You do not have to solve the problems, but the equation for each would be helpful. Even an equation for part of the question, anything helps

4) [8 points total] In class we analyzed the optimal income distribution under four important assumptions: 0 Social welfare is strictlytadditively utilitarian. O Individuals have identical utility functions that depend only on income. O The marginal utility of income diminishes with income. I The total amount of income is xed. 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 = U; + U3, where U; is Ebenezer's utility and U; is Bob's utility. They have identical utility functions, where the marginal of utility of income diminishes with income: MU; = 3460 0.0er and no, = 3450 0.025, where l E is Ebenezer's income and l; 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 shouldbe taxed and transferred to Bob so that eachhas $50,000 of income. a. [2 points] Suppose that social welfare is not strictly utilitarian. Ebenezer is kind of a jerk, so society places twice as much weight onBob's utility: W = U; + 2 - 0;. When you take a dollar from Ebenezer, social welfare decreases by MU; But when you give that dollar to Bob, social welfare increases by 2 . MUa. What is the optimal distribution of income now? (Hint: The social welfarenction has changed from what we considered in class. But there's still only $100,000 togo around; thereire, it is still the case that l; + l; = 100,000.) Pageloffl b. [2 points] Go backto assuming that social welfare is strictly utilitarian. When you take a dollar 'om Ebenezer, social welfare decreases by MU;, and whenyou give that dollar to Bob, social welfare increases by M 0;. However, Ebenezer likes money more than Bob. Ebenezer's marginal utility of income is still M U; = 34-60 0.021;, but Bob's is lower and givenby MU; = 2860 0.021'3. What is the optimal distribution of income now? (Hint: Unlike the example in class, the two people have di'erent marginal utility inctions. But there is still only $00,000 to go around; therefore, it is still the case that l; + I; = 100,000.) c. [2 points] Go backto assuming that Ebenezer and Bob have the same utility functions. However, both have a constant (not diminishing) marginal utility ofincome givenby MU; = MU; : 3460. What is the optimal distribution of income now'?I (Hint: the two people have a constant marginal utility of income, so they amount ofutilit)! that they gain (or lose) with an increase (or decrease) in income doesn 't depend on how much income they have. What does that imply about the change in social welfare when we take a dollar from li'i.'JenezeramiI glue it to Bob?) (1. [2 points] Go backto assuming that Ebenezer and Bob have the same diminishing marginal utility inction. (That is, MU; = 3460 0.021; and MU; = 3460 0.02l;.} However, the total amount ofincome is no longer xed. Every $1.00 transferred to Bob costs Ebenezer $1.20, where the additional $0.20 of cost comes in the form of administrative costs. When you give a dollar to Bob, social welfare increases by M U 3. But in order to get that dollar, $1.20 is taken from Ebenezer, which implies that social welfare decreases by 1.2 * MU;. What is the optimal distribution of income now? (Hint: there a total of $100,000 to go around only ifwe let Ebenezer keep everything. For every $1.00 that me transzr to Bob, the total amount of wealth available decreases by $0.20; therefore, i; + I; : 100,000 0.2 . 1;.)