The term "inelasticity" describes consumer demand, rather than the price. As a good becomes increasingly scarce, the price will rise. But when demand is highly "elastic," the price rise will be mitigated by consumers changing their behavior or seeking a substitute good (in this scenario, by relocating their residence away from NYC). In other words, demand elasticity means that consumers can decide not to pay the higher price because they have alternative options.
In the case of NYC housing, the demand is inelastic, meaning that the price increases stick because enough people do not have any acceptable alternative to paying it (likely because employment in their chosen profession requires being physically located in or near Manhattan).
In the case of NYC housing, the demand is inelastic, meaning that the price increases stick because enough people do not have any acceptable alternative to paying it (likely because employment in their chosen profession requires being physically located in or near Manhattan).