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Well, taxes are high in NYC - you'll probably only have around $1000 after taxes.
Food is expensive in the city (even if you only eat on $10 a day, that's $300 a month) and you also need to budget for commuting costs (a monthly subway card is $104) as well as monthly utilities - your phone (if you have an iPhone, that's $85), plus electric, and maybe gas and water bills, depending on your landlord (don't forget cable, too, if you want to watch TV and have internet access! Internet alone is going to be like $45, which of course you can split up with roommates).
You could do it, but you'd be miserable and would never have any money to enjoy life in the city - no drinks at the bar, no eating out, no movies, no concerts (other than free ones in the Parks in the summer!), etc.