Question

Suppose that an economist has a utility function U =
(Income)^{0.25}. Her income is $65K a year, but there is a
10 percent chance of becoming ill and making only $57K.

(a) What is her expected utility if she does not have
insurance?

(b) What is the actuarially fair insurance premium?

(c) How much is she willing to pay for insurance?

Answer #1

a. The calculation of expected utility (U) is as follows:

0.9 × (Income)^{0.25} + 0.1 ×
(Income)^{0.25}

0.9 × ($65,000)^{0.25} + 0.1 × ($57,000)^{0.25}
= 15.91.

b. The actuary fair insurance premium = 0.1 × ($65,000 - $57,000) = 0.1 × $8,000 = $800.

c. The amount that she is willing to pay for insurance is calculated as follows:

Expected Utility (U) = (Income)^{0.25}

or, 15.91 = (Income)^{0.25}

or, Income = $64,074.

Therefore, the amount that she is willing to pay for insurance = Income - Expected Utility Income under no insurance = $65,000 - $64,074 = $926.

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