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Health care quantity q gives a patient a benefit equal to q", where 0

Health care quantity q givcs a patient a benefit equal to where O < < l. The cost Of q units Of health care quantity is loq. The social surplus is defined to be benefit less cost Of quantity: qa loq. i) What is the quantity, say that maximizes social surplus? Suppose that a physician chooses the quantity q for the patient. The physician is responsible for the quantity cost, and is paid a fixed p. ii) What is the physician's profit when he chooses q unit Of health care quantity for the patient? NOW the physician's utility depends on profit, say (the solution to ii) above) and the patient's benefit in the following form: Irla x qn. That is, his utility is the product Of profit raised to the power I and patient benefit raised to the power a. iii) Given the fee p, what is the physician's quantity choice that maximizes his utility? iv) Find the fee at which the physician's utility-maximizing quantity Will equal , the solution in i). Finally, suppose that the phyician is only rcsponsible for half Of the cost, so when he chooses quantity q his cost is only 5q. v) Find another fee under which the physician's utility-maximizing quantity is also