Question

Charlotte and Wilber are two agents in a two-agent, two-commodity pure exchange economy where apples and bananas are the two commodities. Charlotte loves apples and hates bananas. Her utility function is Ucu, b) = u 5 , where a is the number of apples she consumes and b in the number of bananas she consumes. Wilber likes both apples and bananas. His utility function is Uca, b) = a +2Vb. Charlotte has an initial endowment of no apples and 8 bananas. Wilber has an initial endowment of 16 apples and 8 bananas. a. On the graph below, mark the initial endowment and label it E. Use red ink to draw the indifference curve for Charlotte that passes through this point. Use blue ink to draw the indifference curve for Wilbur that passes through this point. b. If Charlotte hates bananas and Wilber likes them, how many bananas can Charlotte be consuming at a Pareto optimal allocation? On the diagram, use black ink to mark the locus of Pareto optimal allocations of apples and bananas between Charlotte and Wilbur. We know that a competitive equilibrium allocation must be Pareto optimal and the total consumption of each good must equal the total supply, so we know that at a competitive equilibrium, Wilbur must be consuming how many bananas? If Wilbur is consuming this number of bananas, his marginal utility for bananas will be and his marginal utility of apples will be If apples are the numeraire, then the only price of bananas at which he will want to consume exactly 16 bananas is - In competitive equilibrium for this Charlotte-Wilbur economy, Wilbur will consume bananas and apples, while Charlotte will consume bananas and apples. Apples 1 4 8 12 16 Bananas

Answer 1

A.

Apples 16 Wilbur BLACK ILINE UNE bo BLUE LINE 12 16 BANANAS 1CHAR LOTTE

b. Black line is from (0,0) to (0,16) because Charlotte consumes 0 bananas at pareto efficient allocation.

c. Wilbur must be consuming 16 bananas.

(As Charlotte consumes 0, he consumes all 16)

His marginal utility of banana= 1/4

(For Wilbur MUb= 1/√b=1/√16=1/4, MUa=1)

And marginal utility of apples= 1

If apples are numeraire ,price of banana= 1/4

Competitive Equilibrium Wilbur consumes 14 apples and 16 banana and Charlotte consumes 2 apple and 0 banana.