Two equations to understand the monetary question

We use the situations of the euro zone and the OECD as a whole to illustrate our analysis. We begin with two simple equations: The monetary equilibrium (the money supply is equal to demand for money, which varies with the nominal interest rate and nominal GDP); The Fisher relation (the nominal interest rate is equal to the real interest rate plus inflation). First, this explains the current short-term equilibrium: the expansionary monetary policy (increase in the money supply) has driven down the nominal interest rate (to increase demand for money) ; due to the rigidity of inflation, the fall in the nominal interest rate has driven down the real interest rate, boosting activity; But what about the long-term equilibrium ? In the long term, the real interest rate returns to a level that is not determined by monetary policy (rather by structural features of the economy). In the usual version of the model, in the long term, money supply growth determines inflation and inflation determines the nominal interest rate. But there is an alternative version of the model known as “neo-Fisherism”: in the long term, the central bank chooses the nominal interest rate, which determines inflation, and inflation determines money supply growth. In OECD countries, we note that inflation has not reacted to money supply growth , even in the long term . This obviously raises the possibility that the neo-Fisherian model is right.