Question

Happy Goluki likes tea (good 1) and cookies (good 2) Her preferences are represented by the utility function U(q1,q2) (q1)05(q2)0.5, where q1 is the number of cups of tea and q2 is the number of cookies Goluki is given I-$180 that she is allowed to spend as she wishes on tea and cookies. a) Calculate Goluki's optimal bundle if the price of tea is p1=$1 and the price of cookies s p2 $2. Call this bundle A and show it on a diagram (put tea on the horizontal axis). b) Suppose that the price of tea increases to $5 (I know, I can't believe it either, but Melbourne does have some very fancy tea places) and the price of cookies stays the same. Calculate Goluki's new optimal bundle of tea and cookies (call it bundle B). Show it on a diagram from part a) c) Calculate the income required so that Goluki can obtain the same utility as from the original bundle (from part a) but at the new prices. Calculate this new bundle, call it bundle C and show it on the diagram from part a) d) What is the total effect, income effect and substitution effect (on tea) of an increase in price of tea from $1 to $5? Show them on a diagram from part a) e) Goluki's parents are willing to compensate her for an increase in prices. How much extra money would they have to give her to make her as well-off as she was at the old prices? This amount is called the compensating variation. What bundle will she consume? f) Suppose that Goluki's parents gave her enough money, so that she can afford the same combination of tea and cookies, as in part a), at the new prices. Find her new optimal bundle (cal F) in that case and show it on the same diagram as in part d). Is Goluki better off with bundle F or with the bundle chosen in part e)? g) Should Goluki's parents give her extra money as in part e) or in part f)? Briefly explain Please use one diagram for all parts. If any optimal bundle you calculate contains a number of cups of tea or cookies which is not an integer (a whole number), please round to one decimal place (for example, write 4.5 cookies instead of 4.48 cookies, 4.1 instead of 4.05, 4.2 instead of 4.23 and so on). Do the same for values of utility

Answer 1

U = q^1/3_1 + q^1/3_2 MRS = MU_1/MU_2 = 1/3 q^1/3-1_1/1/3 q^1/3-1_2 = q^-2/3_1/q^-2/3_1 = q^2/3_2/q^2/3_1 At optimal, MRS = P_1/P_2 q^2/3_2/q^2/3_1 = 1/4 q^2/3_2 = q^2/3_1/4 = > q^2/3_1 = 4 q^2/3_2 (q^2/3_1)^3/2 = (4 q^2/3_2)^3/2 q_1 = 8q_2 Budget line: - p_1 q_1 + p_2 q_2 = 1 1q_1 + 4 q_2 = 180 8 q_2 + 4 q_2 = 180 12 q_2 = 180 q_2 = 15units \r\n q_1 = 8 times 15 = 120units p_1 = 9 & p_2 = 4 Budget line: - 9q_1 + 4q_2 = 180 optimal: - MRS = 9/4 q^2/3_2/q^2/3_1 = 9/4 q^2/3_2 = 9/4 q^2/3_1 (q^2/3_2)^3/2 = (9/4 q^2/3_1)^3/2 q_2 = (3^2/2^2)^3/2 times q_1 q_2 = 27/8 q_1 Now 9q_1 + 4(27/8)q_1 = 180 9q_1 + 13.5q_1 = 180 22.5q_1 = 180 q_1 = 8units q_2 = 27/8 times 8 = 27 units Income required to purchase original bundle \r\n i.e: A(120, 15) = 120 times 9 + 15 times 4 = 1080 +