# Expected Utility: Optimal Insurance

Expected Utility: Optimal Insurance

A lottery is played in which a person stands to obtain 1 or 10 units of wealth with probabilities and , respectively. A logarithmic utility function of wealth, , is plotted in dark blue. The blue chord represents the expected utility of the lottery, . Insurance that pays out in the event of the unfavorable outcome of obtaining only 1 unit of wealth can be purchased. The cost of the insurance is the price of the insurance times the desired level of . The orange chord represents the expected utility of the insured lottery. The optimal level of insurance yields the maximum expected utility possible, represented by the green dot.

p

1-p

u(x)

E(u(x))=pu(1)+(1-p)u(10)

q

q