The government will not meet its house building target until 2032 and home-ownership is in "freefall", according to the Labour Party.
John Healy warned today pro-Brexit Conservatives were exploiting the failure of their own government to build enough homes to scapegoat immigrants in an effort to win the referendum.
Writing for The Huffington Post UK, the shadow housing minister, said the most recent government statistics show the number of new homes built is down 9%, and 32% below the peak under Labour - six years since David Cameron became prime minister.
Labour pointed to the Conservative governments target of building a million homes over five years set against the average annual growth rate in housing completions between 2009/10 and 2015/16 of just 2.6%.
"When housing policy fails so badly, then it gives an opening for those who want to stoke division and resentment," he said.
Chris Grayling, the Tory leader of the Commons and leading Brexit cabinet minister, recently argued leaving the EU would make it easier for young people to get on the housing ladder as there would be fewer immigrants.
But Healey dismissed Grayling's claim. "It’s now under Conservative ministers that home-ownership is in freefall - with a third of a million fewer under-35s owning a home than in 2010," he said.
"So the Leader of the House should point the finger at his Conservative Cabinet colleagues, not the EU."
Labour pointed to the European Union’s European Investment Bank recent commitment of £1bn of funding for 20,000 new affordable homes across the UK as evidence EU membership helped not hindered young people finding a home.
Healey said: "Conservative failure on housing risks opening the door on crude efforts to blame Europe for domestic policy failures.
"So my plea to voters on Referendum Day is this: don’t let Tory failure on housing fool you. We can fix Britain’s housing crisis, and the European Union can be an ally in doing so."
In an interview with The Guardian, Grayling appealed to young voters to look at the "practical consequences" of leaving the EU.
"If we are bringing a population the size of Newcastle upon Tyne into the country every single year, if we cannot set limits on the number of people that come and work in Britain, then simple maths says it is going to be even more difficult to get on to the housing ladder," he said.
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