CA2BA said:
just outside the city the cost would be lower.
The 'safest' zone would be the Northern Zone, but the most desirable neighbourhoods are just as expensive as in town, and the price of groceries etc will not go down.
You'd have to go pretty far out of the city to notice a difference, and then you probably wouldn't find yourself coming back into town that often. And a lot of the barrios where you would get a low price, you'd want a car and you definitely wouldn't want to have to go out on the street at night to pick up some milk or something you forgot for dinner etc.
A lot of zones outside of the city are just single-family houses, so harder to find an apartment to rent. Renting an apartment in Vicente Lopez, Olivos, Acassuso, La Lucila, Martinez, San Isidro etc is possible, but for a short-term rental I don't think that you're going to find a difference in price (long-term rentals do get a bit of a break).
The other side of the Panamericana is cheaper but also less desirable, and really not interesting for a tourist, especially one on their own -- Carapachay, Munroe, Villa Adelina, etc. -- one good thing about living in those areas I guess is that you'll never encounter foreigners, but you'll also be really stuck in the burbs without a car, and you wouldn't want to have to wait for a bus on your own at night etc.
If you're willing to go to another city entirely, then Cordoba or Rosario are cheaper than Capital.