I’ve read everything so far …
As @Chronos says, if they reach a mutually happy arrangement, that is by definition “fair (enough)” for their purposes. To which I’d add " … if it stays mutually happy".
IMO …
The situation is so wildly asymmetrical that “fairness” is far more about morality and ethics than it is economics. And yes, I wonder if these folks’ maturity is sufficient to bridge that gap.
As a business matter Betty is (probably) asking too much and Dave is offering far too little. The “(probably)” is there on the off chance Dave is living in a studio apartment in a slum, and Betty is living in the nice part of town, such that Dave’s $1600 is barely room rent in Betty’s neighborhood.
The OP was silent on this point, but if these folks are friends also looking for simple companionship or are BF/GF/SO looking for a full-bore couples’ relationship then perhaps the intangible benefits each derives from that part of the cohabitation will also be put on the overall fairness scale in addition to the financial bits of mere indifferently co-existing roommates. Recognizing the difficulty of establishing an “exchange rate” between dollars & emotional [whatevers].
Switching back to economics, another question is the relative income and relative assets of the two parties. “Fairness” is easier the more equal they are on both dimensions. The less equal, the more e.g. 50/50 is less fair and “from each according to their ability” is more fair.
It happens I have a sorta similar thing in my recent past and probably future. Which is a bit of a hijack, but may have some useful parallels to the OP’s friends. I’ll hide them for the benefit of folks not interested in that stuff.
Recent Past
When I moved out on my now ex-wife a couple years ago I ended up rooming for ~4 months with a guy friend who lived alone in the 4-BR paid-off house where he’d raised his now 40-something kids. At the time I moved in, the expectation was that it would not be a permanent situation, but there was no specific end-date in mind either. My original guess was between 6 and 12 months as I decided where on Earth I wanted to live now that I was fully free of all constraint of job or family.
We mutually agreed to look at Zillow for the cost of a room for rent in the same neighborhood and whatever that consensus was, that’s what I’d pay. Turned out to be $1200/mo. Groceries and such were “each buy their own”, but nobody was fussy about who drank whose booze or coffee. Neither one of us cared whit one about the amount of money changing hands, but it seemed the honest way for neither of us to take advantage of the other. Which was both of our goals; neither to take advantage, nor be taken advantage of. Worked great.
Probable Future
Unrelated to that, my current GF and I are drifting towards a co-habit.
I live in a rented apartment in the fun neighborhood we want to live together in. She has a … wait for it … paid off house in a neighborhood we don’t want to live in. My place is too small for two; her place is plenty big for two. So in summary, I’m forced to move to either a bigger place near here, or in with her where she is. And she’s forced to move to my area which she prefers, or avoid the hassle of moving by staying put in her comfortable, but very boring, area.
We’re both similarly financially secure. Similar age too. She has adult self-sufficient kids. I have no kids. Zero interest by either of us in ever marrying, mostly due to phase of life. But this is intended as a permanent serious SO arrangement.
If we do both move to a new joint residence, which is the most likely outcome, the betting is we’ll want to rent, not buy. But that’s not for certain. Yet.
We’ve both given a lot of thought to how this might play out in a fair fashion, and where our redlines are. It’s still far enough in the future we’re not down to negotiating, but we’ve both been trying to draw the outlines of the envelope in our head.