Analayze the optimal income distribution under four important assumptions: Social welfare is strictly/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 fixed.

Under those assumptions, it is socially optimal to transfer income from the rich to the poor until perfect income equality is achieved. How will this be 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:

=+, where is Ebenezer's utility and is Bob's utility. They have identical utility functions, where the marginal utility of income diminishes with income:

=34600.02 and=34600.02,where is Ebenezer's income and 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 of income.

a. 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:=+2. When you take a dollar from Ebenezer, social welfare decreases by. But when you give that dollar to Bob, social welfare increases by 2. What is the optimal distribution of income now?

b. Assuming that social welfare is strictly utilitarian. When you take a dollar from Ebenezer, social welfare decreases by, and when you give that dollar to Bob, social welfare increases by. However, Ebenezer likes money more than Bob. Ebenezer's marginal utility of income is still = 3460 0.02, but Bob's is lower and given by = 2860 0.02. What is the optimal distribution of income now?

C. Go back to assuming that Ebenezer and Bob have the same utility functions. However, both have a constant (not diminishing) marginal utility of income given by = = 3460. What is the optimal distribution of income now?

d. Go back to assuming that Ebenezer and Bob have the same diminishing marginal utility function. (That is, = 3460 0.02 and = 3460 0.02.) 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 administrative costs. When you give a dollar to Bob, social welfare increases by. But in order to get that dollar, $1.20 is taken from Ebenezer, which implies that social welfare decreases by 1.2. What is the optimal distribution of income now?