Long term bitcoin mining profitability

Here's a question for the bitcoin gurus on bitcoin's long term future (i.e. the next decades).

Halving occurs every four years which in practice means that the block reward in BTC decreases exponentially with time. To keep paying for mining costs, bitcoin's price in USD should then increase exponentially. However, bitcoin price in USD has shown to only grow like a power law (google "bitcoin power law"), which is slower than exponential in the long term. I am assuming that hardware capabilities and energy prices will not vary drastically over time and certainly not rapidly enough to compensate for the exponential reduction of the block reward.

This may suggest that there will be a time in the future when the block reward will not be sufficient for the most efficient miners to stay in the game. Are fees going to be what keep miners afloat? What sort of fees will be required?

I would appreciate pointers to relevant economic analyses of these factors.