DESPITE its vote to leave the European Union, plenty of Europeans still seem keen to move to Britain: in eastern European cities such as Kiev and Chisinau leaflets promising “English visas” still flutter. Marion, a lawyer who recently moved to London from Paris, says that Brexit barely featured in her decision. “I guess that emotionally I still find Brexit hard to believe.” Britain’s government, however, is busy thinking of ways to keep them out.

Since June’s referendum result, many have wondered anxiously whether Britain will remain part of the EU’s single market after Brexit. The pound tumbled when Theresa May, the prime minister, said that she planned to leave it. People have worried less, however, about the economic impact of the government’s post-Brexit immigration policy. This is strange: the impact of slashing the number of foreigners allowed into Britain could be as serious as anything that could happen to trade.

In the year to September net migration (immigration minus emigration) was under 300,000, split about evenly between EU and non-EU folk. It has been high by historical standards (see chart) since the mid-2000s, when citizens from new, poorer EU members acquired the right to move to Britain. Despite the continuing influx, net migration into Britain is hardly out of control, at least compared with other rich countries. On average annually it amounts to about three times the attendance at a Manchester United football match. Compared with their population, Ireland, Australia and Canada see far more new arrivals. But British concern about immigration has little to do with raw numbers. Even in 1995, when net migration was well under 100,000, two-thirds of Britons wanted it cut. No reference to immigration appeared on the ballot paper, but politicians believe that the Brexit vote represented a desire to “take back control” of the country’s borders. Since then Mrs May and Amber Rudd, the home secretary, have repeated a long-standing commitment to cut annual net migration to the “tens of thousands”. That will be no easy task. The government will have to count on about 50,000 Britons continuing to quit the country each year. If settling in Europe becomes harder for Britons after Brexit, that may not happen. Even if the rules are changed, the number of non-Britons settling each year, minus the number leaving, would have to fall to around 150,000.

Net migration of family members and refugees is around 70,000. On February 22nd the government largely prevailed in a case in the Supreme Court, allowing it to set tough income requirements on those who want a loved one to join them. The ruling’s wording, however, implies that tightening these rules further will be tricky. Meanwhile, reducing immigration by unskilled workers from outside the EU is difficult since it is almost non-existent, says Jonathan Portes of King’s College London.

About half of the EU nationals emigrating to Britain move into less-skilled jobs. Cutting that sort might reduce net migration by EU workers to 50,000 (a slowing economy is already helping). Halving net migration of foreign students, say by restricting the growth of universities (though that would hamper a lucrative industry), might reduce it to 50,000. But that might still leave total net migration at around 150,000. If the government is serious about hitting its tens-of-thousands target, it may have to restrict skilled migration.

That would sit oddly alongside its recent white paper on Brexit, which promised to “encourage the brightest and the best to come to this country”. And it would weaken Mrs May’s negotiating hand. In 2015 combined net migration from America and India was about 30,000. Cutting that would be awkward for the prime minister, who is desperate to strike post-Brexit trade deals with both.

How would the economy cope if the tens-of-thousands target were reached? Firms reliant on foreigners are worried. Food manufacturers are vulnerable: 40% of such workers are non-British. Skilled industries would also suffer: a quarter of scientific researchers are foreign-born.

It may be for that reason that David Davis, the Brexit secretary, this week hinted that Britain is not about to shut the door even on unskilled EU migrants. Other Brexiteers, however, counter that ending the supply of cheap workers would shake up Britain’s business model for the better. Firms would invest in labour-saving technology, boosting Britain’s low productivity. One study of American tomato-growers finds some evidence to support this thesis. If productivity rose, those workers left behind might see higher wages. Britons might also see less competition for jobs.

But these effects are likely to be small. If the benefits of investing in technology were so great, bosses should have already done so. And many jobs—such as care work—are not easily performed by robots. In these industries, many firms will either become less profitable or go under.

Few economists see lower immigration leading to a wage bonanza for locals. One paper calculates that cutting migration to the tens of thousands could boost wages in industries most affected by it by an imperceptible 0.2-0.6% by 2018.

And these tiny increases would be dwarfed by a slowdown in the wider economy. According to research by Katerina Lisenkova of Strathclyde University, annual net migration of 100,000 would lower GDP per person by 1% in the long term. Others reckon the economic cost of lower migration could match that of the hit to trade from Britain leaving the single market.

The biggest loser from slashing immigration would be the public finances. Native Britons are ageing rapidly; the number who are of working age is shrinking. When counting only native-born folk, Britain has a higher “old-age dependency ratio” (the number of elderly people as a share of those of working age) than that of many European countries, including France, and it is worsening fast. This drives up spending on health care and pensions.

As it stands, the flow of people into and out of Britain tilts the numbers favourably, improving the dependency ratio. Britain exports old, creaky people and imports young, taxpaying ones. More than 100,000 British pensioners live it up in sunny Spain; meanwhile, up to 100,000 working-age Spaniards brave the British cold.

With low net migration, Britain’s elderly would be more burdensome. Workers would need to be taxed more heavily to pay for care for their elders. The government’s fiscal watchdog suggests that by the mid-2060s, with annual net migration of about 100,000, public debt would be roughly 30 percentage points higher than if that figure were 200,000. Taking back control comes with a whopping bill.