Question 1. (35%) On Monday morning Bob discovers mould growing in the bathroom of his apartment. The mould initially covers 2m (two square meters). Should Bob fail to clean the mould, it would grow to 3m on Tuesday, and 5m on Wednesday. Bob's choice is whether to clean the mould on Monday morning, on Tuesday morning, on Wednesday morning, or never. Cleaning the mould on a day in which it covers r square meters has a utility cost of r. Bob's girlfriend Alice is due to come to his apartment for dinner on Wednesday. Should Bob fail to clean the mould by the time Alice arrives for dinner, she would be shocked at the state of his bathroom and dump him, causing him a utility loss of 1000. Bob is a hyperbolic discounter with 8 = 0.5 and 6 = 1. 1. When (if at all) would Bob clean the mould if Bob is naive? Explain carefully. 2. When (if at all) would Bob clean the mould if Bob is sophisticated? Explain carefully. 3. Suppose that in the morning of Monday and Tuesday, Bob has a further option: asking Alice to reschedule and come for dinner earlier in the week. Doing this comes at a utility cost of 0.8 that is incurred on the day the dinner is rescheduled to. For example, on Monday morning Bob can either keep the dinner on Wednesday evening, or reschedule it to one of the two remaining evenings (Monday evening or Tuesday evening, depending on his preference). Suppose Bob decides (on Monday morning) to reschedule the dinner to Tuesday evening, he incurs a cost of 0.8 on Tuesday. The consequences of Alice's visit are as before: if the mould hasn't been cleaned by the time Alice arrives, Bob gets dumped. On which day would Bob end up holding the dinner? Does it matter if he is naive on sophisticated? Explain carefully