Not long ago, British prime ministers used to pride themselves on being a new president's first guest at the White House. Thatcher had kicked off her heels and downed a whisky in Reagan's West Wing even before the inauguration bunting came down. Blair furiously wagged his tail at the mere mention of going walkies on the White House lawn with Clinton and George W. The phlegmatic Brown - with Bush, at least - kept his cool. Being seen to be mates with the architect of the Iraq war had become a clear liability for Labour.

Times have changed. David Cameron has been PM for nearly two years but this is the first time he and his wife have been invited to the White House by Barack Obama (though Cameron did visit alone in July 2010). If evidence is still needed that the "special relationship" is a figment of the British imagination, Obama made showy but token concessions to his honoured guest and even continued campaigning by watching basketball in the swing state of Ohio with a perplexed Cameron in tow.

Cameron may be excused for feeling a little like Debbie Edwards, the Natalie Wood character kidnapped by Comanches in John Ford's The Searchers. Obama and Cameron come from rival tribes and have confronted the world economic crisis in diametrically opposite ways. Obama launched a trillion-dollar Keynesian stimulus to put the US - and the rest of the world - back to work, while Cameron has opted for quickly paying down British debt by slashing the size of the state.

Big gamble

It's too early to tell whose gamble will pay off but on the face of it not only did Obama's huge borrowing consign the Great Recession to the alternative history books but the US is crawling out of the slump faster than Britain. It's a miracle we're not already in a double dip, after taking a scythe to public spending and raising taxes on private spending. But the worst is yet to come.

For the past six months, the US economy has been adding about 250,000 jobs a month and the unemployment rate is 8.3 per cent and falling; in Britain, the jobless rate is 8.4 per cent and rising, with planned public-sector cuts throwing hundreds of thousands on to the dole in the years ahead. Obama and Cameron know their re-election chances depend on the right economic choices but they confined their discussions to Syria, Iran and Afghanistan where there is little new to say, while their unbridgeable differences over how best to promote growth were left unstated.

With the main issue off the table, Obama may be excused for treating the Cameron beano as an amusing irrelevance, one of the amiable chores of being head of state. There were smiles and photo opportunities galore for the impressionable British press aboard Air Force One - the US media have no interest in Cameron and Americans have little idea who he is - but such routine civilities disguise the profound variance in thinking between the two men.

That is because Cameron champions the very economic views held by Republicans in Congress who are frustrating Obama's attempts to govern. Driven by the Tea Party's absolute demand that taxes should not be raised by a cent, the Republican majority in the House of Representatives has ignored Obama's request to borrow half a trillion more dollars to stimulate growth further and brought to an abrupt halt the quotidian give-and-take of civilised political discourse.

It is usual for prime ministers to meet their ideological soulmates in the interstices between formal events but Cameron made no attempt to commune with the Republican leadership on Capitol Hill, nor any of the Republican presidential candidates.

Nor did the PM draw attention to the true "special relationship" he enjoys with his dogmatic allies, the presidential wannabes Mitt Romney, Rick Santorum, Newt Gingrich and Ron Paul. He understands full well that associating with such frauds, flakes and fruitcakes would draw attention in Britain to the black hole at the heart of his experimental economic policy.

Spending time with those who admire and hope to emulate Cameron's example would have reminded voters back home that promising to slash and burn public services does not win elections. It may be far too early to call the November contest but it is telling that after 20 widely watched TV debates in which each Republican candidate has urged the Cameron prescription of small government and paying off the deficit without delay, Obama would, according to New York Times pollsters, still beat Romney, the least scary of the line-up, by 3 points and handily trounce Santorum by 4.

Crazy ideas

The rag, tag and bobtail Republican field is a sharp reminder to Cameron that he, too, failed to be elected. And that had he proposed the major restructuring of society he is now embarked upon and adopted the mean-minded social policies advocated by some of his backbenchers, he would likely have received even fewer votes. He is understandably reluctant to become tainted by the homophobic social conservatism of his American political allies that is scaring off US floating voters and independents.

Among the litany of crazy Republican ideas: Santorum would not just outlaw abortions but allow states to ban contraceptives, too, and he condemns Obama's goal of broadening access to college education as snobbish; Paul would legalise heroin and cocaine - in the interest of freeing the market in all things - while rescinding Lyndon Johnson's civil rights legislation that vastly improved the plight of African Americans; and, among other reactionary positions, Gingrich thinks poor children should be sent out to work.

Breaking bread with a charismatic and compassionate president encourages the notion that Cameron is a decent fellow who is reluctantly obliged to impose austerity as a matter of national emergency. But the Prime Minister must hope against hope that Obama wins in November, for standing shoulder to shoulder with President Romney or, worse, President Santorum would reveal how extreme, shallow and misguided his policies really are.

Nicholas Wapshott's "Keynes Hayek: the Clash that Defined Modern Economics" is published by W W Norton