forecast_tool.py

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()