Risk and Consumption

CRRA with Risky Returns, CARA with Income Risk, and Time-Varying Interest Rates

Alan Lujan

Johns Hopkins University

February 24, 2025

CRRA with Risky Returns

Setup

A consumer with CRRA utility \(u(c) = c^{1-\rho}/(1-\rho)\) holds a single risky asset with lognormal return factor \(\tilde{R}\):

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

The dynamic budget constraint uses market resources \(m_t\):

\[m_{t+1} = (m_t - c_t)\,\tilde{R}_{t+1}\]

No labor income enters. The consumer’s only wealth is the risky portfolio.

The Euler Equation

Start with the standard CRRA Euler equation:

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

Guess a linear consumption rule \(c_t = \kappa\, m_t\). Because market resources appear in both the numerator and denominator, they cancel.

If labor income entered the numerator, this guess-and-verify approach would fail.

Verifying the Guess

Substituting \(c_t = \kappa\,m_t\) and \(m_{t+1} = (1-\kappa)\,m_t\,\tilde{R}_{t+1}\):

\[\begin{aligned} 1 &= \beta\,\mathbb{E}_t\!\left[\tilde{R}_{t+1}\bigl((1-\kappa)\tilde{R}_{t+1}\bigr)^{-\rho}\right] \\ &= \beta\,(1-\kappa)^{-\rho}\,\mathbb{E}_t\!\left[\tilde{R}_{t+1}^{1-\rho}\right] \end{aligned}\]

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

The Exact MPC

The marginal propensity to consume is

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

This is an exact, closed-form result. No approximation is needed because the linear consumption rule and lognormal returns allow the expectation to be evaluated analytically.

Evaluating the Expectation

Since \(\log \tilde{R}_{t+1}^{1-\rho} = (1-\rho)\log \tilde{R}_{t+1}\) is normally distributed, the lognormal moment formula gives

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

The key step is collecting terms: the \((1-\rho)^2\sigma_r^2/2\) from the lognormal formula simplifies with the \(-(1-\rho)\sigma_r^2/2\) from the mean specification.

The Approximate MPC

Substituting into the exact formula and applying Taylor approximations (\(\beta^{1/\rho} \approx \exp(-\rho^{-1}\vartheta)\) where \(\vartheta\) is the discount rate):

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

Three terms, three forces:

Term	Force
\(\tilde{r}\)	Income effect of the return
\(-\rho^{-1}(\tilde{r} - \vartheta)\)	Intertemporal substitution
\(-(\rho-1)\sigma_r^2/2\)	Precautionary saving

Precautionary Saving

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

As risk rises (\(\sigma_r^2\) increases), the MPC falls because the \(-(\rho-1)\sigma_r^2/2\) term is negative for \(\rho > 1\). Falling MPC means the consumer saves more: the precautionary saving motive.

With log utility (\(\rho = 1\)), the risk term vanishes. The covariance between \(\tilde{R}\) and \(u'(c_{t+1})\) exactly offsets the Jensen’s inequality effect, a special case that makes log utility implausibly insensitive to risk.

Code: MPC vs Risk Aversion and Risk

CARA with Income Risk

Setup

A consumer with constant absolute risk aversion (CARA) utility:

\[u(C) = -\frac{1}{\alpha}\,e^{-\alpha C}, \qquad u'(C) = e^{-\alpha C}\]

faces a constant interest rate \(r = R - 1\), budget constraint \(M_{t+1} = (M_t - C_t)R + Y_{t+1}\), and random-walk permanent income:

\[Y_{t+1} = \bar{P}_{t+1} + P_{t+1}, \quad \bar{P}_{t+1} = \Gamma\bar{P}_t, \quad P_{t+1} = P_t + \Psi_{t+1}\]

where \(\Psi_{t+1} \sim \mathcal{N}(0, \sigma_\Psi^2)\).

Perfect Foresight: Additive Growth

Under CARA utility, the Euler equation implies additive changes in consumption levels (contrast with the multiplicative growth under CRRA):

\[C_{t+1} = C_t + \frac{1}{\alpha}\log(R\beta)\]

The intertemporal elasticity of substitution \(\alpha^{-1}\) governs the absolute increment in consumption, not its growth rate.

Solution Under Uncertainty

With normally distributed permanent shocks \(\Psi_{t+1}\), the consumption process that satisfies the FOC is

\[\boxed{C_{t+1} = C_t + \frac{1}{\alpha}\log(R\beta) + \frac{\alpha\sigma_\Psi^2}{2} + \Psi_{t+1}}\]

Verification: substituting into \(1 = R\beta\,\mathbb{E}_t[e^{-\alpha(C_{t+1} - C_t)}]\) and using the lognormal moment formula, all terms cancel to yield \(1 = 1\).

The Precautionary Premium

Define \(\hat{\kappa} = \alpha^{-1}\log(R\beta) + \alpha\sigma_\Psi^2/2\). Then

\[C_{t+1} = C_t + \hat{\kappa} + \Psi_{t+1}\]

The term \(\alpha\sigma_\Psi^2/2\) is the precautionary premium: expected consumption growth is faster than under certainty, because the consumer saves more today to buffer against future income risk. The premium increases in both risk aversion \(\alpha\) and income volatility \(\sigma_\Psi^2\).

The Consumption Function

Imposing the intertemporal budget constraint (PDV of consumption = wealth + PDV of income):

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

Three terms, three forces:

1. \(P_t\): the consumer’s idiosyncratic permanent income level
2. \(\frac{r}{R}[\cdots]\): interest income on total “certain” wealth (human + nonhuman)
3. Precautionary + substitution: the combined effect of impatience and income risk

Two Peculiarities

The MPC out of capital does not depend on impatience.

The marginal effect of bank balances on consumption is \(\partial C_t / \partial B_t = r/R\), the interest income on an extra dollar. Impatience affects the change in consumption over time, but not the level (given the budget constraint).

Precautionary saving is independent of wealth.

The dollar amount saved for precautionary reasons (\(\alpha\sigma_\Psi^2/2\) per period) is the same regardless of the consumer’s resources. This follows from the constant absolute risk aversion property: risk attitudes do not scale with wealth.

Infinite Horizon: Patience and Impatience

The infinite-horizon consumption function simplifies to

\[C_t = \frac{R - (R\beta)^{1/\alpha}}{R}\,W_t\]

where \(W_t = B_t + H_t\) is total wealth. The consumer is:

• Impatient (spends more than income) if \((R\beta)^{1/\alpha} < 1\)
• Balanced (spends exactly income) if \(R\beta = 1\)
• Patient (saves) if \((R\beta)^{1/\alpha} > 1\)

When \(R\beta = 1\), the consumer spends \(\kappa\,W_t = (r/R)\,W_t\): exactly the interest income on total wealth.

The Three Effects of \(r\)

Using the approximation \(C_t \approx (r - \alpha^{-1}(r - \vartheta))\,W_t\) (where \(\vartheta\) is the time preference rate) and writing human wealth as \(H = \bar{Y}\,R/r\):

\[C_t \approx \bigl(\underbrace{r}_{\text{income}} - \underbrace{\alpha^{-1}(r - \vartheta)}_{\text{substitution}}\bigr)\left[B_t + \underbrace{\bar{Y}\,\frac{R}{r}}_{\text{human wealth}}\right]\]

A change in \(r\) affects consumption through all three channels, and the net effect depends on their relative magnitudes.

Code: CARA Consumption Paths

Time-Varying Interest Rates

Setup: Campbell-Mankiw Framework

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

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

The goal is to derive how the consumption-wealth ratio responds to changes in the path of interest rates, separating income and substitution effects.

Log-Linearizing the Budget Constraint

Dividing by \(W_t\) and taking logs:

\[\Delta w_{t+1} \approx r_{t+1} + \log(1 - \exp(c_t - w_t))\]

where lowercase letters denote logs. Define \(x_t = c_t - w_t\) (the log consumption-wealth ratio) and expand around the steady-state value \(\bar{x}\):

\[\Delta w_{t+1} \approx r_{t+1} + \log\xi + (1 - 1/\xi)(c_t - w_t)\]

where \(\xi = 1 - \exp(\bar{x})\) is slightly below one (since \(C/W\) is small).

Forward Iteration

Setting two expressions for \(\Delta w_{t+1}\) equal and iterating forward yields the approximate IBC:

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

This result is purely accounting: no behavioral assumptions yet. It says the log consumption-wealth ratio must equal the discounted value of future returns minus future consumption growth.

Substituting the Euler Equation

A perfect-foresight CRRA consumer satisfies \(\Delta c_{t+1} = \mu + \rho^{-1} r_{t+1}\), where \(\mu = \rho^{-1}\log\beta\) and \(\rho^{-1}\) is the intertemporal elasticity of substitution. Substituting:

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

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

Income vs. Substitution Effects

The coefficient \((1 - \rho^{-1})\) determines the net effect of higher future interest rates:

\(\rho\)	IES \(= \rho^{-1}\)	\(1 - \rho^{-1}\)	Net effect
\(< 1\)	\(> 1\)	\(< 0\)	Substitution dominates: save more
\(= 1\)	\(= 1\)	\(0\)	Effects cancel (log utility)
\(> 1\)	\(< 1\)	\(> 0\)	Income dominates: consume more

When \(\rho > 1\) (the empirically relevant case), higher expected future returns raise the consumption-wealth ratio today: the consumer feels richer.

The Human Wealth Channel

The Campbell-Mankiw result holds current wealth \(w_t\) fixed. But interest rate changes also alter the present value of future labor income:

\[H_t \approx \frac{Y_t}{r - g}\]

A permanent drop in \(r\) has an enormous effect on \(H_t\), and therefore on \(W_t\) itself. Summers (1981) showed that this human wealth channel dominates for most calibrations, making the full effect of interest rate changes much larger than the partial-equilibrium result suggests.

Code: Consumption-Wealth Ratio and Interest Rates

Summary

Key Takeaways (1/2)

• CRRA with risky returns: the MPC out of risky wealth is \(\kappa = 1 - (\beta\,\mathbb{E}[\tilde{R}^{1-\rho}])^{1/\rho}\); the approximate formula reveals three forces: income, substitution, and precautionary saving
• Log utility is special: with \(\rho = 1\) the precautionary term vanishes because the covariance between the return and marginal utility exactly offsets the Jensen’s inequality effect; \(\rho \geq 2\) is a plausible lower bound

Key Takeaways (2/2)

• CARA with income risk: CARA utility produces additive (not multiplicative) consumption changes and a closed-form precautionary premium \(\alpha\sigma_\Psi^2/2\) that is independent of wealth
• Time-varying \(R\): the Campbell-Mankiw decomposition shows the consumption-wealth ratio depends on the coefficient \((1 - \rho^{-1})\) times expected future returns; when \(\rho > 1\) (empirically relevant), the income effect dominates

Connections Across Models

Feature	CRRA + Risky \(R\)	CARA + \(Y\) Risk	Time-Varying \(R\)
Utility	\(c^{1-\rho}/(1-\rho)\)	\(-(1/\alpha)e^{-\alpha C}\)	\(c^{1-\rho}/(1-\rho)\)
Risk source	Return	Income	Interest rate path
Precautionary saving	\(-(\rho-1)\sigma_r^2/2\)	\(\alpha\sigma_\Psi^2/2\)	Through human wealth
Key parameter	IES = \(\rho^{-1}\)	ARA = \(\alpha\)	IES = \(\rho^{-1}\)