8. Consider a market of risk-averse decision makers, each with a utility function U = √I. Each decision maker has an income of $90,000, but faces the possibility of a catastrophic loss of $50,000 in income. Each decision maker can purchase an insurance policy that fully compensates her loss. This insurance policy has a cost of $11,600. Suppose each decision maker potentially has a different probability q of experiencing the loss.

(6%) What is the smallest value of q so that a decision maker purchases insurance?