Last month, just before China let its currency deprecate and its stock market crashed, the San Francisco Fed published a nice Economic Letter by Zheng Liu, "Is China's Growth Miracle Over?"
China's rapid, but decelerating, growth is broadly consistent with the implications of the classic Solow growth model we teach our intermediate macroeconomics students. This model predicts that low-income countries should grow quickly, but growth will slow down as they approach the leading countries, whose per-capita growth is constrained by the rate of technological progress. That is, there should be "convergence" in per capita GDP.
As this chart from the letter shows, China is following a similar path to Korea and Japan.
The idea can be extended to include "human capital" (i.e., knowledge and skills), as Mankiw, Romer and Weil did in a 1992 paper.
While the Solow model gets the broad contours of the growth experiences of Korea, Japan and (it seems so far) China correct (and does pretty well for the US as well), it does miss a couple of big things:
(1) A diminishing marginal product of capital implies that the financial rewards to investing in a low income country should be vastly higher than in high-income countries. In a world where people can invest across borders, this implies a huge incentives for financial flows from high-income to low-income countries, but we do not observe such large net flows. This was the puzzle Robert Lucas noted in a 1990 paper.
(2) While the experiences of some low-income countries is consistent with the convergence hypothesis; in many cases, low-income countries have fallen further behind (or, as Lant Pritchett wrote, "Divergence, Big Time."). From the standpoint of the Solow model, growth "miracles" like those of Korea are to be expected, and the real puzzle is the fact the failure of so many countries to converge.
As Moses Abramovitz pointed out in 1986, it is usually a subset of the low-income countries that are growing fastest. This would suggest there are forces for convergence, but something is preventing them from applying everywhere. Current thinking is that the answer lies in "institutions" - the set of legal rights, culture, and governance which shape the economic environment and incentives for people to take actions within it, including to accumulate capital.
This is where assuming that China will continue to follow in the convergence footsteps of Korea and Japan may be questionable. While China's institutions have gotten it this far, there are reasons to doubt whether they are appropriate for achieving levels of GDP per capita comparable to Korea, Japan and Europe, as this column by Brad DeLong and this by Eduardo Porter discuss. That said, the institutions in the US during its late 19th century industrialization were hardly what an economist would recommend (in particular, corruption was rampant), and yet it somehow managed to take over leadership in per capita GDP from Britain.