<p>Consider the following maximisation problem:
<span class="math-container">\begin{eqnarray*} \max_{x_1, x_2, y} &amp; x_1 + 6\sqrt{y} + \alpha(x_2 + 10 \sqrt{y}) \\ \text{s.t.} &amp; x_1 + x_2 +y = 100 \\ \text{and} &amp; x_1\geq 0, x_2 \geq 0,y \geq 0\end{eqnarray*}</span>
which can be re-written as:
<span class="math-container">\begin{eqnarray*} \max_{x_1, x_2, y} &amp; x_1 + \alpha x_2 + 2\beta \sqrt{y} \\ \text{s.t.} &amp; x_1 + x_2 +y = 100 \\ \text{and} &amp; x_1\geq 0, x_2 \geq 0,y \geq 0\end{eqnarray*}</span>
where <span class="math-container">$\beta = 3 + 5\alpha$</span>.</p>
<p>Clearly, for any <span class="math-container">$\alpha &gt; 0$</span>, any solution to the above problem is Pareto efficient.
We can now find the set of solutions to the above problem:</p>
<p><span class="math-container">\begin{eqnarray*} (x_1^*, x_2^*, y^*)(\alpha) \in \begin{cases} \{(100 - \beta^2,0,\beta^2)\} &amp; \text{if } \alpha &lt; 1 \\ \{(x_1, x_2, y)\in\mathbb{R}^3_+|x_1+x_2=36,y=64\} &amp; \text{if } \alpha = 1 \\ \left\{\left(0,100-\frac{\beta^2}{\alpha^2},\frac{\beta^2}{\alpha^2}\right)\right\} &amp; \text{if } \alpha &gt; 1 \end{cases} \end{eqnarray*}</span></p>
<p>Now we can check for Pareto efficiency of the given options:</p>
<ul>
<li>Option (a) <span class="math-container">$(50,0,50)$</span> is Pareto efficient.  For that consider <span class="math-container">$\beta = \sqrt{50}$</span> or when <span class="math-container">$\alpha = \dfrac{\sqrt{50} -3}{5}$</span>.</li>
<li>Option (b) <span class="math-container">$(90,0,10)$</span> is Pareto efficient.  For that consider <span class="math-container">$\beta = \sqrt{10}$</span> or when <span class="math-container">$\alpha = \dfrac{\sqrt{10} -3}{5}$</span>.</li>
<li>Option (d) <span class="math-container">$(16,20,64)$</span> is Pareto efficient.  For that consider <span class="math-container">$\alpha = 1$</span>.</li>
<li>Option (c) <span class="math-container">$(0,80,20)$</span> is not Pareto efficient, because <span class="math-container">$(0,75,25)$</span> is Pareto superior.</li>
</ul>
<p>Besides the solutions obtained above, there are two more Pareto efficient allocations: <span class="math-container">$\lim_{\alpha\rightarrow 0^+}(x_1^*, x_2^*, y^*)(\alpha) = (91, 0, 9)$</span> and <span class="math-container">$\lim_{\alpha\rightarrow \infty}(x_1^*, x_2^*, y^*)(\alpha) = (0, 75, 25)$</span>. These are the solutions to the following problems:
<span class="math-container">\begin{eqnarray*} \max_{x_1, x_2, y} &amp; x_1 + 6\sqrt{y} \\ \text{s.t.} &amp; x_1 + x_2 +y = 100 \\ \text{and} &amp; x_1\geq 0, x_2 \geq 0,y \geq 0\end{eqnarray*}</span>
and
<span class="math-container">\begin{eqnarray*} \max_{x_1, x_2, y} &amp; x_2 + 10\sqrt{y} \\ \text{s.t.} &amp; x_1 + x_2 +y = 100 \\ \text{and} &amp; x_1\geq 0, x_2 \geq 0,y \geq 0\end{eqnarray*}</span>
respectively.</p>
<p>To see how to find the set of efficient allocations graphically, you can watch this playlist: <a href="https://youtube.com/playlist?list=PLUJGfL_499TJIHv2nw4rn40BbQIe8DChz" rel="nofollow noreferrer">https://youtube.com/playlist?list=PLUJGfL_499TJIHv2nw4rn40BbQIe8DChz</a></p>

CC BY-SA 4.0
2022-05-05T12:41:36.227Z
2022-05-05T13:00:59.277Z
2022-05-05T13:00:59.277Z
Amit
<p>The social welfare maximizing outcome you computed (by equating the sum of marginal benefits to marginal cost) is just <em>one of</em> the Pareto efficient outcomes. Although maximization of joint utilities is a sufficient condition for Pareto efficiency, it is not a necessary condition. Pareto efficiency is defined by the lack of Pareto improvement -- a reallocation of resources to make at least someone better off without making any other worse off. </p>

<p>Option C is not Pareto efficient because a Pareto improvement exists. At <span class="math-container">$(0,80,20)$</span>, 
<span class="math-container">\begin{equation}
u_1(0,20)=0+6\sqrt{20}\approx 26.83 \qquad 
u_2(80,20)=80+10\sqrt{20}\approx124.72. 
\end{equation}</span>
Observe that at this bundle, consumer 2's MU for the good <span class="math-container">$y$</span> is <span class="math-container">$5/\sqrt{20}\approx1.11$</span> and his MU for <span class="math-container">$x$</span> is <span class="math-container">$1$</span>. Therefore, increasing one unit of good <span class="math-container">$y$</span> at the expense of one less unit of <span class="math-container">$x$</span> is utility enhancing for consumer 2. At the same time, one extra <span class="math-container">$y$</span> benefits consumer 1 as well. So we can easily verify that <span class="math-container">$(0,79,21)$</span> is a Pareto improvement over <span class="math-container">$(0,80,20)$</span>:
<span class="math-container">\begin{equation}
u_1(0,21)=0+6\sqrt{21}\approx27.495 \qquad 
u_2(79,21)=79+10\sqrt{21}\approx124.83
\end{equation}</span>
where both consumers experience higher utilities.</p>

<p>In a similar way, you should be able to verify that no Pareto improvements exist for options A, B, and D.</p>

CC BY-SA 4.0
2019-06-08T23:32:40.507Z
2019-06-08T23:32:40.507Z
Herr K.
<p>As part of self study towards an entrance exam, I am solving the following question. </p>

<p>There are two consumers in the economy and two goods, one private and one public. The utilities of the consumers is given by: </p>

<p><span class="math-container">$$u_1(x_1,y) = x_1+6\sqrt y$$</span></p>

<p><span class="math-container">$$u_2(x_2,y = x_2 + 10\sqrt y$$</span></p>

<p>where <span class="math-container">$x_i$</span> is the quantity of the private good and <span class="math-container">$y$</span> is the quantity of the public good consumed. Initial endowments of the private goods: </p>

<p><span class="math-container">$$\omega_1 = 40, \omega_2 = 60$$</span>. </p>

<p>A unit of private good can be converted one-for-one to a public good. </p>

<p>My attempt: </p>

<p>The Pareto Efficient level of public good is determined by the equation: </p>

<p><span class="math-container">$$|MRS_1| + |MRS_2| = MC(G) \quad (1)$$</span></p>

<p>where <span class="math-container">$MRS_1 = \frac{MU_G}{MU_{x_1}}$</span>, <span class="math-container">$MRS_2 = \frac{MU_G}{MU_{x_2}}$</span> and <span class="math-container">$MC(G)$</span> is the marginal cost of providing the public good. </p>

<p>Solving <span class="math-container">$(1)$</span> with the given utility functions I get: </p>

<p><span class="math-container">$$\frac{3}{\sqrt y} + \frac{5}{\sqrt y} = 1$$</span>
<span class="math-container">$$\implies y = 64$$</span></p>

<p>Therefore, the Pareto efficient level of public good is <span class="math-container">$y=64$</span>. </p>

<p>The question asks to select the Pareto Inefficient bundle <span class="math-container">$(x_1,x_2,y)$</span> amongst the following: </p>

<p>A. <span class="math-container">$(50,0,50)$</span></p>

<p>B. <span class="math-container">$(90,0,10)$</span></p>

<p>C. <span class="math-container">$(0,80,20)$</span></p>

<p>D. <span class="math-container">$(16,20,64)$</span></p>

<p>I think all of A, B and C are Pareto Inefficient. </p>

<p>The given answer is C.</p>

<p>Thanks for reading my question.</p>

<pareto-efficiency><public-goods>
Pareto Efficient level of Public Good
Vizag
# Pareto Efficient level of Public Good

(Asked by: username_0 on 2019-06-08)

As part of self study towards an entrance exam, I am solving the following question.

There are two consumers in the economy and two goods, one private and one public. The utilities of the consumers is given by:

$$u\_1(x\_1,y) = x\_1+6\sqrt y$$

$$u\_2(x\_2,y = x\_2 + 10\sqrt y$$

where $x\_i$ is the quantity of the private good and $y$ is the quantity of the public good consumed. Initial endowments of the private goods:

$$\omega\_1 = 40, \omega\_2 = 60$$.

A unit of private good can be converted one-for-one to a public good.

My attempt:

The Pareto Efficient level of public good is determined by the equation:

$$|MRS\_1| + |MRS\_2| = MC(G) \quad (1)$$

where $MRS\_1 = \frac{MU\_G}{MU\_{x\_1}}$, $MRS\_2 = \frac{MU\_G}{MU\_{x\_2}}$ and $MC(G)$ is the marginal cost of providing the public good.

Solving $(1)$ with the given utility functions I get:

$$\frac{3}{\sqrt y} + \frac{5}{\sqrt y} = 1$$
$$\implies y = 64$$

Therefore, the Pareto efficient level of public good is $y=64$.

The question asks to select the Pareto Inefficient bundle $(x\_1,x\_2,y)$ amongst the following:

A. $(50,0,50)$

B. $(90,0,10)$

C. $(0,80,20)$

D. $(16,20,64)$

I think all of A, B and C are Pareto Inefficient.

The given answer is C.

Thanks for reading my question.

**Tags:** pareto-efficiency, public-goods

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## 2 answers

### Answer by username_2 on 2019-06-08. Score: 3 | Accepted answer

The social welfare maximizing outcome you computed (by equating the sum of marginal benefits to marginal cost) is just *one of* the Pareto efficient outcomes. Although maximization of joint utilities is a sufficient condition for Pareto efficiency, it is not a necessary condition. Pareto efficiency is defined by the lack of Pareto improvement -- a reallocation of resources to make at least someone better off without making any other worse off.

Option C is not Pareto efficient because a Pareto improvement exists. At $(0,80,20)$,
\begin{equation}
u\_1(0,20)=0+6\sqrt{20}\approx 26.83 \qquad
u\_2(80,20)=80+10\sqrt{20}\approx124.72.
\end{equation}
Observe that at this bundle, consumer 2's MU for the good $y$ is $5/\sqrt{20}\approx1.11$ and his MU for $x$ is $1$. Therefore, increasing one unit of good $y$ at the expense of one less unit of $x$ is utility enhancing for consumer 2. At the same time, one extra $y$ benefits consumer 1 as well. So we can easily verify that $(0,79,21)$ is a Pareto improvement over $(0,80,20)$:
\begin{equation}
u\_1(0,21)=0+6\sqrt{21}\approx27.495 \qquad
u\_2(79,21)=79+10\sqrt{21}\approx124.83
\end{equation}
where both consumers experience higher utilities.

In a similar way, you should be able to verify that no Pareto improvements exist for options A, B, and D.


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### Answer by username_1 on 2022-05-05. Score: 1

Consider the following maximisation problem:
\begin{eqnarray\*} \max\_{x\_1, x\_2, y} & x\_1 + 6\sqrt{y} + \alpha(x\_2 + 10 \sqrt{y}) \\ \text{s.t.} & x\_1 + x\_2 +y = 100 \\ \text{and} & x\_1\geq 0, x\_2 \geq 0,y \geq 0\end{eqnarray\*}
which can be re-written as:
\begin{eqnarray\*} \max\_{x\_1, x\_2, y} & x\_1 + \alpha x\_2 + 2\beta \sqrt{y} \\ \text{s.t.} & x\_1 + x\_2 +y = 100 \\ \text{and} & x\_1\geq 0, x\_2 \geq 0,y \geq 0\end{eqnarray\*}
where $\beta = 3 + 5\alpha$.

Clearly, for any $\alpha > 0$, any solution to the above problem is Pareto efficient.
We can now find the set of solutions to the above problem:

\begin{eqnarray\*} (x\_1^\*, x\_2^\*, y^\*)(\alpha) \in \begin{cases} \{(100 - \beta^2,0,\beta^2)\} & \text{if } \alpha < 1 \\ \{(x\_1, x\_2, y)\in\mathbb{R}^3\_+|x\_1+x\_2=36,y=64\} & \text{if } \alpha = 1 \\ \left\{\left(0,100-\frac{\beta^2}{\alpha^2},\frac{\beta^2}{\alpha^2}\right)\right\} & \text{if } \alpha > 1 \end{cases} \end{eqnarray\*}

Now we can check for Pareto efficiency of the given options:

* Option (a) $(50,0,50)$ is Pareto efficient. For that consider $\beta = \sqrt{50}$ or when $\alpha = \dfrac{\sqrt{50} -3}{5}$.
* Option (b) $(90,0,10)$ is Pareto efficient. For that consider $\beta = \sqrt{10}$ or when $\alpha = \dfrac{\sqrt{10} -3}{5}$.
* Option (d) $(16,20,64)$ is Pareto efficient. For that consider $\alpha = 1$.
* Option (c) $(0,80,20)$ is not Pareto efficient, because $(0,75,25)$ is Pareto superior.

Besides the solutions obtained above, there are two more Pareto efficient allocations: $\lim\_{\alpha\rightarrow 0^+}(x\_1^\*, x\_2^\*, y^\*)(\alpha) = (91, 0, 9)$ and $\lim\_{\alpha\rightarrow \infty}(x\_1^\*, x\_2^\*, y^\*)(\alpha) = (0, 75, 25)$. These are the solutions to the following problems:
\begin{eqnarray\*} \max\_{x\_1, x\_2, y} & x\_1 + 6\sqrt{y} \\ \text{s.t.} & x\_1 + x\_2 +y = 100 \\ \text{and} & x\_1\geq 0, x\_2 \geq 0,y \geq 0\end{eqnarray\*}
and
\begin{eqnarray\*} \max\_{x\_1, x\_2, y} & x\_2 + 10\sqrt{y} \\ \text{s.t.} & x\_1 + x\_2 +y = 100 \\ \text{and} & x\_1\geq 0, x\_2 \geq 0,y \geq 0\end{eqnarray\*}
respectively.

To see how to find the set of efficient allocations graphically, you can watch this playlist: <https://youtube.com/playlist?list=PLUJGfL_499TJIHv2nw4rn40BbQIe8DChz>


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