<blockquote>
  <p>Prove that $\Pi_j(x_j)$ is strictly concave in $x_j$.</p>
</blockquote>

<p>$\Pi_j(x_j) = G(x_j) + F\left(\frac{x_j}{y}\right) =G(x_j) + F\left(R_j(x_j)\right) $</p>

<p>Differentiating it we get</p>

<p>$\Pi_j'(x_j) =G'(x_j) + F'\left(R_j(x_j)\right)R_j'(x_j) $</p>

<p>Differentating $\Pi_j'(x_j)$, we get</p>

<p>$\Pi_j''(x_j) =G''(x_j) + F''\left(R_j(x_j)\right)(R_j'(x_j))^2 + F'\left(R_j(x_j)\right)R_j''(x_j) &lt; 0 $</p>

<p>Therefore, $\Pi_j$ is concave.</p>

<blockquote>
  <p>In a symmetric Nash equilibrium, determine the number of hours devoted to academic work. Call it $x^{**}$.</p>
</blockquote>

<p>Symmetric Nash equilibrium where everyone choose $x^{**}$ satisfy :</p>

<p>$G'(x^{**}) + F'\left(1\right)R_j'(x^{**}) = 0 $</p>

<blockquote>
  <p>Determine if $x^{**}$ is greater than, less than, or equal to $x^*$? </p>
</blockquote>

<p>Note that $x^*$ satisfy :</p>

<p>$G'(x^{*}) = 0$ </p>

<p>and $x^{**}$ satisfy :</p>

<p>$G'(x^{**}) + F'\left(1\right)R_j'(x^{**}) = 0$</p>

<p>Combining the two, we get</p>

<p>$F'\left(1\right)R_j'(x^{**}) = G'(x^{*}) - G'(x^{**})$</p>

<p>Since $F'\left(1\right) &gt; 0$ and  $R_j'(x^{**}) &gt; 0$ holds, we obtain</p>

<p>$G'(x^{*}) - G'(x^{**}) &gt; 0$</p>

<p>Since $G$ is concave, we get $x^* &lt; x^{**}$.</p>

<blockquote>
  <p>Compare the students' welfare at $x^*$ and $x^{**}$.</p>
</blockquote>

<p>$\Pi_j(x^{**})=G(x^{**}) + F(1) = G(x^{**}) &lt; G(x^{*})$</p>

<p>Therefore, students are worse off at $x^{**}$ compared to $x^{*}$.</p>

<blockquote>
  <p>Find the tax $\theta^* &gt; 0$ on academic work, that ensures that $x^{**}= x^{*}$.</p>
</blockquote>

<p>Symmetric Nash equilibrium where everyone choose $x^{**}$ in the presence of tax $\theta$ satisfy :</p>

<p>$G'(x^{**}) + F'\left(1\right)R_j'(x^{**}) - \theta = 0 $</p>

<p>To determine the sign of $\partial x^{**}/\partial \theta$, we differentiate the above condition with respect to $\theta$ :</p>

<p>$\displaystyle G''(x^{**})\frac{\partial x^{**}}{\partial \theta} + F'\left(1\right)R_j''(x^{**})\frac{\partial x^{**}}{\partial \theta} - 1 = 0 $</p>

<p>This yields</p>

<p>$\displaystyle \frac{\partial x^{**}}{\partial \theta}  = \frac{1}{G''(x^{**})+F'\left(1\right)R_j''(x^{**})} &lt; 0 $</p>

<p>Finally, we want to find $\theta^*$ that ensures $x^{**} = x^*$. To do so, we'll use</p>

<p>$G'(x^{**}) + F'\left(1\right)R_j'(x^{**}) - \theta = 0 $</p>

<p>and substitute $x^{**} = x^*$ in above to get</p>

<p>$\theta^* = F'\left(1\right)R_j'(x^{*})$</p>

CC BY-SA 4.0
2018-06-12T00:43:45.007Z
Amit is it possible to have a look at this question. It has to do with subgame perfect equilibrium. I am so confused. Thank you in advance. https://economics.stackexchange.com/questions/22413/subgame-perfect-equilibrium-and-expected-profit
Stefanos Makridis
CC BY-SA 4.0
2018-06-12T02:20:22.870Z
Dear Amit, I also post this question with Bertrand subgame perfect equilibrium. If you have time and if you can, it would be helpful to have a look.Thank you in advance.https://economics.stackexchange.com/questions/22415/bertrand-game-nash-equilibrium
Stefanos Makridis
CC BY-SA 4.0
2018-06-16T12:23:42.707Z
Amit I have one last question. Is it possible to have a  look if you have time ? https://economics.stackexchange.com/questions/22473/maximization-under-two-different-states-of-the-world
Stefanos Makridis
CC BY-SA 4.0
2018-06-08T10:37:17.793Z
2018-06-08T10:37:17.793Z
Amit
<p><a href="https://i.sstatic.net/aVQQg.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/aVQQg.jpg" alt="first page"></a></p>

<p><a href="https://i.sstatic.net/OMvqG.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OMvqG.jpg" alt="second page"></a></p>

<p>I have a problem regarding the last two questions of this exercises because I can not understand how to set up my model in order to find the optimal tax the optimal quantity to be as there was not a tax. </p>

<p><a href="https://i.sstatic.net/OSU8U.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OSU8U.jpg" alt="My thoughts"></a></p>

<p><a href="https://i.sstatic.net/4HnTC.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4HnTC.jpg" alt="My thoughts"></a></p>

<p>Please share your ideas with me. Thanks.</p>

<microeconomics><game-theory><self-study>
Finding the optimal tax
Stefanos Makridis
# Finding the optimal tax

(Asked by: username_0 on 2018-06-06)

[![first page](https://i.sstatic.net/aVQQg.jpg)](https://i.sstatic.net/aVQQg.jpg)

[![second page](https://i.sstatic.net/OMvqG.jpg)](https://i.sstatic.net/OMvqG.jpg)

I have a problem regarding the last two questions of this exercises because I can not understand how to set up my model in order to find the optimal tax the optimal quantity to be as there was not a tax.

[![My thoughts](https://i.sstatic.net/OSU8U.jpg)](https://i.sstatic.net/OSU8U.jpg)

[![My thoughts](https://i.sstatic.net/4HnTC.jpg)](https://i.sstatic.net/4HnTC.jpg)

Please share your ideas with me. Thanks.

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

---

## 1 answers

### Answer by username_1 on 2018-06-08. Score: 4 | Accepted answer

> Prove that $\Pi\_j(x\_j)$ is strictly concave in $x\_j$.

$\Pi\_j(x\_j) = G(x\_j) + F\left(\frac{x\_j}{y}\right) =G(x\_j) + F\left(R\_j(x\_j)\right) $

Differentiating it we get

$\Pi\_j'(x\_j) =G'(x\_j) + F'\left(R\_j(x\_j)\right)R\_j'(x\_j) $

Differentating $\Pi\_j'(x\_j)$, we get

$\Pi\_j''(x\_j) =G''(x\_j) + F''\left(R\_j(x\_j)\right)(R\_j'(x\_j))^2 + F'\left(R\_j(x\_j)\right)R\_j''(x\_j) < 0 $

Therefore, $\Pi\_j$ is concave.

> In a symmetric Nash equilibrium, determine the number of hours devoted to academic work. Call it $x^{\*\*}$.

Symmetric Nash equilibrium where everyone choose $x^{\*\*}$ satisfy :

$G'(x^{\*\*}) + F'\left(1\right)R\_j'(x^{\*\*}) = 0 $

> Determine if $x^{\*\*}$ is greater than, less than, or equal to $x^\*$?

Note that $x^\*$ satisfy :

$G'(x^{\*}) = 0$

and $x^{\*\*}$ satisfy :

$G'(x^{\*\*}) + F'\left(1\right)R\_j'(x^{\*\*}) = 0$

Combining the two, we get

$F'\left(1\right)R\_j'(x^{\*\*}) = G'(x^{\*}) - G'(x^{\*\*})$

Since $F'\left(1\right) > 0$ and $R\_j'(x^{\*\*}) > 0$ holds, we obtain

$G'(x^{\*}) - G'(x^{\*\*}) > 0$

Since $G$ is concave, we get $x^\* < x^{\*\*}$.

> Compare the students' welfare at $x^\*$ and $x^{\*\*}$.

$\Pi\_j(x^{\*\*})=G(x^{\*\*}) + F(1) = G(x^{\*\*}) < G(x^{\*})$

Therefore, students are worse off at $x^{\*\*}$ compared to $x^{\*}$.

> Find the tax $\theta^\* > 0$ on academic work, that ensures that $x^{\*\*}= x^{\*}$.

Symmetric Nash equilibrium where everyone choose $x^{\*\*}$ in the presence of tax $\theta$ satisfy :

$G'(x^{\*\*}) + F'\left(1\right)R\_j'(x^{\*\*}) - \theta = 0 $

To determine the sign of $\partial x^{\*\*}/\partial \theta$, we differentiate the above condition with respect to $\theta$ :

$\displaystyle G''(x^{\*\*})\frac{\partial x^{\*\*}}{\partial \theta} + F'\left(1\right)R\_j''(x^{\*\*})\frac{\partial x^{\*\*}}{\partial \theta} - 1 = 0 $

This yields

$\displaystyle \frac{\partial x^{\*\*}}{\partial \theta} = \frac{1}{G''(x^{\*\*})+F'\left(1\right)R\_j''(x^{\*\*})} < 0 $

Finally, we want to find $\theta^\*$ that ensures $x^{\*\*} = x^\*$. To do so, we'll use

$G'(x^{\*\*}) + F'\left(1\right)R\_j'(x^{\*\*}) - \theta = 0 $

and substitute $x^{\*\*} = x^\*$ in above to get

$\theta^\* = F'\left(1\right)R\_j'(x^{\*})$

> **Comments:**
> **username_0**: Amit is it possible to have a look at this question. It has to do with subgame perfect equilibrium. I am so confused. Thank you in advance. https://economics.stackexchange.com/questions/22413/subgame-perfect-equilibrium-and-expected-profit  
> **username_0**: Dear Amit, I also post this question with Bertrand subgame perfect equilibrium. If you have time and if you can, it would be helpful to have a look.Thank you in advance.https://economics.stackexchange.com/questions/22415/bertrand-game-nash-equilibrium  
> **username_0**: Amit I have one last question. Is it possible to have a  look if you have time ? https://economics.stackexchange.com/questions/22473/maximization-under-two-different-states-of-the-world  

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