Problem Set 4 Solutions 1. Senator Deal proposes to offer a choice to future retirees: if you retire before age 70, the benefits are calculated on the last 35 years of income; if you retire at age 73, however, you receive benefits calculated on only the last 15 years of income. Which option are high-income workers likely to choose? Low-income workers? Why? A high-income worker may not benefit by much if he delays retirement until age 73, and he would lose three years of benefits. He is likely to choose the earlier retirement age. Assuming no major work interruptions, which is perhaps a more reasonable assumption for a high-wage earner than a low-wage earner, his benefits will be calculated based on his wage since he was in his mid-thirties. These are likely to be fairly-high-earning years, as they begin a decade after a person would have completed his education. Because of the regressive nature of benefit calculations, the higher wages of the last 15 years would yield a low marginal benefit. High-wage earners are also better able to save for retirement in other ways, so they may be able to afford retiring three years earlier. Low-wage earners will be more likely to delay retirement until age 73. They would lose three years of benefits, but their benefits, once they do retire, will be higher if their income is higher in the last 15 years of work. This option will be particularly attractive if these workers had some low- or zero-earning years over the course of their working lives. In addition, calculated benefits are a higher percent of average monthly wage for these workers, so they stand to lose less by working more years. 2. Consider an economy that is composed of identical individuals who live for two periods. These individuals have preferences over consumption in periods 1 and 2 given by U = ln(C1) + ln(C2). They receive an income of 100 in period 1 and an income of 50 in period 2. They can save as much of their income as they like in bank accounts, earning an interest rate of 10% per period. They do not care about their children, so they spend all their money before the end of period 2. Each individual’s lifetime budget constraint is given by: C1+ C2/(1 + r) = Y1 + Y2/(1+ r) Individuals choose consumption in each period by maximizing lifetime utility subject to this lifetime budget constraint. a. What is the individual’s optimal consumption in each period? How much saving does he or she do in the first period? Individuals solve max U = ln(C1) + ln(C2) subject to C1+ C2/(1.1) = 100 + 50/(1.1) Rearrange the budget constraint C2= 110 + 50 – 1.1 C1 and plug into the maximand: max U = ln(C1) + ln(160 – 1.1 C1).Then take the derivative and set it equal to zero: 1/ C1= 1.1/(160 – 1.1 C1), or 2.2 C1 = 160. So C1 ≈ 72.7, and savings 100 – C1 ≈ 27.3. The optimal consumption in the second period is then 50 + 1.1(100 – C1) = 80. b. Now the government decides to set up a social security system. This system will take $10 from each individual in the first period, put it in the bank, and transfer it to him or her with interest in the second period. Write out the new lifetime budget constraint. How does the system affect the amount of private savings? How does the system affect national savings (total savings in society)? What is the name for this type of social security system? Individuals now solve: max U = ln(C1) + ln(C2) subject to C1+ C2/(1.1) = 100 – 10 + (50+10(1.1))/(1.1) Rearranging the budget constraint gives C2 = 160 – 1.1 C1 again—so the consumption levels are the same as in a. Since after-tax income is 10 lower in period 1, however, this means that private savings falls by 10 per individual. Total savings is unchanged, however, since the increased savings through the government exactly offsets the decreased private savings. This is an example of a funded social security system: the money needed for second period benefits is saved in the first period c. Suppose instead that the government uses the $10 contribution from each individual to start paying out benefits to current retirees (who did not pay in to a social security when they were working). It still promises to pay current workers their $10 (plus interest) back when they retire using contributions from future workers. Similarly, it will pay back future workers interest on their contributions using the contributions of the next generation of workers. An influential politician says: “This is a free lunch: we help out current retirees, and current and future workers will still make the same contributions and receive the same benefits, so it doesn’t harm them, either.” Do you buy this argument? If not, what is wrong with it? The Senator makes it seem as if we can pay benefits to current retirees without ever paying for it, but there’s (usually) no such thing as a free lunch, which makes us think someone is harmed by this policy. The key observation is that this policy lowers the national savings rate as compared with b. With lower national savings, the economy will grow less quickly, and future generations will, in fact, be worse off. 3. Consider two households, the Smiths and the Joneses. The Smiths are a two-earner household: both Dick and Jane Smith work and earn the same amount each year. The Joneses are a one-earner household: Sally Jones works while Harry Jones is a homemaker and stay-at-home dad. Use the way spousal benefits are treated in the Social Security system to address the following:a. How do the relative rates of return on Social Security payroll taxes compare for the two families? All else equal, the Joneses will get a better return on their payroll tax contribution. This is because only Sally pays in to the Social Security system, but Harry is still entitled to a 50% spousal benefit (and a full spousal benefit if Sally dies first). Both the Smiths had payroll taxes withheld, and neither gets a free benefit like Harry. (This is complicated by the progressivity of the system for converting AIME into PIA, however.) b. After the kids go off to college, Harry considers taking a small part-time job. How might the Social Security system of taxes and benefits affect his decision? Harry will have to pay payroll taxes if he works, but his work is unlikely to increase his net Social Security benefit since he will be unlikely to earn enough to have a large enough personal Social Security benefit to make it worth taking his own benefit instead of half of his wife’s.