First, for anyone who needs a refresher, core inflation is a measure that excludes goods with very volatile prices, generally food and energy. As for those prices that are included in the core measure, Krugman says:
The key thing about these less flexible prices -- the insight that got Ned Phelps his Nobel -- is that because they aren't revised very often, they're set with future inflation in mind. Suppose that I'm setting my price for the next year, and that I expect the overall level of prices -- including things like the average price of competing goods -- to rise 10 percent over the course of the year. Then I'm probably going to set my price about 5 percent higher than I would if I were only taking current conditions into account.
And that's not the whole story: because temporarily fixed prices are only revised at intervals, their resets often involve catchup. Again, suppose that I set my prices once a year, and there's an overall inflation rate of 10 percent. Then at the time I reset my prices, they'll probably be about 5 percent lower than they "should" be; add that effect to the anticipation of future inflation, and I'll probably mark up my price by 10 percent -- even if supply and demand are more or less balanced right now.
Now imagine an economy in which everyone is doing this. What this tells us is that inflation tends to be self-perpetuating, unless there's a big excess of either supply or demand. In particular, once expectations of, say, persistent 10 percent inflation have become "embedded" in the economy, it will take a major period of slack -- years of high unemployment -- to get that rate down. Case in point: the extremely expensive disinflation of the early 1980s.
In short, since the goods in the basket for core inflation have very sticky prices. So when they do move, it really maters, since it's a strong indication that inflation expectations are changing. And when it comes to inflation, expectations are very important -- a dollar is only worth what the market dictates.
I think this logic makes a great deal of sense, but I'm not completely convinced that food and energy prices are easily ignored. Energy, in particular, matters along every step of the way in the supply chain. Producers need energy for making a good; their shipper burns energy getting it to retailers; stores need power to stay open. If energy prices are increasing, then costs increase. The wholesale prices, shipping and retail prices should all reflect that.
When costs increase, firms must either raise their prices or cut their other costs in order to maintain their profit margins. It's true that these prices will are sticky, but if costs have already been deeply cut (like they have been over the past few years), then firms might be more likely to feel forced to raise their prices. And that's not so much going to affect inflation expectation as much as the "catchup" necessary to get the price level right.
Don't get me wrong: this kind of has to do with expectations too. Firms will care more about those energy price increases if they don't believe they're temporary. So expectations still play a part. But if energy prices rise and remain high for a prolonged period, then price level increases for core goods would be unavoidable, no matter what expectations were.
That doesn't mean core isn't the most important measure for inflation -- it is. But I think the inflation trend of those goods with less stable prices, like energy, is also important for the reason I explained. If a systematic increase in energy prices prevails, then I don't see how greater core inflation is ultimately avoided.
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