There are two basic answers:
1. The FOMC majority believes that its statutory dual mandate commands the Fed to use the tools at its disposal even though internal and external factors may limit their impact. If there are forces such as mortgage lending constraints and business and household uncertainty about taxes, regulation and health care costs that are gumming up the "monetary transmission mechanism," well then, the Fed just needs to push all the harder on its policy levers.
An ancillary point is that the FOMC majority believes the economy's woes are primarily "cyclical," which is to say because of a shortfall of aggregate demand, and thus amenable to monetary policy.
2. If the normal interest rate channel through which lower rates are supposed to work is clogged, that doesn't render monetary policy impotent in the eyes of the FOMC majority. There are other channels through which unconventional easing measures can work that policymakers would rather not talk about publicly — namely the exchange rate channel. Lower rates and rate expectations should, in theory lower the value of the dollar, making U.S. goods more competitive in global markets and boost net exports.
The Fed also hopes that by holding rates very low, investors will put their money into riskier assets such as stocks, yielding a positive "wealth effect" on the economy.
Well, what's next?
The Fed, in its limited experience with QE has never done an "open-ended" asset purchase program. QE1 and QE2 involved pre-announced and well-defined large amounts of bond buying with a fixed end date. QE3, as designed, seems like a sensible departure and likely garnered support it might not otherwise have had because of its flexibility.
For one thing, it parallels more conventional monetary policymaking in which, in normal times, the FOMC decides meeting by meeting how much to adjust the federal funds rate. It will enable the FOMC to honestly say it is not wedded to a predetermined amount of easing and can terminate the bond buying, or extent and enlarge it, depending on the circumstances.
How much or how little the FOMC ultimately will decide to do is anyone's guess. It will depend, among other things, on the election’s outcome, the steepness of the fiscal cliff, the success of the European Central Bank's latest bond-buying adventure and the magnitude of the Chinese slowdown.
One thing is for sure, until such time as the composition and/or leadership of the FOMC changes significantly and so long as it has tools left to use, the Fed will keep plugging away at stimulating the economy.