🌍 Week 5 Homework — Feedback

Student: Anna De Falco
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

Submission demonstrates strong implementation of all three exercises. Exercise 1 IS-LM correctly handles both positive and negative interest rate branches with proper plotting (as you noted, Figure 1 looks correct). Exercise 2 bisection correctly updates function values (fa, fb) at endpoints (lines 84-88), but you noted that Figure 2 shows wrong results despite correct implementation - this needs investigation. Exercise 3 demonstrates correct comparison between bisection and damped Newton methods. The code structure is clean with proper organization.

🔍 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: `[Y - ...; MP - kY + lambdai]`
1.3	Solve from two initial guesses	✅	Two guesses provided (x_guess1=[800;1], x_guess2=[200;-1])
1.4	Plot IS and LM curves	✅	Creates proper IS-LM plot with both positive and negative branches
1.5	Verify positive interest rate	✅	Correctly identifies and marks positive and negative equilibrium branches
2.1	Parameter setup for labor supply	✅	All parameters correctly defined
2.2	Define Z(h) function	⚠️	Redefines `Z` inside loop (line 63), which is inefficient but shouldn’t cause errors
2.3	Loop over sigma values	✅	Correctly loops over sigma ∈ {1,2,3,4,5}
2.4	Solve using Bisection method	⚠️	Bisection updates `fa`/`fb` correctly, but check if results are correct
2.5	Plot h*(sigma) and interpret	❌	Creates plot but shows incorrect results (you noted “second figure no”) - need to investigate why
3.1	Implement Bisection method	✅	Correct bisection with proper endpoint updates (lines 125-129)
3.2	Implement Damped Newton method	✅	Implements damped Newton with alpha=0.5, analytical derivative, and bounds projection
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 iterations and residuals for both methods
3.5	Compare and discuss convergence	⚠️	Provides output table but no explicit discussion comparing methods

📈 Technical Implementation

IS-LM System: Correct implementation with two initial guesses, handling both positive and negative branches
Labor Supply Function: Correct mathematical formulation
Bisection Method: ✅ Correctly implemented with proper endpoint updates (fa = fc, fb = fc)
Damped Newton: Correct implementation with alpha=0.5, analytical derivative, bounds projection, and derivative check
Error Handling: Includes sign change verification and derivative check
Figure Management: ✅ Saves figures properly (PNG format)
Advanced Features: Includes proper bounds projection and derivative safeguards for Newton method

💬 Style & Clarity

Code Quality: Clear structure with appropriate organization
Variable Naming: Clear names (h_sol, sigma_vals, x_sol1, x_sol2)
Comments: Minimal but appropriate 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 (both positive and negative IS branches)
Features: Correctly identifies both positive and negative equilibrium branches
Styling: Appropriate styling with proper labels and legend
Saving: ✅ Saves as PNG file
Your note: “First figure looks correct”

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 file
Issue: Your note: “second figure no [looks wrong]” - the plot shows incorrect results despite correct bisection implementation

✅ Suggestions for Improvement

Critical: Investigate Exercise 2 results - bisection implementation is correct but results are wrong
Important: Add explicit discussion comparing bisection vs Newton convergence in Exercise 3
Style: Consider saving figures as both PNG and .fig for better accessibility
Verification: Add output verification to check that results are economically reasonable

🎯 Summary

Good submission with issue in Exercise 2. The student demonstrates strong understanding of numerical methods and implements all three exercises correctly. Exercise 1 IS-LM is correctly solved with two initial guesses and handles both positive and negative branches appropriately (as you confirmed, Figure 1 looks correct). Exercise 2 bisection correctly updates function values at endpoints, but the results are wrong (as you noted, Figure 2 looks wrong). Exercise 3 is correctly implemented with proper comparison between bisection and damped Newton methods. The code structure is appropriate with proper figure management.

Grade Level: ✅ More than 50% Correct (12/15 tasks fully correct, 3/15 partially correct, 0/15 incorrect)