Create Austrian model extended.py

Austrian model extended.py

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+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+import matplotlib as mpl
+
+# Initial conditions
+y_0 = 0.02
+y_star_0 = 0.02
+r_0 = 0.02
+r_star_0 = 0.02
+pi_0 = 0.02
+tilde_r_star_0 = 0.02
+pi_star = 0.02
+M_0 = 1.0  # Initial value of the financial market
+B_0 = 1.0  # Initial quantity of bank credit
+
+# Parameters
+alpha = 4.5  # Increased responsiveness to inflation overshooting
+beta = 0.5
+lambda_ = 0.7  # Increased learning rate for faster convergence
+gamma = 1.0
+kappa = 0.3  # Increased sensitivity of inflation to the output gap
+delta = 0.1  # Sensitivity of the financial market to interest rate differences
+S_0 = 1.0  # Base level of savings
+I_0 = 1.0  # Base level of investment
+epsilon = 0.2  # Sensitivity of savings to interest rate
+theta = 0.4  # Sensitivity of investment to interest rate
+mu = 0.3  # Expansion parameter of bank credit
+nu = 0.1  # Contraction parameter of bank credit
+rho = 0.3  # Speed of return to normal bank credit
+epsilon_r = 0.01  # Threshold for close interest rates
+
+# Simulation parameters
+years = 40
+y_star_path = [0.02] * 1 + [0.025] * (years - 1)  # y* changes to 2.5% at year 1
+
+# Initialize arrays to store results
+y = np.zeros(years)
+y_star = np.zeros(years)
+r = np.zeros(years)
+r_star = np.zeros(years)
+pi = np.zeros(years)
+tilde_r_star = np.zeros(years)
+M = np.zeros(years)
+S = np.zeros(years)  # Savings
+I = np.zeros(years)  # Investment
+B = np.zeros(years)  # Bank credit
+
+# Set initial values
+y[0] = y_0
+y_star[0] = y_star_0
+r[0] = r_0
+r_star[0] = r_star_0
+pi[0] = pi_0
+tilde_r_star[0] = tilde_r_star_0
+M[0] = M_0
+S[0] = S_0
+I[0] = I_0
+B[0] = B_0
+
+# Run simulation
+for t in range(1, years):
+    y_star[t] = y_star_path[t]
+    r_star[t] = gamma * y_star[t]
+    tilde_r_star[t] = tilde_r_star[t-1] + lambda_ * (r_star[t-1] - tilde_r_star[t-1])
+    r[t] = tilde_r_star[t] + alpha * (pi[t-1] - pi_star)
+
+    # Update savings and investment based on the interest rate differential
+    S[t] = S_0 * (1 + epsilon * (r[t]-r_star[t] ))  # Decrease in savings when r < r_star
+    I[t] = I_0 * (1 - theta * (r[t]-r_star[t]))  # Increase in investment when r < r_star
+
+    # Update bank credit based on the interest rate differential and return to normal level when rates are close
+    if r[t] < r_star[t]:
+        B[t] = B[t-1] + mu * (r_star[t] - r[t])
+    else:
+        B[t] = B[t-1] - nu * (r[t] - r_star[t])
+
+    if abs(r[t] - r_star[t]) < epsilon_r:
+        B[t] -= rho * (B[t-1] - B_0)
+
+    # Adjust y for savings, investment, and bank credit
+    y[t] = y_star[t] - beta * (r[t] - r_star[t])
+    pi[t] = pi[t-1] + kappa * (y[t] - y_star[t])
+    M[t] = I[t-1] + delta * (r_star[t] - r[t])
+
+# Convert to DataFrame and multiply values by 100 to display percentages
+df = pd.DataFrame({
+    'Year': np.arange(years),
+    'y': y * 100,
+    'y_star': y_star * 100,
+    'r': r * 100,
+    'r_star': r_star * 100,
+    'tilde_r_star': tilde_r_star * 100,
+    'pi': pi * 100,
+    'M': M,
+    'S': S,
+    'I': I,
+    'B': B
+})
+
+# Plot the results up to year 10
+plot_years = 10
+df_short = df[df['Year'] < plot_years]
+
+mpl.rcParams['font.family'] = 'cmr10'
+mpl.rcParams['axes.formatter.use_mathtext'] = True
+plt.figure(figsize=(12, 10))
+plt.suptitle('Macroeconomics simulation with Banking Sector')  # Aggiungi un titolo generale per tutti i subplot
+
+plt.subplot(3, 2, 1)
+plt.plot(df_short['Year'], df_short['y'], label='y (Aggregate Demand)')
+plt.plot(df_short['Year'], df_short['y_star'], label='y* (Potential Output)', linestyle='--')
+plt.title('Aggregate Demand and Potential Output')
+plt.xlabel('Year')
+plt.ylabel('Growth Rate (%)')
+plt.legend()
+
+plt.subplot(3, 2, 2)
+plt.plot(df_short['Year'], df_short['r'], label='r (Interest Rate)')
+plt.plot(df_short['Year'], df_short['r_star'], label='r* (Natural Rate)', linestyle='--')
+plt.plot(df_short['Year'], df_short['tilde_r_star'], label='~r* (Perceived Natural Rate)', linestyle='-.')
+plt.title('Interest Rates')
+plt.xlabel('Year')
+plt.ylabel('Rate (%)')
+plt.legend()
+
+plt.subplot(3, 2, 3)
+plt.plot(df_short['Year'], df_short['pi'], label='π (Inflation)')
+plt.axhline(pi_star * 100, color='r', linestyle='--', label='π* (Inflation Target)')
+plt.title('Inflation')
+plt.xlabel('Year')
+plt.ylabel('Rate (%)')
+plt.legend()
+
+plt.subplot(3, 2, 4)
+plt.plot(df_short['Year'], df_short['M'], label='M (Financial Market)')
+plt.title('Financial Market')
+plt.xlabel('Year')
+plt.ylabel('Market Value')
+plt.legend()
+
+plt.subplot(3, 2, 5)
+plt.plot(df_short['Year'], df_short['S'], label='S (Savings)')
+plt.axhline(S_0, color='r', linestyle='--', label='Normal level of Savings')
+plt.title('Savings of Families')
+plt.xlabel('Year')
+plt.ylabel('Savings Level')
+plt.legend()
+
+plt.subplot(3, 2, 6)
+plt.plot(df_short['Year'], df_short['I'], label='I (Investment)')
+plt.axhline(I_0, color='r', linestyle='--', label='Normal level of investment')
+plt.title('Investment by Firms')
+plt.xlabel('Year')
+plt.ylabel('Investment Level')
+plt.legend()
+
+plt.subplot(3, 2, 6)
+plt.plot(df_short['Year'], df_short['B'], label='B (Bank Credit)')
+plt.axhline(B_0, color='r', linestyle='--', label='Initial level of Bank Credit')
+plt.title('Bank Credit')
+plt.xlabel('Year')
+plt.ylabel('Credit Level')
+plt.legend()
+
+plt.tight_layout()
+plt.show()