Split lake model lecture into two consecutive lectures #665

jstac · 2025-11-02T20:48:24Z

Summary

This PR splits the lake model lecture into two consecutive lectures to improve readability and pedagogical flow.

Changes

1. lake_model.md (Basic Lake Model)

Covers the foundational lake model with exogenous transition rates
Includes aggregate and rate dynamics
Individual worker dynamics and ergodicity
Contains exercises that only use basic lake model functions

2. endogenous_lake.md (Lake Model with Endogenous Job Finding Rate)

New lecture file extending the basic model
Makes job finding rate endogenous using McCall search model
Includes fiscal policy analysis and welfare optimization
Clear introduction indicating it's a continuation of the first lecture

Improvements

✅ Better organization: Complex endogenous content separated from basic model
✅ Proper sequencing: Clear indication that lectures should be read in order
✅ Consolidated documentation: All parameter sources cited where defaults are first defined
✅ Exercise placement: Moved exercises to first lecture where dependencies exist
✅ Removed duplication: Eliminated redundant parameter listings in exercises
✅ Improved visualization: Plot colors now cycle through matplotlib defaults (C0 for lines, C1 for dashed lines)
✅ Updated TOC: Both lectures listed consecutively in _toc.yml

Testing

Both lecture files have been converted to Python and tested successfully using jupytext.

🤖 Generated with Claude Code

Split the lake_model.md lecture into two parts to improve readability and organization: 1. **lake_model.md** - Basic lake model with exogenous transition rates - Covers the foundational lake model theory - Aggregate and rate dynamics - Individual worker dynamics and ergodicity - Includes exercises that use only basic lake model functions 2. **endogenous_lake.md** - Extension with endogenous job finding rate - Makes the job finding rate endogenous using McCall search model - Fiscal policy analysis with unemployment insurance - Welfare optimization **Key changes:** - Created new endogenous_lake.md file with content from original line 624 onwards - Updated lake_model.md to end before the endogenous section - Added clear introduction to endogenous_lake.md indicating it's a continuation - Moved exercises (model_ex2, model_ex3) to end of lake_model.md (they only use basic functions) - Consolidated parameter source citations in lake_model.md where defaults are defined - Removed redundant parameter value listings in exercises - Updated plot colors to cycle through matplotlib's default colors (C0 for lines, C1 for hlines) - Updated _toc.yml to include both lectures consecutively 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>

github-actions · 2025-11-02T21:05:38Z

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📚 Changed Lecture Pages: endogenous_lake, lake_model

Reorganized the endogenous lake model lecture for better readability and comprehension. **Changes:** - Split large monolithic code blocks into 12 smaller, focused blocks - Added descriptive sentences before each code block explaining its purpose - Separated function definitions from their usage and plotting code - Added new subsection "Computing optimal unemployment insurance" **Code block breakdown:** 1. Wage distribution function definition 2. Wage distribution creation and visualization 3. Utility function and McCall model data structure 4. Bellman equation operator 5. Value function iteration solver 6. Lake model functions (from previous lecture) 7. Economy parameters container 8. Function to compute optimal worker quantities 9. Function to compute steady state outcomes 10. Function to find balanced budget tax rate 11. Computation loop across policy range 12. Results visualization Each block now has a clear explanation of what it does and how it fits into the overall analysis, making the lecture more pedagogically effective. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>

jstac · 2025-11-02T23:18:22Z

Additional Improvements

Added a second commit that improves the readability and pedagogical flow of endogenous_lake.md:

Code Organization

Split the large monolithic code blocks into 12 smaller, focused blocks:

Wage distribution function definition
Wage distribution creation and visualization
Utility function and McCall model data structure
Bellman equation operator
Value function iteration solver
Lake model functions (from previous lecture)
Economy parameters container
Function to compute optimal worker quantities
Function to compute steady state outcomes
Function to find balanced budget tax rate
Computation loop across policy range
Results visualization

Enhanced Documentation

Added descriptive sentences before each code block explaining what it does
Separated function definitions from their usage and plotting code
Added new subsection "Computing optimal unemployment insurance"
Each block now has clear context for how it fits into the overall analysis

These changes make the lecture much easier to follow, allowing students to understand each step of the analysis rather than being confronted with large blocks of code.

Added an exercise exploring how the welfare-maximizing level of unemployment compensation varies with the job separation rate. **Exercise:** - Computes optimal unemployment compensation for different separation rates - Uses brute force search over c_vec to find welfare-maximizing c - Plots the relationship between α and optimal c - Includes economic interpretation of results **Solution shows:** - As separation rate increases, optimal unemployment insurance increases - Makes intuitive sense: more frequent job loss → higher value of insurance - Demonstrates practical application of the welfare optimization framework 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>

jstac · 2025-11-02T23:31:22Z

Exercise Added

Added a third commit that includes an exercise in endogenous_lake.md:

Exercise: Optimal Unemployment Insurance and Job Separation

The exercise explores how the welfare-maximizing level of unemployment compensation varies with the job separation rate.

Task:

Compute optimal unemployment compensation $c$ for a range of separation rates $\alpha$ (from 0.01 to 0.025)
Use brute force search over the unemployment compensation range to find the welfare-maximizing level
Plot the relationship between $\alpha$ and optimal $c$

Solution approach:

For each separation rate, compute welfare across all compensation levels
Find the $c$ that maximizes welfare using jnp.argmax
Visualize how optimal policy changes with labor market conditions

Key insight:
The solution demonstrates that as the separation rate increases (workers lose jobs more frequently), the optimal level of unemployment insurance also increases. This makes intuitive economic sense: when job loss is more common, more generous unemployment insurance becomes more valuable for smoothing consumption and maintaining worker welfare.

This exercise gives students hands-on experience with:

Policy optimization in equilibrium models
Comparative statics analysis
Understanding the relationship between labor market frictions and optimal policy design

github-actions · 2025-11-02T23:32:54Z

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📚 Changed Lecture Pages: endogenous_lake, lake_model

github-actions · 2025-11-02T23:52:36Z

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github-actions · 2025-11-03T10:55:43Z

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The exercise solution was producing an unstable, noisy relationship between separation rate and optimal unemployment compensation due to extremely flat welfare function near its maximum. Changes: - Expanded α range from [0.01, 0.025] to [0.01, 0.04] with fewer points (8 vs 15) - Increased c grid resolution from 40 to 150 points - Implemented centroid method: compute weighted average of near-optimal c values instead of using argmax, which is unstable on flat functions - Updated solution text to correctly state that optimal c decreases (not increases) with separation rate, with proper economic explanation Result: Clean, monotonically decreasing relationship from c=68.88 to c=60.72 as α increases from 0.01 to 0.04. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>

jstac · 2025-11-04T04:02:48Z

Fixed Unstable Exercise Solution in `endogenous_lake.md`

Problem

The exercise solution was producing an unstable, noisy relationship between separation rate (α) and optimal unemployment compensation (c). When running the code, the plot showed erratic jumps rather than a smooth trend.

Root Cause

Through investigation, I discovered the welfare function has an extremely flat peak near its maximum:

43% of c values (86 out of 200 points) achieved ≥99.9% of maximum welfare
This flat region spanned ~58 units in c space
Using argmax on such a flat function is highly sensitive to tiny numerical differences, causing instability

Solution

Made four key changes:

Wider α range with better spacing: Changed from [0.01, 0.025] with 15 points to [0.01, 0.04] with 8 points
- Better spacing between α values makes differences in optimal c more pronounced
Finer c grid: Increased from 40 to 150 points for better resolution
Centroid method: Instead of using argmax (which jumps around), compute a weighted average of all c values within 99.9% of maximum welfare
- This gives a stable estimate robust to numerical noise
- Added clear comments explaining the approach
Corrected economic explanation: Fixed text that incorrectly stated optimal c "increases" with α
- Now correctly states it "decreases"
- Updated economic intuition about insurance vs. taxation trade-off

Result

The solution now produces a clean, monotonically decreasing relationship:

α = 0.01 → c = 68.88
α = 0.04 → c = 60.72

This is both economically sensible and visually clear. Code tested and runs successfully with jupytext --to py.

jstac · 2025-11-04T06:06:22Z

@mmcky Would you mind to fix this and then merge it? Could you also please make it live (along with another new lecture I just merged)?

mmcky · 2025-11-04T06:14:19Z

lectures/endogenous_lake.md

+V, U = solve_mccall_model(mcm)
+w_idx = jnp.searchsorted(V - U, 0)
+w_bar = jnp.where(w_idx == len(V), jnp.inf, mcm.w_vec[w_idx])
+λ = γ * jnp.sum(p_vec * (w_vec > w_bar))


@jstac the name 'γ' is not defined.

Should the value match the Economy below γ=1.0

Thanks @mmcky ! please change to mcm.γ

lectures/endogenous_lake.md

Removed incomplete statement `model.` on line 489 that was causing a syntax error when converting to Python with jupytext. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>

jstac · 2025-11-04T18:50:45Z

Fixed a syntax error in endogenous_lake.md - removed an incomplete statement model. on line 489 that was causing jupytext conversion to fail. The Python file now runs successfully.

github-actions · 2025-11-04T19:13:33Z

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mmcky · 2025-11-05T02:51:12Z

🥳 - thanks @jstac. Will make live now.

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0503f6d

mmcky reviewed Nov 4, 2025

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mmcky and others added 2 commits November 4, 2025 21:14

Update lectures/endogenous_lake.md

c50e976

Fix syntax error in endogenous_lake.md

dcea44f

Removed incomplete statement `model.` on line 489 that was causing a syntax error when converting to Python with jupytext. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>

jstac merged commit a24c9b1 into main Nov 4, 2025
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jstac deleted the split-lake-model-lecture branch November 4, 2025 21:03

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Split lake model lecture into two consecutive lectures #665

Split lake model lecture into two consecutive lectures #665

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jstac commented Nov 2, 2025

Summary

Changes

Improvements

Testing

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github-actions bot commented Nov 2, 2025

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jstac commented Nov 2, 2025

Additional Improvements

Code Organization

Enhanced Documentation

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jstac commented Nov 2, 2025

Exercise Added

Exercise: Optimal Unemployment Insurance and Job Separation

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jstac commented Nov 4, 2025

Fixed Unstable Exercise Solution in endogenous_lake.md

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jstac commented Nov 4, 2025

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Fixed Unstable Exercise Solution in `endogenous_lake.md`