1. A patient lives for two periods, 1 and 2. Her well-being in period 2 depends...

     

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1. A patient lives for two periods, 1 and 2. Her well-being in period 2 depends on her state of health, 20, where la grnumbers me an better he alt h as well as s o me health-related action t20 which is taken in period 1, but has a health impact in period 2. The patient derives utility from two sources. First, she gets instrumental utility in period 2 from having her heath behavior match her health state. Formally, her in- strumental utility isst. This means that in terms of instru nental utility, it is optimal to align the action perfectly with the state to set ts. As an example, lower values of t could represent taking health problems more seriously, for instance by having a better diet or exercising; then, the above instrumental utility means that amoreseñoushealthconditioncalls for amore seriousresponse (Ifsimportant to notice that in this model the action, t, does not affect the healthstate, it just a ffects the person's wellbeing, given their health state.) Second, the patient derives anticipatory utility in period 1 from her beliefs about her health condition in period 2. The patient's initial belief is that with probabilityp = 0.5 her health state will be s1 =25, and withprobabilityl - p = 0.5 it will be s₂ = 36. Her anticipatory utility, which depends on her expected health state given her beliefs, is 20₁√p³₁+(1-P)³₂- The patient's expected total utility in period 1, which combines expected instrumental utility in period 2, plus anticipatory utility in period 1, is thus 20 √/p3, +(1-P) ³₂ -p s₁ -t - (1-p) |s, - t. V In period 1, the patient has the option of visiting a doctor to get diagnosed. The visit is free, and will enable her to know the true future value of with certainty. (In other words, her belief about P will go from P = 0.5 to either P = 0 or p= 1. Ifs hedos not visit the doctor, she will not learn any information about s, and will keep believing that the two states are equally likely, p=0.5. After deciding whether to go to the doctor, and after getting the diagnosis if she does go, the patient then chooses what health action, t, totake. (a) Write the patient's expected total utility in period 1, as a function of t,ifs he does not visit the doctor. What does she choose? What is her expected total utility given the optimal t? (b) Write the patient's expected total utility as a function of t if she visits the doctor and gets a bad diagnosis, p = 1, so her future healthstate is s₁ =2 5. What t does she choose? What is her utility given the optimal t? (c) Repeat the exercise in part (b) for the case where the patient visits the doctor and gets a good diagnosis, P= 0,soherfuture healthstateiss, = 36. (d) Write the patient's expected total utility from deciding to visit the doctor, not knowing which diagnosis she will get. This is the weighted sum of the utilities in parts (b) and (c), with the weights equal to the probabilities of the two possible diagnoses. Will the patient choose to visit the doctor? (e) Now suppose that s1 = 0, so that the patient's possible negative diagnoss is more serious. The other possibility is still s, -36, withthe two healthstatesstill being equally likely. Using the same steps as in parts (a) through (d), solve for whether the patient goes to the doctor. (f) Conventional economic wisdom says that when information is more important for making decisions such as above, when a patient's problem is potentially more serious a person is more likely to seek out that information. Thus, simply making information available about health risks and the fect of health behaviors is an optimal public policy. How does the consideration of anticipatory utility alter this conventional public-policy wisdom? 2. Consider the "Freddy" model of the representativeness heuristic from class, and le= 8. Suppe Freddyo ber ves quarters of perfor na me by mu alfundna rager Helga. Helga may be skilled, mediocre, or unskilled. A skilled mutual fund manager has a 3/4 chance of beating the market each quarter, a mediocre manager has a 1/2 chance of beating the market each quarter, and an unskilled manager has al/4 chance of beating the market each quarter (and Freddy knows all this). In reality, the performance of a manager is independent from quarter to quarter. (a) Suppose first that Freddy thinks Helga is mediocre. What does Freddy think is the probability that Helga beats the market in the first quarter? Suppose that she does actually beat the market in the first quarter. What does Freddy think is the probability she does it again? Suppose that she beats the market again. What does Freddy think is the probability that she will do so a third time? (b) How do the three probabilities in part (a) relate to each other? What phenomenon does this reflect? (c) Now suppose that Freddy does not know whether Helga is skilled, mediocre, or unskilled. He has just observed three consecutive quarters of below-market per- formance by Helga. Can he conclude which type of manager Helga is? Can he rule out any type? Explain the intuition. (d) How many more quarters of below-market performance does Freddy need to ob- serve to be sure of Helga's type? (e) This part asks you to derive what Freddy concludes about the proportion of skilled, mediocre, and unskilled managers in the population when he observes the per- formance of a large sample of mutual-fund managers over two quarters. Suppose that in reality allmanagersaremediocre i. What proportion of managers will have two above-market performances? Two below-market performances? Mixed performances? This is what Freddy ob- serves. Gelif ii. Suppose Freddy thought that the proportion of skilled, mediocre, and un- skilled managers in the population was 9, 1-29, anda, respectivdy What does Freddy expect should be the proportion of managers who show two above-market performances in a row? iii. Given your answers to the previous two parts, what does Freddy deduce is the proportion of skilled managers in the population? Give an intuition for your answer. culty of explaining to a basketball (f) Explain intuitively how part (e) relates to the fan that there is no such thing as a hot hand. 1. A patient lives for two periods, 1 and 2. Her well-being in period 2 depends on her state of health, 20, where la grnumbers me an better he alt h as well as s o me health-related action t20 which is taken in period 1, but has a health impact in period 2. The patient derives utility from two sources. First, she gets instrumental utility in period 2 from having her heath behavior match her health state. Formally, her in- strumental utility isst. This means that in terms of instru nental utility, it is optimal to align the action perfectly with the state to set ts. As an example, lower values of t could represent taking health problems more seriously, for instance by having a better diet or exercising; then, the above instrumental utility means that amoreseñoushealthconditioncalls for amore seriousresponse (Ifsimportant to notice that in this model the action, t, does not affect the healthstate, it just a ffects the person's wellbeing, given their health state.) Second, the patient derives anticipatory utility in period 1 from her beliefs about her health condition in period 2. The patient's initial belief is that with probabilityp = 0.5 her health state will be s1 =25, and withprobabilityl - p = 0.5 it will be s₂ = 36. Her anticipatory utility, which depends on her expected health state given her beliefs, is 20₁√p³₁+(1-P)³₂- The patient's expected total utility in period 1, which combines expected instrumental utility in period 2, plus anticipatory utility in period 1, is thus 20 √/p3, +(1-P) ³₂ -p s₁ -t - (1-p) |s, - t. V In period 1, the patient has the option of visiting a doctor to get diagnosed. The visit is free, and will enable her to know the true future value of with certainty. (In other words, her belief about P will go from P = 0.5 to either P = 0 or p= 1. Ifs hedos not visit the doctor, she will not learn any information about s, and will keep believing that the two states are equally likely, p=0.5. After deciding whether to go to the doctor, and after getting the diagnosis if she does go, the patient then chooses what health action, t, totake. (a) Write the patient's expected total utility in period 1, as a function of t,ifs he does not visit the doctor. What does she choose? What is her expected total utility given the optimal t? (b) Write the patient's expected total utility as a function of t if she visits the doctor and gets a bad diagnosis, p = 1, so her future healthstate is s₁ =2 5. What t does she choose? What is her utility given the optimal t? (c) Repeat the exercise in part (b) for the case where the patient visits the doctor and gets a good diagnosis, P= 0,soherfuture healthstateiss, = 36. (d) Write the patient's expected total utility from deciding to visit the doctor, not knowing which diagnosis she will get. This is the weighted sum of the utilities in parts (b) and (c), with the weights equal to the probabilities of the two possible diagnoses. Will the patient choose to visit the doctor? (e) Now suppose that s1 = 0, so that the patient's possible negative diagnoss is more serious. The other possibility is still s, -36, withthe two healthstatesstill being equally likely. Using the same steps as in parts (a) through (d), solve for whether the patient goes to the doctor. (f) Conventional economic wisdom says that when information is more important for making decisions such as above, when a patient's problem is potentially more serious a person is more likely to seek out that information. Thus, simply making information available about health risks and the fect of health behaviors is an optimal public policy. How does the consideration of anticipatory utility alter this conventional public-policy wisdom? 2. Consider the "Freddy" model of the representativeness heuristic from class, and le= 8. Suppe Freddyo ber ves quarters of perfor na me by mu alfundna rager Helga. Helga may be skilled, mediocre, or unskilled. A skilled mutual fund manager has a 3/4 chance of beating the market each quarter, a mediocre manager has a 1/2 chance of beating the market each quarter, and an unskilled manager has al/4 chance of beating the market each quarter (and Freddy knows all this). In reality, the performance of a manager is independent from quarter to quarter. (a) Suppose first that Freddy thinks Helga is mediocre. What does Freddy think is the probability that Helga beats the market in the first quarter? Suppose that she does actually beat the market in the first quarter. What does Freddy think is the probability she does it again? Suppose that she beats the market again. What does Freddy think is the probability that she will do so a third time? (b) How do the three probabilities in part (a) relate to each other? What phenomenon does this reflect? (c) Now suppose that Freddy does not know whether Helga is skilled, mediocre, or unskilled. He has just observed three consecutive quarters of below-market per- formance by Helga. Can he conclude which type of manager Helga is? Can he rule out any type? Explain the intuition. (d) How many more quarters of below-market performance does Freddy need to ob- serve to be sure of Helga's type? (e) This part asks you to derive what Freddy concludes about the proportion of skilled, mediocre, and unskilled managers in the population when he observes the per- formance of a large sample of mutual-fund managers over two quarters. Suppose that in reality allmanagersaremediocre i. What proportion of managers will have two above-market performances? Two below-market performances? Mixed performances? This is what Freddy ob- serves. Gelif ii. Suppose Freddy thought that the proportion of skilled, mediocre, and un- skilled managers in the population was 9, 1-29, anda, respectivdy What does Freddy expect should be the proportion of managers who show two above-market performances in a row? iii. Given your answers to the previous two parts, what does Freddy deduce is the proportion of skilled managers in the population? Give an intuition for your answer. culty of explaining to a basketball (f) Explain intuitively how part (e) relates to the fan that there is no such thing as a hot hand.