An economy has two people, Charlie and Doris.There are two goods, apples and bananas.Charlie has an initial endowment of 3 apples and 8 bananas.Doris has an initial endowment of 6 apples and 4 bananas.Charlie's utility function is U(A_C, B_C)  A_CB_C, where A_C is his apple consumption and B_C is his banana consumption.Doris's utility function is U(A_D, B_D)  A_DB_D, where A_D and B_D are her apple and banana consumptions.At every Pareto optimal allocation,

A) Charlie consumes 9 apples for every 12 bananas that he consumes.
B) Charlie consumes more bananas per apple than Doris does.
C) Doris consumes equal numbers of apples and bananas.
D) Charlie consumes the same number of apples as Doris.
E) Doris consumes 6 apples for every 4 bananas that she consumes.

According to the utility functions, Charlie's marginal rate of substitution between apples and bananas is equal to the ratio of their prices (1:1), which is given by the MRSAB= MUA_CMUB_C=1 In other words, at every Pareto optimal allocation, Charlie consumes apples and bananas in a ratio of 1:1.
Therefore, if Charlie consumes 9 apples, he must consume 9 bananas as well to reach his highest level of satisfaction. But his initial endowment does not allow him to do that. In fact, if he consumes 9 apples, he only has 2 bananas left for consumption. Similarly, if he consumes 12 bananas, he only has 1 apple left.
Therefore, the only Pareto optimal allocation that satisfies Charlie's utility function is where he consumes 3 apples and 4 bananas. Consequently, he consumes 9 apples for every 12 bananas that he consumes, which is (A) the correct answer.