Sometimes I write posts about things I don't understand and want to understand. (Or don't understand very well and want to understand better.) This is one of those posts.
Suppose the government nationalises all the commercial banks, and merges them into the Bank of Canada. And suppose the Bank of Canada abolishes paper currency. So everyone has a chequing account at their local branch of the Bank of Canada, and they use that chequing account to buy and sell everything else. No other financial institution is allowed to offer an account with chequing (or debit card) privileges, just like no other financial institution is currently allowed to print paper currency that could compete with the Bank of Canada's paper currency.
In that imaginary Canada the Bank of Canada has two monetary policy instruments. First, it can vary the quantity of money by buying or selling IOU's (bonds, loans, etc.). Second, it can vary the rate of interest it pays on chequing account balances (and it can make that rate of interest negative if it wishes). [It could also vary the limits placed on overdrafts; and could also vary the spread between the interest rate charged on overdrafts and paid on positive balances. But let's set all those other instruments aside.]
I understand how monetary policy would work in that imaginary Canada (at least, I think I do). Increasing the quantity of money (holding the interest rate paid on money constant) shifts the LM curve to the right/down. Increasing the rate of interest paid on holding money (holding the quantity of money constant) shifts the LM curve left/up. Done.
It's a crude model of an artificial economy. But it's a helluva lot better than a simple New Keynesian model where money (allegedly) does not exist and the central bank (somehow) sets "the" nominal interest rate (on what?).
What I do not understand well enough is how well that thought-experiment translates into the real Canada, where only the commercial banks issue chequing account money for the public, and only the Bank of Canada issues paper currency, and paper currency always pays 0% interest. But the Bank of Canada still has those same two instruments: it can buy and sell IOU's; and it can vary the interest rate paid on the commercial banks' chequing accounts at the Bank of Canada.
[Remember that commercial banks can pay interest in kind, like when they reduce fees for balances above some amount. And remember that when a commercial bank makes a loan it is buying the borrower's IOU.]