Why would that desperate person be willing to pay the xN premium needed to cover the idle time? Why would there be exactly one desperate person in the market? Y

Why would that desperate person be willing to pay the xN premium needed to cover the idle time? Why would there be exactly one desperate person in the market? You need a curve, charging more as the space fills up. Desperate renters don't have infinite money. People with money are the people who are not desperate, because they have flexible options.