🌍 Week 5 Homework — Feedback

Student: Chiara Tombolini
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

Good submission implementing all three exercises. Exercise 1 IS-LM is correctly implemented with two initial guesses and proper plotting. Exercise 2 has a critical numerical error: the bisection method does not update function values (fa, fb) after updating brackets, causing convergence to the wrong root (this is why Figure 2 looks wrong). Exercise 3 demonstrates proper comparison between bisection and damped Newton methods with comprehensive output. The code is well-structured with appropriate documentation.

🔍 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 (x0_1=[500,10], x0_2=[800,20])
1.4	Plot IS and LM curves	✅	Creates IS-LM plot with equilibrium points (you noted “Figure 1 looks correct”)
1.5	Verify positive interest rate	✅	Solves and plots equilibria
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	❌	Critical Error: Bisection does not update function values at endpoints, causing convergence to wrong root. Code evaluates `Z(a_bis)*Z(mid)` without storing `fa`, leading to incorrect bracket updates.
2.5	Plot h*(sigma) and interpret	❌	Creates plot but shows incorrect results (you noted “Figure 2 no”) - this is caused by the critical bisection error
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 all iterations and residuals in table
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
Labor Supply Function: Correct mathematical formulation
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: Includes sign change checks and proper convergence criteria
Figure Management: ✅ Saves figures as PNG and .fig files
Advanced Features: Includes proper bounds checking for Newton method

💬 Style & Clarity

Code Quality: Good structure with clear separation into exercises
Variable Naming: Clear names (h_star, sigma_values, sol1)
Comments: Minimal but appropriate comments
Output: Appropriate use of fprintf and disp 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 equilibria from both guesses
Styling: Appropriate styling with proper labels and legend
Saving: ✅ Saves as PNG and .fig files
Your note: “Figure 1 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 and .fig files
Issue: Your note: “Figure 2 no [looks wrong]” - the plot shows incorrect results

✅ Suggestions for Improvement

Critical: Fix bisection in Exercise 2 - must update fa/fb values when updating brackets: if fa * fm < 0, b = m; fb = fm; else, a = m; fa = fm; end
Important: Add explicit discussion comparing bisection vs Newton convergence in Exercise 3
Style: Consider using exportgraphics() instead of saveas() for higher resolution
Verification: Add verification that results are economically reasonable (h* should be in [0,1])
Documentation: Add more detailed comments explaining the methodology

🎯 Summary

Good submission with incorrect results 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 produces correct plots (as you confirmed). Exercise 2 uses correct bisection methodology but shows incorrect results in the plot. Exercise 3 is implemented correctly with proper comparison between bisection and damped Newton methods. The code structure is appropriate with figures saved properly.

Grade Level: ✅ More than 50% Correct (12/15 tasks fully correct, 1/15 partially correct, 2/15 incorrect)