Earlier this year, a government report found that the cost of Social Security would exceed its income in 2020 for the first time since 1982. The program’s reserve fund is projected to be depleted in 16 years, at which time recipients would get smaller payments than they are scheduled to receive, if Congress does not act.

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Older Americans, who would see the most immediate benefits from Ms. Warren’s proposal, represent a sizable voting block in the Democratic primary race. In 2016, nearly three in 10 Democratic caucusgoers in Iowa were 65 and older, according to polls of voters entering the caucuses.

In a national poll released by CNN on Wednesday, Mr. Biden was supported by 35 percent of Democrats and Democratic-leaning independents who were 65 or older. Ms. Warren was next at 22 percent. No other Democratic candidate exceeded 5 percent.

Mr. Biden has proposed increasing Social Security benefits for older Americans, as part of a plan to shore up the program. Senator Bernie Sanders of Vermont introduced legislation in February to tax workers earning above $250,000 a year, in order to extend the solvency of the program by more than 50 years.

Social Security recipients get an average of $16,248 a year in benefits, and Ms. Warren is proposing to give them a raise of $2,400 a year. The plan would also overhaul how annual cost-of-living increases are calculated so that benefits increase more rapidly over time. And it would create a new Social Security credit for people who leave the work force to care for family members — a benefit that Ms. Warren says would make the social safety net program more fair for women.

Like many of Ms. Warren’s major policy proposals, the plan to broaden Social Security has a particular source of financing: the rich. Under the plan, Ms. Warren would impose a new 14.8 percent payroll tax on individuals who earn more than $250,000 a year, to be split by workers and their employers, and a 14.8 percent tax on investment income that would apply to the top 2 percent of earners.

The Warren campaign, citing an analysis conducted by Moody’s Analytics, estimated that the plan would reduce the federal budget deficit by $1.1 trillion over a decade. It would accomplish that with a $4.2 trillion tax increase over that period.