The stock-to-flow (S2F) model is the quantitative framework for monetary hardness developed by Saifedean Ammous and operationalized as a Bitcoin price model by the pseudonymous analyst Plan B in his 2019 article "Modeling Bitcoin's Value with Scarcity." S2F is a ratio — existing supply divided by annual new supply; high S2F means a good is hard, since new issuance cannot meaningfully dilute the standing supply. Gold's historical S2F of ~60 made it the hardest commodity money; Bitcoin's S2F was ~50 after the 2020 halving, ~110 after 2024, and approaches infinity as the halving schedule drives flow toward zero. As a conceptual framework, S2F is load-bearing for the Austrian-Bitcoin case for monetary hardness — Ammous's framing made "hardness" quantitative rather than rhetorical. As a price model, however, Plan B's S2F has substantial post-2021 empirical problems and is now superseded in serious analytical work by the Power Law model of Giovanni Santostasi and Stephen Perrenod. The note distinguishes the two roles: S2F as hardness measure is foundational; S2F as price prediction is, on the empirical record, broken.
Why this note matters
S2F appears across the Bitcoin economic framework in two distinct roles that get conflated in popular discussion but should be kept separate. First, as a conceptual measure of monetary hardness — the ratio that lets us say with precision why gold is harder than silver, why silver is harder than copper, and why Bitcoin is now harder than gold. Second, as a specific price model proposing that Bitcoin’s market value tracks its S2F ratio on a power-law curve. The first role is intellectually load-bearing for the entire hard-money tradition. The second role has been empirically problematic since 2021 and is no longer considered reliable by careful analysts.
The note exists to (1) make the S2F-as-hardness-measure framework explicit and load-bearing, (2) treat Plan B’s price model substantively without endorsing it, (3) engage the substantive analytical critiques — Coppola’s cointegration argument, Santostasi’s time-vs-S2F argument, and the post-2021 empirical record — that have superseded the price model in serious analytical work, and (4) name the Power Law model as the analytical successor while preserving the hardness framework’s contribution.
The ratio: stock divided by flow
Stock-to-flow is the simplest possible measure of supply scarcity:
The interpretation is intuitive. If stock is 1,000 units and flow is 50 units per year, S2F is 20 — it would take 20 years of production at current rates to double the supply. Higher S2F means new issuance cannot meaningfully dilute the existing stock; lower S2F means it can.
Approximate historical S2F values for major commodity monies, drawn from Saifedean Ammous’s The Bitcoin Standard:
| Commodity | Stock-to-flow ratio |
|---|---|
| Gold | ~60 |
| Silver | ~22 |
| Platinum | ~0.4 |
| Palladium | ~1.1 |
| Copper | ~0.3 |
| Bitcoin (post-2020 halving) | ~50 |
| Bitcoin (post-2024 halving) | ~110 |
| Bitcoin (post-2028 halving) | ~220 (projected) |
The key historical observation: the highest-S2F good was always the dominant monetary good. Gold’s monetary status was not coincidental — it was a structural consequence of gold being far harder than its competitors. When the U.S. abandoned the gold standard in 1971 (see Bretton Woods and the Nixon shock), the world transitioned from a monetary system anchored to a high-S2F good to one anchored to fiat currencies whose effective S2F approaches the central bank’s discretion — that is, structurally low.
Ammous’s framing: hardness as quantitative
The conceptual contribution of S2F to the Austrian-Bitcoin tradition comes through Saifedean Ammous’s The Bitcoin Standard (2018). Before Ammous, “hard money” was a tradition-loaded term — the Austrians had spent a century arguing for sound money on the grounds that fiat regimes inflate and gold doesn’t, but the analytical framework was largely qualitative.
Ammous’s move was to operationalize hardness as a specific ratio with cross-commodity comparison. This had three immediate consequences:
- Hardness became measurable. Hard money could be defined precisely. Disputes about whether silver, copper, or seashells were “hard enough” to function as money became empirical disputes about ratios.
- Bitcoin’s hardness became visible. Bitcoin’s pre-2020 S2F was below gold’s; its post-2020 S2F was comparable; its post-2024 S2F surpassed gold; and its asymptotic S2F goes to infinity. The trajectory toward maximum hardness is encoded in the halving schedule (see The halving - Mechanism and Bitcoin fixed supply and issuance schedule).
- The hard-money argument acquired a Bitcoin-native dimension. Where Mises (see Mises and the theory of money) and Rothbard (see Rothbard and sound money) had argued from theoretical first principles, Ammous could now point to specific numbers showing Bitcoin’s structural superiority on the most-load-bearing monetary dimension.
This framing is now universal across Bitcoin economics writing. It is the most-cited and most-load-bearing analytical move in The Bitcoin Standard and arguably the single most consequential analytical contribution to contemporary Bitcoin economic thought.
Plan B’s price model
In March 2019, the pseudonymous Dutch analyst Plan B published “Modeling Bitcoin’s Value with Scarcity” on Medium. The article took Ammous’s hardness framework one significant step further: it argued that Bitcoin’s market price itself follows a power-law function of its S2F ratio.
The proposed relationship, fit on monthly data from 2009 through 2019:
Or equivalently, plotted on a log-log scale, Bitcoin’s market cap appears to fall on a straight line as a function of S2F. The model predicted dramatic price increases following each halving — flow drops by half, S2F roughly doubles, and the modeled market cap rises by a factor of approximately ten ().
Plan B’s specific 2019 prediction: a post-2020-halving price target of approximately $100,000 per BTC by late 2021.
In the immediate aftermath, the model appeared spectacularly vindicated. Bitcoin’s price tracked the S2F band closely through 2019, 2020, and most of 2021, reaching ~$69,000 by November 2021 — below the explicit target but within the model’s standard-deviation band.
Plan B refined the model in subsequent articles, producing an “S2FX” cross-asset variant treating gold, silver, and Bitcoin as different phase-states of monetary goods, and various refinements treating market cycles and adoption phases.
The empirical problem: 2022 and after
The post-2021 record has been substantially less kind to the price model. Two specific failures stand out: the 2022 drawdown, which fell well below the model’s lower band; and the post-2024 underperformance, which has tracked materially below the model’s central prediction with delayed timing.
The 2022 drawdown
The November 2021 peak at ~15,500 was the first sustained, unambiguous violation of the S2F model’s predicted band. The drawdown to ~$15,500:
- Fell well below any reasonable reading of the model’s lower 1-σ band.
- Persisted for a substantial portion of 2022 and 2023, not for a brief excursion that the model’s natural variability could absorb.
- Coincided with no change in the supply schedule — S2F was unchanged from the trajectory the model was tracking. The price moved while the model variable didn’t.
The standard Plan B response was that the 2022 drawdown was an “outlier” or a “fluctuation around the trend” — but this response is structurally problematic. A model with predictive content must specify what observations would falsify it. If a year-long drawdown to roughly 22% of the model’s prediction is still “consistent with” the model, the model has no falsifiable content.
A separate defense — that the 2022 drawdown was driven by exogenous shocks (the Luna/Terra collapse, FTX implosion, generalized risk-off macro environment) the S2F model could not have anticipated — is plausible as far as it goes, but it points to a structural problem: any model that requires “exogenous shocks” to explain its largest forecast errors is admitting that the model only describes the trend in the absence of shocks. The entire purpose of a price model is to forecast through shocks, not around them.
The post-2024 underperformance
The April 2024 halving produced the most-watched test of the S2F model’s predictive content. The model’s prediction (in its various versions through 2024) was a post-halving acceleration consistent with prior cycles, with price targets ranging from ~1 million per BTC by late 2025 or early 2026.
The actual record (through mid-2026):
- Bitcoin appreciated meaningfully post-halving, consistent with the model’s directional prediction.
- The magnitude has been materially below S2F predictions — roughly an order of magnitude below the central price target.
- The timing has been delayed by quarters relative to prior-cycle analogues — the model’s implicit assumption that each cycle behaves like a scaled version of the prior cycle has not held.
This is the most decisive empirical evidence against the S2F price model as currently specified. A model that predicts an order-of-magnitude move that doesn’t materialize is not within any reasonable uncertainty band. The “next cycle” defense — that the target will be hit by the 2028 halving rather than by late 2026 — is structurally similar to the indefinite postponement of predictions that characterizes failed forecasting frameworks generally. The original model specified timing; pushing the timing out by years without re-specifying the model is unfalsifiability in disguise.
What this implies
The honest reading is that the price model is structurally broken while the hardness framework remains intact. These are distinct claims: hardness is a measure of supply scarcity; the price model is a specific hypothesis about how that measure maps to market prices. The first survives the second’s failure.
The Power Law alternative
The most serious successor framework is the Power Law model developed independently by Giovanni Santostasi (see Giovanni Santostasi) and Stephen Perrenod (see Stephen Perrenod). Where S2F treats Bitcoin’s market cap as a function of its supply-side scarcity, the Power Law model treats Bitcoin’s price as a power-law function of time, with the underlying driver being adoption — specifically, network effects (see Network effects and Metcalfe’s Law) and the broader monetization S-curve (see Monetization S-curve).
The Power Law has several analytical advantages over S2F:
- Better empirical fit through 2026. The Power Law accommodates the post-2021 price action substantially better than S2F.
- A causal mechanism rooted in adoption rather than scarcity. Bitcoin’s price rises because more people use it, not because new supply is small. This is consistent with how monetary goods actually monetize historically — through adoption-driven demand, not through supply restriction.
- Convergence with mainstream technology-adoption frameworks. The Power Law is structurally similar to S-curves for past transformative technologies, anchoring Bitcoin’s trajectory in a broader empirical pattern rather than a Bitcoin-specific scarcity argument.
- Honesty about cycle dynamics. The Power Law explicitly accommodates the four-year halving cycles (see The halving - Mechanism) as oscillations around a long-term trend rather than as the primary driver.
The Power Law does not refute the hardness framing — it complements it. Bitcoin’s hardness is a precondition that makes monetization possible; the monetization trajectory itself follows the adoption dynamics the Power Law captures.
What S2F got right (and still gets right)
Even granting the price model’s failure, the S2F framework made several contributions that retain their force:
- The quantitative formalization of hardness. Before Ammous, the hard-money tradition was qualitative; after, hardness was a measurable property with cross-commodity comparability. This contribution is foundational and unlikely to be revised.
- The structural prediction of the post-2024 hardness inversion. The S2F framework correctly predicted that Bitcoin would surpass gold in hardness around 2024-2025. This empirical prediction was confirmed and has substantive implications for the monetization thesis.
- The link between supply schedule and monetization trajectory. Even if the specific price model is broken, the broader claim that Bitcoin’s pre-programmed scarcity creates monetization tailwinds is preserved by the Power Law and other adoption-based frameworks.
- The popularization of hardness thinking. Plan B’s model, whatever its empirical failures, brought the hardness framework to a much broader audience than Ammous’s book alone would have reached.
The model has been an important pedagogical tool even where its specific predictions don’t hold. Many readers came to the Austrian-Bitcoin framework through Plan B and only later engaged the deeper theoretical infrastructure.
How to use S2F in current analysis
For practical analytical work in 2026 and beyond, the recommended treatment:
- Use S2F as a hardness measure, consistent with Ammous’s framing in The Bitcoin Standard. The ratio remains the right way to think about supply-side monetary properties.
- Do not use S2F as a price model. The empirical record after 2021 is too damaging. Use the Power Law model (see The Power Law model and Long-term price models and cycles) for price-trajectory thinking.
- Cite Plan B’s contribution carefully. Acknowledge the model’s pedagogical importance and its specific empirical failures. Don’t conflate the price model with the hardness framework.
- Engage critics charitably. Coppola’s and Santostasi’s critiques of the Plan B model are substantive; engaging them honestly strengthens the case for the underlying hardness framework rather than weakening it.
Counter-arguments and tensions
The cointegration critique (Coppola)
The earliest and most analytically rigorous critique came from Frances Coppola (see Frances Coppola) in 2019 and 2020, building on a general statistical observation about regressions involving non-stationary time series.
The argument in compact form. Both Bitcoin’s price and its stock-to-flow ratio have risen monotonically over the historical sample available to Plan B in 2019. When two time series are non-stationary and trending, a regression of one on the other will produce an apparently strong fit even when no underlying causal relationship exists. This is the classic “spurious regression” problem identified by Granger and Newbold in 1974 and is a fundamental issue in time-series econometrics.
The cointegration critique points out that:
- Plan B’s regression treats Bitcoin price and S2F as causally related without testing for the alternative that both are driven by a third variable (time, adoption, market cycle) or that the apparent relationship is purely statistical artifact.
- Working in first differences (looking at changes rather than levels) eliminates most of the explanatory power. If the underlying relationship were causal, this transformation should preserve the relationship; the fact that it doesn’t is diagnostic of spurious correlation.
- The R² values cited in defense of S2F are inflated by the non-stationarity. The relevant statistical question is not “does this regression fit” but “does this regression fit better than a null model that respects the time-series structure” — and the answer is, by Coppola’s analysis, no.
The technical sophistication. Coppola’s critique is not amateur statistical objection. It draws on standard cointegration testing methodology and applies it carefully to the S2F data. The methodology is the same one used in empirical macroeconomics for half a century to distinguish genuine economic relationships from time-series artifacts.
Plan B’s response. Plan B argued that the S2F-price relationship satisfies cointegration tests under specific specifications. This response was widely considered analytically inadequate: the specifications under which the relationship survives are not the ones standard practice would use, and the broader question of whether the model has a falsifiable form remained unaddressed.
Why the post-2021 record matters here. If the relationship had been genuinely causal in the 2009-2021 sample, the model would have continued to fit through 2024 and 2026 without parameter re-estimation. Instead, the post-2021 record requires either (1) re-estimating the parameters substantially, (2) widening the uncertainty bands until the model is unfalsifiable, or (3) acknowledging that the original fit was spurious. None of these is consistent with the original strong-causal claim.
The honest response. The cointegration critique is, on its terms, correct. The defense is not that Coppola is wrong about the statistics; it is that the underlying monetary mechanism (hardness → demand for monetary services → price) is real even if the specific functional form S2F predicts is not. The Power Law framework handles the cointegration question more honestly.
The time-vs-S2F critique (Santostasi)
The second critique, developed by Giovanni Santostasi (see Giovanni Santostasi) and complementary work by Stephen Perrenod (see Stephen Perrenod), takes a more constructive form: it proposes a successor framework rather than just a refutation.
The core observation. The S2F variable is itself a deterministic function of time, since Bitcoin’s halving schedule is fully pre-programmed. Specifically, S2F can be written as an exponential function of the halving epoch, which is itself a step function of time. Any model that fits Bitcoin price to S2F can therefore be re-expressed as a model fitting price to time.
The question becomes: does the model fit better when expressed in S2F terms or in time terms? Santostasi’s empirical work shows that:
- The pure time-based power law fits the empirical record better than the S2F-based power law over the full available sample.
- The time-based model handles the halvings as oscillations around a trend rather than as the primary driver. This is more consistent with how the price actually moves: halvings produce cycles, not trend-changes.
- The causal story under the time-based framework is adoption-driven — more time means more users means more demand for the asset — which is consistent with how monetary goods historically monetize.
The Power Law model formal statement. Bitcoin’s price tracks where is time since inception and is approximately 5.7 (Santostasi’s fit; other analyses find similar exponents). The model captures both the long-term trend and provides a framework for understanding the cyclical deviations around the trend. See The Power Law model.
The structural advantage. The time-based framing is consistent with how analogous technology-adoption curves work historically (see Network effects and Metcalfe’s Law). Bitcoin is not unique in following a Power Law trajectory — most transformative technologies do. This anchors the Bitcoin case in a broader empirical pattern rather than relying on Bitcoin-specific scarcity arguments.
The relationship to S2F-as-hardness. Critically, the Power Law critique does not claim that Bitcoin’s hardness is irrelevant. It claims that the hardness is a precondition that enables monetization, but the trajectory of monetization is governed by adoption dynamics rather than by the supply schedule directly. This preserves Ammous’s contribution while replacing Plan B’s specific operationalization.
This is the most consequential analytical critique because it suggests a successor framework rather than just a refutation. The right response is not to defend S2F as a price model but to acknowledge that Power Law is the better framework and S2F-as-hardness is the surviving conceptual contribution.
The “S2F predicts halving cycles correctly” defense
The argument: Defenders point out that the broad shape of post-halving cycles — appreciation following each halving, with subsequent drawdown and consolidation — has been roughly consistent with S2F predictions through 2020.
Response: The cycle pattern is real but is also predicted by adoption-based frameworks, network-effect dynamics, and supply-shock economics generally. S2F’s specific functional form is not necessary to explain the cycles; many frameworks predict halving-related price action. The post-2024 cycle has been particularly hard on S2F-specific predictions about magnitude and timing.
Within-Bitcoin critique: the framework anchored harmful expectations
The argument: Some within-Bitcoin critics (including elements of the on-chain analytics community — see James Check) argue that the S2F price model has been actively harmful to Bitcoin investor behavior by anchoring expectations on unrealistic price targets and producing disappointment when those targets are missed. The model gave a generation of holders unrealistic mental models for cycle timing and magnitude.
Response: This is fair as a critique of how the model was used culturally, but it doesn’t bear on the analytical question. A model can be both intellectually serious and pedagogically harmful. The right response is to be honest about the model’s limitations going forward, not to dismiss it retroactively.
The “hardness model is also broken” objection
The argument: Some critics push further than Coppola or Santostasi to claim that the hardness framework itself is the problem — that Bitcoin’s value should track some measure of utility or adoption rather than supply scarcity, and that the whole Ammous synthesis is built on a category error.
Response: This is a much stronger claim than the Coppola or Santostasi critiques, and it conflicts with the broader monetary-theoretical case for hard money (see Hard money vs fiat money, Mises and the theory of money). The hardness framework doesn’t claim that price equals hardness; it claims that hardness enables monetization. The Power Law model is consistent with the hardness framework — it specifies the dynamics through which a hard money’s value evolves. The hardness-is-irrelevant claim conflicts with the empirical record of historical monetary goods, which uniformly favored harder commodities.
The defenses are sometimes right
The argument: Not every defense of S2F is bad-faith unfalsifiability. Some defenses correctly point out that the model captures real features of Bitcoin’s halving cycles, that the 2009-2021 fit was genuinely strong, and that the model’s failures are concentrated in specific identifiable regimes.
Response: Fair. The post-2021 underperformance does coincide with substantial regime change — central bank tightening, institutional adoption, ETF launches, ETF flows, fiscal-dominance dynamics — that the original model couldn’t have anticipated. The honest reading is that the model worked in one regime and fails in another, which is informative but doesn’t rescue it as a general-purpose price predictor.
Within-Bitcoin: the framework didn’t deserve this much engagement
The argument: Some serious Bitcoin economists argue that the S2F price model never deserved as much attention as it received, and that engaging it extensively now is itself a form of legitimizing a framework that should have been quickly recognized as unrigorous.
Response: Partially right but procedurally wrong. The S2F price model received broad attention; it shaped a generation of investor expectations; and engaging it rigorously is necessary regardless of whether one wishes it had received less attention initially. The intellectual responsibility is to engage the framework that exists, not the framework one wishes had been engaged.
Open questions for further development
- What does it actually mean for a model to be “broken” rather than “violated within its uncertainty band”? Plan B’s defense relied on increasingly elastic uncertainty interpretations; the meta-question of when to consider a model falsified is genuinely difficult and not unique to S2F.
- If the price model is broken, why did the 2019-2021 fit appear so strong? Was it pure coincidence, was the cointegration explanation sufficient, or was there a genuine but unstable relationship that broke down with regime change?
- How should the hardness framework be updated when Bitcoin’s S2F approaches infinity asymptotically? At some point the ratio becomes practically meaningless and we need different language for “the hardest possible monetary good.”
- How should other Bitcoin-economic arguments that lean on hardness as a quantitative measure (the Bitcoin vs. gold comparison in Bitcoin vs gold, for instance) be adjusted to reflect the surviving framework rather than the broken model?
- Does the failure of S2F-as-price-model undermine Ammous’s broader Austrian-Bitcoin synthesis, or is the synthesis robust to this specific failure? The strongest reading is that Ammous’s framework is independent of Plan B’s specific operationalization, but this deserves examination.
- What is the appropriate epistemic stance toward Plan B’s continued public defense of the model? The community is split between those who consider the model thoroughly falsified and those who continue to invest in its predictions.
- If the Power Law model becomes the consensus framework, what are its falsification criteria? Santostasi’s framework should be held to the same epistemic standard the S2F critics applied to Plan B’s.
- How should Bitcoin investors who anchored on S2F predictions update their priors? The cohort that bought near the 2021 peak based on $100K-by-late-2021 expectations and rode through the 2022 drawdown has specific psychological and financial damage that the community has not fully reckoned with.
Canonical sources for this note
Foundational framing
- The Bitcoin Standard, Saifedean Ammous (2018) — chapters on hardness, salability, and the S2F framework as monetary measure
- Principles of Economics, Carl Menger (1871) — Mengerian salability framework that S2F operationalizes
- Human Action, Ludwig von Mises (1949) — Misesian framework for monetary hardness
The price model
- “Modeling Bitcoin’s Value with Scarcity,” Plan B (March 2019) — the original article proposing the price model
- “Bitcoin Stock-to-Flow Cross Asset Model,” Plan B (April 2020) — the S2FX refinement treating Bitcoin, gold, and silver as monetary phase-states
- Plan B’s ongoing Twitter and Medium output through 2026
Critical engagements
- Various Frances Coppola articles and tweets — the cointegration critique
- Coppola’s Coppola Comment blog posts engaging Plan B directly
- Giovanni Santostasi, “The Bitcoin Power Law Theory” and related papers — the Power Law alternative
- Stephen Perrenod, various Substack writings on Power Law model — co-developer’s framing
- Santostasi and Perrenod, “A Mechanistic Derivation of the Bitcoin Price Power Law: Network Adoption Dynamics and Generalised Metcalfe Scaling” — Scientific Bitcoin Institute paper grounding the Power Law alternative
- Various Nico Smid (Digital Asset Research) analyses — empirical engagement with S2F’s predictions
- BitMEX Research, various S2F analyses — institutional-style empirical engagement
- Granger and Newbold (1974), “Spurious regressions in econometrics” — the foundational statistical paper on the cointegration issue Coppola applies
Bitcoin-side defenses and updates
- Plan B’s articles addressing the post-2021 underperformance
- Various Saifedean Ammous podcasts engaging the price-model vs. hardness-framework distinction
- Various Lyn Alden engagements with the model
Related notes
- Hard money vs fiat money — the broader framework S2F operationalizes
- Bitcoin fixed supply and issuance schedule — the supply schedule encoded in S2F’s denominator
- The halving - Mechanism — the four-year mechanism driving S2F’s trajectory
- Bitcoin vs gold — the comparison most thoroughly using the hardness framework
- Monetization S-curve — adoption-side framework complementing supply-side analysis
- Network effects and Metcalfe’s Law — the demand-side framework underlying the Power Law
- Bitcoin as emergent money — broader emergence framework
- Store of value vs medium of exchange vs unit of account — phase framework S2F supports
- Austrian economics foundations — methodology
- Carl Menger — Mengerian salability foundation
- Saifedean Ammous — originator of the hardness-as-quantitative framing
- Plan B — creator of the specific price model
- Giovanni Santostasi — Power Law alternative
- Stephen Perrenod — Power Law co-developer
- Frances Coppola — the cointegration critique
- Vijay Boyapati — monetization framework complementing hardness
- Lyn Alden — empirical macroeconomist engaging the framework
- James Check — on-chain analyst critical of S2F-anchored expectations
- The Power Law model — the successor framework
- Long-term price models and cycles — sub-MOC for the price-models area