Suppose a student asks an artificial intelligence (Al) the following question: "If there were a perfectly competitive insurance market in which each household could buy and sell insurance for any realized value of future stochastic income without any financial constraints other than the flow-of-funds constraint, what condition would each household's consumption satisfy in equilibrium?"
Suppose the Al replies as follows: "In a perfectly competitive insurance market, households would face the same price for each insurance policy. The price of each insurance policy would equal the expected present discounted value of the future benefit of the policy in each possible state of the world. Each household would buy and sell insurance policies to maximize its expected utility, subject to its flow-of-funds constraint. This implies that _______would be equalized across households in each period in equilibrium, regardless of the realization of stochastic income."
Answer the following questions.
(Hint: To answer the questions, you can modify the consumption CAPM model covered in the lecture into a consumption insurance model in the following way. Suppose each household maximizes the same lifetime utility function as in the consumption CAPM model. Each household receives constant income and stochastic income in each period. Assume that the values of constant income differ across households. The stochastic income for each household takes one of two possible values, high or low, in each period. Assume households are equally split into two permanent types, A and B. There are also two possible states in each period: in one state, the value of stochastic income for each type-A household is high while that for each type-B household is low; and in the other state, the distribution of stochastic income for the two types of households is the opposite. The probability of each state is $50 \%$ in each period regardless of the history of realized states in prior periods. For each state, there is an insurance policy that will pay a unit of goods to the policyholder if the state is realized in the next period. In each period, households can buy and sell arbitrary units of the insurance policy for each state in the next period at a perfectly competitive price, given the flow-of-funds constraint for each household. The competitive price of each insurance policy is determined by the market clearing condition--that is, the supply and the demand for the insurance policy for each state in the next period are equal in each period--in equilibrium. There is no other asset than the insurance policies. Households hold rational expectations (i.e., each household can correctly predict the value of its consumption at each state of stochastic income in the next period.) You can assume that the non-negativity constraint on consumption never binds.)
