The duo are adapting Fey's 2004 film which starred Lindsay Lohan, Rachel McAdams, Amanda Seyfried and Lacey Chabert as a cliquey gang of mean high-school girls.
The film also starred Lizzy Caplan, Amy Poehler and Fey herself as a teacher, reported Billboard.
"We definitely want to see some kind of shape by the end of the summer," Richmond said.
But he said there would still be plenty of work to do once he, Fey and lyricist Nell Benjamin felt they had finished.
The musical is eagerly-awaited by fans of the film, which won several MTV Movie Awards and Teen Choice Awards in 2005 and has since gained cult status among many.
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