Refactor McCall lecture: Emphasize persistent and transitory wage shocks (#691)

* Complete Job Search IV lecture: Add utility function and fitted VFI implementation

Updated mccall_fitted_vfi.md to implement the continuous wage offer model with CRRA utility.

Key changes:
- Added CRRA utility function u(c, γ) = (c^(1-γ) - 1)/(1-γ) to mathematical formulation
- Updated Model class to include ρ, ν, and γ parameters
- Implemented Monte Carlo integration for computing conditional expectations (Pv_u)(w)
- Updated Bellman operator T() to use u(w, γ) and u(c, γ)
- Added get_greedy() function for computing optimal policy
- Fixed all model unpacking throughout code
- Implemented compute_expectation() using w' = w^ρ * exp(ν * z) with standard normal draws
- Added Exercise 3: Exploring reservation wage as function of risk aversion γ
- Reformatted text: each sentence on separate line for better version control

Mathematical consistency:
- Code now matches theory where wages and unemployment compensation enter through utility function
- Monte Carlo approximation: (Pv_u)(w) ≈ (1/N) Σ v_u(w^ρ exp(ν z_i))
- Proper JAX implementation with interpolation for fitted value function iteration

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

* Rename and refactor McCall lecture to emphasize persistent and transitory wage shocks

Renamed mccall_correlated.md to mccall_persist_trans.md to better reflect the lecture's focus on the decomposition of wages into persistent and transitory components, distinguishing it from the earlier mccall_model_with_sep_markov.md lecture.

Key changes:
- Updated title to "Job Search V: Persistent and Transitory Wage Shocks"
- Rewrote Overview section to:
  - Emphasize the persistent-transitory decomposition as the key innovation
  - Add references to baseline model (mccall_model) and Job Search III (mccall_model_with_sep_markov)
  - Explain why we return to permanent jobs (to isolate wage dynamics effects)
  - Note use of fitted value function iteration from Job Search IV
- Replaced 'jr' abbreviation with explicit 'jax.random' throughout for clarity
- Refactored draw_τ function:
  - Renamed to draw_duration for clarity
  - Extracted as standalone JIT-compiled function with explicit parameters
  - Prevents unnecessary recompilation when model parameters change
  - compute_unemployment_duration now serves as a clean wrapper
- Simplified JobSearchModel class to Model
- Changed Model instantiation to use positional arguments instead of keyword arguments
- Fixed grammatical errors throughout the text (added missing commas, articles, etc.)
- Updated _toc.yml to reflect filename change

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* fix: syntax for code-cell error

* Convert mccall_persist_trans.md to MyST format

Used jupytext to convert the file from markdown to MyST format:
- Changed format_name from "markdown" to "myst" in YAML frontmatter
- Converted all ```python code blocks to ```{code-cell} ipython directives
- Removed extra "jupyter:" wrapper level in frontmatter
- Updated format_version to match other lecture files (0.13)

This ensures the file is properly processed as a MyST markdown file
consistent with all other lectures in the repository.

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

* Convert mccall_model_with_sep_markov.md to MyST format

Used jupytext to convert Job Search III lecture from markdown to MyST format:
- Changed format_name from "markdown" to "myst" in YAML frontmatter
- Converted all ```python code blocks to ```{code-cell} ipython3 directives
- Removed extra "jupyter:" wrapper level in frontmatter
- Updated format_version to match other lecture files (0.13)

This was the only remaining lecture file in the repository using plain
markdown format instead of MyST. All lectures are now consistent.

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

* Add missing bibliography entries for MaCurdy1982 and Meghir2004

Added two bibliography entries referenced in mccall_persist_trans.md:
- MaCurdy (1982): "The use of time series processes to model the error
  structure of earnings in a longitudinal data analysis"
- Meghir & Pistaferri (2004): "Income variance dynamics and heterogeneity"

These are classic references on wage dynamics in labor economics, used to
support the persistent-transitory decomposition approach in Job Search V.

This fixes the build warnings:
  WARNING: could not find bibtex key "MaCurdy1982"
  WARNING: could not find bibtex key "Meghir2004"

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

* Remove duplicate section in mccall_fitted_vfi.md

Removed a duplicated section that repeated the Bellman operator explanation
and P operator definitions. The section starting with "In the discrete case,
we ended up iterating on the Bellman operator" was appearing twice (lines
97-135 and 140-177), along with a duplicate "### Fitting" header.

Removed the duplicate at lines 140-177, keeping only the first occurrence.

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

* Replace \RR with \mathbb{R} in mccall_fitted_vfi.md

Changed the LaTeX command for the real numbers from \RR to the standard
\mathbb{R} notation for consistency with mathematical typesetting
conventions.

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

* Remove incomplete exercise about mean-preserving spread

Deleted the second exercise (mfv_ex2) which asked about mean-preserving
spread and had an incomplete solution placeholder. The exercise was about
exploring how reservation wage varies with volatility while holding mean
constant.

Renumbered the third exercise (mfv_ex3 about risk aversion) to become
the new second exercise (mfv_ex2).

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---------

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

Author: 3 people

lectures/_static/quant-econ.bib

@@ -2711,3 +2711,25 @@ @article{fischer2024improving
   journal={arXiv preprint arXiv:2410.16076},
   year={2024}
 }
+
+@article{MaCurdy1982,
+  title={The use of time series processes to model the error structure of earnings in a longitudinal data analysis},
+  author={MaCurdy, Thomas E.},
+  journal={Journal of Econometrics},
+  volume={18},
+  number={1},
+  pages={83--114},
+  year={1982},
+  publisher={Elsevier}
+}
+
+@article{Meghir2004,
+  title={Income variance dynamics and heterogeneity},
+  author={Meghir, Costas and Pistaferri, Luigi},
+  journal={Econometrica},
+  volume={72},
+  number={1},
+  pages={1--32},
+  year={2004},
+  publisher={Wiley Online Library}
+}

lectures/_toc.yml

@@ -67,7 +67,7 @@ parts:
   - file: mccall_model_with_separation
   - file: mccall_model_with_sep_markov
   - file: mccall_fitted_vfi
-  - file: mccall_correlated
+  - file: mccall_persist_trans
   - file: career
   - file: jv
   - file: odu

lectures/mccall_fitted_vfi.md

@@ -132,7 +132,7 @@ $$
 
 where $\psi$ is the standard normal density.
 
-Here we are thinking of $v_u$ as a function on all of $\RR_+$.
+Here we are thinking of $v_u$ as a function on all of $\mathbb{R}_+$.
 
 
 ### Fitting
@@ -473,33 +473,6 @@ This makes economic sense: when the value of being unemployed rises (through hig
 ```{exercise}
 :label: mfv_ex2
 
-Let us now consider how the agent responds to an increase in volatility.
-
-To try to understand this, compute the reservation wage when the wage offer distribution is uniform on $(m - s, m + s)$ and $s$ varies.
-
-The idea here is that we are holding the mean constant and spreading the support.
-
-(This is a form of *mean-preserving spread*.)
-
-Use `s_vals = jnp.linspace(1.0, 2.0, 15)` and `m = 2.0`.
-
-State how you expect the reservation wage to vary with $s$.
-
-Now compute it - is this as you expected?
-```
-
-```{solution-start} mfv_ex2
-:class: dropdown
-```
-
-Maybe add an exercise that explores a pure increase in volatility.
-
-```{solution-end}
-```
-
-```{exercise}
-:label: mfv_ex3
-
 Create a plot that shows how the reservation wage changes with the risk aversion parameter $\gamma$.
 
 Use `γ_vals = jnp.linspace(1.2, 2.5, 15)` and keep all other parameters at their default values.
@@ -508,7 +481,7 @@ How do you expect the reservation wage to vary with $\gamma$? Why?
 
 ```
 
-```{solution-start} mfv_ex3
+```{solution-start} mfv_ex2
 :class: dropdown
 ```
 

lectures/mccall_model_with_sep_markov.md

@@ -1,16 +1,14 @@
 ---
-jupyter:
-  jupytext:
-    default_lexer: ipython3
-    text_representation:
-      extension: .md
-      format_name: markdown
-      format_version: '1.3'
-      jupytext_version: 1.17.2
-  kernelspec:
-    display_name: Python 3 (ipykernel)
-    language: python
-    name: python3
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.17.2
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
 ---
 
 (mccall_with_sep_markov)=
@@ -22,7 +20,7 @@ jupyter:
 </div>
 ```
 
-
++++
 
 # Job Search III: Search with Separation and Markov Wages
 
@@ -104,6 +102,7 @@ $$
     \right]
 $$
 
++++
 
 ## Computational Approach
 
@@ -121,6 +120,7 @@ $$
 
 2. Substitute into the unemployed agent's Bellman equation to get:
 
++++
 
 $$
     v_u(w) = 
@@ -137,6 +137,7 @@ $$
 
 The optimal policy turns out to be a reservation wage strategy: accept all wages above some threshold.
 
++++
 
 ## Code
 
@@ -334,6 +335,7 @@ plt.show()
 
 Can you provide an intuitive economic story behind the outcome that you see in this figure?
 
++++
 
 ## Employment Simulation
 
@@ -492,7 +494,7 @@ This is because she uses the wage $w$ from her last job to draw a new wage offer
 via $P(w, \cdot)$, and positive correlation means that a high current $w$ is
 often leads a high new draw.
 
-
++++
 
 ## The Ergodic Property
 
@@ -545,6 +547,7 @@ As a result, we can study steady-state unemployment either by:
 
 Often the second approach is better for our purposes, since it's easier to parallelize.
 
++++
 
 ## Cross-Sectional Analysis