ge_arrow

• Series: lecture-python.myst

• File: lectures/ge_arrow.md

• Audit date: 2026-08-26

• Corpus snapshot: e25fdf2345

• Categories audited: writing, math, code, figures, references, links, admonitions (JAX out of scope)

• Overall score: 7.3 / 10

• Priority: HIGH

Score breakdown

Category	Score	One-line note
Writing	3/10	qe-writing-006 ×15; qe-writing-005 ×6; qe-writing-002 ×5, +5 more.
Math	3/10	qe-math-010 (proposed) ×5; qe-math-003 ×11; qe-math-004 ×5, +1 more.
Code	7.5/10	qe-code-001 ×6.
JAX	out of scope	JAX rules target lecture-jax.
Figures	7.5/10	qe-fig-003 ×1; qe-fig-005 ×1; qe-fig-008 ×1, +1 more.
References	10/10	no mechanical violations detected.
Links	10/10	no mechanical violations detected.
Admonitions	10/10	no mechanical violations detected.

Issues

Critical

None found.

High severity

• [qe-code-001] (reviewer) — Follow PEP8 unless closer to mathematical notation. Count: 6. Lines: 940, 1110, 1126, 1130, 1188, 1287. Example: three dead locals in wealth_distribution - n = self.n, Q = self.Q, y, ys = self.y, self.ys at 940-942 - are assigned and never read (F841); print ( has a space before the parenthesis at 1110, 1188 and 1287 (E211); and ex2.P[0,1] / ex2.P[1,0] at 1126 and 1130 omit the space after the comma (E231).

• [qe-math-003] — Use square brackets for matrix notation. Count: 11. Lines: 540, 550, 566, 597, 607, 623, 627, 721, 731, 754, …. Example: array used as matrix.

• [qe-math-004] — Do not use bold face for matrices or vectors. Count: 5. Lines: 67, 187, 704, 711, 765. Example: {\bf.

• [qe-math-010 (proposed)] — Blackboard \mathbb{P}, \mathbb{E}, \mathbb{V} with braces. Count: 5. Lines: 113, 299, 393, 446, 462. Example: non-blackboard \Pr.

• [qe-writing-002] (reviewer) — Keep writing clear, concise, and valuable. Count: 5. Lines: 33, 443, 623, 754, 802. Example: large blocks are repeated verbatim rather than referred back to. The ‘Inputs’ bullet at 443-447 reproduces 296-300 word for word, display included; the column vectors \(y^k\) and \(y\) are displayed three times (566-571, 623-632, 754-763); the two Remarks at 636-641 reappear almost unchanged at 767-772; and the five-step algorithm at 677-687 is re-listed at 802-813 with only the equation labels changed. Line 33 is a single 50-word bullet that chains a utility assumption, a representative-consumer claim, a pricing-kernel formula and an ordering result together.

• [qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 6. Lines: 528, 583, 636, 641, 767, 772. Example: bold is used as a directive label six times - ‘Key finding:’ at 528 and ‘Remark:’ at 583, 636, 641, 767 and 772 - none of which is a definition. MyST has {note} for exactly this, and the file already uses it correctly once at 346-349 for the resolvent operator.

• [qe-writing-006] — Capitalize lecture titles properly. Count: 15. Lines: 148, 242, 287, 318, 355, 373, 435, 486, 588, 643, …. Example: H2 Title Case: ‘Recursive Formulation’ (Formulation).

• [qe-writing-008] — Remove excessive whitespace between words. Count: 142. Lines: 19, 27, 33, 35, 37, 39, 50, 52, 58, 62, …. Example: 2 spaces.

Medium severity

• [qe-fig-003] — No matplotlib embedded titles. Count: 1. Lines: 1249. Example: .set_title.

• [qe-math-009] (reviewer) — Choose simplicity in mathematical notation. Count: 4. Lines: 187, 201, 311, 449. Example: plain TeX and decorative constructions are used where standard LaTeX is shorter: \hskip.5cm at 187, {\beta u'_k(c_{t+1}^k) \pi(s_{t+1}|s_t) \over u'_k(c_t^k)} at 201-202 instead of \frac, and an invented braced operator {\textrm{Price}}\{\cdot\} at 311. Most pervasive is \left( \right) wrapped around single-symbol arguments - y^k\left(s\right), u\left(c\right), c\left(s\right), \left[1, \ldots, K\right] - dense from 449 through 520 and again at 540-570, 601-611 and 651-657, where plain (s) and \{1,\ldots,K\} would read better.

• [qe-writing-001] — Use one sentence per paragraph. Count: 4. Lines: 583, 636, 767, 772. Example: 2 sentences in one paragraph.

• [qe-writing-003] (reviewer) — Maintain logical flow. Count: 2. Lines: 72, 584. Example: line 72 warns the reader in a parenthesis that the notation is unreliable - ‘(Sometimes we inadvertently reverse the recording order and denote a history as \(s^t = [s_t, s_{t-1}, \ldots, s_0]\))’ - which leaves every later history expression ambiguous rather than fixing the convention. Line 584 points forward in prose, ‘See the section on a Finite Horizon Economy below’, to a section actually titled ‘## Finite Horizon’ (701), where a {ref} to a label would carry the reader there.

• [qe-writing-004] — Avoid unnecessary capitalization in narrative text. Count: 2. Lines: 584. Example: mid-sentence ‘Horizon’.

• [qe-writing-007] (reviewer) — Use visual elements to enhance understanding. Count: 2. Lines: 641, 1357. Example: the lecture’s headline result - state-variable degeneracy, that every agent’s continuation wealth returns to zero whenever the Markov state returns to \(s_0\) (263-280, 641, 772) - is asserted three times and never drawn, although a simulated \(\{\psi_t^k\}\) path along a Markov sample path shows it in one panel. In the finite-horizon example the text says at 1350-1352 that ψ and J come back as sequences ordered from \(t=T\) to \(t=0\), and then prints them as raw arrays at 1357-1365; a sequence indexed by \(t\) is a line plot. The lecture has exactly one figure in 1387 lines (1242-1252).

Low severity

• [qe-fig-001] — Do not set figure size unless necessary. Count: 1. Lines: 1243. Example: figsize=.

• [qe-fig-005] — Descriptive figure names for cross-referencing. Count: 1. Lines: 1242. Example: code-cell figure without mystnb figure metadata.

• [qe-fig-008] — Use lw=2 for line charts. Count: 1. Lines: 1247. Example: plot() without lw=.

Strengths

• The law of iterated values is derived in deliberate parallel with the law of iterated expectations (380-433), the two chains of equalities lined up term for term, so the analogy the section is about is visible in the layout rather than only asserted.

• Every formula the algorithm needs is labelled and then cited by {eq} at the point of use: eq:Qformula (521) at 679 and 805, eqn:alphakform (668) at 681, eq:continwealth (618) at 685, eqn:optport (652) at 687, and eq:vv, eq:w, eq:ww (749, 787, 795) at 807-811.

• One RecurCompetitive class covers both the infinite-horizon and the finite-horizon economy (858-905), and the finite-horizon results are then checked numerically against the infinite-horizon ones at \(T = 10000\) (1370-1387) - the convergence claim at 1368 is verified rather than stated.

• The resolvent operator is defined in a {note} at the moment it first appears (346-349), and the same object is then recognised in the continuation-wealth and value-function formulas later, which is the thread the introduction promises at 50.

• Code identifiers use unicode Greek matching the mathematics - γ, β, α, ψ, λ, μ, δ (862-863, 946, 959, 1229, 1265-1267) - with 0 qe-code-002 violations across a 1387-line lecture.

Recommended actions

1. Fix the class’s dependence on notebook globals: __init__ uses a module-level n at 891-902, pricing_kernel uses a module-level P at 927, and continuation_wealths uses n and K at 955, so RecurCompetitive only works because every example redefines K, n and P first (1030, 1086, 1259, 1312). Use self.n, self.K, self.P and delete the dead locals at 940-942.

2. Sentence-case the 15 Title Case headings at 148, 242, 287, 318, 355, 435, 486, 588, 643, 701, … (qe-writing-006) and fix the mid-sentence capitals at 584 (qe-writing-004).

3. Correct the index and operator slips: \sum_i should be \sum_k at 143; ‘for all \(i\)’ should be ‘for all \(k\)’ at 263, 268 and 271; the last line of the law-of-iterated-values derivation at 431 writes \(E V(\cdot)\) where the claim at 419 is about \(V\); Prob( at 360 should be \mathbb{P}; and the two apostrophe transposes at 187 and 473 should be ^\top (qe-math-002).

4. Cut the duplication: make 443-447 a reference back to 296-300, display \(y^k\) and \(y\) once instead of at 566, 623 and 754, and merge the two algorithm lists (677-687, 802-813) and the two Remark pairs (636-641, 767-772).

5. Modernise the math markup: convert the 11 \left[\begin{array} displays at 540, 550, 566, 597, 607, 623, 627, 721, 731, 754, … to bmatrix (qe-math-003); replace {\bf S}, {\bf T} at 67, 187, 704, 711 and 765 (qe-math-004); use \frac at 201, \quad at 187, \mathbb{P} at 299 and 446 (qe-math-010 (proposed)); and drop the \left(\right) padding around single arguments.

6. Add the two missing exhibits: a simulated continuation-wealth path showing the recurrence claimed at 641 and 772, and a plot of the finite-horizon ψ and J sequences instead of the raw arrays at 1357-1365; then give the existing figure mystnb metadata (1242), move set_title at 1249 into its caption, drop figsize at 1243 and set lw=2 at 1247.

7. Sweep the typos and the whitespace: ‘pricing kernal’ at 1054, 1109, 1117, 1121, 1286 and 1341; the method name value_functionss (981); ‘Engle curves’ for Engel at 504; ‘\(s^\tau\)as’ at 79; ‘the following a string of inequalities’ at 422 (they are equalities); the duplicated comment ‘when the initial state is state 2’ at 1355 and 1362; the four two-sentence paragraphs at 583, 636, 767 and 772; and the 142 double-space runs.