6) [8 points total] In class we analyzed the optimal income distribution under four important assumptions: i. Social welfare is strictly/additively utilitarian. ii. 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+U3. where HE is Ehenezer's utility and HE is Bob's utility. They have identical utility functions. where the marginal of utility of income diminishes with income: MUS : 3460 0.0215 and MUB : 34-60 0.0213. where [E is Ebenezer's income and [B is Bob's income. Finally. Ebenezer is endowed with $100000 of wealth. whereas Bob has $0. Under the four assumptions listed above. half of Ebenezer's wealth should be taxed and transfen'ed to Bob so that each has $50,000 of income. a. [2 points] Suppose that social welfare is not strictly utilitarian. Ebenezer is kind of ajerk, so society places twice as much weight on Bob's utility: W : UE + 2 ' 113. When you take a dollar from Ebenezer, social Welfare decreases by M UE. But when you give that dollar to Bob. social welfare increases by 2 - M U 3. What is the optimal distribution of income now\".7 (Him: The social nelfaremctiun has changed from What we considered in class. But there's still only $100,000 to go around; therefore, it is still the case that I; + IB : 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 M UE : 3460 0.0215, but Bob's is lower and given by M U3 2 2860 7 0.0213. What is the optimal distribution of income now? (Hint: Unlike the example in class, the two people have dilarent marginal utility functions. But there '5 still only $100,000 to go mound; therefore, it is still the case that [E + [E = 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 [1,; : M U3 : 3460. What is the optimal distribution of income now? (Him: The two people [mine a com rant marginal utility of income, whereas we considered a diminishing marginal utility of income in class. But there '3' still only $100,000 to go around; therefore, it is still the care 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 7 0.02[E andMUE : 3460 7 0.0215.) However. the total amount of income is no longer fixed. Every $1.00 transferred 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 U3. But in order to get that dollar. $1.20 is taken from Ebenezer. which implies that social welfare decreases by 1.2 - M UE. What is the optimal distribution of income now\".7 (Hint: Unlike the example in class, the two people have dihrent marginal itiilitirrnctiotts. But there '5 still only $100,000 to go around; therefore, it is still the case that [E + [E = 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 [1,; : M U3 : 3460. What is the optimal distribution of income now? (Him: The two people [tune a com rant marginal utility of income, whereas we considered a diminishing marginal utility of income in class. But there '3' still only $100,000 to go around; therefore, it is still the care that IE +15 = 100,000.) [2 points] Go back to assuming that Ebenezer and Bob have the same diminishing marginal utility function. (That is. MUE = 3460 7 0.02lE andMU'B = 3460 7 0.0213.) However. the total amount of income is no longer fixed. Every $1.00 transferred 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 U3. But in order to get that dollar. $1.20 is taken from Ebenezer. which implies that social welfare decreases by 1.2 - M UE. What is the optimal distribution of income now\".7 Page 3 of4 (Hint: There is only a total Uf$100, 000 to go around )fwe let Ebenezer keep everything. For every $1. 00 that we transfer to Bob, the total amount ofwerrlth arailabie decreases by $0. 20; therefore, I a + I B : 100.000 7 0.2 - 13.)
