The problem is simple. However, I'm having trouble deriving the price vector for the system. What exactly am I doing wrong/missing out?
$$ U_A = \min(x,y), \hskip 20pt U_B = x+y. $$
Initial endowments $$ (x_A,y_A)=(100,100), \hskip 20pt (x_B,y_B)=(50,0). $$
Putting A on the bottom left, the contract curve will be a line with slope 1 from A, from point (0,0) to (100,100), A's endowment point. [Seen from A's perspective.] All points on this contract curve are points of 'tangency' between A's kinked indifference curves, and B's linear indifference map.
$x_A= 100(p_x+p_y)/p_x $
$x_B = 0$ (if $p_x>p_y$)
$x_B = 50$ (if $p_x<p_y$)
A cannot benefit from more than (100,100), while B will benefit from as much of either good as she can get: i.e. endowment point lies on contract. Thus, the endowment point can be the competitive equilibrium allocation.
The question is, what will be the equilibrium price vector.
If we keep the price vector as (1,1), X can sell and buy back endowment, and so can B. So, (1,1) makes the endowment point affordable. However, so does any price vector where $p_x<p_y$ (ensuring 0 demand for Y from B.)
Does a unique price vector (ratio) exist for this system? How do we derive it? And what am I doing wrong?
Any help (resources on CE problems of this kind) will be wonderful. Thanks.