CompEc 2025/2026

Computational Economics
Università degli Studi di Firenze

SYLLABUS

Official Syllabus from University of Florence website

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A TENTATIVE PROGRAM

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TOPICS

1. Introduction to Computational Economics

Notes on Introduction to Computational Economics
Notes on Income Expenditure Model

Script - Static

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""" @author: Giorgio Ricchiuti @website: www.grarchive.net Description Income-Expenditure Model (static) see notes https://grarchive.net/wp-content/uploads/2022/10/NotesIE.pdf """ import numpy as np import matplotlib.pyplot as plt import sympy as sym def static_IE(Z,z): y=sym.Symbol('y') eqn=sym.solveset(Z+z*y-y,y) print(eqn) Z=1000 z=0.8 static_IE(Z, z) Check=Z/(1-z) print('Check: {}'.format(Check))

Script - Dynamic Model

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""" @author: Giorgio Ricchiuti @website: www.grarchive.net Description Income-Expenditure Model (dynamical model) see notes https://grarchive.net/wp-content/uploads/2022/10/NotesIE.pdf """ import numpy as np import matplotlib.pyplot as plt def IE(N, x0,Z,z,phi): x=np.zeros(N) x[0]=x0 y=np.linspace(0, 50000, 11) R=Z+z*y plt.plot(y,R, color="black") plt.plot(y,y) plt.grid(True) plt.axhline(color="black", lw=0.5) plt.axvline(color="black", lw=0.5) for t in range(N-1): x[t+1]=x[t]+phi*(Z+z*x[t]-x[t]) plt.plot([x[t],x[t]],[x[t],x[t+1]], color="blue") plt.plot([x[t],x[t+1]],[x[t+1],x[t+1]], color="red") plt.title("Dynamic IE Model") plt.show() return x x = IE(20,10000,15000,0.53,1) plt.plot(x) plt.title("Time series for Incomce") plt.show

2. HAMS and Stylized Facts in FM

Lecture Notes on HAMS and Stylized Facts in Financial Markets

3. Logistic Map

Python Scripts:

Logistic Function

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#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ Created on Fri Mar 10 11:33:34 2023 @author: giorgio """ import numpy as np import matplotlib.pyplot as plt def logistic_map(N, x0, lamb): x=np.zeros(N); x[0]=x0; #let's make a draw of the logistic function z=np.linspace(0,1,101) y=lamb*z*(1-z) plt.plot(z,y,color="black", label="y=lamb*z*(1-z)" ) #a loop to calculate the evolution for t in range(N-1): x[t+1]=lamb*x[t]*(1-x[t]) plt.plot([x[t],x[t]],[x[t], x[t+1]], color='blue') plt.plot([x[t],x[t+1]],[x[t+1], x[t+1]], color='blue') plt.plot([0,1.2],[0,1.2], color='black', linestyle='dashed', linewidth=0.5) return x ''' Figures with different initial conditions ''' f= logistic_map(35, 0.3, 3.9) plt.title('Logistic Map 1') plt.grid(True) plt.show() g= logistic_map(35, 0.3001, 3.9) plt.title('Logistic Map 2') plt.grid(True) plt.show() plt.plot(f, color='blue') plt.plot(g, color='red') plt.xlabel('time') plt.ylabel('x') plt.title('Time series of x') plt.show() ''' Figures for different values of lambda ''' f= logistic_map(35, 0.3, 1) plt.title('Logistic Map 1') plt.grid(True) plt.show() g= logistic_map(35, 0.3, 3.2) plt.title('Logistic Map 2') plt.grid(True) plt.show() h= logistic_map(35, 0.3, 3.9) plt.title('Logistic Map 3') plt.grid(True) plt.show() plt.plot(f, color='blue', label='1') plt.plot(g, color='red', label='3.2') plt.plot(h, color='green', label='3.9') plt.legend(loc=3) plt.xlabel('time') plt.ylabel('x') plt.title('Time series of x') plt.show() """ All figures with subplots """ fig=plt.figure() ax1=fig.add_subplot(321) x = logistic_map(35, 0.3, 2) ax2=fig.add_subplot(322) plt.plot(x) ax3=fig.add_subplot(323) x1 = logistic_map(35, 0.3, 3.3) ax4=fig.add_subplot(324) plt.plot(x1) ax5=fig.add_subplot(325) x2 = logistic_map(35, 0.3, 3.9) ax6=fig.add_subplot(326) plt.plot(x2) #plt.title('Time series 3') plt.show()

Bifurcation py 1

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#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ @author: giorgio """ import numpy as np import matplotlib.pyplot as plt def logistic_eq(r,x): return r*x*(1-x) # Create the bifurcation diagram def bifurcation_diagram(seed, n_skip, n_iter, step=0.0001, r_min=0): print("Starting with x0 seed {0}, skip plotting first {1} iterations, then plot next {2} iterations.".format(seed, n_skip, n_iter)); # Array of r values, the x axis of the bifurcation plot R = [] # Array of x_t values, the y axis of the bifurcation plot X = [] # Create the r values to loop. For each r value we will plot n_iter points r_range = np.linspace(r_min, 4, int(1/step)) for r in r_range: x = seed; # For each r, iterate the logistic function and collect datapoint if n_skip iterations have occurred for i in range(n_iter+n_skip+1): if i >= n_skip: R.append(r) X.append(x) x = logistic_eq(r,x); # Plot the data plt.plot(R, X, ls='', marker=',') plt.ylim(0, 1) plt.xlim(r_min, 4) plt.xlabel('r') plt.ylabel('X') plt.show() return R, X, x R, X, x=bifurcation_diagram(0.2, 100, 5) R, X, x= bifurcation_diagram(0.2, 100, 10) R, X, x= bifurcation_diagram(0.2, 100, 10, r_min=2.8)

Bifurcation py 2

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#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ @author: giorgio """ import numpy as np import matplotlib.pyplot as plt # Create the bifurcation diagram def bifurcation_diagram(N): step=0.001 x=np.zeros(N) X=np.zeros((int(1/step+1),N)) r_min=0 r_max=3.95 x[0]=0.01 r_range = np.linspace(r_min, r_max, int(1/step+1)) flag=-1 for i in r_range: flag = flag + 1 for t in range(N-1): x[t+1]=i*x[t]*(1-x[t]) #print(x) X[flag:] = x #print(X) T_last = 350 Nr =int(N - T_last) plt.show() for i in range(len(r_range)): plt.plot(r_range[i]*np.ones((T_last)), X[i, Nr:N], linestyle='dashdot', linewidth=0.01, markersize=0.1, marker=',', color='black') return x x=bifurcation_diagram(400)
Bibliography:
1. Simple mathematical models with very complicated dynamics (Robert May, Nature 1976)
2. 2.Hommes, C. (2013), Behavioural Rationality and Heterogenous Expectations, ed. Cambridge University Press (Chapter 2)

4. Heterogeneous Fundamentalists and Imitative Processes

Python scripts:

Dynamics

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""" @author: Giorgio Heterogeneous Fundamentalists and Imitative Processes Naimzada A. and Ricchiuti G. (2008) Applied Mathematics and Computation, Volume 199, Issue 1, 15 May 2008, Pages 171-180. https://doi.org/10.1016/j.amc.2007.09.061 """ import numpy as np import matplotlib.pyplot as plt def fundamentalist(N, p0, a, F1, F2): p=np.zeros(N) p[0]=p0 n1=np.zeros(N) n2=np.zeros(N) n1[0]=1-((F1-p[0])**2)/((F1-p[0])**2+(F2-p[0])**2) n2[0]=1-n1[0] z=np.linspace(0,5,501) y=z+a*((F1-z)*(F2-z)*(F1+F2-2*z)/((F1-z)**2+(F2-z)**2)) plt.plot(z,y,color='black') plt.plot(z,z) plt.grid(True) plt.axhline(color='black', lw=0.5) plt.axvline(color='black', lw=0.5) for t in range(N-1): p[t+1]=p[t]+a*((F1-p[t])*(F2-p[t])*(F1+F2-2*p[t])/((F1-p[t])**2+(F2-p[t])**2)) plt.plot([p[t],p[t]],[p[t],p[t+1]]) plt.plot([p[t],p[t+1]],[p[t+1],p[t+1]]) n1[t+1]=1-((F1-p[t+1])**2)/((F1-p[t+1])**2+(F2-p[t+1])**2) n2[t+1]=1-n1[t+1] return p, n1,n2 """ Figures """ fig=plt.figure() ax1=fig.add_subplot(321) p, n1, n2=fundamentalist(100, 3.6, 1, 1, 4) ax2=fig.add_subplot(322) plt.plot(p) ax1=fig.add_subplot(323) p, n1, n2=fundamentalist(100, 3.6, 2.1, 1, 4) ax2=fig.add_subplot(324) plt.plot(p) ax1=fig.add_subplot(325) p, n1, n2=fundamentalist(100, 3.6, 2.8, 1, 4) ax2=fig.add_subplot(326) plt.plot(p) plt.show() p, n1, n2=fundamentalist(100, 3.6, 3.1, 1, 4) plt.show() plt.plot(p) plt.show() p, n1, n2=fundamentalist(100, 3.6, 3.1, 1, 4) plt.show() fig=plt.figure() ax1=fig.add_subplot(321) ax1=fig.add_subplot(211) plt.plot(n1) plt.ylabel("quotas") plt.plot(n2) ax1=fig.add_subplot(212) plt.plot(p) plt.ylabel("price") plt.xlabel("time") plt.show()

Bifurcation Diagram

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#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ @author: giorgio """ import numpy as np import matplotlib.pyplot as plt def fundamentalist(N, a,F1,F2,p): return p+a*((F1-p)*(F2-p)*(F1+F2-2*p))/((F1-p)**2+(F2-p)**2) # Create the bifurcation diagram def bifurcation_diagram(seed, n_skip, n_iter, step=0.001, a_min=1.5): print("Starting with x0 seed {0}, skip plotting first {1} iterations, then plot next {2} iterations.".format(seed, n_skip, n_iter)); # Array of A values, the x axis of the bifurcation plot A = [] # Array of P_t values, the y axis of the bifurcation plot P = [] N=n_skip+n_iter F1=1 F2=5 # Create the A values to loop. For each A value we will plot n_iter points a_range = np.linspace(a_min, 3, int(1/step)) for a in a_range: p = seed; # For each A, iterate the function and collect datapoint if n_skip iterations have occurred for i in range(n_iter+n_skip+1): if i >= n_skip: A.append(a) P.append(p) p = fundamentalist(N, a,F1,F2,p) # Plot the data plt.plot(A, P, ls='', marker=',', color='red') plt.ylim(0, 7) plt.xlim(a_min, 3) plt.xlabel('alfa*beta') plt.ylabel('P') for a in a_range: seed=4.5 p = seed; # For each A, iterate the logistic function and collect datapoint if n_skip iterations have occurred for i in range(n_iter+n_skip+1): if i >= n_skip: A.append(a) P.append(p) p = fundamentalist(N, a,F1,F2,p) # Plot the data plt.plot(A, P, ls='', marker=',', color='blue') plt.ylim(0, 7) plt.xlim(a_min, 3) plt.xlabel('alfa*beta') plt.ylabel('P') plt.show() bifurcation_diagram(0.8, 200,10)
Bibliography:

Naimzada A. and Ricchiuti G. (2008), Applied Mathematics and Computation, Volume 199, Issue 1, 15 May 2008, Pages 171-180

TOOLS

Python (project, anaconda)
LateX (project, example)

CompEc 2025/2026

Computational Economics Università degli Studi di Firenze

SYLLABUS

A TENTATIVE PROGRAM

SUGGESTED READINGS

TOPICS

1. Introduction to Computational Economics

Script - Static

Script - Dynamic Model

2. HAMS and Stylized Facts in FM

3. Logistic Map

Logistic Function

Bifurcation py 1

Bifurcation py 2

4. Heterogeneous Fundamentalists and Imitative Processes

Dynamics

Bifurcation Diagram

TOOLS

Computational Economics
Università degli Studi di Firenze