Why does raising interest rates work? Why does the federal raising interest rates actually have an effect on the economy? The typical answer is that it reduces money supply by incentivizing saving and increasing cost of loans.

But couldn't market participants just react by adjusting FX rates, inflation, etc. by the appropriate amount? Then even though the money supply is reduced, the real rates are unchanged, resulting in no net effect. What are the factors preventing this from actually happening?
The typical answer is that it reduces money supply by incentivizing saving and increasing cost of loans.


This is backwards.  The Federal Reserve (the United States' central bank) sets a target for the federal funds rate.  The rate itself is a weighted average based on market activity.  The Fed encourages the average rate to match the target by buying and selling bonds in what are called open market operations.  Buying bonds puts money into circulation, increasing the money supply.  Selling bonds takes money out of circulation.  The Fed changes the money supply to change the interest rate, not the other way around.  


  But couldn't market participants just react by adjusting FX rates, inflation, etc. by the appropriate amount?


Only if all (or even most) of the participants worked together.  If only one or a small portion of participants attempt this, then it creates arbitrage opportunities for the remainder.  Currency exchange rates and inflation are both set based on market actions.  They aren't so easy to adjust.