What Mark said. I have never wedged a housed joint because it will not work like you hope it will. What will happen is the joist will shrink and the wedge will fall out. If it is important that the joint appear tight I will step the housing in so you cannot see the gap when the frame shrinks. ( I also lag or oly screw the joist so it does not fall out of the housing )
Wedge material should be matched to the job it is doing. If it is used to assemble a joint it might be better for it to be softer than the wood being assembled so you do not split your work. I cut a hammerhead splice and ruined it when the wedge sheared off the head and dang near split the morticed piece because the wedge face was not parrallel to the grain.
Bake
