The Labour Party will be under enough pressure as it is, to add all sorts of social objectives to any monetary policy agreement it would make. House affordability would be one of them, especially as, under Wheeler, the Reserve Bank has taken on the added task of regulating mortgage lending.
It has done so in the name of financial stability rather than social policy.
It was not the Reserve Bank's responsibility to help first-home seekers find affordable houses but it was its role to worry about the banking sector's exposure to over-stretched borrowers.
Wheeler and his advisers became so concerned about mortgage borrowing on very low deposits that they introduced limits on loan-to-value ratios.
The first "LVRs" made no exemption for first-home seekers although Prime Minister John Key publicly urged the governor to make an exemption.
To his credit, Wheeler resisted, maintaining the bank's statutory independence from political influence.
But the first LVRs slowed Auckland house prices for only a month or two and in November, 2015, a second round of LVRs were tougher and aimed exclusively at investment house purchases in Auckland.
Those are being given the credit for the present slowing of the market, although it may have more to do with trading banks taking their own decisions to reduce residential property lendingmid-way through last year.
Under Wheeler, the Reserve Bank and trading banks fell out of step last year for the first time since 1989 when monetary policy set out to be transparent and predictable. The central bank kept lowering its base interest rate when trading banks wanted to raise their lending rates.
Wheeler was trying to keep New Zealand interest rates, and the dollar's exchange rates, from getting out of line with countries which were lowering their rates in the belief their flat economies needed inflation.
Since Wheeler had been undershooting his 1-3 per cent inflation target for some time, he had a technical justification for taking the official cash rate down to the record low where it remains.
But with continuing growth in this economy, the rate is probably too low.
He held the rate yesterday despite the US Federal Reserve beginning to lift its rate against the prospect of inflationary tax cuts and spending by President Donald Trump.
The next few years are going to be difficult and the next governor will need to be bold.