China's economy grew at only 7.6% last quarter (China Daily). A rule of thumb tells us that if this rate of growth continues the economy will double in size roughly every 10 years. At $8,400 per capita GDP China's out put per person is roughly 1/5 of the United States. Doubling every 10 years would put China at $16,800 GDP per capita in 2022 and 33,600 in 2032 and given current growth rates it would then pass the US sometime before I retire.
Of course this assumes China will keep growing at its current rate. The main model of economic growth theory (the Solow model) predicts that growth rates slow down once an economy is larger. This happens because each additional machine we have adds less than the previous one (for example last week I had no computer due to this computer failing, going from zero to one computer increased my productivity a lot, giving me a second computer when i have another that works doesn't increase my productivity that much.) This is also called diminishing returns.
Thinking about this made me recall something I read on a prediction by Paul Samuelson in the book Why Nations Fail the well known economist and textbook author predicted in 1973 that the USSR would have a GDP per capita equal to the US by somewhere between 1990 and 2015.
Brad Delong has an older post on how using the standard Solow model even if we assume that communism under China or the USSR has lower productivity (that is the same number of machines per person produces less) that a communist country could catch up to the US by essentially forcing savings and in the model any savings is used to buy more machines thus increasing output.
So we should watch not only China's growth but also changes in consumption and savings.
Finally, it is my duty as an economist to say it isn't necessarily bad if China has a higher GDP per capita than the US.