For a few days over Christmas, a massive effort by charities such as Crisis and the volunteers they recruit tries to offer something like festive cheer for thousands of people who might otherwise have been sleeping out on the cold, wet streets of Britain’s cities. Crisis at Christmas – Europe’s largest volunteer-run event – last year mobilised more than 10,500 volunteers and helped over 4,500 homeless people in centres in London, Birmingham, Coventry, Edinburgh and Newcastle. This year, they expect even more people to need their support. The generosity of individuals on this scale is a fine thing. But it is no answer to the perfect storm that now howls round housing policy, driving the surge in homelessness and the number of families in temporary accommodation.

There is an easy, obvious, explanation for why this perfect storm has built up; and there is a less obvious but more immediate one. The easy, obvious explanation is the catastrophic shortage of new homes. That drives up prices while at the same time banks demand deposits of at least 5%. That often means upwards of £7,500, a fortune out of reach to many potential buyers. So they swell the demand for renting.

The less obvious but now most pressing reason for the storm is this: since 2010 there has been a ferocious squeeze on welfare. By March 2016 almost £15bn had been cut out of the work and pensions budget. Not all of the cuts fall on direct support for housing. But people are often tipped into homelessness by, say, a benefit sanction or rent arrears. Forthcoming changes are likely to make the situation worse. Single under-35s will only be entitled to support at the shared accommodation rate, and in many parts of the UK housing support would probably not be enough to be able to afford somewhere to rent. Child benefit will be restricted to the first two children, and for out-of-work families all benefits have already been capped at £23,000 in London, £20,000 in the rest of the country. Families who have three children and can’t find a job, according to the Chartered Institute of Housing, will by next April be likely to struggle to find social housing they can afford anywhere in England. The gap between what benefits will pay for, and what rents cost, has grown so large that in many places only about a 10th of homes in the private rented sector are within reach. Meanwhile, for many people in work, pay has barely increased since 2008.

In a notorious passage in his memoir of the coalition government, the former Liberal Democrat leader Nick Clegg claimed that George Osborne once said: “I don’t understand why you keep going on about the need for more social housing – it just creates Labour voters.” Even if the story is apocryphal, there are pages of statistics showing how severe, and how predictable, is the cumulative consequences of the cuts. Theresa May’s government has slowed some changes, but the trend remains the same. A backbench bill – most unusually sponsored by all the MPs on the cross-party communities and local government committee – that will put more obligations on local councils to intervene earlier with help and advice for people at risk of homelessness, now has government support. But a third of all those who end up in temporary accommodation are people who were just about managing until they got to the end of their tenancy and the landlord wanted to put the rent up. Ending assured tenancies in the private sector is the single biggest factor behind families seeking help. Meanwhile, homelessness charities report lettings agents gathering wannabe tenants together and going round the room looking for the highest bidder. Some inner-London councils with families facing homelessness send them miles away: Westminster council was recently criticised by the supreme court for sending a family to Milton Keynes, without making any attempt to find out if they would be able to manage.

This is a crisis that won’t go away after Christmas. In the short term, the solution may be less difficult than it first appears. There is a hint that the government sees this, in a new £50m fund to encourage agencies to work together locally. However sceptical voters are about the old mantra of joined-up government, when the cost of renting is driving up the cost of housing support, and trying to cut the cost of housing support is becoming a trigger for homelessness, it is time all the departments with a responsibility – Treasury, Work and Pensions and Communities and Local Government – sat down together, and fixed it.

• This article was amended on 22 December 2016 to correct and clarify details about restrictions on benefits.