3 Adverse Selection - Akerlof's Model of Lemons Suppose there is a consumer who wishes to purchase health insurance from an insurance company. Let the consumer be denote by i have

utility U₁ = -P+X -X if they purchase insurance, where P is the price for insurance and X is the dol- lars of medical spending consumer i requires. Notice that this insurance plan will cover all medical costs. Essentially, the consumer receives utility U₁ = -P from purchasing insurance. Suppose that the consumer receives utility U₁ = -X if they do not purchase insurance. Let the insurance company be denoted by j and receive utility U₁ = P - E(X) from selling the insurance plan. Essentially, the insurance company receives price P from the consumer, but must pay the expected value of medical costs E(X). The insurance company has the option to not sell an insurance plan, at which point the insurance company will earn U₁ = 0. Suppose there is a market determined price of an insurance plan that is P = 40. Suppose the medical spending of the consumer X is uniformly distributed on the interval (0, 100) a Will the insurance company sell an insurance plan to the consumer in equilib- rium? Please give a thorough explanation of the consumer's and insurance company's behavior and preferences, and why this leads to the result you find. (HINT: Follow the procedure of Akerlofs model. First, figure out for which values of X will a consumer purchase the insurance plan, or in other words when does the consumer get higher utility from purchasing insurance than not? Then consider the insurance company's problem. What do they an- ticipate E(X) to be, based on the consumer's behavior? Will the insurance company find it worthwhile to sell the insurance plan under this belief?) b Based on your answer to the previous question, should the government im- plement a policy to the market between the consumer and insurance compa- nies to ensure a better equilibrium? Why or why not? Which policy, if any. should be implemented? c Suppose the insurance company's utility function is now U = P - 2 *E(X). What is the interpretation of the multiplier 2 in the insurance company's ob- jective function? What is the new outcome in equilibrium? Why does the equilibrium change?