<p>Let <span class="math-container">$y_i$</span> be the contribution of agent i towards the production of <span class="math-container">$y$</span></p>
<p><span class="math-container">$y=\sqrt{\sum y_i}$</span></p>
<p>So when <span class="math-container">$y=4$</span>
<span class="math-container">$\Rightarrow \sum y_i=16 \Rightarrow \sum x_i = 14$</span></p>
<p>The set of pareto efficient allocations is <span class="math-container">$\{(x_1,x_2):x_1+x_2=14,x_1 \in [0,10], x_2 \in [0,20]\}$</span></p>
<p>Private consumption of the goods will be the same as the endowments and the provision of public good will be <span class="math-container">$0$</span> at the equilibrium.</p>
<p>Reason:</p>
<p><span class="math-container">$max_{{x_1,y_1}: x_1 + y_1 = 10} x_1 + \sqrt{\sum y_i}$</span></p>
<p><span class="math-container">$\Rightarrow \frac{39}{4} + y_{2} = x_{1}$</span></p>
<p><span class="math-container">$max_{(x_2, y_2):x_2+y_2=20} x_2 \sqrt{\sum y_i}$</span></p>
<p><span class="math-container">$\Rightarrow x_2= \frac{1}{3}(2y_1 + 40)$</span></p>
<p>Let the price of the private good be <span class="math-container">$1$</span>. Given the utility function of agent <span class="math-container">$1$</span> this implies
<span class="math-container">$y_1=y \Rightarrow y_2=y^2 - y$</span>
Clearly the above system of equations has no real solution.
So equilibrium does not exist.</p>
<p><strong>Edit</strong>: The equilibrium will be when <span class="math-container">$x_1=10, y_1=0, x_2=\frac{40}{3},y_2=\frac{20}{3}$</span>
<strong>Credits</strong>: Agrim Rana</p>

CC BY-SA 4.0
2022-08-15T10:38:00.063Z
I see. Thank you for your great answer!
studentp
CC BY-SA 4.0
2022-08-15T05:36:16.807Z
2022-08-15T07:02:05.030Z
2022-08-15T07:02:05.030Z
Pallak Goyal
<p>Set of feasible allocations in this economy is:</p>
<p><span class="math-container">$\mathcal{F}=\{(x_1,x_2,y)\in\mathbb{R}^3_+|x_1+x_2+y^2=30\}$</span></p>
<p>This set can also be represented in graph in the following way:</p>
<p><a href="https://i.sstatic.net/E0wrC.png" rel="noreferrer"><img src="https://i.sstatic.net/E0wrC.png" alt="enter image description here" /></a></p>
<p>To determine (interior) efficient allocations, we can first calculate MRS of individual 2 whose utility is <span class="math-container">$u_2(x_2,y)=x_2y$</span>, and get:</p>
<p><span class="math-container">$\text{MRS}_2 = \frac{y}{x_2}$</span></p>
<p>Now we can rewrite <span class="math-container">$u_1(x_1, y)=x_1+y$</span> as <span class="math-container">$u_1^{\text{ADJ}}(x_2,y) = u_1(30-x_2-y^2, y)=30-x_2-y^2 + y$</span>. We have just written <span class="math-container">$u_1^{\text{ADJ}}$</span> as a function of <span class="math-container">$x_2$</span> and <span class="math-container">$y$</span>, by taking into account the feasibility constraint. We now calculate MRS of individual 1 using <span class="math-container">$u_1^{\text{ADJ}}(x_2,y)$</span>, and get:</p>
<p><span class="math-container">$\text{MRS}_1^{\text{ADJ}} = \frac{1}{2y-1}$</span></p>
<p>(Interior) efficient allocations satisfy the condition:</p>
<p><span class="math-container">$\text{MRS}_1^{\text{ADJ}}=\text{MRS}_2$</span> which gives: <span class="math-container">$x_2 = 2y^2-y$</span></p>
<p>Here is the set of efficient allocations in the graph:</p>
<p><a href="https://i.sstatic.net/DTYt0.png" rel="noreferrer"><img src="https://i.sstatic.net/DTYt0.png" alt="enter image description here" /></a></p>
<p>Clearly, there is no (interior) feasible Pareto efficient allocation that satisfy <span class="math-container">$y=4$</span>. We can also check this by looking at the condition <span class="math-container">$x_2 = 2y^2-y$</span> which yields <span class="math-container">$x_2=28$</span> when we plug in <span class="math-container">$y=4$</span>, and that is not feasible.</p>
<p>To show that there is no efficient allocation at which <span class="math-container">$y=4$</span> holds at the edges also, we just need to consider two allocations:</p>
<ul>
<li><span class="math-container">$(x_1,x_2,y)=(0,14,4)$</span>. Observe that <span class="math-container">$(x_1,x_2,y)=(1,20,3)$</span> is Pareto Superior.</li>
<li><span class="math-container">$(x_1,x_2,y)=(14,0,4)$</span>. Observe that <span class="math-container">$(x_1,x_2,y)=(21,0,3)$</span> is Pareto Superior.</li>
</ul>
<p>Therefore, there is no Pareto efficient allocation where 4 units of public good are produced.</p>

CC BY-SA 4.0
2023-04-12T22:12:56.083Z
2023-04-12T22:34:32.997Z
2023-04-12T22:34:32.997Z
Amit
<p>Consider a 2 person 2 good economy where there is a private good <span class="math-container">$x$</span> and a public good <span class="math-container">$y$</span>. Agent 1 has an endowment of 10 units of the private good and Agent 2 has an endowment of 20 units of the private good. Initially, there is no public good in the economy. In order to produce <span class="math-container">$y$</span> units of the public good, <span class="math-container">$y^2$</span> units of the private good should be used. That is, the cost function is <span class="math-container">$c(y)=y^2$</span>.</p>
<p>Utility functions of the agents are as follows;</p>
<p><span class="math-container">$$u_1(x_1,y)=x_1+y$$</span> <span class="math-container">$$u_2(x_2,y)=x_2y$$</span></p>
<p>Firstly, I need to find the Pareto efficient allocations where 4 units of public good are produced.</p>
<p>Secondly, I need to find the private consumption of both agents as well as the public good level at the equilibrium.</p>
<p>—-</p>
<p>In other to find the Pareto efficient allocations, I maximize the sum of payoffs of both agents</p>
<p><span class="math-container">$$max\{ u_1(x_1,y)+ u_2(x_2,y)-c(y)\}$$</span></p>
<p>First order condition with respect to <span class="math-container">$y$</span> is <span class="math-container">$6+x_2-2y=0$</span></p>
<p>So, I found <span class="math-container">$x_2=2\times 4-6=2$</span></p>
<p>But my attempt is not true. I cannot do the correct solution. I will appreciate if you help me to solve the parts of this question.</p>

CC BY-SA 4.0
2022-08-14T22:20:37.680Z
Hi! 1. What textbook did you read, what mathematical condition are you using to determine Pareto-efficiency? 2. Why do you think your attempt is incorrect? If you are sure it is incorrect, why include it?
Giskard
CC BY-SA 4.0
2022-08-14T23:18:02.783Z
I remember this definition. I have an answer key but it only includes its results at the end. And my result doesn’t coincide with that of the answer key. I included my solution because when I post the question without any attempt, my post is either closed or downvoted. Due to rules, I add my wrong answer. @Giskard
studentp
CC BY-SA 4.0
2023-04-13T03:58:52.707Z
@studentp Are/Were you interested in the Lindahl equilibrium or competitive equilibrium?
Amit
<microeconomics><game-theory><self-study><pareto-efficiency>
Finding the Pareto efficient allocations
studentp
# Finding the Pareto efficient allocations

(Asked by: username_0 on 2022-08-14)

Consider a 2 person 2 good economy where there is a private good $x$ and a public good $y$. Agent 1 has an endowment of 10 units of the private good and Agent 2 has an endowment of 20 units of the private good. Initially, there is no public good in the economy. In order to produce $y$ units of the public good, $y^2$ units of the private good should be used. That is, the cost function is $c(y)=y^2$.

Utility functions of the agents are as follows;

$$u\_1(x\_1,y)=x\_1+y$$ $$u\_2(x\_2,y)=x\_2y$$

Firstly, I need to find the Pareto efficient allocations where 4 units of public good are produced.

Secondly, I need to find the private consumption of both agents as well as the public good level at the equilibrium.

—-

In other to find the Pareto efficient allocations, I maximize the sum of payoffs of both agents

$$max\{ u\_1(x\_1,y)+ u\_2(x\_2,y)-c(y)\}$$

First order condition with respect to $y$ is $6+x\_2-2y=0$

So, I found $x\_2=2\times 4-6=2$

But my attempt is not true. I cannot do the correct solution. I will appreciate if you help me to solve the parts of this question.

**Tags:** microeconomics, game-theory, self-study, pareto-efficiency

> **Comments:**
> **username_1**: Hi! 1. What textbook did you read, what mathematical condition are you using to determine Pareto-efficiency? 2. Why do you think your attempt is incorrect? If you are sure it is incorrect, why include it?  
> **username_0**: I remember this definition. I have an answer key but it only includes its results at the end. And my result doesn’t coincide with that of the answer key. I included my solution because when I post the question without any attempt, my post is either closed or downvoted. Due to rules, I add my wrong answer. @username_1  
> **username_2**: @username_0 Are/Were you interested in the Lindahl equilibrium or competitive equilibrium?  

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## 2 answers

### Answer by username_2 on 2023-04-12. Score: 5 | Accepted answer

Set of feasible allocations in this economy is:

$\mathcal{F}=\{(x\_1,x\_2,y)\in\mathbb{R}^3\_+|x\_1+x\_2+y^2=30\}$

This set can also be represented in graph in the following way:

[![enter image description here](https://i.sstatic.net/E0wrC.png)](https://i.sstatic.net/E0wrC.png)

To determine (interior) efficient allocations, we can first calculate MRS of individual 2 whose utility is $u\_2(x\_2,y)=x\_2y$, and get:

$\text{MRS}\_2 = \frac{y}{x\_2}$

Now we can rewrite $u\_1(x\_1, y)=x\_1+y$ as $u\_1^{\text{ADJ}}(x\_2,y) = u\_1(30-x\_2-y^2, y)=30-x\_2-y^2 + y$. We have just written $u\_1^{\text{ADJ}}$ as a function of $x\_2$ and $y$, by taking into account the feasibility constraint. We now calculate MRS of individual 1 using $u\_1^{\text{ADJ}}(x\_2,y)$, and get:

$\text{MRS}\_1^{\text{ADJ}} = \frac{1}{2y-1}$

(Interior) efficient allocations satisfy the condition:

$\text{MRS}\_1^{\text{ADJ}}=\text{MRS}\_2$ which gives: $x\_2 = 2y^2-y$

Here is the set of efficient allocations in the graph:

[![enter image description here](https://i.sstatic.net/DTYt0.png)](https://i.sstatic.net/DTYt0.png)

Clearly, there is no (interior) feasible Pareto efficient allocation that satisfy $y=4$. We can also check this by looking at the condition $x\_2 = 2y^2-y$ which yields $x\_2=28$ when we plug in $y=4$, and that is not feasible.

To show that there is no efficient allocation at which $y=4$ holds at the edges also, we just need to consider two allocations:

* $(x\_1,x\_2,y)=(0,14,4)$. Observe that $(x\_1,x\_2,y)=(1,20,3)$ is Pareto Superior.
* $(x\_1,x\_2,y)=(14,0,4)$. Observe that $(x\_1,x\_2,y)=(21,0,3)$ is Pareto Superior.

Therefore, there is no Pareto efficient allocation where 4 units of public good are produced.


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### Answer by username_3 on 2022-08-15. Score: 1

Let $y\_i$ be the contribution of agent i towards the production of $y$

$y=\sqrt{\sum y\_i}$

So when $y=4$
$\Rightarrow \sum y\_i=16 \Rightarrow \sum x\_i = 14$

The set of pareto efficient allocations is $\{(x\_1,x\_2):x\_1+x\_2=14,x\_1 \in [0,10], x\_2 \in [0,20]\}$

Private consumption of the goods will be the same as the endowments and the provision of public good will be $0$ at the equilibrium.

Reason:

$max\_{{x\_1,y\_1}: x\_1 + y\_1 = 10} x\_1 + \sqrt{\sum y\_i}$

$\Rightarrow \frac{39}{4} + y\_{2} = x\_{1}$

$max\_{(x\_2, y\_2):x\_2+y\_2=20} x\_2 \sqrt{\sum y\_i}$

$\Rightarrow x\_2= \frac{1}{3}(2y\_1 + 40)$

Let the price of the private good be $1$. Given the utility function of agent $1$ this implies
$y\_1=y \Rightarrow y\_2=y^2 - y$
Clearly the above system of equations has no real solution.
So equilibrium does not exist.

**Edit**: The equilibrium will be when $x\_1=10, y\_1=0, x\_2=\frac{40}{3},y\_2=\frac{20}{3}$
**Credits**: Agrim Rana

> **Comments:**
> **username_0**: I see. Thank you for your great answer!  

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