🌍 Week 5 Homework — Feedback

Student: Klea Lushaj
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

The student demonstrates understanding of numerical methods and submits all three exercises with detailed output and discussion. Exercise 1 IS-LM implementation has a minor LM equation sign issue and plotting range cuts off the positive interest rate equilibrium (plots Y only up to 700). Exercise 2 has issues with the Z(h) function implementation and likely produces incorrect results. Exercise 3 is implemented correctly with appropriate bisection and damped Newton methods and includes detailed analysis. The code structure is good with clear separation into three files.

🔍 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	⚠️	System formulation has wrong sign on LM equation
1.3	Solve from two initial guesses	✅	Two guesses provided (x0_1=[500;5], x0_2=[400;10])
1.4	Plot IS and LM curves	⚠️	Creates plot but Y range [300,700] cuts off positive solution
1.5	Verify positive interest rate	✅	Verifies positive interest rate
2.1	Parameter setup for labor supply	✅	All parameters correctly defined
2.2	Define Z(h) function	⚠️	Function has potential issues with vectorization
2.3	Loop over sigma values	✅	Correctly loops over sigma ∈ {1,2,3,4,5}
2.4	Solve using Bisection method	✅	Implements bisection correctly with sophisticated corner solution detection (chooses boundary with smaller	Z	)
2.5	Plot h*(sigma) and interpret	⚠️	Creates plot but shows incorrect results (you noted “ex 2 seems 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 all iterations and residuals in table
3.5	Compare and discuss convergence	✅	Provides detailed discussion and visualization of convergence

📈 Technical Implementation

IS-LM System: ⚠️ LM equation has wrong sign: MP - k*x(1) + lambda*x(2)] should be MP - k*x(1) + lambda*x(2) or MP - (k*x(1) - lambda*x(2))
Labor Supply Function: Function definition may have issues with vectorization
Bisection Method: Correct implementation with proper sign change logic (< 0)

Corner Solutions: ✅ Excellent - Sophisticated corner solution detection that identifies which boundary (h=0 or h=1) is closer to being a root by comparing

Z(a)

Z(b)

Damped Newton: Correct implementation with alpha=0.5 and analytical derivative
Error Handling: Includes sign change checks, proper bounds handling, and advanced corner solution diagnostics
Figure Management: ✅ Saves figures as PNG files
Advanced Features: Includes detailed discussion, interpretation, visualization of convergence comparison, and comprehensive corner solution reporting

💬 Style & Clarity

Code Quality: Good structure with clear separation into three files (exone.m, extwo.m, exthree.m)
Variable Naming: Clear names (h_star, sigma_values, x_star_1)
Comments: Good comments explaining methodology
Output: Appropriate use of fprintf with detailed tables and analysis
Organization: Clear separation into three exercises in separate files

📊 Visual Output Assessment

Figure 1: IS-LM Equilibrium ⚠️

Layout: Plot with IS and LM curves and equilibrium point
Features: Shows both positive and negative interest rate branches
Styling: Appropriate styling with proper labels and legend
Saving: ✅ Saves as PNG file (Figurex1.png)
Issue: Your note: “Although she plots Y up to 700 so missing the positive solution for i” - the plotting range is too narrow

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 value annotations
Saving: ✅ Saves as PNG file (Figurex2.png)
Issue: Your note: “Ex 2 seems wrong” - the plot shows incorrect results

Figure 3: Convergence Comparison ✅

Layout: Plot comparing iteration counts between methods
Features: Shows bisection vs Newton for different starting guesses
Styling: Appropriate styling with proper labels and annotations
Saving: ✅ Saves as PNG file (Figurex3.png)
Quality: Good visualization of convergence comparison

✅ Suggestions for Improvement

Critical: Fix Exercise 1 LM equation sign - should be MP - (k*x(1) - lambda*x(2)) not MP - k*x(1) + lambda*x(2)
Critical: Expand Exercise 1 plotting range to Y up to 900 or 1000 to show the positive interest rate equilibrium
Critical: Check Exercise 2 Z(h) function definition - there may be issues with the formulation
Important: Verify that Exercise 2 results are economically reasonable (h* should be in [0,1])
Style: The code uses sigma=1 in Exercise 3 instead of sigma=2 as intended
Verification: Run Exercise 2 with correct Z(h) function to verify results

🎯 Summary

Good submission with minor errors. The student demonstrates understanding of numerical methods and implements all three exercises with detailed output and discussion. Exercise 1 has a minor LM equation sign issue and plotting range issue. Exercise 2 shows incorrect results likely due to problems with the Z(h) function. Exercise 3 is implemented correctly with proper comparison between bisection and damped Newton methods, including visualization and detailed discussion. The code is well-organized with clear separation into three files and comprehensive output.

Grade Level: ✅ More than 50% Correct (10/15 tasks fully correct, 4/15 partially correct, 1/15 incorrect)