What is the unemployment rate of Niue?

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10.7% (2006)

I don't know where this 10% came from. If it came from the Niue Statistics department then it probably measures the amount of working age people that don't work for the government in which case it is an abomination.
There is no real unemployment in Niue. Subsistence agriculture and fishing are gainfully undertaken by a very small percentage of those not employed by the government or private sector. In a population of 1200 there wouldn't be 1% who don't partake in any of these forms of employment.
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