Consider an economy that is composed of individuals who live for two periods, and who have the following preferences over consumption in period 1 (C1) and period 2 (C2): U: 11013) + ln(C2} These individuals all receive an income I\" in period 1, and no income in period 2. They can save as much of their period 1 income as they like in bank accounts, earning a real interest rate equal to r. a} How much will a typical individual choose to save under this system? [Hint First nd the budget constraint relating consumption in the second period to consumption in the rst period. Solve for optimal consumption in period 1, and then back out optimal savings] b} Now, the government decides to set up a social security system. This system will take a (small) amount, I", from each individual in the rst period, put it in the bank, and transfer it to them (with interest} in the second period. What does this do to the amount of savings done by individuals? What does it do to total savings in society? What do we call this type of social security system? c) Now assume that instead of putting the money from the tax (1") in the bank, the government simply transfers it from the current young to the current old in any period (keep assuming that T is small). Assume that the population grows at n% per period, and that there is no wage growth (3:0). (What do we call this type of social security system?) Solve for consumption in period 1 for individuals who enter period 1 after this social security system is already in place. How much will individuals choose to save in period 1 under this system? [Hint there are different returns on private investment and on taxes paid to the social security system. Your answer should he in terms of the problem's parameters]. d} How do savings by individuals under the system in c} compare to savings by individuals under the systems in a} and b} under the following three cases: D n=r ii) rt > 1' iii) n '1 1' Explain your ndings in each case. How do total societal savings in c} compare to total societal savings in a) and h)