How much does it cost to live in paris each month?

depends on your cost of living.
eveyrhting is expensive and prime location is going to be higher in cost.
your best bet is finding a roomate because its so much more expensive to live on your own.
good luck with your travels and journeys.

I'll give you approximate prices.
Renting in the 16th is probably quite costly because it is considered to be one of the posh areas. A small studio will cost at least 700 euros a month probably including charges.
Food is quite expensive again. I think more than even in London. Depends how you feed yourself of course, but I doubt you'll want to live off bread and pasta for long. I'd say at least between 50 and 70 euros a week. Internet/phone another 30 euros minimum (per month though). Transport etc I don't know you could check the RATP website and it depends on your needs. Let's say 100 euros.
If you want to be very economical you could potentially live on a small budget and still manage to entertain yourself thanks to numerous exhibitions/museums/ street activities, but obviously most of us like to eat out once in a while, buy clothes and random things.
As an example I know a mother who rented 30m2 in the 10th arrondissement with her teenage daughter and she had a salary of 2000 euros a month. They still managed to go on holiday abroad and despite the small accommodation I can't say that they were poor but then they also had some savings.
Honestly have at least 1200 euros. You won't be well off or anything but you'll survive ok. Then obviously things like (most) education and healthcare are almost free.
About the 16th arrondissement my sole warning is that it is quite a dead area. People don't really go out much but tend to stay indoors in their luxurious apartments/mansions. It may seem like a caricature but I have a fairly wealthy friend who's lived there all her life and she always tells me about how boring the arrondissement is. You have much more things happening elsewhere. The 19th/20th and parts of the 18th (Montmartre-Barbes) are to be avoided at certain times and you should be careful about shopping for food or else there. Anywhere else is fine and just apply the same principles of caution as you would in any big city.
I personally really like the area around Bastille, it's not really touristy but you can find so many nice and affordable bars. It has quite a young hip crowd too: architects, designers etc...
Sorry it's a bit random what I give you but my point is unlike many answers which I'm sure will warn you that unless you have 2000-3000 or more a month you won't survive is a bit exaggerated. Beside living in Paris is nice in itself, you won't have to put up with the amount of time spent in transport like people living in the close suburbs.

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You will need approximately 1500 to 2000 Euro's a month.
Good Luck

This site
http://www.numbeo.com
will give you an idea as to the cost of living there.
Good Luck