Economists Emmanuel Saez and Gabriel Zucman of the University of California, Berkeley are advising Elizabeth Warren’s presidential campaign and drawing generous media attention for their assertion that the U.S. tax system is flat—that the middle class and poor pay as great a share of their income in taxes as the rich. They’re wrong, and three huge mistakes underlie their analysis.

The biggest mistake is to focus on gross, not net, taxes. They ignore transfer payments, like Social Security, which are disproportionately paid...