Consider an overlapping generations economy in which people are endowed with 100 goods when young and with nothing when old. Population grows at the rate n. Capital pays the gross rate of return of x. Utility is given by In(c_1,t) + In(c_2,t+1) If people have no income when old, this implies that young people want to save half of their after-tax income when young.
a. Suppose there is no government expenditure, taxation, or government debt. What is the budget constraint of a young person born in period 1? Of an old person in period 2? Combine these to find the lifetime budget constraint of a young person born in period 1.
b. Using the utility function, find a person's choice of first-period consumption, second-period consumption, 

savings, and capital holdings.
c. Find an expression for gross domestic product (GDP) in period 2.
d. Now suppose that, in period 1, the government decides to issue bonds of a one-period maturity worth 10 goods per young person, the proceeds of which will be offered as a gift to the king of a distant country. In period 2, government expenditure reverts to 0, and the outstanding debt (both principal and interest) is rolled over into a new debt issue. What is the budget constraint of a young person born in period 1? Of an old person in period 2? Combine these to find the lifetime budget constraint of a young person born in period 1.
e. Compare the consumption pattern of this generation with that found in part b. Compare the savings of this generation with that found in part
b. Compare the capital of this generation with that found in part b.
f. Find an expression for GDP in period 2. Why does GDP differ from that found in part c? What will the real value be of debt per young person issued in period 2?
g. Can the policy to introduce the one-time government expenditure be financed in such a way that the consumption of no future generation is affected? For what parameter values is this possible?
h. As an alternative to the bond finance of part d, suppose the one-time expenditure is financed with taxes of 10 goods per young person in period 1. What is the budget constraint of a young person born in period 1? Of an old person in period 2? Combine these to find the lifetime budget constraint of a young person born in period 1.
i. Compare the consumption pattern of this generation with that found in part b. Compare the savings of this generation with that found in part
b. Compare the capital of this generation with that found in part b.
j. (advanced) Use calculus and the budget constraints found in part a to verify the assertion that, if people have no income when old, young people want to save half of their after-tax income when young.