Immigration. It's not that immigrants buy up the houses (opposites, we tend to live cramped up in flat shares). It's that they support landlords profits (with cream on top of not knowing your rights) and have overall pressure on market.
Building should at least match immigration/population growth.
Also not building dense enough. Kiwis always tout "I wouldn't wanna live in a house without a backyard" - well I don't wanna rent forever either. Given choice of no housing and shitty housing I choose shitty housing.