Risk and Consumption

How does risk affect consumption decisions? Part I derives the exact marginal propensity to consume for a CRRA consumer whose only asset has a risky return. The lognormal distributional assumption yields a closed-form MPC, and a Taylor approximation decomposes it into income, substitution, and precautionary saving components. Part II switches to CARA utility with normally distributed income shocks, producing a tractable model where precautionary saving is additive and independent of wealth. Part III introduces time-varying interest rates into a CRRA framework and derives the Campbell-Mankiw decomposition (Campbell & Mankiw (1989)) relating the consumption-wealth ratio to expected future returns.

import numpy as np
import matplotlib.pyplot as plt
import sympy as sp
import pandas as pd
from collections import namedtuple

# Parameters for the CRRA risky-return model
CRRARiskModel = namedtuple(
    'CRRARiskModel',
    ['beta', 'rho', 'r_tilde', 'sigma_r']
)

# Parameters for the CARA income-risk model
CARAModel = namedtuple(
    'CARAModel',
    ['R', 'beta', 'alpha', 'sigma_psi', 'Gamma', 'T']
)

# Parameters for the Campbell-Mankiw model
CampbellMankiwModel = namedtuple(
    'CampbellMankiwModel',
    ['beta', 'rho', 'xi', 'r_bar']
)

Part I: CRRA with Risky Returns¶

A consumer with CRRA utility $u(c) = c^{1-\rho}/(1-\rho)$ holds a single risky asset. The return factor $\tilde{R}_{t+1}$ is lognormally distributed:

\log \tilde{R}_{t+1} \sim \mathcal{N}\!\left(\tilde{r} - \sigma_r^2/2,\; \sigma_r^2\right),

(1)

which ensures that $\mathbb{E}[\tilde{R}_{t+1}] = e^{\tilde{r}}$ . The dynamic budget constraint is $m_{t+1} = (m_t - c_t)\tilde{R}_{t+1}$ , where $m_t$ denotes market resources and no labor income enters.

Guess-and-Verify for the MPC¶

The Euler equation is

1 = \beta\,\mathbb{E}_t\!\left[\tilde{R}_{t+1}\left(\frac{c_{t+1}}{c_t}\right)^{-\rho}\right].

(2)

Postulate a consumption rule $c_t = \kappa\,m_t$ , where $\kappa$ is a constant MPC. Substituting $c_{t+1} = \kappa\,m_{t+1} = \kappa(1-\kappa)m_t\,\tilde{R}_{t+1}$ into (2), the $m_t$ terms cancel (they appear in both numerator and denominator raised to the same power). After simplification:

\kappa = 1 - \left(\beta\,\mathbb{E}_t[\tilde{R}_{t+1}^{1-\rho}]\right)^{1/\rho}.

(3)

This is an exact, closed-form result. The guess-and-verify approach works because market resources are the only source of wealth. If labor income appeared in the budget constraint, $m_t$ would not cancel and a linear consumption rule would fail.

# Symbolic derivation of the exact MPC
from sympy.printing.mathml import mathml

def show(expr):
    ml = mathml(expr, printer='presentation')
    print(f'<math display="block" xmlns="http://www.w3.org/1998/Math/MathML">{ml}</math>')

beta_s, rho_s, sig_s, r_s = sp.symbols('beta rho sigma_r r', positive=True)
kappa_s = sp.Symbol('kappa')

# E[R^(1-rho)] for lognormal R
E_R_1_rho = sp.exp((1 - rho_s) * r_s - rho_s * (1 - rho_s) * sig_s**2 / 2)

# Exact MPC
mpc_exact = 1 - (beta_s * E_R_1_rho)**(1/rho_s)
mpc_exact_simplified = sp.simplify(mpc_exact)

print("**Lognormal expectation** $\\mathbb{E}[\\tilde{R}^{1-\\rho}]$:")
print()
show(E_R_1_rho)
print()
print("**Exact MPC:**")
print()
show(sp.Eq(kappa_s, mpc_exact_simplified))

Evaluating the Lognormal Expectation¶

Since $\tilde{R}^{1-\rho}$ is also lognormally distributed with $\log \tilde{R}^{1-\rho} = (1-\rho)\log \tilde{R}$ , the standard lognormal moment formula yields

\mathbb{E}_t[\tilde{R}_{t+1}^{1-\rho}] = \exp\!\left[(1-\rho)\tilde{r} - \rho(1-\rho)\sigma_r^2/2\right].

(4)

The simplification involves collecting terms: $(1-\rho)^2\sigma_r^2/2$ from the lognormal formula combines with $-(1-\rho)\sigma_r^2/2$ from the mean specification to produce $-\rho(1-\rho)\sigma_r^2/2$ .

The Approximate MPC¶

Applying Taylor approximations ( $\beta^{1/\rho} \approx \exp(-\rho^{-1}\vartheta)$ where $\beta = 1/(1+\vartheta)$ ) to the exact formula gives

\kappa \approx \tilde{r} - \rho^{-1}(\tilde{r} - \vartheta) - (\rho - 1)\sigma_r^2/2.

(5)

Three terms correspond to three economic forces:

Income effect ( $\tilde{r}$ ): higher returns generate more income, raising consumption
Substitution effect ( $-\rho^{-1}(\tilde{r} - \vartheta)$ ): higher returns make future consumption cheaper, encouraging saving
Precautionary saving ( $-(\rho-1)\sigma_r^2/2$ ): greater risk lowers the MPC (more saving) for $\rho > 1$

When $\sigma_r^2 = 0$ , the formula reduces to the perfect foresight result $\kappa = \tilde{r} - \rho^{-1}(\tilde{r} - \vartheta)$ .

# Compare exact and approximate MPC formulas
def exact_mpc(rho, r_tilde, sigma_r, beta):
    """Exact MPC for CRRA consumer with lognormal risky return."""
    E_R = np.exp((1 - rho) * r_tilde - rho * (1 - rho) * sigma_r**2 / 2)
    return 1 - (beta * E_R)**(1 / rho)

def approx_mpc(rho, r_tilde, sigma_r, vartheta):
    """Approximate MPC from Taylor expansion."""
    return r_tilde - (r_tilde - vartheta) / rho - (rho - 1) * sigma_r**2 / 2

params = CRRARiskModel(beta=1/1.04, rho=3.0, r_tilde=0.04, sigma_r=0.15)
vartheta = 1/params.beta - 1

rho_grid = np.linspace(1.01, 8.0, 200)
fig, axes = plt.subplots(1, 2, figsize=(10, 4))

# Panel 1: MPC vs risk aversion
mpc_ex = [exact_mpc(rho, params.r_tilde, params.sigma_r, params.beta) for rho in rho_grid]
mpc_ap = [approx_mpc(rho, params.r_tilde, params.sigma_r, vartheta) for rho in rho_grid]
axes[0].plot(rho_grid, mpc_ex, lw=2, label='Exact')
axes[0].plot(rho_grid, mpc_ap, '--', lw=2, label='Approximation')
axes[0].set_xlabel(r'Relative risk aversion $\rho$')
axes[0].set_ylabel(r'MPC $\kappa$')
axes[0].set_title(r'MPC falls as risk aversion rises ($\sigma_r = 0.15$)')
axes[0].legend(frameon=False, fontsize=8)
axes[0].grid(True, alpha=0.3)

# Panel 2: MPC vs return volatility
sig_grid = np.linspace(0, 0.30, 200)
mpc_ex_sig = [exact_mpc(params.rho, params.r_tilde, s, params.beta) for s in sig_grid]
mpc_ap_sig = [approx_mpc(params.rho, params.r_tilde, s, vartheta) for s in sig_grid]
axes[1].plot(sig_grid, mpc_ex_sig, lw=2, label='Exact')
axes[1].plot(sig_grid, mpc_ap_sig, '--', lw=2, label='Approximation')
axes[1].set_xlabel(r'Return volatility $\sigma_r$')
axes[1].set_ylabel(r'MPC $\kappa$')
axes[1].set_title(r'MPC falls as risk rises ($\rho = 3$)')
axes[1].legend(frameon=False, fontsize=8)
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

The left panel shows that more risk-averse consumers save more (lower MPC). The right panel shows the precautionary saving motive: as return volatility increases, the MPC falls. The approximation tracks the exact formula closely for moderate parameter values but diverges at extreme risk aversion or volatility.

# Tabulate the decomposition of the approximate MPC
rows = []
for rho in [1.0, 2.0, 3.0, 5.0]:
    income = params.r_tilde
    substitution = -(params.r_tilde - vartheta) / rho
    precautionary = -(rho - 1) * params.sigma_r**2 / 2
    total = income + substitution + precautionary
    exact = exact_mpc(rho, params.r_tilde, params.sigma_r, params.beta)
    rows.append({
        'rho': rho,
        'Income': round(income, 4),
        'Substitution': round(substitution, 4),
        'Precautionary': round(precautionary, 4),
        'Approx MPC': round(total, 4),
        'Exact MPC': round(exact, 4),
    })
df_decomp = pd.DataFrame(rows)
df_decomp

The table confirms that the precautionary term vanishes at $\rho = 1$ (log utility) and grows in magnitude with risk aversion. For $\rho = 5$ , the precautionary component is -0.045, a substantial reduction in the MPC driven entirely by risk.

Part II: CARA with Income Risk¶

The Problem¶

A consumer with CARA utility $u(C) = -(1/\alpha)\,e^{-\alpha C}$ (where $\alpha > 0$ is the coefficient of absolute risk aversion) maximizes

\max \;\mathbb{E}_t\!\left[\sum_{s=t}^{T} \beta^{s-t}\,u(C_s)\right]

(6)

subject to $M_{t+1} = (M_t - C_t)R + Y_{t+1}$ , where the interest rate $r = R - 1$ is constant. Income has two components: a deterministic trend $\bar{P}_{t+1} = \Gamma\bar{P}_t$ and a random-walk deviation $P_{t+1} = P_t + \Psi_{t+1}$ , with $\Psi_{t+1} \sim \mathcal{N}(0, \sigma_\Psi^2)$ .

Perfect Foresight: Additive Growth¶

Under CARA, the Euler equation implies additive consumption changes:

C_{t+1} = C_t + \alpha^{-1}\log(R\beta).

(7)

This contrasts sharply with CRRA, where consumption changes are multiplicative (affecting the growth rate). Under CARA, the intertemporal elasticity of substitution $\alpha^{-1}$ determines the absolute dollar increment in consumption per period.

# Symbolic derivation of the CARA Euler equation
alpha_s, R_sym, beta_sym = sp.symbols('alpha R beta', positive=True)
C_t, C_t1 = sp.symbols('C_t C_{t+1}')

# u'(C) = exp(-alpha*C)
# Euler: u'(C_t) = R*beta*u'(C_{t+1})
# exp(-alpha*C_t) = R*beta*exp(-alpha*C_{t+1})
# -alpha*C_t = log(R*beta) - alpha*C_{t+1}
# C_{t+1} - C_t = (1/alpha)*log(R*beta)

growth = sp.Eq(C_t1 - C_t, sp.log(R_sym * beta_sym) / alpha_s)
print("**Perfect foresight Euler equation under CARA utility:**")
print()
show(growth)
print()
print("Consumption changes by a *constant dollar amount* each period, not a constant *rate*.")

Solution Under Uncertainty¶

With normally distributed permanent income shocks, the consumption process

C_{t+1} = C_t + \alpha^{-1}\log(R\beta) + \alpha\sigma_\Psi^2/2 + \Psi_{t+1}

(8)

satisfies the Euler equation under uncertainty. To verify, substitute (8) into $1 = R\beta\,\mathbb{E}_t[\exp(-\alpha(C_{t+1} - C_t))]$ and use the fact that for $Z \sim \mathcal{N}(\mu, \sigma^2)$ , $\mathbb{E}[e^Z] = e^{\mu + \sigma^2/2}$ :

\begin{aligned} 1 &= R\beta\,\exp(-\alpha^2\sigma_\Psi^2/2)\,\exp(\alpha^2\sigma_\Psi^2/2)\,\exp(-\log R\beta) \\ &= R\beta \cdot (R\beta)^{-1} = 1. \quad \checkmark \end{aligned}

(9)

# Symbolic verification of the CARA solution under uncertainty
sigma_s = sp.Symbol('sigma_Psi', positive=True)
Psi_s = sp.Symbol('Psi_{t+1}')

delta_C = sp.log(R_sym * beta_sym) / alpha_s + alpha_s * sigma_s**2 / 2 + Psi_s
exp_neg_alpha_dC = sp.exp(-alpha_s * delta_C)

# Taking expectations: E[exp(-alpha*Psi)] = exp(alpha^2*sigma^2/2)
# so exp(-alpha*delta_C) without the shock part:
deterministic = sp.exp(-alpha_s * (sp.log(R_sym * beta_sym) / alpha_s + alpha_s * sigma_s**2 / 2))
shock_expect = sp.exp(alpha_s**2 * sigma_s**2 / 2)  # E[exp(-alpha*Psi)]

product = sp.simplify(R_sym * beta_sym * deterministic * shock_expect)

print("**Verification:** $R\\beta \\cdot \\exp(-\\alpha \\cdot \\text{det. part}) \\cdot \\mathbb{E}[\\exp(-\\alpha\\Psi)] =$")
print()
show(product)

The Precautionary Premium¶

Define $\hat{\kappa} = \alpha^{-1}\log(R\beta) + \alpha\sigma_\Psi^2/2$ . The consumption process becomes $C_{t+1} = C_t + \hat{\kappa} + \Psi_{t+1}$ . The term $\alpha\sigma_\Psi^2/2$ is the precautionary premium (Caballero (1990)): it makes expected consumption growth faster than under certainty, reflecting additional saving today to buffer against future income risk. Two properties distinguish it from the CRRA case:

The premium is additive (a constant dollar amount per period), not multiplicative
It does not depend on wealth or income level, because CARA risk attitudes are constant in absolute terms

# Visualize the precautionary premium
params_cara = CARAModel(R=1.04, beta=0.96, alpha=2.0, sigma_psi=0.05, Gamma=1.02, T=60)

fig, axes = plt.subplots(1, 2, figsize=(10, 4))

# Panel 1: Consumption paths under different sigma
np.random.seed(42)
sigma_vals = [0.0, 0.03, 0.06, 0.10]
for sig in sigma_vals:
    khat = np.log(params_cara.R * params_cara.beta) / params_cara.alpha + params_cara.alpha * sig**2 / 2
    C = np.zeros(params_cara.T)
    C[0] = 10.0
    for t in range(1, params_cara.T):
        noise = np.random.normal(0, sig) if sig > 0 else 0.0
        C[t] = C[t-1] + khat + noise
    axes[0].plot(C, lw=1.5, label=rf'$\sigma_\Psi = {sig}$')

axes[0].set_xlabel('Period')
axes[0].set_ylabel('Consumption $C_t$')
axes[0].set_title('Higher uncertainty raises the consumption path (precautionary saving)')
axes[0].legend(frameon=False, fontsize=8)
axes[0].grid(True, alpha=0.3)

# Panel 2: Precautionary premium as function of sigma and alpha
sig_range = np.linspace(0, 0.12, 100)
for a in [1.0, 2.0, 4.0]:
    premium = a * sig_range**2 / 2
    axes[1].plot(sig_range, premium, lw=2, label=rf'$\alpha = {a}$')

axes[1].set_xlabel(r'Income volatility $\sigma_\Psi$')
axes[1].set_ylabel(r'Precautionary premium $\alpha\sigma_\Psi^2/2$')
axes[1].set_title('Premium is quadratic in volatility, linear in risk aversion')
axes[1].legend(frameon=False, fontsize=8)
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# Table: precautionary premium for different combinations
rows = []
for a in [0.5, 1.0, 2.0, 4.0]:
    row = {'ARA (alpha)': a}
    for sig in [0.02, 0.05, 0.10]:
        row[f'sigma={sig}'] = round(a * sig**2 / 2, 6)
    rows.append(row)

df_premium = pd.DataFrame(rows)
df_premium.columns = [r'ARA ($\alpha$)'] + [rf'$\sigma_\Psi = {s}$' for s in [0.02, 0.05, 0.10]]
df_premium

The Consumption Function¶

Imposing the intertemporal budget constraint, the infinite-horizon CARA consumption function is

C_t = P_t + \frac{r}{R}\!\left[B_t + \frac{\bar{P}_t}{1 - \Gamma/R}\right] - r\!\left(\frac{\alpha^{-1}\log(R\beta) + \alpha\sigma_\Psi^2/2}{(1 - R)^2}\right),

(10)

where $P_t$ is the idiosyncratic permanent income level, $B_t$ is beginning-of-period bank balances, and $\bar{P}_t$ is the deterministic income trend.

Two notable features:

The MPC out of capital is $r/R$ , the interest income on an extra dollar. It does not depend on impatience or risk aversion.
The precautionary saving amount is independent of wealth. The dollar amount saved for precautionary reasons is the same whether the consumer is rich or poor.

# Visualize the CARA consumption function: C vs B for different sigma
r = params_cara.R - 1
P_bar, P_level = 1.0, 0.0
B_grid = np.linspace(0, 20, 200)

fig, axes = plt.subplots(1, 2, figsize=(10, 4))

for sig in [0.0, 0.05, 0.10]:
    khat = np.log(params_cara.R * params_cara.beta) / params_cara.alpha + params_cara.alpha * sig**2 / 2
    human_wealth = P_bar / (1 - params_cara.Gamma / params_cara.R)
    precaut_term = r * khat / (1 - params_cara.R)**2
    C = P_level + (r / params_cara.R) * (B_grid + human_wealth) - precaut_term
    axes[0].plot(B_grid, C, lw=2, label=rf'$\sigma_\Psi = {sig}$')

axes[0].plot(B_grid, r / params_cara.R * B_grid + P_bar, 'k--', lw=1, alpha=0.5, label='Interest income')
axes[0].set_xlabel(r'Bank balances $B_t$')
axes[0].set_ylabel(r'Consumption $C_t$')
axes[0].set_title('CARA consumption function: parallel shifts with risk')
axes[0].legend(frameon=False, fontsize=8)
axes[0].grid(True, alpha=0.3)

# Panel 2: MPC out of bank balances is constant r/R
axes[1].plot(B_grid, np.full_like(B_grid, r / params_cara.R), color='C0', lw=2)
axes[1].set_xlabel(r'Bank balances $B_t$')
axes[1].set_ylabel(r'MPC $\partial C / \partial B$')
axes[1].set_xlim(0, 20)
axes[1].set_ylim(0, 0.08)
axes[1].set_title(f'MPC out of capital is constant: $r/R = {r/params_cara.R:.4f}$')
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

The consumption function shifts down in parallel as income risk increases. The slope $\partial C / \partial B = r/R$ is identical across all risk levels, confirming that the MPC out of capital does not depend on uncertainty.

Patience and Impatience¶

The infinite-horizon consumption function can be written as $C_t = \frac{R - (R\beta)^{1/\alpha}}{R}\,W_t$ , where $W_t = B_t + H_t$ is total wealth. The consumer is impatient (spending more than income) when $(R\beta)^{1/\alpha} < 1$ , and patient when $(R\beta)^{1/\alpha} > 1$ . When $R\beta = 1$ , the consumer spends exactly the interest income on total wealth: $C_t = (r/R)\,W_t$ .

# MPC out of total wealth as function of R*beta for different alpha
Rbeta_grid = np.linspace(0.90, 1.10, 200)
fig, ax = plt.subplots(figsize=(8, 4))

for a in [1.0, 2.0, 5.0]:
    coeff = (params_cara.R - Rbeta_grid**(1/a)) / params_cara.R
    ax.plot(Rbeta_grid, coeff, lw=2, label=rf'$\alpha = {a}$')

ax.axhline(r / params_cara.R, color='gray', ls='--', lw=0.8, label=r'$r/R$ (balanced)')
ax.axvline(1.0, color='gray', ls='--', lw=0.8)
ax.set_xlabel(r'$R\beta$')
ax.set_ylabel('MPC out of total wealth')
ax.set_title('Impatient ($R\\beta < 1$): MPC exceeds $r/R$')
ax.legend(frameon=False, fontsize=8)
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

The Three Effects of $r$ ¶

Using the approximation $C_t \approx (r - \alpha^{-1}(r - \vartheta))[B_t + \bar{Y}R/r]$ (where $\vartheta = 1/\beta - 1$ is the time preference rate), three channels connect the interest rate to consumption:

Income effect (first $r$ ): higher $r$ increases the payout rate on total wealth
Substitution effect ( $\alpha^{-1}(r - \vartheta)$ ): higher $r$ makes future consumption cheaper
Human wealth effect ( $\bar{Y}R/r$ in the brackets): higher $r$ reduces the present value of future labor income

# Decompose the three effects of r on consumption
Y_bar = 1.0  # constant income
tau = 1/params_cara.beta - 1  # time preference rate
alpha_val = params_cara.alpha

r_grid = np.linspace(0.01, 0.10, 200)
R_grid = 1 + r_grid

fig, ax = plt.subplots(figsize=(8, 4))

# Total consumption at each r
B_val = 5.0
for a in [1.0, 2.0, 5.0]:
    C_approx = (r_grid - (r_grid - tau) / a) * (B_val + Y_bar * R_grid / r_grid)
    ax.plot(r_grid * 100, C_approx, lw=2, label=rf'$\alpha = {a}$')

ax.set_xlabel('Interest rate $r$ (%)')
ax.set_ylabel('Consumption $C_t$')
ax.set_title(f'Consumption vs. interest rate ($B = {B_val}$, $\\bar{{Y}} = {Y_bar}$)')
ax.legend(frameon=False, fontsize=8)
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

Part III: Consumption with Time-Varying Interest Rates¶

The Campbell-Mankiw Framework¶

An infinite-horizon representative agent holds total wealth $W_t$ (human plus nonhuman). The return factor $R_{t+1}$ is riskless but varies over time:

W_{t+1} = (W_t - C_t)\,R_{t+1}.

(11)

The goal is to express the consumption-wealth ratio as a function of expected future interest rates, separating income and substitution effects (Campbell & Mankiw (1989)).

Log-Linearized Budget Constraint¶

Dividing (11) by $W_t$ , taking logs, and applying a Taylor expansion around the steady-state log consumption-wealth ratio $\bar{x} = \bar{c} - \bar{w}$ :

\Delta w_{t+1} \approx k + r_{t+1} + (1 - 1/\xi)(c_t - w_t),

(12)

where $\xi = 1 - \exp(\bar{x})$ is a constant slightly below one (since the consumption-wealth ratio $C/W$ is small), $k$ is a constant that depends on $\xi$ , and lowercase letters denote logs.

Forward Iteration¶

Setting two expressions for $\Delta w_{t+1}$ equal and iterating forward produces the approximate intertemporal budget constraint:

c_t - w_t = \sum_{j=1}^{\infty} \xi^j(r_{t+j} - \Delta c_{t+j}) + \frac{\xi k}{1 - \xi}.

(13)

This result is pure accounting: it holds for any consumption path consistent with the budget constraint. The log consumption-wealth ratio today must equal the discounted value of future returns minus future consumption growth. Higher expected future returns (holding consumption growth fixed) require higher consumption today.

# Show the geometric sum structure symbolically
xi_s, r_s_sym, dc_s = sp.symbols('xi r_{t+j} Delta_c_{t+j}', positive=True)
j_s = sp.Symbol('j', positive=True, integer=True)

sum_expr = sp.Sum(xi_s**j_s * (r_s_sym - dc_s), (j_s, 1, sp.oo))
print("**Approximate IBC** (geometric discounting at rate $\\xi$):")
print()
print("$c_t - w_t =$")
show(sum_expr)
print()
print(r"$+ \;\xi k/(1-\xi)$")

Substituting the Euler Equation¶

For a CRRA consumer with $u(c) = c^{1-\rho}/(1-\rho)$ , the log Euler equation under perfect foresight is

\Delta c_{t+1} = \mu + \rho^{-1} r_{t+1},

(14)

where $\mu = \rho^{-1}\log\beta$ . Substituting into (13) and collecting terms:

c_t - w_t = (1 - \rho^{-1})\sum_{j=1}^{\infty}\xi^j\,r_{t+j} + \frac{\xi(k - \mu)}{1-\xi}.

(15)

The coefficient $(1 - \rho^{-1})$ on expected future returns governs the net effect of interest rate changes on the consumption-wealth ratio.

# Impulse response of consumption-wealth ratio to interest rate shocks
params_cm = CampbellMankiwModel(beta=0.96, rho=3.0, xi=0.97, r_bar=0.04)

rho_vals = [0.5, 1.0, 2.0, 5.0]
T_irf = 40

fig, axes = plt.subplots(1, 2, figsize=(10, 4))

# Panel 1: Response to a persistent interest rate shock (phi = 0.8)
phi = 0.8
for rho in rho_vals:
    coeff = 1 - 1/rho
    r_shock = np.array([0.01 * phi**j for j in range(T_irf)])
    cw_path = np.zeros(T_irf)
    for t in range(T_irf):
        cw_path[t] = coeff * sum(params_cm.xi**k * r_shock[t + k]
                                  for k in range(1, T_irf - t))
    axes[0].plot(cw_path * 100, lw=2, label=rf'$\rho = {rho}$')

axes[0].set_xlabel('Period')
axes[0].set_ylabel(r'$c_t - w_t$ response (pp)')
axes[0].set_title(r'Response to persistent interest rate shock ($\phi = 0.8$)')
axes[0].legend(frameon=False, fontsize=8)
axes[0].grid(True, alpha=0.3)

# Panel 2: Coefficient (1 - 1/rho) vs rho
rho_range = np.linspace(0.3, 8, 200)
coeff_range = 1 - 1/rho_range
axes[1].plot(rho_range, coeff_range, lw=2, color='C0')
axes[1].axhline(0, color='gray', lw=0.8, ls='--')
axes[1].axvline(1, color='gray', lw=0.8, ls='--')
axes[1].fill_between(rho_range[rho_range < 1], coeff_range[rho_range < 1],
                      alpha=0.15, color='C1', label='Substitution dominates')
axes[1].fill_between(rho_range[rho_range > 1], coeff_range[rho_range > 1],
                      0, alpha=0.15, color='C0', label='Income dominates')
axes[1].set_xlabel(r'Risk aversion $\rho$')
axes[1].set_ylabel(r'Coefficient $1 - \rho^{-1}$')
axes[1].set_title('Income vs. substitution effects of interest rate changes')
axes[1].legend(frameon=False, fontsize=8, loc='lower right')
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

Income vs. Substitution Effects¶

The coefficient $(1 - \rho^{-1})$ determines how the consumption-wealth ratio responds to expected future interest rates:

Risk aversion $\rho$	IES $= \rho^{-1}$	Coefficient	Interpretation
$< 1$	$> 1$	Negative	Substitution dominates: higher $r$ lowers $c/w$
$= 1$	$= 1$	Zero	Income and substitution exactly cancel (log utility)
$> 1$	$< 1$	Positive	Income dominates: higher $r$ raises $c/w$

For the empirically relevant case $\rho > 1$ , higher expected future interest rates raise the consumption-wealth ratio: the consumer feels richer and spends more today.

# Tabulate the effects for several values of rho
rows = []
for rho in [0.5, 1.0, 2.0, 3.0, 5.0]:
    ies = 1/rho
    coeff = 1 - ies
    if coeff < 0:
        interp = 'Substitution dominates'
    elif coeff == 0:
        interp = 'Effects cancel'
    else:
        interp = 'Income dominates'
    rows.append({
        'rho': rho,
        'IES': round(ies, 2),
        '1 - 1/rho': round(coeff, 2),
        'Interpretation': interp,
    })
df_effects = pd.DataFrame(rows)
df_effects

The Human Wealth Channel¶

The Campbell-Mankiw result holds the level of total wealth $w_t$ fixed. But human wealth responds to interest rates:

H_t \approx \frac{Y_t}{r - g},

(16)

where $g$ is the income growth rate. A permanent decline in $r$ can increase $H_t$ enormously when $r - g$ is small. Summers (1981) showed that this human wealth channel often dominates the direct income and substitution effects in general equilibrium.

# Human wealth sensitivity to interest rate changes
Y, g = 1.0, 0.02
r_grid = np.linspace(0.025, 0.10, 200)
H = Y / (r_grid - g)

fig, ax = plt.subplots(figsize=(8, 4))
ax.plot(r_grid * 100, H, lw=2, color='C0')
ax.set_xlabel('Interest rate $r$ (%)')
ax.set_ylabel(r'Human wealth $H = Y/(r-g)$')
ax.set_title(f'Human wealth is highly sensitive to $r$ when $r - g$ is small ($g = {g}$)')
ax.axvline(g * 100, color='gray', ls='--', lw=0.8, label=f'$g = {g*100:.0f}\\%$')
ax.legend(frameon=False, fontsize=9)
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

As $r$ approaches $g$ , human wealth diverges. Even moderate interest rate changes produce large swings in $H_t$ . This is why a full analysis of interest rate effects on consumption must account for the human wealth channel, not just the direct substitution and income effects on the right-hand side of the Campbell-Mankiw formula.

Exercises¶

Solution to Exercise 1

beta_val = 1/1.04
rho_val = 3.0
r_tilde_val = 0.04
vartheta_val = 1/beta_val - 1

rows = []
for sig in [0.0, 0.05, 0.10, 0.15, 0.20, 0.25]:
    ex = exact_mpc(rho_val, r_tilde_val, sig, beta_val)
    ap = approx_mpc(rho_val, r_tilde_val, sig, vartheta_val)
    err = abs(ex - ap)
    rows.append({
        'sigma_r': sig,
        'Exact MPC': round(ex, 6),
        'Approx MPC': round(ap, 6),
        'Error (pp)': round(err * 100, 2),
    })

df_ex1 = pd.DataFrame(rows)
df_ex1

The approximation error exceeds 0.5 percentage points around $\sigma_r \approx 0.20$ . For moderate volatility ( $\sigma_r \leq 0.15$ ), the linear approximation is quite accurate.

Solution to Exercise 2

np.random.seed(42)
R_val, beta_val, alpha_val, sigma_val = 1.04, 0.96, 2.0, 0.05
C0, n_paths, T = 10.0, 500, 100

khat_theory = np.log(R_val * beta_val) / alpha_val + alpha_val * sigma_val**2 / 2

all_growth = []
for _ in range(n_paths):
    C = np.zeros(T)
    C[0] = C0
    for t in range(1, T):
        C[t] = C[t-1] + khat_theory + np.random.normal(0, sigma_val)
    dC = np.diff(C)
    all_growth.extend(dC.tolist())

mean_growth = np.mean(all_growth)
print(f"Theoretical expected growth:  {khat_theory:.6f}")
print(f"Simulated mean growth:       {mean_growth:.6f}")
print(f"Difference:                  {abs(khat_theory - mean_growth):.6f}")

The simulated mean consumption growth matches the theoretical prediction $\hat{\kappa}$ to within sampling noise, confirming (8).

Solution to Exercise 3

xi_val, rho_val = 0.97, 3.0
coeff = 1 - 1/rho_val
T_irf = 40
r_bar = 0.04
shock = 0.01

fig, ax = plt.subplots(figsize=(8, 4))
for phi in [0.0, 0.5, 0.9]:
    # Interest rate path after shock
    r_path = np.zeros(T_irf + 50)
    r_path[1] = shock
    for t in range(2, len(r_path)):
        r_path[t] = phi * r_path[t-1]

    # c_t - w_t at each t
    cw_response = np.zeros(T_irf)
    for t in range(T_irf):
        cw_response[t] = coeff * sum(
            xi_val**j * r_path[t + j] for j in range(1, 50)
        )

    ax.plot(cw_response * 100, lw=2, label=rf'$\phi = {phi}$')

ax.set_xlabel('Period')
ax.set_ylabel(r'$\Delta(c_t - w_t)$ (pp)')
ax.set_title(r'Higher persistence amplifies the consumption-wealth response')
ax.legend(frameon=False, fontsize=9)
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

With $\phi = 0$ (no persistence), the response is a single-period blip. With $\phi = 0.9$ , the shock persists, and the consumption-wealth ratio responds about $1/(1-\phi\xi) \approx 7.7$ times more on impact. Persistent interest rate shocks are far more consequential for consumption.

References¶

Campbell, J. Y., & Mankiw, N. G. (1989). Consumption, Income, and Interest Rates: Reinterpreting the Time Series Evidence. NBER Macroeconomics Annual, 4, 185–216.
Caballero, R. J. (1990). Consumption Puzzles and Precautionary Savings. Journal of Monetary Economics, 25(1), 113–136.
Summers, L. H. (1981). Capital Taxation and Accumulation in a Life Cycle Growth Model. American Economic Review, 71(4), 533–544.