Paul Krugman explains why the Fed focuses on core rather than headline inflation. I've made this argument a different way, but it's basically the same argument. Boiled down, the argument is that the problem prices are the sticky prices, not the prices like those on food and energy that are flexible, and you want an index (core inflation) that best highlights the problem, i.e. that includes only the problem prices and throws out the ones that can be left to adjust on their own (I prefer measures that trim out the most volatile prices or use other mechanisms to extract the slowly adjusting prices rather than simply tossing out food and energy, but that's a technical point).
Here's how Krugman explains the intuition (the exact story is model dependent, but the intuition almost always comes down to indexing the problem prices so that they stand out, and leaving the other prices to take care of themselves, and expectations of inflation are a key part of the story). But more important than the explanation for why the Fed focuses on core inflation is the conclusion Krugman comes to, that the Fed shouldn't let worries about inflation interfere with stabilization of the financial sector and the broader economy:
Embedded vs. non-embedded inflation, by Paul Krugman: The big economic debate of the moment is whether the Fed and its peers have made a terrible mistake by focusing on staving off financial crisis while more or less ignoring the rise of inflation. “Inflation is rising and it seems the world’s central banks have critically misjudged the situation,” says Wolfgang Munchau in the FT. I’ve heard even more apocalyptic views from some serious people in the last couple of weeks. As it happens, I don’t agree. And I thought it’s worth spelling out why — especially because many if not most of the participants in this debate don’t explain their premises very clearly. So: when is it appropriate to get very concerned about inflation, and when is it OK to assume that a rise in prices is a temporary shock that will pass? The answer is that inflation becomes a big problem if it becomes “embedded” in the economy, which makes it hard to restore more or less stable prices. But how does inflation get embedded? [Here's how...]
So now you've read Krugman and you are confused. I've talked about core measures of price inflation, yet Krugman is worried about wages adjusting sluggishly. How can the two views be reconciled? There is a larger theory that says to include any price that adjusts slowly, whether it's an input price or an output price, in Fed's monetary policy rule. Here's Michael Woodford:
The theory also provides important insights into the question of which price index or indexes it is more important to stabilize. Again, the answer depends on the nature of the nominal rigidities. If prices are adjusted more frequently in some sectors of the economy than in others, then the welfare-theoretic loss function puts more weight on variations in prices in the sectors where prices are stickier... This provides a theoretical basis for seeking to stabilize an appropriately defined measure of "core" inflation rather than an equally weighted price index. .... Similarly, if wages are sticky as are goods prices, as implied by many empirical ... models, then instability in the rate of growth of a broad index of nominal wages results in distortions similar to those created by variations in goods price inflation. If [adjustments in] wages are staggered ..., then the welfare-theoretic loss function includes a term proportional to the squared rate of goods price inflation and another term proportional to the squared rate of wage inflation each period. In this case, optimal policy involves a tradeoff between inflation stabilization, nominal wage growth stabilization, and output-gap stabilization...
I should also add that, under some theoretical formulations, asset prices should also be part of the Fed's monetary policy decision rule. However, for the most part, asset prices exhibit a high degree of flexibility so little is lost from leaving those prices out. However, as I've argued before, an exception is housing prices - they can exhibit downward rigidity - and this is a reason to consider including housing prices in decisions about the direction of monetary policy, something that would cause the Fed to lean against housing price bubbles as they are inflating.
One final note. Some people will disagree that the Fed should worry about wage inflation, i.e. they will say that the Fed moves to suppress wages whenever workers begin to realize gains and hence works against their interests. But I think that wage inflation that exceeds productivity growth hurts workers in the long-run. Thus, it's something to avoided.