HW 4 -Beavin - hw#4 PDF

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Unit 4: Homework ECO 765 Suppose Bob has income of $18,000. There is a 20% chance that Bob will get sick and have to spend $10,000 of his income on a treatment. Suppose Bob’s income-utility relationship is given by: Where I am Bob’s income.푈(퐼) = √퐼a. Use the facts above to complete the table below ...

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Unit 4: Homework ECO 765 1. Suppose Bob has income of $18,000. There is a 20% chance that Bob will get sick and have to spend $10,000 of his income on a treatment. Suppose Bob’s income-utility relationship is given by: Where I am Bob’s income. � (�) = √� a.

b.

Use the facts above to complete the table below to find Bob’s utility for various levels of income, along with his marginal utility associated with increases in his income. Use the given values to help you get started. What do you notice about the shape of this curve? a. The curve of the utility function implies that this person is risk averse, according to the text a sure amount would always be preferred over a risky bet having the same expected value 160.0 140.0 120.0 Utility

100.0 80.0 60.0 40.0 20.0 0.0 0

5000

10000

15000

20000

Wealth

c.

b. What does this imply about Bob’s attitudes towards risk? Bob is Risk Averse (Risk Avoiding).

Utility

Marginal Utility = change in utility per extra $2,000

0

0.0

0.0

2000

44.7

44.7

4000

63.7

18.5

6000

77.5

14.3

8000

89.4

11.9

10000

100.0

10.6

12000

109.5

9.5

14000

118.3

8.8

16000

126.5

8.2

18000

134.2

7.7

20000

141.4

7.22

22000

148.3

6.88

Income

2.

3.

4.

What is Bob’s expected income, given that he faces a chance of illness? What is the actuarially fair premium for Bob’s circumstance?  Bob's income=$18,000  probability of illness=0.2  probability of being healthy=0.8  expenses during illness=$10,000  Bob's expected income=0.2*($18,000-$10,000)+0.8*($18,000)  = 0.2*$8,000+0.8*$18,000  = $1,600+$14,400  =$16,000  Actuarially fair premium=0.2*$10,000=$2,000 How happy is Bob if he doesn’t buy any insurance? (Hint: expected utility!) Compare this to how happy Bob is if he can buy full insurance at the actuarially fair premium....will Bob want to buy the insurance?  b. Bob's expected utility with insurance=0.2 sqrt18000-2000 +0.8√18000  =0.2* √16000 + 0.8√18000  =0.2*126.5+0.8*134.2=132.66  Bob's expected utility with no insurance=0.2*√8000 + 0.8√18000  =0.2*89.4+0.8*134.2  =125.24  As Bob's expected utility is more when he takes insurance of fair premium so, he should take definitely take the insurance policy

Complete the following table to gain further insights and then find the maximum Bob would be willing to pay for full insurance. Does this suggest that an insurance market can exist? Probability of getting sick

Expected Income 1

8000

Utility of expected income

Expected utility

89.4

89.4

0.9

9000

94.9

93.9

0.8

10000

100.0

98.4

0.7

11000

104.9

102.9

0.6

12000

109.5

107.3

0.5

13000

114.0

111.8

0.4

14000

118.3

116.3

0.3

15000

122.5

120.7

0.2

16000

126.5

125.2

0.1

17000

130.4

129.7

0

18000

134.2

134.2

5.

Illustrate Bob’s situation graphically, showing all of the most important features. You don’t have to show every single data point...just enough to convey the main ideas (you can do this by hand or create a graph in Excel)

136.0

126.0

116.0

106.0

96.0

86.0 8000

10000

12000

14000

Utility of expected income

6.

16000

18000

Expected utility

Suppose an insurance company offers a standard contract to everyone. The policy charges a premium of $2,100 and pays out $8,500 to you if you get sick. From Bob’s perspective, is this contract fair? Full? Explain using the definitions of fair and full insurance contracts in the text. 

This is considered a fair contract, as it covers some aspect of the damage, but it doesn’t cover the entire damage. As fair insurance covers some percentage of the damage made but not the entire amount fully only up to $8500.

7.

The Akerlof model and its lessons can be directly applied to health insurance markets. In health insurance markets, who is analogous to the car buyers? The car sellers. What would it mean for the health insurance market to “unravel”? Explain what this would look like.  The insurance firms are analogous to car buyers and the insurance customers are the car sellers. This market would unravel if no insurance company is willing to offer an insurance contract at any premium for fear of attracting the sickest population to their plans. This is similar to buyers refusing to buy cars at any price for fear of buying care that consumers will buy bad cars.

8.

Who is harmed by asymmetric information in insurance markets? Who is helped? Explain. 

9.

Insurance companies are harmed by the asymmetric information, in the health insurance market, the patient has more knowledge of their medical needs which gives them an advantage to buying the health insurance. Policy holders are helped by this and insurance companies are harmed.

Suppose private markets offered insurance contracts that committed people to lifetime insurance. For instance, suppose you purchase a policy when you are young, the premiums are fixed for life, but you must commit to holding the policy your entire life. The insurance company must pay your medical bills for life. What are pros and cons of such a policy? Do you think these sorts of contracts would work in the real world? Why or why not?



These policies definitely have both pros and cons, this policy can lead to a fall in the cost of the fixed premium year after year, or a premium amount as bob ages A pro would be for instance, if Bob signs a contract young and healthy, he will pay the premiums every year for the rest of his life, no matter if Bob is diagnosed with a chronic disease or if he is perfectly healthy. If Bob remains extremely healthy throughout his life, then that means he may want to change insurance companies to pay a smaller premium but will be unable to drop the insurance policy. The reason this would not work in the real world is because the nature of the contract also prevents competition, because these two customers are stuck with their insurer for life then no other company can get their business....