I read this story in 2004, in a photocopied handout for a class, but I'm sure it was published long before then. Though the plot is pretty dark, the tone of the writing is lighthearted.
A lonely man finds a magic toy gun that makes people disappear. Once he realizes what it does, he gradually eliminates people he dislikes. Unfortunately, he's very unsociable and dislikes most people he meets, so he gradually eliminates the population of his entire city. Finally, he goes on a series of dates with different women, but find ridiculous superficial reasons to eliminate each one with the gun.
After everyone in the city is gone, he shoots himself with the gun and vanishes. The city gradually repopulates, and at the end of the story, a child finds the gun and tries shooting someone with it. As he pulls the trigger, the story ends like:
Nothing happened. He was too young to have any really strong desires yet.
the gun can actually do anything you desire. This guy wanted to make everyone disappear, so that's what the gun did.
I think the story also mentions the man's childhood, but this may be from a different story. He was very ugly as a baby. As a joke, his family put their dog in his bassinet and showed it to their relatives. The relatives couldn't tell the difference between the dog and the ugly baby (and actually thought the dog looked better), so the family just kept the baby hidden and passed off the dog as their family's infant son.