Saving with uncertain future income

Consider a consumer who lives for three periods: youth, middle age, and old age. When young, the consumer earns $20,000 in labor income. Earnings during middle age are uncertain; there is a 50% chance that the consumer will earn $40,000 and a 50% chance that the consumer will earn $100,000. When old, the consumer spends savings accumulated during the previous periods. Assume that inflation, expected inflation, and the real interest rate equal zero. Ignore taxes for this problem.

a. What is the expected value of earnings in the middle period of life? Given this number, what is the present discounted value of expected lifetime labor earnings? If the consumer wishes to maintain constant expected consumption over her lifetime, how much will she consume in each period? How much will she save in each period?

b. Now suppose the consumer wishes, above all else, to maintain a minimum consumption level of $20,000 in each period of her life. To do so, she must consider the worst outcome. If earnings during middle age turn out to be $40,000, how much should the consumer spend when she is young to guarantee consumption of at least $20,000 in each period? How does this level of consumption compare to the level you obtained for the young period in part (a)?

c. Given your answer in part (b), suppose that the consumer's earnings during middle age turn out to be $100,000. How much will she spend in each period of life? Will consumption be constant over the consumer's lifetime? (Hint: When the consumer reaches middle age, she will try to maintain constant consumption for the last two periods of life, as long as she can consume at least $20,000 in each period.) d. What effect does uncertainty about future labor income have on saving (or borrowing) by young consumers?