Pareto set with Cobb-Douglas and Leontief preferences

Question

If $U_A(x_A,y_A)=x_Ay_A$ and $U_B(x_B,y_B)=min(x_B,y_B)$ and the total endowments are (8,4), is the Pareto set given by the line joining the kinks of B (black line shown in the diagram)? Shouldn't the bottom left origin and the flat portion on the horizontal axis also be Pareto efficient? Suppose we are at the bottom left origin, there is no way to improve one person's utility without hurting the other. By this definition, the origin should be included in the Pareto set. Please point out any flaw in the logic.

Answer 1

I believe you are correct. The points for which Ya = 0 and \begin{equation} 0\le Xa \le 4 \end{equation} will be all Pareto efficient points.

Proof: Consider an allocation like (2,0). The indifference curve through this point for Individual A is the positive x-axis. (Look at the graph for the IC of Individual B)

• Increasing the satisfaction level of A would require us to move to a point where Ya>0 but this would reduce the satisfaction level of B.
• Increasing the satisfaction level of B would require us to move to a point where Ya<0 but this is not possible as we cannot have a negative allocation of product y for individual A.

Hence, all such points are also Pareto efficient in addition to the line of kinks for individual 2.

Answer 2

The bottom right origin is actually not in the Pareto set. At that point, $(x_A,y_A)=(8,0)$, so $U_A(x_A,y_A)=0$. Similarly, $(x_B,y_B)=(0,4)$, so $U_B(x_B,y_B)=0$.

As an example, $B$ could give one unit of $y$ to $A$ and thereby raise $A$'s utility to 8 without hurting $B$'s own utility (which would remain zero). As a matter of fact, any allocation other than the top left origin is a Pareto improvement over the bottom right origin.