[rational_expectations] Update content and styling

[xuanguang-li]

This pull request makes a series of stylistic and clarity improvements to the lectures/rational_expectations.md file. There are no code changes, as the control problem is solved using quantecon.LQ, for which numpy works well.

Key changes:

Mathematical and technical clarifications:

• Clarified the meaning of $Y_{t+1}$ in different places.
• Updated the Euler equation to use the belief function $H$ consistently, and added an equivalent form for clarity.
• Added an explanation for the failure of contractivity using Blackwell's condition, improving conceptual understanding.
• Clarified and corrected the law of motion for the case of a unit measure of firms in the first exercise.

Formatting and reference improvements:

• Converted links to internal documentation references
• Added a proper definition block.
• Changed all section and subsection headings to sentence case

[github-actions]

📖 Netlify Preview Ready!

Preview URL: https://pr-764--sunny-cactus-210e3e.netlify.app (f2c363d)

📚 Changed Lecture Pages: rational_expectations

[jstac]

🤖 Status note for a future session — from a maintainer investigation on 2026-07-08 into why open-PR previews 404. Context only, not instructions.

Netlify preview: https://pr-764--sunny-cactus-210e3e.netlify.app/ currently returns 404.

Why previews are down (repo-wide findings)

1. This branch is stale — 94 commits behind main. A preview build compiles the whole site from this branch. This branch's lectures/house_auction.md still has unpinned !pip install prettytable, which now breaks on a wcwidth incompatibility. main fixed this on 2026-06-28 by pinning prettytable<3.18 (#939). This alone fails any rebuild of this branch until it's updated to main.

2. The arviz failure was a red herring — do NOT pin arviz or rewrite plotting. A 2026-07-07 rebuild also failed in ar1_bayes/ar1_turningpts with an arviz_plots figsize ValueError. That was a transient bug in an intermediate arviz-plots 1.x release, already fixed in arviz 1.2.0. Verified locally on a clean latest-stack venv: the real az.plot_trace(trace) cell (pymc + numpyro InferenceData) runs green. The lectures use only 1.x-compatible arviz APIs (plot_trace, summary, from_numpyro, compare).

Recommended first step for this PR

Update this branch to main (merge or rebase — pulls in #939 plus ~94 other commits), then let CI rebuild. On today's latest libraries the site builds clean, so the preview should return. house_auction is the known blocker; updating also picks up other since-merged fixes — rebuild and address any remaining per-lecture failures. Verify with:

curl -sI https://pr-764--sunny-cactus-210e3e.netlify.app/rational_expectations.html

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This PR touches: rational_expectations.md. Last CI build: success@2025-12-14. Branch: 94 commits behind main as of 2026-07-08.

[github-actions]

📖 Netlify Preview Ready!

Preview URL: https://pr-764--sunny-cactus-210e3e.netlify.app

Commit: 20870f7

📚 Changed Lectures

• rational_expectations

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Build Info

• Workflow: Build Project [using jupyter-book]