2. Consider the model of insurance introduced in class. Suppose that an individual has a Bernoulli utility given by u(x) = ln(x), initial income I > 0, and a p > 0 chance of an accident that results in a loss of 0 < d < I. As before, let case (state) 1 be the accident case, and assume that there is an insurance contract (a, b) available where a > 0 is the net payout if an accident occurs and 6 > 0 is the premium. (a) Suppose that the insurance company offers an insurance contract that yields pos- itive expected profit that is proportional to the size of the net payout. That is, suppose ET=ca where c > 0 is profit per unit of net payout. Determine the relationship between a and b satisfying this profit condition. (b) Determine the relationship between 21 and 22 for any level of insurance purchased. (c) Assume that c is such that our agent will optimally purchase a positive amount of insurance (i.e. interior solution). Determine the optimal size of net payout purchased (a*), the associated premium (6*), and the associated consumption levels in each case (x and x2). (d) Illustrate this optimal insurance purchase in a graph. Does our individual use insurance to remove all risk? Interpret this result in terms of the individual's risk premium and the insurance company's profit. 1