PH226A Health Economics Homework

This homework assignment focuses on risk aversion and the demand for insurance in health economics. The assignment includes several questions. Question 1 explores expected values of income and utility functions, and involves creating a chart. Question 2 presents scenarios with different individuals (Peter, Tim, Jay) and analyzes whether a standard insurance contract is fair for each, based on their expected income with and without insurance. Question 3 delves into risk aversion, examining the characteristics of utility functions for risk-averse, risk-neutral, and risk-affine individuals and their preferences for insurance contracts. The solutions provided demonstrate calculations of expected income and utility, analysis of fairness of insurance contracts, and explanations of risk preferences and their relationship to utility functions.

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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

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.

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 Uti lity 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.