$5\mathrm{.}18$ In a two$-$good, two$-$consumer economy, utility functions are

$u^1(x_1,x_2)=x_1(x_2{)}^2,$

$u^2(x_1,x_2)=(x_1{)}^2{x}_2.$

Total endowments are $(10,20).$

$($a$)$ A social planner wants to allocate goods to maximise consumer $1\text{'}$s utility while holding con$-$

sumer $2\text{'}$s utility at $u^2=\frac{8000}{27}.$ Find the assignment of goods to consumers that solves the

planner's problem and show that the solution is Pareto efficient.

$($b$)$ Suppose, instead, that the planner just divides the endowments so that $e^1=(10,0)$ and $e^2=$

$(0,20)$ and then lets the consumers transact through perfectly competitive markets. Find the

Walrasian equilibrium and show that the WEAs are the same as the solution in part $($a$).$