Week 6

Week 6 – Optimization & Calibration

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

1. Understand the difference between unconstrained and constrained optimization.
2. Use MATLAB’s built-in optimization functions (fminsearch, fmincon).
3. Implement simple search methods (e.g., golden section search) manually.
4. Calibrate model parameters to match economic targets.
5. Assess goodness of fit for calibration exercises.

Suggested Readings

• Greenwood & Marto, Numerical Methods for Macroeconomists, Ch. 3 (Optimization) – focus on unconstrained search and constrained optimization.

In-Class Activities

• Implement golden section search to maximize a quadratic function.
• Use fminsearch to minimize a negative Cobb–Douglas utility function $U(c, l) = c^{\alpha} l^{1-\alpha}$.
• Constrained optimization example with fmincon:
  • Maximize utility subject to budget and time constraints.
• Calibration exercise:
  • Target steady-state labor supply = 0.3 of time endowment.
  • Choose preference parameter $\alpha$ to match target.

Homework / Practice

• Calibrate a simple Solow model parameter (e.g., savings rate) to match observed output growth over time.
• Compare results from:
  • Manual search (looping over candidate values).
  • Built-in MATLAB optimization functions.
• Plot the objective function over a parameter range and mark the optimum.

Files

• Slides Week 6 – Slides from lecture.
• codes_week6 on Matlab online
  • week6.m – Example script from lecture.
  • solve_lstar.m – Helper function for in class example.
  • make_fig_opt_curve.m – File to reproduce figure for: “Visualizing Optimization” slide.
  • make_fig_golden_vs_fminsearch.m – File to reproduce figure for: “Comparing Search Methods” slide.
  • week6_challenge_starter.m – Starter file for the challenge.

Homework submission

• Submit your homework here
• Please upload your homework as a single zip file. Access it’s now open with any email address! But remember to name your file with your full name and or student ID.
• The submission should include the .m file used to produce the results.
• You can modify you submission until the beginning of week 7.