What is the unit of account vs. the unit of exchange and how has Bitcoin created a problem?

The unit of account is what you write down in a ledger to say, “Magne has $1000”. This is what banks do. They credit your account as having some amount of money.

The unit of exchange is how you exchange the money you have with someone else. This can be done in two ways. It can be done by accounting, as in a wire transfer or use of a credit or debit card at a point of sale terminal.

The unit of exchange can also be a physical thing, classically, it is a coin. Coin tends to be difficult to produce, it can be hoarded or lost, and there is only one of each of them.

To more fully understand why hard-currency can be a problem read this paper. It will step you through a simple system that shows you why hard currency is so problematic, and then translate that into a continuous equation. https://arxiv.org/abs/1506.08231

What bitcoin has done is to create an electronic version of a hard-currency coin. There is only one of each bitcoin token. That token can be used to exchange. But, you cannot do it by accounting methods. You can only do it by sending the token to its new owner, in an electronic version of giving a coin to someone else.

It is possible for a bitcoin exchange to keep your bitcoins for you, and send you statements of what you have. However, what bitcoin agents do is act as vault-keepers. And until that vault is opened and they take out your coins and give them to you, you really don’t know if they are there. The clients of Mt. Gox found that out.

Banking began by virtualizing coins and bullion. Vault keepers arose to keep the hard currency safe. And vault keepers sent letters of credit to each other. This was a better system than sending a cart with gaurds and a chest full of cash that could be stolen. Over time, vault keepers issued redeemable notes, and certain letters became something like checks. Those notes and letters were so much more convenient. Among other things, if stolen, the letters could be stopped. And then they started making loans as letters of credit.

Eventually, vault keepers realized that they had more outstanding loan value than they had vault contents to back it up. But nobody seemed to care and things went better than they had before. Banking was born.

Notice something crucial here. Banks must virtualize hard currency if they are going to operate. Because a vault keeper makes money on fees. But a bank makes its money charging interest on its loans.

This can’t be done with bitcoin.

If you want to reiterate this, then go back and read the paper. It’s all there. https://arxiv.org/pdf/1312.2048.pdf