# 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 ?

• What do budget constraints have to do with Pareto optimal allocations? Commented Dec 19, 2020 at 23:54
• I don’t understand what you mean. I wrote the budget constraint @MichaelGreinecker Commented Dec 20, 2020 at 8:36
• The budget constraint relates to who owns what. Pareto efficiency does not. Using something clearly unrelated is a non-starter. Commented Dec 20, 2020 at 8:54
• For example, for consumer 1 $xP_x +yP_y= P_x e_x + P_y e_y$ where e represents endowment for a good. @MichaelGreinecker Commented Dec 20, 2020 at 10:15
• What has that to do with Pareto efficiency? Do you know the definition of Pareto efficiency? Commented Dec 20, 2020 at 11:56

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 $$x_1y_1 \geq 9$$ .
• Where did $x_1y_1 \ge 9$ come from? Also, the dashed line doesn't work, it decreases agent 2's utility, so none of those trades could ever occur. The contract curve is the line $x_1=y_1$ in the intersection of the shaded region you have and to the right of the blue indifference curve through the endowment point. Commented Dec 31, 2020 at 23:45
• @AmaanM Contract curve $\neq$ Pareto set. The dashed line is the Pareto set, not the contract curve Commented Jan 1, 2021 at 2:15