Pareto set with Cobb-Douglas and Leontief preferences 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.
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.