What happens to capitalism when global wealth inequality shrinks?

[Art]

That's a totally ridiculous assertation.
Right now, about 1/3 of the world is so wealthy that nearly all the products made for this 1/3 of the world must be made in another 1/3 of the world that is poorer. This 1/3 of the world (China, India, and South East Asia, which are actually far more than 1/3 of the world's population) are rapidly growing because of the immense amounts of wealth being pumped into them by the currently rich Western World. Why is raising the living standard of the remaining 1/3 anything but inevitable?

Whilst I see merit in your central premise that over time it is likely nations will reach a global equilibrium I question your assertion regarding a harmonisation of wealth within these nations.
Statistics show that in all capitalist countries wealth gravitates to the wealthy even when as in France and Britain there is a 'socialist' government in charge. Taking Britain as an example the current figure is approx. 3% of the population own 97% of the wealth.
Given this huge disparity there will always be an underclass to exploit. With no cheap international labour pool to fish in I would imagine capitalism would simply revert to pre-globalization days (which is a relatively recent phenomena anyway) in which each nation simply exploits it's own underclasses as it does today. Wage competition has not eradicated the need for western capitalist countries to impose minimum wage levels to prevent the worst excesses of exploitation.
The effects of no longer having cheap international labour available will actually be fairly marginable. It is perhaps worth bearing in mind that for most manufactured items labour costs account for less than 10% of total production costs and so given the increase in transport to get the goods to market the bottom line saving to manufacturers from out-sourcing is typically in the low single digits.

 

[Smurf]

I honestly don't see "capitalism" ever taking over the whole world. I think the only way you'd be able to do that is if you find a way to constantly, systematically purge out the lower classes.

Hey, at least we'd have the population under control.