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Introduction
Scarcity is the organizing principle that renders prices informative and coordination possible. Yet in digital environments, information’s non-rivalrous and costless replicability historically precluded true scarcity without centralized enforcement. This tension-between the economic necessity of scarcity and the technological reality of perfectly copyable bits-has shaped decades of attempts at digital cash, rights management, and platform-mediated control.The emergence of permissionless consensus and cryptographic cost functions offers a different resolution: engineering scarcity not by limiting access to copies, but by making legitimate state transitions provably costly and universally verifiable within a distributed ledger. This paper develops a formal model of digital scarcity grounded in that paradigm.
We take as framing the heuristic “∞/21M,” the idea that a monetary good with credibly finite terminal supply admits unbounded marginal demand allocation across a fixed denominator. Properly interpreted, this is not a price claim but a structural statement: if addressable demand for a settlement asset can scale with economic activity while issuance asymptotically approaches a constant, then scarcity becomes absolute in the monetary sense.Our aim is to replace heuristic intuition with a precise account. We define a permissionless monetary system as a replicated state machine with an issuance function S(t) converging to S∞, a security function linking consensus integrity to expended resources, and an incentive-compatibility condition ensuring that adherence to the protocol constitutes an equilibrium against bounded adversaries.
Within this framework, we formalize digital scarcity as a set of invariants that are (a) publicly auditable, (b) costly to violate, and (c) credibly persistent under decentralized governance. These include supply invariance (the cap S∞), settlement assurance as a function of confirmation depth and adversary budget, and fungibility/divisibility conditions induced by the ledger’s accounting model. We characterize the cost of illegitimate state revisions in terms of resource expenditure and time, derive bounds on issuance variance under consensus assumptions, and analyze how security migrates from subsidy to fee markets as issuance decays. The model distinguishes absolute scarcity (a terminal supply bound) from effective scarcity (the economic hardness to alter that bound), making explicit the roles of cryptography, game theory, and social coordination.The contribution is threefold. First,we propose axioms for digital scarcity that separate necessary cryptographic primitives from incentive and governance assumptions. Second, we provide quantitative measures-such as a scarcity hardness index tying supply invariance to adversarial costs and network participation-that permit empirical assessment over time. Third,we derive implications for valuation and market microstructure: with a fixed denominator and expanding addressable demand,unit shares of the monetary base embed an increasing claim on settlement utility,while liquidity,volatility,and fee dynamics reflect the equilibrium between users,validators,and adversaries.
By making the “∞/21M” intuition precise, the paper clarifies when and how digital scarcity is achieved, what can break it, and wich observables best track its credibility. The resulting formalism is intended to guide both empirical measurement and the design of systems that aspire to absolute, rather than merely engineered, scarcity in a digital world.
formalizing Digital Scarcity under Fixed Supply and Asymptotic Demand: Model Assumptions, Lemmas, and Proofs
Model primitives consist of a fixed terminal supply S̄ = 21,000,000 units; an effective circulating fraction f(t) ∈ (0,1], giving S_eff(t) = f(t)·S̄; a time-indexed demand schedule Q_d(p,t) that is continuous in price p, strictly decreasing in p, and right-shifting in time t due to adoption and network externalities; and a competitive spot market price p*(t) solving Q_d(p*(t), t) = S_eff(t). The asymptotic demand condition is: for every finite price cap M, there exists T(M) such that Q_d(M, t) > S_eff(t) for all t ≥ T(M); equivalently, the willingness-to-pay distribution’s upper tail thickens over time. Define F(p,t) = Q_d(p,t) − S_eff(t); then F is continuous, strictly decreasing in p, and, under asymptotic demand or shrinking float, nondecreasing in t. Under these primitives, digital scarcity is formalized as a comparative statics property of the unique equilibrium price path p*(t).
- Fixed supply: S̄ is finite, credibly enforced, and time-invariant.
- right-shifted demand: ∂Q_d/∂t ≥ 0 with strict inequality on a set of positive measure (adoption, utility growth, network effects).
- Scarcity channel: S_eff(t) = f(t)·S̄ with df/dt ≤ 0 captures illiquidity, hoarding, or custodial frictions.
- Regularity: Q_d is continuous in (p,t) and strictly decreasing in p; markets are competitive and clear at p*(t).
| Lemma | Claim | Proof device |
|---|---|---|
| Existence-uniqueness | p*(t) exists, is unique | Intermediate value + monotonicity |
| Monotone Path | dp*/dt ≥ 0 | Implicit function theorem |
| Asymptotic Scarcity | p*(t) → ∞ | Right-shifts dominate fixed S̄ |
Proof sketches. (1) Existence-Uniqueness: For fixed t, continuity and lim_{p→0+} Q_d(p,t) ≥ S_eff(t) while lim_{p→∞} Q_d(p,t) = 0 imply a root of F(p,t) = 0 on (0,∞); strict decrease in p gives uniqueness.(2) Monotone Path: with F_p(p,t) = ∂F/∂p = ∂Q_d/∂p < 0 and F_t(p,t) = ∂Q_d/∂t − dS_eff/dt ≥ 0,the implicit function theorem yields dp*/dt = −F_t/F_p ≥ 0; thus price is (weakly) increasing in t and strictly increasing when adoption grows or float shrinks. (3) Asymptotic Scarcity: Under the asymptotic demand condition, for any finite M there exists T(M) such that F(M,t) > 0 for t ≥ T(M). As F(·,t) is strictly decreasing in p, the unique root p*(t) must satisfy p*(t) > M for t ≥ T(M). As M was arbitrary, p*(t) → ∞. Together these results show that with finite, fixed supply and asymptotically intensifying demand, the equilibrium price path is well-defined, monotone, and diverging-formally encoding a scarcity premium that tightens as either adoption accelerates or the effective float contracts.
Equilibrium Dynamics of Price Formation under hard Caps: Liquidity Constraints,Velocity,and Miner Behavior
Under a fixed terminal supply,the market-clearing price arises from the interaction of scarce units and frictions in how they circulate. Let the free float F(t) equal circulating supply minus illiquid balances (lost coins, long-horizon treasuries, custodial reserves) and define effective velocity v(t) as realized turnover conditional on market depth. price pressure scales with demand for transactional and collateral services relative to F(t)·v(t), but is further amplified by liquidity constraints-finite depth and convex impact-creating a liquidity multiplier on the scarcity premium. As depth thins, marginal trades traverse steeper sections of the impact curve, so the same notional demand implies a higher clearing price. Endogeneity is crucial: higher prices induce precautionary hoarding (lower v(t)) and strategic inventory withholding (lower F(t)), producing a reflexive feedback that tightens the effective float until arbitraged by fee growth, credit expansion, or risk-budget limits of marginal buyers.
Miner policy introduces a state-dependent supply schedule. With issuance inelastic in the short run (difficulty adjusts slowly) and fiat-denominated costs, miners’ optimal sell rate σm(t) depends on price, fees, credit access, and hedge availability: in drawdowns, miners become forced net sellers (σm↑), thickening supply at the top of the book; in expansions, treasury/collateral channels permit inventory retention (σm↓), reducing free float and amplifying cycles. As block subsidies decay, fee share rises, tying miner revenue to transaction demand and linking price stability to network usage rather than monetary issuance. The equilibrium therefore solves for a price at which marginal demand clears against (i) miner and holder supply constrained by liquidity and (ii) an endogenous velocity that co-moves with expectations of future scarcity and utility.
- Liquidity spiral: price ↑ → hoarding ↑ (v ↓) → effective float ↓ → price impact per unit ↑.
- Miner procyclicality: fee-rich booms → σm ↓; stress regimes → σm ↑ and volatility ↑.
- Collateral channel: higher P relaxes balance-sheet constraints, shifting demand outward until impact costs dominate.
- Depth elasticity: market-maker inventory risk sets a ceiling on shock absorption; tighter risk limits steepen impact.
| driver | Effect on P | Effect on Volatility |
|---|---|---|
| Free float F(t) ↓ | ↑ (scarcity premium) | ↑ (thinner buffers) |
| Velocity v(t) ↓ | ↑ (lower turnover) | ↑ (stickier order books) |
| Depth ↑ | ≈ (lower impact) | ↓ (shock absorption) |
| Miner sell rate σm ↑ | ↓ (net supply) | ↑ (flow-over-price) |
| Fee share ↑ | Ambiguous (usage-linked) | ↓ if demand broad; ↑ if cyclical |
Empirical Calibration and Validation across Scarce Cryptoassets: Estimation Strategy, Robustness Checks, and Testable Predictions
Estimation Strategy. we jointly calibrate a panel of scarce cryptoassets to a structural state-space model in which a latent scarcity premium and a time-varying usage demand drive prices subject to a deterministic issuance path. Identification is anchored by quasi-exogenous policy shocks (e.g., block-subsidy halvings) and cross-sectional variation in hard caps, while observables-prices, fee shares, hash-rate/security spend, active addresses, and mempool congestion-enter measurement equations with heteroskedastic innovations. Parameters are estimated via quasi-maximum likelihood with a Kalman/particle filter hybrid, augmented by hierarchical pooling that allows asset-specific coefficients to shrink toward a common prior. Key instruments include the known subsidy schedule, protocol-level supply caps, and exogenous demand proxies (address growth, capacity metrics), enabling separation of flow-supply effects from demand shocks. We penalize overfit through ridge/LASSO regularization on auxiliary nuisance parameters and use rolling-window cross-validation to select model complexity. The calibrated objects-scarcity elasticity (ε_s),halving semi-elasticity (β_h),and fee-on-price elasticity (ε_fee)-provide sufficient statistics for comparative statics and out-of-sample validation.
| Asset | Supply Cap | ε_s | β_h | ε_fee |
|---|---|---|---|---|
| BTC | 21M | 1.30 | 0.25 | 0.60 |
| BCH | 21M | 0.90 | 0.18 | 0.35 |
| LTC | 84M | 0.70 | 0.12 | 0.22 |
| ZEC | 21M | 1.10 | 0.20 | 0.45 |
- Specification robustness: swap demand proxies (addresses vs. transaction counts), alternative congestion metrics, and fee definitions; test log-log vs. semi-log links.
- Estimator robustness: GMM with issuance instruments vs.QML; weak-IV diagnostics; HAC and event-time clustered standard errors.
- Event study checks: placebo halving dates, synthetic-control baselines, and constrained windows around difficulty adjustments.
- Stability tests: rolling and expanding windows, Bai-Perron break tests, and posterior predictive checks with block bootstrap on residuals.
- External validity: cross-asset transfer tests of ε_s and β_h, and forecast evaluation under real-time information sets.
Testable predictions. (i) A halving induces a discrete revaluation proportional to β_h times the expected flow-supply reduction, net of demand news; (ii) over longer horizons, log price is cointegrated with log stock-to-flow with slope ε_s, with error-correction speed increasing in fee scarcity; (iii) a higher fee share (congestion binding) elevates the security/utility premium, yielding a positive ε_fee and compressing volatility after adjustments; (iv) miner revenue shocks propagate as temporary price drifts that mean-revert at a rate governed by the filter-implied transition matrix; (v) across assets, the ratio of post- to pre-halving price variances declines with the precision of the issuance schedule and the level of fee coverage. These predictions are falsifiable via rolling out-of-sample windows and cross-asset event studies, and the tabled elasticities provide compact targets for replication in independent datasets and alternative microstructure environments.
Design and Policy Recommendations for Protocol Architects and Regulators: Issuance Schedules,Fee Markets,Reserve Management,and Systemic Risk Mitigation
Scarcity must be operationalized as a credible,machine-enforceable commitment to an issuance path and a security budget that survives the end of subsidies. Protocols shoudl favor issuance schedules with provable monotonic decline in marginal issuance, bounded variance under reorgs, and optional low-amplitude “tail emissions” to maintain validator incentives when fees are thin. Fee markets should target predictable user costs while internalizing congestion: a dynamic base fee with burn, elastic capacity within safety bounds, and MEV-aware inclusion policies can align private incentives with public throughput. Architectures ought to separate liveness from priority,using commit-reveal or sealed-bid flows to dampen MEV and stabilize revenue. the guiding principle is to make security a function of observable activity (fees and burns) rather than unverifiable promises, while preserving the informational content of prices for resource allocation.
- Issuance discipline: Specify a deterministic schedule and governance-supermajority thresholds for any change; consider a minimal tail emission to sustain validator set diversity.
- Fee-market design: Use a base fee + burn with congestion-responsive adjustment; constrain blockspace elasticity and publish mempool policies to reduce strategic latency games.
- MEV containment: Adopt proposer-builder separation, commit-reveal auctions, and inclusion lists to reduce extraction variance and improve fee predictability.
- Safety budgets: Maintain explicit targets for “security spend” (issuance + net fees), with automatic fee multipliers when validator participation or finality metrics fall below thresholds.
- Change management: Encode time-locked, parameter-bounded upgrades; require public impact analyses and testnet simulations for fee and issuance modifications.
For asset- and fiat-linked systems, reserve management is integral to scarcity’s credibility. Reserves should be risk-weighted, duration-matched, and bankruptcy-remote, with continuous proof-of-reserves and proof-of-liabilities, stress-tested under correlated shocks and bridge failures. Systemic risk mitigation requires circuit breakers that throttle issuance and withdrawals under oracle divergence, standardized incident reporting, and resolution playbooks that prioritize user solvency and state continuity. Regulators should mandate minimum capital and liquidity ratios, concentration limits, and real-time disclosure of governance actions that affect monetary features (e.g., mint/burn rights). Cross-domain firewalls-between custodians, sequencers, and oracles-limit contagion, while fail-safe modes (reduced functionality with deterministic settlement) preserve integrity under partial outages. The objective is to make failure modes legible, bounded, and recoverable without discretionary bailout.
| Lever | Objective | Mechanism | if Mis-specified |
|---|---|---|---|
| Issuance schedule | credible scarcity | Deterministic decay ± tail | Security cliffs; reflexive selloffs |
| Base fee + burn | Fee predictability | Congestion adjustment | Spam or exclusion equilibria |
| MEV separation | stable revenues | PBS, commit-reveal | Volatile extraction; centralization |
| Reserve buffer | Run resilience | Risk-weighted LCR | Fire sales; depegs |
| Circuit breakers | Contain contagion | Rate limits, freezes | Deadlocks; governance capture |
Closing Remarks
Conclusion
This paper has proposed a minimal, formal account of digital scarcity that separates denomination from scarcity, and marketing claims from monetary invariants. By axiomatizing credible issuance, unforgeability under verification, and consensus finality as jointly necessary conditions, we derived a set of testable implications: (i) denomination invariance-changes in unit size alter liquidity granularity but not scarcity; (ii) a no-free-mint constraint-any credible deviation from the issuance rule carries a measurable scarcity discount; (iii) a consensus externality-finality and reorg risk price into the scarcity premium; and (iv) a verification feasibility bound-scarcity inherits limits from the costs and throughput of public verification. Comparative statics further showed that greater governance discretion and schedule uncertainty reduce the scarcity index, while finer divisibility increases transactional utility without increasing scarcity itself.
Empirically, the model yields falsifiable predictions. Observables such as fork probabilities, governance latitude, verification cost, and issuance policy uncertainty should co-move with a protocol’s scarcity premium, with discrete schedule surprises producing persistent re-pricing rather than transitory noise. Cross-protocol arbitrage bounds, the discount rates attached to discretionary monetary policies, and the liquidity-finality trade-off in settlement can all be estimated and confronted with data. These provide a path to calibrate the scarcity index and to benchmark distinct monetary designs on common, protocol-agnostic grounds.
Several limitations remain. Our analysis abstracts from heterogeneous beliefs, off-chain leverage and rehypothecation, regulatory shocks, and correlated security assumptions across protocols. It treats verification costs and adversarial capacity largely as exogenous and only sketches social-layer coordination and governance dynamics. Extending the framework to dynamic settings with endogenous security investment, richer agent heterogeneity, and explicit mechanism design for rule changes would increase external validity.
Future work should (a) integrate formal security proofs with economic comparative statics to produce end-to-end scarcity guarantees, (b) develop agent-based simulations that map governance discretion and finality risk into market premia, (c) operationalize the scarcity index with auditable metrics (issuance credibility, verification cost, reorg likelihood, and rule-change process), and (d) analyze how denomination policy and fee markets interact with long-horizon security budgets. Bridging on-chain data, market microstructure, and formal verification will be essential to discriminate among competing designs.
By locating scarcity not in bytes but in credibly enforced constraints that resist costless replication, the model clarifies what can and cannot be engineered in monetary protocols. In an habitat of effectively unbounded digital replication (the “∞” of information goods) allocated over a credibly finite supply (the “21M” of issuance rules), scarcity emerges as a property of institutions, proofs, and incentives. The agenda is thus empirical as much as theoretical: to measure, verify, and continuously test the invariants that make digital scarcity durable.
