Problem 1 Optimal demand and price changes (41 points) Let Zoe's preferences over consumption bundles z = (1, z2) be represented by the utility function u (1, T2) = T172. Zoe has income m = 24. The price of z; is one dollar, that is po = 1. The price of good 11is p; = 4. 1. Carefully showing your work, solve for Zoe's optimal consumption bundle and explictly state the following: (a) State Zoe's optimal consumption bundle z*. (b) State Zoe's utility in her optimal consumption bundle u(z*). 2. The price of good 1 increases to pj = 9. All else remains the same. (a) Solve for Zoe's new optimal consumption bundle, z'. (b) Solve for Zoe's utility level in her new optimal bundle u(z'). (c) Comment on how the good 1 price change from 4 to 9 has impacted demand for goods 1 and 2, respectively. Include discussion of income and substitution effects in your comments. 3. Calculate how much income Zoe would require so as to be perfectly compensated for the good 1 price increase from 4 to 9. Denote this compensating income level by 7. Strucure your answer as follows [note that question (b) can be answered even if (a) has not been]: (a) For the determination of the compensating income level, find the income level 7 such that when Zoe optimally chooses consumption subject to income m and good 1 price py = 9, she will have exactly the same utility as she did originally, u(z*). (b) Denote by & the optimal consumption bundle subject to income M and and good 1 price py = 9. Using the three bundles z*, 2/, and Z define the income and substitution effects associated with the good 1 price increase