🌍 Week 5 Homework — Feedback

Student: Sebastiano Bacchi
Assignment: Solving Nonlinear Equations in Economics

✅ Overall Assessment

Result: ✅ More than 50% Correct

Submission implements all three exercises with good organization and figure saving. However, there are critical errors: (1) Exercise 1 uses only one initial guess instead of two, (2) Exercise 2 bisection has the same critical error affecting multiple students - does not update function values at endpoints, and (3) Exercise 3 bisection also has this error. Exercise 3 damped Newton is correctly implemented. The code structure is organized with proper figure management, but the numerical errors need correction.

🔍 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: `[Y - ...; kY - lambdai - MP]`
1.3	Solve from two initial guesses	❌	Critical Error: Only provides ONE initial guess (x_0=[1000;10]) instead of two
1.4	Plot IS and LM curves	✅	Creates proper IS-LM plot with curves
1.5	Verify positive interest rate	⚠️	Finds one equilibrium but doesn’t explicitly search for two or verify both
2.1	Parameter setup for labor supply	✅	All parameters correctly defined (A instead of a, but functionally equivalent)
2.2	Define Z(h) function	⚠️	Unusual formulation using vector input `[h; sigma]` - works but unconventional
2.3	Loop over sigma values	✅	Correctly loops over sigma ∈ {1,2,3,4,5}
2.4	Solve using Bisection method	❌	Critical Error: Does not update function values at endpoints (line 88: `elseif Z([a;sigma])*Z([c;sigma])` re-evaluates Z([a;sigma]) instead of storing `fa`)
2.5	Plot h*(sigma) and interpret	⚠️	Creates plot of Z(h) for different sigma values (not the required h* vs sigma plot). ✅ Handles corner solutions by manually setting h_star(3:5)=1 when no interior root exists
3.1	Implement Bisection method	❌	Same critical bisection error (line 161: re-evaluates Z([a;Sigma]))
3.2	Implement Damped Newton method	✅	Implements damped Newton with alpha=0.5, analytical derivative, derivative check, and bounds protection
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 format
3.5	Compare and discuss convergence	✅	Provides discussion comparing bisection vs Newton convergence

📈 Technical Implementation

IS-LM System: Correct formulation but only uses one initial guess
Labor Supply Function: Unusual formulation with vector input [h; sigma] - unconventional but works
Bisection Method: ❌ Critical error - does not update function values at endpoints, causing wrong root convergence (same error as Pablo Romanella, Michele Nascia, Chiara Tombolini)
Corner Solutions: ✅ Good - Recognizes that for sigma≥3 no interior solution exists and sets h*=1 manually with economic interpretation (“marginal benefit > marginal cost”)
Damped Newton: Correct implementation with alpha=0.5, analytical derivative, derivative check, and finite value checks
Error Handling: Includes derivative check and finite value checks for Newton method
Figure Management: ✅ Saves figures properly in separate figures folder
Advanced Features: Includes proper derivative safeguards, convergence tracking in table format, and economic analysis of corner solutions

💬 Style & Clarity

Code Quality: Excellent organization with separate code and figures folders
Variable Naming: Clear names (h_star, sigmas, x_star)
Comments: Good economic interpretation and discussion
Output: Uses disp and fprintf appropriately with table display
Organization: Excellent - separates code into parts with clear structure
Documentation: Includes economic interpretation of results

📊 Visual Output Assessment

Figure 1: IS-LM Equilibrium ✅

Layout: Plot with IS and LM curves
Features: Shows both curves
Styling: Appropriate styling with proper labels and legend
Saving: ✅ Saves in separate figures folder
Issue: Only one equilibrium point (missing second guess)

Figure 2: Z(h) for Different Sigma Values ⚠️

Layout: Plot showing Z(h) vs h for different sigma values
Features: Shows Z functions for sigma ∈ {1,2,3,4,5}
Styling: Appropriate styling with proper labels
Saving: ✅ Saves in separate figures folder
Issue: Not the required plot - should plot h*(sigma) vs sigma, not Z(h) vs h

✅ Suggestions for Improvement

CRITICAL: Exercise 1 - provide TWO initial guesses as required

CRITICAL: Fix bisection in Exercises 2 and 3 - must update function values when updating brackets:

fa = Z([a; sigma]); fb = Z([b; sigma]);
if fa * fc < 0
    b = c; fb = fc;  % Update function value
else
    a = c; fa = fc;  % Update function value
end

CRITICAL: Exercise 2 - plot h*(sigma) vs sigma, not Z(h) vs h
Important: Verify both equilibria in Exercise 1 (positive and negative interest rate)
Style: Consider defining Z function more conventionally with sigma as parameter rather than vector input

🎯 Summary

Submission with critical numerical errors. The student demonstrates understanding of numerical methods and implements all three exercises with excellent organization (separate code and figures folders). Exercise 1 IS-LM uses only one initial guess instead of two. Exercise 2 has the critical bisection error of not updating function values at endpoints (same error as Pablo, Michele, Chiara), and the plot shows Z(h) vs h instead of h*(sigma) vs sigma. Exercise 3 has the same bisection error but damped Newton is correctly implemented. The code structure is appropriate with proper figure management and table outputs for Exercise 3.

Grade Level: ✅ More than 50% Correct (10/15 tasks fully correct, 2/15 partially correct, 3/15 incorrect)