The RBNZ would be better off not talking about inequality
In a OpEd for Stuff, the Governor of the RBNZ writes that "Unemployment ... underpins income inequality". He appears to be defending the Bank in terms of how its policies may have greatly increased house and stock prices due to keeping interest rates close to zero, thereby exacerbating wealth inequality, by arguing instead that the Bank is trying to keep inequality low by focusing on reducing unemployment.
However, the idea that unemployment "underpins" inequality is wrong. In 2019 the unemployment rate was extremely low in China and the US, at 3.6% and 3.7%, respectively. These two countries also have among the highest rates of inequality in the world. Inequality in China and the US is "underpinned" by enormous wage inequality.
Furthermore, many European countries have very generous unemployment insurance programs to help ensure low inequality. However those schemes are often blamed for pushing unemployment rates upward.
The fact that central banks have no long run influence on the unemployment rate only adds to the confusion around the Governor's comments that it is somehow designing monetary policy with a view to influencing inequality.