They do grow like weeds, maybe you should start cutting some. Look at this price below from a supply company (WSS) for building surf board, $49.99 per plank of Paulownia. They state each plank contains 4.5 bf of Paulownia, this price does not include shipping.
This below is copied from the website:
WSS HAS PAULOWNIA!
Our paulownia is just slightly heavier than balsa but much stronger.
Each plank is 8 ft. long, 6" wide, and 1-1/8" thick
(4.5 board feet per plank)
In the late 70’s and 80’s Paulownia was $2 to $6.00 per ft in the log. Shipping them to Japan. Today you can’t give one away. Back then Paulownia was higher than the best walnut.
If one kind of tree gets red hot on price sell it. It will go down and you may not be able to give it away in a year or two.