It’s ‘A Christmas Carol’. It even opens with the first few lines verbatim, almost:
“Old Marley was as dead as a doornail… but that was seven years ago, and I hadn’t been hired to investigate Marley.” No, this private detective has been building up a dossier on Scrooge’s life, especially his childhood. He hands this over to his mysterious employer on Christmas Eve, is paid and told to forget everything about the case, but he just can’t help but be curious what’s going on. Short, funny, and plays off the original text nicely.
The Case of the Skinflint’s Specters by Brian M. Thomsen
Availability: print only
Word count: 2,800
First published: Christmas Ghosts, edited by Mike Resnick and Martin H. Greenberg, 1993, DAW
Where else to find it: The Dragon Done It, edited by Eric Flint and Mike Resnick, 2008, Baen
Ten to Infinity 