What's Wrong With A Little Asset Price Inflation?

One of the more common responses to the fact that inflation is low is the idea that the inflation is all in asset prices. So, for instance, if someone were to say that all the Fed's post-crisis stimulus didn't result in inflation you might look at stock prices and argue that the price increases all flowed into stocks. That's not necessarily wrong, but I think it needs to be well understood because asset price inflation is generally good for the economy while consumer price inflation could be bad.¹