Dynamic Stochastic General Equilibrium

Materials¶

• Slides: Dynamic Stochastic General Equilibrium

Notebooks¶

The following computational notebook accompanies the source chapters in the Reading Assignment below:

• Dynamic Stochastic General Equilibrium: Brock-Mirman and the Prescott RBC Model: Companion to The Brock-Mirman Stochastic Growth Model and The Prescott Real Business Cycle Model

Learning Objectives¶

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

1. Derive the closed-form consumption rule $\kappa = 1 - \alpha\beta$ in the Brock-Mirman model and identify the three knife-edge assumptions that make it possible

2. Explain why the expected-self steady state coincides with the no-shocks steady state in Brock-Mirman and why this coincidence is special to the linear-in-income consumption rule

3. Derive the log-linear law of motion $k_{t+1} = \log(\alpha\beta) + a_t + \alpha k_t$ and interpret the AR(1) coefficient as capital’s share

4. Set up the Prescott RBC household problem with Cobb-Douglas preferences over consumption and leisure, and derive the intratemporal FOC $w\ell/c = \zeta/(1-\zeta)$

5. Calibrate the leisure weight $\zeta$ from long-run time-use data and explain why Cobb-Douglas is the unique preference class consistent with the lack of trend in hours

6. State the hours volatility puzzle, distinguish the two candidate mechanisms (labor supply elasticity vs interest rate channel), and explain why each fails to match the data

Key Concepts¶

• Brock-Mirman closed form: with 100 percent depreciation, log utility, and Cobb-Douglas production, the consumption rule is linear in income with MPC $\kappa = 1 - \alpha\beta$ and the $A_{t+1}$ terms cancel exactly inside the Euler equation

• Stochastic steady state: three definitions (no-shocks limit, ergodic mean, expected-self point) coexist in the literature; they coincide in Brock-Mirman but generally differ in richer models, and the ergodic mean requires a stationary shock process to exist

• Cobb-Douglas preferences over consumption and leisure: $u(c, \ell) = (c^{1-\zeta}\ell^\zeta)^{1-\rho}/(1-\rho)$ is the unique class that keeps the budget share of leisure constant when wages change, consistent with the long-run facts documented by Ramey and Francis (2009)

• Leisure weight calibration: under the assumption $c \approx wn$, the intratemporal FOC implies $\ell = \zeta$, giving $\zeta \approx 2/3$ for a 40-hour work week with 8 hours of sleep per day

• Leisure Euler equation: $\widehat{\ell}_{t+1} \approx -\widehat{w}_{t+1} + (r_{t+1} - \theta)$ where $\theta \equiv -\log\beta$ is the time preference rate (distinct from depreciation $\delta$); hours respond to wage and interest rate movements

• Hours volatility puzzle: the RBC model matches $\sigma_y$ at 84 percent of the US data but under-predicts $\sigma_n$ by half; resolving the gap via a large labor supply elasticity contradicts micro evidence, and resolving it via the interest rate channel implies counterfactual consumption comovement

• Calibration vs estimation: Summers (1986) argues that Prescott’s moment-matching procedure is a rhetorical exercise without standard errors or a formal rejection region; this methodological debate shapes the DSGE literature through to New Keynesian extensions

Reading Assignment¶

Final Project¶

Looking Ahead¶