If a boy completes the Wolf badge early, may he begin working on the Bear badge?

No. In the Cub Scout program, all boys in a den work toward a badge that is geared to their level of development. If the Wolf badge is completed before the end of the program year, a boy may work on electives to earn Arrow Points, but he may not begin working on the requirements for the Bear badge. His work on the Bear badge will begin the next program year, when he graduates into a Bear den.