Why did the UK vote to leave the EU? For conservatives, Brexit was about regaining parliamentary sovereignty. For socialists it was about escaping the single market. For still more it was a chance to punish David Cameron and George Osborne. But supreme among the causes was the desire to reduce immigration.

For years, as the government repeatedly missed its target to limit net migration to "tens of thousands", the EU provided a convenient scapegoat. The free movement of people allegedly made this ambition unachievable (even as non-European migration oustripped that from the continent). When Cameron, the author of the target, was later forced to argue that the price of leaving the EU was nevertheless too great, voters were unsurprisingly unconvinced.

But though the Leave campaign vowed to gain "control" of immigration, it was careful never to set a formal target. As many of its senior figures knew, reducing net migration to "tens of thousands" a year would come at an economic price (immigrants make a net fiscal contribution of £7bn a year). An OBR study found that with zero net migration, public sector debt would rise to 145 per cent of GDP by 2062-63, while with high net migration it would fall to 73 per cent. For the UK, with its poor productivity and sub-par infrastructure, immigration has long been an economic boon.

When Theresa May became Prime Minister, some cabinet members hoped that she would abolish the net migration target in a "Nixon goes to China" moment. But rather than retreating, the former Home Secretary doubled down. She regards the target as essential on both political and policy grounds (and has rejected pleas to exempt foreign students). But though the same goal endures, Brexit is forcing ministers to reveal a rarely spoken truth: Britain needs immigrants.

Those who boasted during the referendum of their desire to reduce the number of newcomers have been forced to qualify their remarks. On last night's Question Time, Brexit secretary David Davis conceded that immigration woud not invariably fall following Brexit. "I cannot imagine that the policy will be anything other than that which is in the national interest, which means that from time to time we’ll need more, from time to time we’ll need less migrants."

Though Davis insisted that the government would eventually meet its "tens of thousands" target (while sounding rather unconvinced), he added: "The simple truth is that we have to manage this problem. You’ve got industry dependent on migrants. You’ve got social welfare, the national health service. You have to make sure they continue to work."

As my colleague Julia Rampen has charted, Davis's colleagues have inserted similar caveats. Andrea Leadsom, the Environment Secretary, who warned during the referendum that EU immigration could “overwhelm” Britain, has told farmers that she recognises “how important seasonal labour from the EU is to the everyday running of your businesses”. Others, such as the Health Secretary, Jeremy Hunt, the Business Secretary, Greg Clark, and the Communities Secretary, Sajid Javid, have issued similar guarantees to employers. Brexit is fuelling immigration nimbyism: “Fewer migrants, please, but not in my sector.”

The UK’s vote to leave the EU – and May’s decision to pursue a "hard Brexit" – has deprived the government of a convenient alibi for high immigration. Finally forced to confront the economic consequences of low migration, ministers are abandoning the easy rhetoric of the past. Brexit may have been caused by the supposed costs of immigration but it is becoming an education in its benefits.