The article examines economies of scale in Ethereum rollups, highlighting Optimism's decreasing costs per transaction and Mb of data with higher volumes, suggesting efficiency gains. In contrast, Arbitrum shows increasing costs per transaction and Mb with higher volumes, indicating potential inefficiencies. Further research is needed for a more nuanced and comprehensive understanding.

Economies of scale describe the phenomenon where producing more units of a product or service on a larger scale generally leads to lower average input costs per unit. This is due to the inverse relationship between the per-unit fixed cost and the quantity produced. As the quantity of output increases, the per-unit fixed cost tends to decrease, making production more cost-efficient with scale. In short, economies of scale make the product marginally more effective with each incremental input.

In blockchains, economies of scale mean lower costs per transaction as the output (or better, the throughput, so how many transactions the blockchain can process in a given amount of time) increases. For rollups, this efficiency could also apply to the amount of data posted to Ethereum and the cost per Mb of data posted. Chris Ahn writes:

Unfortunately, blockchains typically exhibit diseconomies of scale, where incremental usage makes the network marginally more expensive to use. Because blockchains have competitive fee markets for blockspace, more usage drives up the cost of blockspace.

Is it really so? This short article analyses the existence of cost-efficiency economies of scale in the two major Ethereum rollups, Arbitrum and Optimism, controlling 50% and 25% of the market share, respectively. The analysis focuses on producer economies of scale, achieved by companies or products when the average cost per unit of production decreases as the scale of operations increases, rather than user economies of scale, enjoyed by end users when the cost per unit of the product or service decreases as the scale of consumption increases.

Data are extrapolated and elaborated from Dune Analytics, specifically from contributors @0xKofi and @MSilb7. The analysis covers 2023, from January 1st to December 31st.

Rollups Operational Costs

From the user perspective, every optimistic rollup transaction has two cost components: an L2 (execution) fee and an L1 (data) fee. At a high level, the L2 fee is the cost to execute the transactions on the L2, and the L1 fee is the estimated cost to publish transactions on L1 (in a rollup batch). While the L2 execution fee resembles the traditional fees on Ethereum (gas cost * gas price), the L1 data fee is unique to optimistic rollups because the transaction data for all rollup transactions must be published to Ethereum (or, in the future, other Data Availability layers, possibly making this fee component cheaper).

• L2 execution fee. This is the amount paid to the rollup nodes as compensation for computational and storage costs incurred in processing transactions, much like gas fees on Ethereum. Currently, the node (Sequencer) is a single, centralized entity in most rollups, run by OP Foundation on Optimism and Offchain Labs on Arbitrum. Rollup nodes charge lower transaction fees since L2s have higher processing capacities and aren't faced with network congestions that force validators on Ethereum to prioritize transactions with higher fees. Importantly, in Optimism, the Sequencer only gets the priority fees, while the base fee is burned due to EIP-1559 implementation. On the contrary, Arbitrum currently implements a first-come-first-served policy and doesn’t have a public mempool, meaning there is no priority fee.

• The L1 data fee covers the expense of publishing transaction data on L1, essentially representing the cost of ensuring Ethereum-equivalent security. Optimistic rollups publish transaction data and block headers (consisting of the previous block header hash, state root, batch root) to Ethereum as calldata. Optimistic rollups can mitigate the cost of writing data to L1 by consolidating multiple transactions into a single batch, thereby spreading the gas costs across multiple transactions. The L1 security fee is directed towards Ethereum block producers. Overall, the L1 fee depends on the size of the transaction in bytes, the current gas price on Ethereum, and several small parameters. Publishing data is expensive and makes up 80-90% of L2 users' transaction fees. In the case of Arbitrum, ArbOS dynamically adjusts the pricing of L1 gas to ensure that the amount collected in L1 gas fees closely aligns with the long-term costs that need to be covered.

L1 Fees Formulas

Tx Data Gas Cost in Optimism is the gas cost of the transaction given its size after compression. Fixed Overhead cost in Optimism is similar to Rewards in Arbitrum, and Dynamic Overhead is similar to Dynamic Adjustments.

It’s important to highlight that while L2 execution fees are a cost for users, they can be considered revenue for the L2 since the Sequencer is currently a single identity managed by the rollup itself, meaning the entirety of the L2 execution fees accrues to the L2’s treasury (Optimism uses these funds to support public goods). Therefore, when analyzing the cost structures and dynamics from the L2 perspective, only L1 data fees are considered.

The scatter plots below represent the relationship between the number of transactions versus the cost per transaction and the amount of posted calldata in Mb versus the cost per Mb.

From a visual perspective, Optimism displays an inverse relationship between each couple of variables, while the same doesn’t happen for Arbitrum. A deeper analysis with correlation coefficients confirms that the variables considered have opposite correlations in the two rollups.

In Optimism, a moderate negative correlation exists (-0.25) between the amount of data posted to Ethereum (in Mb) and the cost per Mb, indicating that as the amount posted increases, the cost per Mb decreases. This suggests potential economies of scale or efficiency gains with larger data volumes. Additionally, a negative correlation (-0.36) between the number of transactions processed and the cost per transaction suggests that as transaction volumes increase, the cost per transaction tends to decrease, implying further economies of scale or operational efficiencies with higher transaction volumes. Conversely, in Arbitrum, a moderate positive correlation (0.37) between the amount of data posted to Ethereum and the cost per Mb suggests that as data volume increases, the cost per Mb tends to increase, potentially indicating diseconomies of scale compared to Optimism. Similarly, a positive correlation (0.26) between the number of transactions and the cost per transaction implies that as transaction volumes rise, the cost per transaction tends to increase, reflecting potential differences in cost structures or operational dynamics that may lead to diseconomies of scale.

Considerations

Economies of scale can arise from various factors, including optimization of network infrastructure, spreading fixed costs, and implementing batching strategies, among others. A more comprehensive dataset could enhance our understanding of the underlying forces driving these correlations. For instance, different fee structures might be another contributing factor. Take, for example, Optimism, which calculates the transaction size (distinguishing between zero and non-zero bytes, leading to different costs akin to Ethereum) along with associated gas costs. In contrast, Arbitrum dynamically computes the L1 fee based on an estimated transaction size (without distinguishing between zero and non-zero bytes, but charging a uniform 16 gas per byte) and gas cost, incorporating dynamic and flexible additional components. Furthermore, in Arbitrum, a transaction incurs L1 gas fees only if it is part of a sequencer batch and not sent directly to the L2 contract on the L1, whereas this distinction does not seem to apply to Optimism.

In addition to expanding the dataset and considering additional variables, a more rigorous research methodology, such as regression analysis, could help determine whether a causal relationship exists between the variables rather than just a correlation.

Finally, it could be interesting to extend the analysis to other rollups. Base, which gained considerable steam despite representing only about 3% of the market share, is not considered in this analysis as it’s built on the OP Stack and only officially launched on August 9th, 2023. Nonetheless, including Base in future examinations may yield valuable insights, as initial observations suggest that it may exhibit even stronger economies of scale than Optimism.