(25 points) Assume there is a small country where individuals' life is di- vided into two periods. Each individual 3' has a utility function over con sumption in periods 1 and 2 given by Um- = (0901) / 3+(Iog(022)) / 2, where Cl and 02 are period 1 and period 2 consumption levels. In period 1, each individual works full-time and earns an income of $250 while in period 2 they are older and work part-time which pays them half a full-time worker while the wage of a fulltime worker has increased by %20 from period 1 to period 2. Let us take is = 0.07 as the interest rate paid by the banks for savings which is also equal to the discount factor. (a) (b) (d) If private transition of money between the generations is not possible, how much of its money each individual will spent before the end of period 2? If they maximize their lifetime utility subject to their lifetime budget constraint, what is the optimal level of C in each period? How much does each individual save in each period? (7 pts) A social security system has been introduced to the economy. It takes %15 of each individual's income in period 1, and pays it back to him with interest (It) in period 2. What is the new lifetime budget constraint? What is the effect of this social security system on private savings? How does the system affect total savings in society? (7 pts) Assume that the social security requires the individuals to retire in period 2. Therefore they will not have any income in period 2, instead they receive pension 5 which is equal to 97615 of their income in period 1 plus the interest. What is the new optimal consumption in each period? How much do the individuals save? (7 pts) Compare your results in part c with the results of part a, how do the individuals' consumption and saving change? Explain. (4 pts)