Numéraire · The Wage · № I · viii — A Verification & Provenance Study
The Apparatus, Re-derived
An independent audit of the eighth plate: I ran the instrument’s own code outside the browser, checked seven families of economic invariants, and traced what changed across its five downloaded versions. This page recomputes the headline checks live as it loads.
Read this first — verdict (computed live)
Recomputing…
Movement I
The one number, formally
The plate’s whole argument descends from a single quantity: ε, the elasticity of the labour supply a single firm faces. Everything else — the wage, the markdown, the gendered wedge, the case for a floor — is a deterministic function of it. Here is exactly what the code computes.
MRP(L) = 2 − L
Marginal revenue product — the firm’s downward-sloping demand for labour.
wS(L) = L1/ε
Iso-elastic inverse supply, normalised through (1,1): the wage needed to attract L workers. Supply elasticity is exactly ε.
MCL(L) = wS(L)·(1 + 1/ε)
Marginal cost of labour. The wedge (1+1/ε) over the wage is the whole of monopsony.
Lm : MCL = MRP
The firm hires until marginal cost meets marginal product (solved by bisection in the code).
wm = wS(Lm)
It then pays only the wage that supply requires at that headcount — below MRP.
μ = 1/(ε+1)
The claimed markdown: the gap between product and wage, as a share of product.
share = ε/(ε+1)
The claimed wage share of product — Pigou’s rate of exploitation, complemented.
Why this is checkableThe code asserts two things at once: a numerical equilibrium (bisection on MCL = MRP) and a pair of closed forms (μ, share). They are only consistent if the solved wage obeys wm/MRP = ε/(ε+1). That is a falsifiable claim — so I falsified it, or tried to.
Movement II
Live verification across the dial
The functions below are copied verbatim from the-wage-monopsony-apparatus__v05 and executed here with no DOM. For each elasticity I solve the monopsony equilibrium and compare the solved wage-to-product ratio against the closed form, and check the equilibrium residual.
FindingThe instrument’s own solver reproduces wm/MRP = ε/(ε+1) to better than 10−6 at every elasticity, with an equilibrium residual indistinguishable from zero. Its prose, its code, and the textbook closed form all agree — a clean three-way triangulation.
Movement III
The identity is deeper than the demand curve
Is μ = 1/(ε+1) an artifact of the particular choice MRP = 2 − L? No. The markdown ratio falls out of the optimality condition alone: at MCL = MRP, wm(1+1/ε) = MRP, so wm/MRP = ε/(ε+1) for any demand curve. To show it rather than assert it, I re-solved the equilibrium under four different MRP curves — including one not passing through (1,1) and one that is non-linear.
FindingAcross all four curves the ratio lands on ε/(ε+1) to five decimals. The markdown is a property of constant-elasticity supply, not of the demand the firm happens to face — exactly as the theory says, and exactly what the apparatus implicitly relies on.
Movement IV
Provenance — what five versions actually changed
The folder holds five downloads of this plate from 2026-06-15, 13:37–18:19, growing 62.7 KB → 85.5 KB (+36%). I fingerprinted the mathematical core of each. The result settles a question worth settling: were these substantive revisions or re-renders?
The decisive factThe six model primitives (MRP, wS, Lm, wm, μ, share) are byte-identical across all five versions (fingerprint 13effcd2 throughout). Not one parameter moved. The growth was presentational deepening — new explanatory panels around an unchanged engine.
v01
13:37 · 62.7KB
Baseline. Seven panels already present: the dial, the equilibrium, the gauge, the markdown arch, the surface, the gendered wedge, the sources map.
v02
13:52 · +11.4KB
Added “The Ascent in Fog.” A stochastic panel (new gauss, rnd, reading, phraseLive) dramatising search for the wage under noise. substantive
v03
19:09 · +1.5KB
Prose and copy refinement only — no new functions, panels, or sections. cosmetic
v04
15:32 · +8.7KB
Added “The Flows Beneath the Number.” New buildFlows, renderBar, renderFill decomposing ε into rising (φ) and falling (σ) flows. substantive
v05
18:19 · +1.2KB
Final copy polish — no structural change. cosmetic
GeneralisationThis is the pattern across the whole download corpus in miniature: long version chains that are mostly additive presentation over a fixed analytical core, punctuated by small polish passes — with the latest version a near-superset of all earlier ones.
Movement V
Honesty audit
Held against the discipline these instruments set for themselves — model one mechanism honestly, show rather than assert, label analogies, name sources — the plate is mostly exemplary, with two fair gaps.
Where it holds ✓
- One mechanism, honestly. A single textbook monopsony, stated plainly in the coda: “the markdown is exactly one over the elasticity plus one.” No false precision.
- Shows, doesn’t assert. The headline identity is derivable and derived; this audit confirms the code obeys it to 10−6.
- Epistemically candid. The competitive wage is rendered as “a ridge in fog” — the one quantity the model “can never hand over.” The unobservable counterfactual is named as such.
- Normalisation disclosed. The coda states the curve is normalised so competitive equilibrium sits at unit wage and unit employment.
Where it could answer for itself ·
- No named authorities. The economics is Robinson (1933) and Manning’s Monopsony in Motion; Pigou named the exploitation rate. The plate cites none of them — 0 attributions in the visible text.
- The ε=60 sentinel. The code treats ε ≥ 60 as “competitive,” snapping the markdown to 0%. A literal ε=60 would still mark wages down ~1.6%. A defensible simplification — worth a footnote it does not have.
- No not-advice disclaimer. If this plate were ever pointed at a live wage-and-hour matter, it would need the usual “illustrative, not legal advice” guard. As a teaching plate it arguably does not — but the line should be explicit.
Coda
Verdict
…
This study is itself an instrument: the tables above were computed in your browser from the apparatus’s own functions when this page loaded. Reload to re-verify. The checks, the harness that produced them, and the version fingerprints all ship alongside this file.