Midterm Study Guide

Materials¶

• Slides: Asset Pricing

This study guide covers the key concepts, results, and equations from Lectures 03 through 07. Derivations are omitted; focus on understanding what each result says and when it applies.

Intertemporal Choice (Lecture 03)¶

The Fisher Two-Period Model¶

• A consumer allocates resources across two periods to maximize $u(c_1) + \beta\,u(c_2)$

• The intertemporal budget constraint: the PDV of spending equals nonhuman wealth $b_1$ plus human wealth $h_1 = y_1 + y_2/R$

• The Euler equation:

• Perturbation argument: at the optimum, the consumer is indifferent between consuming one more unit today and saving it for tomorrow

CRRA Utility and the IES¶

• CRRA utility: $u(c) = c^{1-\rho}/(1-\rho)$, marginal utility $u'(c) = c^{-\rho}$

• Consumption growth: $(c_2/c_1) = (R\beta)^{1/\rho}$

• The IES is $1/\rho$: how strongly the consumption ratio responds to a change in $R$

• Fisherian separation: consumption growth depends on $R$ and $\beta$, not on the timing of income

Three Effects of a Change in $R$¶

• Substitution: future consumption is cheaper $\Rightarrow$ save more ($c_1 \downarrow$)

• Income: higher return on savings $\Rightarrow$ richer ($c_1 \uparrow$)

• Human wealth: PDV of future labor income falls ($c_1 \downarrow$)

• Summers (1981): the human wealth effect dominates quantitatively for most consumers

Labor Supply¶

• Cobb-Douglas preferences keep the expenditure share on leisure constant as wages rise

• The model predicts strong labor supply responses to predictable wage variation; the data show near-constant hours

• This is the “small intertemporal elasticity of labor supply” puzzle

The OLG Model¶

• Young workers save, old retirees consume; no bequests; two generations alive at any time

• With log utility, the saving rate $\beta/(1+\beta)$ is independent of $R$

• Capital accumulation: $k_{t+1} = \mathcal{Q}\,k_t^\varepsilon$ converges to steady state $\bar{k} = \mathcal{Q}^{1/(1-\varepsilon)}$

• Golden Rule: $f'(\bar{k}^{**}) = n$ maximizes steady-state per-capita consumption

• The competitive equilibrium can be dynamically inefficient (too much capital)

Consumption Theory (Lecture 04)¶

The Envelope Condition¶

• In the $T$-period Bellman equation, the envelope theorem gives:

• Marginal value of resources = marginal utility of consumption

• This extends the Euler equation $u'(c_t) = R\beta\,u'(c_{t+1})$ to any finite or infinite horizon

Perfect Foresight CRRA¶

• Consumption grows at the absolute patience factor $\Phi = (R\beta)^{1/\rho}$ every period

• Three conditions for a well-behaved infinite-horizon solution:

  • AIC ($\Phi < 1$): consumption falls over time (absolute impatience)

  • FHWC ($G < R$): human wealth is finite

  • RIC ($\Phi/R < 1$): the PDV of desired consumption is finite

• The consumption function is linear in overall wealth $o_t = b_t + h_t$:

• The GIC ($\Phi < G$) determines whether the wealth-to-income ratio is falling

The Random Walk of Consumption¶

• Quadratic utility + $R\beta = 1$ $\Rightarrow$ $\mathbb{E}_t[c_{t+1}] = c_t$

• Hall (1978): no lagged variable should predict consumption changes

• Powerful because it is model-free (no need to specify the income process)

• Fails with CRRA utility or when $R\beta \neq 1$

• Quadratic utility has $u''' = 0$, so the random walk model rules out precautionary saving

The Consumption Function and Muth’s Insight¶

• With permanent ($\psi_t$) and transitory ($\theta_t$) shocks:

• MPC out of transitory income: $r/R \approx 5\%$

• MPC out of permanent income: 1 (one for one)

• The Keynesian $c = \alpha_0 + \alpha_1 y$ is meaningless without specifying which type of shock changed income

Advanced Consumption Models (Lecture 05)¶

Habit Formation¶

• Past consumption raises a reference point; higher habits make any given $c_t$ less satisfying

• The consumer accounts for the cost of raising future habits, which pushes consumption down relative to the standard model

• Modified Euler equation:

• With $u(c,h) = f(c - \alpha h)$, consumption growth is positively autocorrelated: $\Delta\log c_{t+1} \approx \text{const} + \alpha\,\Delta\log c_t$

Durable Goods¶

• The stock $d_t$ yields utility; expenditure $x_t = d_t - (1-\delta)d_{t-1}$ is a flow

• Intratemporal condition:

• The marginal utility of the durable is lower than that of the nondurable because it yields service over multiple periods

• With Cobb-Douglas utility, $d/c = \gamma$ is constant

• Spending is volatile: a small consumption adjustment of size $\epsilon$ multiplies durable expenditure by a factor of $(\epsilon + \delta)/\delta$

Quasi-Hyperbolic Discounting (Laibson)¶

• An extra discount factor $\delta_h < 1$ ($\approx 0.7$) applies to the step from “now” to “all of the future”

• The present-biased consumer consumes more than the standard consumer

• The distortion scales with the MPC: high-MPC consumers (young, poor, constrained) suffer the most from present bias

• Low-MPC consumers (wealthy, long horizon) are barely affected

Risk and Consumption (Lecture 06)¶

CRRA with Risky Returns¶

• With a single risky asset (lognormal), no labor income, and $c_t = \kappa\,m_t$:

• Approximate MPC decomposes into three forces:

• More risk ($\sigma_r^2 \uparrow$) lowers the MPC $\Rightarrow$ more saving (for $\rho > 1$)

• Log utility ($\rho = 1$) is a knife-edge where risk does not affect consumption

CARA with Income Risk¶

• CARA utility $u(C) = -(1/\alpha)e^{-\alpha C}$ yields additive consumption changes

• Consumption under uncertainty:

• The precautionary premium $\alpha\sigma_\Psi^2/2$ does not depend on wealth

• The MPC out of bank balances is $r/R$, independent of impatience

Campbell-Mankiw (Time-Varying $R$)¶

• Log-linearization relates the log consumption-wealth ratio to future log interest rates:

• $(1 - \rho^{-1})$ governs the net effect:

  • $\rho < 1$: substitution dominates (save more)

  • $\rho = 1$: effects cancel

  • $\rho > 1$: income dominates (consume more)

• The human wealth channel $H_t \approx Y_t/(r-g)$ can dwarf the direct effects

Asset Pricing (Lecture 07)¶

The Lucas Tree Model¶

• An endowment economy: identical consumers hold identical trees; output is exogenous fruit

• Market clearing: $c_t = d_t$ (all fruit is eaten; you cannot plant more trees)

• The stochastic discount factor prices every asset:

• Asset price as PDV of dividends:

• With log utility: $P_t = d_t/\vartheta$ (income and substitution effects cancel exactly)

• Gordon formula (constant growth): $P_t \approx d_t/(r - g)$

• Prices follow a martingale: all known information is already in the price

The Fallacy of Composition¶

• Any individual can save one more dollar and earn $\mathbf{R}$

• If everyone saves one more dollar, aggregate fruit is unchanged; the market clears through higher tree prices, not more output

• This distinction between individual and aggregate is central to the Lucas model

Portfolio Choice¶

• CARA: optimal dollar investment $S = \phi/(\alpha\sigma^2)$ is independent of wealth (Buffett and Homer Simpson hold the same dollar position, which is implausible)

• CRRA (Merton-Samuelson): optimal portfolio share is independent of wealth:

• Surprising: portfolio risk falls when asset risk rises, because the consumer cuts exposure so aggressively

• After portfolio optimization, the MPC becomes:

The Consumption CAPM (C-CAPM)¶

• The expected excess return on any asset satisfies:

• Only consumption covariance matters for pricing; the asset’s own variance is irrelevant

• Procyclical assets (pay off when $c$ is high, $u'$ is low): must offer higher returns

• Countercyclical assets: expensive, low returns (they are insurance)

The Equity Premium and Riskfree Rate Puzzles¶

• Equity premium puzzle: $0.08 \approx \rho \times 0.004$ requires $\rho \approx 20$

• Riskfree rate puzzle: $0.015 \approx 0.01/\rho$ requires $\rho < 1$

• The two puzzles demand opposite values of $\rho$; this is what makes the joint puzzle so hard

• Proposed resolutions: habit formation, rare disasters, long-run consumption risk, limited participation

Rational Bubbles¶

• The Euler equation admits $P_t = P_t^* + B_t$ where $B_t$ grows at rate $R^f$

• Blanchard (1979): stochastically bursting bubbles grow faster than $R^f$ while alive to compensate for crash risk; every such bubble eventually bursts

• Arguments against: no negative bubbles, reproducible assets cap prices, risk aversion makes bubbles harder to sustain, GE prevents the bubble from exceeding total wealth

Upcoming¶