0
$\begingroup$

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 ?

$\endgroup$
5
  • $\begingroup$ What do budget constraints have to do with Pareto optimal allocations? $\endgroup$ Dec 19 '20 at 23:54
  • $\begingroup$ I don’t understand what you mean. I wrote the budget constraint @MichaelGreinecker $\endgroup$
    – 1190
    Dec 20 '20 at 8:36
  • $\begingroup$ The budget constraint relates to who owns what. Pareto efficiency does not. Using something clearly unrelated is a non-starter. $\endgroup$ Dec 20 '20 at 8:54
  • $\begingroup$ 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 $\endgroup$
    – 1190
    Dec 20 '20 at 10:15
  • $\begingroup$ What has that to do with Pareto efficiency? Do you know the definition of Pareto efficiency? $\endgroup$ Dec 20 '20 at 11:56
2
$\begingroup$

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 $x_1y_1 \geq 9$ . enter image description here

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$
    – Amaan M
    Dec 31 '20 at 23:45
  • $\begingroup$ @AmaanM Contract curve $\neq$ Pareto set. The dashed line is the Pareto set, not the contract curve $\endgroup$
    – Brennan
    Jan 1 at 2:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.