代写EC 979 Health Economics 2024 Problem Set 2调试R语言程序

日期：2024-08-22 06:26

Health Economics 2024

EC 979

Problem Set 2

1. (10 marks each part)

A doctor needs to decide on the treatment to provide to their patients. There are two possible treatments: A and B. Each patient that the doctor sees suffers from one (and only one) of two illnesses: a or b. The probability that a patient suffers from illness a is 0.5 and the probability that they suffer from illness b is 0.5. Assume that treatment A is the “right” treatment for patients with illness a and treatment B is the “right” treatment for patients with illness b. More specifically, if the patient receives the “right” treatment for their illness, the probability that the treatment will succeed is 0.75 and the probability that the treatment will fail is 0.25, whereas if the patient receives the “wrong” treatment, the treatment will fail with certainty. In order for the doctor to diagnose the patient’s illness, they need to invest effort that has a cost of 10. If the doctor invests that effort, they will correctly diagnose the patient’s illness, whereas if they do not, the cost they incur will be 0 but they will not be able to diagnose the patient’s illness. Assume that the doctor can choose whether or not to exert the effort, and if they do not, they will choose one of the two treatments randomly.

Assume that there is a health plan that contracts with doctors to treat patients. The doctor’s objective is to maximize their expected utility.  Assume that the doctor’s utility from treating a patient is given by:

where w is the payment the doctor receives for each patient and Ce is the doctor’s cost of effort (that is Ce = 0 if the doctor exerts no effort and Ce = 10 if the doctor exerts the effort). Finally, assume that the doctor can always refuse to treat a patient and obtain an expected utility of U = 40 from some other activity.

(a) Assume that neither the doctor’s effort, nor the treatment outcome is verifiable, and thus the plan can only offer the doctor a lump sum payment per patient. What is the lowest lump sum at which the doctor will agree to treat patients? What is the probability that the treatment will succeed in this case?  

(b) Assume now that treatment outcome is verifiable, but the doctor’s effort is not (that is, the payment to the doctor can depend on whether or not the treatment succeeded but it cannot depend on the doctor’s effort). The health plan’s objective is that the doctors will provide the “right” treatment to every patient at the lowest cost to the health plan. Describe the contract that the health plan should offer the doctor in this case.  

(c) Assume now that the doctor’s effort is also verifiable. Here too assume that the health plan’s objective is that the doctors will provide the “right” treatment to every patient at the lowest cost to the health plan. Describe the contract that the health plan should offer the doctor in this case.  

(d) Assume now that not only the doctor, but also the health plan can invest the effort (incurring the same cost as the doctor) in order to diagnose the patient’s illness and then inform. the doctor. For each of the three cases above, should the health plan invest that effort?

2. (15 marks each part)  

An individual can be either healthy or sick.  With probability 0.75 the individual will be healthy and in this case their utility function will be given by , where c is their level of consumption, and with probability 0.25 the individual will be sick, and their utility will be given by:   where x is the amount of health services they consume. Assume that the market price of each unit of consumption is 1 and that of each unit of health care is also 1. The individual’s income is I = 100 and they can choose any level of consumption they wish (subject to their budget constraint) but they can choose only one of three levels of health services: x= 0 or x=36 or x=64 (that is, they cannot choose any other level of x).

Assume that individuals choose their level of health services only if and after they become ill.  

(a) What level of health care services will the individual purchase if they are not insured?

(b) What is the actuarially fair premium for full insurance?

(c) Is there a moral hazard problem if the individual purchases full insurance?

(d) Will the individual purchase full insurance at the actuarially fair premium if they can choose only between being fully insured or not being insured at all?