In Downey, building permits come from the city's Building and Safety Division, which also checks plans and does the inspections, and permits are issued at the Building counter in City Hall. The city's own wording is that, generally, any alteration or repair to an existing building needs a permit, and insulation isn't on its named list, so before anything else I check with the division for your address, and when a job calls for a permit we take care of it before work starts, which can push the start date back. A contractor pulling a permit in Downey needs a current City of Downey business license, so it's fair to ask anyone you're considering about it, us included.
What moves the price is the size of the attic, how many inches are already up there and what has to come out before anything new goes in. If the old insulation is dry and just thin, sealing the gaps and topping it up is usually the better buy, and if it's soaked or full of droppings it comes out first, which adds cost and time. Foam costs more because it seals air while it insulates, so I usually only suggest it when the ductwork runs through the attic. I write all of this down at the estimate in the order it has to happen, the same way we plan jobs back home in East Los Angeles and over in El Monte.