Question

Suppose that Elizabeth has a utility function U= (or U=W^(1/3) ) where W is her wealth and U is the utility that she gains from wealth. Her initial wealth is $1000 and she faces a 25% probability of illness. If the illness happens, it would cost her $875 to cure it.

1. What is Elizabeth’s marginal utility when she is well? And when she is sick? Is she risk-averse or risk-loving?
2. What is her expected wealth with no insurance?
3. What is her expected utility with no insurance?
4. What is the actuarially fair premium (expected value of my loss)?
5. What is the most I would be willing to pay to shed the risk?

Tip: using a general graph of the certain and the expected utility could be helpful

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Answer 1

U= W^(1/3)

W= 1000

P(illness)= 25% (1/4)

P(well) = 75% (3/4)

Cost of cure= $875

MU= dU/dW = 1/3*W^(-2/3) = 1/300 (when she is well)

=1/25 (when she is sick)

Expected value when she is well= 3/4*1/300= 1/400

Expected value when she is sick= 1/4*1/25= 1/100

She is risk-averse.

Expected wealth with no insurance= 1000*3/4= 750

Expected utility with no insurance= 750^(1/3)

The expected value of loss= 875*(1/4)= 218.75