Hagrid keeps his broken wand. Breaking seems a mode of punishment and not just an inconvenience.
Can a wizard buy a new wand if the old one is broken?
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It fully depends on the reasoning behind the breaking of the wand. Upon accidental breaking of the wand a replacement can indeed be ordered. However Hagrid had his wand broken as a punishment for being expelled from hogwarts. He is not allowed to practice magic as he never graduated.
I should also mention that he keeps his broken wand inside his umbrella and still uses magic.
Mentions of broken wands being replaced that i can remember are:
Neville longbottom, he got a new wand after breaking his dads old wand.
Ron breaks his wand on the womping willow in CoS and gets a replacement by the PoA.
Why not? Ron got Charlie's old wand when he started at Hogwarts. One or 2 other examples are available. I expect wizards in good standing can purchase a new wand.
Ron bought a new wand with the remainder of the Weasley's windfall after they visit Egypt in the Prisoner of Azkaban.