Like thousands of other single women living in Bushwick, Brooklyn, Brigette, 24, collects Medicaid and food stamps. Unlike most of her neighbors, she's white and a college graduate—the kind of welfare recipient rarely considered in debates over public assistance. Brigette, whose parents and two sisters run a restaurant in rural Vermont, got her B.A. in film from Bard College, a top-tier liberal arts school in upstate New York. She moved to New York City about two years ago to pursue experimental filmmaking. As young self-styled bohemians have always done, she found a neighborhood with cheap rent and cobbled together a...