Fix bugs and improve JS III

[jstac]

Summary

This PR fixes several bugs in the McCall model with separation lecture and improves the pedagogical structure:

• Bug fixes: Fixed syntax errors including extra parenthesis, undefined variable w (should be w_vals), and incorrect use of jnp.max for scalar comparison
• Pedagogical improvement: Added a first-pass solution method that directly iterates on both value functions before introducing the more efficient approach
• Verification test: Added a test comparing both solution methods to verify they produce the same reservation wage
• Exercise enhancement: Provided specific guidance on which c values to use in the exercise

Test plan

• Converted the markdown file to Python using jupytext
• Ran the Python file successfully without errors
• Verified both solution methods produce identical reservation wages (difference: 0.00e+00)
• Confirmed all plots and simulations run correctly

🤖 Generated with Claude Code

[jstac]

Bug Fix: Syntax Errors

Fixed several critical syntax errors that prevented the code from running:

1. Line 269: Removed extra closing parenthesis in T_e function
2. Line 269: Changed u(w, γ) to u(w_vals, γ) - the variable w was undefined
3. Line 295: Changed jnp.max(error_u, error_e) to jnp.maximum(error_u, error_e) for proper scalar comparison
4. Line 323-325: Added missing initial conditions v_u_init and v_e_init when calling solve_model_first_pass

All these fixes are necessary for the code to run successfully.

[jstac]

Pedagogical Improvement: Two-Stage Solution Approach

The lecture now presents two solution methods in sequence:

First Pass (Lines 237-326):

• Introduces a straightforward approach that directly iterates on both v_u and v_e
• Implements separate Bellman operators T_u and T_e
• Easier for students to understand as it directly follows the theory
• Includes a road test with plots showing the value functions

Improved Efficiency (Lines 351+):

• Shows how to eliminate v_e and iterate only on v_u
• Demonstrates optimization techniques in computational economics
• Uses JAX compilation for better performance

Verification (Lines 467-471):
Added a test that confirms both methods produce identical results:

Reservation wage (first method):  1.365155
Reservation wage (second method): 1.365155
Difference: 0.00e+00

This helps students understand that efficiency improvements don't change the solution, just the computation speed.

[jstac]

Exercise Enhancement

The exercise at the end now provides much clearer guidance for students (lines 919-922):

Before:

Create a plot that investigates more carefully how the steady state cross-sectional unemployment rate changes with unemployment compensation.

After:

Create a plot that investigates more carefully how the steady state cross-sectional unemployment rate changes with unemployment compensation.

Try a range of values for unemployment compensation c, such as c = 0.2, 0.4, 0.6, 0.8, 1.0.
For each value, compute the steady-state cross-sectional unemployment rate and plot it against c.

What relationship do you observe between unemployment compensation and the unemployment rate?

This gives students:

• Specific values to try (removes ambiguity)
• Clear instructions on what to compute
• A thought-provoking question to guide their analysis

These hints match the solution that was already provided, making the exercise more accessible.

[github-actions]

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📚 Changed Lecture Pages: mccall_model_with_sep_markov