Economics 321: Public Economics Prof. Marika Cabral Fall 2017 UT Austin Problem Set 4 Solution 1. [Ch 13] 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. The payroll tax will tend to discourage him from working. c. Suppose that both families have retired and have started to receive Social Security benefits. By what fraction will these benefits fall for each of these families if one member of the household dies? What implications does this have for relative consumption smoothing in these two households? Total Social Security income will fall by 50% for the Smiths since the two Smiths originally had equal benefits and one of the two benefits is lost upon death. Total Social Security income will fall by only 1/3 for the Joneses. Prior to the death, they received 150% of Sally’s benefit (Sally’s benefit plus a 50% spousal benefit). After death, the survivor receives only 100% of Sally’s benefit. The steeper drop-off for the Smiths may make it harder for them to consumption smooth. 2. [Ch 13] Lalaland is an extremely stable country with 200,000 residents, half of whom are young workers and half of whom are retirees. At the end of each “year,” the 100,000 retirees die, the 100,000 young workers retire, and 100,000 new young workers are born. Workers earn a total of $5,000 for the year. Lalaland operates a “pay as you go” social security system, where each current worker is taxed $2,500 and the revenue collected is used to pay a $2,500 pension to each retiree. The neighboring country, Gogovia, is larger and more dynamic. Gogovia has an active stock market that Lalalandians can invest in and earn a 10% rate of return. It also has an active banking sector, which will gladly lend the Lalalandian government money, charging them 10% interest per year. Lalaland is considering moving to a system of personal accounts, where each Lalalander would take her $2,500 and invest it in Gogovian markets (and earn a much higher rate of return!). The government would borrow $250 million ($2,500 × 100,000) from Gogovian bankers to pay for current retirees. It would then tax retirees each year by just enough to pay the interest on this debt. Would this new system be better or worse for Lalaland?The new system would be neither better nor worse for Lalanland. In fact, it is an entirely equivalent system. The interest due on the debt would be 10% × ($250m) = $25m, so taxes would have to be $25m/100,000 = $250 per retiree. This is exactly enough to offset the higher returns Lalalanders would earn in the stock market. 3. [Ch 13] 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 – 1.1C1 and plug into the maximand: max U = ln(C1) + ln(160 – 1.1C1). Then take the derivative and set it equal to zero: 1/C1 = 1.1/(160 – 1.1C1), or 2.2C1 = 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 – 1.1C1 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