Question

1. Suppose there are only two people, Simon and Casey, who must split a fixed income of $100. For Simon, the marginal utility of income is MU_S = 400 - 2I_S, while for Casey, marginal utility is MU_C = 400 - 6I_C, where I_S and I_C are the amounts of income to Simon and Casey, respectively.

1. What is the optimal distribution of income if the social welfare function is additive?
2. What is the optimal distribution of income if the social welfare function is W= I_S + 8 I_C?
3. What is the optimal distribution of income if the society only values the utility of Simon? Comment on your answers.
4. What is the optimal distribution of income if the society only values the utility of Casey? Comment on your answers.
5. Comment on how your answers change under question (a) if MU_S = 400 and MU_C = 400.

Answer 1

a) To maximize W, set marginal utilities equal; the constraint is  I_S + I_C = 100

So, 400 - 6I_C = 400 - 2I_S

Substituting I_C = 100- I_S gives us 2I_S = 6(100-I_S)

Therefore,  I_C = $25  I_S = $75

b) Society's overall welfare is the sum of the utility of each individual i.e. Simon and Casey in this case.

400 - 2I_S +  400 - 6I_C = I_S + 8 I_C

Substituting Is = 100- I_cgives us 800=14 I_C + 3(100- I_C)

Therefore,  I_C = $45.45 I_S = $54.55

c) If only Simon matters, then give all the money to Simon until MU_s = 0 (unless all the money in the economy is exhuasted first)

So,  400 - 2I_S = 0 hence I_s =$200

Comment: Giving all the money to Simon is optimal. Infact, we would like to give him upto $200.

d) If only Casey matters, then give all the money to Casey until MU_C = 0 (unless all the money in the economy is exhuasted first)

So, 400 - 6I_C = 0 hence, I_C = $66.67

Comment: Giving any more money to Casey will make her marginal utility negative, which is not optimal. We don't care whether the remaining money is given to Simon or not.

e) MU_S = MU_C = 400., for all levels of income. Hence the society is indifferent among all distributions of income.