Application of Actuarially Fair Premiums to Health Insurance

Transcription

1 Application of Actuarially Fair Premiums to Health Insurance Ben Segel-Brown ( ) Econ 4460, November 2011 If sick, The budget function for a sick person is Finding the derivative Finding the maximum by setting the derivative to 0 Thus, it the sick person maximizes consumption equivalents; he will utilize medical services equal to 20 less double the cost of medical services.

2 a) Assuming initially that p is the same as P, the amount charged by providers for each unit of health services and that P=4, find the utilities of both well and ill persons, and compute the expected utility in the population, when the value of the parameter a is 0.5. This is the uninsured person s utility if they are well. If sick, It is given that We can find q by subbing this value into the maximization equation from a) Subbing these values (p=4, q=12) into the equation for x 2 above Assuming a = 0.5 This is the uninsured person s utility if they are sick.

3 To find expected utility for the population, we calculate the weighted sum of the utilities for each case This is the expected utility of the uninsured population. b) Now consider an insurance plan that pays 100% of the cost of a sick person s utilization of health services. Find the amount of health services that a sick person will use in this case. Continuing to assume P=4, find the actuarially fair premium that must be charged for such a plan. From a) we know that q = 20 p. Since p = 0, q = 20 A sick person will consume 20 units of health services when they are fully insured If P=4, then the costs for these 20 units will be The insurance costs will be 80 for each person that gets sick The actuarially fair premium is equal to the expected payout The actuarially fair premium would be 32

4 c) Continuing to assume that a=.5, find the expected utility of a person covered by such an insurance plan, taking into account that the values of x in each state has to be reduced by the amount of the insurance premium the person has to pay (continue to assume that the premium is actuarially fair). Without computing its value, what can you say about the gain from insurance when the person is covered by this plan? Substituting a =0.5 into the utility function we get that With the 100% insurance at an AFP of 32 This is the insured person s utility if they are well. If sick, From part c, p=0, q = 20 This is the insured person s utility if they are sick. Utility for sick (and well) if insured > EU if uninsured > > 26 The gains from insurance must be negative because both sick and well people have lower utilities than the expected utility of the uninsured. d) Answer the same questions as in b), c), and d), but now assuming that a=2. Comment on your answer. If a = 2

5 Returning to part b) (uninsured people) with this new utility function This is the uninsured person s utility if they are well. As before (given q, p and the relationships between them are unchanged) if sick, This is the uninsured person s utility if they are sick. This is the expected utility of the uninsured population. Because P, q, and the relationship between them and x has not changed, sick people will still consume 20 units of health care creating an actuarial fair premium of 32 This is the insured person s utility if they are well. As before for the fully insured (Given q, p and the relationships between them are unchanged p = 0, q = 20, AFP = 32, if sick:

6 This is the insured person s utility if they are sick This is the expected utility of the insured population. EU Insured > EU Uninsured > With the new utility function expected utility is higher for the insured so there are positive gains from insurance. e) Now consider an alternative insurance plan. Specifically, suppose the person is covered by a plan that does not pay for her health care costs, but simply transfers an amount of 64 to a person when she is ill. For a person covered by such a plan, what is the optimum of health services q that she will utilize when she is ill? Find the actuarially fair premium for this plan. The budget function for a sick person is Finding the derivative Finding the maximum by setting the derivative to 0

7 This shows that the transfer has not distorted demand (ie, there is no budget effect). Assuming that P=4 still holds The actuarially fair premium is equal to the expected payout, in this case the transfer The actuarially fair premium would be f) Returning now to the assumption that a=0.5, find the expected utility of a person covered by the plan described in f), again keeping in mind that the values of x in the two states will reflect both the premium payment and the transfer of money to sick individuals under the plan. Compare the expected utility with your answers in a) and d), and comment on the comparison. This is the insured person s utility if they are well. If sick, This is the insured person s utility if they are sick This is the expected utility of the uninsured population. EU of uninsured < EU of insured with 64 payout <

8 8.73 < The gains from insurance are positive with a lump sum payment whereas they were negative even for those who got sick with the 100% of health costs insurance. This occurs because the lump sum payment does not distort prices while the 100% of health costs insurance causes people to overconsume health care so everyone pays high premiums. Price of health care Graphical Representation of Insurance Inefficiency MPC P Cost per sick person (Uninsured or receiving lump sum) Additional Cost per sick Person (100% Insurance) MB Q a Quantity of Health Care