With the same transaction volume (assuming all simple balance transfers), "market cap" as Bitcoin, cost of processing a transaction is negligible (it has to be pretty low for nodes to be able to sync in the first place), types of transactions, etc., and assuming value transfers that happen on one blockchain could equally well take place on the other, we observe the following for non-segwit Bitcoin with 1MB block size:
- The present gas limit of ~6.7M and a gas cost of 21k gas for a value transfer puts a limit of 319 transactions per block. With an average block time of 30s, this means that ~6k transactions can occur every 10 minutes (Bitcoin's block time). We can continue to use a 30s estimate because of EIP649 in Byzantium.
- Bitcoin transaction sizes depend on the number of inputs required for a transaction. Fewer transactions fit on-chain if many inputs are required for a transaction. Fragmentation will decrease the number of transactions that fit in a block. The upper limit on transactions in the best case is ~5k transactions per 10 minutes.
- Bitcoin seems to be hitting a bit over 2k transactions per block in practice (which was also achieved during "stress testing" of the Bitcoin network).
Given the assumption of "all-else equal", the Ethereum supply of transactions in a 10 minute window is ~3x more than that achieved by Bitcoin -- but could be raised/lowered by miners. With a higher supply and lower demand, we'd expect to see lower transaction costs on Ethereum. Note that I've assumed transactions on the blockchain are not elastic to the point where transactions fall immediately to zero above a particular price.
There is further downward pressure on price in Ethereum: with Bitcoin, the cost of setting a transaction fee too low is that you'll need to wait another 10 minutes for the next block causing over-spending. With Ethereum, with a block every 30s, one has 20 opportunities to reprice one's transaction before a Bitcoin block is expected to be mined.
Thus, we'd expect Ethereum to have lower transaction costs due to
- the ability to increase availability of transaction "slots",
- higher raw throughput of transactions in protocol (~6k vs. up to ~5k and ~2k in practice),
- the account balance model used by Ethereum vs. the transaction inputs model used by Bitcoin for determining spending, and
- faster block times (coupled with the ability to replace transactions with higher fees)