And it's good to target the stickiest prices; and bad to target the more flexible prices.
It means that recessions under inflation targeting can last as long as it takes for the stickiest prices to change. Which is bad. And it's especially bad for us old macroeconomists, who remember that the whole point of New Keynesian macroeconomics, when it began in the 1970's, was to show that a good feedback rule for monetary policy, unlike a k% money growth rule, meant that recessions did not have to last as long as it took firms to change prices.
Some prices are stickier than others. Here's a simple extreme example to illustrate my point:
Half the firms change their prices every n periods; so a fraction 1/n of them change their price each period. Think of n as a big number, so 1/n is a small number. And they have to announce any price changes one period in advance, to warn their customers (this assumption doesn't matter much). The other half the firms change their prices every period, (without advance warning).
The economy is humming along normally, with all prices constant. Then a negative shock hits aggregate demand. The central bank does not have a crystal ball, so does not see the shock coming. It only learns of the shock when it sees all the flexible price firms cut their prices 10%. How does the central bank respond in the following period?
Constant Price Level Target. The central bank wants the price level to go back to where it was before the shock. So it loosens monetary policy just enough to offset the shock fully, so the flexible price firms raise their prices back to where they were before. The sticky price firms, since they know the central bank will do this, don't change their prices at all. The recession is over after one period (as long as it takes for the central bank to recognise the shock and respond to it).
0% Inflation Target. Since half the firms have cut their prices by 10%, the average price level has fallen by 5%. The central bank lets bygones be bygones, and wants the price level to remain at its new lower level, so inflation stays at 0% thereafter. The central bank knows that every period a fraction 1/n of the sticky price firms will cut their prices by 5%, and to offset this it loosens monetary policy just a little so that the flexible price firms raise their prices by (5/n)%. After n periods all firms have prices that are 5% lower than originally. The economy slowly recovers from the recession, but does not fully recover until all firms have adjusted their price to the new level of aggregate demand. Which means the recession lasts for n periods. After n/2 periods, for example, 3/4 of firms's prices are where they want them to be, but 1/4 of firms still have prices that are too high, so their sales and output is too low.
Intuition. Inflation targeting ignores the levels of prices. It only looks at changes in prices. It wants all price changes to average out to 0% (for a 0% inflation target). So firms that change their prices twice as frequently get twice the weight in the average that the central bank targets. But firms who change their prices twice as frequently should only get half the weight; because they can adjust to aggregate demand shocks twice as easily, and have only half the need for the central bank to adjust aggregate demand so they don't have to change prices.
Other off-topic stuff: A Price Level Target also acts as an automatic stabiliser (for a central bank that sets a nominal interest rate). Because the expected inflation in the second period means the real interest rate falls in the first period. So the shock to aggregate demand is partly offset even if the central bank doesn't see it. And an NGDP level path target would be better still, if there are supply shocks, especially if the Short-Run Phillips Curve is flattish. But all that other stuff is beyond this post.