In stonej writes:

If a flight is sold out on coach and only a few first class tickets
remain how often do airlines let the first class tickets go for a
coach price? Seems it is better to fill a seat with a paying
customer than to let it sit empty.

Never. The coach section is (usually) oversold anyway; so they'll
upgrade some people out of coach (status FF members, full fare payers,
etc), and then put the overbooked coach buyers into the freed up coach
seats.