PoW/Bitcoin has the same dynamics as gold mining under a gold standard. There is a return to spending a marginal resource in bitcoin mining, vs something else. As you add more resources to bitcoin mining, the return becomes low compared to the "something else" options, and no more is invested. You reach some equilibrium.
There's no reason that equilibrium would be the entire economy; most likely, some small fraction of it. But note, network costs scaling with the size of the global economy is not a point in Bitcoin's favor!
Proof of work does, in a way, rely on the entire world spending enough computational work on the blockchain that it becomes infeasible for any one person to spend more. So it does have much of the dynamics of an arms race.
Edit: The same logic explains why the problem will affect most attempts to have an "eco-friendly" PoW Blockchain, like Bram Cohen's proposal:
https://techcrunch.com/2017/11/08/chia-network-cryptocurrenc...
That just shifts the arms race to something else: in that case, on spending economic resources on faster ways of doing a lookup. Still should reach equilibrium at a stable fraction of the economy and produce the same waste, although perhaps in the form of hyper-optimized DB lookups and the related hardware and labor for it, not energy.