The following Python script reproduces the savings simulation, the cumulative cost of inflation, the equivalent cost in working hours, and the decline in the real wage.
import numpy as np
import matplotlib.pyplot as plt
# --------------------------------------------------
# Model parameters
# --------------------------------------------------
annual_interest_rate = 0.01
annual_inflation_rates = {
"2% inflation": 0.02,
"5% inflation": 0.05,
"10% inflation": 0.10,
}
consumption = 0.90
hours_per_month = 160
simulation_months = 36
# --------------------------------------------------
# Monthly rates
# --------------------------------------------------
monthly_interest_rate = (
(1 + annual_interest_rate) ** (1 / 12) - 1
)
monthly_inflation_rates = {
label: (1 + annual_rate) ** (1 / 12) - 1
for label, annual_rate in annual_inflation_rates.items()
}
# --------------------------------------------------
# Savings function
# --------------------------------------------------
def savings(t, monthly_inflation):
"""
Calculate the household's financial position
at the end of month t.
A negative value represents a financing shortfall
rather than positive savings.
"""
financial_position = 0.0
for month in range(1, t + 1):
price_level = (1 + monthly_inflation) ** month
nominal_consumption = consumption * price_level
financial_position = (
1
+ financial_position * (1 + monthly_interest_rate)
- nominal_consumption
)
return financial_position
# --------------------------------------------------
# Cumulative inflation cost
# --------------------------------------------------
def inflation_cost(t, monthly_inflation):
"""
Calculate the cumulative additional expenditure
required to maintain the initial level of real
consumption through month t.
The result is expressed in monthly-wage units.
"""
if t == 0:
return 0.0
geometric_sum = (
(1 + monthly_inflation)
* ((1 + monthly_inflation) ** t - 1)
/ monthly_inflation
)
return consumption * (geometric_sum - t)
# --------------------------------------------------
# Inflation cost in working hours
# --------------------------------------------------
def inflation_cost_hours(t, monthly_inflation):
"""
Convert the cumulative inflation cost from
monthly-wage units into working hours.
"""
return (
inflation_cost(t, monthly_inflation)
* hours_per_month
)
# --------------------------------------------------
# Real-wage change
# --------------------------------------------------
def real_wage_change(t, monthly_inflation):
"""
Calculate the proportional change in the purchasing
power of a constant nominal wage after t months.
"""
return (
1 / (1 + monthly_inflation) ** t
- 1
)
# --------------------------------------------------
# Savings exhaustion
# --------------------------------------------------
def first_negative_month(monthly_inflation, maximum_months=600):
"""
Return the first month in which the household's
financial position becomes negative.
Return None if the position remains non-negative
during the selected horizon.
"""
for month in range(1, maximum_months + 1):
if savings(month, monthly_inflation) < 0:
return month
return None
# --------------------------------------------------
# Time horizon
# --------------------------------------------------
time = np.arange(
0,
simulation_months + 1
)
# --------------------------------------------------
# Plot 1: Household financial position
# --------------------------------------------------
for label, monthly_rate in monthly_inflation_rates.items():
financial_positions = [
savings(month, monthly_rate)
for month in time
]
plt.plot(
time,
financial_positions,
label=label
)
plt.axhline(
y=0,
linewidth=0.8,
linestyle="--"
)
plt.xlabel("Month")
plt.ylabel("Financial position in monthly-wage units")
plt.title("Household Financial Position over Time")
plt.legend()
plt.tight_layout()
plt.show()
# --------------------------------------------------
# Plot 2: Cumulative inflation cost
# --------------------------------------------------
for label, monthly_rate in monthly_inflation_rates.items():
cumulative_costs = [
inflation_cost(month, monthly_rate)
for month in time
]
plt.plot(
time,
cumulative_costs,
label=label
)
plt.xlabel("Month")
plt.ylabel("Cumulative cost in monthly-wage units")
plt.title("Cumulative Cost of Inflation")
plt.legend()
plt.tight_layout()
plt.show()
# --------------------------------------------------
# Plot 3: Inflation cost in working hours
# --------------------------------------------------
for label, monthly_rate in monthly_inflation_rates.items():
cumulative_hours = [
inflation_cost_hours(month, monthly_rate)
for month in time
]
plt.plot(
time,
cumulative_hours,
label=label
)
plt.xlabel("Month")
plt.ylabel("Equivalent working hours")
plt.title("Cumulative Cost of Inflation in Working Hours")
plt.legend()
plt.tight_layout()
plt.show()
# --------------------------------------------------
# Numerical results
# --------------------------------------------------
print("Monthly rates")
print("-" * 50)
print(
f"Monthly interest rate: "
f"{monthly_interest_rate:.8f}"
)
for label, monthly_rate in monthly_inflation_rates.items():
print(
f"{label}: {monthly_rate:.8f}"
)
print()
for label, monthly_rate in monthly_inflation_rates.items():
print(label)
print("-" * len(label))
for month in (12, 36):
financial_position = savings(
month,
monthly_rate
)
cumulative_cost = inflation_cost(
month,
monthly_rate
)
equivalent_hours = inflation_cost_hours(
month,
monthly_rate
)
wage_change = real_wage_change(
month,
monthly_rate
)
print(
f"Month {month}:"
)
print(
f" Financial position: "
f"{financial_position:.3f} "
f"monthly-wage units"
)
print(
f" Cumulative inflation cost: "
f"{cumulative_cost:.3f} "
f"monthly-wage units"
)
print(
f" Equivalent working hours: "
f"{equivalent_hours:.2f}"
)
print(
f" Real-wage change: "
f"{wage_change:.2%}"
)
exhaustion_month = first_negative_month(
monthly_rate
)
if exhaustion_month is None:
print(
" The financial position does not become "
"negative within the selected horizon."
)
else:
print(
f" First negative financial position: "
f"month {exhaustion_month}"
)
print()
After 12 months, the cumulative cost of inflation expressed in equivalent working hours is approximately:
After 36 months, the corresponding cumulative costs are approximately:
The household's financial position first becomes negative at approximately:
A negative financial position represents the financing shortfall that would arise if the household continued maintaining the same level of real consumption after exhausting its savings.