Risk and Consumption

Materials¶

• Slides: Risk and Consumption

Notebooks¶

The following computational notebook is a companion to the reading assigned in Lecture 05:

• Risk and Consumption: Companion to Consumption out of Risky Assets, Consumption with CARA Utility, and Dynamics of Consumption with Time-Varying R

Learning Objectives¶

By the end of this lecture, students will be able to:

1. Apply the guess-and-verify method to derive the exact MPC out of risky wealth under CRRA utility with lognormal returns

2. Decompose the approximate MPC into income, substitution, and precautionary saving components

3. Explain why log utility ($\rho = 1$) is a knife-edge case where risk does not affect the consumption level

4. Derive the closed-form consumption process under CARA utility with normally distributed permanent income shocks

5. Identify the precautionary premium in the CARA model and explain why it is independent of wealth

6. Describe the three channels through which interest rates affect consumption in the CARA framework (income, substitution, human wealth)

7. Use the Campbell-Mankiw log-linearization to express the consumption-wealth ratio as a function of expected future interest rates

8. Explain when income effects dominate substitution effects based on the intertemporal elasticity of substitution

Key Concepts¶

• Guess-and-verify method: Postulating a linear consumption rule $c_t = \kappa\,m_t$ and showing it satisfies the Euler equation; works when market resources cancel but fails when labor income enters

• Precautionary saving (CRRA): The $-(\rho - 1)\sigma_r^2/2$ term in the approximate MPC formula; consumers with $\rho > 1$ save more as return risk increases

• CARA utility: Constant absolute risk aversion $u(C) = -(1/\alpha)e^{-\alpha C}$; produces additive rather than multiplicative consumption changes, and the precautionary premium $\alpha\sigma_\Psi^2/2$ does not depend on wealth

• Precautionary premium: The additional expected consumption growth under uncertainty relative to perfect foresight; reflects the extra saving needed to self-insure against income shocks

• Campbell-Mankiw decomposition: Log-linearization of the budget constraint that expresses the consumption-wealth ratio as a discounted sum of future interest rates minus consumption growth

• Income vs. substitution effects: The coefficient $(1 - \rho^{-1})$ governs the net response of consumption to interest rate changes; when $\rho > 1$ the income effect dominates

• Human wealth effect: Interest rate changes alter the present value of future labor income $H_t \approx Y_t/(r-g)$, a channel that can dominate the direct income and substitution effects

Reading Assignment¶

Homework¶