Week 10

Week 10 – Stochastic Dynamics: Business Cycle Models

Learning Outcomes By the end of this week, students will be able to:

1. Formulate a stochastic neoclassical growth model.
2. Represent productivity shocks as an AR(1) process.
3. Discretise the shock process using Tauchen’s method.
4. Solve the model using policy function iteration (Coleman operator) and the endogenous grid method (EGM).
5. Simulate business cycle dynamics from the solved model.

Suggested Readings

• Greenwood & Marto, Numerical Methods for Macroeconomists, Ch. 9 (Stochastic dynamics).

In-Class Activities

• Define the stochastic growth model: [ y_t = e^{z_t} k_t^\alpha, \quad z_{t+1} = \rho z_t + \epsilon_t, \quad \epsilon_t \sim N(0, \sigma_\epsilon^2) ]
• Tauchen’s Method: Implement a function to discretize the continuous AR(1) shock into a Markov Chain.
• Solution Methods:
  • Solve using the Coleman Operator (Time Iteration on Euler Equation).
  • Solve using the Endogenous Grid Method (EGM) and observe the speed increase.
• Challenge: Precautionary Savings:
  • Run the model with $\sigma_\epsilon \approx 0$ (Deterministic).
  • Run the model with $\sigma_\epsilon = 0.1$ (High Uncertainty).
  • Compare the savings policy $k’(k)$ to see how risk affects behavior.

Homework / Practice

• Task 1: Quantifying Precautionary Savings
  • Compare the long-run average capital stock in a deterministic model ($\sigma_\epsilon \approx 0$) vs. a stochastic model ($\sigma_\epsilon = 0.04$).
  • Calculate the “Precautionary Savings Premium” (percentage increase in capital due to uncertainty).
• Task 2: Effect of Risk Aversion
  • Solve the model for two different values of the risk aversion parameter: $\sigma = 1$ (Log Utility) and $\sigma = 5$.
  • Plot the policy functions $k’(k)$ for both cases on the same graph.
  • Discuss how risk aversion affects the savings decision.

Files

• Slides Week 10 – Lecture slides.
• codes_week10 on Matlab online
  • tauchen.m – Helper function for discretization.
  • coleman.m – Policy function iteration solver.
  • egm.m – Endogenous Grid Method solver.
  • simulation_stoch.m – Simulation script.

Homework submission

• Submit your homework here
• Please upload your homework as a single zip file. Access is open with any email address! Remember to name your file with your full name and/or student ID.
• The submission should include the .m files used to produce the results.
• You can modify you submission until 9am on Monday of week 11.