No book on economics in recent times has received such a glowing initial reception as Thomas Piketty’s “Capital in the Twenty-First Century.” He remains a hero on the left, but the honeymoon may be drawing to a sour close as evidence mounts that his numbers don’t add up.

Mr. Piketty’s headline claim is that capitalism must result in wealth becoming increasingly concentrated in fewer hands to a “potentially terrifying” degree, on the grounds that the rate of return to capital exceeds the rate of economic growth. Is there any empirical evidence to back up this sweeping assertion? The data in his book—purporting to show a growing inequality of wealth in France, the U.K., Sweden and particularly the United States—have been challenged. And that’s where the story gets interesting.

In late May, Financial Times economics editor Chris Giles published anessay that found numerous errors in Mr. Piketty’s data. Mr. Piketty’s online “Response to FT” was mostly about Europe, where the errors Mr. Giles caught seem minor. But what about the U.S.?

Mr. Piketty makes a startling statement: The data in his book should now be disregarded in favor of a March 2014 Power Point presentation, available online, by Mr. Piketty’s protégé, Gabriel Zucman (at the London School of Economics) and his frequent co-author Emmanuel Saez (of the University of California, Berkeley). The Zucman-Saez estimates, Mr. Piketty says, are “much more systematic” and “more reliable” than the estimates in his book and therefore “should be used as reference series for wealth inequality in the United States… (rather than the series reported in my book).”

Zucman-Saez concludes that there was a “large increase in the top 0.1% wealth share” since the 1986 Tax Reform, but “no increase below the top 0.1%.” In other words, all of the increase in the wealth share of the top 1% is attributed to the top one-tenth of 1%—those with estimated wealth above $20 million. This is quite different from the graph in Mr. Piketty’s book, which showed the wealth share of the top 1% (which begins at about $8 million, according to the Federal Reserve’s Survey of Consumer Finances) in the U.S. falling from 31.4% in 1960 to 28.2% in 1970, then rising to about 33% since 1990.

In any event, the Zucman-Saez data are so misleading as to be worthless. They attempt to estimate top U.S. wealth shares on the basis of that portion of capital income reported on individual income tax returns—interest, dividends, rent and capital gains.

This won’t work because federal tax laws in 1981, 1986, 1997 and 2003 momentously changed (1) the rules about which sorts of capital income have to be reported, (2) the tax incentives to report business income on individual rather than corporate tax forms, and (3) the tax incentives for high-income taxpayers to respond to lower tax rates on capital gains and dividends by realizing more capital gains and holding more dividend-paying stocks. Let’s consider each of these issues:

  • Tax reporting. Tax laws were changed from 1981 to 1997 to require that more capital income of high-income taxpayers be reported on individual returns, while excluding most capital income of middle-income savers and homeowners. This skews any purported increase in the inequality of wealth.

For example, interest income from tax-exempt municipal bonds was unreported before 1987—so the subsequent reporting of income created an illusory increase in top incomes and wealth. Since 1997, by contrast, most capital gains on home sales have disappeared from the tax returns of middle-income couples, thanks to a $500,000 tax exemption. And since the mid-1980s, most capital income and capital gains of middle-income savers began to vanish from tax returns by migrating into IRAs, 401(k)s and other retirement and college savings plans.

Balances in private retirement plans rose to $12.4 trillion in 2012 from $875 billion in 1984. Much of that hidden savings will gradually begin to show up on tax returns as baby boomers draw them down to live on, but they will then be reported as ordinary income, not capital income.

Tax law changes, in summary, have increased capital income reported at the top and shifted business income from corporate to individual tax returns, while sheltering most capital income of middle-income savers and homeowners. Using reported capital income to estimate changing wealth patterns is hopeless.

  • Switching from corporate to individual tax returns. When individual tax rates dropped from 70% in 1980 to 28% in 1988, this provoked a massive shift: from retaining private business income inside C‑corporations to letting earnings pass through to the owners’ individual tax returns via partnerships, LLCs and Subchapter S corporations. From 1980 to 2007, reports the Congressional Budget Office, “the share of receipts generated by pass-through entities more than doubled over the period—from 14 percent to 38 percent.” Moving capital income from one tax form to another did not mean the wealth of the top 1% increased. It simply moved.
  • Tax rates and capital gains. There were huge, sustained increases in reported capital gains among the top 1% after the capital-gains tax was reduced to 20% from 28% in 1997, and when it was further reduced to 15% in 2003. Although more frequent asset sales showed up as an increase in capital income, realized gains are no more valuable than unrealized gains so realization of gains tells us almost nothing about wealth. Similarly, a portfolio shift from municipal bonds, coins or cash into dividend-paying stocks after the tax on dividends fell to 15% in 2003 might look like more capital income when it was merely swapping an untaxed asset for a taxable one.

In his book, Mr. Piketty constructed estimates of top wealth shares, decade by decade, melding and massaging different kinds of data (estate tax records, the Federal Reserve’s Survey of Consumer Finances). These estimates are suspect in their own right; but as we now learn from Mr. Piketty’s response to Mr. Giles, we can ignore them.

Yet Mr. Piketty’s preferred alternative, the Zucman-Saez slide show, is also irreparably flawed as a guide to wealth concentration. Mr. Piketty’s premonition of soaring U.S. wealth shares for the top 1% finds no credible support in his book or elsewhere.