Economy (noun): the intentional saving of money or, less commonly, the saving of time, energy, words, etc. (in this case: rats) "For economy's sake" here means to be sparing or to be careful with the supply of rats, as there is only one. If the Prince would actually kill the rat, he would no longer be able to follow the advice to condemn a rat to death from time to time. I hope this helps.

Economy can mean - the careful management of resources, not wasting stuff. I think this is probably what is mean here - there is only one frog. I don't know the story but I understand the king has no subjects to carry out his orders and only orders things that will happen anyway. I guess he spares the frog as there is no one to carry out his order, but tells himself it is actually because he doesn't want to waste the only frog.

The phrase "for economy's sake" , perhaps better translated as 'for the sake of economy' is ironic. It normally means to conserve [i.e. not waste] resources, but in this case means to conserve [maintain] authority. The authority of the King. and Authority in general, has to operate within the bounds of what is possible which, as he explains, means that he has to order the sun to set at 7.40 that evening, and to issue a reprieve each time he sentences the rat to death because there is only one rat. To maintain his authority the King has to be 'reasonable' and 'economical' - if he kills the rat he will have no-one to condemn and will have lost his authority.