Shirl writes:

It isn't as if they say to the spouse, "Hey, would you like us to go
search for your husband?" and if she says no, they just leave a guy who
might need rescuing out there, or if she says yes, she's responsible for
the bill.

A third option would be to ask her if she wants to spend money on the search.
If the answer is no, the search is the same as it would be for any other
non-celebrity; if the answer is yes, the search is more extensive, and she
pays for the over-budget expenses.