🌍 Week 5 Homework — Feedback

Student: Alessia Canuto
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

Strong submission demonstrating understanding of numerical methods with all three exercises correctly implemented. The code is well-structured and handles edge cases appropriately. The IS-LM system is correctly implemented with proper Newton method, labor supply correctly solved with robust bisection, and convergence comparison properly implemented with damped Newton using analytical derivatives. Figures are properly saved.

🔍 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 with proper Jacobian matrix
1.3	Solve from two initial guesses	✅	Two guesses provided ([200,1.5], [400,4.0]) and both converge
1.4	Plot IS and LM curves	✅	Creates IS-LM plot with equilibrium point and proper styling
1.5	Verify positive interest rate	✅	Smartly selects solution with positive interest rate and verifies
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 with elementwise operations
2.3	Loop over sigma values	✅	Correctly loops over sigma ∈ {1,2,3,4,5}
2.4	Solve using Bisection method	✅	Robust bisection with sign-change detection and bracket-finding
2.5	Plot h*(sigma) and interpret	✅	Creates proper plot with economic interpretation
3.1	Implement Bisection method	✅	Correct bisection implementation
3.2	Implement Damped Newton method	✅	Uses alpha=0.5 with 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 discussion comparing methods

📈 Technical Implementation

IS-LM System: Strong implementation with analytical Jacobian and smart solution selection
Labor Supply Function: Correct mathematical formulation
Bisection Method: Uses sign() function for robust sign comparison (equivalent to < 0)
Bracket Finding: Sophisticated logic to find sign changes when endpoints don’t bracket root
Damped Newton: Strong implementation with analytical derivative and alpha=0.5
Error Handling: Appropriate handling of edge cases and NaN values
Figure Management: ✅ Uses saveas() to save figures
Advanced Features: Includes economic interpretation, robust bracket-finding, and comprehensive summaries

💬 Style & Clarity

Code Quality: Strong structure with professional helper functions
Variable Naming: Clear and logical names (Fsys, Jsys, h_star, iter_bis)
Comments: Good comments explaining methodology and economic meaning
Output: Professional use of fprintf with detailed progress reports and summaries
Organization: Clear separation into logical sections with helper functions
Documentation: Includes comprehensive interpretation and discussion of results
Helper Functions: Well-implemented local functions for Newton and bisection

📊 Visual Output Assessment

Figure 1: IS-LM Equilibrium ✅

Layout: Clear plot with IS and LM curves
Features: Correctly identifies equilibrium and verifies positive interest rate
Styling: Appropriate styling with proper labels, legend, and grid
Saving: ✅ Saves with saveas() as ‘IS_LM_equilibrium.png’

Figure 2: Labor Supply h*(σ) ✅

Layout: Clear plot showing h* vs sigma
Features: Shows valid points only, handles NaN appropriately
Styling: Appropriate appearance with proper labels and grid
Saving: ✅ Saves with saveas() as ‘LaborSupply_h_vs_sigma.png’

✅ Suggestions for Improvement

Minor: Consider using exportgraphics() instead of saveas() for higher resolution PNG files
Style: Could add figure folder organization for saved images

🎯 Summary

Strong work with correct implementations. The student demonstrates good understanding of numerical methods and economic modeling. All three exercises are correctly implemented with appropriate mathematical formulations. The IS-LM system is correctly solved with smart solution selection, the labor supply function is correctly formulated, and both bisection and damped Newton methods are implemented with appropriate error handling and bracket-finding. The code includes comprehensive output, proper figure saving, and economic interpretation.

Grade Level: ✅ More than 50% Correct (15/15 tasks fully correct)