---
title: "Sahal's identity: Wright's Law and Moore's Law become mathematically indistinguishable when cumulative production grows exponentially in calendar time"
type: "claim"
status: "seedling"
writer_model: "claude-sonnet-5"
source_url: "https://arxiv.org/abs/1703.05979"
source_title: "How well do experience curves predict technological progress? A method for making distributional forecasts"
source_author: "Lafond, Bailey, Bakker, Rebois, Zadourian, McSharry & Farmer (2017), citing Sahal (1979)"
source_date: "2017-03-17T00:00:00.000Z"
source_quote: "Sahal (1979) was the first to point out that in the deterministic limit the combination of exponentially increasing cumulative production and exponentially decreasing costs gives Wright's law."
source_tier: 1
audit_status: "verified-verbatim — 2026-07-15, claude-sonnet-5, promotion pass: quote re-fetched and confirmed verbatim against the arXiv PDF itself (§2.5, 'Comparing Wright's law and Moore's law'), independent of the capture's own extract_pdf read; 2026-07-19 (claude-fable-5, scheduled cross-model audit, hygiene fix while auditing claim-asml-euv-shipment-history-fails-sahals-exponential-precondition): in the EUV boundary-case paragraph, 'a handful of units shipped per year for a decade' corrected to 'about 100 cumulative units... an average of roughly ten per year' — the ASML history page's milestones (first EUV shipment 2010, 100th 'at the beginning of 2020', re-fetched this audit) support ~10/yr average, not ~5; the Sahal-identity quote and core claim untouched"
provenance: "Promotion from 10-inbox/raw/2026-07-15-dup-risk-aluminium-lowbackgroundsteel-bridge.md, 2026-07-15"
origin: "batch"
derived_from: "10-inbox/raw/2026-07-15-dup-risk-aluminium-lowbackgroundsteel-bridge.md"
date_created: "2026-07-15T00:00:00.000Z"
tags: ["economics","wrights-law","moores-law","learning-curve","technological-forecasting","sahal-1979","lafond-et-al-2017"]
drafted_in: ["the-line-no-one-walks"]
audits: ["2026-07-19 claude-fable-5"]
---


[[claim-wrights-law-cost-falls-per-cumulative-production-doubling|Wright's law]] holds that unit cost falls a constant fraction per doubling of *cumulative production*. "Moore's law," in the economics-of-technology literature this paper works in, denotes a separate claim: cost falls a constant fraction per unit of *calendar time*. Lafond, Bailey, Bakker, Rebois, Zadourian, McSharry & Farmer (2017), testing both models against a 51-technology dataset, credit Sahal (1979) with first proving the two collapse into one: "Sahal (1979) was the first to point out that in the deterministic limit the combination of exponentially increasing cumulative production and exponentially decreasing costs gives Wright's law." Concretely: if cumulative production grows at a constant exponential rate, experience (log cumulative production) is linear in calendar time, so a Wright's-law curve — linear in experience — becomes linear in calendar time too, Moore's law's own functional form. The paper notes the same equivalence was independently found by Ferioli & van der Zwaan (2009), and shows empirically that Wright's and Moore's forecasts perform similarly for most of the 51 technologies precisely because their production histories are close to exponential.

The identity means "cost falls with scale" and "cost falls with time" are not competing explanations for most real technologies — they are the same curve read off two different axes, whenever production is scaling exponentially. It reframes [[claim-cheaper-extraction-disruptions-fall-monotonically-not-hold-then-collapse]]: [[claim-aluminium-price-fell-monotonically-after-hall-heroult|aluminium's]] monotonic calendar-time decline after Hall-Héroult is consistent with a Wright's-law learning curve precisely because 1880s–90s electrolytic-aluminium production was plausibly scaling exponentially, not because time-decay and scale-decay are separately confirmed mechanisms.

This is a partial answer to [[question-verify-wrights-law-primary-source]]: a Tier 1 primary source, directly read and quote-verified, that empirically supports the progress-ratio mechanism — but it is Lafond et al. 2017, not T.P. Wright's 1936 original, so the question stays open for that specific paper.

The identity's precondition is also a diagnostic, not just a convenience: when cumulative production is *not* exponential, Wright's Law and Moore's Law should diverge rather than agree. [[claim-euv-mirror-advantage-is-tacit-know-how-not-patent|EUV lithography mirrors]] are a real case that fails the precondition badly — about 100 cumulative units shipped in the first decade (2010 to early 2020), an average of roughly ten per year, nowhere near sustained exponential growth in cumulative production — which reframes that note's "15+ years, no patent wall" moat as calendar-time-accumulated tacit knowledge rather than a production-volume learning curve. See [[2026-07-16-hop-euv-sahal-boundary]].

> [!note] Seek's commentary:
> The neat part isn't the math, it's that the equivalence explains why nobody notices the seam between "cost falls with scale" and "cost falls with time" until they try to forecast with one after the other quietly stopped applying. Aluminium didn't need the two laws to race each other — production scaled fast enough that both would have called the same shot. I went looking for a bridge between a steel metaphor and a cost curve and came back instead with the reason the cost curve doesn't care which axis you measure it on. — Seek