The FED in the US has a target inflation rate and target employment. Given that mandate they set the rates which drives the rest of the interest rates on bonds, mortgages etc. Currently they have been aggressively bumping them to tame inflation. Assume a year from now inflation is back to the range they need, does the FED need to cut rates? And if so, is there some rule of thumb where the range must be? It sounds to be as pendulum clock, they bring rates up, slow the economy and we get a recession, then they need to lower the rates to stimulate it and pretty much it is likely to overheat. Is there a "stable" state at some point? Or can there be? Perhaps a relationship say when the inflation is at X, the fed fund is expected to be about Y. I can see from the market data the rates are expected to go down but why is such expectation? What if we get the inflation under control without recession and the FED job would be just to maintain it, so far higher rates did not tank the economy, so why does it need to lower it?
Assume a year from now inflation is back to the range they need, does the FED need to cut rates?
That depends on the state of the economy. Inflation does depend on interest rates but it also depends on other variables. Fed is obliged to behave in a way that is consistent with its mandate for full employment and low inflation, so whether Fed cuts the rates depends on how well those objectives are achieved.
Is there a "stable" state at some point?
There isn't a stable point because economy is constantly perturbed by shocks. Central bank behavior, when it comes to interest rate, can be described by Taylor rule. There are different versions of this rule but basic textbook version is given by (see Romer Advanced Macroeconomics pp 544):
$$i_t=r_n+\theta (π_t−π^*)+\gamma (y_t−y_n)$$
where $i_t$ is Fed's interest rate, $r_n$ is the 'natural' rate of the real interest rate, $\pi_t$ is current inflation, $\pi^*$ is Fed's desired level of inflation. $y_t$ is log of real output, $y_n$ is log of natural rate of output, and finally $\theta$ and $\gamma$ are parameters that measure the intensity Fed cares about inflation or output.
Now the reason why Fed has to constantly change interest rates is that:
- $r_n$ changes, and Fed has only very little influence over natural rate of real interest rate. If $r_n$ changes Fed has to change its policy. This rate is primarily determined by things such as long run return to capital, peoples time preference and these things depend on things like demographics, or rate at which new technologies are invented etc (Brand et al 2018).
- When it comes to $\pi_t$ Fed does exercise some control over inflation rate, but it is not like Fed has some 'inflation thermostat'. Inflation depends also on some other factors than Fed's direct policy, and in addition to that Fed might pursue too loose or too tight monetary policy by accident leading to too much/little inflation.
- $\pi^*$ can be in principle freely chosen, its the central bank target inflation rate. However, Fed has mandate from congress to keep inflation down, that is traditionally being interpreted as inflation being about 2% (see Fed). So even though in principle $\pi^*$ can be exactly chosen by CB in practice Fed's hands are tight and $\pi^* \approx 2\%$
- y_t is the log of current level of real output. Fed does not have direct control over business cycle. They do have some influence over it due to nominal rigidities but its not like recessions and expansions would not happen in absence of Fed. Recessions and expansions are natural part of economic cycle and so far there never was any society that would manage to avoid economic cycles altogether.
- $y_n$ is the log of natural rate of output. Natural rate of output is typically interpreted as a rate of output at which inflation is not accelerating (NAIRU), and natural level of output is completely out of the hands of Fed. This is determined by things like technology, factor/resource endowments and so on.
Hence, unless congress would change Fed's mandate to just ignoring the economy and keeping interest rate constant, with its current mandate interest rate would be constant only if long run returns to capital never change, people's time preference never changes, if there is no business cycle, no economic growth, no exogenous shocks to inflation due to lets say supply issues, no technological progress or discovery of new resources. This is clearly completely unrealistic.
What if we get the inflation under control without recession and the FED job would be just to maintain it
No this still would not work. ECB has mandate to only keep inflation low (even though recently de facto they behave like if they have dual mandate), but ECB still has to change interest rates when $\pi_t$ or $r_n$ changes, even if they would not care about output at all. This is because $r_n$ is not directly controllable by ECB, and controlling $\pi_t$ through monetary policy is not like choosing preferred temperature on thermostat as inflation depends also on some other factors and even when it comes to monetary policy the relationship can be sometimes stronger and other times weaker depending on various factors (e.g. economy being in liquidity trap etc).
Hence even if we would get inflation under control, and even if congress would change Fed's mandate to only care about inflation, Fed would still have to fiddle with interest rates.
Only if congress would change Fed's mandate to keep interest rates constant or some other esoteric mandate (e.g. always target current inflation and do not care about output) you will see no change in interest rates over long periods of time. However, this would effectively ban Fed from trying to manage economy by reducing the severity of business cycle.