1. Labor Income Tax. (35 points) Suppose individuals get utility from consumption and disutility from working according to the following utility function: el-a "(cl)= 1 - 0 where c and / denote consumption and hours of work, respectively, and a > 0 is a parameter. Moreover, assume that the hourly wage is given by w in terms of the consumption goods. (a) Set up the maximization problem of the individuals and solve for the optimal choices of con- sumption and labor (c*, I'). (5 points) (b) Now, suppose that labor income is taxed at rate v. Solve for the new optimal choices of labor and consumption (c,, I,). (5 points) (c) Does the introduction of the tax induce an increase or a decrease in the individual's labor supply through the substitution effect? Briefly explain. (5 points) (d) Does the introduction of the tax induce an increase or a decrease in the individual's labor supply through the income effect? Briefly explain. (5 points) (e) Discuss which effect dominates. Hint: It depends on the parameter a. (5 points) (f) Solve for the tax rate that maximizes government's tax revenues for o = -. Explain why the revenue-maximizing tax rate is strictly less than 100%. (10 points)2. Social Security. (35 points) Individuals live for two periods: working period and retirement period. Individuals maximize a discounted sum of flow utilities i.e. V = ve + Bye,. They receive an income of / = 105 in the working period and no income in the retirement period. At the beginning of the working period, individuals decide how much to consume and save in the working period. They can save in private pension funds for the interest rate of ~ = 10%. Assume that / = 0.91. The agents do not leave any bequests at the end of the retirement period. (a) Set up the agent's problem. Clearly specify the objective function and the 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: 0.91 x 1.1 = 1. (10 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 10%. Individuals can also save in private pension funds. (c) Set up the 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. (10 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)3. Full vs Asymmetric Information in Insurance Markets. (Based on Gruber, Chapter 12, Question 12) (20 points) There are two types of drivers on the road today. Speed Racers have a 5% chance of causing an accident per year, while Low Riders have a 1% chance of causing an accident per year. There are twice as many Speed Racers as there are Low Riders. The cost of an accident is $12, 000. (a) Suppose an insurance company knows with certainty each driver's type. What premium would the insurance company charge each type of driver? (10 points) (b) Now suppose that there is asymmetric information, so the insurance company does not know with certainty each driver's type. Would insurance be sold if drivers self-report their types to the insurance company and no information at all is known about individual driver's types? If you are uncertain whether insurance would be sold, explain why. (10 points) 4. Social Security Reform. (Based on Gruber, Chapter 13, Question 17) (30 points) Read textbook's Section 13.4 "Social Security Reform". Then, for each of the reforms in the follow- ing list, briefly discuss the pros and cons of the reform, paying attention in particular to efficiency implications (through potential behavioral responses to the change) and equity implications (who wins and who loses). (a) Increase the number of years used to calculate benefits from 35 to 40. (b) Reduce benefits for beneficiaries with high asset levels (wealth). (c) Add new state and local government workers to the pool of covered workers (i.e., they pay payroll taxes now and receive benefits when they are old). (d) Gradually increase the normal retirement age (NRA) from 67 to 70 in the next few years