Economist Eric Crampton estimates that Labour's abolition of the youth wage has cost 12,350 young people jobs.I'd be more comfortable saying that the abolition of the youth wage has most likely cost between eight and twelve thousand jobs. The numbers since mid 2009 have ranged from about seven thousand to about thirteen thousand more kids unemployed than we'd expect given the adult unemployment rate. 12,350 is the number that drops out of the last quarter, but StatsNZ warned about data problems with that quarter's HLFS given data collection problems in Canterbury.
The "most likely" matters too as I don't have a design that allows for strong causal statements. I'd put 20:1 that youth minimum wage increases are to blame for current excess youth unemployment, but it's always possible that something else did it. Something else that affected kids (and not adults) and that hit with the right timing and that has a good body of theory predicting such an effect and that would have been large enough to have had that effect. I can't think of what else meets the bill, but maybe I'm unimaginative.