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Currently, fees in Bitcoin are quite high ($1-$30) to move anything under $50 USD. If Bitcoin is worth 1 million dollars one day, how much would an on-chain fee be? Assuming blocks are going to be full of course. Will there ever be a block size increase to solve this scalability problem? Supposedly, lightning network solves it, but fees there would also be high, no? What are some proposed solutions to this problem that work on Bitcoin layer 1?

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I think no one can read the future but some things can be said with high confidence and I will try to point them out.

First I think it the conversion to USD does at best make sense for cosmetics thus I will do most computations in the more natural form of satoshis and confirm only at the end if necessary.

if the average feerate in a block is x sat/Byte and with 1 MB blocks we will have a total fee of x Million satoshi per block. This is equivalent to x * 0.01 BTC.

Assuming 2500 tx per block we would end up having x * 4,000 sats per transaction. (I guess with taproot, schnoor and signature aggregation we could stuff in a few more tx.)

as 1 sat / byte is the absolute minimum fee these 4,000 sats would at a 1 million usd / BTC boil down to $40 per transaction which would have to be multiplied by the feerate x which is any number larger or equal to 1.

I see it from the perspective of Lightning Network rather than from the perspective of making on chain payments. It seems like a reasonable fee to open your channels and be able to transact as often and frequently as you wish. As far as I know it is currently not clear how fees on the Lightning Network would evolve.

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Currently fees in Bitcoin is quite high ($1-$30) to move anything under $50 USD.

According to https://mempool.space its ~$10 based on average native segwit transaction of 140 vBytes

mempool.space

If Bitcoin is worth 1 million dollars one day, how much would an on chain fee be?

If BTCUSD is 1,000,000 USD right now, above fee would be $200 assuming similar demand for block space, same block weight, no other development or improvement until then like schnorr or something that reduces transaction size, less people moving to LN or Sidechains for quick transfers involving less amounts, minimum default fee rate stays 1 sat/vByte etc.

Assuming blocks are going to be full of course. Will there ever be a block size increase to solve this scalability problem?

Maybe

Blocksize increase is simple to initiate, difficult to get consensus and almost impossible to manage decentralization with simple blocksize increase. Few things mentioned in below links:

https://bitcoin.stackexchange.com/a/101788/

https://www.youtube.com/watch?v=CqNEQS80-h4

Supposedly lightning network solves it, but fees there would also be high no?

Which fees are you talking about here? Fees to close channels or transact over LN? Fees for transactions over LN will be low and fees to open/close channels can be managed afaik but not the best person to answer questions about LN.

Related question: Is there a solution to paying high fees when opening and closing lightning channels once we hit a fee only market?

What are some proposed solutions to this problem that works on Bitcoin layer 1?

  1. Consider Layer 1 as the base layer of decentralized network, other layers on top of it and pay fees according to the demand for block space in it.

  2. Use Layer 2 for transactions that involve less amounts and cannot wait for confirmations.

  3. Use latest bitcoin technologies when implemented over layer 1 and don't delay them for years like segwit usage by many of the exchanges and wallets.

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