Lecture 08, Notebook 11: OLG Exercise — Closed-Form Savings Rates and Lifecycle Profiles

RUN_MODE = "smoke"  # one of: "smoke", "teaching", "production"
SEED = 0

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(SEED)  # exercise is deterministic but seed for completeness
plt.rcParams['font.size'] = 13
if RUN_MODE == "smoke":
    # Silence figures in CI runs but keep them rendered in the notebook itself.
    plt.rcParams['figure.max_open_warning'] = 0

A = 6
beta = 0.7

# TODO: Compute savings rates for each cohort h = 1, ..., A-1
# savings_rates = []
# for h in range(1, A):
#     beta_h = ...
#     savings_rates.append(beta_h)

# Don't forget agent A has savings rate 0
# savings_rates.append(0.0)

# TODO: Plot savings rate vs age (bar chart)
# plt.bar(range(1, A+1), savings_rates)
# plt.xlabel('Age h')
# plt.ylabel('Savings rate')
# plt.title('Analytical Savings Rates by Age')

# Which agent saves the most? The least? Why?

A = 6
beta = 0.7

# Compute savings rates for each cohort h = 1, ..., A-1
savings_rates = []
for h in range(1, A):
    beta_h = beta * (1 - beta**(A - h)) / (1 - beta**(A - h + 1))
    savings_rates.append(beta_h)

# Agent A consumes everything (dies next period)
savings_rates.append(0.0)

# Print savings rates
for h, sr in enumerate(savings_rates, 1):
    print(f"  Agent h={h}: savings rate = {sr:.4f}")

# Bar chart
fig, ax = plt.subplots(figsize=(7, 4))
ages = np.arange(1, A + 1)
bars = ax.bar(ages, savings_rates, color='steelblue', edgecolor='black')
ax.set_xlabel('Age $h$')
ax.set_ylabel('Savings rate $\\beta_h$')
ax.set_title('Analytical Savings Rates by Age ($A=6$, $\\beta=0.7$)')
ax.set_xticks(ages)
ax.set_ylim(0, 0.7)
plt.tight_layout()
plt.show()

print("\nYoung agents save the most because they have the longest remaining")
print("horizon. The oldest agent saves nothing — they die next period.")

# Assume steady-state prices
r = 0.15
w = 1.0

# TODO: Simulate one lifecycle
# Agent 1 earns wage w, saves k2 = beta_1 * w
# Agent 2 earns r*k2, saves k3 = beta_2 * (r*k2)
# ... and so on until agent A consumes everything

# income = [w]
# capital = []
# consumption = []
# for h in range(A):
#     c_h = (1 - savings_rates[h]) * income[h]
#     consumption.append(c_h)
#     if h < A - 1:
#         k_next = savings_rates[h] * income[h]
#         capital.append(k_next)
#         income.append(r * k_next)

# TODO: Plot consumption vs age and capital vs age (2 subplots)

r = 0.15
w = 1.0

# Simulate the lifecycle
income = [w]
capital = []
consumption = []

for h in range(A):
    c_h = (1 - savings_rates[h]) * income[h]
    consumption.append(c_h)
    if h < A - 1:
        k_next = savings_rates[h] * income[h]
        capital.append(k_next)
        income.append(r * k_next)

# Print lifecycle
print("Lifecycle simulation (r={}, w={}):".format(r, w))
for h in range(A):
    cap_str = f", k_{h+2} = {capital[h]:.4f}" if h < A - 1 else " (dies)"
    print(f"  Age {h+1}: income = {income[h]:.4f}, c = {consumption[h]:.4f}{cap_str}")

# 2-panel plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
ages = np.arange(1, A + 1)

# Left panel: consumption lifecycle
ax1.plot(ages, consumption, 'o-', color='steelblue', linewidth=2, markersize=8)
ax1.set_xlabel('Age $h$')
ax1.set_ylabel('Consumption $c_h$')
ax1.set_title('Lifecycle Consumption')
ax1.set_xticks(ages)

# Right panel: capital holdings
# Capital k_h is held at the start of age h (for h=2,...,A)
ax2.plot(ages[1:], capital, 's-', color='firebrick', linewidth=2, markersize=8)
ax2.set_xlabel('Age $h$')
ax2.set_ylabel('Capital $k_h$')
ax2.set_title('Capital Holdings by Age')
ax2.set_xticks(ages)

plt.tight_layout()
plt.show()

print("\nConsumption falls over the lifecycle because capital income (r*k)")
print("shrinks rapidly — each generation saves a smaller fraction of a")
print("smaller income. With r < 1, capital decumulates quickly.")

# TODO: Loop over beta_values = [0.5, 0.7, 0.9]
# For each, compute the full lifecycle (savings rates, capital, consumption)
# Plot all 3 consumption profiles on the same axes
# What happens as beta increases?

# beta_values = [0.5, 0.7, 0.9]
# A = 6
# r = 0.15
# w = 1.0
#
# for beta_val in beta_values:
#     # Step 1: compute savings rates
#     sr = []
#     for h in range(1, A):
#         ...
#     sr.append(0.0)
#
#     # Step 2: simulate lifecycle
#     inc = [w]
#     cons = []
#     for h in range(A):
#         ...
#
#     # Step 3: plot
#     plt.plot(range(1, A+1), cons, 'o-', label=f'beta={beta_val}')

beta_values = [0.5, 0.7, 0.9]
A = 6
r = 0.15
w = 1.0
colors = ['#e74c3c', 'steelblue', '#2ecc71']

fig, ax = plt.subplots(figsize=(8, 5))

for beta_val, color in zip(beta_values, colors):
    # Step 1: compute savings rates
    sr = []
    for h in range(1, A):
        beta_h = beta_val * (1 - beta_val**(A - h)) / (1 - beta_val**(A - h + 1))
        sr.append(beta_h)
    sr.append(0.0)  # oldest agent

    # Step 2: simulate lifecycle
    inc = [w]
    cons = []
    for h in range(A):
        c_h = (1 - sr[h]) * inc[h]
        cons.append(c_h)
        if h < A - 1:
            k_next = sr[h] * inc[h]
            inc.append(r * k_next)

    # Step 3: plot
    ax.plot(range(1, A + 1), cons, 'o-', color=color, linewidth=2,
            markersize=8, label=f'$\\beta = {beta_val}$')

ax.set_xlabel('Age $h$')
ax.set_ylabel('Consumption $c_h$')
ax.set_title('Effect of Patience on Lifecycle Consumption')
ax.set_xticks(range(1, A + 1))
ax.legend()
plt.tight_layout()
plt.show()

print("Key insight: More patient agents (higher beta) consume less when")
print("young and save more, building up capital. However, since r < 1,")
print("the capital income still declines — but more patient agents have")
print("a flatter consumption profile (closer to consumption smoothing).")