Hotels in Paris are costly; prohibitively expensive particularly when you're trying to travel on a budget. Renting an apartment often ends up being a more cost effective option, especially if you snag a small kitchen. The Paris experience will be different, to be sure, but you can hit up farmer's markets and specialty shops and gather provisions for a home-cooked meal - another way to keep costs down.
AirBnB has a fantastic selection of Paris apartments but if you prefer to work with small businesses, Haven in Paris is one of the short-term apartment rental companies I trust and recommend most. They have a number of studios in their collection which are ideal for singles and couples, each in fantastic locations. The photo above was taken from one of their 1 bedroom apartments in the 18th arrondissement (Montmartre) that I visited recently. It has a stunning view, a small balcony with a table for dining and offers visitors the feeling of living like a local, albeit briefly. What more could you want?