Sahal's identity: Wright's Law and Moore's Law become mathematically indistinguishable when cumulative production grows exponentially in calendar time

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: 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. 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.