The best way to reduce Unemployment is to decrease the lending rate (I.e. increase inflation). The best way to reduce Inflation is to increase the lending rate (and thus increase unemployment). Now, other factors come into play with inflation and unemployment, but those are long term in nature (Or the result of Congressional spending as in the stimulus program).
Now, you must understand the Federal Reserve (Often called the "FED") best weapon against inflation is its control of short term interest rates do to the fact the FED can lower or raise such rates as the FED sees fit. Longer term interest rates (and Inflation rates) is more market driven and thus outside the hands of the Federal Reserve. The problem is such short term interest rates also impact unemployment (again in the short term i.e. less then one year, time period). Long term unemployment (do to structural changes in the economy) is generally outside the control of interest rates and this outside the control of the Federal Reserve.
Thus to take away from the Federal Reserve its duty to keep unemployment low, is to strip the Federal Reserve of the ability (and more the desire) to inflate the economy when that is needed to drive down short term unemployment rates.
Remember the Federal Reserve had almost no control over long term inflation rates, Interest Rates or Unemployment rates. The FED's main area of control is short term (less then one year in duration) interest rates which directly affects short term Inflation and unemployment rates (and does so inversely, i.e. increase interest rates leads to less employment and thus more unemployment and less inflation, decrease interest rates leads to increase business investment, which leads to increase inflation and increase employment (and this decrease unemployment).
http://en.wikipedia.org/wiki/Phillips_curvehttp://www.questia.com/googleScholar.qst;jsessionid=D3174908C132567F1CE769289970145A.inst1_1a?docId=5000302642http://www.economicshelp.org/2008/02/essays-on-inflation-and-unemployment.htmlThe problems the US is facing today is NOT short term (less then a year in duration) but long term (It dates from at least 2008 when this recession began). Such long term economic problems have to be addressed by Congress not the Fed, and Congress does NOT want to address it (The Voters want Unemployment to go down, but the people who donate to campaigns wants interest rates near zero). Congress is feeling the push from two directions, unemployment and Inflation. In the 1970s Congress opt to reduce Unemployment even if that meant long term inflation (and policy continued under Reagan and Bush Sr, through at much lower rates of inflation then in the 1970s but 3-4 times what the inflation level is right now). (Inflation was as high as 18% in the late 1970s but stayed above 6-7 % till wheel into Clinton's second term when the effect of Cutting back the Defense Budget balanced the budget and saw unemployment hit almost Zero with Inflation almost Zero).
My point is long term (Duration more then one year) interests, Inflation and Unemployment is beyond the Control of the Federal Reserve, but the result of deficit spending authorized by Congress. Today, we are seeing the result of Bush Jr's Tax Cuts, which increased the Deficits beyond all reason. Thus Congress wants to blame someone other then itself for today's mess (And this is more true of the GOP then the Democrats for the Democrats only control Congress the last four years as the full affect of Bush's Tax Cut hit).
Thus the GOP wants to get the FED to concentrate on Inflation, even if that means a vast increase in unemployment. A vast increase in unemployment will lead to lower wages and thus saving business money. That it will destroy the Country is something the GOP will NOT accept i.e. not accept for it goes against the GOP economic views NOT that the GOP wants to destroy the country, the GOP holds the theory is sound, reality does not exist.
Thus the GOP desire to get the FED to concentrate on Inflation, unemployment is "nature" in their view and should NOT be a concern of the Government.