2. Social Security. ['35 points} Individuals live for two periods: working period and retirement period. Individuals maximize a discounted sum of ow utilities i.e. U = E + . They receive an income of I = 1115 in the working period and no income in the retirement period. At the beginning of the working periodr 1 individuals decide how much to consume and save in the working period. They can save in private pension funds for the interest rate of r = 10%. Assume that H = [191. The agents do not leave any bequests at me end of the retirement period. {a} Set up die agent's problem. Clearly specify the objective function and die constraints. {5 points) [b] Solve the agent's problem. Explain the steps and numerically find optimal consumption and savings. You can use the following approximation: 1191 x 1.1 s: 1. fir] points} Now suppose that the government provides mandatory social security. Government taxes income at rate 15% and returns the collected funds with a return equal to l%. Individuals can also save in private pension funds. {c} Set up due agent's problem in the presence of the social security program. Clearly specify the objective function and the constraints. Explain how it differs from the problem in part {a}. (.5 points} [d] Solve the agent's problem in the presence of the social security program. Explain the steps and numerically find optimal consumption and savings. 1' 143 points} [e] Compare the optimal solution in the absence and presence of the social security program. How does the program affect private savings? (.5 points)