CompEc 2023/2024

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. Stylized Facts in Financial Markets

Notes

3. Notes on Python

Python.org Beginners Guide

4. 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)

5. 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

6. Two Dimensional System – Lotka-Volterra Model

Python Scripts:

Jacobian Matrix of Lotka-Volterra (plus Determinant and Trace)

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import numpy as np import matplotlib.pyplot as plt import sympy as sym from sympy import * from sympy.matrices import Matrix x=symbols('x') diff=diff(x**2, x) print(diff) def Jacobian(v_str, f_list): vars = sym.symbols(v_str) f = sym.sympify(f_list) J = sym.zeros(len(f),len(vars)) for i, fi in enumerate(f): for j, s in enumerate(vars): J[i,j] = sym.diff(fi, s) return J A=Jacobian('x y', ['(1+r)*x-s*x**2-b*x*y','c*y*x-e*y']) print(A) B=A.det() print(B) C=A.trace() print(C)

Lotka-Volterra

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#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ Created on Thu Nov 26 22:03:13 2020 @author: gr Model Lotka-Volterra X_{t+1}= (1+r) X_t-sX^2_{t}-bX_{t}Y_{t}\\ Y_{t+1}= Y_{t}(cX_t-e) """ import numpy as np import matplotlib.pyplot as plt def lotka_volterra(N, y0, x0, r, s, b, e, c): x=np.zeros(N) #prey y=np.zeros(N) #predator y[0]=y0 #initial condition x[0]=x0 #initial condition for t in range(N-1): x[t+1]=(1+r)*x[t]-s*(x[t]**2)-b*x[t]*y[t] y[t+1]=y[t]*(c*x[t]-e) plt.scatter(x[N-1000:N],y[N-1000:N], color='red', s=1) plt.xlabel("prey") plt.ylabel("predator") return x, y parameters = { "x0": 1.5, "y0": 0.5, "r" : 0.5, "s" : 0.21, #0.18,0.16, 0.125 "b" : 1, "e" : 0.6, "c" : 1, "N" : 2000, } x,y= lotka_volterra(**parameters) plt.show() plt.plot(x[0:100], label="prey") plt.plot(y[0:100], label="predator") plt.legend(bbox_to_anchor=(-0.4, 1), loc='upper left', borderaxespad=0.1) plt.show() plt.hist(x, 50, density=True, facecolor='b', alpha=0.75) plt.ylabel('Probability') plt.title('Histogram of Prey') plt.grid(True) plt.show() plt.hist(y, 50, density=True, facecolor='g', alpha=0.75) plt.ylabel('Probability') plt.title('Histogram of Predator') plt.grid(True) plt.show()

Steady States for the Lotka-Volterra Model

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#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ Created on Thu Mar 16 20:18:54 2023 @author: Giorgio Ricchiuti @website: www.grarchive.net Steady States for the Lotka-Volterra Model """ import sympy as sym #from sympy import * def Lotka_Volterra_SS(): y,x, s,c, r, b, e=sym.symbols('y, x, s, c, r, b, e') system=sym.solve([(1 + r)*x-s*x**2-b*x*y-x, y*c*x-y*e-y], (x,y))#, force=True, manual=True, set=True ) print(system) system=Lotka_Volterra_SS()
Bibliography:
1. Hommes, C. (2013), Behavioural Rationality and Heterogenous Expectations, ed. Cambridge University Press (Chapter 3)

7. The Discrete Choice Model

Bibliography:
1. Hommes, C. (2013), Behavioural Rationality and Heterogenous Expectations, ed. Cambridge University Press (Chapter 6.1,6.2,6.3)
2. Train K, (2009), Discrete Choice Methods with Simulation (Chapter 1)

8. Heterogeneous beliefs and routes to chaos in a simple asset pricing model

William A. Brock, Cars H. Hommes, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, Journal of Economic Dynamics and Control, Volume 22, Issues 8–9, 1998, Pages 1235-1274, ISSN 0165-1889,
Galanis G., Lustenhouwer J., Ricchiuti G. (2023), ` A fundamendalists’ profits paradox?’ (slides)
Python Code

Script

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# -*- coding: utf-8 -*- """ @author: giorgio BH1998 with two types of agents """ import numpy as np import matplotlib.pyplot as plt import statsmodels.api as sm from statsmodels.graphics import tsaplots def BH1998two(N, beta,b1, b2, g1, g2): x=np.zeros(N) # difference between price and optimal price y=np.zeros(N) # delayed difference between price and optimal price n1=np.zeros(N) # quotas of first group n2=np.zeros(N) # quotas of second group z=np.zeros(N) pi1=np.zeros(N) pi2=np.zeros(N) U1=np.zeros(N) U2=np.zeros(N) V=np.zeros(N) s=np.zeros(N) R=1.1 # risk-free C=1 # cost of the fundamental predictor w=0 #fitness measure persistence n10=0.2 delta=0.01 V[0]=0 x[0]=0.21 x[1]=0.22 y[0]=0.2 n1[0]=n10 n2[0]=1-n10 pi1[0]=0 # profits first group pi2[0]=0 # profits second group U1[0]=0 #Utility 1 U2[0]=0 #Utility 2 z[0]=0 # plt.axhline(color='black', lw=0.5) # plt.axvline(color='black', lw=0.5) meanX=np.zeros(N) for t in range(1,N-1): #compute profits pi1[t]=(x[t]-R*x[t-1])*(g1*y[t-1]+b1-R*x[t-1])-C pi2[t]=(x[t]-R*x[t-1])*(g2*y[t-1]+b2-R*x[t-1]) #compute utility U1[t]=(1-w)*pi1[t]+w*U1[t-1] U2[t]=(1-w)*pi2[t]+w*U2[t-1] #fraction z[t]= np.exp(beta*U1[t])+np.exp(beta*U2[t]) n1[t+1]=(delta*n1[t]+(1-delta)*np.exp(beta*U1[t]))/z[t] n2[t+1]=1-n1[t+1] s[t+1]=n1[t+1]*(g1*x[t]+b1)+n2[t+1]*(g2*x[t]+b2) #update deviation price from fundamental price x[t+1]=s[t+1]/R V[t+1]=(x[t+1]-x[t])/x[t] y[t]=x[t-1] meanX=np.mean(x) meanpi1=np.mean(pi1) meanpi2=np.mean(pi2) diffpi=pi1-pi2 #difference between profits of fund and profits of chartists diffmeanpi=meanpi1-meanpi2 #difference mean profits return x, y, s, pi1, pi2, n1, n2, U1, U2, V, meanX, meanpi1, meanpi2, diffpi, diffmeanpi """ Moving Average Function """ def window(size): return np.ones(size)/float(size) x, y, s, pi1, pi2, n1, n2, U1, U2, V, meanX, meanpi1, meanpi2, diffpi, diffmeanpi = BH1998two(10000, 3.7, b1=0, b2=0, g1=0, g2=1.2) plt.plot(x, label=r'x_t') plt.plot(np.convolve(x,window(50),'same'),'r', label=r'rolling 50t') plt.plot(np.convolve(x,window(500),'same'),'b', label=r'rolling 100t') plt.title("Deviation from Fundamental Value") plt.legend() plt.xlabel('Time') plt.ylabel('X') plt.xlim([9000, 9500]) plt.ylim([-0.2, 3]) plt.show() plt.plot(x) plt.title("Deviation from Fundamental Value") plt.xlabel('Time') plt.ylabel('X') plt.xlim([9000, 10000]) plt.ylim([-0.2, 3]) plt.show() plt.plot(n1, label='Fundamentalists') plt.plot(n2, label='Chartists') plt.xlabel('Time') plt.ylabel('Fraction of agents') plt.title("Time series of Quotas") plt.legend() plt.xlim([9400, 9500]) plt.ylim([-0.1, 1.1]) plt.show()

8.bis Empirical Validation and Lab Experiments

Gusella F, Ricchiuti G. (2023) Endogenous Cycles In Heterogeneous Agent Models: A State-Space Approach (slides, text)

9. Network: An Introduction

Lecture Notes

10. Topology Measures

Lecture Notes

11. Asset price dynamics with heterogeneous beliefs and local network interactions

Valentyn Panchenko, Sergiy Gerasymchuk, Oleg V. Pavlov,
Asset price dynamics with heterogeneous beliefs and local network interactions, Journal of Economic Dynamics and Control, Volume 37, Issue 12, December 2013, Pages 2623-2642
Lecture Notes

12. Interaction in Financial Markets

Simone Berardi and Gabriele Tedeschi, (2017) `From banks’ strategies to ﬁnancial (in)stability‘, International Review of Economics & Finance, Volume 47, January 2017, Pages 255-272

TOOLS

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

CompEc 2023/2024

Computational Economics Università degli Studi di Firenze

SYLLABUS

A TENTATIVE PROGRAM

SUGGESTED READINGS

TOPICS

1. Introduction to Computational Economics

2. Stylized Facts in Financial Markets

3. Notes on Python

4. Logistic Map

5. Heterogeneous Fundamentalists and Imitative Processes

6. Two Dimensional System – Lotka-Volterra Model

7. The Discrete Choice Model

8. Heterogeneous beliefs and routes to chaos in a simple asset pricing model

8.bis Empirical Validation and Lab Experiments

9. Network: An Introduction

10. Topology Measures

11. Asset price dynamics with heterogeneous beliefs and local network interactions

12. Interaction in Financial Markets

TOOLS

Computational Economics
Università degli Studi di Firenze