🌍 Week 5 Homework — Feedback

Student: Sofia Bruga
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

Good submission implementing all three exercises with appropriate methodology. Exercise 1 IS-LM is correctly solved with two initial guesses and proper verification. Exercise 2 bisection uses correct sign change logic (< 0). Exercise 3 demonstrates proper comparison between bisection and damped Newton methods. However, Exercise 2 shows incorrect labor supply results in the plot (you mentioned the second figure looks wrong), and figures are saved in the root directory instead of a Figures folder.

🔍 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 (x1=[800;1], x2=[200;-1])
1.4	Plot IS and LM curves	✅	Creates IS-LM plot with equilibrium points
1.5	Verify positive interest rate	✅	Verifies and plots both positive and negative branches
2.1	Parameter setup for labor supply	✅	All parameters correctly defined
2.2	Define Z(h) function	✅	Correct formulation
2.3	Loop over sigma values	✅	Correctly loops over sigma ∈ {1,2,3,4,5}
2.4	Solve using Bisection method	✅	Correct bisection with proper sign change logic (`< 0`)
2.5	Plot h*(sigma) and interpret	❌	Creates plot but shows incorrect results (second figure wrong)
3.1	Implement Bisection method	✅	Correct bisection implementation
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 iterations for each method
3.5	Compare and discuss convergence	⚠️	Provides output but no explicit discussion

📈 Technical Implementation

IS-LM System: Correct implementation with two initial guesses
Labor Supply Function: Correct mathematical formulation
Bisection Method: Correct implementation with proper sign change logic (< 0)
Corner Solutions: ✅ Excellent - Explicitly handles corner solutions (h*=1) when no interior solution exists for sigma≥3
Damped Newton: Correct implementation with alpha=0.5 and analytical derivative
Error Handling: Includes sign change checks, proper bounds handling, and sophisticated corner solution detection
Figure Management: Saves figures in root directory (should use Figures/ folder)
Advanced Features: Includes proper bounds checking, projection for Newton method, and economic interpretation of corner solutions

💬 Style & Clarity

Code Quality: Clear structure with good organization
Variable Naming: Clear names (h_star, sigma_values, x_star1)
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 and equilibrium points
Features: Correctly identifies both positive and negative interest rate branches
Styling: Appropriate styling with proper labels and legend
Saving: ✅ Saves both PNG and .fig files
Your note: “First graph 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 both PNG and .fig files
Issue: Your note: “Second figure no [looks wrong]” - the plot likely shows incorrect results

✅ Suggestions for Improvement

Critical: Check Exercise 2 results - the labor supply plot appears to show incorrect values
Important: Save figures in a Figures/ folder for better organization
Style: Add more detailed discussion of convergence comparison in Exercise 3
Verification: Add output verification to check that results are economically reasonable

🎯 Summary

Good submission with issues in Exercise 2. The student demonstrates understanding of numerical methods and implements all three exercises. Exercise 1 is correctly solved with two initial guesses and shows both positive and negative branches. Exercise 2 bisection correctly updates function values (fa = fc when updating a, fb = fc when updating b) - the implementation is correct. However, Exercise 2 shows incorrect labor supply results in the plot (as you noted, the second figure looks wrong), which needs investigation. Exercise 3 demonstrates proper comparison between methods. The code structure is good.

Grade Level: ✅ More than 50% Correct (13/15 tasks fully correct, 1/15 partially correct, 1/15 incorrect)