baby farm
n.— «Among many of the phrases used to describe her for her despicable crime, the 33-year-old was called a “baby farm” because she appeared to deliberately increase her brood to gain more state aid.» —“Reforms hark back to roots of the welfare state” by Laura Devlin EDP24 (Norwich, England, United Kingdom) Dec. 11, 2008. (source: Double-Tongued Dictionary)
Further reading
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Why does the term vegetarian cause so much confusion? Some people assume it means avoiding red meat but still eating chicken. And is there a term for a vegetarian who also eats fish and poultry? Plus, screwball comedies from the 1930s mix slapstick...
I’ve heard the word “baby factory” in this context but never baby farm.