Add transient income innovation to IFP model and wealth inequality analysis (#745)

* teaked timing discussion in ifp_egm

* Add transient income innovation to IFP model and wealth inequality analysis

This commit modifies the income fluctuation problem (IFP) model to include
a transient IID income shock, following the specification in ifp_advanced.

Changes to ifp_egm.md:

1. Income process modification:
   - Changed from Y_t = exp(Z_t) to Y_t = exp(a_y * η_t + Z_t * b_y)
   - Added IID shock η_t ~ N(0,1) to create transient income fluctuations
   - Updated Euler equation to integrate over η shocks using Monte Carlo
   - Kept interest rate R constant throughout (as specified)

2. Implementation updates:
   - Added parameters a_y=0.2, b_y=0.5, shock_draw_size=100
   - Updated NumPy implementation with numba.jit optimization
   - Updated JAX implementation with separate utility functions
   - Modified simulation code to draw η shocks during dynamics
   - Updated all code cells to unpack new IFP parameters

3. New wealth inequality analysis section:
   - Added Gini coefficient computation
   - Added top 1% wealth share computation
   - Analyzed how inequality varies with interest rate r
   - Tested 12 interest rate values from 0 to 0.015
   - Generated plots showing Gini and top 1% vs interest rate

4. Performance optimization:
   - Added @numba.jit decorators to u_prime, u_prime_inv, Y, and K_numpy
   - Created separate JAX versions to avoid numba/JAX conflicts

Changes to ifp_advanced.md:

- Updated shock_draw_size from 50 to 100 in both Numba and JAX implementations

The modifications maintain consistency with ifp_advanced while keeping R
constant in ifp_egm. The transient income shock creates more realistic
wealth inequality through increased precautionary savings.

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

* Refactor ifp_egm: use JAX throughout and improve code organization

- Convert Gini coefficient and top share calculations from NumPy to JAX
- Embed utility functions (u_prime, u_prime_inv) and income function (y)
  inside K_numpy and K operators to simplify function signatures
- Introduce z_prime variable for better readability in nested loops
- Fix y_bar(k) to correctly implement mathematical definition of expected
  labor income conditional on current state
- Use vmap for vectorization in y_bar computation
- Remove redundant y_mean vector in favor of direct y_bar(k) calls

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

* Fix simulate_household to use embedded y function

- Add local y function inside simulate_household to replace removed y_jax
- Maintains consistency with refactoring pattern used in K_numpy and K
- All tests pass successfully

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

* Improve ifp_egm lecture: add JAX dependency, fix grammar, enhance documentation

- Add jax to pip install requirements
- Improve y_bar function docstring with clearer mathematical notation
- Fix grammar and consistency in introduction
- Add spacing in K_numpy function for readability

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

---------

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

Author: jstac

lectures/ifp_advanced.md

@@ -376,7 +376,7 @@ class IFP:
                  b_r=0.0,
                  a_y=0.2,
                  b_y=0.5,
-                 shock_draw_size=50,
+                 shock_draw_size=100,
                  grid_max=10,
                  grid_size=100,
                  seed=1234):
@@ -665,7 +665,7 @@ def create_ifp_jax(γ=1.5,
                    b_r=0.0,
                    a_y=0.2,
                    b_y=0.5,
-                   shock_draw_size=50,
+                   shock_draw_size=100,
                    grid_max=10,
                    grid_size=100,
                    seed=1234):