please answer the qusetions

2 Aggregate Demand Curves (4 points) Ariadne and her three sisters all have the same preferences, described by the utility function U(:r,y) = 323:4. The budget constraint for each sister is I5 = 33,511: + pyy, where I3 is the sis- ter's income. The demand functions of each sister are given by $3 = ISA/(3px) and pg = 21's ,1' (3391,). (a) What is the aggregate demand function for good 3:, if Ariadne has an income of $30 and her three sisters Bianca, Clara, and Diana have incomes of $60, $90, and $120, respectively? (1) (b) What is the aggregate demand at a price of p; = 10? (1) 1Hint: Figure out how man),r (small) units 1m of y she has to buyr to keep utilityr constant. (c) How does aggregate demand for good a: change if the income of each sister increases by 10%? (1) (d) How does aggregate demand for good a: change if the price of good 3; increases by 10%? (1) The utility function is given by U (3:,y) = 55:92. The resulting demand functions are therefore 3; = I/(3pm) and y = 21/(3py). (a) What is the maximum utility the consumer can achieve as a function of I , p3, and py? [1) (b) What is the minimum the consumer needs to spend to achieve a given level of utility I? as a function of pI, and pg? [1) (c) The initial income is $576, initial prices are 392 : 3 and p3 : 4. What is the quantity of good a: consumed, 3:0? (1) [(1) Now the price of good a: rises to p; : 64/9, while the price of good '9 remains the same. What is the new quantity of good as consumed, cc"? (1) (e) What income would the consumer need at the least, in order to be as well off under the new prices as under the old ones? (3) (f) If the consumer had that amount of money, how much would she consume of good a: (call it 13?)? (1) (g) What are the total, income, and substitution effects of the price change on her demand of 1:? (2)