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2. Suppose individuals live for two periods and have preferences over consumption in period 1 and period 2 given by: U : 111(01) + ln(C'2), where 0'1 is consumption in period 1 and 02 is consumption in period 2. The lifetime budget constraint is given by C2 Y2 C = Y . 1 + 1 + T 1 + 1 + 1" Y1(= 100) is labor income in period 1, Yg(= 50) is labor income in period 2, and r (2 0.10) denotes the interest rate. Individuals choose 01 and 02 in order to maximize lifetime utility subject to this lifetime budget constraint. (a) Calculate an individual's optimal consumption in each period. Calculate the level of sav ings in the rst period. [2 marks] (b) Assume the government sets up a public pension system. This system will tax each indi vidual $10 in period 17 invests the revenue at a rate of return 7' (2 0.10), and transfers the funds back to the individual with interest in period 2. i. What is the name for this type of public pension system? [1 mark] ii. Write out the new lifetime budget constraint and calculate the impact of the system on the amount of private savings. [2 marks] iii. Calculate the impact of the system on national savings (:private savings+government savings) [2 marks] (c) Now assume that the existence of the public pension system results in individuals retiring in period 2, so individuals receive no labor income in period 2. i. Solve for the individuals revised optimal consumption in each period. [2 marks] ii. Calculate the new level of private and national savings. Does it differ from the level of savings in part (b), and if so, Why (explain intuitively)? [2 marks]