There are two states of the world: Bad(B) and Good (G) with probabilities 0.5 and 0.5, respectively. Alice's preferences over lotteries satisfies the axioms of expected utility theorem and her utility from her wealth w is w. Her initial wealth is w = 100. If bad state occurs, she looses L = 75 and her wealth is 25, and if the good state occurs her wealth remains 100. Alice has to decide how much insurance x [0, 100] she should buy. Let [0.5, 1] be the rate at which one unit of insurance can be purchased. Solve for the value of such that Alice's optimal level of insurance is zero at the level of you find, and for all values lower than the value you find, Alice will purchase positive amount of insurance, and for all higher values, Alice will not purchase insurance. Show your workings. [Hint: You can use the first order condition as we do in the two lectures we covered insurance example to find the value.]