I don't agree with this statement. The author is using today's metric for wealth (GDP per capita) to compare post-renaissance world to ancient empires, concluding that they were not rich enough.
In ancient times, the idea of GDP/capita was unheard of. What mattered was whether you could create an army (and pay them a decent amount) to attack other nations, plundering them so you could have even more money to raise an even bigger army. The examples the author mentions (Romans, Persians, Mongols) had this initial money to begin with.