In an infinite horizon representative agent model with rational expectations, wealth maximization equals maximizing the discounted expected stream of real consumption over the infinite time horizon. Even if the current accumulator is finitely lived, they may have bequest motives as pointed out by Barro, somewhat excessively. I see nothing of theoretical substance to this point. This is not generally answered and depends on relations with other economies, the nature of technologies, and a lot of other things. Lucas says it is a continuum of outcomes, pick your preferences.
Path dependence says indeed there are thresholds.
The latter looks more plausible to me, even if we do not have complete explanations of what determines them.