George W. Bush is at record lows in the opinion polls at home. He is widely despised outside his own country as well. But when he walks into the room at Wednesday’s Group of Eight leading industrialised nations summit in Germany, there will still be no doubt about who is top dog.

Just think of his relationship with the other leaders there. At various times he has given Angela Merkel, the German chancellor, a back-rub (subsequently apologised for); hollered “Yo Blair!” at his closest ally; and pronounced himself satisfied with the state of Vladimir Putin’s soul. This is classic top dog behaviour – all the more so because the condescension to other world leaders is so instinctive and unstudied.

Mr Bush’s demeanour reflects the fact that America’s position as the world’s sole superpower remains unchallenged. Despite the Iraq war – despite everything – we are still in the “unipolar moment”. The US may as well enjoy it. By 2020 that moment is likely to be over.

Global power has many dimensions to it: economic, military, cultural, technological, intellectual and institutional. Today the US can still claim full-spectrum dominance. It is the world’s largest economy. It spends more on the military than the next 14 countries combined. Its popular culture – from films to music to fast food – has a global reach that is unrivalled. Its universities are the best in the world. US companies have led the information technology revolution. After victory in the cold war, there is still no coherent alternative to the political and economic ideas (capitalism and democracy) associated with the US. Most of the world’s most important institutions are either based in America (the United Nations, the International Monetary Fund, the World Bank), or dominated by Americans (Nato).

Yet every one of these forms of dominance is under challenge. The most obvious challenge is economic. The rise of China is now so rapid that Goldman Sachs recently revised its projection for the moment at which the Chinese economy will be larger than that of the US. The bank now thinks that China will overtake the US by 2027 (in real dollar terms), rather than by 2035 as previously predicted. China already has the fourth largest economy in the world and by 2020 its economy will be bigger than that of every member of the G8, except the US.

India is also catching up fast. Goldman Sachs estimates that by 2025 its economy will be bigger than six of the G8 members and by 2050 the Indian economy will also be larger than that of the US.

Sheer economic size will inevitably translate into political power. It is already anachronistic that the Chinese and Indians are taking part in the G8 only as part of an “outreach” meeting with developing economies. If the G8 is serious about reaching an agreement on climate change, the Indians and Chinese should be there as equal partners. And what is true of climate change will be true across a whole range of other issues, as economic power shifts east.

When China is the world’s largest market and source of savings, international businessmen and bankers will have to pay attention to Chinese regulators – in the same way that they now worry about America’s Securities and Exchange Commission or Federal Reserve. The threat of exclusion from Chinese markets – already fairly dire – will make politicians around the world jump.

A huge economy will have more money to spend on the most basic form of “hard power”: the military. Chinese military spending is already large enough for some analysts to think that it could challenge US primacy in Asia, particularly if there were a clash over Taiwan.

It will be a long time before any other country can aspire to the global military reach of the US. But the Iraq war has shown that all that dazzling firepower is not as useful as many Americans assumed. How can it be, if the US is unable to control a third-world country of 27m people?

Iraq has also damaged America’s ability to lead the world in other ways. US prestige has plummeted – a fact reflected in opinion polls around the world. The election of a new president and the closure of the Guantánamo prison camp may help recover some ground. But America’s own willingness to lead has also taken a knock. Blithe faith that democracy, free markets and the US army will sweep all before them is giving way to a warier and more cynical approach to the world.

Even American technological and intellectual leadership has been affected by Mr Bush’s foreign policy failures. The security backlash after the attacks of September 11, 2001 has damaged the ability of US universities to attract the world’s best minds. This shift comes at a particularly bad time for the US – just as the Chinese and Indians have begun to invest seriously in research and development.

The erosion of America’s power to lead is well under way. But it would be a mistake to assume that US leadership will simply end on the day – 20 years from now – when China’s economy overtakes that of the US.

Size is not everything. Even when the Chinese economy is larger than that of the US, the average American will still be far richer than the average Chinese. Combine riches with political freedom and it is likely that the “American dream” will remain much more attractive than Chinese reality for many years to come – sustaining the cultural and intellectual power that is a vital part of America’s ability to lead.

As the reality of an increasingly powerful China sinks in, bashing the US may also become a less popular pastime. Critics of the US who regard its foreign policy as amoral and nationalistic should meet the Chinese.

Sheer inertia will also help the Americans. Institutions and patterns of thought built during the American century may persist long after its economic supremacy has passed. ÛS leadership is well entrenched in many of the world’s most important institutions. The fact that the Chinese seem ambivalent about assuming the responsibilities of global leadership could also help. One reason the meeting on Wednesday is not the G9 is that the Chinese government is reluctant to join – fearing, perhaps, that the “developed” nations will gang up on China.

So the G8 remains largely a white, western club. But no future US leader will ever preside there with quite the swagger of Mr Bush.

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