## Question from Past Microeconomics Qualifying Exam

Fall 2003 - Section II, Question one, George Mason University

Suppose your EU=w0.4, and insurance is sold at 20 percent more than actuarially fair rate. Your uninsured income is \$100,000 with p=.5, and \$50,000 with p=.5. Solve for your optimal quantity of insurance, i.

At actuarially fair rates, purchasing \$i of insurance will cost 1.2(0.5)i = 0.6i. In order to solve for the optimal quantity of insurance, we need to maximize EU in the following setup:

max EU, for EU=0.5(100,000 - 0.6i)0.4 + 0.5(50,000 - 0.6i + i)0.4

max EU, for EU=0.5(100,000 - 0.6i)0.4 + 0.5(50,000 + 0.4i)0.4

(differentiating with respect to i)

0.5(0.4)(-0.6)(100,000 - 0.6i)-0.6 + 0.5(0.4)(0.4)(50,000 + 0.4i)-0.6 = 0

(-0.12)(100,000 - 0.6i)-0.6 + (0.08)(50,000 + 0.4i)-0.6 = 0

(after some algebra)

(100,000 - 0.6i)-0.6 = (2/3)(50,000 + 0.4i)-0.6

100,000 - 0.6i = 1.2754(50,000 + 0.4i)

36228.8 = 1.11i

i = 32638.5

This risk-averse agent will purchase \$32,638.50 worth of insurance.

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