My general reaction to this discussion is that such a situation isn't
possible and that even were one to achieve this perfect balance that it
would soon fall out of balance.
I have to agree with you here. In order for labor costs to equalize the world would have to converge toward a common standard of living, and I think it must be apparent that we are a long, long way from achieving that. Also, to the extent that a "common standard of living" is understood to mean a First World standard, there are good reasons to believe that will never happen. Baring some really revolutionary technological breakthrough for example it's unlikely that the world has sufficient energy reserves for China's 1.1 billion people to achieve the same energy consumption profile as the average North American (note that it's entirely possible that in the future the world will converge toward a LOWER mean standard of living, but people generally expect the developing world to catch up with the developed one, not the other way around, although the latter scenario is arguably more likely). Note as to Emily's point of a person in a Third World country making $15 / hr, a convergence of living standards has to occur simultaneous with a convergence of wages, rather than one preceding the other.
Wages are only part of the story however. In order for production costs to be equalized across countries (which I believe is what we are actually talking about) a myriad of other factors need to be roughly equal, as Ben points out. Furthermore no one has yet mentioned the importance of public policy. Even if all production costs were strictly equal, individual governments could gain competitive advantage but adopting business friendly policies like loosening environmental standards, suppressing organized labor, and manipulating it's exchange rate (note: China for example is actually doing all these things as we speak).
Perhaps one could argue that the entire notion of a country would have
to cease to exist for this to be possible since the very notion of a
country leads to the formation of a group identity.
Funny you should mention that, because political scientists have been predicting the eminent demise of the nation state since the 1930s (Cf. David Mitrany's The Progress of International Government, 1933). The best example of this theory in practice is the European Union, though as even that example demonstrates nation states have proven far more durable than political scientists anticipated.
That having been said though there is one important exception that is profoundly important for understanding the dynamics of modern political economy: for the world's elites the nation state is already mostly irrelevant. By that I mean they have little meaningful attachment -residential, professional, educational, cultural- to any one particular country. They are truly "citizens of the world" who often have far more in common (most obviously in terms of patterns of consumption) with elites from other countries than with fellow citizens in their own. What happens when a cosmopolitan, trans national elite attains self consciousness? The social contract that underpins the stability of the nation state begins to unravel, because the elite at the top of the socio economic pyramid no longer recognize a community of interest between themselves and the less privileged members of their own societies. Rather, they increasingly identify their interests with their peers in other countries and against those who are lower on the pyramid, regardless of the two nominally inhabiting the same society. They champion policies like globalization and neo liberalism intended to cement their pre eminence, and it's all the easier for them to do so because while the locus of their identity has shifted from (crudely speaking) nation to class, a similar shift has not occurred for those lower down the ladder, for whom the nation state is still the primary foundation of their identity. Those people, failing to realize that their elites no longer recognize any sort of mutual obligation toward them, extend to them a degree of goodwill that is undeserved and increasingly abused -as when President Obama talks about the need for "shared sacrifice" to explain the hardships endured by ordinary Americans thanks to the greed and hubris of its financial elites, even as he transfers trillions of taxpayer dollars to those elites to ensure that they, unlike ordinary people, will not have to suffer any consequences for their own folly.