🌍 Week 5 Homework — Feedback

Student: Simone Iudice
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

Good submission implementing all three exercises with clear documentation and appropriate methodology. The IS-LM implementation is correct, Exercise 2 bisection is properly implemented, and Exercise 3 demonstrates understanding of convergence comparison. However, Exercise 1 is missing a second initial guess as required, and Exercise 2 has incomplete sign change logic that could affect convergence.

🔍 Task-by-Task Check

Task	Description	Status	Notes
1.1	Parameter setup for IS-LM	✅	All parameters correctly defined (C0, c, I0, beta, T, G, MP, k, lambda)
1.2	IS-LM system definition	✅	Correct formulation: `[Y-(C0+c(Y-T)+I0-betai^2+G); MP-(kY-lambdai)]`
1.3	Solve from two initial guesses	❌	Issue: Only one initial guess provided (x0=[900;2]). Missing second guess requirement.
1.4	Plot IS and LM curves	✅	Creates IS-LM plot with equilibrium point and proper styling
1.5	Verify positive interest rate	✅	Verifies positive interest rate with conditional check
2.1	Parameter setup for labor supply	✅	All parameters correctly defined (w, tau, a, g, phi, chi)
2.2	Define Z(h) function	✅	Correct formulation: `c(h)^(-sigma)(1-tau)w - chi*(1-h)^phi`
2.3	Loop over sigma values	✅	Correctly loops over sigma ∈ {1,2,3,4,5}
2.4	Solve using Bisection method	⚠️	Correct bisection logic but uses `Z(a0)*Z(c) < 0` without updating values
2.5	Plot h*(sigma) and interpret	✅	Creates plot and provides economic interpretation
3.1	Implement Bisection method	✅	Correct bisection implementation
3.2	Implement Damped Newton method	✅	Implements damped Newton with alpha=0.5 and numerical derivative
3.3	Test from multiple starting guesses	✅	Tests from three guesses (0.2, 0.5, 0.8)
3.4	Record iterations and residuals	✅	Creates results table with all metrics
3.5	Compare and discuss convergence	✅	Provides detailed discussion comparing methods

📈 Technical Implementation

IS-LM System: Correct implementation using fsolve() with proper formulation
Labor Supply Function: Correct mathematical formulation with appropriate bounds [1e-6, 1-1e-6]
Bisection in Exercise 2: Issues with sign change logic - evaluates Z(a0) and Z(c) without updating these values in the loop
Bisection in Exercise 3: Correct bisection implementation
Damped Newton: Correct implementation with alpha=0.5 and numerical derivative
Error Handling: Includes sign change checks and proper convergence criteria
Figure Management: ✅ Saves figures in ‘Figures/’ folder
Advanced Features: Includes economic interpretation and detailed discussion

💬 Style & Clarity

Code Quality: Clear structure with comprehensive comments
Variable Naming: Clear and logical names (h_star, sigma_values, xstar)
Comments: Good comments explaining methodology and economic interpretation
Output: Appropriate use of fprintf and table display
Organization: Clear separation into three exercises with proper headers
Documentation: Includes detailed economic interpretation and convergence analysis

📊 Visual Output Assessment

Figure 1: IS-LM Equilibrium ✅

Layout: Plot with IS and LM curves and equilibrium point
Features: Correctly identifies equilibrium and verifies positive interest rate
Styling: Appropriate styling with proper labels and legend
Saving: ✅ Saves with saveas() in ‘Figures/’ folder

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 and grid
Saving: ✅ Saves with saveas() in ‘Figures/’ folder

✅ Suggestions for Improvement

Critical: Exercise 1 requires two different initial guesses to solve from (currently only one provided)
Important: Fix bisection loop in Exercise 2 - need to update a0 or b0 values, not just check Z(a0)*Z(c) < 0 without updating the interval endpoints
Style: Consider adding projection to keep h in [0,1] in the Newton method
Verification: Add more detailed output for Exercise 2 showing iteration count and residual for each sigma

🎯 Summary

Good submission with minor issues. The student demonstrates understanding of numerical methods and implements all three exercises. The IS-LM implementation is correct, Exercise 3 is implemented with appropriate bisection and damped Newton methods, and figures are properly saved. However, Exercise 1 is missing the second initial guess requirement, and Exercise 2’s bisection loop has a logic issue that could affect convergence (evaluating Z(a0) without updating it in the loop).

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