In Problem 4, 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 3 units of quiche and 7 units of wine and Barbie's endowment were 6 units of quiche and 7 units of wine, then at any Pareto optimal allocation where both persons consume some of each good, A) Ken would consume 3 units of quiche for every 7 units of wine. B) Ken would consume 9 units of quiche for every 14 units of wine that he consumes. C) Barbie would consume twice as much quiche as Ken. D) Barbie would consume 6 units of quiche for every 7 units of wine that she consumes. E) None of the above.

Question

Quizplus · Accepted Answer

In a Pareto optimal allocation, both individuals must consume some of each good. Let's assume that Ken consumes x units of quiche and y units of wine, while Barbie consumes (3-x) units of quiche and (14-y) units of wine. 
The total utility in this case is U = xy + (3-x)(14-y).
To maximize U, we need to take the partial derivative of U with respect to x and y and set them equal to zero:
dU/dx = y - (14-y) = 2y-14 = 0  --> y=7
dU/dy = x - (3-x) = 2x-3 = 0  --> x=3/2
So Ken would consume 3/2 units of quiche for every 7 units of wine. Simplifying, we can say that Ken would consume 9 units of quiche for every 14 units of wine. Therefore, the correct answer is B.

In Problem 4, Ken's Utility Function Is U(QK, WK) =

In Problem 4, Ken's Utility Function Is U(Q_K, W_K) =