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The problem isn't a limit in general but that 1MB provides so little transaction capacity that, under any meaningful adoption scenario, it will push all individual users off the blockchain to rely on trusted third parties. 1MB is insufficient for end to end direct user access, but it is sufficient for a robust inter-‘bank’ settlement network.

If the cap is not raised to some higher limit; allowing a larger number of users to maintain direct access, then individuals will be priced out of the blockchain. When that happens Bitcoin becomes yet another network with no direct (peer) access; like FedWire, SWIFT, and other private closed transfer networks.  There is no realistic scenario where a network capped permanently at 1MB can have meaningful adoption whilst still maintaining direct access to the blockchain by individuals. To be clear, by ‘direct access’ I mean both parties transacting directly on-chain without relying on an intermediary or trusted third party.
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If the transaction demand exceeds capacity by a magnitude or more it will lead to direct users being replaced by trusted third parties acting as aggregators.   There are a lot of disadvantages to centralized services but they are more efficient and if the artificial limit is kept it will play right into the advantages of centralized services.  A third party can facilitate instant off-chain transactions.  If demand outstrips the very limited capacity it will force transactions off-chain using trusted third parties which I will call processors. 

Today these processors would include online wallet providers, exchanges, merchant payment service providers, and services where the user maintains a balance under the control of the merchant (casino, poker room).  In time even traditional financial companies and banks could be processors.  The one thing they all have in common is that customers of these services do not have direct control over private keys and do not directly make transactions on the network.  The processor has control over the private keys, keeps the Bitcoins in reserve and maintains an internal ledger of user transactions and balances.   Processors can trivially facilitate transactions between two customers of their service.   Since they control the private keys they simply update the ledger balances of the two customers and many of them do this today.   

However transactions can still occur off-chain even if they involve customers of different processors.  The process is not trustless but risk can be managed.  Two processors would aggregate all the transactions which occur between their customers and then periodically settle the difference.  When a payment from a customer of one processor is made to a customer of another processor the sending processor will notify the receiving processor.  Both processors will update their internal ledgers.  Over time this will result in an accumulated balanced owed by one processor to the other which can be settled with a single on-chain transaction.  The key thing is that there is a one to many relationship between the settlement transactions and underlying customer transactions.   While the 1MB block limit does not provide sufficient capacity for a large number of direct user transactions, third party processors can facilitate a very large number of off-chain transactions using a smaller number of on-chain settlements.  Blocks of a finite size can support a nearly unlimited amount of off-chain transaction capacity with the limitation that it involves the use of trusted third parties.

You can't competing with a 'bank'
You might be considering the point above but dismissing it because you still 'could' submit a transaction to the network.  The processors described above don't have the ability to close the network but a network that you have technical access to but which is uneconomical is effectively no access at all.  The current block size realistically limits capacity to no more than two to four transactions per second.  Two transactions per second is roughly 64 million transactions per year.  A finite number of transactions can't support an infinite number of direct users.  Say at some point there are ten million users and they wish to make two transactions per month.  That is 240 million transactions but only 64 million will fit in the blocks.  What happens to the excess.  If third party processors are attractive the difference will be handled by them. When you consider a settlement network would allows these third party processors to offer instant, 'no risk' transactions at significantly lower fees than on chain transactions the excess demand will be processed off-chain.   If the network continues to grow the profitability of these companies will grow and that will lead to more third party companies.   Those settlement transactions allow more off-chain transactions but at the same time compete with direct user transactions for the limited on-chain capacity.  In a settlement network the upper limit on the number of settlements required grows exponentially with the number of trusted peers.  Just two hundred trusted peers (crypto banks) performing hourly settlements would fill the blocks, all the blocks perpetually.  There may be billions of 'Bitcoin' transactions but they would be nothing more than updates on centralized ledgers.  The blockchain would just handle the settlement between massive financial service providers.  As these entities are collecting fees, and on chain transactions are a necessity of doing business, they can and will pay far more than you to ensure timely inclusion in a block.   When you as an individual have been reduced to a position where you must outbid a 'bank' for space in the blockchain then you have effectively lost access to the blockchain.
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Now when demand for transactions exceeds what is possible who do you think can pay more in fees?  You or a bank?  If transaction demand exceeds capacity then some transactions will not make it into a block.  Those paying the highest fee will be the ones who retain access to the blockchain and those unable or unwilling to will be excluded.  It is delusional to think it will be the 'banks' that suffer in a situation like that.

The reported 7tps transaction capacity does not exist.
There is a myth that without raising the limit the network could handle 7tps.  It can't.  The limit is 1MB per block the actual transaction capacity depends on the average transaction size and realistically that provides no more than 2 to 4 tps.
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1MB can not support a sufficient number of direct users even if transaction frequency is very low
One argument made by those favoring the cap is that Bitcoin doesn't need to be used as a transactional currency to be successful.  Users could primarily acquire Bitcoins as a way to secure store of value (savings) and continue to use other currencies for routine purchases.  Bullion and other stores of value have a much lower velocity than transactional currencies.  This means a block of the same size could support more more users.  While the user of a non-transaction currency may not make dozens of transactions a day, meaningful access would at least require access on the order of dozens of transactions per year.  If your savings or brokerage account restricted you to only one deposit per quarter and one withdrawal per year I don't think you would find that acceptable.  Future users of Bitcoin will not find it any more acceptable if they are forced to transaction as infrequently.


As you can see even with an average transaction frequency of just once a week or once a month the network can't support more than a token number of users.  When someone advocates a permanent cap of 1MB what they are saying is I think Bitcoin will be great if it is never used by more than a couple million users making less than one transaction per month.  Such a system will never flourish as a store of value as it is eclipsed by alternatives which are more inclusive.  To support even 100 million direct users making an average of one transaction every two weeks would require a throughput of 82 tps and an average block size of 20 to 40 Megabytes.
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On a transaction fee basis.
Currently the cost of the network is roughly $300 million annually. The users of the network are collectively purchasing $300 mil worth of security each year.  If users paid $400 million the network would be more secure and if they paid $200 million it would be less secure. Today the majority of this cost is paid indirectly (or subsidized) through the creation of new coins but it is important to keep in mind the total unsubsidized security cost.  At 2 tps the network the unsubsidized cost per transaction would be about $5. At 100 tps it would be $0.05.
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Conclusion
Restricting the size of blocks to 1MB permanently is great if you are a major financial services company.   You could co-opt a very robust network, act as a trusted intermediary and force direct users off the chain onto centralized services.  For the same reasons, it is a horrible idea if you even want to keep open the possibility that individuals will be able to participate in that network without using a trusted third party as an intermediary.