I'm looking for the name of a short black and white movie in which people were given an allotted amount of time at birth, and used that time to buy things. For example, a can of beans in the store would be priced something like 23 minutes, and that amount would be deducted from one's remaining time. People could perform services for each other and settle the transaction using time remaining. A key part of the story involves the person going to the central temporal authority to petition that his mother be given more time.
This looks like it might be The Price of Life, a 1987 short film produced by Chanticleer Films and originally aired on the U.S. cable channel, Showtime.
The Price of Life was a 38-minute short film (story by Stephen Tolkin and Michel Monteaux) in which a time account is physically linked to every infant at birth, with death automatic when the balance drops to zero. An elite upper-class is portrayed as living hundreds of years or more. The protagonist is given a certain amount of time as an infant, and as a young boy adds days and years to his time account by buying valuables from people and selling them to visiting tourists from the rich enclave. After his sister dies after gambling away her time, the protagonist (now a young man) sets out on a journey to the enclave of "the Old Ones" in order to save the life of his mother, who is (literally) running out of time. He gets there and meets a beautiful older woman who co-opts him into the immortal lifestyle.
The full movie can be found here, on Vimeo.
It wasn't a "short, black & White" movie, but this is very much like the premise of In Time, which starred Justin Timberlake.
Humans had been genetically engineered -- they no longer aged (protagonist's mother looked 25, so did the protagonist), but had a display on their arm of the remaining time in their life. Time was literally money -- you got paid for your job in hours (hopefully much more an 1:1), you spent time for rent, bills, and groceries -- and if you ran completely out of time, you died, right there on the spot.
I think you are looking for a short film called "The Price of Life" produced in 1987 by Stephen Tolkin. While its not in B&W it does sound like what your referring to in plot and premise.
In the future money is not the most valuable commodity -- not stocks or shares. Instead the currency of choice is time and the populace are able to trade hours, days and weeks off their lives in exchange for services and goods. Into this world is born baby Zachary -- who grows into a little boy with a game plan and a scheme to make more time; whereas his sister swaps weeks for sweets etc as if it didn't matter. As adults, Zachary has continued this path, but so has his sister and he finds himself trying to challenge the rules to try and save her and his mother.
Also why is the off-topic answer getting up-votes but mine is getting down-votes?