Improve cake eating time iteration lecture: fix alpha parameter handling and code style

[jstac]

Summary

This PR improves the cake_eating_time_iter.md lecture with several important fixes:

• Fix alpha parameter inconsistency: Functions f and f_prime now take α as an explicit parameter instead of capturing it from global scope
• Fix spelling and grammatical errors: Corrected hyphenation, spacing, and grammar issues
• Improve code readability: Unpacked model attributes at the beginning of code blocks for cleaner code

Test plan

• Converted to Python using jupytext
• Ran the Python file successfully
• Verified both test cases converge correctly

🤖 Generated with Claude Code

[jstac]

Detailed Changes

1. Fix Alpha Parameter Inconsistency

Problem: The parameter α appeared in the Model definition and create_model function, but when f and f_prime were used, they relied on a global variable captured in their closure rather than the α stored in the model.

Solution:

• Changed f = lambda k: k**α to f = lambda k, α: k**α
• Changed f_prime = lambda k: α * k**(α - 1) to f_prime = lambda k, α: α * k**(α - 1)
• Updated euler_diff to pass α when calling these functions: f(x - c, α) and f_prime(x - c, α)

This ensures the α parameter stored in the Model is actually used, making the code consistent with how other parameters like β, μ, and ν are handled.

2. Spelling and Grammatical Corrections

Fixed several issues:

• Line 123: Removed extra space: "with respect to $x$, and then" (was "with respect to $x$, and then")
• Line 128: Changed "first order condition" to "first-order condition" (compound adjectives should be hyphenated)
• Line 182: Removed extra space and inappropriate period after "that solves"
• Line 197: Removed double space: "The important thing" (was "The important thing")
• Line 506: Changed "first order conditions" to "first-order conditions"
• Line 543: Added article: "Use the same production function" (was "Use same production function")

3. Code Readability Improvements

Replaced repeated model.attribute access with unpacked variables at the beginning of code blocks:

Example (lines 477-481):

# Before:
σ_init = np.copy(model.grid)

# After:
# Unpack
grid = model.grid

σ_init = np.copy(grid)

Example (lines 487-496):

# Before:
ax.plot(model.grid, σ, ...)
ax.plot(model.grid, σ_star(model.grid, model.α, model.β), ...)

# After:
# Unpack
grid, α, β = model.grid, model.α, model.β

ax.plot(grid, σ, ...)
ax.plot(grid, σ_star(grid, α, β), ...)

This makes the code cleaner and more consistent with Python best practices.

Testing

All changes have been tested:

• ✅ Converted to Python using jupytext --to py cake_eating_time_iter.md
• ✅ Ran the Python file successfully
• ✅ Both test cases (log utility and CRRA utility) converge correctly

[github-actions]

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📚 Changed Lecture Pages: cake_eating_time_iter