$.$ Consider the same setup as in Questions $1$ and $2$ of Exercise $4($Lecture #$06),$

which is the pure exchange economy with two goods $($apples and oranges$).$ There

are two agents A and B with utility functions given by uA$($xA$,$yA$)=$ x$1/3$y$2/3$ AA

and uB$($xB$,$yB$)=$ x$2/3$y$1/3$ respectively. For an explanation of the notations, see BB

Lectures #$06$ and #$07.$ Let the initial endowment be such that agents A and B both have $6$ apples and $6$ oranges.

$($a$)$ Consider the allocation xA $=2,$ yA $=16/3,$ xB $=10,$ yB $=20/3.$ Show that the marginal rate of substitution $($MRS$)$ of the two agents is equal at this allocation. You may use the answers from Exercise $4.$

$($b$)$ Is it possible to find prices to make the allocation in $($a$)$ a competitive equi$-$ librium allocation without changing the initial endowment? Why or why not? You may use the answers from Exercise $4.$

$($c$)$ Is it possible to find prices to make the allocation $($a$)$ a competitive equilibrium allocation by changing only the initial endowment of the amount of apples to each agent, while leaving the amount of oranges unchanged? Why or why not? You may use an Edgeworth box to help illustrate your answer.

$($d$)$ Is it possible to find prices to make the allocation $($a$)$ a competitive equilibrium allocation by changing only the initial endowment of the amount of oranges to each agent, while leaving the amount of apples unchanged? Why or why not? You may use an Edgeworth box to help illustrate your answer.