Banking is a subset of finance. Money and finance go together. Money and banking go even more together.
But they don't have to go together. Maybe they didn't ought to go together. Finance is unstable. Banking is even more unstable than the rest of finance. A bank makes promises it knows it might not be able to keep. A bank is an accident waiting to happen.
Unstable money is a very bad thing, much worse than unstable finance or unstable banking. Why do we have an economy in which money is linked to unstable finance? Why do we have an economy in which money is linked to the most unstable part of finance? Isn't this a really stupid sort of monetary system to have?
If we had a monetary system in which money and finance, or money and banking, were separate, then financial instability would just be a spectator sport for monetary economists. You could bail out the banks, or not. Just like you could bail out the carmakers, or not. And us monetary economists would shrug our shoulders and stay out of it, and let the finance guys and industrial economics guys argue it out among themselves. If a financial crash didn't cause an excess demand for money, and the resulting recession, only the microeconomists would care. Sure, there might be some structural unemployment, as workers switched from finance and investment industries to producing consumer goods instead, but monetary policy can't do much about that anyway.
But instead we have a monetary system in which finance, especially banks, and things that work like banks, are heavily interwoven. So if banks go bust money disappears and people can't buy and sell all sorts of things that have nothing to do with finance. If all our cars went bust at the same time (which they don't) we could at least walk to the supermarket.
Money is a medium of exchange and medium of account. It's got nothing to do with borrowing and lending. Forget that "standard of deferred payment" rubbish.
I can imagine an economy with money and no finance. People never borrow and lend. It's tabu, or they all have exactly the same intertemporal preferences and production possibilities so won't want to borrow or lend. But they still use monetary exchange, because barter is such a PITA. They use gold, cowrie shells, or bits of paper for money. But the bits of paper are not IOUs for anything. They are just irredeemable bits of paper.
I can imagine an economy with finance and no money. It's a barter economy, so any good can be traded directly for any other good. But people still borrow and lend apples, cars, land, labour, wheat, whatever. I can even imagine banks in a barter economy (unless you insist on defining "bank" as a financial intermediary whose liabilities are money). A wheat bank lets you withdraw wheat on demand whenever you want some quickly. That doesn't mean that wheat is the medium of exchange or unit of account. Wheat banks would need to keep wheat reserves to cope with fluctuations in the demand for wheat. There could be runs on wheat banks; people would rush to withdraw their wheat first if they thought their bank were illiquid or insolvent, so they might not be able to withdraw their wheat quickly, or ever. Bakers might be worried about runs on wheat banks, but it wouldn't be a macroeconomic problem.
Money and banking don't have to go together. Money and finance don't have to go together. But they do go together. Why?
You can use wheat to bake bread. You can't use a promise to pay wheat to bake bread. You can use a Bank of Canada $20 bill as a medium of exchange. And sometimes, you can use a promise to pay a Bank of Canada $20 bill as a medium of exchange. A promise to pay money can itself sometimes be used as money. If that promise to pay is made by a bank, like the Bank of Montreal, it very often is used as money.
The Bank of Canada issues money that is irredeemable. The Bank of Canada can exchange it for something else if it likes, but it doesn't have to. It doesn't promise to redeem it for anything. It may choose to redeem it, usually for government bonds, but in a quantity or exchange rate that the Bank of Canada chooses to meet its macroeconomic objectives. A $20 Bank of Canada note is not an IOU. It is not a liability of the Bank of Canada. We only say it's a liability to prevent the accountants getting upset when they try to balance the books of the Bank of Canada.
The Bank of Canada is not a bank. It is more like a closed end mutual fund, where the fund managers can choose whether or not to sell assets and redeem units when the price of their units falls. And even that analogy doesn't work well. It's like a closed end mutual fund where the dividends on its assets, minus operating costs, are given to the owners of the mutual fund (the government in this case) and not paid out to the unitholders. The unitholders are willing to hold the units, even though the units depreciate at 2% a year, and pay no dividends, because the units are just really handy for doing the shopping.
A central bank on the gold standard (or with a fixed exchange rate) really is a bank. Its money really is an IOU, redeemable in gold. Its money really is a liability. A central bank on the gold standard is more like a money market mutual fund. (It's not exactly the same, since it still pays any dividends from its assets to its owner, and not to the unitholder.) A central bank on the gold standard can suffer a run, and it can be unable to fulfill its promise of redemption in gold. Its liabilities can "break the (gold) buck", just like a money market mutual fund, if it doesn't hold 100% gold reserves.
When we decided that the gold standard was a bad idea, we decided that central banks should not be banks. Going off gold, and abandoning fixed exchange rates, was a decision to separate money and banking. And it was a really good decision. Banking is unstable. Money ought to be stable. So separating money and banking was a good thing. But we didn't go far enough. Central banks are no longer banks. But commercial banks are still banks.
Deep down, I think that all monetary cranks vaguely understand that there's something wrong with having money and banking go together. Banks with 100% reserves aren't really banks either, because those banks have a perfect match of their assets and liabilities. They have just changed paper money into electronic money. It's like holding $100 bills as assets , and issuing more convenient denominations as liabilities, because the central bank can't be bothered to print $10 bills. Banks with 100% reserves are more like money changers than money producers.
But those monetary cranks are standing in front of the train of the History of Finance, trying to get it to stop. Even if they succeed, Finance will switch to cars, and drive right around them.
People would like to borrow to invest in long, risky, illiquid, and complicated projects. And they would like to lend in short, safe, liquid, and simple assets. They want to hold money, only paying interest. Finance tries to give people what they want. Finance tries to convert all assets into money. Finance wants to join money and finance together, because that's what people want. That's the problem.
What to do?
1. What we are doing now. Regulate banks and finance to try to make them more stable, support them when they fail, and let central banks (which aren't banks) take offsetting action when finance fluctuates.
2. Try to prevent people doing what they want, and try to separate money from finance, especially separate money from banking, which is the most unstable part of finance. Chequable stock market mutual funds, maybe? Your cheque would still be worth a fixed number of dollars, but the balance in your chequing account would rise and fall with stock prices.