# Proving Pareto-efficiency with MRS

Given three people with the same utility function:

$$u_A(x_1,x_2)=u_B(x_1,x_2)=u_C(x_1,x_2)=\sqrt{x_1x_2}$$

Prove that the following allocation is Pareto efficient:

$$x_A=(2,2),\: x_B=(3,3),\: x_C=(1,1)$$

I read my prof's answer to this question which said: The allocation is Pareto efficient as it can be shown that the marginal rates of substitution are equal.

I have only seen that this is a necessary condition of Pareto efficiency not that it is sufficient. Is that the case? If not, how can I show that the allocation is Pareto efficient?

• When all the utility functions are concave and continuous everywhere and differentiable in the interior of the feasible set, then the following is true: An interior feasible allocation is Pareto efficient if and only if MRS of all the individuals are equal at that allocation. Therefore, In the above problem, the condition that MRS are equal is also sufficient.
– Amit
Jun 22 at 13:21
• In the above result, it is also required that all $u_i$s are increasing functions i.e. when consumption of both $x$ and $y$ of individual $i$ increases, utility $u_i$ must also increase.
– Amit
Jun 22 at 14:20

One way to check that this allocation is Pareto efficient is using the first welfare theorem. Consider the exchange economy with utility functions $$\begin{eqnarray*} u_i(x_i, y_i) = \sqrt{x_iy_i}\end{eqnarray*}$$ for $$i\in\{A,B,C\}$$.
Suppose endowment $$\omega$$ is given by $$\begin{eqnarray*} \omega_A = (\omega_A^X, \omega_A^Y) = (2,2) \\ \omega_B = (\omega_B^X, \omega_B^Y) = (3,3) \\ \omega_C = (\omega_C^X, \omega_C^Y) = (1,1)\end{eqnarray*}$$ Check that the endowment $$\omega$$ is the competitive equilibrium allocation of the above economy $$\left((u_i, \omega_i)_{i\in\{A,B,C\}}\right)$$ supported by the prices $$(p^X, p^Y) = (1,1)$$, and therefore, it is Pareto efficient.