# Corner Solutions for Pareto Efficiency

I have been reviewing calculations to find walrasian equilibrium and pareto efficient allocations.

Assume we are in an environment with two consumers, $A$ and $B$, and two goods, $x$ and $y$. From this article in wikipedia, we have that a sufficient condition for pareto efficiency is that the marginal rates of substitution are equal for both consumers.

However for practical purposes this seems only useful when utility functions are differentiable. Assuming preferences are continuous and locally non-satiated, is there any "fast and loose" rule for pareto efficiency when utility functions are not differentiable?

For example with leontieff preferences $u(x,y) = \min\{ x,2y \}$ in which case there might be a corner solution.

• Are we in a setting where the non-differentiable utility functions for both consumers are weakly monotonically increasing in both goods and continuous?
– BKay
Dec 17, 2015 at 16:39
• @Bkay Yes (otherwise I think the question becomes somewhat pathological). I will edit the question. Dec 17, 2015 at 16:51
• If the non-differentiability is caused by kinks in the indifference curves (is in the extreme case of Leontieff) then one should be able to calculate a left-hand MRS and a right-hand MRS (being the MRS to the left and right of the kink respectively. Pareto optimality should then reduce to a comparison between these left- and right-hand MRSs. In fact, think about a standard 2-person Edgeworth box. When we say the MRSs should be equal in a standard differentiable, convex case, we are implicitly saying that each agent's left hand MRS should be greater than the other guy's right-hand MRS. Dec 17, 2015 at 17:22

Let $$F$$ be the set of feasible allocations i.e. the points in the Edgeworth-box. Consider any allocation $$((x_1, y_1), (x_2, y_2)) \in F$$, to check for efficiency of this allocation - consider the two sets:
• $$B_1 = \{((x_1', y_1'), (x_2', y_2'))\in F | u_1(x_1', y_1') \geq u_1(x_1, y_1) \}$$ (Upper level set for individual 1)
• $$B_2 = \{((x_1', y_1'), (x_2', y_2'))\in F | u_2(x_2', y_2') \geq u_2(x_2, y_2) \}$$ (Upper level set for individual 2)
If at every allocation $$((x_1', y_1'), (x_2', y_2')) \in B_1\cap B_2$$, we have
• $$u_1(x_1', y_1') = u_1(x_1, y_1)$$ and
• $$u_2(x_2', y_2') = u_2(x_2, y_2)$$
then the allocation $$((x_1, y_1), (x_2, y_2))$$ is Pareto efficient. If not, $$((x_1, y_1), (x_2, y_2))$$ is not Pareto efficient.