(a) (10 points) Maria is a graduate student who likes to spend her limited leisure time of thirty hours a month doing one of two activities: playing tennis (3:), and singing karaoke (y). Each tennis match takes three hours, and each karaoke session takes two hours. Further, suppose that Maria has a xed monetary budget to spend on leisure activities each month. She currently exhausts this entire budget by playing seven tennis matches and attending four karaoke sessions. With this monthly budget, she would also have been able to afford exactly six tennis matches and six karaoke sessions. Assume that both goods are perfectly divisible. i. Write down Maria's money and time constraints as algebraic inequalities. ii. Plot Maria's money and time constraints using dotted lines. Clearly label each con- straint, any axis intercepts, and any points of intersection. Shade in her budget set, using solid lines to indicate where the boundaries of the budget set are. (b) (14 points) Suppose that Maria's preferences over tennis matches and karaoke sessions are such that she likes them exactly equally: she is always willing to trade one tennis match for one karaoke session (and remain exactly as well of as she was before.) i. One valid utility representation of Maria's preferences is u(x, y) = m. a) Show that u) = uz is a positive monotonic transform when u > 0. [3) Apply f to the utility function u, to find another utility representation of Maria's preferences. ii. (1) Compute Maria's marginal utilities for each good using a valid utility representa tion of your choice. ,8) Using your answer to the previous part, or otherwise, prove that Maria's prefer ences are strongly monotone. iii. (1) Find three distinct bundles (my) that give Maria a utility level of 2. ) Hence, or otherwise, show that her preferences are not strictly convex. iv. a) Explain why Maria's optimal consumption bundle must lie on the outer boundary of her budget set. 3) Compute Maria's marginal rate of substitution. 7) Find Maria's optimal consumption bundle. E3317 points) Suppose that Maria has just won the lottery, so that her money constraint is now irrelevant. We disregard it for the remainder of this question. Suppose irther that the duration of a tennis match were to decrease to 90 minutes. i. What would Maria's new optimal consumption of tennis be? Use the Hicks-Allen decomposition to determine the income and substitution effects that comprise the price effect. -' W must the duration of a tennis match be so that all bundles on Maria's budget. ' . awe optimal simultaneously. "