Explain why there is little adverse selection in Employer based health insurance programs, compared to the non-group private market for health insurance. 2. Suppose that in a country with a generous unemployment insurance system, we see longer unemployment spells than in another that is less generous. Does this clearly indicate the program is too generous, and should be reduce? Explain why or why not. 3. Consider an insurance market with two types of people; healthy and sick. Both have wealth $10,000, and face a possible loss of $8,400, and ? have utility U (c) = c. Healthy people have a probability ph = .20 of suffering this loss, while sick people have a probablity ps = .40. (a) Calculate the certainty equivalent for the lottery each type of person faces. 5 points. (b) Calculate the most each time of person would pay for full insurance. 5 points. (c) Calculate the actuarially fair price for each type of person. 5 points. (d) If insurance companies can tell these types of people apart, what prices should they charge? 5 points. Suppose insurance companies cannot tell these types of people apart, but know that proportion ? = .90 of the population is healthy, while 1 ? ? = .10 is sick. (a) Is a pooling equilibrium possible? 5 points. Suppose there is a third type of person, who is very sick, and will incur the loss with certainty p = 1. (a) Calculate the most this person would pay for full insurance. 5 points. (b) Calculate the actuarial fair price for this type of person. 5 points. Suppose there is proportion of very sick people, were is small. There are therefore ?(1 ? ) = .9(1 ? ) healthy people, (1 ? ?)(1 ? ) = .1(1 ? ) sick people, and very sick people. 1 (a) What will happen in this market as rises from 0? (b) How large does have to be for the market to start to unravel? 4. Suppose that doctors must choose a level of care, x, for each of their patients. The cost of such care is c(x) = 5x2 + 5. The benefit of care (translated into dollars) is f (x) = 90x ? 10x2 . Suppose doctors have utility functions given by U (x) = (1 ? ?)P (x) + ?f (x) ? c(x) where P (x) is the monetary payment the doctor receives from treating a patient with x units of care. Doctors care about a weighted average of their own monetary revenue from care P (x) and the patients benefit f (x), with the weight given by the parameter ? ? [0, 1]. (a) What is the optimal level of care x? ? (b) Suppose that the Medicare system begins with retrospective payments. Under this scheme, doctors choose whichever level of x they feel is appropriate, and then the Medicare program gives the doctors $40 for every unit of x that they prescribe. i. Under these incentives, what amount of care will each doctor prescribe? How does your answer depend on ?? To what does ? = 0 correspond? To what does ? = 0 correspond? ii. In general, is the level of care too high or too low (the answer may depend on ?)? Why? For the remainder of this problem, set ? = 0. (a) Now suppose that the Medicare system switches to a prospective payment system. For our purposes, lets say that the prospective payment system compensates doctors by a fixed payment ? provided that x > x , where x  is some level set by the Medicare administrators. That is, the prospective payment system gives doctors a fixed dollar amount per patient, but requires that doctors provide at least a certain amount of care. i. If the Medicare prospective payment system wants doctors to provide precisely the optimal level of care, what will they set as x ? ii. Provided that Medicare wants doctors to care for all patients (and not send patients away), what should they set as ?? 2 iii. Demonstrate mathematically how doctors have limited incentive to provide care beyond x . iv. Show that the total health benefits under this system are actually slightly smaller than those for under retrospective payment. (b) Now suppose that that there are two types of patients divided into two diagnosis related groups (DRGs). Some patients are serious cases (S), and the other patients are not so serious (N). The cost of care is as above. Suppose further that the Medicare system compensates doctors with ?N = 10f orxN ? 1; and piS = 60f orxS ? 3. i. Calculate a doctors profits for each type of patient. ii. If doctors can re-label patients describe what will happen. iii. How didDRG Creep undo some of the gains of prospective payments, both in this model and in reality?

 

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