They'd make less money [1]. Demand for shelter is close to inelastic, but demand for shelter in Atlanta is not.
Consider the extreme: they mandate rents of $1mm a month. Assuming no government interference, they might land a couple tenants at that price. But the vast majority would balk, thereby leaving them with less profit.
You can realistically use something like this to maybe bump rents a bit, but you cannot use it to radically alter markets, because it is supply and demand that set market prices.
Price fixing doesn't eliminate supply and demand entirely or make demand elasticity vanish.
What it eliminates is real competition among sellers ("fixing" the supply curve).
It raises prices - it doesn't magically make any asking price feasible.
Not finding demand to suddenly ask for $2M for a two bedroom house in Kirkwood is not some proof "hey, supply and demand is working just fine".
You're correct in one technical sense and wrong in another. (Either way, a good discussion.)
In a competitive market, price and quantity are set by the marginal incremental cost (MIC) and average total cost (components of the supply curve) and the demand curve.
Monopolists, however, ignore the demand curve. Their quantity produced is entirely set by MIC and marginal incremental revenue (MIR) [1]. In a very real sense, the market is no longer about supply and demand--it's about the producer's costs and revenue. (Your technical win is in the MIR being related to the demand curve.)
[1] https://en.wikipedia.org/wiki/Monopoly_price#/media/File:Mon...
That doesn't make sense, to obtain the MIR monopolists have to have an implicit consideration of the demand curve. Monopolists can't push the price off the demand curve without pointlessly leaving arbitrage opportunities (or burning money, I suppose) and their revenue depends on price and quantity.
There is a fun thought experiment too if we consider a silly degenerate case, imagine the demand curve has a practically impossible shape and spikes to infinite price at 10 widgets. Any logically sound strategy will tell the monopolist to produce 10 widgets because profit would be infinite. So they can't ignore a demand curve in theory because at a minimum they have to care that it is not a degenerate case.
And I that’s also a possible answer your question, on why they are leaving money on the table. To avoid inviting political action.
The monopolist doesn't have to follow this rule: they set the price to whatever makes the most profits and leave the price there regardless of fluctuations in housing supply.
no doubt you can find things to quibble with this too, but it's more accurate to say "price is set by supply and demand" means that the quantity sold is at the "market clearing price". Any other price will leave unsold inventory or unmet demand.
the point being that market manipulations will shift the supply or perhaps the demand curves.
Monopoly pricing leaves unmet demand in favour of higher expected profits.
Which would be illegal.