<blockquote>
<p>(I) A monopolist has a cost function <span class="math-container">$c(q)=q$</span>. It faces the following demand function <span class="math-container">$ q(p)=100/p$</span> for <span class="math-container">$p\le 20$</span> and <span class="math-container">$q(p)=0$</span> for <span class="math-container">$p\ge 20$</span>. What are the profit maximizing price and output.</p>
</blockquote>
<p>Given the demand function
<span class="math-container">\begin{eqnarray*} q(p) = \begin{cases} \frac{100}{p} &amp; \text{if } p \leq 20 \\ 0 &amp; \text{if } p &gt; 20\end{cases} \end{eqnarray*}</span></p>
<p>So, revenue as a function of quantity <span class="math-container">$(q)$</span> is
<span class="math-container">\begin{eqnarray*} r(q) = \begin{cases} 100 &amp; \text{if } q \geq 5 \\ 0 &amp; \text{if } q &lt; 5\end{cases} \end{eqnarray*}</span></p>
<p>Given that the cost is <span class="math-container">$c(q) = q$</span>, profit as a function of quantity <span class="math-container">$(q)$</span> is
<span class="math-container">\begin{eqnarray*} \pi(q) = r(q) - c(q) = \begin{cases} 100 - q &amp; \text{if } q \geq 5 \\ -q &amp; \text{if } q &lt; 5\end{cases} \end{eqnarray*}</span></p>
<p>We simply solve the following problem to find the optimal quantity of the monopolist:
<span class="math-container">\begin{eqnarray*} \max_{q \geq 5} \ 100 - q \end{eqnarray*}</span></p>
<p>Solving it we get the optimal quantity <span class="math-container">$(q^m)$</span> and the corresponding price <span class="math-container">$(p^m)$</span> chosen by the monopolist as
<span class="math-container">\begin{eqnarray*} q^m &amp; = &amp; 5 \\ p^m &amp; = &amp; 20\end{eqnarray*}</span></p>
<blockquote>
<p>(ii) a monopolist has a cost function <span class="math-container">$c(q)=\alpha q$</span> where alpha is fixed marginal cost. It’s demand function has a constant price elasticity of demand whose value is <span class="math-container">$-3$</span>. The government imposes a tax of 6 dollar per unit of output. By how much will the monopolist price rise?</p>
</blockquote>
<p>Given that the demand function has a constant price elasticity whose value is <span class="math-container">$-3$</span>, it is of the form:
<span class="math-container">$$q(p) = \beta p^{-3}$$</span>, where <span class="math-container">$\beta &gt; 0$</span> is a constant.</p>
<p>Now profit maximization problem of the monopolist in terms of price <span class="math-container">$(p)$</span> can be written as :
<span class="math-container">\begin{eqnarray*} \max_{p} \ \ pq(p) - c(q(p)) \end{eqnarray*}</span></p>
<p>Given that the cost is <span class="math-container">$c(q)=\alpha q$</span> and the demand is <span class="math-container">$q(p) = \beta p^{-3}$</span>, the profit maximization problem is :
<span class="math-container">\begin{eqnarray*} \max_{p} \beta p^{-2} - \alpha\beta p^{-3} \end{eqnarray*}</span>
Solving it we get the optimal price as:
<span class="math-container">\begin{eqnarray*} p^m = \frac{3\alpha}{2} \end{eqnarray*}</span></p>
<p>With tax of 6 per unit, the profit maximization problem is
<span class="math-container">\begin{eqnarray*} \max_{p} \beta p^{-2} - \alpha\beta p^{-3} - 6\beta p^{-3}\end{eqnarray*}</span></p>
<p>Solving it we get the optimal price as:
<span class="math-container">\begin{eqnarray*} p^t = \frac{3(\alpha+6)}{2} \end{eqnarray*}</span></p>
<p>Therefore, increase in price due to tax equals :
<span class="math-container">\begin{eqnarray*} \Delta = p^t - p^m = 9 \end{eqnarray*}</span></p>

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2018-06-09T07:49:12.003Z
Dear Amit, We have some questions.when you have time, please look at them? Thanks a lot:)
1. 
https://economics.stackexchange.com/questions/22365/subgame-perfect-equilibrium-question
2. 
https://economics.stackexchange.com/questions/22363/utility-maximization-problem-with-free-disposal
3. 
https://economics.stackexchange.com/questions/22362/pure-exchange-economy-with-free-and-non-free-disposal
4. 
https://economics.stackexchange.com/questions/22382/game-theory-question-with-technology
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CC BY-SA 4.0
2018-06-07T10:02:13.130Z
2022-03-04T11:19:20.260Z
2022-03-04T11:19:20.260Z
Amit
<p>ii.</p>

<p>I don't think the demand function in (i.) applies for this question. Here's what I did:</p>

<p>By Lerner Index, $\displaystyle{\frac{P - MC}{P} = \frac{-1}{E_d}}$</p>

<p>Since $E_d = -3$, we get $P = \frac{3}{2} MC$</p>

<p>We know before tax MC is $c'(q) = \alpha$, and after tax MC is $\alpha + 6$</p>

<p>In this sense, MC increased by 6, so P will probably increase by 9</p>

CC BY-SA 4.0
2018-06-07T09:46:08.217Z
2018-06-07T10:06:02.117Z
2018-06-07T10:06:02.117Z
T. G.
<p>(I) A monopolist has a cost function $c(q)=q$. It faces the following demand function $ q(p)=100/p$ for $p\le 20$ and $q(p)=0$ for $p\ge 20$. What are the profit maximizing price and output. </p>

<p>(ii) a monopolist has a cost function $c(q)=\alpha q$ where alpha is fixed marginal cost. It’s demand function has a constant price elasticity of demand whose value is $-3$. The government imposes a tax of 6 dollar per unit of output. By how much will the monopolist price rise?</p>

<p>My solutions are as follows:</p>

<p>I am not sure whether it is correct or not. Please tell me and show me my mistakes. </p>

<p><a href="https://i.sstatic.net/ost5T.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ost5T.jpg" alt="enter image description here"></a>
<a href="https://i.sstatic.net/OtBnE.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OtBnE.jpg" alt="enter image description here"></a></p>

<microeconomics><self-study><profit-maximization>
Question on profit maximization with government taxation
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# Question on profit maximization with government taxation

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

(I) A monopolist has a cost function $c(q)=q$. It faces the following demand function $ q(p)=100/p$ for $p\le 20$ and $q(p)=0$ for $p\ge 20$. What are the profit maximizing price and output.

(ii) a monopolist has a cost function $c(q)=\alpha q$ where alpha is fixed marginal cost. It’s demand function has a constant price elasticity of demand whose value is $-3$. The government imposes a tax of 6 dollar per unit of output. By how much will the monopolist price rise?

My solutions are as follows:

I am not sure whether it is correct or not. Please tell me and show me my mistakes.

[![enter image description here](https://i.sstatic.net/ost5T.jpg)](https://i.sstatic.net/ost5T.jpg)
[![enter image description here](https://i.sstatic.net/OtBnE.jpg)](https://i.sstatic.net/OtBnE.jpg)

**Tags:** microeconomics, self-study, profit-maximization

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## 2 answers

### Answer by username_1 on 2018-06-07. Score: 1 | Accepted answer

> (I) A monopolist has a cost function $c(q)=q$. It faces the following demand function $ q(p)=100/p$ for $p\le 20$ and $q(p)=0$ for $p\ge 20$. What are the profit maximizing price and output.

Given the demand function
\begin{eqnarray\*} q(p) = \begin{cases} \frac{100}{p} & \text{if } p \leq 20 \\ 0 & \text{if } p > 20\end{cases} \end{eqnarray\*}

So, revenue as a function of quantity $(q)$ is
\begin{eqnarray\*} r(q) = \begin{cases} 100 & \text{if } q \geq 5 \\ 0 & \text{if } q < 5\end{cases} \end{eqnarray\*}

Given that the cost is $c(q) = q$, profit as a function of quantity $(q)$ is
\begin{eqnarray\*} \pi(q) = r(q) - c(q) = \begin{cases} 100 - q & \text{if } q \geq 5 \\ -q & \text{if } q < 5\end{cases} \end{eqnarray\*}

We simply solve the following problem to find the optimal quantity of the monopolist:
\begin{eqnarray\*} \max\_{q \geq 5} \ 100 - q \end{eqnarray\*}

Solving it we get the optimal quantity $(q^m)$ and the corresponding price $(p^m)$ chosen by the monopolist as
\begin{eqnarray\*} q^m & = & 5 \\ p^m & = & 20\end{eqnarray\*}

> (ii) a monopolist has a cost function $c(q)=\alpha q$ where alpha is fixed marginal cost. It’s demand function has a constant price elasticity of demand whose value is $-3$. The government imposes a tax of 6 dollar per unit of output. By how much will the monopolist price rise?

Given that the demand function has a constant price elasticity whose value is $-3$, it is of the form:
$$q(p) = \beta p^{-3}$$, where $\beta > 0$ is a constant.

Now profit maximization problem of the monopolist in terms of price $(p)$ can be written as :
\begin{eqnarray\*} \max\_{p} \ \ pq(p) - c(q(p)) \end{eqnarray\*}

Given that the cost is $c(q)=\alpha q$ and the demand is $q(p) = \beta p^{-3}$, the profit maximization problem is :
\begin{eqnarray\*} \max\_{p} \beta p^{-2} - \alpha\beta p^{-3} \end{eqnarray\*}
Solving it we get the optimal price as:
\begin{eqnarray\*} p^m = \frac{3\alpha}{2} \end{eqnarray\*}

With tax of 6 per unit, the profit maximization problem is
\begin{eqnarray\*} \max\_{p} \beta p^{-2} - \alpha\beta p^{-3} - 6\beta p^{-3}\end{eqnarray\*}

Solving it we get the optimal price as:
\begin{eqnarray\*} p^t = \frac{3(\alpha+6)}{2} \end{eqnarray\*}

Therefore, increase in price due to tax equals :
\begin{eqnarray\*} \Delta = p^t - p^m = 9 \end{eqnarray\*}

> **Comments:**
> **username_0**: Dear Amit, We have some questions.when you have time, please look at them? Thanks a lot:) 1.  https://economics.stackexchange.com/questions/22365/subgame-perfect-equilibrium-question 2.  https://economics.stackexchange.com/questions/22363/utility-maximization-problem-with-free-disposal 3.  https://economics.stackexchange.com/questions/22362/pure-exchange-economy-with-free-and-non-free-disposal 4.  https://economics.stackexchange.com/questions/22382/game-theory-question-with-technology  

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### Answer by username_2 on 2018-06-07. Score: 1

ii.

I don't think the demand function in (i.) applies for this question. Here's what I did:

By Lerner Index, $\displaystyle{\frac{P - MC}{P} = \frac{-1}{E\_d}}$

Since $E\_d = -3$, we get $P = \frac{3}{2} MC$

We know before tax MC is $c'(q) = \alpha$, and after tax MC is $\alpha + 6$

In this sense, MC increased by 6, so P will probably increase by 9


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