The aphorism “₿ = ∞/21M” has emerged as a concise articulation of a monetary intuition: when an asset that provides monetary services has a credibly capped supply, it’s relative price can, in principle, scale without bound as the scope of monetization expands. Despite its rhetorical appeal, this statement lacks a precise interpretation within standard monetary theory. This paper proposes a formalization of “₿ = ∞/21M” as a boundary condition in a class of finite-supply monetary equilibria, embedding a strictly limited, perfectly divisible, durable bearer asset into otherwise conventional models of money-cash-in-advance, money-in-utility, Lagos-Wright, and search-theoretic frameworks. By treating the 21 million cap as a terminal stock constraint and the “∞” as asymptotically unbounded demand for liquidity services (via adoption, transaction intensity, or balance-sheet demand), we derive conditions under which price paths, intertemporal choices, and expectations are well-defined, determinate, and stable.
The approach separates three channels that collectively determine the valuation of a finite-supply money: (i) its liquidity services and convenience yield relative to choice assets, (ii) the possibility cost of holding balances as governed by the interest rate and risk premia, and (iii) the velocity and breadth of monetization, which jointly map real activity and financial demand into desired real balances. The fixed cap serves as a boundary condition that replaces customary seigniorage-driven equilibria with stock-limited ones, shifting price formation from flow-based to stock-based logic. Within rational expectations, the no-arbitrage and transversality conditions restrict admissible price paths: the finite supply permits bubble-like components only insofar as they are supported by liquidity premia, leading to testable relationships between expected appreciation, convenience yield, and interest rates. Intertemporally, a credibly finite supply implies a deflationary drift conditional on adoption and velocity, altering Euler equations for consumption-saving decisions and affecting welfare, wealth distribution, and the coexistence of credit.
The paper makes four contributions.First, it formalizes “₿ = ∞/21M” as an equilibrium boundary condition that nests standard monetary models while accommodating a hard-cap constraint, thereby clarifying the determinacy and stability of price levels when seigniorage is zero. second, it characterizes price formation under finite supply, deriving comparative statics that link expected appreciation to interest rates, convenience yield, and adoption dynamics, and it identifies when price paths are pinned by fundamentals versus when they admit liquidity-supported bubble components. Third, it analyzes intertemporal choice in a deflationary-cap regime, quantifying how real balances, velocity, and portfolio choice adjust as the opportunity cost of holding the monetary asset varies. Fourth, it yields falsifiable predictions: the term structure of expected returns around supply schedule changes; velocity-rate comovement; adoption-driven variance in long-horizon price dynamics; and basis behavior in derivatives consistent with a convenience yield that varies with scarcity and depth of monetization.
By providing a model-driven interpretation of a popular heuristic, the analysis connects a fixed-supply monetary asset to established monetary theory and asset pricing, offering transparent conditions under which its price can scale with the breadth of monetization while remaining disciplined by rational expectations. The framework organizes empirical tests that can discriminate between liquidity-premium valuation and purely speculative dynamics, thereby rendering the “∞/21M” intuition operational and falsifiable.
Boundary condition formalization of ₿ = ∞/21M and its effects on price formation under a perfectly inelastic money supply with modeling guidelines and calibration targets
Formalization. Treat the expression ₿ = ∞/21M as a boundary condition in a general equilibrium with a perfectly inelastic money stock: the effective supply M̄ ≤ 21M is fixed exogenously, while the price of the monetary asset pₜ is the Lagrange multiplier that clears the cash-in-advance/liquidity constraint. let aggregate nominal demand for monetary balances Dₜ arise from microfoundations (money-in-utility,cash-in-advance,or inventory-theoretic motives) and let the free float Fₜ ≤ M̄ be the subset of supply available to transact. Then a reduced-form price identity is pₜ = χₜ·Dₜ/Fₜ, where χₜ encodes state-dependent intertemporal preference for inventory (convenience yield), risk, and microstructure frictions; with Dₜ unbounded and Fₜ bounded, lim Dₜ→∞ pₜ = ∞, formalizing the boundary. Under perfect inelasticity, equilibrium price absorbs demand shocks and hoarding shifts thru nonlinear impact: convexity increases as Fₜ thins and as χₜ rises with expected scarcity (lower velocity). The rational-expectations closure requires a transversality condition on discounted marginal utility of money and a no-arbitrage restriction linking expected Bitcoin returns to the risk-free rate minus convenience yield plus risk premia; deflationary expectations embed via E[pₜ₊₁/pₜ] = 1 + rₜ − ψₜ + ρₜ. This yields testable implications for (i) state-dependent price elasticity, (ii) velocity-endogenous amplification, and (iii) asymmetric price responses to positive vs. negative net inflows when Fₜ is small.
- Modeling guideline: Separate effective supply M̄ from free float Fₜ; the latter is the relevant state for price impact and must be endogenized via portfolio choice of heterogeneous cohorts.
- Demand block: Specify Dₜ from microfoundations (e.g., cash-in-advance with stochastic transaction needs) and allow χₜ to depend on expected scarcity, liquidity services, and risk (term structure consistent).
- Velocity channel: Endogenize Vₜ via inventory costs and expectations; model Vₜ as a function of χₜ and uncertainty to capture hoarding-induced convexity in pₜ.
- Microstructure link: Map net fiat inflows to pₜ using an impact kernel with state-dependent depth (Kyle-λ or quadratic impact) parameterized by Fₜ and off-chain liquidity.
- Expectations and closure: Impose a transversality condition and cointegration between price, adoption, and liquidity services; estimate χₜ with futures basis, lending rates, and order-book depth.
- Identification: Use exogenous issuance schedule as an instrument; decompose shocks into adoption,liquidity,and risk,and test for predicted convexity and asymmetry across regimes of Fₜ.
Calibration targets. Empirical discipline arises from aligning the state vector (M̄, Fₜ, Vₜ, χₜ) with observables. Calibrate M̄ as 21M net of losses; infer Fₜ from long-/short-term holder cohorts and custody data; proxy Vₜ with age-adjusted turnover; and estimate χₜ from the convenience yield (lending spreads, futures basis) and risk premia (option-implied volatility). Liquidity depth and cross-venue frictions anchor the impact kernel; adoption metrics anchor Dₜ’s secular trend; and the discount curve closes intertemporal conditions. The boundary condition implies that as adoption (and thus Dₜ) scales faster than Fₜ, estimated price elasticity shrinks and impact convexity rises-both are directly testable via regime-dependent impulse responses. Parameter stability should be checked pre/post issuance halvings and under liquidity regime shifts to ensure that χₜ and impact parameters are state-contingent rather than time-invariant.
| Target | Definition in model | Empirical proxy |
|---|---|---|
| Effective supply (M̄) | 21M minus unrecoverable stock | UTXO dormancy/loss heuristics |
| Free float (Fₜ) | Tradable subset of M̄ | LTH/STH cohorts; custody balances |
| Velocity (Vₜ) | Turnover of monetary balances | adjusted transfer volume; CDD |
| Demand (Dₜ) | Nominal demand for balances | Adoption S-curve; net fiat inflows |
| Convenience yield (ψₜ) | Utility of holding balances | Futures basis; lending rates |
| Risk premium (ρₜ) | Compensation for risk | Option-implied vol; realized skew |
| Impact depth (λₜ) | Price response to flows | Order-book depth within 10 bps |
| Liquidity share | on-/off-exchange availability | Venue balances; spreads |
Intertemporal choice liquidity constraints and optimal portfolio allocation in finite supply economies with recommendations for hedging consumption smoothing and leverage
In a finite-supply monetary regime, the intertemporal allocation problem is governed by a binding settlement constraint: agents optimize lifetime utility while balancing the shadow value of immediacy against the expected appreciation of a scarce asset. When the liquidity constraint tightens,the marginal utility of cash-like claims rises,compressing the optimal risky share despite higher expected excess returns. A tractable policy rule is to separate the portfolio into a high-certainty liquidity tranche and a duration tranche exposed to the scarcity premium, with state-contingent hedging to stabilize marginal utility of consumption. Practical diagnostics and instruments include:
- Liquidity buffer: hold multi-currency cash and short-duration bills sized to ≥ 12 months of net cash outflows.
- Volatility overlay: dynamic collars or put spreads on the hard asset to cap left tails during funding stress.
- Funding diversification: staggered maturities and non-correlation of collateral sources to reduce rollover risk.
- Convertibility bridges: pre-arranged lines/market-makers to monetize collateral without fire-sale slippage.
- rebalancing discipline: threshold-based rules tied to realized volatility and drawdown, not price levels.
Consumption smoothing is enhanced by matching the duration of cash commitments with the liquidity tranche while letting the scarcity asset express long-horizon growth in real purchasing power; leverage, if any, must be constrained by cash-flow coverage, not by mark-to-market equity.A parsimonious policy set is:
- Barbell allocation: safe cash/bills plus the scarcity asset; avoid intermediate beta that adds correlation without liquidity.
- Countercyclical hedge ratios: raise protection when funding spreads widen or basis dislocates.
- Leverage guardrails: maximum LTV set by worst-case cash flow, with automatic deleveraging triggers and pre-hedged liquidation.
- Tax-aware smoothing: harvest volatility via options rather than spot sales to preserve long-term basis.
| State | BTC weight | Cash buffer | Hedge ratio | Leverage cap |
|---|---|---|---|---|
| Calm, liquid | 40-60% | 6-9 mo | 0-10% | ≤ 0.2x LTV |
| Transition spike | 30-45% | 9-12 mo | 20-35% | ≤ 0.1x LTV |
| Stress, illiquid | 10-25% | 12-18 mo | 40-60% | 0x (unlevered) |
Rational expectations learning dynamics and equilibrium determinacy in finite supply monetary models with testable predictions and identification strategies
Interpreting the boundary condition ₿ = ∞/21M as a hard cap on nominal supply with possibly unbounded claims on real value reframes stability analysis around expectations rather than policy discretion. Under rational expectations, equilibrium paths are determinate when the scarcity premium endogenously adjusts to neutralize self-fulfilling demand surges: the expected real return must contract as the price level rises such that the perceived law of motion is E-stable (in the Evans-Honkapohja sense) and the Blanchard-Kahn eigenvalue configuration yields a unique saddle path. In finite-supply cash-in-advance or search-theoretic environments, determinacy obtains if (i) the semi-elasticity of demand to expected appreciation is bounded below a critical threshold, (ii) the no-bubble transversality condition excludes price paths that outgrow discounted liquidity services, and (iii) the convenience yield absorbs shocks sufficiently quickly to keep the expectation operator contractionary. These conditions jointly suppress sunspot equilibria: expectations can move prices only to the extent they alter the discounted scarcity premium, which in equilibrium cannot drift away from fundamentals imposed by the cap.
- E-stability criterion: learning dynamics converge when the Jacobian of the expectations feedback map has spectral radius below one.
- No-bubble constraint: discounted prices cannot exceed the present value of liquidity services plus terminal scarcity premium.
- Liquidity wedge: higher expected appreciation reduces money demand; determinacy requires this feedback to be dampening, not amplifying.
The boundary condition generates sharp empirical content under adaptive learning versus full-details rational expectations. With a known, deterministic issuance path, forecast errors around supply regime shifts (e.g., halvings) must be orthogonal to public information under rational expectations; systematic drifts imply learning. Identification exploits quasi-exogenous instruments that shift expected liquidity services or mining fundamentals without directly moving preferences. Empirically credible designs include IV-SVAR with sign restrictions, local projections around scheduled supply shocks, and state-space estimation of a time-varying scarcity premium. Testable implications include: (a) a fall in the variance of forecast errors across consecutive issuance epochs, (b) a tightening cointegration between prices and discounted stock-to-flow under learning convergence, and (c) option-implied densities that reprice the tail risk of indeterminacy after regime announcements.
| Hypothesis | Observable | Identification |
|---|---|---|
| E-stability holds | declining forecast error variance across epochs | Halving schedule as instrument |
| No-bubble constraint binds | Bounded price-liquidity ratio | Fee spikes as liquidity shocks |
| Determinacy (no sunspots) | Absence of regime-switch volatility without news | Exchange outages as info shifters |
| Learning vs. RE | Drift in option-implied tails pre/post regime | Narrative timing + local projections |
Empirical design data requirements and policy recommendations for evaluating boundary effects in crypto monetary systems
Identifying boundary effects implied by the formal relation ₿ = ∞/21M demands a design that integrates high-frequency microstructure with block-level state changes and jurisdictional discontinuities. Researchers should instrument for exogenous shocks at structural edges-such as halvings, mempool policy updates, L1/L2 liquidity gates, stablecoin de/pegs, and custody-self-custody transfers-while aligning clocks across venues and ledgers to enable causal designs (RD, event studies, difference-in-differences). Empirical completeness requires triangulating on-chain UTXO dynamics with off-chain order books and payment-channel telemetry to capture price, liquidity, and latency kinks where constraints (supply cap, divisibility granularity, fee market, and regulatory borders) bind.
- On-chain state: UTXO set diffs, coin age/size, realized cap, block template metadata, fee-rate histograms, orphan/uncle-like events, reorgs.
- Mempool microstructure: submission/eviction timestamps, package/RBF flags, feerate ladders, propagation latencies across relays.
- Exchange data: full-depth L2, tick-by-tick prints, cancellations, cross-venue spreads, funding/borrowing rates, options IV surface.
- L2 and routing: channel openings/closures, capacity and policy (base/ppm), HTLC success/failure, path length, rebalance flows.
- Cross-boundary events: bridge flows, stablecoin mint/burn, proof-of-reserves/liabilities attestations, API outages, jurisdictional rule changes.
- Synchronization and governance: nanosecond-level timestamps, canonical symbol/tick-size registry, venue-level survivorship flags, reproducible snapshots.
Policy should codify a public-good data layer that lowers identification error at boundaries where fixed supply meets effectively unbounded divisibility and layered settlement. Mandates and incentives must target verifiable disclosure (proof-of-reserves and liabilities), standardized schemas for on/off-chain microstructure, and privacy-preserving aggregation that balances research utility with user protection. Coordinated registries for protocol and venue events, plus testbed sandboxes for fee-policy and mempool changes, enable ex ante impact assessment of boundary adjustments.
- Open standards: Adopt an Open Crypto Monetary Data standard (OCMDS) for UTXO/mempool/exchange/L2 schemas, with signed metadata and versioning.
- Transparency requirements: Time-locked, Merkle-summed reserves and liabilities; venue-level latency, outage, and tick-size disclosures.
- Event registries: Canonical feeds for halvings, soft/hard-forks, fee-policy revisions, and bridge status; synchronized to PTP/NTP and block heights.
- L2 disclosures: Channel policy snapshots, path reliability metrics, and liquidity maps to quantify L1→L2 boundary spillovers.
- Privacy by design: Differential privacy or secure enclaves for user-level telemetry; open aggregation code and reproducible pipelines.
- Crisis playbooks: Circuit-breaker coordination for de/peg events and mempool congestion; minimum data continuity SLAs during stress.
| Variable | Boundary | Freq. | Source |
|---|---|---|---|
| Feerate ladder | Blockspace/fee | per second | Mempool relays |
| Depth at NBBO | Venue microstructure | Tick-level | Exchanges |
| Channel capacity | L1/L2 liquidity | Minute | Lightning graph |
| Mint/Burn flow | Stablecoin peg | Hourly | Issuer chains |
| Reorg rate | Finality edge | Per block | Node logs |
Closing Remarks
In sum, treating ₿ = ∞/21M as a boundary condition rather than a slogan maps absolute scarcity into a well-defined constraint on equilibrium selection.Within this framework, price formation reflects the interaction of an inelastic long‑run supply with time‑varying money demand, intertemporal substitution, and a scarcity (or convenience) yield. The resulting Euler restrictions connect expected appreciation, real rates, and liquidity services, while rational expectations discipline narratives about “digital gold” by imposing testable cross‑market no‑arbitrage relationships. The model thereby translates heuristic claims about finite supply into falsifiable propositions about volatility, velocity, fee dynamics, term premia, and the microstructure of a market organized around an exogenous cap.Several limitations qualify our results. Our baseline abstracts from heterogeneous liquidity needs, credit creation and collateral reuse, security‑budget feedbacks via fees and hashpower, and the institutional competition from fiat, stablecoins, and off‑chain claims. Network effects, layer‑2 settlement frictions, and regulatory shocks can also shift money demand in ways our reduced forms only partially capture. These omissions suggest concrete extensions: embedding the scarcity constraint in search‑theoretic or cash‑in‑advance environments; allowing heterogeneous agents with borrowing constraints and collateralized leverage; integrating miner behavior and fee market equilibria; and modeling adoption as an S‑curve that interacts with the convenience yield.The empirical agenda is equally clear. The theory implies moment conditions and event‑study designs around halvings, protocol changes, and regulatory announcements; cross‑sectional tests using futures basis, options‑implied premia, and funding rates; and time‑series restrictions linking velocity, fee levels, and hoarding/spending margins. Structural estimation-via GMM or Bayesian methods-can recover the convenience yield and its cyclicality,while microstructure evidence can verify predicted convex price impact under inelastic supply.Rejection of these joint restrictions would falsify the framework; confirmation would calibrate the magnitude of scarcity’s role in monetary substitution.By formalizing ₿ = ∞/21M as an equilibrium boundary, we move from rhetoric to measurement. The implication is not that finite supply guarantees any particular trajectory, but that it pins down the map from expectations and frictions to prices, quantities, and welfare. this provides a disciplined basis for evaluating policy interactions, assessing systemic complementarities with existing monetary arrangements, and, ultimately, situating a strictly capped asset within modern monetary economics.
