<p>I'll be using more convenient notation with fewer subscripts. Here is the question:</p>
<blockquote>
<p>Given an economy with 20 identical consumers, one private good <span class="math-container">$x$</span> and
one public good <span class="math-container">$g$</span>. The public good is produced using the private
good as input with the following technology <span class="math-container">$g=f(x_0)=x_0^{1/2}$</span>, where
<span class="math-container">$x_0$</span> is units of private good employed in producing public good. The utility function of the consumer <span class="math-container">$i$</span>
is: <span class="math-container">$$u(g,x_i)=\ln(g)+\ln(x_i)$$</span> The individual endowment of private
good is <span class="math-container">$\omega_i=1$</span> each. Each consumer owns <span class="math-container">$5\%$</span> of the firm that
produces public good. Find Lindahl equilibrium.</p>
</blockquote>
<p>To find Lindahl equilibrium prices <span class="math-container">$p_i=p_j=p&gt;0$</span> for all <span class="math-container">$i,j\in\{1,2,\ldots, 20\}$</span>, we proceed in the following way.</p>
<p>First we'll solve the firm's profit maximisation problem:</p>
<p><span class="math-container">$\displaystyle\max_{(g,x_0)\in\mathbb{R}^2_+}(20p)g - x_0$</span> subject to <span class="math-container">$g\leq \sqrt{x_0}$</span></p>
<p>Solving this problem we get, the supply of public good and demand for the input as follows:</p>
<p><span class="math-container">$(g^s,x_0^d)(p) = (10p,100p^2)$</span></p>
<p>Therefore, optimal Profits of the firm are:</p>
<p><span class="math-container">$\pi(p)=100p^2$</span></p>
<p>Now we solve the consumer <span class="math-container">$i$</span>'s utility maximisation problem:</p>
<p><span class="math-container">$\displaystyle\max_{(g,x_i)\in\mathbb{R}^2_+}\ln g + \ln x_i$</span> subject to <span class="math-container">$pg + x_i=\omega_i+0.05(\pi(p))$</span></p>
<p>i.e. we solve</p>
<p><span class="math-container">$\displaystyle\max_{(g,x_i)\in\mathbb{R}^2_+}\ln g + \ln x_i$</span> subject to <span class="math-container">$pg + x_i=1+5p^2$</span></p>
<p>and get the demand for public good and private good as:</p>
<p><span class="math-container">$(g^d,x_i^d)(p) = \left(\dfrac{1+5p^2}{2p}, \dfrac{1+5p^2}{2}\right)$</span></p>
<p>Setting <span class="math-container">$g^s=g^d$</span>, we can solve for the Lindahl equilibrium price <span class="math-container">$p$</span>:</p>
<p><span class="math-container">$\dfrac{1+5p^2}{2p}=10p$</span></p>
<p>and we get <span class="math-container">$p=\frac{1}{\sqrt{15}}$</span>.</p>
<p>Corresponding demand and supply of public good equals <span class="math-container">$\frac{10}{\sqrt{15}}=\sqrt{\frac{20}{3}}$</span>. Demand for private good <span class="math-container">$x_i$</span> for consumption is <span class="math-container">$\frac{2}{3}$</span> for each of the consumer <span class="math-container">$i\in\{1,2,\ldots, 20\}$</span>, and as input for the production of public good it is <span class="math-container">$x_0=\frac{20}{3}$</span>. So, the aggregate demand for private good is <span class="math-container">$20(\frac{2}{3})+\frac{20}{3}=20$</span>.</p>
<p>This allocation obtained as Lindahl equilibrium is also Pareto efficient because it maximizes the sum of utilities of the 20 consumers subject to the feasibility constraint:</p>
<p><span class="math-container">$\displaystyle\max_{(x_0,x_1,\ldots, x_{20})\in\mathbb{R}^{21}_+} 20 \ln (\sqrt{x_0}) + \sum_{i=1}^{20}\ln (x_i)$</span> subject to <span class="math-container">$\displaystyle\sum_{i=0}^{20}x_i = 20$</span>, where <span class="math-container">$g=\sqrt{x_0}$</span>.</p>

CC BY-SA 4.0
2024-07-24T10:52:21.700Z
2024-07-24T11:16:13.610Z
2024-07-24T11:16:13.610Z
Amit
<p>Let an economy with 20 consumers, one private good $x_2$ and one public good $x_1$. The public good is produced using the private good as input with the following technology $x_1=g(z)=z^{1/2}$, where $z$ is units of private good. TheUtility function of the consumer "i" is: $$U(x_1,x_{2i})=ln(x_1)+ln(x_{2i})$$ the aggregate endowment of private good is $\sum_{i=1}^{20}w_i=20$. My problem is en the section b)</p>

<p>a)Obtain the pareto optimal allocation.</p>

<p>Here, I have the following lagrangian:</p>

<p>$$Max_{x_1,x_{2i},z}20ln(x_1)+\sum_{i=1}^{20}ln(x_{2i})+\lambda_1(40-\sum_{i=1}^{20}x_{2i}-z)+\lambda_2(z^{1/2}-x_1)$$</p>

<p>with CPO's:</p>

<p>$$x_1:20/x_1=\lambda_2$$ $$x_{2i}:1/x_{2i}=\lambda_1$$ $$z:-\lambda_1+\frac{1}{2}\lambda_2z^{-1/2}=0$$ $$\lambda_1: 40=\sum_{i=1}^{20}x_{2i}+z$$ $$\lambda_2:z^{1/2}=x_1$$</p>

<p>Where the solution $x_{2i}*=4/3$, $x_1*=\sqrt{\frac{40}{3}}$, $z*=\frac{40}{3}$ is pareto optimal.</p>

<p>b)Obtain the Lindahl equilibrium allocation with $·p_i=p_j \hspace{0.3cm} \forall i\neq j$. Here each consumer maximize his utility function subject to a restriction that has to be with a personalized price; and the firm which produce the public good maximize benefits. So the problem of consumer is: 
$$Max_{x_1,x_{2i}}ln(x_1)+ln(x_{2i})+\lambda_1(w_i-x_{2i}-p_ix_1)$$</p>

<p>The CPO's:</p>

<p>$$x_1:1/x_1=\lambda_1p_i$$ 
$$x_{2i}:1/x_{2i}=\lambda_1$$ 
$$\lambda_1:(w_i=x_{2i}+p_ix_1)$$</p>

<p>So, from the first and the second CPO: $x_{2i}=x_1p_i$ substituting in the third
$x_{2i}=\frac{w_i}{2}$ Here is my problem. In order to obtain the Pareto allocation in a) $x_{2i}=\frac{w_i}{2}=\frac{4}{3}$ so , it must be $w_i=8/3$ for each consumer $i$ but $\sum_{i=1}^{20}w_i=20\frac{8}{3}=\frac{160}{3} \neq 40$ So What is happening?</p>

CC BY-SA 4.0
2024-07-24T10:51:54.203Z
Your Pareto optimal allocation is incorrect. It is not possible for every individual to consume $\frac{4}{3}$ units of private good as the total amount of private good available is  $1$ unit per individual out of which some will also be used to produce public good.
Amit
<general-equilibrium><pareto-efficiency><public-economics><public-goods>
About Lindahl equilibrium and Pareto Optimality
Marcos Fabian C
# About Lindahl equilibrium and Pareto Optimality

(Asked by: username_0 on 2016-10-10)

Let an economy with 20 consumers, one private good $x\_2$ and one public good $x\_1$. The public good is produced using the private good as input with the following technology $x\_1=g(z)=z^{1/2}$, where $z$ is units of private good. TheUtility function of the consumer "i" is: $$U(x\_1,x\_{2i})=ln(x\_1)+ln(x\_{2i})$$ the aggregate endowment of private good is $\sum\_{i=1}^{20}w\_i=20$. My problem is en the section b)

a)Obtain the pareto optimal allocation.

Here, I have the following lagrangian:

$$Max\_{x\_1,x\_{2i},z}20ln(x\_1)+\sum\_{i=1}^{20}ln(x\_{2i})+\lambda\_1(40-\sum\_{i=1}^{20}x\_{2i}-z)+\lambda\_2(z^{1/2}-x\_1)$$

with CPO's:

$$x\_1:20/x\_1=\lambda\_2$$ $$x\_{2i}:1/x\_{2i}=\lambda\_1$$ $$z:-\lambda\_1+\frac{1}{2}\lambda\_2z^{-1/2}=0$$ $$\lambda\_1: 40=\sum\_{i=1}^{20}x\_{2i}+z$$ $$\lambda\_2:z^{1/2}=x\_1$$

Where the solution $x\_{2i}\*=4/3$, $x\_1\*=\sqrt{\frac{40}{3}}$, $z\*=\frac{40}{3}$ is pareto optimal.

b)Obtain the Lindahl equilibrium allocation with $·p\_i=p\_j \hspace{0.3cm} \forall i\neq j$. Here each consumer maximize his utility function subject to a restriction that has to be with a personalized price; and the firm which produce the public good maximize benefits. So the problem of consumer is:
$$Max\_{x\_1,x\_{2i}}ln(x\_1)+ln(x\_{2i})+\lambda\_1(w\_i-x\_{2i}-p\_ix\_1)$$

The CPO's:

$$x\_1:1/x\_1=\lambda\_1p\_i$$
$$x\_{2i}:1/x\_{2i}=\lambda\_1$$
$$\lambda\_1:(w\_i=x\_{2i}+p\_ix\_1)$$

So, from the first and the second CPO: $x\_{2i}=x\_1p\_i$ substituting in the third
$x\_{2i}=\frac{w\_i}{2}$ Here is my problem. In order to obtain the Pareto allocation in a) $x\_{2i}=\frac{w\_i}{2}=\frac{4}{3}$ so , it must be $w\_i=8/3$ for each consumer $i$ but $\sum\_{i=1}^{20}w\_i=20\frac{8}{3}=\frac{160}{3} \neq 40$ So What is happening?

**Tags:** general-equilibrium, pareto-efficiency, public-economics, public-goods
> **Comments:**
> **username_1**: Your Pareto optimal allocation is incorrect. It is not possible for every individual to consume $\frac{4}{3}$ units of private good as the total amount of private good available is  $1$ unit per individual out of which some will also be used to produce public good.  

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## 1 answers

### Answer by username_1 on 2024-07-24. Score: 1

I'll be using more convenient notation with fewer subscripts. Here is the question:

> Given an economy with 20 identical consumers, one private good $x$ and
> one public good $g$. The public good is produced using the private
> good as input with the following technology $g=f(x\_0)=x\_0^{1/2}$, where
> $x\_0$ is units of private good employed in producing public good. The utility function of the consumer $i$
> is: $$u(g,x\_i)=\ln(g)+\ln(x\_i)$$ The individual endowment of private
> good is $\omega\_i=1$ each. Each consumer owns $5\%$ of the firm that
> produces public good. Find Lindahl equilibrium.

To find Lindahl equilibrium prices $p\_i=p\_j=p>0$ for all $i,j\in\{1,2,\ldots, 20\}$, we proceed in the following way.

First we'll solve the firm's profit maximisation problem:

$\displaystyle\max\_{(g,x\_0)\in\mathbb{R}^2\_+}(20p)g - x\_0$ subject to $g\leq \sqrt{x\_0}$

Solving this problem we get, the supply of public good and demand for the input as follows:

$(g^s,x\_0^d)(p) = (10p,100p^2)$

Therefore, optimal Profits of the firm are:

$\pi(p)=100p^2$

Now we solve the consumer $i$'s utility maximisation problem:

$\displaystyle\max\_{(g,x\_i)\in\mathbb{R}^2\_+}\ln g + \ln x\_i$ subject to $pg + x\_i=\omega\_i+0.05(\pi(p))$

i.e. we solve

$\displaystyle\max\_{(g,x\_i)\in\mathbb{R}^2\_+}\ln g + \ln x\_i$ subject to $pg + x\_i=1+5p^2$

and get the demand for public good and private good as:

$(g^d,x\_i^d)(p) = \left(\dfrac{1+5p^2}{2p}, \dfrac{1+5p^2}{2}\right)$

Setting $g^s=g^d$, we can solve for the Lindahl equilibrium price $p$:

$\dfrac{1+5p^2}{2p}=10p$

and we get $p=\frac{1}{\sqrt{15}}$.

Corresponding demand and supply of public good equals $\frac{10}{\sqrt{15}}=\sqrt{\frac{20}{3}}$. Demand for private good $x\_i$ for consumption is $\frac{2}{3}$ for each of the consumer $i\in\{1,2,\ldots, 20\}$, and as input for the production of public good it is $x\_0=\frac{20}{3}$. So, the aggregate demand for private good is $20(\frac{2}{3})+\frac{20}{3}=20$.

This allocation obtained as Lindahl equilibrium is also Pareto efficient because it maximizes the sum of utilities of the 20 consumers subject to the feasibility constraint:

$\displaystyle\max\_{(x\_0,x\_1,\ldots, x\_{20})\in\mathbb{R}^{21}\_+} 20 \ln (\sqrt{x\_0}) + \sum\_{i=1}^{20}\ln (x\_i)$ subject to $\displaystyle\sum\_{i=0}^{20}x\_i = 20$, where $g=\sqrt{x\_0}$.


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