Please answer questions below:

1 Insurance Consider an agent with wealth w who faces probability p of incurring a loss L. She can insure against this loss by buying a policy that will pay out in the event the loss occurs. A policy that will pay a in the event the loss occurs costs qa dollars. 1. Write an expression for her wealth if the loss occurs and another expression for her wealth in the event that it does not occur. 2. Suppose her utility function is isolelastic, meaning we can write it u(w ) = why - 1 1 -y (1) Where y is the constant of relative risk aversion. Assume that y 2 0 and y * 1. Given that her goal is to pick the optimal level of insurance coverage a, write her optimization problem. 3. Find the first order condition of the optimization problem. 4. Check the second order condition - are the first order conditions sufficient for a solution? 5. Use the first order condition to solve for the optimal quantity of insurance a. 6. Based on your formula for a, how does a change in y affect the amount of insurance purchased? Assume q 2 p, meaning the insurance is priced to make at least zero profits. (Hint: you can think of your formula for a as a function of y - take the derivative of a(y) with respect to y to see how a changes when y changes). 7. Explain what you've found in words. Does it make sense intuitively?2 Portfolio Choice Consider an agent with a wealth to who has to decide how to invest it. She has two choices: to keep in in cash, in which case it will neither grow nor shrink, or to invest it in a startup. There is a 5% chance the startup will grow to 50x the initial investment and a 95% chance it will fail, meaning she will lose her entire investment in it. Her goal is to pick a quantity a to invest in the asset. 1. Write an expression for her wealth in the event that the startup succeeds and in the event that it fails. These are the two \"states of the world' we are interested in for this problem. . Assume her utility function is given by ln(w) and that she is an expected utility maximizer. Write her optimization problem. . Find the rst order condition of the optimization problem. . Use the rst order condition to the optimization problem to nd a formula for the optimal startup investment. . Now, suppose that instead of being an expected utility maximizer, she makes decisions in a way that is consistent with cumulative prospect the- ory. Discuss how reference points (you'll also have to decide what reference point should be), probability weighting, and loss aversion might each affect her decision