CHAPTER 7 7-1. Evaluate the validity of the following claim: The increasing wage gap between highly educated and less educated workers will itself generate shifts in the labor market over the next decade. As a result of these responses, much of the excess gain currently accruing to highly educated workers will soon disappear. (discussion question) 7-2. What effect will each of the following proposed changes have on wage inequality? (a) Indexing the minimum wage to inflation. (b) Increasing the benefit level paid to welfare recipients. (c) Increasing wage subsidies paid to firms that hire low-skill workers. (d) An increase in border enforcement, reducing the number of illegal aliens entering the country. 7-3. From 1970 to 2000, the supply of college graduates to the labor market increased dramatically, while the supply of high school (no college) graduates shrunk. At the same time, the average real wage of college graduates stayed relatively stable, while the average real wage of high school graduates fell. How can these wage patterns be explained? (discuss using graphs) 7-4. (a) Is the presence of an underground economy likely to result in a Gini coefficient that over-states or under-states poverty? over-state (discuss) (b) Consider a simple economy where 90 percent of citizens report an annual income of $10,000 while the remaining 10 percent report an annual income of $110,000. What is the Gini coefficient associated with this economy? 0.45 (c) Suppose the poorest 90 percent of citizens actually have an income of $15,000 because each receives $5,000 of unreported income from the underground economy. What is the Gini coefficient now? 0.349 7-1
7-6. Consider an economy with the following income distribution: each person in the bottom quartile of the income distribution earns $15,000; each person in the middle two quartiles earns $40,000; and each person in the top quartile of the income distribution earns $100,000. (a) What is the Gini coefficient associated with this income distribution? 0.327. (b) Suppose the bottom quartile pays no taxes, the middle two quartiles pay 10 percent of its income in taxes, and the top quartile pays 28 percent of its income in taxes. Two-thirds of all tax money is redistributed equally to all citizens in the form of military defense, government pensions (social security), roads/highways, etc. The remaining one-third of tax money is disturbed entirely to the poorest quartile. What is the Gini coefficient associated with this redistribution plan? Would you consider this tax and redistribution plan to be a particularly aggressive income redistribution policy? First, we must determine total taxes: 500(40,000)(0.10) + 250(100,000)(0.28) = $ 9 million are paid in total taxes. This comes from each person in the lowest quartile paying $0; each person in the middle two quartiles paying $4,000 in taxes; and each person in the top quartile paying $28,000 in taxes. Of this $9 million, $6 million is repaid to all people equally in the form of defense, social security, road, etc. So, each person claims $6,000 of this pool of taxes. The remaining one-third of taxes, which equals $3 million, is divided among the poorest quartile. Thus, the poorest 250 people each receive an income transfer of $12,000. New incomes, therefore, are $33,000 for the bottom quartile, $42,000 for the middle two quartiles, and $78,000 for the top quartile. Given these new numbers, the analysis from part (a) can be repeated. The bottom quartile receives $8.25 million of income, or 16.9% of $48.75 million. The middle two quartiles receive $21 million of income, or 43.1% of $48.75 million. The top quartile receives $19.5 million of income, or 40.0% of $48.75 million. And the Gini coefficient is 0.173245. This appears to be a fairly substantial redistribution scheme as it cuts the Gini coefficient in half. It also has the richest person earning 2.36 times that of the poorest person (78/33) whereas this ratio was 6⅔ times (100/15) without the redistribution. 7-7. The two points for the international income distributions reported in Table 7-1 can be used to make a rough calculation of the Gini coefficient. Use a spreadsheet to estimate the Gini coefficient for each country. Which three countries have the most equal income distribution? Which three countries have the most unequal income distribution? The three countries with the most inequality are Chili (0.288), Guatemala (0.266), and the Dominican Republic (0.244). The three countries with the most equality are Sweden (0.014), Hungary (0.014), and Germany (0.021). It should be emphasized that these are very crude measures as they rely on only two points in the income distribution. 7-2
7-8. Consider the following (highly) simplified description of the U.S. wage distribution and income and payroll tax schedule. Suppose 50 percent of households earn $40,000, 30 percent earn $70,000, 15 percent earn $120,000, and 5 percent earn $500,000. Marginal income tax rates are 0 percent up to $30,000, 15 percent on income earned from $30,000 to $60,000, 25 percent on income earned from $60,000 to $150,000, and 35 percent on income earned in excess of $150,000. There is also a 7.5 percent payroll tax on all income up to $80,000. (a) What is the marginal tax rate and average tax rate for each of the four types of households? What is the average household income, payroll, and total tax bill? What percent of the total income tax is paid by each of the four types of households? What percent of the total payroll tax bill is paid by each of the four types of household? Marginal tax rates: households earning $40,000 is 15 percent + 7.5 percent = 22.5 percent. households earning $70,000 is 25 percent + 7.5 percent = 32.5 percent. households earning $120,000 is 25 percent + 0 percent = 25 percent. households earning $500,000 is 35 percent + 0 percent = 35 percent. Income tax bill: households earning $40,000 is 15 percent of $10,000 = $1,500. households earning $70,000 is $4,500 + 25 percent of $10,000 = $7,000. households earning $120,000 is $4,500 + 25 percent of $60,000 = $19,500. households earning $500,000 is $27,000 + 35 percent of $350,000 = $149,500. Payroll tax bill: households earning $40,000 is 7.5 percent of $40,000 = $3,000. households earning $70,000 is 7.5 percent of $70,000 = $5,250. households earning $120,000 is 7.5 percent of $80,000 = $6,000. households earning $500,000 is 7.5 percent of $80,000 = $6,000. Total tax bill and average tax rates: households earning $40,000 is $4,500 => ATR = 11.25 percent. households earning $70,000 is $12,250 => ATR = 17.5 percent. households earning $120,000 is $25,500 => ATR = 21.25 percent. households earning $500,000 is $155,500 => ATR = 31.10 percent. Average tax bills over all households are: Income:.5(1,500) +.3(7,000) +.15(19,500) +.05(149,500) = $13,250 Payroll:.5(3,000) +.3(5,250) +.15(6,000) +.05(6,000) = $4,275. Total: $17,525. Percent of the total income tax collected by the government that is paid by each household group: $40,000 households pay.5(1,500) / 13,250 = 5.67 percent. $70,000 households pay.3(7,000) / 13,250 = 15.85 percent. $120,000 households pay.15(19,500) / 13,250 = 22.08 percent. $500,000 pay.05(149,500) / 13,250 = 56.42 percent. 7-3
Percent of the total payroll tax collected by government that is paid by each household group: $40,000 households pay.5(3,000) / 4,275 = 35.09 percent. $70,000 households pay.3(5,250) / 4,275 = 36.84 percent. $120,000 households pay.15(6,000) / 4,275 = 21.05 percent. $500,000 pay.05(6,000) / 4,275 = 7.02 percent. (b) What is the Gini coefficient for the economy when comparing after-tax incomes across households? (Hint: assume there are 1,000 households in the economy.) What happens to the Gini coefficient if all taxes were replaced by a single 20 percent flat tax on all incomes? Suppose there were 1,000 households. Total after-tax income is 500($35,500) + 300($57,750) +150($94,500) +50($344,500) = $66,475,000. The cumulative gross income shares of the four income groups, therefore, are: 500($35,500) / $66.47m = 26.7 percent. 26.7 percent + 300($57,750) / $66.47m = 52.8 percent. 52.8 percent + 150($94,500) / $66.47m = 74.1 percent. 74,1 percent + 50($344,500) / $66.47m = 100.0 percent. The area under the Lorenz curve, therefore, is (.5) (½)(.267) + (.3)[.267+(½)(.528.267 )] + (.15)[.528+(½)(.741.528 )] +(.05)[.741+(½)(1.741 )] = 0.6675 + 0.11925 + 0.95175 + 0.043525 = 0.3247. The Gini coefficient, therefore, is [.5..3247 ] /.5 =.3506. Suppose there was a flat tax of 20 percent. Total after-tax income is 500(.8)($40,000 + 300(.8)($70,000) +150(.8)($120,000) +50(.8)($500,000) = $67,200,000. The cumulative gross income shares of the four income groups, therefore, are: 500($32,000) / $67.2m = 23.8 percent. 23.8 percent + 300($56,000) / $67.2m = 48.8 percent. 48.8 percent + 150($96,000) / $67.2m = 70.2 percent. 70.2 percent + 50($400,000) / $67.2m = 100.0 percent. The area under the Lorenz curve, therefore, is (.5) (½)(.238) + (.3)[.238+(½)(.488.238 )] + (.15)[.488+(½)(.702.488 )] +(.05)[.702+(½)(1.702 )] =.0595 +.1089 +.08925 +.04255 =.3002. The Gini coefficient if there was a 20 percent flat tax, therefore, would be [.5..3002 ] /.5 =.3996. (c) A presidential candidate wants to remove the cap on payroll taxes so that every household would pay payroll taxes on all of its income. To what level could the payroll tax rate be reduced under the proposal while keeping the total amount of payroll tax collected the same? Suppose there are 1,000 households: 500 pay $3,000, 300 pay $5,250, and 200 pay $6,000. The total payroll tax receipts, therefore, are $4,275,000, while total income is $84 million. Thus, generating $4.275 million of tax on $84 million of income requires a tax rate of 5.089 percent. 7-4
7-10. Ms. Aura is a psychic. The demand for her services is given by Q = 2,000 10P, where Q is the number of one-hour sessions per year and P is the price of each session. Her marginal revenue is MR = 200 0.2Q. Ms. Aura s operation has no fixed costs, but she incurs a cost of $150 per session (going to the client s house). (a) What is Ms. Aura s yearly profit? $6,250 per year. (b) Suppose Ms. Aura becomes famous after appearing on the Psychic Network. The new demand for her services is Q = 2500 5P. Her new marginal revenue is MR = 500 0.4Q. What is her profit now? $153,125. (c) Advances in telecommunications and information technology revolutionize the way Ms. Aura does business. She begins to use the Internet to find all relevant information about clients and meets many clients through teleconferencing. The new technology introduces an annual fixed cost of $1,000, but the marginal cost is only $20 per session. What is Ms. Aura s profit? Assume the demand curve is still given by Q = 2500 5P. $287,000. (d) Summarize the lesson of this problem for the superstar phenomenon. (discuss) 7-11. Suppose two households earn $40,000 and $56,000 respectively. What is the expected percent difference in wages between the children, grandchildren, and great-grandchildren of the two households if the intergenerational correlation of earnings is 0.2, 0.4, or 0.6? (straightforward) 7-12. Suppose 50 percent of a population all receive an equal share of p percent of the nation s income, where 0 p 50. (a) For any such p, what is the Gini coefficient for the country? 0.5 p. (b) For any such p, what is the 90 10 wage gap? (1 p)/p. 7-5
7-13. Consider two developing countries. Country A, though quite poor, uses government resources and international aid to provide public access to quality education. Country B, though also quite poor, is unable to provide quality education for institutional reasons. The distribution of innate ability is identical in the two countries. (a) Which country is likely to have a more positively skewed income distribution? Why? Plot the hypothetical income distributions for both countries on the same graph. (discuss) (b) Which country is more likely to develop faster? Why? Plot the hypothetical income distributions in 20 years for both countries on the same graph. (discuss) 7-14. File sharing software threatens the music industry in part because artists will not be fully compensating for their recording of songs. Suppose that the government decides that file sharing software products are legal anyway. (a) The almost immediate result will be that artists start earning very little money for their recordings but they continue to earn money for live performances. How will income change for the music industry? How does your answer relate to the superstar phenomenon? Musicians who are also great performers will continue to make new music and will continue to earn a lot of money. Musicians who are good at making new music but not good at giving live performances will see a reduction in their income. For example, rumor has it that Steely Dan and the Steve Miller Band are fairly horrible in concert, although their music is loved by many people. These bands would struggle to earn money under the policy. Other artists that are known to put on great shows (The Rolling Stones) will continue to earn a lot of money. Thus, the public policy doesn t choose which superstars will continue to do well; it simply redefines what talents are going to be rewarded. (b) Although one would expect lower prices to benefit the music-listening public if the government decides that file sharing software products are legal, but in what way(s) could the music-listening public also be hurt from the policy? The most obvious way such a policy would hard the music-listening public is that fewer people make music as it is less profitable. Thus, the public will have fewer choices of music to listen to. This is analogous to if the government removed the patent system from new drugs, which would results in fewer new drugs being developed. 7-6