Bitcoin is cherished by its supporters and administrators because of the fact that it will be always limited in supply. The world will never see more than 21 million bitcoins regardless of the population and the need will be met by the price of Bitcoin which can be split up to more than 8 decimals hypothetically. Once all of them are mined, no new bitcoins will be created unless there comes a change in the protocol to meet the user requirements.
Bitcoin has a resemblance to gold in many ways because both are available in fixed supply and must be extracted and put into use. Just like gold is set a standard by banks at which paper money is created, if bitcoin gets standardized we will be seeing similar trends.
Keeping all that in mind, one cannot ignore the fact that the transaction fee will be the only source of income for the miners when all the bitcoins are mined. But will that be enough? A question so difficult to answer because it is not easy to predict how much will be transaction fee in the long run and how will the cost of mining correlate to it.
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It is a speculation that the mining chips will be so small and efficient that it will be easy to install them in electronic devices which will make the transaction fee a reasonable reward for the miners. However, a rise in the transaction fee is not very probable at the moment as the community is focused on the enhancement of block size to scale up the bitcoin network.
This is an alarming situation for the miners who have to rely only on transaction fees as bigger block size means transactions will always be registered at low cost and new blocks will not be added to the system as upsizing the block means much more transactions can be recorded on the block than before.
When no new blocks are available, transactions may get dropped because of space issues on blocks and this causes a rise in transaction fee as users would be willing to pay high for making payments but it would be an overall disaster for the currency if majority of transactions start getting dropped.
To conclude, apparently it may seem that miners will run out of money when all bitcoins are mined but in theory, it creates a way for the miners to survive on the transaction fees.