Inflation and monetary policy through the lens of the Fisher relationship

In the long run, the Fisher relationship implies that: Nominal interest rate   =   Real interest rate   +   Inflation In contemporary economies (we look at the cases of the United States, the euro zone and Japan) , central banks control nominal interest rates , even in the long run , but not the money supply. So inflation can no longer be said to result from the money supply chosen by central banks. If the nominal interest rate is controlled, and if it is kept at a low level by the central bank, the question is then how the effects of this low nominal interest rate are shared between the real interest rate and inflation: For neo-Fisherians, the real interest rate is determined in the long run by the features of the real economy (technological progress, the savings-investment equilibrium) and the entire fall in the nominal interest rate turns into a fall in inflation; But empirical analysis shows that in the long run, monetary policy continues to have an effect on real interest rates (which also depend on the savings-investment equilibrium). A portion of the fall in nominal interest rates results in a fall in real interest rates; the rest in a fall in inflation. But even with this approach, a policy of persistently low nominal interest rates leads to low inflation in the long run; One remaining possibility of course is an exogenous inflationary shock (caused by commodity prices or a change in wage formation) , which, as central banks decide to react to it , drives up nominal interest rates . But if central banks keep nominal interest rates low despite this inflationary shock, it will inevitably be temporary.