CHICAGO—The nation’s leading economists are suffering an identity crisis as many of the institutions they helped build and causes they advanced have come in for public scorn and rejection at the ballot box.

The angst was on display this weekend at the annual conference of the American Economic Association, the profession’s largest gathering. The conference is a showcase for agenda-setting research, a giant job fair for the nation’s most promising young economists and, this year, the site of endless discussion about how to...