Inequality doesn't actually mean anything. It's just a status marker.
You can see here there's no actual, material inequality in the first world.
as the economy develops, the majority of households have crossed the income threshold (for appliance ownership)
The inequality that matters is stuff like rent and food, and those aren't related to a Gini index. ...
Is it always a trade off between efficiency and equity?
No there is not always trade-off between efficiency and equity, but equity does not simply mean government spends some money on something in presence of progressive tax system (that is extremely naïve view and there are myriad of examples where programs that do that exacerbate inequality).
Equity is ...