# 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? – Michael Greinecker Dec 19 '20 at 23:54
• I don’t understand what you mean. I wrote the budget constraint @MichaelGreinecker – B11b Dec 20 '20 at 8:36
• The budget constraint relates to who owns what. Pareto efficiency does not. Using something clearly unrelated is a non-starter. – Michael Greinecker Dec 20 '20 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 – B11b Dec 20 '20 at 10:15
• What has that to do with Pareto efficiency? Do you know the definition of Pareto efficiency? – Michael Greinecker Dec 20 '20 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. – Amaan M Dec 31 '20 at 23:45
• @AmaanM Contract curve $\neq$ Pareto set. The dashed line is the Pareto set, not the contract curve – Brennan Jan 1 at 2:15

Pareto optimality requires that, after the agents trade once to a new allocation on the contract curve (set of Pareto optimal points), they each maintain at least the same utility they had prior to the trade, there are no potential "next" trades that make either agent better off, and there is no wastage in the economy (all goods are consumed). None of these require a price, so the budget constraint is unnecessary in solving this problem.

$$U_1(e_1) = x_1y_1 = 1*4 = 4 ,$$ $$U_2(e_2) = \min\{x_2y_2, 4\} = \min\{4*1,4\} = 4 .$$

So, for any potential point on the contract curve, we must have

$$U_1(x_1,y_1) \ge 4 \Rightarrow x_1y_1 \ge 4 ,$$ $$U_2(x_2,y_2) \ge 4 \Rightarrow \min\{x_2y_2,4\} \ge 4 \Rightarrow x_2y_2 \ge 4 .$$

Note, since we know $$U_2(x_2,y_2) \ge 4$$ for any Pareto optimal solution, we are able to simplify $$U_2$$ and ignore the $$\min\{\cdot\}$$ operator in this case:

$$U_2 = \min\{x_2y_2, 4\} = x_2y_2 .$$

Next, we need to guarantee that there are no potential trades following a move to the contract curve. For an interior solution, we just need to set the marginal rates of substitution equal to each other (our tangency condition):

$$MRS_1 = MRS_2$$

$$\frac{\partial{U_1} / \partial{x_1}}{\partial{U_1} / \partial{y_1}} = \frac{\partial{U_2} / \partial{x_2}}{\partial{U_2} / \partial{y_2}}$$

$$\frac{y_1}{x_1} = \frac{y_2}{x_2}$$

Note, in this case, at any point along the edge of the Edgeworth box, we'd have $$U_1 = U_2 = 0 < 4$$, so no points along the edge can be on the contract curve.

Finally, our no wastage conditions:

$$x_1 + x_2 = e_1^x + e_2^x = 5 ,$$ $$y_1 + y_2 = e_1^y + e_2^y = 5 .$$

Substituting these into our tangency condition, we have

$$\frac{y_1}{x_1} = \frac{5-y_1}{5-x_1}$$ $$\Rightarrow x_1 = y_1$$

Combining our tangency condition with our minimum utility conditions from above, we have that our contract curve from Agent 1's perspective is the set

$$\{(x_1,y_1)\in [0,5]^2 : x_1 = y_1 \cap x_1y_1 \ge 4 \cap (5-x_1)(5-y_1) \ge 4 \}$$