3. A patient lives for two periods. 1 and 2. Her wellheing in period 2 depends on her state of health as well as some action t 1:- 1] taken in period 1. Suppose the patients state of health can he represented by a real muuber a :3 I}. For example. higher numbers could represent better health and higher potential lifetime utility. The patient's initial belief is that s = s1 = 25 with probability onehalf and a = s; = 3,5 with probability one-half. The patient derives utility from two sources. First. she would like to take the appropriate action. Formally. if her state of health is a and she takes action 1': her \"instrumental utility" is |s t|. This means that in terms of instrumental utility. it is optimal to align the action perfectly with the state to set E = 5. Because lower values of 3 represent worse health. corresponding low vaJues of 1' could represent taking health problems more seriously, for instance by having a better diet or exercising; in other words. a more serious health condition calls for a more serious response. Second. the person derives anticipatory utility from her beliefs. Her anticipatory utility depends on the average state of health given her heliefs. Specically, if she thinks the probability of the state 31 is p. then her anticipatory utility is 213w ps1 + {l p}.sg. The patient's overall utility. which combines expected instru mental utility and anticipatory utility. is then 2\" p51+{1pie -if'l-'?1-i'I-{1-Pl|52 fl The patient has the option of visiting a doctor in period 1 to get diagnosed. The visit is free. If she visits the doctor, she will know the true 5 with certainty. If she does not visit the doctor: she does not learn any information about .s. and will keep believing that the probability of state at is onehalf. If she visits the doctor. she can choose t after the visit. (a) [4pts] Write the patient's expected utility as a function of t if she does not visit the doctor. What t or range of t does she choose? What is her expected utility given the optimal t? (b) [4pts] Write the patient's utility as a function of t if she visits the doctor and learns that her state of health is s, = 25. What t does she choose? What is her utility given the optimal t? (c) [4pts] Repeat the exercise in part (b) for the case when the patient visits the doctor and learns that her state of health is $2 = 36. (d) [4pts] Write the patient's total expected utility from visiting the doctor. This is the weighted sum of the utilities in parts (b) and (c), with the weights equal to the probabilities of the two possible states of health. Does the patient visit the doctor? (e) [10pts] Now suppose that $1 = 0, so that the patient's possible problem is more serious. The other possibility is still $2 = 36, with the two health states still being equally likely. Using the same steps as in parts (a) through (d), solve for whether the patient goes to the doctor. (f) [4pts] Conventional economic wisdom says that when information is more im- portant for making decisions such as above, when a patient's health problem is potentially more serious-a person is more likely to seek out that informa- tion. How does the consideration of anticipatory utility qualify this insight