https://blog.ethereum.org/2014/07/05/stake/
https://blog.ethereum.org/2015/01/10/light-clients-proof-sta...
https://blog.ethereum.org/2015/01/28/p-epsilon-attack/
Achievement unlocked!
If there's a chain fork (which is the important corner case that Proof of Work is excellent at dealing with), there is no incentive for a Proof of Stake holder not to "vote" on both forks. This reduces network security substantially.
It also leads to the rich gaining more and more wealth over time.
Proof of Work leads to moderate energy consumption, but it's negligible compared to the utility of a Bitcoin.
I wonder how much CO2 it takes to make around $250 in cash?
Similarly, rising prices is an effect of inflation, and not inflation itself.
Your confusion is understandable: over the last decade or so, several high profile "macro-economists" have subtly but suddenly began misusing these terms as you just did, so much so that dictionaries and textbooks have begun to follow suit.
In any case, most proof-of-stake coins inflate per their mechanism, and do so at a generally faster pace than proof-of-work coins. Moreover, proof-of-stake advocates believe that their coins' mechanisms favor a healthier and more predictable distribution of units, rather than the volatile arms race of proof-of-work.