For years, Democrats and even some Republicans such as former President Donald J. Trump have called for closing the so-called carried interest loophole that allows wealthy hedge fund managers and private equity executives to pay lower tax rates than entry-level employees.

Those efforts have always failed to make a big dent in the loophole — and the latest proposal to do so also faltered this week. Senate leaders announced on Thursday that they had agreed to drop a modest change to the tax provision in order to secure the vote of Senator Kyrsten Sinema, Democrat of Arizona, and ensure passage of their Inflation Reduction Act, a wide-ranging climate, health care and tax bill.

An agreement reached last week between Senator Chuck Schumer, the majority leader, and Senator Joe Manchin III, Democrat of West Virginia, would have taken a small step in the direction of narrowing carried interest tax treatment. However, it would not have eliminated the loophole entirely and could still have allowed rich business executives to have smaller tax bills than their secretaries, a criticism lobbed by the investor Warren E. Buffett, who has long argued against the preferential tax treatment.

The fate of the provision was always in doubt given the Democrats’ slim control of the Senate. And Ms. Sinema had previously opposed a carried interest measure in a much larger bill called Build Back Better, which never secured the 50 Senate votes needed — Republicans have been unified in their opposition to any tax increases.

Had the legislation passed in the form that Mr. Schumer and Mr. Manchin presented it last week, the shrinking of the carried interest exception would have brought Democrats a tiny bit closer to realizing their vision of making the tax code more progressive.

What is carried interest?

Carried interest is the percentage of an investment’s gains that a private equity partner or hedge fund manager takes as compensation. At most private equity firms and hedge funds, the share of profits paid to managers is about 20 percent.

Under existing law, that money is taxed at a capital-gains rate of 20 percent for top earners. That’s about half the rate of the top individual income tax bracket, which is 37 percent.

The 2017 tax law passed by Republicans largely left the treatment of carried interest intact, after an intense business lobbying campaign, but did narrow the exemption by requiring private equity officials to hold their investments for at least three years before reaping preferential tax treatment on their carried interest income.

What would the Manchin-Schumer agreement have done?

The agreement between Mr. Manchin and Mr. Schumer would have further narrowed the exemption, in several ways. It would have extended that holding period to five years from three, while changing the way the period is calculated in hopes of reducing taxpayers’ ability to game the system and pay the lower 20 percent tax rate.

Senate Democrats say the changes would have raised an estimated $14 billion over a decade, by forcing more income to be taxed at higher individual income tax rates — and less at the preferential rate.

The longer holding period would have applied only to those who made $400,000 per year or more, in keeping with President Biden’s pledge not to raise taxes on those earning less than that amount.

The tax provision echoed a measure that was initially included in the climate and tax bill that House Democrats passed last year but that stalled in the Senate. The carried interest language was removed amid concern that Ms. Sinema, who opposed the measure, would block the overall legislation.

Why hasn’t the loophole been closed by now?

Many Democrats have tried for years to completely eliminate the tax benefits private equity partners enjoy. Democrats have sought to redefine the management fees they get from partnerships as “gross income,” just like any other kind of income, and to treat capital gains from partners’ investments as ordinary income.

Such a move was included in legislation proposed by House Democrats in 2015. The legislation would also have increased the penalties on investors who did not properly apply the proposed changes to their own tax filings.

The private equity industry has fought back hard, rejecting outright the basic concepts on which the proposed changes were based.

“No such loophole exists,” Steven B. Klinsky, the founder and chief executive of the private equity firm New Mountain Capital, wrote in an opinion article published in The New York Times in 2016. Mr. Klinsky said that when other taxes, including those levied by New York City and the state government, were accounted for, his effective tax rate was between 40 and 50 percent.

What would the change have meant for private equity?

The private equity industry has defended the tax treatment of carried interest, arguing that it creates incentives for entrepreneurship, healthy risk-taking and investment.

The American Investment Council, a lobbying group for the private equity industry, described the proposal as a blow to small business.

“Over 74 percent of private equity investment went to small businesses last year,” said Drew Maloney, chief executive of the council. “As small-business owners face rising costs and our economy faces serious headwinds, Washington should not move forward with a new tax on the private capital that is helping local employers survive and grow.”

The Managed Funds Association said the changes to the tax code would hurt those who invested on behalf of pension funds and university endowments.

“Current law recognizes the importance of long-term investment, but this proposal would punish entrepreneurs in investment partnerships by not affording them the benefit of long-term capital gains treatment,” said Bryan Corbett, the chief executive of the association.

“It is crucial Congress avoids proposals that harm the ability of pensions, foundations and endowments to benefit from high-value, long-term investments that create opportunity for millions of Americans.”

Jim Tankersley contributed reporting.