In the most difficult economy in a generation, middle-income and poor Americans are hurting the worst. Congress is tied in knots, barely able to pass even the most basic measures to help.

That has put pressure on the one arm of government with the power and the flexibility to try to boost ordinary Americans’ fortunes: the Federal Reserve. But the limited policies the Fed has at its disposal mostly put money in the hands of the affluent, at least through their direct effects. The affluent, in turn, are less likely than most to spend that money in the wider economy.

That may be a key reason that a series of dramatic steps by the central bank has not done more to raise living standards for American workers.

The Fed has aimed to strengthen growth and lower joblessness by pumping cash into the economy, buying vast amounts of government bonds using newly printed money.

The bond purchases have pushed up the stock market, in which the wealthy are much more heavily invested than the poor and the middle class. The bond purchases also have helped lower mortgage rates, and the affluent are more likely to buy a home — and have bigger homes to refinance — than those of lesser means.

At the same time, Fed bond purchases tend to weaken the dollar, driving up the cost of imported oil — and the poor spend a higher proportion of their incomes on gasoline than the rich.

If Fed policies succeed in invigorating the economy, millions of people looking for work — or worried about losing it — would be among the big winners, and leaders of the central bank see a need to do whatever they can to try to get the overall economy back on track.

Obstacles to success

But the success of those policies are limited by their very nature. The Fed, as a central bank, largely acts through bond market purchases and interest rate changes that do not equally affect segments of society.

A wide range of research shows that the poorer people are, the more likely they are to spend any new money they get, which keeps it circulating through the economy. Wealthier people are more likely to save it, which does little to foster economic activity. In 2010, middle-income families — those making about $46,000 a year — spent 91 percent of their after-tax income. The upper 20 percent, those who make an average of $157,000, spent 62 percent.

“The gains in the stock market have gone to families that are more likely to save them than would be the case if the same money went to families that are living hand to mouth,” said Karen Dynan, a senior fellow at the Brookings Institution who formerly studied household finances at the Fed. “Tight lending standards have meant that . . . the homeowners most able to refinance into lower-rate mortgage loans right now are also the most likely to pocket their savings rather than spend them.”

In trying to boost the economy through bond purchases, the Fed is using one of the few tools it has left. But Fed leaders have acknowledged that the program’s effectiveness is uncertain and have vigorously debated the risks involved. The technique that the central bank traditionally prefers to goose economic growth — reducing interest rates — is no longer an option because rates are already near zero.

“The Fed can pump an extra $2 trillion into the economy, but it can’t control where it goes,” said Dhaval Joshi, chief European strategist at BCA Research, who has examined the effects of similar policies in Britain. “We’re not saying it didn’t help the economy, but we’re saying that if you look at who it benefits, you see that it fuels stock prices and corporate profits but isn’t having much impact on wages and employment.”

This dynamic could be just one of several factors limiting the power of the Fed’s dramatic initiative. Fed economists have been examining these obstacles in recent months. Other issues include problems in the U.S. mortgage market, which have stopped many prospective borrowers from taking advantage of lower interest rates to buy a home, and the failure of banks to quickly pass on low rates to borrowers.

The unequal short-term effects of the bond program, known as quantitative easing, on income distribution are not particularly troubling even for those analysts and officials who often worry about income inequality in the United States.

“The main effects of quantitative easing look to me to be on encouraging growth and weakening the dollar, and both of those are necessary and important for everybody,” said Jared Bernstein, a senior fellow at the Center on Budget and Policy Priorities and former economic adviser to Vice President Biden.

Some liberals argue that if the Fed’s bond-buying program is having less of an effect than hoped — whether because of unequal short-term effects or other factors — that’s an argument for making the program even larger.

Effects of QE2

To see how the Fed’s latest efforts have affected different parts of the population, look at the effects of the decision in 2010 to buy $600 billion in Treasury bonds, known as the second round of quantitative easing, or QE2.

Fed Chairman Ben S. Bernanke raised the possibility of such a move in late August 2010, and it came to look ever more likely over the following months. By the time the second round of bond purchases was announced in November 2010, financial markets had soared, in part reflecting the expected effect of the policy. The Standard & Poor’s 500-stock index rose 16 percent in that span.

Here’s how the massive bond program can affect the stock market. Investors, who are displaced from the bond market by the Fed’s purchases, have to put their money somewhere, and much of that money moves into stocks, increasing demand and thus stock prices.

And who benefits when stock prices rise? According to the most recent Fed figures, the top 10 percent of U.S. income earners — those making more than $148,000 a year — are heavily invested in retirement funds and the stock market directly. In this group, 92 percent had investments in a retirement fund and 53 percent had direct investments in the stock market.

The picture is completely different at the bottom for households making less than $23,000 a year. Only 15 percent have a retirement account, and 5 percent own stocks directly.

That’s not to say that a bull stock market only helps the rich. Rising stock prices can make Americans feel richer and more confident, prompting consumers to spend more money and stores in turn to hire more workers. But the first-order benefits go to the affluent, while the poor rely on these ripple effects.

The Fed’s QE2 program also helped drive mortgage rates down. Before it was announced, the interest rate on a 30-year fixed-rate mortgage averaged 4.5 percent. Afterward, it was 4.2 percent. That lower rate is worth big bucks to anyone who was either buying a house or was in a position to refinance an existing mortgage. Among the top 10 percent of the income group, 73 percent had a mortgage in 2009. The typical amount owed was $230,000, meaning that a family that refinanced could have saved about $500 a year because of QE2.

By comparison, in the bottom 20 percent of earners, only 14 percent had a mortgage, and the typical amount they owed was about $42,000. That means that most poor Americans couldn’t benefit from lower mortgage rates. And those that could saved only about $90 a year if they refinanced.

Still, any steps that spur the building of new homes could translate into construction jobs, and construction workers have been among those hit hardest by unemployment.