How short a time horizon is needed to motivate catch-up growth?

A few centuries ago, the ratio between the per capita income of the richest country and poorest country was maybe five to one.  Today it is maybe one hundred to one.

The classic example of economic catch-up is given by East Asia in the mid-twentieth century, starting with Japan.  In those days it was possible to obtain near-parity with the West in about thirty to thirty-five years.  In other words, as a young man you could see near-parity before you retired and you could see near-parity for your grandchildren.  You could see your children making it halfway there, even before they are entering the workforce.

What if, in the future, for the remaining poor countries, the West (and East Asia) is so rich that catch-up takes seventy years?  One hundred years?  Will any poor country be bothered?  Won't it all seem too far off to be worth the trouble?  (Catch-up growth takes lots of hard work and savings and sacrifices of previous social norms.)  Or do you believe in a technology-transfer Solow model where the maximum possible rate of catch-up growth keeps on growing?  One hundred years from now, will it be plausible to imagine catch-up growth of twenty or thirty percent a year?