They have no purpose, save for use on construction paper. Which no one ever has, other than kindergartens. Construction paper is the affirmative action of the drawing paper world. White crayons just sit there in the box, lonely and unwanted, like the scrawny kid with glasses in gym class dodgeball.
The white crayon gets its comeuppance only late in life, when its brethren have been broken and worn down to stubs. Then it stands proudly in its virgin glory, untested but free of abuse. "You pompous bastards," the white crayon boasts with a booming crayon cackle. "Look at you now. Just look at you."