SOLVE ONLY QUESTION 3

Question 1 (10 points) Alice enjoys goods m and :9 according to the utility function We y) = m3/4y1/4- (a) Find Alice's marginal rate of substitution at the point (3, 2). (b) Draw the indifference curve through the point (3, 2). (c) On the same diagram, draw Alice's budget line if pa, = $4, pg = $2, and I = $16. (d) What is the s10pe of the budget line? Does the budget line have any common points with the indifference curve? How can you interpret your answer to the last question? (e) Find the Marshallian demand functions, 92039,, py, I) and yoz, py,I) by writing the Lagrangian and nding its critical points. (f) What is Alice's demand for each of the goods at pz 2 $4, 13,, = $2, and I = $16? What is the highest utility level, U *, that Alice can achieve if pa; = $4.10,, = $2, and I = $16? (g) Derive Alice's indirect utility function. Question 2 (10 points) Alice enjoys goods 1: and y according to the utility function \"(in y) = 323/4111\"- (a) Using your answer to Question 1(g) and duality between utility maxi- mization and expenditure minimization problems, calculate Alice's ex- penditure function for each level of utility, U\". (b) Suppose that the price of good 2: becomes 1);, = $3.50 and pg remains unchanged. By how much should Alice's income change to provide her the same level of utility as she received when pa; was $4? 2 (c) Using your answer to Question 1(e) and duality between utility maxi- mization and expenditure minimization problems, nd the compensated demand functions $3013, pg, U\") and yc(pm,py, U\"). (d) Find the substitution effects for both goods when the price of good :1: drops from 191 = $4.00 to p; = $3.50. Question 3. (10 points) Assume that the score you can get on the midterm or practice test is a random variable which can take only two values - high (H) or low (L). Set the utility 11.0?) = 1 and u(L) = 0. The score that you get on the practice test depends on your choice of eort 0 g e g 1. To be more specic, with probability 6, you will get H, and with probability 1 8, you will get L on this homework. You (dis)utility of eort is e2, so, after you complete the practice test, your utility will be either 1 e2 (with probability 8), or 82 (with probability 1 8). Next, assume that your performance on the midterm depends on the score that you achieve on the practice test, and no further e'ort is required. Namely, if you get H on the practice test, you get H on the midterm with probability (1;; e (O, 1); and if you get L on the practice test, you get H on the midterm With probability (1;, E (0, 1). Further, assume that qL