Update Job Search III: Add utility function to mathematical formulation

[jstac]

Summary

This PR updates the Job Search III lecture (mccall_model_with_sep_markov.md) to properly apply the CRRA utility function throughout the code implementation.

Key changes:

• Added utility function transformation u(c, γ) = (c^(1-γ) - 1)/(1-γ) to all value function computations
• Updated Bellman operator T() to use u(w_vals, γ) and u(c, γ) instead of raw values
• Modified get_greedy() policy function to apply utility transformation
• Updated plotting code to compute utility-transformed accept/reject values
• Added γ parameter to model unpacking in all relevant functions
• Fixed syntax error (missing comma) in create_js_with_sep_model() parameter list

Benefits:

• Code now correctly matches the mathematical formulation in the Bellman equations
• Wages and unemployment compensation properly enter value functions through utility function
• Maintains consistency between mathematical theory and implementation

Test plan

• Tested mccall_model_with_sep_markov.py script runs successfully
• Converted back to markdown using jupytext
• All code examples maintain correct functionality
• Mathematical formulation and code implementation now aligned

🤖 Generated with Claude Code

[jstac]

Implementation Details

Updated the lecture to ensure mathematical consistency between the theory and code:

Mathematical Formulation (Bellman Equations)

The lecture correctly specifies that wages and unemployment compensation enter through a utility function:

v_e(w) = u(w) + β[α(Pv_u)(w) + (1-α)v_e(w)]
v_u(w) = max{v_e(w), u(c) + β(Pv_u)(w)}

Code Changes

Previously, the code was using raw wage values w and c directly. Now it properly applies the CRRA utility function:

Before:

accept = d * (w_vals + α * β * P @ v)
reject = c + β * P @ v

After:

accept = d * (u(w_vals, γ) + α * β * P @ v)
reject = u(c, γ) + β * P @ v

This ensures the code matches the theoretical model, where risk aversion (controlled by γ) affects decision-making through the utility function rather than using linear utility implicitly.

[jstac]

Additional Update: Job Search IV (Fitted VFI) Lecture

I've now completed the Job Search IV: Fitted Value Function Iteration lecture (mccall_fitted_vfi.md) with the same utility function approach.

Mathematical Updates

Added CRRA Utility to Bellman Operator:
The continuous wage case now correctly uses the utility function in the Bellman operator:

(Tv_u)(w) = max{
    (1/(1-β(1-α))) * (u(w) + αβ(Pv_u)(w)),
    u(c) + β(Pv_u)(w)
}

Monte Carlo Integration:
Implemented the conditional expectation using Monte Carlo with standard normal draws:

(Pv_u)(w) = ∫ v_u(w^ρ exp(ν z)) ψ(z) dz
          ≈ (1/N) Σ v_u(w^ρ exp(ν z_i))

Code Implementation

1. Utility function: Added u(c, γ) with CRRA form
2. Model structure: Extended to include ρ, ν, γ parameters and z_draws for Monte Carlo
3. Bellman operator T(): Now uses u(w_grid, γ) and u(c, γ) throughout
4. Expectation computation: Implemented compute_expectation(w) that:
   • Generates next period wages: w_next = w**ρ * jnp.exp(ν * z_draws)
   • Interpolates value function at these points
   • Returns the mean over Monte Carlo draws
5. Policy function: Added get_greedy() using utility transformation
6. New exercise: Shows how reservation wage decreases with risk aversion γ

Key Features

• Fitted VFI: Combines value function iteration with linear interpolation
• JAX implementation: Fully compatible with JAX's JIT compilation
• Mathematical consistency: Code matches theory where utility mediates all consumption values
• Clean formatting: Each sentence on separate line for better git diffs

The lecture now provides a complete treatment of continuous state space job search with risk aversion, demonstrating how fitted value function iteration handles infinite-dimensional problems through interpolation and Monte Carlo integration.

[jstac]

@mmcky Can you please trigger a rebuild of this when you've merged your changes and the system is stable again?

[mmcky]

@jstac the latex build is reporting a missing equation reference

/home/runner/_work/lecture-python.myst/lecture-python.myst/lectures/mccall_fitted_vfi.md:146: WARNING: equation not found: bell2mcmc

• Should this reference the bellman equation in VFI section?

[jstac]

Ready to Merge

This PR is ready to merge. I've fixed the build error that was preventing the documentation from building.

Build Fix Applied

Fixed missing equation reference in mccall_fitted_vfi.md

Error: WARNING: equation not found: bell2mcmc

Fix: Added the label :label: bell2mcmc to the Bellman operator equation and converted it from $$ delimiters to the {math} directive to support cross-references.

Merge Strategy

This PR should be merged before PR #691. PR #691 builds on top of this PR and includes:

• The new Job Search V lecture (persistent/transitory wage shocks)
• Additional bibliography entries
• Table of contents updates

After this PR is merged, PR #691 will need a simple rebase onto main, which should go smoothly with no conflicts. ✅