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

CC BY-SA 4.0
2022-06-22T13:51:57.203Z
2022-06-22T13:51:57.203Z
Amit
<p>Given three people with the same utility function:</p>
<p><span class="math-container">$$
u_A(x_1,x_2)=u_B(x_1,x_2)=u_C(x_1,x_2)=\sqrt{x_1x_2}
$$</span></p>
<p>Prove that the following allocation is Pareto efficient:</p>
<p><span class="math-container">$$
x_A=(2,2),\: x_B=(3,3),\: x_C=(1,1)
$$</span></p>
<p>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.</p>
<p>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?</p>

CC BY-SA 4.0
2022-06-22T13:21:25.843Z
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
CC BY-SA 4.0
2022-06-22T14:20:50.813Z
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
<microeconomics><utility><pareto-efficiency><resource-allocation>
Proving Pareto-efficiency with MRS
Yeet
# Proving Pareto-efficiency with MRS

(Asked by: username_0 on 2022-06-22)

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?

**Tags:** microeconomics, utility, pareto-efficiency, resource-allocation

> **Comments:**
> **username_1**: 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.  
> **username_1**: 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.  

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## 1 answers

### Answer by username_1 on 2022-06-22. Score: 7 | Accepted answer

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.


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