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Capital Costs Protect the Network: Mining Gap Dynamics in Proof-of-Work Blockchains Under a Stochastic Mempool

Hasan Selçuk Kılıç, Nail Akar

arXiv:2610.22451v1cs.CR

TL;DR

Fee-dominated Proof-of-Work networks can have idle periods when transaction fees do not cover miners’ operating costs, raising security concerns. The paper models the high-fee backlog with a Lindley-type recursion and derives the mining-gap distribution. It finds that the expected gap grows with hashpower and approaches a full round as hashpower becomes unbounded, while capital costs keep equilibrium hashpower finite.

  • Problem

    As block subsidies decline, transaction fees may not cover miners’ costs, raising concerns about participation and network security.

    Source: §1 · p. 2

  • Method

    The paper models the high-fee backlog with a Lindley-type recursion that has a reflecting barrier at zero and a floor at the mining threshold.

    Source: §6.1 · p. 15

  • Results

    Expected mining gap grows strictly with hashpower and tends to a full round as hashpower becomes unbounded; capital costs keep equilibrium hashpower finite.

    Source: §15 · p. 36

  • Takeaways & Limitations

    Revenue available immediately keeps miners active, while revenue that accumulates only after a delay can make waiting optimal even as the idle phase weakens the network.

    Source: §15 · p. 37

    The non-atomic model excludes participants with a non-negligible share of hashpower, such as large mining pools.

    Source: §3.3 · p. 8

Abstract

from arXiv · show

Proof-of-Work blockchains increasingly rely on transaction fees to keep miners engaged as block subsidies decline. An immediate outcome of this is that miners may sit idle at the start of a round until enough high-fee transactions accumulate in order to make mining worthwhile. This idle period is known as the mining gap. We model the high-fee transaction backlog as a Lindley-type recursion under a two-tier fee market and use its stationary distribution to characterize the mining gap. Under Bitcoin's difficulty-adjustment mechanism, the expected gap grows strictly with the hashpower active on the network and consumes the entire round as that hashpower grows without bound. A free-entry condition determines a unique long-run equilibrium hashpower. Capital cost caps the equilibrium hashpower and prevents the mining gap from occupying the entire round. We further characterize how the block subsidy, transaction fees, and operating cost shape the equilibrium hashpower and the mining gap.

1 Introduction

The paper models mining gaps arising from fee backlogs that persist across rounds and examines their distribution, security, and equilibrium consequences. It considers how block rewards, operating and capital costs, and participants’ history-based hashpower choices shape these dynamics.

1 Introduction

  • Mining gaps leave the honest chain stationary while attackers can mine immediately, creating opportunities for double-spending, selfish mining, and fee sniping without majority hashpower.The gap occurs when accumulated fees cannot cover operating costs; unlike operating costs, capital costs continue while hardware is idle.Source: §1 · pp. 2–3
  • The paper formalizes miners’ long-run profit-rate objective and shows that activating as soon as reward rate covers operating cost is weakly dominant for each negligible miner.The rule is myopic because it ignores future rounds, yet no alternative strategy yields a higher long-run profit rate regardless of other miners’ strategies.Source: §1 · p. 3
  • The paper derives a closed-form stationary mining-gap distribution from a two-tier fee market and a Lindley-type backlog recursion that preserves high-fee transaction carryover.The recursion has a floor reflecting that mining starts only after enough high-fee transactions accumulate.Source: §1 · p. 3
  • As aggregate hashpower rises, the expected mining gap grows strictly and approaches a full round; positive capital costs keep equilibrium hashpower finite and prevent that outcome.A larger block subsidy shortens the gap and raises equilibrium hashpower.Source: §1 · p. 4
  • With free entry and estimated costs, Bitcoin’s gap is projected to begin in 2044 or 2056, or not before 2070 if fees fall in bitcoin; by 2070 it lasts about 2.5 minutes per ten-minute round.Without miner entry or exit after 2026, a gap would arise after one or two halvings unless bitcoin’s price rose enough to offset them.Source: §1 · p. 4

2.1 Gap Games

  • Unlike most prior gap models, the paper retains stochastic fee carryover across blocks and analytically derives its dynamics and stationary behavior rather than relying on reset-to-zero rounds or simulation.Existing security analyses that need gap duration take it as an input; this paper derives its distribution and studies its response to hashpower and network economics.Source: §1 · p. 3, §2.1 · p. 5, §2.3 · p. 6

3.3 The Player Space

  • The model treats miners as a non-atomic population, so individual decisions cannot change aggregate outcomes; it excludes large miners and their size-dependent strategies.A model combining negligible miners with a few large ones is left for future research.Source: §3.3 · p. 8

3.4 The Mempool

  • The analysis assumes a healthy, fork-free network and a two-tier mempool: high-fee transactions arrive over time, low-fee transactions fill unused block capacity, and the backlog persists across epoch boundaries.The two-tier reduction supports tractability but does not represent every individual fee; including forks or many fee levels is left for future work.Source: §3.4 · pp. 9–10

3.5 Block Reward

  • High-fee arrivals raise block rewards linearly before saturation, up to Rmax; additional arrivals beyond block capacity do not increase fee revenue.Blocks include available Type-1 transactions up to capacity and fill any remaining capacity with Type-0 transactions.Source: §3.5 · p. 10

3.6 Costs

  • Operating costs are paid only while hashpower is active, whereas positive capital costs apply whether it operates and therefore do not affect activation decisions.Operating costs may vary across participants; the passage imposes only measurability and positivity at this stage.Source: §3.6 · pp. 10–11

3.7 Strategies

  • Participants choose their active hashpower fraction using only publicly observed history up to the current time, without access to future information.Strategies map public histories to active fractions and must be non-anticipative.Source: §3.7 · p. 11
  • Aggregate active hashpower is at most total hashpower, with equality only when every participant operates at full capacity.The population strategy profile assigns a strategy to each participant and defines aggregate active hashpower across participants.Source: §3.7 · p. 11

4 Miner Objective and the Myopic Activation Rule

The paper formulates mining as a long-run profit-rate problem and shows that, for individually negligible miners, a myopic profitability threshold is weakly dominant. The analysis assumes a stabilized fee regime with a stable high-fee backlog.

4 Miner Objective and the Myopic Activation Rule

  • The model evaluates miners by long-run expected profit per unit time, rather than optimizing a single round in isolation.For negligible participants, maximizing that objective is equivalent to maximizing expected profit in each round separately.Source: §4 · p. 12
  • Because a negligible miner cannot change aggregate network states, its long-run objective decomposes into per-round profit maximization.The backlog trajectory, round durations, and subsequent winning probability remain invariant to that participant’s actions.Source: §4 · p. 12
  • Because reward is non-decreasing over time, the optimal strategy has one continuous active phase beginning when net profit becomes non-negative.The miner activates at the threshold and remains active for the rest of the round.Source: §4 · pp. 12–13
  • The myopic threshold rule is weakly dominant: activate when the reward rate covers operating cost, regardless of other participants’ strategies.Under the rule, the miner remains active until the round ends.Source: §4 · p. 13

4.1 Operating Regime and Stability Restrictions

  • The analysis assumes a stabilized fee regime with some profitable miners and a stable high-fee backlog; transition periods are out of scope.The arrival rate is restricted to levels that maintain backlog stability across rounds.Source: §4.1 · pp. 13–14

5 Aggregate Behavior Under a Common Cost

The aggregate analysis assumes a common operating cost, producing a shared mining threshold. That threshold determines when the population activates as Type-1 transaction supply accumulates.

5 Aggregate Behavior Under a Common Cost

  • The aggregate analysis imposes a common operating cost, although the individual myopic rule permits heterogeneity; heterogeneous costs would spread activation thresholds and require a different backlog analysis.Participants with lower costs would begin mining before those with higher costs.Source: §5 · p. 14
  • The threshold is the supply level where reward first covers operating cost in the non-saturated regime, floored at zero when an empty mempool already suffices.In that zero-threshold case, no mining gap forms.Source: §5 · p. 14
  • With common costs, miners share a threshold, so aggregate hashpower switches from idle to active at one total Type-1 supply level.The population acts as a single unit apart from a zero-mass set of participants.Source: §5 · pp. 14–15

6 Backlog Dynamics

Section 6 models the high-fee transaction backlog within an epoch and derives its stationary distribution and the resulting mining-gap distribution. Carryover makes the gap random, with both zero-gap and maximum-gap outcomes possible.

6.1 The Recursion

  • Mining begins when the backlog reaches threshold s1; after a block, up to sb transactions are cleared and the remainder carries into the next round.The resulting Lindley-type recursion has a reflecting barrier at zero and a floor at s1 because mining never begins below that threshold.Source: §6.1 · p. 15

6.2 Stationary Backlog Distribution

  • Under λ/μ < sb and s1 < sb, the backlog has a unique stationary distribution and converges to it from every initial condition.Its tail is Pr(X > x) = e^-K(sb-s1+x) for x ≥ 0, with an atom p0 at zero.Source: §6.2 · p. 16

6.3 Mining Gap Distribution and Expectation

  • The stationary mining gap has atoms at zero and at its maximum s1/λ, with masses e^-Ksb and p0, respectively.The zero-gap atom corresponds to carryover backlog already meeting the threshold; the maximum-gap atom corresponds to an empty mempool.Source: §6.3 · pp. 16–17
  • Unlike a resetting mempool, carryover makes the gap random and shortens its mean; as K grows, the shortening term vanishes and the backlog behaves as if it reset each block.With a resetting mempool, every round starts empty and the gap is s1/λ in every round.Source: §6.3 · p. 17

7 The Difficulty Adjustment Map and Stability

Section 7 connects the expected round length to Bitcoin’s difficulty-adjustment map and identifies its invariant winning probability. It also gives conditions for the invariant point’s local stability.

7.1 The Mean Round Length

  • The mean round length m(w) strictly decreases with winning probability because the mining threshold falls and aggregate block-finding rate rises.It combines the expected gap with an active phase of expected duration 1/μ.Source: §7.1 · pp. 17–18

7.2 The Protocol Map

  • In the unclamped region, the protocol map adjusts winning probability according to realized average round length, correcting block times that differ from target B.The analysis assumes the clamp is inactive; it binds only when the network is far from its invariant point.Source: §7.2 · pp. 18–19

7.3 The Invariant Winning Probability

  • When λB < sb, there is a unique invariant winning probability w* satisfying m(w*) = B.The mean round length is continuous and strictly decreasing, tending to sb/λ at the lower end and to zero as w grows without bound.Source: §7.3 · p. 19

7.4 Local Stability

  • The invariant point is locally asymptotically stable when ε(w*) > -2 and unstable when ε(w*) < -2; without a gap, ε(w*) = -1.With a gap, stability depends on equilibrium values; at large aggregate hashpower, the limiting condition is that the fee premium over a target round exceed baseline block revenue.Source: §7.4 · p. 20

8 Security Metrics

Section 8 defines three measures of network security: mining-gap vulnerability, aggregate hashpower, and hashwork. They capture different attack dimensions and are coupled by difficulty adjustment.

8.1 Mining Gap

  • Expected mining gap E[Tg] measures vulnerability because honest miners are idle during the gap, increasing adversarial success probability for less than half of total hashpower.The gap violates the continuous-honest-participation assumption in Nakamoto’s classical double-spending analysis for a fraction approximately E[Tg]/B of each round.Source: §8.1 · p. 21

8.2 Aggregate Hashpower

  • Aggregate hashpower γ measures resistance to hashpower-based attacks because a higher total makes a given share more costly to acquire in hardware and energy.A higher aggregate hashpower also raises the absolute hashpower behind any given network fraction.Source: §8.2 · p. 21

8.3 Hashwork

  • Expected hashwork W measures brute-force attack cost, while difficulty adjustment couples it to aggregate hashpower and gap length.In the illustrated scenarios, catching up requires just over 33% of combined hashpower in Blocks 1 and 2, versus just over 20% in Block 3.Source: §8.3 · p. 22

9 Aggregate Hashpower and the Vanishing Active Phase

Under the invariant difficulty-adjustment mechanism, mining gaps emerge above a hashpower threshold and grow with aggregate hashpower; positive capital costs instead yield a unique finite free-entry equilibrium.

9.2 The Aggregate Active Block-Finding Rate vs The Aggregate Active Hashpower

  • Above γ0, the active block-finding rate μ* and expected gap both increase strictly with aggregate hashpower γ.The difficulty adjustment lowers the winning probability as hashpower rises, leading miners to wait for a higher fee threshold and leaving less time for active mining.Source: §9.2 · p. 23, §9.3 · p. 24

9.1 When Does a Mining Gap Form?

  • For λB < s_b, a mining gap exists at invariant winning probability if and only if aggregate hashpower γ exceeds γ0.At or below γ0, baseline block revenue covers operating costs and no gap forms.Source: §9.1 · p. 23, §9.3 · p. 24

9.3 The Vanishing Active Phase

  • As γ grows without bound, μ* grows without bound and the expected gap approaches B, so the active mining phase vanishes.The hashwork W = γ/μ* increases toward C.Source: §9.3 · pp. 23–24
  • As the gap approaches B, an adversary’s required absolute hashpower rises while its required share of total hashpower vanishes.The passage gives the exact requirement as γ_adv > W/B.Source: §9.3 · p. 24
  • The full-round gap limit presumes difficulty adjustment settles at invariant winning probability w*; for large γ, that stability depends on fee-premium revenue being sufficiently large.When baseline block revenue is larger, the invariant probability is unstable for large γ; cited work finds oscillations across epochs for a mempool without carryover.Source: §9.3 · p. 24

10 Free Entry and the Equilibrium Hashpower

  • Positive capital cost produces a unique finite free-entry equilibrium γ*, where entry occurs below equilibrium and exit above it.With zero capital cost, expected profit remains positive for every γ, so no finite equilibrium exists; Figure 5 depicts the revenue-cost intersection at γ*.Source: §10 · p. 25, Figure 5 · p. 26

11 Preventing the Mining Gap

The no-gap equilibrium depends on baseline revenue and costs relative to the fee premium. In the gap regime, subsidy, fees, and costs have distinct effects on the gap, hashwork, and equilibrium hashpower.

11 Preventing the Mining Gap

  • A no-gap equilibrium occurs if and only if the equilibrium hashpower γ* does not exceed γ0; high baseline reward or capital cost favors this regime, while high fee premium or operating cost pushes toward a gap.The relevant parameter comparisons hold r1 fixed when r0 increases and r0 fixed when α increases.Source: §11 · p. 26
  • Higher capital cost can eliminate the mining gap but lowers equilibrium aggregate hashpower, whereas a higher block subsidy helps prevent the gap and supports more hashpower.Table 1 summarizes parameter effects on gap prevention and equilibrium hashpower.Source: §11 · p. 27, Table 1 · p. 27

12 Reducing the Mining Gap

  • In the gap regime, a higher block subsidy lowers the honest-miner activation threshold, shrinks the gap, and raises both hashwork and equilibrium aggregate hashpower.Table 2 summarizes parameter effects in the gap regime.Source: §12 · p. 27, Table 2 · p. 28
  • A higher low-tier fee r0 decreases the gap and increases hashwork, but its effect on equilibrium aggregate hashpower depends on the cost structure.The direction changes at threshold Γ★.Source: §12 · p. 27, Table 2 · p. 28
  • A higher fee premium α lengthens the gap while raising both hashwork and equilibrium aggregate hashpower.The paper attributes the longer gap to difficulty adjustment shortening the active phase as more hashpower enters.Source: §12 · p. 28, Table 2 · p. 28
  • Higher capital cost shortens the gap but reduces equilibrium aggregate hashpower and honest hashwork per block.It lowers the equilibrium aggregate block-finding rate by reducing hashpower entry.Source: §12 · p. 28, Table 2 · p. 28
  • Higher operating cost lengthens the gap and lowers both hashwork and equilibrium aggregate hashpower.The passage gives higher electricity prices as an example of higher operating cost.Source: §12 · p. 28, Table 2 · p. 28

13 A Numerical Illustration for Bitcoin

The Bitcoin illustration follows subsidy halvings from 2026 to 2070 under fixed-hashpower and free-entry settings, across scenarios for prices and fees. It finds that free entry and capital costs can postpone the mining gap, while gap length and stability depend on costs, fees, and the scenario.

13 A Numerical Illustration for Bitcoin

  • The illustration compares fixed hashpower with yearly free-entry equilibrium from 2026 to 2070, varying price and fee scenarios as the subsidy halves.The scenarios illustrate the model and are not forecasts of prices, fees, or mining technology.Source: §13 · p. 28

13.1 Inputs Without Units

  • The model’s inputs reduce to unit-free ratios, including hashpower relative to the no-gap threshold or, under free entry, the operating-to-capital-cost ratio Γ.The two reported security measures are expected gap length and the hashpower share above which an uninterrupted attacker outpaces the honest chain on average.Source: §13.1 · pp. 28–29

13.2 Estimating the Inputs

  • The 2026 estimates set Γ = 1.5 and γ/γ0 = 0.60, both below the no-gap threshold; estimated inputs come from Bitcoin block data and a two-tier approximation of its mempool.Without the subsidy, the estimated ν is 0.29; the authors note that the two-tier approximation makes estimates prone to error.Source: §13.2 · pp. 29–30

13.3 Free Entry Postpones the Gap

  • With free entry at Γ = 1.5, the gap begins in 2056 under S0 and S2, in 2044 with growing fees, and not before 2070 when fees fall in BTC; fixed hashpower reaches a gap in 2028 under S0.Under free entry, the ratio reaches 2.5 by 2070 under S0 and S2, while it remains below one through 2070 under S1.Source: §13 · p. 28, §13.3 · p. 31, Figure 6 · p. 32

13.5 The Length of the Mining Gap

  • With fixed hashpower under S0, the gap reaches 9.9 minutes by 2070 and the adversary needs 1.1% of total hashpower; under free entry at Γ = 1.5, it reaches 2.4 minutes and 43%.For free entry at Γ = 3, the gap reaches 4.0 minutes by 2070 and the adversary needs 37% of total hashpower.Source: §13.5 · pp. 32–33, Figure 8 · p. 34

13.4 Capital Costs Protect the Network

  • With fees constant in BTC, the gap begins in 2064 at Γ = 0.5, 2056 at Γ = 1.5, 2052 at Γ = 3, and 2044 at Γ = 10; growing fees bring these dates forward.When the subsidy approaches zero, the model predicts a gap in every scenario when Γ exceeds 0.29.Source: §13.4 · pp. 31–32, Figure 7 · p. 33

13.6 Stability Is Not Guaranteed

  • The invariant point is stable when the Γ = 1.5 gap begins, but unstable at the Γ = 10 crossing; under fixed hashpower, it remains unstable until 2059.The analysis is local and reports the invariant point in unstable years rather than modeling the dynamics around it.Source: §13.6 · pp. 33–34

13.7 Limits of the Illustration

  • The numerical results depend on a two-tier approximation of a many-fee mempool, the fee regime, and long-run free entry, while sunk hardware costs can make actual adjustment gradual.In congested early 2024, ν without the subsidy was about 1.0 rather than 0.29.Source: §13.7 · p. 34

14 Discussion

The discussion identifies scope boundaries in the cost, backlog, difficulty-adjustment, and fee assumptions. The conclusion argues that hashpower, capital cost, subsidy, and fee policy shape mining gaps and other security metrics differently.

14 Discussion

  • The difficulty-adjustment analysis is local, so unstable invariant points and their possible cross-epoch oscillations require analysis of backlog, entry, and difficulty jointly.The paper leaves that joint analysis for future work.Source: §14 · p. 35
  • The security interpretation is qualitative: expected gap length is an input for attack analysis, while attack success probabilities remain for future work.During a gap, an adversary mining continuously faces no honest competition.Source: §14 · p. 36
  • Mining gap, hashwork, and equilibrium aggregate hashpower do not move in concert under the studied parameter changes, so evaluations must specify the target metric and its cost.The discussion points to Tables 1 and 2.Source: §14 · p. 36

15 Conclusion

  • Unbounded aggregate hashpower makes the gap approach the full round, while capital cost caps equilibrium hashpower and prevents that limit in the model.With Bitcoin data, cheap hardware brings the gap forward to 2044 and lengthens it to six and a half minutes by 2070; high relative capital cost delays it until 2064 and keeps it below half a minute through 2070.Source: §15 · p. 36
  • Among the studied parameters, only the block subsidy shortens the gap while raising both equilibrium hashpower and computational work per block; a higher fee floor also shortens gaps but raises users’ transaction costs.The effect of a higher fee floor on equilibrium hashpower depends on the network’s cost structure.Source: §15 · pp. 36–37
  • The results are an optimistic limit because the model excludes forks and assumes difficulty settles at a steady puzzle difficulty.The authors expect real networks to fare no better as the subsidy shrinks.Source: §15 · p. 37

EC.2 The Myopic Activation Rule

The appendix establishes a myopic activation rule as weakly dominant and derives the stationary backlog and mining-gap distribution. The backlog follows a Lindley-type queue recursion, yielding a unique stationary law under the stated stability condition.

EC.2 The Myopic Activation Rule

  • The rule that activates a miner once current reward rate covers operating cost is weakly dominant for every miner in the non-atomic population.No alternative strategy earns a higher long-run profit rate regardless of other miners’ strategies.Source: EC.2 The Myopic Activation Rule · pp. 42–43

EC.3 The Backlog and the Mining Gap

  • The stationary backlog has an atom at zero and an exponential density above zero; its law is the unique stationary law and the backlog converges to it from every initial condition.The density coefficient is determined by normalization and the stationary equation.Source: EC.3 The Backlog and the Mining Gap · pp. 44–45
  • The high-fee backlog’s excess over the activation threshold follows a Lindley recursion with negative-mean increments, giving a unique stationary distribution and convergence from any initial condition.The recursion corresponds to queue waiting time with deterministic interarrival times and exponential service times.Source: EC.3 The Backlog and the Mining Gap · pp. 44–45
  • The stationary mining gap has an atom at zero, another at s1/λ, and a distribution and mean given by the backlog law.The two atoms have probabilities e^-Ks_b and p0, respectively.Source: EC.3 The Backlog and the Mining Gap · p. 45

EC.4 The Difficulty Adjustment Map and Stability

The difficulty-adjustment map has a unique invariant winning probability, and its local stability depends on the elasticity of mean round length. At high hashpower, the backlog approaches a resetting-mempool regime for the elasticity result.

EC.4 The Difficulty Adjustment Map and Stability

  • The mean round length is continuous and strictly decreasing in winning probability, yielding a unique invariant value where mean round length equals the target B.The map’s invariant point is locally stable when elasticity lies between −2 and 0, and unstable below −2.Source: EC.4 The Difficulty Adjustment Map and… · p. 46
  • At the invariant point, local stability holds when ε(w*) is between −2 and 0; values below −2 imply instability.The criterion follows from the derivative of the difficulty-adjustment map at its fixed point.Source: EC.4 The Difficulty Adjustment Map and… · p. 46
  • As hashpower grows without bound, the backlog-empty probability tends to one and the gap-case elasticity approaches the resetting-mempool expression −1 − (R_b + s_b r_0)/((r_1 − r_0)λB).This is the elasticity obtained for a mempool that resets after every block.Source: EC.4 The Difficulty Adjustment Map and… · p. 47

EC.5 Aggregate Hashpower

The appendix characterizes equilibrium aggregate hashpower as operating cost relative to miner revenue. Above the threshold cost ratio, equilibrium winning probability, gap threshold, and expected gap rise with hashpower, while aggregate hashpower approaches a finite cap.

EC.5 Aggregate Hashpower

  • For γ≤γ0, uniqueness gives w*=1/(γB), m(w*)=B, and no gap; for γ>γ0, μ*=H⁻¹(c̄opγ) is continuous and strictly increasing in γ.For γ>γ0, E[Tg]=B⁻¹/μ* and W=γ/μ* are continuous and strictly increasing in γ.Source: EC.5 Aggregate Hashpower · p. 48
  • As γ grows without bound, the gap threshold approaches λB, expected gap approaches B, and aggregate hashpower W approaches the finite cap C from below.The gap therefore approaches the full target round length in this limit.Source: EC.5 Aggregate Hashpower · pp. 47–48

EC.6 Free Entry and the No-Gap Condition

Free entry yields a unique long-run equilibrium hashpower, with capital cost ensuring the equilibrium remains finite. The paper also characterizes when the equilibrium has a mining gap and how parameters change its length.

EC.6 Free Entry and the No-Gap Condition

  • Capital cost makes expected profit cross zero exactly once, determining a unique equilibrium hashpower; without capital cost, profit has no zero.Expected profit is strictly decreasing in hashpower and tends to a negative limit because capital cost is positive.Source: EC.6 Free Entry and the No-Gap Condition · p. 48
  • The equilibrium has no gap exactly when c̄_op/c̄_cap ≤ R0/(αλB); in that case, γ* = (R0 + αλB)/((c̄_cap + c̄_op)B).Otherwise, the equilibrium lies in the gap regime.Source: EC.6 Free Entry and the No-Gap Condition · p. 48
  • In the no-gap equilibrium, γ* rises with R_b and α, and falls with c̄_cap and c̄_op.At fixed r1, increasing r0 raises R0 + αλB because its derivative is D > 0.Source: EC.6 Free Entry and the No-Gap Condition · p. 49

EC.7 Parameter Effects in the Gap Regime

  • In the gap regime, a higher block subsidy or low-tier fee shortens the gap, while a higher fee premium lengthens it.The subsidy effect is stated in p28; the low-tier-fee and fee-premium effects are stated in p31 and p19, respectively.Source: EC.7 Parameter Effects in the Gap Regime · pp. 49–50
  • In the gap regime, higher capital cost shortens the gap, whereas higher operating cost lengthens it.These effects follow from the respective signs of the equilibrium change in μ*.Source: EC.7 Parameter Effects in the Gap Regime · p. 50

EC.8 Estimating the Fee Inputs

Bitcoin block data are used to estimate fee inputs for the two-tier model. Robust fits yield baseline fee and Type-1 arrival estimates, while a model approximation overstates observed transaction carryover.

EC.8 Estimating the Fee Inputs

  • 0.0045 BTC is the fitted fee intercept, and the slope implies (r1 − r0)λB = 0.0154 BTC per target round.The robust fit uses 737 blocks whose predecessor was found within two minutes and whose own interval was at most five minutes.Source: EC.8 Estimating the Fee Inputs · p. 51, Figure EC.1 · p. 52
  • At p = 20%, the estimated Type-1 arrival rate is λ = 73,000 vB per minute, with ξ = λB/s_b = 0.73.The rate is inferred from Type-1 transaction size growth; estimates remain between 70,000 and 75,000 vB per minute through p = 100%.Source: EC.8 Estimating the Fee Inputs · p. 51
  • The two-tier model predicts about 225,000 vB of Type-1 carryover, above the observed average of about 160,000 vB.The authors identify treating all Type-1 transactions alike as an approximation behind the difference.Source: EC.8 Estimating the Fee Inputs · p. 51
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