The 2016 Summer Olympics have profoundly altered the city, yielding a revitalized port; a new subway line; and a flush of municipal projects, big and small, that had long been on the wish list of city planners.

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RIO DE JANEIRO — There have been cost overruns and complaints about spending billions on a mega-event when teachers have gone unpaid. Critics say upscale areas have been favored at the expense of slum dwellers. A pledge to clean up Rio de Janeiro’s polluted bay went unfulfilled, while the promise of law and order now feels like a cruel taunt in the face of rising crime.

But despite the criticism, the 2016 Summer Olympics in Rio have profoundly altered this city of 6 million, yielding a revitalized port; a new subway line; and a flush of municipal projects, big and small, that had long been on the wish list of city planners.

“If we set aside our political passions, it’s plain to see that the Olympics have created an enormous legacy for Rio,” said Pedro Correa do Lago, a historian, economist and former president of Brazil’s national library. “These are improvements that might have otherwise taken 20 or 30 years to realize.”

To many, it has become an article of faith that the modern Olympics are a drain on public coffers, a sop to corporate interests and a vanity project for glory-seeking leaders hoping to burnish their legacies and their nations’ standing on the world stage.

Brazil is no different. Born seven years ago in the heady days of an economic boom, these games were initially seen as a triumphal capstone for a newly ascendant global power. Instead, as the country suffered through its worst recession in decades, the Olympics became an emblem of government waste and political hubris — and a target for protesters who dogged the Olympic torch relay as it wended its way across the country.

But experts say the Olympics also served as a powerful catalyst for urban revitalization, spurring infrastructure projects, financed with taxpayer money and private investment, that will enhance the lives of Rio’s residents.

Nearly 100 miles of rapid bus lanes have slashed commuting times for thousands of working poor. Four tunnels have been built, and a 17-mile light-rail system opened in June. A new subway line, the system’s first major expansion in decades, began operating four days before the opening ceremony.

The city said it had sped up the construction of more than 400 schools and health clinics in impoverished neighborhoods, part of what the mayor called a revitalization spurred by the Olympics.

Still, critics say the games have delivered uneven benefits, favoring upscale areas like Barra da Tijuca, the site of the Olympic Village, while ignoring hundreds of poor communities where residents live in poorly-built housing that lacks basic sanitation.

“The Olympics have led to displacement, gentrification and sweet deals for real-estate developers and construction companies,” said Theresa Williamson, executive director of Catalytic Communities, an advocacy group for the city’s favelas.

But while acknowledging the dire state of Rio’s public finances — the underfunded schools and hospitals, the unpaid government salaries and the unmitigated misery of its hilltop favelas — some experts say the Olympics will provide benefits for years to come.

Eduardo Paes, Rio’s hard-charging mayor, who has aspirations of higher office, is quick to swat away criticism of the games, calling the event a once-in-a-generation opportunity to lure investment to a city where fortunes have waned in the nearly six decades since the national capital moved to Brasília from Rio.

“No one ever said the Olympics were going to solve all of the city’s problems,” Paes said. “But we used the games as a good excuse to get a lot of things done, things that have been the dream of mayors for 50 years.”

He noted that the $12 billion budget for the Olympics was significantly lower than the expenses of other recent host cities — the roughly $15 billion spent on the 2012 London Games and the $51 billion Russia lavished on the 2014 Winter Olympics in Sochi.

More important, Paes said, much of the money came from the private companies that built the Olympic Village and the Olympic golf course, as well as those that renovated the city’s port, a project that includes a 2-mile waterfront promenade and two new museums.

Overall, he said, the city has built 75,000 units of affordable housing since 2009, although some estimates suggest that nearly as many people, most of them poor, lost their homes to projects related to the Olympics.

Critics dispute some of Paes’ figures, pointing out that cost overruns will most likely bring the final cost of the games to $20 billion. Others note that the 3,600 apartments that make up the Olympic Village will end up as homes for the rich, and that the golf course, which required filling in protected wetlands, will serve only the wealthy.

“Yes, the Olympic Village will be something for rich people,” Paes said. “But there’s no shame in that.”

In a report issued in May, Moody’s said the Games would have a negligible impact on the city’s ailing economy but that the $7 billion in transportation-related spending was money well spent.

That assessment stood in stark contrast to the benefits seen in the estimated $11 billion Brazil spent hosting the 2014 World Cup, which left behind a constellation of 12 new or renovated stadiums, most of which are not used regularly.

Bent Flyvbjerg, an Oxford University economist and the lead researcher on a study that examined Rio’s Olympic finances, said the actual amount spent on sports venues was most likely $4.6 billion, 51 percent over budget.

That amount, Flyvbjerg said, placed Rio somewhere in the middle of host cities that have exceeded their spending projections.

“All governments try to take the most convenient truth and spin it for their own purposes,” he said. “I guess if I was doing PR for the mayor of Rio, I’d also say we’re doing better than the previous three Olympics.”

In recent years, Oslo, Boston and Munich, bowing to popular opposition, have dropped their Olympic ambitions. Over the past three decades, nearly every city that has hosted the Olympics has lost money, and few expect Rio to recoup the billions of dollars spent preparing for an event that lasts just weeks.

The amount of money lost to waste and corruption may never be known. Sergio Cabral, the former governor who helped land the Olympics, has been accused of demanding millions in bribes. It also remains to be seen whether the 12 Olympic venues intended to become schools or community sports centers will end up as white elephants.

But city officials say the Olympics helped move the needle on infrastructure plans that had languished for years.

In some poor neighborhoods, the Olympics served as a cudgel to speed the overhaul of public clinics that had been plagued by long waits and poor service. At one, in the troubled City of God favela, software now streamlines the triage process, a cheery ombudsman takes complaints and a new app lets supervisors track how long doctors spend with each patient — or whether they take inordinately long lunch breaks.

“It’s like night and day,” said Elizabeth Rezende, 61, a retired maid waiting to get her electrocardiogram results after experiencing chest pain. “The other emergency hospitals are so chaotic.”

Then there is Meu Porto Maravilha, or My Wonderful Port, the historic waterfront that, for decades, was cut off from downtown Rio by a hulking elevated highway, its 19th-century warehouses left to molder. Plans to rehabilitate the port, first put forth in the 1980s, had long been stymied by a lack of money and insufficient political will.

The $2.5 billion rehabilitation, much of it financed through the sale of air rights from adjacent properties and tax incentives to developers, included demolishing the viaduct and funneling traffic through a new 3-mile tunnel.

Over the next decade, the developers plan to build 500 apartments that they say will be affordable to residents of a nearby favela. Many of these residents are descendants of the half-million African slaves who first arrived in Brazil at Valongo Wharf. The wharf’s recently unearthed foundations are scheduled to become part of a museum that will also include a forgotten slave cemetery.