I am smarter than Jake, but he's richer than (me or I).
What should come as the object?
I am smarter than Jake, but he's richer than (me or I).
What should come as the object?
As an object of the preposition than, it's me.
I am smarter than Jake, but he's richer than me.
Than can also function as a conjunction:
I am smarter than Jake, but he's richer than I am.
Am is often omitted but that is considered acceptable: