is it desirable to pursue a goal of zero unemployment
No, the unemployment in the market will remain at the point of natural unemployment, natural unemployment in the market is a sum of structural and frictional unemployment together, if we try to reduce it to zero the government will need to introduce a very expansionary policy in the market and that might lead to inflation, but again when the wages settle back to the real rate then the unemployment will be high again.
reducing the unemployment to zero may lead to inflation spiralling out of control.
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