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13. Suppose the social benefit of an amount of health care J: is f(x) = 901: 10x2, and the social cost is 05) = 5x2 + 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 specified as . > - P (x) = 7r if x _ x O otherw15e where 7! is a payment amount and f is a specic 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 i? [This is \"a math problem.\"] (c) Show that doctors have no incentive to provide more care than 3?. ((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. : (e) Now suppose that that there are two kinds of patients, \"serious\" cases (S) and \"less serious\" cases (L). The cost of care is as above. Suppose that the health insurance system compensates doctors with 711 = 10 for an: 2 1 and Rs = 60 for x5 2 3. i. What are the doctor's per-patient profits for each type of patient? ii. If doctors could relabel patients as S or L, what do you think would hap- pen