I think the opposite is self-evident. The idea that private management and allocation of capital can do better than government management of the economy is ridiculous. This has been proven many times over by post-industrial history, and is also immediately obvious if you've ever been to the airport. Or if you know anything about the inner workings of a complex modern commercial apparatus, such as any investment bank, or Enron.
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Sarcasm is when you repeat something without negating it, but indicating that you really think the opposite (negation). Saying “private sector management doesn’t work” and meaning it is not sarcasm.