THIS FALL, PRESIDENT Obama and Congress will have another chance to strike a grand bargain on the deficit. If it happens, a new gauge of inflation that would both reduce federal benefits, like Social Security, and raise taxes, will be at the heart of it. Only 13 states would be affected more than Massachusetts, with its 900,000 residents older than 65 years.

The new inflation index is known as the chained consumer price index, or chained CPI. Obama’s willingness to adopt it as part of his budget proposal is a concession to Republicans and is aimed at finding a deal to replace the across-the-board budget cuts known as the sequester, which Congress adopted in 2011 and that begin to take effect this year.

Washington’s partisans have mostly aligned in familiar camps. Most Democrats and left-leaning groups oppose the new index, saying the chained CPI will hurt low-income people and the elderly. Most Republicans and conservative groups say it’s the one thing they like in Obama’s budget. But the camps are starting to break down, demonstrating just how thorny the effort to reach a grand bargain remains and how difficult it is to forecast an outcome.

In a deal’s favor, the liberal opposition is no longer universal. This spring, Robert Greenstein, a former Newton South High School history teacher who’s now the president of the Center on Budget and Policy Priorities, an influential liberal think tank, wrote a paper in which he said that he could support the chained CPI if it was part of an even-handed budget deal.

On the other hand, conservative support for the deal is weakening. Andrew Biggs, a resident scholar at the American Enterprise Institute and a former top aide to former Ways and Means Committee Chairman Bill Thomas of California, has issued his own treatise arguing that his Republican colleagues would be foolish to take Obama’s deal.

Obama stresses that the chained CPI is part of the balanced approach to deficit reduction that he stressed in his campaign. His budget, for instance, would couple adoption of the chained CPI with big tax increases on the wealthy. He would allow wealthier taxpayers to deduct only $28 from their tax liability for every $100 in itemized deductions. Currently, they can deduct $35. And he would require that people earning more than $1 million pay at least $300,000 of it in federal taxes.

That makes Republicans wary. They say they approved all the new taxes they can stomach in last year’s deal averting the so-called fiscal cliff, which eliminated the Bush era tax cuts for individuals earning more than $400,000 and couples earning $450,000.

Democrats like the tax increases but mostly reject the chained CPI. Those members of the Massachusetts delegation who have expressed a view on the new inflation measure say they don’t like it, even those who profess a desire for a balanced approach to deficit reduction. None of the Massachusetts representatives has endorsed it, and only one—Springfield’s Richard Neal—had not made up his mind as of press time.

In defeating Scott Brown in her run for the Senate last fall, Elizabeth Warren said both spending cuts and tax increases would be necessary to replace the sequester. But she argued at the time that the cuts should come from agriculture subsidies and defense, not from Social Security. She’s been among the most outspoken in opposing the chained CPI. She says the new index amounts to a betrayal of seniors who’ve paid into the Social Security system their entire lives expecting a certain benefit. And she says the cuts hit close to home. Her brother David Herring, a former small businessman, now lives on his monthly check of $1,100.

Sen. Elizabeth Warren says the new index is a betrayal of seniors.

Ed Markey made his opposition to the chained CPI a key talking point in his Senate campaign against Republican Gabriel Gomez, who wanted to use the new index to help balance the budget.

For all members of the Massachusetts delegation, the pressure to block the proposal from interest groups, ranging from the AARP to the AFL-CIO, is intense.

Still, the new index is viewed by many economists as a more accurate inflation gauge because it takes into account the propensity of consumers to adjust their buying habits to combat inflation. A shopper, for instance, might buy more apples and fewer oranges, if the price of oranges were rising quickly.

The inflation rate using the chained CPI is typically 0.3 percentage points lower than under the current inflation gauge. Isabel V. Sawhill, an economist at the centrist Brookings Institution who focuses on anti-poverty programs, calls it “a corrective” to what has been a small but automatic increase in real benefits for many years.” But some economists on the left say the chained CPI is not a fair inflation measure for the elderly, who spend more of their money on health care, a service whose price traditionally rises faster than for other goods and the costs for which can’t be easily avoided.

These critics say they see no reason to cut a deal now that involves Social Security—the program faces no imminent funding shortfall—and would prefer to wait for Democrats to retake the House. At that point, they’d push for lifting the amount of income taxed for Social Security, currently $113,700, and for limiting federal tax breaks for retirement saving programs that mostly benefit the well-to-do, such as 401(k) plans and individual retirement accounts. They say any deal centered on the chained CPI would only embolden Republicans to go after bigger entitlement cuts, such as raising the Social Security retirement age. “It just puts blood in the water,” says Monique Morrissey, an economist with the Economic Policy Institute, a nonpartisan Washington think tank.

But Greenstein of the Center on Budget and Policy Priorities says a deal that includes the chained CPI, but also a broader package of tax increases and budget cuts, may be the best deal liberals are going to get. “A lot of progressives hope that eventually there will be unified progressive control of the government, just as conservatives hope they hold the House, take the Senate, and maybe win the White House in 2016,” he says. “I look at things like the [House Budget Committee Chairman Paul] Ryan budget and frankly am nervous about rolling the dice.”

Ryan, a Wisconsin Republican who was Mitt Romney’s vice presidential running mate, would partly privatize Social Security by allowing workers under 55 to invest part of their Social Security taxes in private investment accounts.

Even if there’s no Republican resurgence, Greenstein notes, current law is bad enough. After cutting more than $100 billion from federal agency budgets this year, the sequestration is slated to cut more than $1 trillion more over the next eight years, slashing budgets for anti-poverty programs Greenstein cherishes like Head Start, student aid, and low-income housing assistance. If Republicans simply hold on to 41 seats in the Senate, they can block any effort to replace the sequester.

Another enticement for Greenstein is that Obama’s proposal mitigates the effect of the chained CPI on poor seniors living on Social Security. When retirees reach age 76, their checks would increase gradually over the next 10 years, eventually adding $750 per year, to make up for some of the money lost to lower annual cost of living adjustments. Retirees who make it to 95 would get another bump.

Those increases would especially help the poor. According to Greenstein’s calculations, a lower-middle-income retiree would lose a few hundred dollars a year in benefits until reaching his early 80s, when his benefits would actually be greater than under current law. Middle-income retirees would absorb a cut of less than 2 percent. A wealthier retiree could expect a cut somewhat larger than 2 percent. Still, it’s real money. One liberal interest group fighting the changes found that the average recipient would lose $4,642 by age 75 and $15,615 by 95.

At the same time, the chained CPI would increase tax revenues. The IRS each year adjusts tax brackets for inflation, as well as the amount of money Americans receive in tax credits and deductions. Those adjustments would shrink, pushing more people into higher income brackets. The Congressional Budget Office figures the chained CPI would bring in an additional $123 billion for the Treasury over the next 10 years. During that decade, the benefit cuts and tax increases would be close to equal, but after that the tax increases would grow more quickly.

Republican leaders in Congress have mostly endorsed the chained CPI, as have conservative think tanks, such as the Heritage Foundation. But Biggs is trying to upset the good feelings among his fellow Republicans, arguing that the proposal doesn’t do much to preserve Social Security’s solvency while conceding too much on taxes. The Social Security cuts are limited, he notes, by the lifespan of recipients. “On the tax side, a lower inflation measure raises tax rates year after year forever,” he says.

Like Morrissey, Biggs worries about opening the barn door. By agreeing to tax increases in this instance, he says Republicans might also go along with the other tax increases Obama wants.

At the same time, with the old age funding increases envisioned by Obama, Biggs figures the chained CPI will only solve about 15 percent of the long-term deficit facing Social Security, forcing Congress to come back later to reach a broader deal.

He argues that Republicans should pursue a more far-reaching overhaul now, aimed at convincing Americans to work longer and to save more for retirement. Some ideas he suggests are reducing Social Security payments for people who retire in their early or mid-60s, while increasing annual cost-of-living adjustments to encourage people to work longer, or eliminating payroll taxes entirely for workers approaching retirement age, providing another incentive to keep working.

“I definitely don’t favor waiting [to reach a deal],” he says, “since the longer we wait the harder the problems are to solve. And I definitely believe in bipartisan agreements, since that’s the only way to get something passed. I just don’t think the current chained CPI offer is a very good one.”