Standardize notation: Use capitals for random variables in Job Search I-V (#704)

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

Updated notation in Job Search lectures I-V to use uppercase letters for
random variables and lowercase for realizations:

- Job Search I (mccall_model.md): w_t → W_t in 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
- Job Search V (mccall_persist_trans.md): w_t → W_t, z_t → Z_t, y_t → Y_t
  for wage components

Lowercase notation remains for realizations in sums (e.g., sum_w v(w) q(w))
and for standard normal shocks (zeta_t, epsilon_t).

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Co-Authored-By: Claude <noreply@anthropic.com>

* Clarify utility function notation to avoid confusion with unemployment compensation

Changed utility function parameter from 'c' to 'x' to distinguish between:
- The consumption argument of the utility function u(x)
- The unemployment compensation parameter c

This prevents confusion when we write expressions like u(c), where c is
the specific value of unemployment compensation being passed to the
utility function.

Updated in:
- 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)

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Co-Authored-By: Claude <noreply@anthropic.com>

* Improve Job Search IV: Add explanations, simulation, and cross-sectional analysis

Major improvements to Job Search IV (Fitted Value Function Iteration):

1. Notation and clarity:
   - Changed utility function from u(c, γ) to u(x, γ) to avoid confusion with
     unemployment compensation parameter c
   - Fixed random variable notation: w → W_t in model description
   - Changed z_i → Z_i in Monte Carlo formula
   - Updated Model constructor from named to positional arguments
   - Renamed variables in T and get_greedy: accept → v_e, reject → continuation_values
   - Standardized naming: h_star → h, w_star → w_bar

2. Added explanations:
   - Explained the explicit form of (Pv_u)(w) with derivation
   - Changed "From the equation above" to "Recall that we have"
   - Added explanation before Bellman operator about Monte Carlo integration

3. Updated plot styling:
   - Simplified plot to match Job Search III style
   - Show only continuation value h and employment value v_e
   - Updated labels and explanation text

4. Added Simulation section:
   - Implemented update_agent function with job separation logic
   - Added simulate_employment_path function
   - Created three-panel plot showing employment status, wage path, and
     cumulative unemployment rate
   - Properly handles Markov wage dynamics with separation

5. Added Cross-Sectional Analysis subsection:
   - Vectorized update function for parallel agent simulation
   - JIT-compiled simulation using lax.fori_loop
   - Cross-sectional unemployment histogram
   - Comparison of time-average vs cross-sectional unemployment rates

6. Added missing import: from functools import partial

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Co-Authored-By: Claude <noreply@anthropic.com>

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Co-authored-by: Claude <noreply@anthropic.com>

Author: jstac