Central banks could well find themselves faced with the dilemma of having to decide to which goal they should give priority. The fact that bubbles tend to occur in periods of low inflation only accentuates that potential conflict. And, last but not least, there is the question of “moral hazard”.
Central banks would need to respond in perfect symmetry to overshooting and to undershooting stock prices. Financial markets would become severely distorted if market participants were to come to the conclusion that the central bank would let them enjoy returns when the markets are up but bail them out once the market goes downhill.