Kens utility function is U(Q K, W K)= Q K W K and Barbies utility function is U(Q B, W B)= Q B W B.If Kens initial endowment were 3 units of quiche and 11 units of wine and Barbies endowment were 6 units of quiche and 11 units of wine, then at any Pareto optimal allocation where both persons consumed some of each good,
A)Ken would consume 3 units of quiche for every 11 units of wine that he consumed.
B)Barbie would consume twice as much quiche as Ken.
C)Ken would consume 9 units of quiche for every 22 units of wine that he consumed.
D)Barbie would consume 6 units of quiche for every 11 units of wine that she consumed.
E)None of the above.

Question

Quizplus · Accepted Answer

In a Pareto optimal allocation, the marginal rate of substitution (MRS) for both individuals must be equal. That is, MRS(QK,WK) = MRS(QB,WB). Using the utility functions given, we can calculate the MRS for each individual:

MRS(K) = (MU(QK)/MU(WK)) = (10/WK)/(QK/2) = 20/(QK * WK)
MRS(B) = (MU(QB)/MU(WB)) = (10/WB)/(QB/2) = 20/(QB * WB)

At a Pareto optimal allocation where both persons consume some of each good, the MRS for each individual must be equal. Setting MRS(K) = MRS(B), we get:

20/(QK * WK) = 20/(QB * WB)

Simplifying and solving for QB/QK, we get:

QB/QK = WK/WB

Using the initial endowments, we can calculate:

QB/QK = (6/11)/(3/11) = 2

Therefore, Barbie will consume twice as much quiche as Ken. Since Ken's initial endowment of quiche was 3 units and Barbie's was 6 units, this means Ken must consume 9 units of quiche and Barbie must consume 18 units of quiche. Given that their total endowment of wine is 22 units, Ken must consume 9 units of wine and Barbie must consume 13 units of wine. Therefore, Ken would consume 9 units of quiche for every 22 units of wine that he consumed.

Kens Utility Function Is U(Q K, W K)= Q K