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.

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.

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.”