Please solve the question clearly providing a step by step explanation on how to do it. Do everything the question asks you to do. Do not miss anything out.

This problem set tells the story of three friends Alice, Bob, who are consumers, and C arol, who owns a firm. Alice, and Bob have preferences over two goods, Dumplings and Free time. Each of them have Ti available hours each day with i=A,B. Carol's firm produces dumplings using two inputs, L abour and K apital with respective prices w and r. Denote the market price of dumplings as p. 2. [22 points] Suppose that the price of dumplings increases from p=1 to p=2, what is Alice's optimal consumption of dumplings and labour supply under the new price? What is the total effect on her dumplings consumption and labour supply of the change in p ? Represent both the old and new choices graphically with consumption of dumplings on the vertical axis and free time on the horizontal. Find the number of available hours TA under which Alice's optimal choices under (p,w)=(1,1) would be just affordable under (p,w)=(2,1). Then compute Alice's optimal consumption of dumplings and labour supply if her amount of available hours were replaced by TA and (p,w)=(2,1). Use this bundle to decompose the total change in her dumpling consumption and labour supply into the substitution and the income effects. Are dumplings normal or inferior