7. For example, there are two conditions faced by a consumer, 1 and 2. State 1 occurs with probability , and indicates the consumer's wealth for each state .

(a) If the consumer is strictly risk averse and 1 = 2, show that the insurance company can provide insurance that keeps his wealth constant, both in conditions 1 and 2, so that he is better off and the insurance company gets the expected profit.

(b) Suppose there are many consumers and there are many insurance companies and the proper allocation is such that the wealth of each consumer is constant, both under conditions 1 and 2. Suppose also that in this case, some consumers can insure others. Show that the same level of wealth for consumers and the benefits expected by insurance companies can be achieved with a reasonable allocation where no other insured consumer.