Week 5

Week 5 – Solving Nonlinear Equations in Economics

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

1. Understand the difference between analytical and numerical solutions.
2. Implement root-finding algorithms (bisection, Newton–Raphson) in MATLAB.
3. Apply numerical methods to economic equilibrium problems.
4. Check for convergence and interpret stopping criteria.
5. Compare performance of different root-finding approaches.

Suggested Readings

• Greenwood & Marto, Numerical Methods for Macroeconomists, Ch. 2 (Nonlinear equations).

In-Class Activities

• Implement the bisection method.
• Implement Newton–Raphson method, including derivative calculation.
• Compare number of iterations required for each method.
• Apply root-finding to a labor–leisure choice problem:
  • Solve for optimal labor supply given parameters.

Homework / Practice

• Solve a non-linear IS–LM equilibrium problem.
• Study the labor–leisure choice problem with varying preferences.
  • Write a function that implements both bisection and Newton–Raphson methods.
  • Run both methods:
    • Record number of iterations.
    • Compare solutions and discuss convergence differences.
• Study the comparative statics of the labor–leisure choice problem.

Files

• Slides Week 5 – Slides from lecture.
• week5.m – Example script from lecture.
• codes_week5 on Matlab online
  • week5_newton_frames.m – Figure showing the convergence of the Newton–Raphson method.
  • week5_bisection_plot.m – Figure showing the convergence of the bisection method.
  • week5_newton_damping_comparison.m – Figure showing the comparison of the Newton–Raphson method with different damping factors.
  • week5_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 6.