Expectations and the Neutrality of Interest Rates

50 years ago, Bob Lucas (1972a) published his pathbreaking analysis of the neutrality and temporary non-neutrality of money. But our central banks set interest rate targets, and do not even pretend to control money supplies. How is inflation determined under an interest rate target?
We finally have a complete theory of inflation under interest rate targets, that mirrors the long-run neutrality and frictionless limit of monetary theory: Inflation can be stable and determinate under interest rate targets, including a k percent rule or a peg. The zero bound era is confirmatory evidence. Uncomfortably, long-run neutrality means that higher interest rates eventually raise inflation.
With a Phillips curve, we have some non-neutrality as well: Higher nominal interest rates raise real rates and lower output. A model in which higher nominal interest rates temporarily lower inflation is a harder task. I exhibit one such model, which adds long-term debt, but has several shortcomings. A better model, and empirical understanding, is as crucial to today’s research agenda as Lucas (1972a) was in its day.
Much of this is contentious. The issues are crucial for policy: Does an end to inflation require interest rates substantially above current inflation? Do central bank interest rate hikes, without contemporaneous fiscal tightening, raise or lower inflation? We do not have well-grounded, widely-agreed answers to these questions. Given the state of knowledge, a bit of humility is in order.