If an individual with initial wealth w that is facing a random risk X that takes values è with probability p and value zero with probability 1 - p. If the individual does not take insurance, his wealth will be w - X. If he takes insurance, his wealth will be w - a,
where a is the insurance premium. Suppose that the investor has utility function

u(y) = 1 - exp(-ãy)

1. How can I show that the certainty equivalent of not taking insurance is = 1 - exp(-ãw)[p exp(ãè) + 1 - p]. and that the certainty equivalent of taking insurance is? = 1 - exp(-ãw) exp(ãè).

2. What is the largest premium that the investor will be willing to pay?

3. How does the premium change as the parameter ã increases?