Earlier this year, Mr Alexander said New Zealand had a shortage of dwellings.
New Zealand needed a further 28,000 dwellings, he said.
That has been lifted to 39,000 after the Christchurch earthquake in February and because of the time needed to get consents issued.
"Our view for the past couple of years has been that at some point, probably this year, we will see increasing pressure on rents as a result of this shortage. This pressure we opined would be greatest in Auckland because the housing cycle of the 2000s involved the strongest buying and excess construction happening in the regions and not in our biggest city. They had their boom in the 1990s.
"Our pick is that this cycle will once again belong to Auckland."
Mr Alexander believed the tightening of the rental market would translate into greater awareness of the shortage, and discussion of a lack of property would become common.
The BNZ also picked that from late this year, house prices would be rising again because of the interaction between the worsening shortage of property, and not just economic growth but also a potentially rapidly tightening labour market.
"So how do we find ourselves now?
First, the awareness of a housing shortage is so great that the Minister of Finance [Bill English] has decided to duplicate the work already done by the commerce select committee over 2007-08 and hold another inquiry into New Zealand housing costs."
This time, the inquiry would be run by the newly formed Productivity Commission.
Although it was early days, there was growing evidence of accelerating growth in rents. That was one element strongly evident in the monthly Barfoot and Thompson report released on Wednesday, he said.
Barfoot and Thompson noted that in their sphere of activity - about one-third of the Auckland residential property market - average rents for newly tenanted properties in March came in at $434 a week, from $402 in February.
That was a rise of 8.2% on a year ago and the greatest one-month increase on record, Mr Alexander said.
"There is a tendency for this measure to fluctuate $10 and sometimes $20 on a non-sustained basis, so we see a very high probability that in April the average rent will decrease. But a fall equal to March's rise seems unlikely, given the strengthening anecdotal feedback regarding tenants' bidding rents up to secure properties."
Average house prices were again rising in Auckland.
As buyers came out of the woodwork, potentially waves of sellers would appear.
First would be those who actually had their properties on the market.
Next, there would be one or two waves of those who might have tried to sell two to three years ago but gave up and had not worried very much because interest rates had been so low, he said.
Included in the second group of groups would be the small investors who, through losing the ability to claim depreciation and offset cash-flow losses against other income, would want to quit the market.
"It is because of this wave theory we think the Auckland housing market will not simply go on a straight line upward," Mr Alexander said.
