Proving Pareto-efficiency with MRS

Question

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?

Answer 1

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.

Yet another way to check for Pareto efficiency of the above allocation is to show that this allocation also maximises the sum of the utilities of the three individuals subject to the feasibility constraint.