Just need help with Question 3 and Question 4

Problem 1 (Crime and Punishment) (30 pts.) Sally is parking her car in a two-hour parking spot. She can either pay for parking at a price p or not pay for parking. She has $M. In the two hours that she is gone, there is a probability # that traffic control will come and may give her a ticket. If Sally has paid for parking, she will not receive a ticket. If she has not paid for parking, she will receive a fine of 6 > 0 . Her utility function is u(x) = r and thus, she is "risk-neutral". Finally, M > > > p. For all these questions, we will keep the actions of traffic control as exogeneous so our game below will be simplified: Monitor(7) Not Monitor (1 - 7) Pay for Parking M - P M - P Not Pay for Parking M - 6 M 1. Find the value of a where Sally is indifferent between paying for parking and not paying (5 pts.) Suppose Sally is indifferent for paying for parking and not paying for parking. Now, assume her utility function is u(c) = = cl -0 - 1 1 - 0 -. o is the coefficient for relative risk aversion. Observe, when o = 0, then the utility function can be expressed as u(c) = c - 1 or just u(c) = c. In other words, she's risk neutral as before. 2. Determine the indifference condition for her expected utility when o > 0? (5 pts.) 3. Suppose that Sally is indifferent between paying for parking and not. Holding the other parameters fixed, describe mathematically or explain the intuition when (15 pts): (a) The fine increases o increases. (b) The probability of being caught a increases. (c) The parking ticket price p increases. (d) (Extra Credit +5 points) Find a condition on a where when Sally's income M increases, she is more likely to pay for parking. Assume that o = 1, M = 10p, 6 = 2p. (e) (Extra Credit +5 points) Find a condition on a where when Sally's risk-aversion o increases, she is more likely to pay for parking. Assume that o = 1, M = 100, 6 = 20, p = 1. Hint: You may need the identity 2 b = b In(b) 4. Lastly, let's think about real life a bit more. Suppose 6 > M, intuitively how might this break down? (Hint: I am looking for something creative here) (5 pts.)