Which is more expensive? Overtime or replacing a house.
I don't know.
They would usually have patrol on perimeter, plus SWAT, plus command, plus medics. For a short-term event you can take some guys from patrol and nearby departments, but if something is going to take days or weeks then other departments are going to pull their officers and you will need to get guys back on the road because presumably other crime doesn't stop all of a sudden just because you've got an emergency.
A serious SWAT callout with shots fired like this one could easily pull fifty or a hundred cops into an area. If you kept that kind of presence going for any length of time I'm guessing it could cost $50k or more per day on salary. That buys you a house pretty quickly in some places, and in others it would take a few days.
Of course once the bad guy is settled in and reasonably quiet you could trim manpower way, way back and leave maybe a dozen cops to watch the house and handle traffic. You'd probably have to. Of course, then those cops are going to have to spend as much time managing property owners trying to come in as they do keeping an eye on the guy in the house.
On a per-event basis it's probably going to usually be cheaper to wait a few days or couple of weeks for the food and water to run out and get the guy to come out. That said, if the criminals know that cops are just going to wait them out you'll probably see this kind of thing happening over and over again. If the criminal happens to have the luck to get barricaded in a Mormon's house then you'd better hope they don't like napping on couches and eating out of the pantry because they will be there a while.
Why don't they just set fire to the house and pay the damages later?
Because fire kills indiscriminately and can spread.
Regardless, I don't think they should be able to walk away and leave the homeowner on the hook for what they did.
Agreed.