Is the Moon Carbon Poor?

The earth and the moon are made up of very similar stuff. There is carbon on the earth's surface, but we're not a hundred percent sure why. One theory is that carbon tends to form very volatile compounds,. things that would end up in an atmosphere if the moon could keep one. A moon could be like, huge, like our moon. Or, you know, it's ok to have a moon that's like a kilometer in size.