Suppose that every consumer is born and lives for three periods: youth, middle age, and old age. During youth, annual income is $40,000. However, earnings during middle age are uncertain; there is...
Suppose that every consumer is born and lives for three periods: youth, middle age, and old age. During youth, annual income is $40,000. However, earnings during middle age are uncertain; there is a 50% chance that he will earn $80,000 and a 50% chance he will earn $140,000. When old, the consumer spends savings accumulated during the previous periods. Assume no inflation as well as inflation expectation and real interest rate = 0. Assume also that earnings are not taxed.
Now suppose the consumer wishes to maintain a minimum consumption level of $40,000 in each stage of his life. To do so, he must consider the worst outcome. If earnings during middle age turn out to be $80,000, how much should the consumer spend in each period to guarantee consumption of at least $40,000 in each period?
The first period is almost trivial: Income is guaranteed to be $40,000, desired consumption is $40,000---so take in $40,000 and spend all $40,000.
It's the second and third period where it gets interesting. We are guaranteed to make at least $80,000 (if only), and have a 50% chance of making $140,000.
Furthermore, we are highly risk-averse; we want to guarantee that we will have enough to consume at least $40,000 in every period.
This means that we need to save at least $40,000 in the second period that we can carry on into the third period.
If we make $80,000, that means we spend $40,000--so we have just enough.
If we make $140,000, we could spend up to $100,000, but actually we would probably want to split it more evenly, spending $70,000 in the second period and $70,000 in the third period.
We might think we'd also want to propagate this backward, and split the overall $180,000 we made over our lifetime into $60,000 in each period; but the problem with that is that in the first period we won't know that we are going to make $140,000 in the second period. So since we are so risk-averse that we want to guarantee enough saving for retirement, we can't afford to spend any more than $40,000 in the first period, even if we had the access to credit necessary to do so. In the worst-case scenario, we'll only make $120,000 all together, so we need to spend $40,000 in each period.
Thus, the optimal strategy, if we end up making the higher income, involves spending much more in the second and third periods than in the first period---which is exactly what we tend to observe in real life.