Consider an economy that is composed of identical individuals who live for two periods. Individuals maximize their lifetime utility: = ₁^0.5₂^0.5

1 is consumption in the first period (working period)

2 is consumption in the second period (retirement period)

In each period there are N workers and N retirees. Each individual receives an income of $500 in period 1 (Y1=500) from full-time work and $200 in period 2 (Y2=200) from part-time work. They can save and earn an interest, the interest rate is 10%. The price of consumption in each period is normalized to one.

a) Derive the intertemporal budget constraint.

b) Find the utility maximizing levels of 1, 2 and savings

c) Now suppose that the government institutes social security. The government takes $25, saves it and gives it back with interest r in period 2. How is this social security system called? What is the new equilibrium level of private savings? Is there crowding out? How does this type of social security system affect national savings?

d) Now suppose that the government decides to change the system so that the money from the current generation ($25) is given as benefits to the older generation. How is this social security system called? What is the new equilibrium level of savings? Is there crowding out? How does this type of social security system affect national savings?