Speculative, as Tyler Cowen would say.
When people play the Dictator Game Ultimatum Game [thanks Alex] they don't play the way economists would predict. The Nash Equilibrium is for the first player to propose a 99:1 division of the pie take-it-or-leave-it offer, and the second player to accept. Because 1% is better than nothing, so the second player will accept, and the first player knows this.
But proposals like that tend to get rejected. And 50:50 splits tend to be common. (But IIRC non-WEIRD people do play Nash, but I can't find a link.)
50% is a magic number, if there are two players. (I wonder what would happen if the Dictator Game were played with three apples, or one bottle of scotch and one bottle of beer?)
49 is not a magic number. But any number can become a magic number by custom and convention. And the 49th parallel has become a magic number for Canadians and Americans. Each would get very upset if the other crossed that line on the map. 49 has become a focal point. The 49th parallel is like a sticky price, that doesn't change at all when relative bargaining power changes a little.
0% change can also be a magic number. I won't get upset if my coffee shop raises its price by 0%.
2% per year is not a magic number. But it can become a magic number by custom and convention. I won't get upset if my coffee shop raises its price by 2% per year. Because that's the custom and convention in Canada. The Bank of Canada has been targeting 2% inflation for 20 years, and we've gotten used to it. 2% inflation has become "quasi-constitutional". If 2% is now a magic number for the Bank of Canada, it wouldn't be surprising if 2% is a magic number for all of us.
The game played between a monopolistically competitive firm and its customers is not exactly the same as the Dictator Ultimatum Game, because it's not a zero-sum game. But like the Dictator Ultimatum Game, the firm moves first, and makes a take-it-or-leave-it offer when it sets the price. The customer decides whether to accept or reject that offer. Haggling is not normal.
If 2% has become a magic number, the Short Run Phillips Curve can become rather flat at around 2% for quite a wide range. Because the representative firm's demand curve is kinked at 2% (more elastic above 2% than below), so it has two Marginal Revenue curves at the kink -- for price increases above and below 2%. "Expectations are well-anchored" as the Bank of Canada loves to say, but "expectations" can also mean "norms". And it would take a long time at a very high or very low unemployment rate before the custom and convention eventually breaks down. The vertical Long Run Phillips Curve is rather thick for a rather long time.
The Very Short Run Phillips Curve won't be exactly horizontal at 2%. There are always "special circumstances" that allow custom and convention to be set aside temporarily. And some markets are much more like auction markets where both sides can haggle over the price. Like oil prices. But auction prices can only depart so much for so long from the 2% inflation customary and conventional core.
If I am right about this, then inflation targeting sowed the seeds of its own destruction, by creating a magic number for inflation. Divine Coincidence (where minimising the deviation of inflation from target coincides with minimising the output gap) works perfectly with a Calvo Phillips Curve. Because Calvo's fairy touches firms at random, so the sample of firms that do change prices are exactly representative of the population of firms that cannot change prices and so allow output to deviate instead. A non-random fairy messes up Divine Coincidence a bit. A 2% magic number will destroy Divine Coincidence; it destroys the very signal that central banks use to tell whether monetary policy is tight or loose. Like Goodhart's Law. All it can respond to is noise.
As I said, speculative.