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 Script:
Dynamics

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#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ @author: giorgio Fundamentalist price dynamics with two fundamental values. The price updating rule follows: P_{t+1} = P_t + alpha * [(F1 - P_t)(F2 - P_t)(F1 + F2 - 2*P_t)] / [(F1 - P_t)^2 + (F2 - P_t)^2] This is a nonlinear map where the price is attracted toward two 'fundamental' values F1 and F2. The parameter alpha controls the speed of adjustment. Depending on alpha, the system can converge to a fixed point, oscillate, or exhibit chaotic behaviour. """ import numpy as np import matplotlib.pyplot as plt def fundamentalist(N, p0, a, F1, F2): """ Simulate fundamentalist price dynamics and plot the cobweb diagram. The cobweb diagram visualises iteration of the map by alternating vertical segments (price update step) and horizontal segments (mapping the new price back to the 45-degree line), making it easy to spot convergence, cycles, or chaos. Parameters ---------- N : int — number of time steps p0 : float — initial price a : float — adjustment speed (alpha) F1 : float — first fundamental value F2 : float — second fundamental value Returns ------- p : ndarray of shape (N,) — simulated price path """ # Initialise price array and set starting price p = np.zeros(N) p[0] = p0 # --- Cobweb diagram --- # Evaluate the map f(z) over a fine grid for plotting 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", label="Map f(z)") # The nonlinear map plt.plot(z, z, color="gray", label="45° line") # Identity line y = z plt.grid(True) plt.axhline(color="black", lw=0.5) # Horizontal axis plt.axvline(color="black", lw=0.5) # Vertical axis # --- Simulate and draw cobweb --- for t in range(N - 1): # Apply the price update rule 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) # Vertical segment: move from (p_t, p_t) up/down to (p_t, p_{t+1}) plt.plot([p[t], p[t]], [p[t], p[t + 1]], color="blue") # Horizontal segment: move from (p_t, p_{t+1}) across to (p_{t+1}, p_{t+1}) plt.plot([p[t], p[t + 1]], [p[t + 1], p[t + 1]], color="red") # Dashed 45-degree reference line scaled to the price range plt.plot( [0, max(p)], [0, max(p)], color="black", linestyle="dashed", linewidth=0.5 ) return p # ------------------------------------------------------------------ # Parameter dictionary (not used directly by the function calls below, # kept here as a reference for the baseline calibration) # ------------------------------------------------------------------ par = { "N": 100, # Number of iterations "p0": 3.6, # Initial price "a": 1, # Adjustment speed "F1": 1, # First fundamental value "F2": 4, # Second fundamental value } # ================================================================== # Figure 1 — All six panels in a single 3×2 grid # Rows vary alpha to illustrate the transition from convergence # to oscillation to chaos; left column = cobweb, right = time series # ================================================================== fig = plt.figure() ax1 = fig.add_subplot(321) # Cobweb, alpha = 1.0 (convergence) p = fundamentalist(100, 3.6, 1, 1, 4) ax2 = fig.add_subplot(322) # Time series, alpha = 1.0 plt.plot(p) ax3 = fig.add_subplot(323) # Cobweb, alpha = 2.1 (period-2 cycle) p1 = fundamentalist(100, 3.6, 2.1, 1, 4) ax4 = fig.add_subplot(324) # Time series, alpha = 2.1 plt.plot(p1) ax5 = fig.add_subplot(325) # Cobweb, alpha = 2.8 (complex dynamics) p2 = fundamentalist(100, 3.6, 2.8, 1, 4) ax6 = fig.add_subplot(326) # Time series, alpha = 2.8 plt.plot(p2) plt.show() # ================================================================== # Figures 2–5 — Individual cobweb + time-series pairs # Useful for presentation or paper figures; alpha increases across # figures to show the route to chaos # ================================================================== # --- alpha = 1.0: convergence to an interior fixed point --- fig = plt.figure() ax1 = fig.add_subplot(211) p = fundamentalist(100, 3.6, 1, 1, 4) ax2 = fig.add_subplot(212) plt.plot(p) plt.title("Time series 1 (alpha = 1.0)") plt.show() # --- alpha = 2.1: period-2 oscillation --- fig = plt.figure() ax3 = fig.add_subplot(211) p1 = fundamentalist(100, 3.6, 2.1, 1, 4) ax4 = fig.add_subplot(212) plt.plot(p1) plt.title("Time series 2 (alpha = 2.1)") plt.show() # --- alpha = 2.8: higher-period or chaotic dynamics --- fig = plt.figure() ax5 = fig.add_subplot(211) p2 = fundamentalist(100, 3.6, 2.8, 1, 4) ax6 = fig.add_subplot(212) plt.plot(p2) plt.title("Time series 3 (alpha = 2.8)") plt.show() # --- alpha = 2.98: deeper into the chaotic regime --- fig = plt.figure() ax5 = fig.add_subplot(211) p2 = fundamentalist(100, 3.6, 2.98, 1, 4) ax6 = fig.add_subplot(212) plt.plot(p2) plt.title("Time series 4 (alpha = 2.98)") plt.show()
Bibliography:

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

4.1 Dynamic effects of increasing heterogeneity in financial markets

Python Script

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#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ Short classroom version of the Naimzada-Ricchiuti (2009) model. Use this file in class when you want: 1. A clean price map 2. A cobweb diagram 3. A short price time series The goal is not full analysis. The goal is to show the mechanism clearly. """ import numpy as np import matplotlib.pyplot as plt # ============================================================ # 1. CORE MODEL # ============================================================ def price_map(p, alpha, beta, gamma, F1, F2): """ One-dimensional price map. p_{t+1} = p_t + alpha*beta * [ (F2 - p_t) - (F2 - F1)/(1 + exp(gamma*(F2-F1)*(2*p_t - F1 - F2))) ] """ delta_F = F2 - F1 logistic_term = delta_F / (1.0 + np.exp(gamma * delta_F * (2.0 * p - F1 - F2))) return p + alpha * beta * ((F2 - p) - logistic_term) def simulate_model(T, p0, alpha, beta, gamma, F1, F2): """ Simulate the price dynamics for T periods. """ p = np.zeros(T) p[0] = p0 for t in range(T - 1): p[t + 1] = price_map(p[t], alpha, beta, gamma, F1, F2) return p # ============================================================ # 2. SIMPLE PLOTS FOR CLASS # ============================================================ def plot_map_and_cobweb(alpha, beta, gamma, F1, F2, p0, xmin=5, xmax=9, ngrid=600, steps=30): """ Plot the map, the 45-degree line, and a cobweb path. """ x = np.linspace(xmin, xmax, ngrid) y = price_map(x, alpha, beta, gamma, F1, F2) plt.figure(figsize=(7, 5)) plt.plot(x, y, label="Map") plt.plot(x, x, "k--", linewidth=1, label="45-degree line") p = p0 for _ in range(steps): p_next = price_map(p, alpha, beta, gamma, F1, F2) plt.plot([p, p], [p, p_next], color="blue", linewidth=1) plt.plot([p, p_next], [p_next, p_next], color="red", linewidth=1) p = p_next plt.xlabel("p_t") plt.ylabel("p_{t+1}") plt.title(f"Cobweb diagram (p0 = {p0})") plt.grid(True) plt.legend() plt.show() def plot_price_series(p, title="Price series"): """ Plot a price time series. """ plt.figure(figsize=(8, 4)) plt.plot(p) plt.xlabel("Time") plt.ylabel("Price") plt.title(title) plt.grid(True) plt.show() # ============================================================ # 3. THREE SHORT CLASSROOM EXPERIMENTS # ============================================================ def experiment_1_single_equilibrium(): """ Experiment 1: Low heterogeneity / moderate reaction. The dynamics move toward a single stable outcome. """ alpha = 1.1 beta = 1.0 gamma = 0.8 F1 = 7.0 F2 = 8.0 p0 = 6.5 T = 60 print("\nExperiment 1: single stable outcome") print(f"alpha={alpha}, beta={beta}, gamma={gamma}, F1={F1}, F2={F2}, p0={p0}") plot_map_and_cobweb(alpha, beta, gamma, F1, F2, p0, xmin=5, xmax=9, steps=20) p = simulate_model(T, p0, alpha, beta, gamma, F1, F2) plot_price_series(p, title="Experiment 1: price convergence") def experiment_2_two_initial_conditions(): """ Experiment 2: Higher heterogeneity. Compare two nearby initial conditions. This helps illustrate multiple long-run outcomes. """ alpha = 1.1 beta = 3.0 gamma = 0.8 F1 = 6.0 F2 = 8.0 p0_a = 6.2 p0_b = 7.8 T = 80 print("\nExperiment 2: two initial conditions") print(f"alpha={alpha}, beta={beta}, gamma={gamma}, F1={F1}, F2={F2}") plot_map_and_cobweb(alpha, beta, gamma, F1, F2, p0_a, xmin=5, xmax=9, steps=25) plot_map_and_cobweb(alpha, beta, gamma, F1, F2, p0_b, xmin=5, xmax=9, steps=25) p_a = simulate_model(T, p0_a, alpha, beta, gamma, F1, F2) p_b = simulate_model(T, p0_b, alpha, beta, gamma, F1, F2) plt.figure(figsize=(8, 4)) plt.plot(p_a, label=f"p0 = {p0_a}") plt.plot(p_b, label=f"p0 = {p0_b}") plt.xlabel("Time") plt.ylabel("Price") plt.title("Experiment 2: dependence on initial conditions") plt.grid(True) plt.legend() plt.show() def experiment_3_flip_and_complexity(): """ Experiment 3: Stronger market-maker reaction. This can generate oscillations and more complex dynamics. """ alpha = 1.1 beta = 4.5 gamma = 0.8 F1 = 6.0 F2 = 8.0 p0 = 6.2 T = 120 print("\nExperiment 3: oscillations / flip dynamics") print(f"alpha={alpha}, beta={beta}, gamma={gamma}, F1={F1}, F2={F2}, p0={p0}") plot_map_and_cobweb(alpha, beta, gamma, F1, F2, p0, xmin=5, xmax=9, steps=40) p = simulate_model(T, p0, alpha, beta, gamma, F1, F2) plot_price_series(p, title="Experiment 3: oscillatory dynamics") # ============================================================ # 4. MAIN # ============================================================ if __name__ == "__main__": # In class, run one experiment at a time. # At home, you can run all three. experiment_1_single_equilibrium() experiment_2_two_initial_conditions() experiment_3_flip_and_complexity()
Bibliography:
1. Naimzada A. and Ricchiuti G. (2009) ‘Dynamic Effects of Increasing Heterogeneity in Financial Markets’, Chaos Solitons and Fractals, Volume 41, Issue 4, August 2009, Pages 1764-1772
2. Naimzada A. and Ricchiuti G. (2012) ‘A Note on Biased Fundamentalists’, Chaos Solitons and Fractals, Volume 45, Issue 3, March 2012, Pages 224 – 228

5. 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)

6. 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)

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

BH with Three Strategies
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()
Notes on `A fundamendalists’ profits paradox?’
Bibliography:
1. Hommes, C. (2013), Behavioural Rationality and Heterogenous Expectations, ed. Cambridge University Press (Chapter 6.1,6.2,6.3)
2. 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
3. Galanis G., Lustenhouwer J., Ricchiuti G. (2023), `A fundamendalists’ profits paradox?’

8. Empirical Validation and Lab Experiments

Lecture notes on Lab Experiments (second part of the presentation)
Lecture notes on Gusella and Ricchiuti 2024
Lecture Notes Delli Gatti, Gusella, Ricchiuti (2025)
Bibliography:
Mikhail Anufriev and Cars Hommes, (2012) Evolutionary Selection of Individual Expectations and Aggregate Outcomes in Asset Pricing Experiments, American Economic Journal: Microeconomics Vol. 4, No. 4, pp. 35-64

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

Delli Gatti D., Gusella F., Ricchiuti G. (2025) `Endogenous vs exogenous fluctuations: unveiling the impact of heterogeneous expectations’, Macroeconomic Dynamics Volume 29, e125

9. Network: An Introduction

Lecture notes:
Bibliography:
1. Mark Newman (2018) Networks, An introduction, ed. Oxford University Press

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

Lecture Notes
Bibliography:
1. 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

11. Interaction in Financial Markets

Lecture Notes
Bibliography:

1. Berardi S, Tedeschi G. (2017) `From banks’ strategies to financial (in)stability‘, International Review of Economics & Finance, Volume 47, Pages 255-272,

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

2. HAMS and Stylized Facts in FM

3. Logistic Map

4. Heterogeneous Fundamentalists and Imitative Processes

4.1 Dynamic effects of increasing heterogeneity in financial markets

5. Two Dimensional System – Lotka-Volterra Model

6. The Discrete Choice Model

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

8. Empirical Validation and Lab Experiments

9. Network: An Introduction

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

11. Interaction in Financial Markets

TOOLS

Computational Economics
Università degli Studi di Firenze