Why would a landlord leave a house vacant for months at extremely high rent instead of lowering the price?

Usually the landlord has not done the real math on what a vacancy costs. The headline rent is easy to fixate on, and an empty month is income you never recover, on top of turn costs like paint, cleaning, carpet, and repairs. Add those up and holding out for a bit more monthly rent can take a very long time to break even. If chasing a higher rent adds a few hundred dollars a month but the vacancy and turn cost several thousand, getting back to even can take a year or more of that higher rent. What most landlords actually benefit from is a deep pool of qualified applicants who can move in quickly at fair market rent. That is worth more than squeezing out the last few dollars, and overpricing tends to stretch the vacancy. It can also land a tenant who feels overcharged and treats the place accordingly. Some owners hold the line because they own the property free and clear and feel no monthly pressure. That is still a real opportunity cost, just a less visible one. If you are the prospective renter, it rarely hurts to make a reasonable offer below asking on a place that has clearly been sitting.