Economics of Uncertainty and Insurance

Transcription

1 Economics of Uncertainty and Insurance Hisahiro Naito University of Tsukuba January 11th, 2013 Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

2 Introduction This section talks about the economics under uncertainty and insurance First, it introduces the expected utility theory and then study the demand for insurance Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

3 Expected Utility Theory People buy insurance to prepare for the evens that does not happen with 100 percent probability. Examples are tra c accidence, health insurance for traveling in foreign countries, insurance for re etc. We need to study the behavior under uncertainty. Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

4 Expected Utility Theory (2) von-neumann and Oscar Morgenstein show that the human behavior under uncertainty can be explained as maximization of the expected utility maximization. Not that expected utility maximization is di erent from maximization of the utility of expected value. Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

5 Expected Utility Theory (3) Consider the following examples. Let the von-nueman Morgenstein utility function be u(c)(nm utility function) Assume that person s income become x 1 with probability p 1 and x 2 with probability 1 p 1. The expected value of income is p 1 x 1 + (1 p 1 )x 2 The utility from the expected value is u(p 1 x 1 + (1 p 1 )x 2 ) The expected utility is p 1 u(x 1 ) + (1 p 1 )u(x 2 ) Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

6 Expected Utility Theory (4) NM utility function u(c) is called risk-averse when u(c) is concave NM utility function is risk lover when u(c) is convex NM utility function is risk-neutral when u(c) is linear. Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

7 Expected Utility Theory (5) To see why, consider the following event. Assume that with probability of 0.5, this person has income 90. With probability of 0.5, this person has income, 110. The expected value of income is 100. How about the expected utility? Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

8 Expected Utility Theory (6) MN function is concave Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

9 Expected Utility Theory (7) NM utility function is convex Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

10 Expected Utility Theory NM function is linear Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

11 Expected Utility Theory Consider the following examples. Assume that u(c) = log c Then, ln(100)= ln(90)+0.5ln(110)= Thus, this consumer prefer certain event with the same expected value. On the other hand, assume that u(c) = c 2 Then, = = Thus, this consumer prefers uncertain events with the same expected values. Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

12 Expected Utility Theory In general, it is believed that people are risk-averter. Thus, it is reasonable to assume that NM utiltiy function is concave. In general, expected utility is written as p(s)(x s ) all s where p(s) is the probability that even s happens x s is the income when event s happens Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

13 Law of Large Number Now consider pooling of income risk. Suppose that there are two person. Person i has income x i which is random variable. The expected value of x i,e (x i ) = µ. The variance of x i is σ 2. Also assume that the uctuation of person A s income is independent of person B s income. In other words, cov(x 1, x 2 ) = 0. Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

14 Law of Large Number Now, suppose that person A and person B agree that they sum their income and divided it by them. This implies that each person income becomes x A+x B 2 The expected value of x A+x B 2 is E ( x A 2 ) + E ( x B 2 ) = µ 2 + µ 2 = µ The variance of x A +x B 2 = var( x A 2 + x B 2 ) = 1 4 var(x A) var(x B ) cov(x A,x B ) = 1 2 σ2 The expected value is the same and the variance becomes smaller. When there are N person and N persons s income is independent. Then the variance of shared income is Thus, the variance become smaller as N increases. This property is called the Law of Large Number. σ 2 N Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

15 Insurance Demand Law of large number is the motive of the supply side of insurance How about the demand side? Consider a case that an individual initially has $100. With probability p 1 some accident happen. If accident happens, his income x 1 which is lower than With probability 1 p 1, his income is still 100. Suppose that there is an insurance. The price of insurance(called premium) is pr. If a person purchase one unit of this insurance, he get $1 when he has an accident. Let y be the amount of insurance that he purchases. How much does he purchase? Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

16 Insurance Demand The consumer will maximize the following utility function and choose optimal y : p 1 u(x 1 pr y + y) + (1 p 1 )u(100 pr y) Now consider the expected pro t of insurance company Expected pro t of selling one unit of insurance is pr p (1 p 1 ) 0 If free entry is allowed for insurance company, the expected pro t should be equal to zero in the long run. Thus, pr = p 1 1 This is called actuarially fair premium. In other words, actuarially fair premium is the premium that will make the expected pro t of insurance company equal to zero. Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

17 Insurance Demand The optimal amount of insurance purchase must satisfy the following FOC: p 1 u 0 (x 1 pr y + y) (1 pr) + (1 p 1 )u(100 pr y) ( pr) = orp 1 u 0 (x 1 pr y + y) (1 pr) = (1 p 1 )u(100 pr y) pr When the insurance premium is fair, pr = p 1. Thus p 1 u 0 (x 1 + (1 p 1 ) y) (1 p 1 ) = (1 p 1 )u(100 p 1 y) p 1 u 0 (x 1 + (1 p 1 ) y) = u(100 p 1 y) x 1 + (1 p 1 ) y = 100 p 1 y y = 100 x 1 Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

18 Insurance Demand Thus, this individual purchases the insurance 100 x 1 units. With this insurance, when the accident happens, his income is x 1 (100 x 1 )pr x 1 = 100 p 1 (100 x 1 ) Without the accident, his income is 100 pr (100 x 1 ) = 100 p 1 (100 x 1 ) Thus, this consumer will choose the full insurance. This holds in general As long as insurance premium is fair and consumers are risk averse, consumers will choose the full insurance so that the consumption in di erent events become the same Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

19 State Dependent Consumption It is useful to analyze the demand of insurance in the state dependent consumption model Expected utility model nicely ts into the state dependent consumption In the economy, there are two state of nature Accident state and non-accidnet state. Consumption is contingent on those two state Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

20 State Dependent consumption(2) Accident state is denoted as subscript a and non-accident is denoted as subscript n. Consumption of accident state is denoted as c a. p(a) is the probability that the accident happens. 1 p(a) is the probability that accident does not happen u(c) is NM utility function. Assume that consumer is risk-averse which implies that u(c) is concave Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

21 State dependent consumption Expected utility is de ned as p(a)u(c a ) + (1 p(a))u(c n ) Because u(c) is concave, the indi erence curve of (c a, c n ) is convex to the origin. It is useful to calculate the MRS at the 45 degree line. Measure the c n on the horizontal line. Measure c a on the vertical line Then MRS is (1 p(a))u 0 (c n ) p(a))u 0 (c a ) At the 45 degree line, c n = c a. The slope of MRS=(1-p(a))/p(a) Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

22 Budget set Let x be the initial income. D is the damage in the case of accident. Let pr be the premium of the one unit of insurance. y be the amount of insurance c n = x pr y c a = x D pr y + y = x D + (1 pr)y c a = x D + (1 pr)(x c n ) 1 pr Assume that the free entry of insurance market. In this case, pr = p(a) The expected pro t of insurance company is equal to zero pr y + p(a) y = 0 p(a) = pr Thus, the budget constraint becomes Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

23 Budget set (2) The budget set pass through (x-d, x) The absolute value of the slope is (1-p(a))/p(a) Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

24 Expected Utility maximization From the graph, the utility maximization is achieved at the 45 degree line c n = c a Full insurance is achieved Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

25 Expected Utility maximization From the graph, the utility maximization is achieved at the 45 degree line c n = c a Full insurance is achieved Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

26 Problems in Insurance Market Generally, there are two types of problems in the insurance market This is due to the asymmetric information between insurer and insuree Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

27 Moral Hazard Often, insurance market, insurer does not completly observe the behavior of the insuree This unobservability cause the so called moral hazard in insurance market Due to the moral hazard problem, private insurance companies introduce some mechanism to reduce the moral hazard behavior. Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

28 Adverse selection Consider an insurance market where there are several times of consumers: high-risk, middle-risk, low-risk. The probablity that accident happens is relatively low to low risk consumers. The probability that accident happens is relatively high to high risk consumer. Middle risk consumer is between two types Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

29 Adverse Selection (2) Suppose that initially three types of consumer joins the insurance and insurance company cannot distinguish three types. Then, insurance company will change the premium based on the average of those three groups. However, such an insurance is not attractive for low risk people. Thus, low risk consumer will opt out. Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

30 Adverse Selection (3) Then, insurance premium will go up. Then, the premium is not attractive even for middle risk people. Thus, the middle risk consumer will opt out. As a result, only the high risk consumers remain in the market. Other consumers will opt out and the market for those consumers do not exit. This is called adverse selection Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31

31 Public Policy Implication of Moral Hazard and Advese Selection Adverse Selection justi es the public intervention of the insurance market Why? Moral hazard problem suggests that full insurance is the best policy Why? Hisahiro Naito (University of Tsukuba) Economics of Uncertainty and Insurance January 11th, / 31