1. The original version expressed a trade-off in terms of the level of inflation vs. the level of the unemployment rate. From about 1953-1968, the trade-off was approximately: inflation + unemployment = 7 percent.
2. In the 1970's Phillips Curve modelers "took another derivative." They said that the relationship was between the change in the rate of inflation and the level of the unemployment rate.
3. Also in the 1970s, there emerged a split in the causal interpretation of the Phillips Curve. The Keynesian interpretation was that the causality runs from unemployment to the change in the inflation rate. When unemployment is high, inflation declines. When unemployment is low, inflation rises. The monetarist, or Chicago, interpretation was that the causality runs from a change in the inflation rate to the level of unemployment. When inflation goes up, unemployment is low. When inflation goes down, unemployment is high.
4. James Tobin, in his 1972 presidential address for the American Economic Association, offered a simple story for the Phillips Curve. Suppose that we have two industries, and the wage in industry A needs to rise relative to industry B in order to reach a new full employment equilibrium. If nominal wages are flexible, no problem. If nominal wages are sticky downward, then the only way to get the relative wage to its proper level is for nominal wages in industry A to rise. Thus, to get the industry shift to occur and avoid unemployment, the economy needs inflation.
5. In the most recent recession, my impression is that private sector wages were not completely sticky downward. Public sector wages have been sticky downward, however.
6. I think that if you do not assume downward stickiness of wages, the recent data are difficult to explain. That is, if you just think there is a relationship between the change in the inflation rate and the level of unemployment, then by now the inflation rate in the U.S. ought to be well below zero. However, with downward stickiness of wages, you can argue that the high unemployment is not having much effect on the inflation rate.
7. I believe that nominal wages are sticky downward on a case-by-case basis. That is, the wage of worker X in firm Y is unlikely to be to be cut. However, worker X can get a wage cut by losing a job and switching to a different firm. Moreover, firm Y can cut its wages by lowering the wage it pays to new workers, or by outsourcing some of its work to a lower-wage firm. So I am not so convinced that the case-by-case stickiness implies aggregate stickiness.
8. Since 2008, we have had very high unemployment, and the reduction in inflation has been miniscule. I see this as the worst performance for the Phillips Curve since the 1970s.