In A Raisin in the Sun, how do power and authority change hands over the course of the play?

This is an excellent question. As you go through the play,
it is clear that by the end of the play the major shift of power that has occurred has
been the transfer from the matriarchal Mama to her son, Walter. Note how at the
beginning of the play Walter tries to do everything he can to persuade Mama to give him
the money from his father's insurance cheque to invest in a liquor business. However, at
the end of Act One, it is clear who is the real head of the family according to
Walter:

What
you want me to say you done right for? You the head of the family.
You run our lives like you want to. It was your money and you did what you wanted with
it. So what you need for me to say it was all right
for?

However, perhaps because
of this speech and the way that Walter feels Mama has "butchered his dreams," in Act Two
Mama has a change of heart, placing him in charge of the rest of the money and giving
him the headship of the family:

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And from now on any penny that come out of it or
that go in it is for you to look after. For you to decide... It ain't much, but it's all
I got in the world and I'm putting it in your hands. I'm telling you to be the heard of
this family from now on like you supposed to
be.

Of course, as the rest of
the play goes on to show, this was perhaps not the wisest of decisions, as Walter is
quick to lose the money. However, ironically, this gives him the chance to assert his
authority at the end when he refuses the cash-offer from Lindner to not move into their
new house.