Question

- 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.

- What is the optimal distribution of income if the social welfare function is additive?
- What is the optimal distribution of income if the social
welfare function is W= I
_{S}+ 8 I_{C}? - What is the optimal distribution of income if the society only values the utility of Simon? Comment on your answers.
- What is the optimal distribution of income if the society only values the utility of Casey? Comment on your answers.
- 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_{c}gives 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.

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Question 11 pts
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