I believe most folk are worried, and the media has been referring to it as a "housing crisis". Notably the price increase has been limited to Auckland though (and Queenstown, but Queenstown is the NZ equivalent of Aspen, Colorado; which is to say it's a bit of a millionaire's club).

Unfortunately the currently in power government has done a lot to court this outcome, and so has mostly been avoiding doing anything about it. I think they did recently step in to make it harder for Chinese investors to purchase property in NZ though.