One of the winning factors in London's bid to host the 2012 Games was the pledge to celebrate those traditions that Britain shares with the Olympic movement: the diversity, the global outlook, the open-to-all inclusivity. The promise was not an empty one. With the Games now upon us, it can be seen in everything from the location in east London, to the make-up of Team GB, to the 70-day torch relay that criss-crossed the nation.

It is also evident, albeit in a different form, in the Global Investment Summit (the first of 17 over the coming weeks) that opened on Thursday with a rousing speech from the Prime Minister full of assurances about "getting behind" British business.

There are those who carp that there is something slightly grubby about using the Olympic Games to drum up trade. Not so. All eyes are on London; as many as 4,000 top executives, and scores of heads of state, will descend on the capital for the Games. At any time such a concentration of decision-makers would be an opportunity to make the most of. With the economy in such dire straits, the case is stronger still.

So far, so good. Except that the heady rhetoric only exposes, more starkly than ever, one of the most glaring contradictions at the heart of the Government's thinking. One hand may be hanging out the sign saying "open for business", but the other – imposing ever tighter immigration rules – is busy bolting the door. Indeed, the stated aim of slashing net migration by more than half by 2015 could hardly send out a clearer message to growing global companies, to highly skilled workers and to highly motivated students alike: you are not welcome here.

Such concerns are not new. The Government has been under sustained pressure to change course, from businesses and universities in particular, ever since the policy was introduced. But the strain is increasing. Only this week a Whitehall report, leaked to this newspaper, listed immigration policy as far and away the biggest concern for potential investors. And with the economy worsening, there are growing signs that the penny may, finally, have dropped. The Prime Minister, for example, was full of reassurances on the subject at Thursday's summit.

For all the warm words, however, the Government is in a nasty political bind. Even as Britain's need for economically productive migrants intensifies, the worsening outlook only adds to public concerns about immigration. Having so wrong-headedly nailed his colours to the mast, David Cameron cannot now afford to be seen – by either his party or the public – as going soft on the issue.

The immigration "cap" is, therefore, a tricky policy to dump. But there is at least one reasonable compromise available, if only the Prime Minister would take it. University chancellors have warned en masse that tighter visa requirements are encouraging many foreign students to look elsewhere. To resolve the situation, they should simply be removed from the net migration tally. No matter that the Home Secretary is resistant, fearful she will be charged with fiddling the figures. Any opportunity to boost the education sector, attract the talented and correct the Government's damaging stance on immigration should be seized forthwith.

While desirable, an exemption for foreign students would still only be a start. This week's GDP statistics point to the worst double-dip recession in 50 years. Whether the fault of the euro crisis or not, the Government should be doing all it can to boost growth – and that means Mr Cameron must extricate himself from the toxic politics of immigration. Restrictive policies are not only a travesty of the traditional British openness being celebrated with such flair at the Olympics. They are also precisely the opposite of what the economy needs.