I’ll offer one thought, from the standard New Keynesian model of money, interest and prices. In this model, the fact that money confers liquidity services is captured, rather crudely, by assuming that we are all like coin collectors and get pleasure from holding real money balances. JP’s thought experiment is undertaken in this model by wondering what happens if you slowly reduce the amount of pleasure all of us coin collectors get.
Woodford explains the title of his textbook ‘interest and prices’, in particular, the virtual absence of money, in his opening salvos, by pointing out that central bank control over interest rates, and thereby over the whole economy, does not depend on this ‘pleasure’ being of a certain magnitude. In fact, the simple version of the model he studies is one in which this pleasure is conjectured to be infinitely small. Monetary policy’s leverage here depends on money’s role as a unit of account.
Looked at this way, the tendency for the economy to get stuck in a liquidity trap in response to shocks to the natural rate of interest (for example) should not depend on the liquidity convenience (pleasure!) we get from money.
Another way we can interpret JP’s question, though is this.
Suppose that a shock to the natural rate comes along, requiring an extended period of interest rates at the zero bound [and perhaps other stimulus too], to return inflation slowly to target. But then on top of that there is some monetary innovation lowering the liquidity services to money. [Apple Pay?!] What would happen then? Looked at through this model, nothing at all. Because the fact that interest rates are already at zero shows that these liquidity service benefits have already long been exhausted.
Here I have assumed that the monetary reform does not change the asymptote of these liquidity service benefits as real balances get very very large. All that happens is that the monetary reform changes the profile of those marginal service benefits at finite values of real balances.
What if the reform that happens mid-way through responding to the natural rate shock did change the asymptote, however? For example, what if the reform involved abolishing higher denomination notes, which increases the costs of managing cash? I’m not absolutely sure here, without actually doing this properly, but I hazard this guess. Nothing happens, much, if central banks hold interest rates where they are, except that they have to allow real balances to decline via a reduction in nominal money quantities. The reform, however, will create an opportunity for the central bank to lower rates if it wants to, in case the old floor to rates [previously zero, now lowered] implied an amount of stimulus that was less than optimal.
The more interesting question – in fact the question that JP Koning actually posed – is what happens in the real world, of course. But you’ll have to go elsewhere for answers to questions like that.