The overwhelming majority of city-financed housing has consisted of smaller apartments — studios, one- and two-bedrooms — in part because city officials see the greatest need for them based on demographic patterns, and because many developers say the city provides subsidies for projects in a way that does not encourage building larger apartments. The shortage of housing for bigger families has been exacerbated because many of the existing apartments with three or more bedrooms in the city’s public housing stock currently have only one or two occupants.

The struggles of these families come as those who have long applauded the efforts of the mayor, who has been credited with overseeing the city’s largest expansion of affordable housing since the 1980s, look more closely at the results.

“We’ve learned as a community that having a big number to shoot for is important,” said Bernie Carr, the coordinator for Housing First, a coalition of groups. “But if you focus too much on it, it can be easy to lose a sense of where are the specific needs.”

The issue has also raised the question of whether it is fair to focus limited resources on a small minority. Just 11 percent of the city’s 3.1 million households have five people or more, with the largest share in the Bronx, Brooklyn and Queens, according to an analysis of census data by Queens College. Citywide, the average household size is 2.5.

“It’s not the city’s job to give open-ended subsidies and reward people for having more members in the family,” said Nicole Gelinas, a senior fellow at the Manhattan Institute for Policy Research. “It is responsible behavior not to have children until you can reasonably support them.”

When the city approved a plan in 2009 for a large percentage of multibedroom apartments in a new affordable housing complex in Brooklyn, a coalition of housing and community groups sued, arguing that the larger units would unfairly favor Hasidic residents over blacks and Latinos and that it was preferable to have more smaller units than fewer large ones. In 2011, a State Supreme Court justice issued an order blocking the plan.

Still, Councilman Stephen T. Levin, who represents part of Williamsburg, said the city had an obligation to help large families living in overcrowded conditions.

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“I’ve had people come to me crying,” said Mr. Levin, who receives a call nearly every day about this issue. “There’s got to be a way we can address the affordable-housing needs of large families. It’s certainly not for the government to essentially discriminate against families because of their size.”

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For large families, ample space has long been hard to find in almost any price range. Of the city’s 2.1 million rental apartments in 2011, which included both market-rate and subsidized housing, only 14 percent were three-bedroom units and 2 percent were four-bedroom units or larger, according to the Furman Center for Real Estate and Urban Policy at New York University.

Many of these are in aging housing projects that were built when large families were more common. Of the 178,889 apartments managed by the New York City Housing Authority, 22 percent were three-bedroom, and 4 percent were four-bedroom or larger. Those apartments, however, do not open up often, with three-bedrooms occupied for an average of 24.8 years, and four-bedrooms for 22.9 years, compared with 10.1 years for studios and 16.1 years for one-bedrooms.

Housing authority officials said many of the multibedroom apartments were unavailable for large families despite being “underutilized”; currently, one or two people occupy 14,597 three-bedroom apartments, 1,354 four-bedroom apartments and 159 five-bedrooms. They said that those tenants would eventually be moved to smaller apartments, but that the process was slow because even smaller units had long waiting lists: about 91,000 people for studios or one-bedrooms, 62,000 for two-bedrooms, and 14,000 for three-bedrooms or larger.

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The city’s expansion of affordable housing has been overseen primarily by the Department of Housing Preservation and Development, which has provided billions of dollars in subsidies to private developers who design and carry out projects that were reviewed by the agency. By the department’s count, 124,418 of the 165,000 promised housing units had been completed through fiscal year 2011. Of those, 65 percent — 81,393 units — were preservations of existing units, and 35 percent — 43,025 units — were new construction.

But it is not clear how many of those units were three-bedroom apartments or larger because the department was unable to provide a complete breakdown of apartments by unit size. Instead, it supplied that data for only 65,796, units, or 53 percent, saying it had only recently started tallying such information.

The partial data showed that large units had been a low priority in new building projects: 3,660 three-bedrooms, 57 four-bedrooms, no five- or six-bedrooms.

“Our goal is to target our resources as efficiently and effectively as possible,” a spokesman for the housing department, Eric Bederman, said in a statement. “While data shows that households of four people or less make up the vast majority of the population, we have worked to ensure that we are balancing our city’s needs while also serving a diversity” of family sizes.

Many developers said, however, that the city’s longtime practice of awarding subsidies for affordable-housing projects based on the total number of units planned, regardless of size, had discouraged building larger apartments. A developer would receive more money by packing a building with studios or one-bedrooms than with larger apartments that take up more space but still count as single units.

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In the Bronx, for instance, a 2008 project was originally designed with 28 units, of which 5 were to be three-bedrooms. But the developer, PWB Management Corporation, eventually eliminated the three-bedrooms to squeeze in more of the smaller apartments to bring the total unit count to 32. Peter Bourbeau, a co-owner of the company, said the additional four units netted another $240,000 in subsidies — at $60,000 per unit — and increased the rental income “to put us over the hump in terms of making the financing work.”

Developers and current and former city housing officials said the practice of awarding money by number of units, and not unit size, was widely used.

RuthAnne Visnauskas, a deputy housing commissioner, said the city was open to building larger units and could provide subsidies case by case, if needed. “These are complicated negotiations, and we do make exceptions,” she said.

Chana Leibowitz said she had no choice but to continue living in the 880-square-foot, two-bedroom Williamsburg apartment where she married her husband, Michael, two decades ago. Now they share it with their six children.

“We’re trapped,” Ms. Leibowitz, said. “We’re hoping every day for a miracle.”

Mr. Tounkara, the father of six, who does not have a high school diploma, said he moved to the Bronx in 1996 for a better life than he had in his native Mali. His wife, Assetou, followed four years later, and they had six children.

“I like kids, so I make more,” he said. “My culture has a lot of kids.”

The Tounkaras, who together earn about $1,700 a month, said they had been on the waiting list for the city’s housing projects for more than four years. For now, their family sleeps in one room the size of two parking spaces, with two children in their double bed, and two more on a blanket on the floor. “It’s too hard,” Mrs. Tounkara said. “I don’t know what we’re going to do.”