Our central banks set interest rates, and do not even pretend to control money supplies. How do interest rates affect inflation? We finally have a complete economic theory of inflation under interest rate targets and unconstrained liquidity. Its long-run properties mirror those of monetary theory: Inflation can be stable and determinate under interest rate targets, including a peg, analogous to a k-percent rule.
Uncomfortably, stability means that higher interest rates eventually raise inflation, just as higher money growth eventually raises inflation. Sticky prices generate some short-run non-neutrality: Higher nominal interest rates can raise real rates and lower output. A model in which higher nominal interest rates temporarily lower inflation, without a change in fiscal policy, is a harder task. I exhibit one such model, but it paints a more limited picture than standard beliefs. Generically, without a change in fiscal policy, monetary policy can only move inflation from one time to another.
The last decade has provided a near-ideal set of natural experiments to distinguish the principal theories of inflation. Inflation did not show spirals or indeterminacies at the long zero bound. The large monetary-fiscal expansion of the covid era produced a temporary spurt of inflation. The same money unleashed in quantitative easing had no such effect.