Find pareto optimal allocations Consumer 1: $U_1(x_1,y_1)=x_1y_1$
Consumer 2:$ u_2(x_2,y_2)=min\{x_2y_2 , 4\} $
Initial endowments e1=(1,4) and e2=(4,1)
I want to find Pareto optimal allocations and show its edgeworth box
My solution
First I solve utility maximization
For consumer 1,
$MRS=P_x/P_y$ And budget constraint $x_1P_x + y_1P_y= P_x+4P_y$
$x_1^*=(P_x+4P_y)/2P_x$
$y_1^*=(P_x+4P_y)/2P_y$
For consumer 2
At optimum, $x_2y_2=4$
Budget constraint $x_2P_x+y_2P_y= 4P_x+P_y$
Then $x_2[(4P_x+P_y)/P_y -P_x/P_y x_2]=4$
Feasibility constraint
$$x_2=5-x_1$$
$$y_2=5-y_1$$
Then I solve together
$$(5-x_1)(5-y_1)=4$$
When I insert $x_1$ and $y_1$ I will obtain
$$(9P_x-4P_y)(6P_y-P_x)=16P_xP_y$$
From this point I cannot proceed the solution
How can I find the Pareto optimal allocations ?
Set of Pareto efficient allocations is given by the dashed line in the Edgeworth Box. It is the set of feasible allocations satisfying $y_1 = x_1$ and $x_1y_1 \geq 9$
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