demand is studied only associated with fundamental and chartist demands, indeed, as already mentioned, being thought to absorb excess demand, alone it loses reasonability from the economic point of view. In all the simulations it holds *v* = 5, *δ* = *<sup>v</sup>* <sup>2</sup> and *d* = 2. The main discussion is about parameters *a*, *b*, *c*, *dm*,*e* which give the impact of each demand and past prices on the dynamics.
