The remain camp’s strategy for winning the EU referendum is like a one-legged stool. It relies too much on a single argument – scaring undecided voters about the economy – when others are necessary too.

There is admittedly a lot to worry about, given the EU accounts for half of Britain’s trade and there’s no reason to suppose we’d get nearly as good access to its market if we quit. The process of leaving would be so disruptive that it could cause an immediate recession. There would also be long-term damage. The Treasury’s estimate today that, by 2030, households could be £4,300 worse off than if we stay is a reasonable ballpark figure.

But this “economic fear” leg needs to be balanced by two others. The remain camp needs to make more of the non-economic risks of leaving. These too are real. We would destabilise the whole EU if we quit and that would boomerang on Britain. The only major global figure who would rub his hands is Vladimir Putin. We’d also risk breaking up the UK: Scotland would have another independence referendum and border controls would have to be imposed between Northern Ireland and the Republic of Ireland.

Stronger In Europe, the official campaign, gives the impression of being a well oiled machine that lacks horsepower

The third leg should be the positive arguments to enthuse the remain camp’s core base. It is a mistake to think this referendum is all about the undecideds, important though they are. The result will also be determined by how many of those who have made up their minds actually come out to vote.

On this score, the leave camp’s shock troops are passionate. The remain camp needs to fire up its supporters so they not only vote, but persuade their friends and family to follow suit. Brussels, of course, has serious problems – underlined by its handling of the migrant and euro crises. But Britain isn’t part of the border-free Schengen area or the eurozone.

By contrast we can use our influence in Europe to become a Greater Britain on the world stage. Our strong economy means we are in a good position to push the EU to complete its single market and cut free trade deals with the rest of the world. In other words, we can lead Europe. Why would we want to throw away that opportunity?

Let’s not forget free movement either. EU migration to Britain enriches us culturally and economically. Meanwhile, our citizens get the opportunity to study, work and retire abroad. These positive arguments may not work with the undecideds. But something like this is needed to put fire in the belly of the remain camp’s supporters.

A rejigged strategy is not enough. The remain camp needs new voices to make its case. It has been relying too much on David Cameron to ride to the rescue. This was never a good idea. It is even less so now, given how the Panama Papers saga has knocked trust in the prime minister.

Cameron is still an asset to the campaign. But he should be just one voice among many. We need to hear more from Gordon Brown, Paddy Ashdown and John Major. We also need some fresh faces, such as Labour’s Emma Reynolds and the Conservatives’ Amber Rudd – smart, sensible women who know their stuff. And we need to hear from non-politicians who feel passionately that Britain should stay in the EU. This is, after all, the people’s referendum, not a general election. The BBC and other broadcasters should make time to hear them too.

Stronger In Europe, the official campaign, meanwhile gives the impression of being a well-oiled machine that lacks horsepower. It has failed to rebut the myriad myths peddled by the leave camp. How, for example, has it allowed its rivals to get away day after day with telling the lie that we send £350m a week to Brussels?

Why can’t it quash the fiction that we’ll easily get a trade deal with our former partners when, in fact, we need the EU more than it needs us? How did the campaign fail to ram home that the Brussels attacks were a reason for staying in the EU, not quitting? And why did it allow its rivals to suggest we could do more to save Port Talbot if we went solo, when we would then be far less able to stand up to Chinese dumping?

Its approach is too slow and too gentle. It needs to rev up its communications and get some more heavyweights on its team. Even if it can’t stop Boris Johnson and Nigel Farage spouting nonsense, it should ensure that broadcasters have counter-arguments at their fingertips so they can make the Brexiteers squirm when they do.

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To be fair, the remain camp is taking some of these points on board. Last week David Miliband explained how Brexit would turn Britain from a global fireman into an arsonist. Barack Obama may make similar points when he visits.

Jeremy Corbyn’s belated decision to back “in” has also helped broaden the campaign beyond Cameron. The trade union movement is finally fired up. Last week also saw the remain camp’s first good rebuttal. When the leave camp tried to suggest that quitting the EU would be good for the NHS, Labour MPs and trade unions hit back hard. They argued that the idea that the likes of Boris Johnson and Michael Gove would protect the NHS was a joke, and that leaving the EU would cause so much economic damage that there would be less money left for the health service, not more.

But the remain side has wasted time. Given how high the stakes are, it desperately needs to put its foot on the accelerator.