Club convergence is a general property among open economies

Here is a short expression describing countries' tendency to form into different clubs, each with similar levels of income and growth. The situation is that growth is determined by the accumulation of a factor (capital, human capital, technology, or anything else, or an aggregate of the quantities) and when countries have a factor amount a little behind the level in another country they can receive further transfers of the factor from the other country. The situation might describe countries becoming increasingly technologically advanced and as a result becoming a target for foreign direct investment with high knowledge content. The analysis helps to examine convergence among large numbers of open economies, a subject on which I conjectured in this post.

Suppose we consider two countries closest to each other in the levels of an income-per-capita-determining factor K, and suppose that the leading country one has growth K1' of its factor K1. The apostrophe denotes time differentiation. Following country two has growth K2' + p.K1' of its factor K2, where p>=0 denotes the proportion of country one's growth that is transferred to country two, and K2' denotes the growth due to other factors (including internal accumulation and accumulation due to other countries' influences - we will abbreviate this to internal accumulation).

The ratio (K2' + p.K1')/K1' is greater than one if country two is catching up with country one in the factor and hence income. Rearranging, we have the condition K2'/K1'>1-p, ie the internal factor accumulation of K2 divided by the internal accumulation of K1 exceeds the proportion of K1's growth from which K2 does not benefit. For large K1', country one pulls away from country two. For other accumulation rates (1>K2'/K1'>1-p) a country could have lower internal accumulation than the leading country, but still be catching up by virtue of the transfer. Since the transfer (p>0) occurs only if two countries have factor amounts close to each other, this effect will occur only if K2 is sufficiently close to K1 to begin with, assuming that factor accumulation rises with the factor amount.

If K2'>K1', then country two will overtake country one at some point. Then, all accumulation effects from other countries are equal on both countries. If we take accumulation to be given by K1'=s1.f(K1).(other influences) and K2'=s2.f(K2).(other influences) for s1 and s2 scalars, then the condition for country one's catch up just as country two overtakes it is given by s1/s2>1-p, which is the loosest condition which will apply since f(K) is increasing in K so that if country two pulls away the ratio f(K1)/f(K2) is less than one, so s1/s2>(1-p).(f(K2)/f(K1)) is stricter (assuming other countries' influences are small on both countries or are larger on country two). All countries satisfying the condition s1/s2>1-p and with country two initially following country one tend to stick together in the income and growth rates. Given the importance of international transfers of capital and knowledge, p is plausibly quite large and the condition is weak. Greater openness should increase p and lead to increased sizes of clubs.