Joe and Delia trick each other in this story for much the same reason that Jim and Della do what they do in "The Gift of the Magi." They do it for love.
Delia wants Joe to believe that she is giving music lessons so he will not feel bad for her. She does not want him to know that she is sacrificing her career for him.
Joe feels the same way about Delia. He does not want her to think that her efforts are in vain. He wants her to think that her music lessons are actually going to support him in his dream.
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