Consider an economy that is composed of identical

individuals who live for two periods. Individuals maximize:

ln c0 + (5/6)ln c1

where ci is consumption in each period i. In each period there are

born N young individuals - so there will be N old individuals next

period, and there is no population growth. Each individual receives

an income of $2200 in period 0 (their 'youth') and an income of $0 in

period 1 (their 'retirement'). Individuals do not want to leave behind

any money. They can save from Period 0 to Period 1 at interest rate

r = :20, and the price of consumption in each period is one.

(a) Write down an individual's lifetime budget constraint, and solve

to nd his consumption choices and private savings. 10 points.

(b) Sketch this solution. 10 points.

(c) Now suppose that the government institutes a social security sys-

tem. The government takes $198 from each agent in their youth

and gives it directly to an old agent that period. In the next

period, they will do the same, and so on.

i. What kind of social security system is this? 5 points.

ii. Write down each individual's lifetime budget constraint. 10

points.

iii. What is each agent's Social Security Wealth? 5 points.

iv. Did this program increase social welfare? Explain why or

why not. 5 points.

(d) Suppose that the population starts growing at 40% per year. In-

comes are unchanged, and agents can still borrow and lend at

interest rate r = :20. The government continues to collect $400

from each member of the young generation, but this will now

result in each worker receiving more than $400 when retired.

i. Write down each individual's new lifetime budget constraint.

10 points.

ii. Did this program increase social welfare? Explain why or

why not. 5 points.