Ken's utility function is U(Q_K, W_K)  Q_KW_K and Barbie's utility function is U(Q_B, W_B)  Q_BW_B.If Ken's initial endowment were 4 units of quiche and 10 units of wine and Barbie's endowment were 8 units of quiche and 10 units of wine, then at any Pareto optimal allocation where both persons consume some of each good,

A) Ken would consume 4 units of quiche for every 10 units of wine.
B) Ken would consume 12 units of quiche for every 20 units of wine that he consumes.
C) Barbie would consume 8 units of quiche for every 10 units of wine that she consumes.
D) Barbie would consume twice as much quiche as Ken.
E) None of the above.

In a Pareto optimal allocation, the MRS of both individuals will be equal.
For Ken, MRS = MUqs/MUws = Q_K/W_K
For Barbie, MRS = MUqs/MUws = Q_B/W_B
At the optimal allocation, both MRS values will be equal and hence we have:
Q_K/W_K = Q_B/W_B
Simplifying it gives:
Q_K/Q_B = W_K/W_B
This implies that Ken and Barbie will consume the goods in the ratio of their marginal utilities.
Using Ken's utility function:
MUqs/MUws = Q_K/W_K
We know that Ken's endowment is 4 units of quiche and 10 units of wine. Hence,
MUqs/MUws = Q_K/W_K = MUqs(4)/MUws(10)
Similarly, using Barbie's utility function, her ratio of consumption will be:
MUqs/MUws = Q_B/W_B = MUqs(8)/MUws(10)
Equating both ratios gives us:
MUqs(4)/MUws(10) = MUqs(8)/MUws(10)
Simplifying this equation gives us MUqs(4) = MUqs(8), which implies that Ken and Barbie would consume quiche in the same ratio as their endowments, which is 4:8 or 1:2.
Therefore, for every 20 units of wine, Ken would consume 12 units of quiche, which gives us choice (B) as the correct answer.