A young boy rides his bike near the site where the Temple Courts public housing complex once stood on May 16, 2013, in Washington. A new study finds that affordable housing is increasingly rare in the District. (Jahi Chikwendiu/Washington Post)

The District’s supply of low-cost rental apartments has continued to plummet in recent years, putting housing out of reach for a greater share of low-wage earners, according to a study by a D.C. think tank.

The number of apartments renting for less than $800 fell about 42 percent, from more than 57,700 in 2002 to 33,400 in 2013, according to an analysis of census data by the D.C. Fiscal Policy Institute, which advocates for low- and moderate-income city residents.

A 2012 study by the institute found a 51 percent drop in low-cost units in the decade that ended in 2010. The institute’s 2012 report had a profound effect on politics in the District, contributing to a new focus on affordable housing under then-Mayor Vincent C. Gray (D). The issue also resonated in last year’s mayoral race, with Muriel E. Bowser (D), the eventual winner, contending that the findings showed that Gray had been too slow to respond to rising income inequality.

[Median rental price for one-bedroom D.C. apartment is $2,000]

Now, the institute’s latest findings appear to present challenges for Bowser. The report concludes that the nation’s capital has almost no apartments left on the open market that rent for less than $800 a month. The number of such units nearly matches the city’s stock of public and heavily subsidized housing, the institute found, meaning that, in effect, only those receiving public assistance are renting for less than $800 a month.

The $800-a-month figure represents about 40 percent of the income of a family of four at the poverty line — $23,800 annually. The lack of available housing extends into higher brackets, with the stock of units renting for between $800 and $1,000 a month also dropping precipitously, the institute found.

“There is virtually no inexpensive housing left in D.C.’s private market,” said Wes Rivers, a policy analyst at the institute. “Without housing assistance, many families have no choice but to devote most of their income for rent.”

The report may help explain the District’s sharp rise in family homelessness over the past two years. More than 700 families are living in the District’s dilapidated shelter on the former D.C. General Hospital campus in Southeast Washington or in motel rooms paid for by the city. The majority are headed by young, single mothers. Some hold part-time or full-time jobs but have told city officials of living for months or years with friends or relatives before wearing out their welcome.

[How high-cost housing conquered D.C. in a single decade]

Last year, Gray’s administration estimated that about 5,000 families in similar situations teeter on the edge of homelessness in the District.

The institute concluded that a large part of the problem is attributable to income stagnation in low-wage jobs over the past decade. As rents rose, average incomes for the bottom 40 percent of renters did not increase at all.

The report warned that the trend would continue to produce poor outcomes for the city’s most vulnerable residents. “Unstable and unhealthy housing puts stress on families that makes it hard for children to focus at school and for parents to keep a job,” Rivers said.

Bowser has pledged to spend $100 million a year on affordable housing. But turning the trend around, she has warned, will take years.