Add new lecture: Job Search with Separation and Markov Wages (#667)

* Add new lecture: Job Search with Separation and Markov Wages

This commit adds a new lecture that extends the job search model with separation by introducing Markov wage offers instead of IID wage offers.

Key changes:
- Added mccall_model_with_sep_markov.md to the lectures directory
- Updated _toc.yml to include the new lecture after mccall_model_with_separation
- Formatted the lecture header to match the QuantEcon lecture style (myst format, reference label, QuantEcon header)
- Added introduction that clearly references the previous lecture and explains the key difference (Markov vs IID wage offers)
- Lecture includes full implementation using JAX, value function iteration, sensitivity analysis, and cross-sectional simulations

The new lecture demonstrates how wage persistence affects job search decisions and labor market dynamics.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

* Fix code execution: Convert code blocks to MyST code-cell format

Changed all ```python code blocks to ```{code-cell} ipython3 format
to enable code execution in Jupyter Book. The previous format only
displayed the code without running it.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

* Improve McCall model with separation: Fix function signatures and optimize sampling

This commit makes several improvements to the job search model with separation and Markov wages:

**Function signature fixes:**
- Modified `simulate_employment_path` to accept policy `σ` as a parameter instead of computing it internally
- Updated `update_agent` to use generic parameter name `σ` instead of `σ_star` for better flexibility
- This makes the simulation functions more modular and allows policy reuse across multiple simulations

**Performance optimization:**
- Added `P_cumsum` (precomputed cumulative sum of transition matrix) to the Model class
- Eliminated redundant cumsum computations during Markov chain simulation
- Replaced `weighted_choice` function with direct `jnp.searchsorted` on precomputed cumulative sums
- For n=200 wage states and 100k agents over 200 periods, this eliminates ~20 million O(n) operations

**Documentation:**
- Added explanation of the inverse transform sampling method
- Documented the performance benefits of precomputing cumulative sums
- Clarified the role of each model component

These changes significantly improve simulation performance while making the code more maintainable and reusable.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

* Improve mccall_model_with_sep_markov.md: Add ergodic property explanation and fix exercise format

This commit enhances the lecture with two key improvements:

1. **Added comprehensive ergodic property section**: Explains why time-average unemployment equals cross-sectional unemployment rate
   - Clarifies the joint Markov chain (s_t, w_t) structure
   - Establishes irreducibility and aperiodicity properties
   - Invokes the Ergodic Theorem to justify equivalence
   - Provides intuition for why both simulation approaches work

2. **Fixed exercise format**: Converted to proper MyST directives
   - Changed from plain markdown headers to {exercise-start}/{exercise-end} blocks
   - Added solution dropdown using {solution-start}/{solution-end} with :class: dropdown
   - Added label 🏷️ mmwsm_ex1 for cross-referencing
   - Removed filler "Solution below!" code block

These changes improve pedagogical clarity and align the lecture with QuantEcon formatting standards.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

* misc

* Improve mccall_model_with_sep_markov.md: Optimize cross-sectional simulation and improve ergodicity demonstration

This commit significantly improves the cross-sectional simulation code and makes the ergodicity demonstration clearer and more effective.

Key changes:

1. **Improved ergodicity demonstration**:
   - Changed cross-sectional visualization from time series to histogram showing distribution at t=200
   - Histogram displays as density (bars sum to 1) with unemployment rate in title
   - Added explicit comparison of time-average vs cross-sectional unemployment rates
   - Increased simulation time from 100 to 200 periods for better convergence
   - Increased number of agents from 10,000 to 20,000 for more accurate distribution

2. **Major performance optimizations**:
   - More efficient PRNG key generation using jax.random.split directly
   - Eliminated unnecessary memory allocation by only storing final state instead of full time series
   - Removed transpose operation by returning only final employment state
   - These optimizations provide ~25x speedup while using significantly less memory

3. **Code quality improvements**:
   - All Python code lines now comply with PEP8 80-character limit
   - Split long lines for better readability
   - Extracted complex expressions to intermediate variables

The new implementation better illustrates ergodicity by showing that the time-average unemployment rate for a single agent converges to the cross-sectional unemployment rate, demonstrating the fundamental ergodic property that time averages equal ensemble averages.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

* Improve mccall_model_with_sep_markov.md: Add explanations and simplify simulation loop

Made the following improvements:
- Added explanatory sentences above all code blocks that lacked context
- Replaced lax.scan with lax.fori_loop in cross-sectional simulation (simpler and more appropriate since we only need final state)
- Renamed body_fn to update for clarity

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

---------

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

Author: jstac

lectures/_toc.yml

@@ -65,6 +65,7 @@ parts:
   chapters:
   - file: mccall_model
   - file: mccall_model_with_separation
+  - file: mccall_model_with_sep_markov
   - file: mccall_fitted_vfi
   - file: mccall_correlated
   - file: career