A Model and Estimation of the Bitcoin Transaction Fee

1. Problem Statement

Bitcoin transaction fees are becoming increasingly critical as the block subsidy (mining reward) declines due to periodic halvings. However, studying fee formation is challenging because standard blockchain data is insufficient; it lacks the unobserved queueing environment (the mempool state) at the moment a transaction fee is chosen. Without knowing the congestion level and the distribution of competing fees, it is difficult to separate the "technology" of confirmation delay from the user's "willingness to pay."

The authors aim to:

1. Develop a structural model of Bitcoin fee choice treating the mempool as a market for scarce blockspace.
2. Estimate the relationship between transaction fees, priority, and network congestion.
3. Quantify how specific institutional mechanisms (like RBF and CPFP) and network states affect fee levels.

2. Data and Methodology

Data Source

The authors constructed a novel, high-frequency mempool panel using a self-run Bitcoin Core node (v30.0.0) operated by the MIT Media Lab Digital Currency Initiative.

• Timeframe: August 3, 2025 – October 9, 2025 (~9 weeks).
• Volume: ~11.1 million confirmed transactions across 1,988 epochs (30-minute intervals).
• Key Observables: Transaction entry/exit times, block inclusion, fee-bumping events (RBF), child-pays-for-parent (CPFP) relationships, and high-frequency snapshots of mempool congestion (size, transaction count, backlog).
• Data Cleaning: Corrected SegWit weight undercounting (merging with mempool.space API data) and collapsed CPFP packages into single "package" observations to reflect true miner priority.

Theoretical Framework

The paper adapts the Huberman et al. (2021) model, treating the mempool as a Vickrey-Clarke-Groves (VCG) mechanism.

• Core Logic: Miners select transactions to maximize revenue per unit of blockspace (fee rate). Users choose fees to minimize the sum of the fee paid and the cost of delay.
• Equilibrium: In equilibrium, fees equal the expected delay externality imposed on lower-priority transactions.
• Key Equation: The model derives a log-linear relationship:
  $$\log(\text{Fee}) \approx \log(\text{Delay Cost}) + \log(\text{Delay Gradient}) + \log(\text{Weight}) + \epsilon$$
  Where the Delay Gradient ($D$) is the marginal reduction in confirmation time achieved by moving up one percentile in the fee queue.

Empirical Strategy (Two-Stage Estimation)

The authors employ a two-stage approach to handle the endogeneity of the delay technology:

Stage 1: Estimating the Delay Technology

• Goal: Estimate the expected confirmation delay ($W$) as a function of fee-rate priority ($p$) and network state ($s$).
• Method: A Random Forest Regressor is trained to predict log-confirmation delay.
• Constraint: To satisfy economic theory (higher priority cannot increase delay), the authors enforce monotonicity using isotonic regression on the forest's predictions.
• Output: The local slope of the delay schedule ($\hat{W}'$), representing the marginal value of priority (the delay gradient).
• Validation: Cross-fitting is used to generate out-of-sample predictions to avoid overfitting bias in the second stage.

Stage 2: Fee Equation Estimation

• Goal: Estimate how fees respond to the estimated delay gradient and transaction characteristics.
• Method: An OLS regression on log-fees with epoch fixed effects (to control for aggregate time shocks) and clustered standard errors.
• Equation:
  $$\log(\text{Fee}_{it}) = \alpha_0 + \alpha_1 \log(\hat{W}'_{it}) + X'_{it}\beta + S'_t\theta + \xi_t + \epsilon_{it}$$
  • $X_{it}$: Transaction controls (RBF flag, CPFP status, size, type).
  • $S_t$: State controls (blockspace utilization, time since last block).
  • $\xi_t$: Epoch fixed effects.
• Impatience Proxy: A re-spend-based proxy for user impatience was tested but found to have limited explanatory power in the baseline.

3. Key Results

Structural Findings

1. Congestion is Dominant: The primary determinant of confirmation delay is network congestion (mempool backlog and blockspace utilization), not just the transaction's own fee rate.
2. The Delay Gradient Effect: The coefficient on the delay gradient ($\alpha_1$) is negative and statistically significant ($-0.046$).
   • Interpretation: When the delay schedule is steep (a small fee increase buys a large time reduction), users pay less for a given priority because the "price" of waiting is high. Conversely, when the schedule is flat (priority doesn't help much), users must bid higher to separate themselves. This confirms the VCG pricing mechanism.
3. Institutional Mechanisms:
   • RBF (Replace-by-Fee): RBF-tagged transactions pay a ~93% premium, signaling high urgency.
   • CPFP (Child-Pays-For-Parent): CPFP packages pay ~15% less than stand-alone transactions of similar apparent priority, reflecting the bundled subsidy.
   • Block Conditions: Fees rise with blockspace utilization and the time elapsed since the last block.

Model Performance

• Cross-Sectional Power: The model explains a substantial share of fee variation within an epoch ($R^2 \approx 0.435$ in-sample).
• Temporal Limitations: The model struggles to predict fee levels across epochs. The structural coefficients capture why Transaction A pays more than Transaction B in the same 30-minute window, but epoch fixed effects are required to explain why the overall fee level shifts over time (driven by unobserved macro shocks).
• Robustness: Results hold across different machine learning learners (Gradient Boosted Trees) and are stable across rolling time windows, though the magnitude of the delay gradient coefficient varies slightly with congestion regimes.

4. Key Contributions

1. Novel Dataset: The first high-frequency, transaction-level mempool dataset that captures the queueing environment (arrivals, exits, and congestion snapshots) rather than just final block inclusion.
2. Structural Identification: Successfully separates the "delay technology" (supply side) from "user preferences" (demand side) using a two-stage machine learning approach with monotonicity constraints.
3. Mechanism Design Validation: Provides empirical evidence that Bitcoin fee markets operate similarly to a VCG mechanism, where fees internalize the delay externality imposed on others.
4. Institutional Insights: Quantifies the economic impact of specific Bitcoin features (RBF, CPFP, SegWit) on fee formation, showing they are priced significantly differently by the market.

5. Significance and Implications

• For Policy and Protocol Design: The findings suggest that as block subsidies decline, the fee market will rely heavily on congestion-driven pricing. Understanding the "delay gradient" is crucial for predicting how fee markets will behave under stress.
• For Wallets and Users: The results validate that fee estimation algorithms should focus on the marginal value of priority (the slope of the delay curve) rather than just absolute congestion levels. It also highlights the cost-benefit trade-offs of using RBF vs. CPFP.
• For Economic Theory: The paper bridges the gap between theoretical mechanism design (VCG) and empirical blockchain data, demonstrating that structural models can be successfully estimated on high-frequency crypto data.
• Limitations: The study is limited to a ~90-day period of relatively stable network operation. The authors note that long-run extrapolation requires caution, and they plan to extend the dataset to capture more extreme congestion regimes and regime-switching behaviors.

In summary, the paper provides a rigorous, data-driven framework for understanding Bitcoin fees, proving that fees are not random but are a rational response to a complex, congestion-dependent priority queue governed by economic incentives.