🌍 Week 5 Homework — Feedback

Student: Pablo Romanella
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

Submission implements all three exercises. Exercise 1 IS-LM is solved with two initial guesses, but both figures show wrong results (as you noted). Exercise 2 bisection has a critical numerical error: the method does not update function values (fa, fb) after updating brackets, causing convergence to the wrong root. Exercise 3 demonstrates the same bisection error. The code structure is clear but the results are incorrect.

🔍 Task-by-Task Check

Task	Description	Status	Notes
1.1	Parameter setup for IS-LM	✅	All parameters correctly defined
1.2	IS-LM system definition	✅	Correct formulation
1.3	Solve from two initial guesses	✅	Two guesses provided (z1=[900;7], z2=[500;1])
1.4	Plot IS and LM curves	❌	Creates IS-LM plot but shows wrong results (you noted “Figure 1 wrong”)
1.5	Verify positive interest rate	❌	Both figures show wrong results
2.1	Parameter setup for labor supply	✅	All parameters correctly defined
2.2	Define Z(h) function	⚠️	Function defined outside loop but redefined inside loop (inefficient)
2.3	Loop over sigma values	✅	Correctly loops over sigma ∈ {1,2,3,4,5}
2.4	Solve using Bisection method	❌	Critical Error: Bisection does not update function values at endpoints (fa/fb), causing wrong root convergence. Code updates `p_a`/`p_b` but not function values at endpoints.
2.5	Plot h*(sigma) and interpret	❌	Creates plot but shows wrong results (you noted “Figure 2 wrong”) - caused by bisection error
3.1	Implement Bisection method	❌	Same critical bisection error - does not update `Z(a_h)`/`Z(b_h)` values
3.2	Implement Damped Newton method	✅	Implements damped Newton with alpha=0.5 and analytical derivative
3.3	Test from multiple starting guesses	✅	Tests from three guesses (0.2, 0.5, 0.8)
3.4	Record iterations and residuals	✅	Records and displays all iterations and residuals
3.5	Compare and discuss convergence	⚠️	Provides output table but no explicit discussion comparing methods

📈 Technical Implementation

IS-LM System: Correct formulation with two initial guesses
Labor Supply Function: Correct mathematical formulation (though defined twice)
Bisection Method: ❌ Critical error - does not update function values at endpoints, causing wrong root convergence
Damped Newton: Correct implementation with alpha=0.5 and analytical derivative
Error Handling: No sign change verification
Figure Management: Saves figures as PNG and .fig files
Advanced Features: Includes bounds projection for Newton method

💬 Style & Clarity

Code Quality: Clear structure with good organization
Variable Naming: Clear names (h_star, sigma_v, zstar1, zstar2)
Comments: Minimal comments
Output: Appropriate use of fprintf to display results
Organization: Clear separation into three exercises

📊 Visual Output Assessment

Figure 1: IS-LM Equilibrium ❌

Layout: Plot with IS and LM curves and equilibrium points
Features: Attempts to identify equilibria from both guesses
Styling: Appropriate styling with proper labels and legend
Saving: ✅ Saves as PNG and .fig files
Issue: Your note: “Both figures show wrong results” - Figure 1 shows incorrect equilibrium points

Figure 2: Labor Supply h*(σ) ❌

Layout: Plot showing h* vs sigma values
Features: Shows labor supply for different risk aversion parameters
Styling: Appropriate styling with proper labels
Saving: ✅ Saves as PNG and .fig files
Issue: Your note: “Figure 2 wrong” - the plot shows incorrect results due to bisection error

✅ Suggestions for Improvement

Critical: Fix bisection in Exercises 2 and 3 - must update function values when updating brackets:

fa = Z(a); fb = Z(b);
if fa * fm < 0
    b = m; fb = fm;  % Update function value
else
    a = m; fa = fm;  % Update function value
end

Critical: Check Exercise 1 results - both equilibria appear incorrect
Important: Add sign change verification before starting bisection
Style: Remove redundant function definitions (Z(h) defined outside and inside loop in Exercise 2)
Documentation: Add more detailed comments explaining the methodology

🎯 Summary

Submission with critical bisection error affecting results. The student demonstrates understanding of numerical methods and implements all three exercises. Exercise 1 solves with two initial guesses but shows incorrect equilibrium points (as you noted, both figures are wrong). Exercise 2 and 3 bisection implementations have the critical error of not updating function values (fa, fb) when updating brackets, causing convergence to wrong roots. Exercise 3 damped Newton is correctly implemented. The code structure is appropriate but results are incorrect.

Grade Level: ✅ More than 50% Correct (9/15 tasks fully correct, 1/15 partially correct, 5/15 incorrect)