Objection 1. It would seem that even a bishop cannot grant indulgences. Because the treasury of the Church is the common property of the whole Church. Now the common property of the whole Church cannot be distributed save by him who presides over the whole Church. Therefore the Pope alone can grant indulgences.
Objection 2. Further, none can remit punishments fixed by law, save the one who has the power to make the law. Now punishments in satisfaction for sins are fixed by law. Therefore the Pope alone can remit these punishments, since he is the maker of the law.
On the contrary, stands the custom of the Church in accordance with which bishops grant indulgences.
I answer that, The Pope has the plenitude of pontifical power, being like a king in his kingdom: whereas the bishops are appointed to a share in his solicitude, like judges over each city. Hence them alone the Pope, in his letters, addresses as "brethren," whereas he calls all others his "sons." Therefore the plenitude of the power of granting indulgences resides in the Pope, because he can grant them, as he lists, provided the cause be a lawful one: while, in bishops, this power resides subject to the Pope's ordination, so that they can grant them within fixed limits and not beyond.
This suffices for the Replies to the Objections.
Source: Summa Theologica (New Advent)