(FORTUNE Magazine) – To friend and foe alike, the U.S. appears to be the unchallenged power in the world. The French call the U.S. a hyperpower. Neoconservatives believe it is a new Rome. But both greatly exaggerate U.S. preeminence--and its staying power. American power rests mainly on advanced technology, which is increasingly available to the whole world. While the economies of China and India are considerably smaller than that of the U.S., China is likely to overtake and India to equal the U.S. economy in size by mid-century. And as the world's economic center of gravity shifts to Asia, U.S. preeminence will inevitably diminish.

Measured in comparable units of purchasing power, the U.S. economy, at $10.7 trillion, is currently almost twice the size of China's ($6.3 trillion) and three times the size of India's ($3.6 trillion). The much higher U.S. per capita income, almost eight times that of China and roughly 11 times that of India, is partially offset by the much larger populations of the two Asian countries. But with per capita incomes in China and India likely to grow more rapidly than in the U.S., thus narrowing the large gaps in living standards, both Asian economies will eventually surpass the U.S.

When poorer countries like China and India are relatively well managed politically and economically, they tend to grow more rapidly than richer countries. The poorer countries have an advantage of relative backwardness, in that they can import the know-how of the leading economies. That occurs through the importation of capital, flows of foreign investment, and the training of scientists and engineers. Technological catch-up after World War II enabled Western Europe to narrow the income gap with the U.S. and let the poorer countries of Southern Europe (Greece, Italy, Portugal, and Spain) narrow the income gap with their richer Northern European neighbors.

Technological catch-up can be frustrated if a lagging country is politically unstable or economically mismanaged. That was the case with Eastern Europe, which languished under communism. Similarly, both China and India squandered the chance for rapid economic growth in the first decades after World War II because of Mao's communism and Nehru's socialism. The trigger for rapid catching up in China was Deng Xiaoping's liberalization of the economy beginning in 1978; in India it was Finance Minister Manmohan Singh's dismantling of the so-called license raj beginning in 1991.

Now both economies are soaring. China's aggregate GNP has grown by about 10% a year since the late 1970s and India's has grown by about 6% a year since 1991, compared with annual U.S. economic growth of about 3.5%. China's and India's soaring growth rates will decline as the income gap with the U.S. narrows. The closer both countries come to the technological levels of the U.S. and other rich countries, the fewer opportunities there will be to import technological know-how. The slowdown is likely to be gradual, however, so both countries, and others like them, will have economic growth greater than that of the U.S. for decades to come.

Econometric evidence suggests that when the income gap between China and the U.S. narrows by half, China's growth advantage will decline by about 1.4 percentage points per year. China's per capita income is currently about 13% of that of the U.S., and per capita GNP in China is growing about 4.6 percentage points per year faster than in the U.S. So when China's per capita income reaches 26% of the U.S. level, the annual growth advantage should fall to about 3.2 percentage points. Still later, when China's per capita income level reaches about half the U.S. level, China's growth advantage will be about 1.8 percentage points.

Extrapolating based on this rule of thumb, China would reach about half the U.S. per capita income level by the year 2050, while its projected population of some 1.4 billion will be about 3.4 times the U.S. population of about 400 million. At that point China's GNP would be about 75% larger than that of the U.S. By a similar calculation, assuming that India's annual per capita income growth now exceeds the U.S. rate by about four percentage points, India's per capita income would reach a little more than a quarter of the U.S. level by 2050, with a population nearly four times as large, leading to an overall economy about the same size as that of the U.S.

Of course war, disease, political instability, and even less tumultuous economic blunders or shifts in population patterns can play havoc with such projections. Yet the trends strongly suggest a future decline in the relative importance in the world of the U.S. economy and its eclipse in size by China and perhaps India. The prediction is undergirded by the clear evidence that China, India, and other fast-growing Asian economies are devoting a rising share of national income to R&D, as they need to do to catch up technologically.

Moreover, the U.S. is facing not only the rise of China and India but also the rise of an integrated Asian economy that links southern Asia, Southeast Asia, and northeastern Asia. Countries throughout Asia are working hard to overcome past strategic animosities in order to create increasingly integrated flows of merchandise, finance, and technology. On the basis of current trends, and as a very rough estimate, this integrated Asian economy could reach about half of world GNP (up from a current one-third), with about 60% of the world's population. The U.S. share of world population will remain about 5%, which means its overall economic weight could slide from more than one-fifth of world GNP today in purchasing-power terms to perhaps half that by 2050.

From an economic point of view the U.S. is likely to be a beneficiary of this process, just as it was a beneficiary of the recovery of Western Europe's economy after World War II. U.S. consumers will benefit from a dazzling increase in the variety of Asian products at low prices. The world as a whole will benefit from an acceleration of scientific and technological advances coming from Asia. One dramatic case in recent years: China's scientists used a traditional herbal remedy to develop antimalaria drugs that will save the lives of millions of Africans and Southeast Asians. If ideological and religious opposition to stem-cell and other biotech research in the U.S. stands in the way of progress, an eager scientific community in Asia could quickly surge ahead in that field.

From a political point of view, the vainglorious vision of the U.S. as the new Rome will be dashed. We got a small glimpse of the coming U.S. eclipse this fall, when President Bush traveled to Asia for an Asia-Pacific economic meeting. The U.S. tried to find Asian allies to bash China over its exchange-rate policies but found no takers, as the other Asian countries sided with China. China's economic diplomacy has already trumped that of the U.S., since China's Asian neighbors clearly understand that China is their engine of growth.

Of course the developing countries of Asia could stumble on their way to prosperity, and some U.S. strategists harbor hopes that Asia's dynamic rise can somehow be contained. Yet we are so interconnected in our fates that any serious Asian crisis would almost surely enmesh the whole world, including the U.S., in dire troubles. Nor could America's 400 million population by mid-century have a prayer of containing Asia's five billion people. Assuming Asia's continued economic success, the 21st century could well be a period of unprecedented prosperity and scientific advance, but one in which the U.S. will have to learn to be one of many successful economies rather than the world's indispensable country.

JEFFREY D. SACHS is director of the Earth Institute and Quetelet Professor of Sustainable Development at Columbia University.