Consider an economy composed of identical individuals who live for two periods. These individuals have preferences over consumption in period 1 and 2 given by U = (c1)0.5 × (c2)0.5. They receive an income of 100 in period 1 (young) and 20 in period 2 (old). They can save as much of their income as they like in bank accounts, earning an interest rate of 10% per period. They do not care about their kids, so they spend all of their money before the end of period 2.

a) Show that their lifetime budget constaint is: c1 +(c2/1+0.1)=100+(20/1+0.1)

b) What is the individual's optimal consumption in each period? How much does she save in the first period?

Now the government decides to set up a social security system. This system will take $10 from each individual in the first period, put it in the bank, and transfer it to them with interest in the second period.

c) Write out the new lifetime budget constraint. How does the system affect the amount of private savings? How does the system affect national savings (total savings in society)? What is the name for this type of social security system?

Now suppose the government decides that it wants to implement a pay-as-you-go social security system instead. It will tax everyone $10 when they are young as in part b. However, here it will be transferred directly to the old at the time. So a given individual who paid a tax of $10 in the first period will receive $10(1 + n + g) in the second period. Assume that the population grows at n = .02 and wages grow at g = .03.

d) What is the new lifetime budget constraint? Solve the individual's new optimal consumption in each period in this case. How does the system affect the amount of private savings? How does the system affect national savings (total savings in society)?