Broadly speaking, there are three approaches to stabilization policies and business cycles:
1. Stop the excesses: In this view, monetary policy is too expansionary at certain times. Some people worry about excessive real GDP growth. Others worry about asset prices bubbles and/or an orgy of lending. In this view, the excessive monetary stimulus leads to a later relapse into recession or depression. Policymakers need to "fix" this problem by being less expansionary during the periods of excess.
2. Stimulate during recessions: In this view the real problem is not RGDP or financial excesses, but shortfalls in economic growth. The problem of recessions and depressions must be fixed with an expansionary monetary policy. But policy should not try to stop economic booms. Keynes is a famous proponent of this view--he argued that we should aim for a permanent boom.
3. Laissez-faire: In this view monetary policy should pay no attention to any of these issues. It should not try to push back against excessive RGDP growth or asset price bubbles or excessive lending, nor should it try to boost the economy during recessions. Rather it should try to provide a stable path for some sort of nominal aggregate that is a proxy for the value of money itself, such as the overall price level or NGDP. Thus the central bank should ignore problems such as recessions and bubbles, and (to use the phrase of Robert Murphy) the economy should be "left alone to sort things out" during a recession.
I would put Friedrich Hayek and myself into the third group. Hayek favored a monetary policy aimed at a stable path for NGDP, and so do I. Perhaps this is all pretty obvious, but when I read other bloggers I often see confusion between NGDP cycles (i.e. nominal or monetary instability issues) and the sort of RGDP cycles that would occur even under a policy of stable NGDP growth (a topic covered by "real business cycle" theory.)
Some people erroneously conclude that those who favor a stable path of NGDP also favor policies aimed at "fixing" recessions or booms. Not so, it is those who claimed monetary policy was too easy in 2004 for financial/realGDP reasons, and those who claim monetary policy was too tight in 1974 for unemployment reasons, who are the activists. I'd call the first groups "conservative activists," and I'd call the second group "left wing activists." Both groups are wrong, and for essentially the same reasons. Even if it were true that monetary policy should try to stabilize some sort of real or financial variable (and it isn't true), you'd want to do so symmetrically. The asymmetric views of these ideologues are the "tell" that they have a different agenda.
The second reason they are wrong is that monetary policy should not be used to fix real problems such as fluctuations in real GDP, fluctuations in real asset prices, or fluctuations in real quantities of lending. Provide a stable path for the value of money and let the economy take care of itself.
PS. I'd like to think Robert Murphy is with Hayek and myself in the third group, but I can't quite tell. Like many other people, Bob Murphy frames the issues in a slightly different way than I do.
PPS. Although I did not know Gary Becker personally, I did take a class from him back during my first year at Chicago (1977-78.) I had many great teachers at Chicago but he was the most brilliant. The opening few paragraphs of his Nobel Prize lecture clear up some misunderstandings regarding his views on rationality and selfishness. I was very sad to hear that he died. Tyler Cowen has a number of very good links relating to Gary Becker.