In a short-run, price level adjusts very slowly. Since central banks control nominal interest rates directly, they directly control also real interest rates in the short term.
Influencing long-term real interest rates is a fairly complicated question. I am actually not sure.
The long-term real interest rate can be increased by increasing the yield on long-term bonds. Yield on long-term bonds is influenced by Central bank's perception about growth & inflation as long-term bond yields tend to converge with the rate of return on capital in the long run. Hence, higher inflation targets, current account imbalances & fiscal deficits could lead to higher real interest rate in the future.
For me (and Hayek, and Wicksell, the one who came with the term "natural interest rate" in the first place), natural interest rates are the rates prevailing in the real capital markets. These are the markets where real capital, that is, saved real purchasing power, is traded. Consequently, in order to directly manipulate the natural interest rate in any exact way, the CB would need to directly determine at least either supply of or demand for real investments.
Unfortunately, today there is no market exclusively dedicated to real purchasing power alone. This is because of the large supply of bank money via bank credit. Therefore, we have no idea of what the natural rate really is anymore.
Traditionally, CBs deal with the reserves market (i.e., market for commercial bank liquidity). Most of the time, CBs are quite capable of influencing real short-term rates in the reserves market through their OMO and standing facility operations, as long as they supply just enough liquidity to meet the demand of commercial banks. Anything beyond or less than this will result respectively in either a too low or too high rate. BTW, this does not mean that CBs can directly manipulate the supply of broad money. This is up to commercial banks to decide, since they are the ones controlling bank credit.
Real long-term rates can be manipulated by the CB, but in a weaker way, by means of unconventional monetary policies that deal with bonds, and even equity and derivative assets. But for a full manipulation of this rate, the CB would need to at least monopolize the capital supply for the whole economy. This could translate into massive inflation as was the case in Latin America in the 1980s.
Anyway, manipulation of reserve market rates does theoretically impact the natural rate. There still is a supply and demand for real purchasing power, but this real purchasing power follows somewhat the orientation given by the reserves market and the broader financial markets. But the impact is indirect and impossible to measure in advance.
Give a look at Claudio Borio's papers on monetary policy.
Frankly, if CBs could really guide the natural rate, we would have no business cycles anymore...
If the financial markets were limited solely to trading of real savings, I believe so. There would still be a need for short-term capital and long-term capital, therefore there would still be a structure of multiple interest rates.
However, for such a situation to emerge, money would need to be fully backed by reserves, and bank credit would consist only of real savings, not of scriptural credit. Of course, in this case there would be no need for CBs, since these are "necessary" only in a situation where scriptural credit exists and commercial banks need fractional reserves.
There are, nevertheless, authors who would say that interest rates could correspond to the natural rate even in the presence of scriptural credit. For these expositions, see the works of George Selgin.
I do not have much to add to the good comments you've already received. The real question is whether or not central banks can affect the underlying real variables in the economy. I suspect in the short run yes, long run: no. However, this is almost tautological on my part as the question presumes the answer, i.e. the short run would be when they can; the long run (by definition), when they can not. I believe a more subtle view, mentioned above, is that a central bank plays many roles in the economy, some of which may affect relative prices of different activities (both short and long run) and thus may have an impact on the equilibrium level of real interest rates (the 'natural' rate--terminology, "natural price" which goes back to Adam Smith). I am thinking specifically of their regulatory role. Through regulation, they may alter the return to private banking thus alter relative prices (if ever so slightly) in the markets. At some point (and in some models incorporating a financial sector) this may lead to a change in "the" natural rate of interest.
The other side of this is that prudential regulation (backed by a "too-big-to-fail" policy, whether implicit or explicit) may reduce uncertainty and volatility in markets and affect "the" natural rate of interest, as well.
I agree that you may want to study some of the earlier authors and papers cited. One wants to start from a general equilibrium oriented approach with these actors in the model and then see what happens under different constraints.
I think one view is to treat liquidity as a productive variable, thus changes in its relative price ought to affect other markets including the productivity of capital. Whether that liquidity is publicly or privately produced, and how substitutable each form is with each other, and then how a certain regulatory regime overlays all of this will depend on the model.
None of the targeting schemes will change the natural rate either in the short or long run, at least as Wicksell used the term. The demand for real capital and the supply of real saving determines this interest rate, which can differ from bank loan rates. According to Wicksell (and Thornton before him), as long the market rate is below the natural rate, prices will be bid upward, i.e. inflation will rise.
This suggests that real variables control the natural rate and the Fed will not have a direct input. The Fed instead will impact the market rate, which can lead to inflation or deflation depending upon whether the market rate is set below or above the natural rate.
If this is indeed what you're asking about, that is, the Wicksellian natural rate, then I don't believe the Fed has much of an impact. Many economists feel that the real economy can be impacted by Fed policy. Maybe, but probably only to make it happen smoother, not more. That is, monetary policy won't make the economy grow a lot faster. But it can make it grind to a temporary lull.
It's better to think of the Fed or any other central bank as providing a good playing surface for the game that's played on the field. The grass is manicured and the surface is in top condition, but it does little to improve the soccer that is happening on the field. However, if the surface is terrible it can create problems. The Fed needs to keep the field in playing condition. After that, the play is on its own.
Those policy rules are all attempting to keep the playing surface in top condition and each will work reasonably well if maintained.
Well, I agree that there are different definitions about the natural rate. Thinking that “natural” means the rate given only by market forces (without government intervention), we would observe that, nowadays, this is impossible because of fiat money (declared by a government). That is, the price of money (interest rate) should always be affected by Central Banks (at least, as conventional theory explains!).