Alice cares about two things: how much leisure time she has (x1) and what mark she gets on tomorrow's Micro exam (x2). Her preferences between these...

Alice cares about two things: how much leisure time she has (x1) and what mark she gets on tomorrow's Micro exam (x2). Her preferences between these goods can be represented by the utility function: u(x1, x2) = x1 + x2 Alice has 16 hours of today to distribute between studying and relaxing (she sleeps the other 8). Studying positively affects Alice's mark up to a certain point, but if she tries to cram too much studying into one day, she will get tired and do worse on tomorrow's exam. Where s represents the hours Alice spends studying, the relationship between her mark and s is as follows: x2 = a √ s − s where a > 0.

(a) Suppose a = 6. At what level of s does continuing to study negatively affect Alice's mark? (b) Suppose a = 6. Write down Alice's constraint: the relationship between x1 and x2 that holds if Alice uses all her 16 hours for either leisure or studying. (c) Suppose a = 6. Find Alice's optimal choice of x1 and x2. (d) For what values of a is Alice's optimal choice a boundary solution in which she spends all 16 hours studying?