In response to the GOP request, I believe the Fed should go forth with operation twist or QE3. Here's why:
1) The economy is in poor shape. Given every monetary policy rule (a rule saying interest rates should be set according to the output gap and inflation) the Fed has examined, interest rates should be zero (see Mankiw's response here).
2) The GOP (and the rest of Congress for that matter) has no busy attempting to influence monetary policy. We've seen how well they have done with fiscal policy. Had Democrats and Republicans been able to work together and cement down policy over the next 5-10 years the economy would be in significant better shape. It is Congresses fault we are here, please don't spread that to the only entity that is able to promote growth.
3) A central banks ability to conduct effective monetary policy depends on their independence from the federal government. To prove they act independent of the federal government they should completely ignore this letter.
4) Critics of the Fed will point to the large injections of money and the risk of high inflation to the economy. Most economists agree, the risks have been subdued. Inflation is low (core inflation for August was 0.2% - annual rate of 2.4%) and the Fed has the tools to reverse inflation very quickly if it shows any signs of increasing.
5) The Fed has a mandate to promote price stability, economic growth, and moderate interest rates. Their policies have been in accordance with their mandate. Even if you want to change the mandate (it has been proposed that the Fed focuses solely on inflation) this necessary change the course of action.
6) The GOP seems concerned about the weak dollar. Clearly they have not looked at the recent data that shows exports are the only component of GDP that has been increasing ($350 billion since the beginning of 2009). I wonder why?
7) Have the low interest rates caused U.S. households to take on more borrowing? Well there's a graph for that:
Total consumer credit outstanding (blue line) has barely increased since the end of the recession. Total revolving credit (credit card debt - red line) continues to decrease. I'd say this is a good thing. People are borrowing in a secured way (cars and homes).
8) This quote is laughable given the actions of Congress:
Ultimately, the American economy is driven by the confidence of consumers and investors and the innovations of its workers. The American people have reason to be skeptical of the Federal Reserve vastly increasing its role in the economy if measurable outcomes cannot be demonstrated
Why don't you do something then.
9) Here are more thoughts (Mark Thoma)