As the Conservatives launch their Brexit gamble, Labour MPs are squabbling among themselves. This is nowhere more true than on immigration. Shadow foreign secretary Emily Thornberry’s attempt to straddle the stark divisions within the party on BBC Radio 4 this morning succeeded merely in emphasising them. While Jeremy Corbyn clings to his politically naive “welcome all comers” attitude, centrist MPs race to reposition themselves, speaking out against EU free movement in a belated attempt to respond to what they heard on the doorstep during the referendum campaign.

These are not people for whom the principle of sovereignty is crucial – in which case their arguments might make some sense. Rather, they believe that limiting immigration as part of any Brexit deal is what they must do to respond to disgruntled voters. They are wrong. While an open-door policy is obviously unworkable, the idea of sacrificing single market membership in order to limit EU migration is equally flawed.

Let’s deal first with a popular canard about the referendum campaign. The notion that the failure of the remain camp can be placed squarely at the feet of Corbyn is, as my colleague John Curtice recently illustrated, untrue. Not only were the demographics stacked against the party (older, white, less-educated voters tended to vote leave, and make up much of the party’s traditional support in its heartlands), but also the remain vote held up better than one might have expected in areas where Labour is strong.

Indeed, one might wonder why Corbyn is given such a hard time. David Cameron failed to convince almost 60% of Conservative voters to back membership, while the proportion of SNP voters who voted leave was virtually identical to that of Labour voters (36%, versus 37%, according to Lord Ashcroft).

Which leaves Labour with the question of how to deal with the fallout. The most important issue in British politics today is to get the Brexit argument straight. Key to this are the debates over free movement. Here again, it is worth being clear about the facts. It was not high levels of migration that prompted people to vote leave – south Staffordshire saw one of the highest leave votes (at 78%), yet less than 1% of its population was born in continental Europe. Indeed, of the 20 areas in the UK with the lowest level of EU migration, 15 voted leave; of the 20 with the highest, 18 voted remain. It was not so much levels of immigration as rapid change in those levels – in places such as Redditch or Lincoln – that seems to have incited discontent.

EU migration has been good for the British economy as a whole. The numbers are clear and unambiguous. That said, there are clearly areas of the country – south Lincolnshire being a topical example in the context of the referendum – where high levels of inward migration place a strain on local services and disrupt local communities.

How, then, to address this problem? One option, clung to by some, can be rapidly dismissed – based, as it is, on the hope that other member states will unanimously agree to allow Britain to remain in the single market while restricting freedom of movement. Another, increasingly espoused even by centrist, staunchly pro-remain Labour figures, is to accept a trade-off between market and free movement and envisage forsaking our membership of the former as a price for restricting the latter.

Yet this would be an act of economic self-harm. Contrary to what senior Labour figures seem to think, the real economic benefit of the EU does not lie in the direct payments we receive from its budget. UK annual receipts from cohesion policy are around £1.6bn, or 0.1% of GDP.

In contrast, the hit to the British economy of leaving the single market could, depending on the relationship we have with it thereafter, be of the order of 1% to 4% of GDP. The IFS has estimated that a fall of 2.1% in GDP (relative to what it would have been had we stayed in) would leave the budget deficit £20bn to £40bn higher than it would otherwise have been. Clearly, payments from the budget are relatively trivial in comparison. And, of course, the very act of reducing migration would exert a further a negative impact on the public finances.

Yet the levers that need to be pulled to address the problems caused by migration – and indeed those generated by globalisation in general – lie in the hands of national government. Investment in housing, in health and education are obvious ways of tackling some of the negative effects that migration might generate. So, too might more rigorous enforcement of the minimum wage, in order to prevent the exploitation of those same migrants at the hands of greedy employers and gang masters.

None of this will be easy. It will require talking to the electorate, rather than blindly reacting to what we think they are saying. And Labour will have to own up to its failure to address many of these issues while in power. Most of all, it will require a Labour government rather than simply a social movement.

Ultimately, however, the choice confronting Labour is one between risking damage to the national economy because of legitimate grievances in a relative small number of areas, or making the case for a progressive answer to the paradoxes of globalisation. The more money at the disposal of the state, the easier it will be to deal with the real problems that a system that has raised over a billion people globally out of poverty and has contributed to the growth of the British economy causes in some parts of the country. Surely, the key to a successful Brexit lies in maintaining as much prosperity as possible and ensuring that the gains from trade and from migration are divided fairly? Isn’t that, after all, what social democracy should be for?