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13. Suppose the social benefit of an amount of health care x is f(x) = 90x 10x2, and the social cost is C(x) = 5:4:2 + 5. Suppose doctors have utility functions U(x) = P(x) C(x) where x is the level of care the doctor provides. P(x) is the monetary payment the doctor receives, and is specied as 71' if x > f P = _ (x) {0 otherwise where n is a payment amount and f is a specific level of care, both set by the insurer. (a) Is this a prospective payment system or a retrospective system? Explain why. (b) If the insurer wants doctors to provide the efficient level of care, what will it set as f? [This is \"a math problem.\"] (c) Show that doctors have no incentive to provide more care than 3?. [You may show this \"mathematically" or in words; just be sure your argument is rigor- ous!] ((1) Which system do you think would lead to more care being offered: the prospec- tive or retrospective system? Explain in two or three sentences. [Note: You do not need the math to answer this part] (e) Now suppose that that there are two kinds of patients, "serious\" cases (8) and "less serious\" cases (L). The cost of care is as above. Suppose that the health insmance system compensates doctors with m = 10 for xL 2 1 and 715 = 60 for x5 2 3. i. What are the doctor's per-patient prots for each type of patient? ii. If doctors could relabel patients as S or L, what do you think would hap- pen