# Efficient allocations with perfect complements

Consider a very simple economy with two agents (A, B) and two goods ($$x$$, $$y$$). Agent A has utility $$u_a = \min \{x_a, 2y_a\}$$; agent B symmetrically has utility $$u_b = \min \{2x_b, y_b\}$$. Suppose that there are 3 units of $$x$$ and 3 units of $$y$$ in total.

It seems that, if $$x_a = 2y_a$$ and endowments are exhausted, then the allocation is efficient. In addition, if $$2x_b = y_b$$ and endowments are exhausted, i.e. $$2(3 - x_a) = 3 - y_a$$, then the allocation is also efficient. Finally, there seem to be some efficient allocations on the boundary of the feasible set. Plotting these yields

Can I confirm that the red lines do indeed depict the set of efficient allocations? I'm not looking for a formal proof, but just a confirmation that I haven't overlooked anything in this problem.

• You can follow the approach posted here. I think it'll help: youtu.be/HzJyV5zBaPg
– Amit
Jan 7 at 16:07