From Worthwhile Canadian Initiative:
In one important respect, what we call "New Keynesian" macroeconomic models are in fact New Gesellian macroeconomic models. That's only in one respect, but it is important.
Silvio Gesell proposed a tax on currency. The higher the tax rate, the faster people would spend that currency. A tax is a negative subsidy. The higher the subsidy rate, the slower people would spend that currency.
Another name for a subsidy on currency is paying interest on currency. If that interest rate is 5%, for every $100 currency you hold, the bank that issued that currency pays you $5 currency per year.
If I hold a $20 banknote, issued by the Bank of Canada, it is as if I have a chequing account at the Bank of Canada with $20 in it. If I buy something from you, and give you that $20 banknote, it is as if I wrote you a cheque for $20, instructing the Bank of Canada to transfer $20 from my account at the Bank of Canada to your account at the Bank of Canada.
Imagine a world where every individual has a chequing account at the central bank. There are no other banks, and no other forms of money. People use cheques drawn on their accounts at the central bank to buy everything. The balance in any individual's chequing account can be either positive or negative. But the central bank puts a limit on how big your overdraft can be, to ensure the no-Ponzi condition is satisfied. You can't have an overdraft bigger than your total human and non-human wealth.
The central bank in that world can choose to pay whatever rate of interest it wants on positive balances, and charge whatever rate of interest it wants on negative balances. It's a purely administrative decision. Suppose the central bank chooses to pay the same rate of interest on positive balances as it charges on negative balances.
Suppose the positive and negative balances in those chequing accounts across all individuals sum to zero. The central bank itself has no assets except the negative balances, and no liabilities except the positive balances, and the two exactly cancel out. If one individual buys something for $20, his account goes down by $20 and the seller's account goes up by $20, so the net balances stay at zero. The central bank earns zero net revenue, because the interest it pays on positive balances exactly matches the interest it charges on negative balances. And suppose the central bank has zero administrative costs. The central bank therefore earns zero profits.
I have just described the monetary system of a New Keynesian macroeconomic model.
Monetary policy in that model is Gesellian, except the tax on central bank money is normally negative. It's a subsidy on positive balances, and a tax on negative balances. The central bank calls that tax rate "the rate of interest". If the central bank wants to speed up spending it reduces that rate of interest; if it wants to slow down spending it increases that rate of interest.
New Keynesians say their models are models of a "cashless" economy. We should ignore them. Forget what they say, and watch what they do with the equations. If there is no "cash", how can the central bank set the nominal rate of interest? If there is no "cash", how can the central bank influence people's spending decisions?