Standardize notation: Use capitals for random variables in Job Search I-V

[jstac]

Summary

Updated notation in Job Search lectures I-V to use uppercase letters for random variables and lowercase for realizations, improving mathematical consistency and clarity.

Changes by lecture:

• Job Search I (mccall_model.md): w_t → W_t for wage offer sequences
• Job Search II (mccall_model_with_separation.md): w_t → W_t for wage offers
• Job Search III (mccall_model_with_sep_markov.md): (s_t, w_t) → (S_t, W_t) for state variables
• Job Search IV (mccall_fitted_vfi.md): Already using correct notation (no changes needed)
• Job Search V (mccall_persist_trans.md): w_t → W_t, z_t → Z_t, y_t → Y_t for wage components

Notation convention:

• Uppercase for random variables (e.g., W_t, Z_t, S_t)
• Lowercase for:
  • Realizations in sums (e.g., sum_w v(w) q(w))
  • Standard normal shocks (e.g., zeta_t, epsilon_t)

This standardization makes it clearer when we're referring to random variables versus their realizations or specific values.

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[jstac]

Summary of Changes

This PR improves notation consistency and adds substantial enhancements to the Job Search lecture series (I-V).

1. Standardized Random Variable Notation (All Lectures)

Changed random variables to uppercase, realizations to lowercase:

• Job Search I: w_t → W_t for wage offer sequences
• Job Search II: w_t → W_t for wage offers
• Job Search III: (s_t, w_t) → (S_t, W_t) for state variables
• Job Search IV: w → W_t in model description, z_i → Z_i in Monte Carlo formula
• Job Search V: w_t → W_t, z_t → Z_t, y_t → Y_t for wage components

Lowercase remains for:

• Realizations in sums (e.g., sum_w v(w) q(w))
• Standard normal shocks (e.g., ζ_t, ε_t)

2. Clarified Utility Function Notation (Lectures II-V)

Changed utility function parameter from c to x to distinguish:

• The consumption argument of utility function u(x)
• The unemployment compensation parameter c

Updated:

• Job Search II: def u(c, γ) → def u(x, γ)
• Job Search III: def u(c, γ) → def u(x, γ)
• Job Search IV: def u(c, γ) → def u(x, γ)
• Job Search V: u(c) = ln(c) → u(x) = ln(x)

3. Major Enhancements to Job Search IV

Improved clarity and explanations:

• Added detailed explanation of the Monte Carlo integral formula for (Pv_u)(w)
• Explained the derivation from W_t = exp(X_t) to w^ρ exp(ν Z_{t+1})
• Changed "From the equation above" → "Recall that we have"
• Added description before Bellman operator about using Monte Carlo integration

Code improvements:

• Renamed variables for clarity: accept → v_e, reject → continuation_values
• Changed Model constructor from named to positional arguments
• Standardized naming: h_star → h, w_star → w_bar
• Added missing import: from functools import partial

Updated visualizations:

• Simplified main plot to match Job Search III style
• Show only continuation value h and employment value v_e
• Added explanation about reservation wage intersection

Added complete Simulation section:

• Implemented update_agent function with proper job separation logic
• Added simulate_employment_path for single agent simulation
• Created three-panel plot showing:
  • Employment status over time
  • Wage path with reservation wage
  • Cumulative unemployment rate
• Properly handles Markov wage dynamics with separation rate α

Added Cross-Sectional Analysis subsection:

• Vectorized update_agent using jax.vmap for parallel simulation
• JIT-compiled _simulate_cross_section_compiled using lax.fori_loop
• simulate_cross_section function for many-agent simulation
• plot_cross_sectional_unemployment for histogram visualization
• Comparison of time-average vs cross-sectional unemployment rates

Verification:

• Converted to Python and ran successfully
• Unemployment rates now realistic (~23-29%) showing proper cycling between employment and unemployment

All changes maintain consistency with the lecture series style and improve mathematical clarity throughout.

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[jstac]

Touching many lectures and looks good so i'll merge.