In an earlier post, I used recent UK experience to suggest looking at other inflation measures besides consumer prices. The two widely published alternatives are the GDP deflator (the price of all that is produced rather than consumed in an economy) and average earnings/wages. The reason I gave for looking at these other inflation measures was purely pragmatic. Some shocks (like value added tax or commodity price changes) could have a significant impact on consumer prices, but it was difficult to know whether their impact is just temporary or whether they could initiate something worse. Looking at output prices and wages would give you a much better handle on that.
But consumer prices are what ultimately matter, right? They are what we care about. Well no, not necessarily. Individuals as consumers do not like consumer prices going up, but wages going up are good for individuals as workers, unless they are someone else’s wages. The first thing we need to do is distinguish between inflation in an abstract sense and movements in real variables. (The two may be associated, of course, which may help explain public attitudes.)
For economists, inflation is the phenomenon where, over some period, all nominal magnitudes are rising together, without any particular implications for real incomes. There are a number of reasons why inflation so defined may be costly. One is that holding money becomes more expensive, because its purchasing power decreases with inflation. In this case a focus on consumer prices is probably correct. But there is another cost of inflation where the source of inflation does matter.
It has been known for some time that inflation goes hand in hand with greater variability in relative prices. To the extent that inflation causes this relative price variability, it is costly, because relative price changes caused by inflation alone distort the market mechanism. One clear reason why this might happen is that many prices are changed infrequently at different times. If the price of good X is changed each April, and the price of good Y is changed each October, then inflation will cause good Y to be too cheap relative to good X in the summer and too expensive in the winter. There is no reason in terms of supply or demand for this pattern in the relative price of X in terms of Y, so if it leads to changes in consumption or production it misallocates resources. The higher the rate of inflation, the greater this misallocation cost.
Is this cost of inflation important? Keynesians certainly believe that slow or 'sticky' price adjustment is pervasive, and is largely responsible for generating persistence in business cycles. The reasons why many prices are sticky appear diverse and complex, but imperfectly competitive markets do seem to be important. In recent years a number of Keynesian economists (following pioneering work by Michael Woodford in particular) have attempted to quantify the costs of the relative price movements generated by inflation and sticky prices, and these calculations suggest that changes in inflation generate significant changes in social welfare through this route. This means that the type and source of inflation is important. If inflation occurs in commodities where prices are changed frequently, then it is much less costly than if inflation occurs in prices that are sticky. In other words, some types of inflation are more costly than others.
This focus on inflation in goods where prices are sticky bears some relation to the idea of focusing on 'core' inflation. It is possible using similar reasoning to argue that the price inflation measure central banks should target is actually output prices (the GDP deflator) rather than consumer prices. (If there are trends in relative prices, then this idea also influences the optimal inflation target: see Alexander Wolman here, for example.) The same type of argument also suggests that central banks should be concerned with wage inflation as well as price inflation. This is because wages are themselves 'sticky', and so wage inflation will generate costs by changing relative wages in a distortionary manner just as price inflation causes distortionary relative price changes. All this is explained clearly, in considerable technical detail, by Michael Woodford here.
To see why this is important, consider the recent impact of higher oil prices. This has a direct and fairly immediate impact on inflation. A hard-line inflation targeter would say that an inflation target is an inflation target, so the monetary authority should try and make non-oil price inflation fall to offset the impact of higher oil prices. However oil prices, and some things they influence like petrol prices, are pretty flexible. If inflation is costly because it induces unnecessary relative price distortions in prices that are sticky, then it makes sense to focus on inflation measures that give much less weight to oil prices than the consumer price index. In other words, core inflation is not just useful because it helps predict longer term trends in actual inflation, it is important because it actually matters more than actual inflation.
 A seminal paper here is Aoki, K (2001), Optimal monetary policy responses to relative-price changes, Journal of Monetary Economics, vol. 48, pp 55-80. See this on the relationship between actual inflation, core inflation and commodity prices in the US.
 For the rationale, and some additional and pretty technical complications, see Kirsanova, Leith and Wren-Lewis (2006) ‘Should Central Banks target consumer prices or the exchange rate?’ Economic Journal, 116, F208–F231.
 : The key paper here is Erceg, Christopher J., Dale W. Henderson, and Andrew T. Levin, “Optimal Monetary Policy with Staggered Wage and Price Contracts,” Journal of Monetary Economics 46: 281-313 (2000).