🌍 Week 5 Homework — Feedback

Student: Felice Urciuoli
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

Good submission with correct IS-LM implementation and proper numerical methods. The student demonstrates strong understanding of the mathematical formulations and implements both bisection and damped Newton methods correctly. However, there are critical issues: (1) Exercise 2 shows incorrect results - h* values approach 1.0 as sigma increases, which is economically unreasonable since higher risk aversion should decrease labor supply, and (2) missing figure saving. The IS-LM system is perfectly implemented, but the labor supply results suggest a fundamental issue with the economic interpretation or parameter values.

🔍 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	✅	Two guesses provided (z0_1=[300;3], z0_2=[150;1.5]) and both converge to same equilibrium
1.4	Plot IS and LM curves	✅	Creates proper IS-LM plot with curves, equilibrium point, and positive branch constraint
1.5	Verify positive interest rate	✅	Correctly implements positive branch constraint with `sqrt(max(num/beta, 0))`
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(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	❌	Critical Error: Wrong sign change logic `fa * fc <= 0` instead of `fa * fc < 0`. Causes convergence to wrong root (h* → 1)
2.5	Plot h*(sigma)	❌	Critical Error: Creates plot but shows incorrect results (h* approaching 1.0 as sigma increases, which is economically unreasonable)
3.1	Implement Bisection method	✅	Correct bisection implementation with proper convergence criteria
3.2	Implement Damped Newton method	✅	Implements damped Newton with alpha=0.5 and proper derivative
3.3	Test from multiple starting guesses	✅	Tests from three starting points (0.2, 0.5, 0.8) as required
3.4	Record iterations and residuals	✅	Records and displays iterations for each method and starting point
3.5	Compare and discuss convergence	✅	Provides clear output comparing bisection vs Newton convergence

📈 Technical Implementation

IS-LM System: Correct formulation and implementation with proper Newton method
Labor Supply Function: Correct economic formulation with proper consumption function
Bisection Method: Good implementation with proper convergence criteria
Damped Newton: Correct implementation with alpha=0.5 damping and derivative protection
Code Structure: Clean organization with clear sections and good comments
Error Handling: Includes protection against small derivatives and proper convergence checks
Figure Management: Creates proper figures but missing exportgraphics() commands
Advanced Features: Implements custom Newton method instead of relying on built-in functions

💬 Style & Clarity

Code Quality: Good structure with clear section headers and comments
Variable Naming: Logical names (z0_1, z0_2, hstars, sigmas)
Comments: Good comments explaining the methodology and economic interpretation
Output: Professional use of fprintf with formatted results and convergence information
Organization: Clear separation into logical sections with proper headers
Documentation: Includes economic interpretation and convergence analysis
Local Functions: Well-implemented local functions for Newton and bisection methods

📊 Visual Output Assessment

Please remember to save the figures in the Figures folder!

Figure 1: IS-LM Equilibrium ✅

Layout: Clean plot showing IS and LM curves with equilibrium point
Features: Correctly implements positive branch constraint for IS curve
Styling: Good styling with proper labels, legend, and grid
Saving: No figure saving in code
Quality: Shows correct economic interpretation

Figure 2: Labor Supply h*(σ) ❌

Layout: Stem plot showing h* vs sigma values
Features: Shows incorrect results (h* approaching 1.0 as sigma increases)
Styling: Clean appearance with proper labels and grid
Saving: No figure saving in code
Issue: Results are economically unreasonable - higher risk aversion should decrease labor supply

✅ Suggestions for Improvement

Critical: Fix bisection sign change logic: change fa * fc <= 0 to fa * fc < 0 in line 121
Critical: The <= condition causes convergence to wrong root when fc = 0 exactly
Important: Add exportgraphics() commands to save all figures automatically
Important: Add sign change verification before starting bisection (check if fa * fb < 0)
Style: Consider adding more detailed economic interpretation of results
Verification: Add verification that solutions satisfy original equations Z(h*) ≈ 0

🎯 Summary

Good work with correct IS-LM implementation but critical bisection algorithm error. The student demonstrates strong understanding of numerical methods and implements the IS-LM system perfectly. The labor supply function formulation is mathematically correct, and the damped Newton method is implemented accurately. However, there’s a critical error in the bisection algorithm: using fa * fc <= 0 instead of fa * fc < 0 for sign change detection. This causes the algorithm to converge to the wrong root (h* → 1) instead of the true interior solution. This is a classic numerical analysis error that leads to economically unreasonable results.

Grade Level: ✅ More than 50% Correct (11/15 tasks fully correct, 2/15 incorrect)

Good submission with critical numerical algorithm error ⚠️