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Homework Assignment

AI Summary

The assignment discusses risk aversion and demand for insurance in a health economics context. It presents expected values of income and utility functions, illustrating the concept of risk aversion through examples involving Peter, Tim, and Jay. The summary also touches on the concepts of convex and concave utility functions, as well as the implications of risk aversion on insurance purchasing decisions.

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Name:

Professor:

Course: PH226A Health Economics

Date:

Homework Assignment: Risk Aversion and Demand for Insurance

Q1a) Expected values of income and utility function:

E(I) = probability of being sick * sick income + probability of being healthy * normal income

= p*IS + (1 – p)*IH

Similarly,

E(U) = p*U(IS) + (1 – p)*U(IH)

Q1b) Chart:

0 100 200 300 400 500 600

0

5

10

15

20

25

30

35

40

45

50

Risk Averse Person Chart

U

IS

E(I)

IH

Income

Utility

M

Actual Utility with Insurance

Delta U

Expected

Loss in

Income

- Expected

Expected

Utility

Professor:

Course: PH226A Health Economics

Date:

Homework Assignment: Risk Aversion and Demand for Insurance

Q1a) Expected values of income and utility function:

E(I) = probability of being sick * sick income + probability of being healthy * normal income

= p*IS + (1 – p)*IH

Similarly,

E(U) = p*U(IS) + (1 – p)*U(IH)

Q1b) Chart:

0 100 200 300 400 500 600

0

5

10

15

20

25

30

35

40

45

50

Risk Averse Person Chart

U

IS

E(I)

IH

Income

Utility

M

Actual Utility with Insurance

Delta U

Expected

Loss in

Income

- Expected

Expected

Utility

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Q2a) Peter:

Expected income = 0.1*0 + 0.9*500 = 450

With insurance, income = 0.1*500 + 0.9*500 – 100 = 400

Thus, Peter loses (450 – 400 =) 50 on buying insurance, in an average year. Therefore, the

insurance is not fair for Peter.

If he does fall sick, income = 500 – 100 = 400.

Q2b) Tim:

Expected income = 0.2*0 + 0.8*500 = 400.

With insurance, expected income = 0.2*500 + 0.8*500 – 100 = 400.

Thus, Tim breaks even in an average year. His expected income does not change after buying

the insurance. Therefore, the standard contract is fair for him.

Q2c) Jay:

Expected income = 0.2*0 + 0.8*1000 = 800.

With insurance, expected income = 0.2*500 + 0.8*1000 – 100 = 800.

So, the income remains protected after buying the insurance. Therefore, the standard contract

is fair for Jay.

Q2d) False: Tim and Jay both protect their expected incomes. So, both gain more than Peter

who loses 50 on buying the standard insurance, on average.

Expected income = 0.1*0 + 0.9*500 = 450

With insurance, income = 0.1*500 + 0.9*500 – 100 = 400

Thus, Peter loses (450 – 400 =) 50 on buying insurance, in an average year. Therefore, the

insurance is not fair for Peter.

If he does fall sick, income = 500 – 100 = 400.

Q2b) Tim:

Expected income = 0.2*0 + 0.8*500 = 400.

With insurance, expected income = 0.2*500 + 0.8*500 – 100 = 400.

Thus, Tim breaks even in an average year. His expected income does not change after buying

the insurance. Therefore, the standard contract is fair for him.

Q2c) Jay:

Expected income = 0.2*0 + 0.8*1000 = 800.

With insurance, expected income = 0.2*500 + 0.8*1000 – 100 = 800.

So, the income remains protected after buying the insurance. Therefore, the standard contract

is fair for Jay.

Q2d) False: Tim and Jay both protect their expected incomes. So, both gain more than Peter

who loses 50 on buying the standard insurance, on average.

Q3a) Diagram of convex utility function:

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

0

50

100

150

200

250

300

Chart of Utility versus Income

Income

Utility

Q3b) Risk aversion and U”:

Insurance products are nearly risk neutral (U” = 0). The expected cost of insurance product is

supposed to be almost zero. In other words, the insurance cost and gains (utility) lie on a

straight line in U-I space.

A risk averse person has concave utility function (U” < 0), which gives higher utility to a

straight-line averaging method (see Figure in Q1a) and prefers lower income when faced with

uncertainty.

A risk affine person has convex utility function (U” > 0), which gives lower utility to a

straight-line averaging. Such a person would like more than fair return on the cost.

Q3c) True: Risk affine person does not prefer actuarially fair, full insurance contract and

would not buy corresponding insurance product.

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

0

50

100

150

200

250

300

Chart of Utility versus Income

Income

Utility

Q3b) Risk aversion and U”:

Insurance products are nearly risk neutral (U” = 0). The expected cost of insurance product is

supposed to be almost zero. In other words, the insurance cost and gains (utility) lie on a

straight line in U-I space.

A risk averse person has concave utility function (U” < 0), which gives higher utility to a

straight-line averaging method (see Figure in Q1a) and prefers lower income when faced with

uncertainty.

A risk affine person has convex utility function (U” > 0), which gives lower utility to a

straight-line averaging. Such a person would like more than fair return on the cost.

Q3c) True: Risk affine person does not prefer actuarially fair, full insurance contract and

would not buy corresponding insurance product.

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