6) [8 points total] In class we analyzed the optimal income distribution under four important assumptions: Social welfare is stiietly/additively utilitarian. Individuals have identical utility functions that depend only on income. The marginal utility of income diminishes with income. The total amount of income is xed. Page 2 of4 Under those assumptions, it is socially optimal to transfer income from the rich to the poor until perfect income equality is achieved. In this problem you will assess how the socially optimal income distribution changes when you relax each of those assumptions. Consider a society with two people, Ebenezer and Bob. Social welfare is given by: W=UE+UB, where [15 is Ebenezer's utility and U5 is Bob's utility. They have identical utility functions, where the marginal of utility of income diminishes with income: Problem Set 1pm GBEQ MUE = 3460 0.0215 and MUB = 3460 0.0er, where IE is Ebenezer's income and 13 is Bob's income. Finally, Ebenezer is endowed with $100,000 of wealth, whereas Bob has $0. Under the four assumptions listed above, half of Ebenezer's wealth should be taxed and transferred to Bob so that each has $50,000 ofiiicome. a. [2 points] Suppose that social welfare is not strictly utilitarian. Ebenezer is kind of a jerk, so society places twice as much weight on Bob's utility: W = UE + 2 ' UB. When you take a dollar from Ebenezer, social welfare decreases by M U5. But When you give that dollar to Bob, social welfare increases by 2 ' M U B. What is the optimal distribution of income now? (Hint: The social welfaremcfion has changedfrom what we considered in class. But there '5 still only $100,000 to go around; therefore, it is still the case that IE + [B : 100,000.) [2 points] Go back to assuming that social welfare is strictly utilitarian. However, Ebenezer likes money more than Bob. Ebenezer's marginal utility of income is still MUE = 3460 0.0215, but Bob's is lower and given by MUB = 2860 0.0213. What is the optimal distribution ofincome now? (Hint: Unlike the example in class, the two people have different marginal utiliiymctions. But there '5 still only $100,000 to go around; therefore, it is still the case that 15 + 13 = 100,000.) [2 points] Go back to assuming that Ebenezer\" and Bob have the same utility functions. However, both have a constant marginal utility of income given by M U E = M U B = 3460. What is the optimal distribution of income now? Problem Sat 1pm GBEQ (Hint: The twopeople have a constant marginal utility ofincome, whereas we considered a diminishing marginal utility of income in class. But there 's still only $100, 000 to go around; therefore, it is still the case that IE +13 = 100,000.) [2 points] Go back to assuming that Ebenezer and Bob have the same diminishing marginal utility function. (That is, MUE = 3460 0.0215 and M05 2 3460 0.0213.) However, the to al amount of income is no longer xed. Every $1.00 transfeired to Bob costs Ebenezer $1.20, where the additional $0.20 of cost comes in the form of excess burden and administrative costs. When you give a dollar to Bob, social welfare increases by M US. But in order to get that dollar, $1.20 is taken from Ebenezer, which implies that social welfare decreases by 1.2 . M U E. What is the optimal distribution of income now? Page 3 of4 (Hint: There is only a total 0f$100, 000 to go around ifwe let Ebenezer keep everything. For every $1.00 that we transfer to Bob, the total amount of wealth available decreases by $0. 20; therefore, IE + [B = 100.000 0.2 - IE.)
