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Which Scarsdale distrcit are you considering? Scarsdale as in the town/village of Scarsdale, or Scarsdale as in the zip code that's actually Edgemont School District?
The drawback with Greenwich is that it is legally defined as a town, but operates more as a small city, with a large school-age population. Thus, it's not that dissimilar from some of the smaller cities in Westchester in terms of population, though the reputation is better, owing to a larger affluent population than a similar community in Westchester.
Scarsdale is a smaller district, geographically, and, as has been mentioned, it's extremely competitive. Edgemont is not as competitive and is even smaller, having only two elementary schools and one combined junior/senior high school. Yet, the district has a good reputation as well, though being a part of the Town of Greenburgh, you don't get the perks of Scarsdale, especially in terms of resident parking at the train station.
One reason why Scarsdale has not come down too far from the peak prices is because of the schools. Families that have to look at their budgets will generally find it more economical to pay slightly more for a house in Scarsdale (and other Westchester towns with great schools) and use the public school system, even with a $30k tax bill. Generally speaking, the taxes are less expensive than private school tuition for a year for one student; so, if there is more than one school-age child in the family, it's less expensive to be in a district like Scarsdale since you get a good school system and town services at a lower annual cost than private schools and property taxes in districts where private schools might be more of an issue.
This, of course, assumes that the houses are not terribly different in price, since when spread over the life of the term of residence, the price differential for Scarsdale actually might be significantly less expensive than a similar house where one would need to use a private school at some point during the school career. New Rochelle is one such area, where it's easy to spend the same amount of money for a house in Scarsdale, but the schools at the middle and high school level are huge and do not perform to the same standard as a Scarsdale, though some of the elementary schools are good. At the middle and high school levels, private school tuitions for more than one child can outpace the difference in price and taxes that one incurred by purchasing in a district like Scarsdale at the outset. So, while the prices may not be overbid as they were during the peak, the area has had a softer deflation from peak pricing, owing to inherent demand factors that keep the area strong among purchasers who can afford the town.
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