<p>The monopolists original problem without the tax was:
<span class="math-container">$$\begin{aligned} 
\max_{p,y} \quad &amp; py(p)-c(y)\\
\textrm{s.t.} \quad &amp; y(p)=\begin{cases} \frac{100}{p} &amp; \text{if } p \leq 20\\ 0 &amp; \text{if } p&gt;20\end{cases} \\
&amp; c(y)=y
\end{aligned}$$</span></p>
<p>Upon substitution the problem becomes:
<span class="math-container">$$\begin{aligned} 
\max_{y\geq 5} \quad &amp; 100-y\\
\end{aligned}$$</span></p>
<p>Clearly, the objective is decreasing output <span class="math-container">$y$</span>, so we set output to the lowest value it can take i.e., <span class="math-container">$y^*=5$</span></p>
<p>Now let us consinder the case with the tax.
The tax imposed is per unit so his new cost function is <span class="math-container">$c(y)=y+1\cdot y=2y$</span></p>
<p>The new problem of the monopolist is:
<span class="math-container">$$\begin{aligned} 
\max_{p,y} \quad &amp; py(p)-c(y)\\
\textrm{s.t.} \quad &amp; y(p)=\begin{cases} \frac{100}{p} &amp; \text{if } p \leq 20\\ 0 &amp; \text{if } p&gt;20\end{cases} \\
&amp; c(y)=2y
\end{aligned}$$</span></p>
<p>using substitution like before gives:
<span class="math-container">$$\begin{aligned} 
\max_{y\geq 5} \quad &amp; 100-2y\\
\end{aligned}$$</span>
Which again gives <span class="math-container">$y^*=5$</span></p>
<p>Since the equilibrium quantity and price remains unchanged, it must be that the consumer surplus is still the same. Clearly the profit of the monopolist has reduced as compared to before due to paying taxes, but as Amit pointed out in the comment, the government's revenue increases by the difference in old and new profits of the monopolist. Therefore, the aggregate social surplus stays the same leading to no deadweight loss</p>

CC BY-SA 4.0
2023-04-25T04:06:22.400Z
2023-04-25T04:06:22.400Z
mynameparv
<p>If the govt. imposes a tax of <span class="math-container">$1$</span> per unit, then the new marginal cost becomes <span class="math-container">$2$</span> per unit.</p>
<p><strong>Profit.</strong> Using <span class="math-container">$p \leq 20 \iff \frac{100}{q} \leq 20 \iff q \geq 5$</span>, we have <span class="math-container">$\pi(q) = 100 - 2q$</span> when <span class="math-container">$q \geq 5$</span> <span class="math-container">$($</span>or <span class="math-container">$p \leq 20)$</span> which is maximized at <span class="math-container">$q^{*} = 5$</span>.</p>
<p><strong>Deadweight loss.</strong> There's no tax-induced deadweight loss in this scenario. As the tax doesn't change the optimal price/quantity, the only thing that changes is a part of the producer's surplus going to the government as tax revenue.</p>

CC BY-SA 4.0
2023-04-25T02:01:50.193Z
2023-04-26T01:09:40.837Z
2023-04-26T01:09:40.837Z
solow supremacy
<blockquote>
<p>A monopolist has cost function <span class="math-container">$c(y) = y$</span> so that its marginal cost is constant at 1 per unit. It faces the following demand curve <span class="math-container">$D(p) = \begin{cases}  \frac{100}{p}, &amp;\text{if}&amp;p ≤ 20 \\0,&amp;\text{if} &amp;p&gt;20 \end{cases}$</span></p>
<p>Find the profit maximizing level of output if the government imposes a per unit tax of Re. 1 per unit, and also the deadweight loss from the tax.</p>
</blockquote>
<p><span class="math-container">$TR=100 \Rightarrow$</span> <span class="math-container">$MR=0$</span> while <span class="math-container">$MC=1$</span> before tax is imposed and <span class="math-container">$2$</span> after tax is imposed. So, I can't even understand how to calculate profit-maximising output from the usual <span class="math-container">$MR=MC$</span> condition.</p>
<p>I can maybe see the profit maximising problem being reduced to cost-minimisation since the total revenue is constant. In which case, firm minimises its cost while earning a positive profit for <span class="math-container">$q=5$</span> for both cases of <span class="math-container">$MC.$</span></p>
<p>But, I still don't understand how to calculate deadweight loss. Could someone please guide me with this problem?</p>

CC BY-SA 4.0
2023-04-24T12:23:46.317Z
Try to solve the problem again by including per unit tax. You can solve the problem in this way: https://economics.stackexchange.com/a/22346/11824
Amit
CC BY-SA 4.0
2023-04-24T13:00:11.507Z
$q*$ and $p*$ do not change while $\pi_{BeforeTax}=100-q=95$ and $\pi_{AfterTax}=100-2q=90.$ Consumer surplus should not change, so is deadweight loss $95-90=5?$
alyosha
CC BY-SA 4.0
2023-04-24T13:32:25.017Z
$5$ is the tax revenue, so we need to add it back to the total surplus. So, there is no deadweight loss from the tax.
Amit
<microeconomics><taxation><monopoly>
Calculating deadweight loss from tax for a monopolist
alyosha
# Calculating deadweight loss from tax for a monopolist

(Asked by: username_0 on 2023-04-24)

> A monopolist has cost function $c(y) = y$ so that its marginal cost is constant at 1 per unit. It faces the following demand curve $D(p) = \begin{cases} \frac{100}{p}, &\text{if}&p ≤ 20 \\0,&\text{if} &p>20 \end{cases}$
>
> Find the profit maximizing level of output if the government imposes a per unit tax of Re. 1 per unit, and also the deadweight loss from the tax.

$TR=100 \Rightarrow$ $MR=0$ while $MC=1$ before tax is imposed and $2$ after tax is imposed. So, I can't even understand how to calculate profit-maximising output from the usual $MR=MC$ condition.

I can maybe see the profit maximising problem being reduced to cost-minimisation since the total revenue is constant. In which case, firm minimises its cost while earning a positive profit for $q=5$ for both cases of $MC.$

But, I still don't understand how to calculate deadweight loss. Could someone please guide me with this problem?

**Tags:** microeconomics, taxation, monopoly

> **Comments:**
> **username_1**: Try to solve the problem again by including per unit tax. You can solve the problem in this way: https://economics.stackexchange.com/a/22346/11824  
> **username_0**: $q*$ and $p*$ do not change while $\pi_{BeforeTax}=100-q=95$ and $\pi_{AfterTax}=100-2q=90.$ Consumer surplus should not change, so is deadweight loss $95-90=5?$  
> **username_1**: $5$ is the tax revenue, so we need to add it back to the total surplus. So, there is no deadweight loss from the tax.  

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## 2 answers

### Answer by username_3 on 2023-04-25. Score: 4 | Accepted answer

If the govt. imposes a tax of $1$ per unit, then the new marginal cost becomes $2$ per unit.

**Profit.** Using $p \leq 20 \iff \frac{100}{q} \leq 20 \iff q \geq 5$, we have $\pi(q) = 100 - 2q$ when $q \geq 5$ $($or $p \leq 20)$ which is maximized at $q^{\*} = 5$.

**Deadweight loss.** There's no tax-induced deadweight loss in this scenario. As the tax doesn't change the optimal price/quantity, the only thing that changes is a part of the producer's surplus going to the government as tax revenue.


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### Answer by username_2 on 2023-04-25. Score: 4

The monopolists original problem without the tax was:
$$\begin{aligned}
\max\_{p,y} \quad & py(p)-c(y)\\
\textrm{s.t.} \quad & y(p)=\begin{cases} \frac{100}{p} & \text{if } p \leq 20\\ 0 & \text{if } p>20\end{cases} \\
& c(y)=y
\end{aligned}$$

Upon substitution the problem becomes:
$$\begin{aligned}
\max\_{y\geq 5} \quad & 100-y\\
\end{aligned}$$

Clearly, the objective is decreasing output $y$, so we set output to the lowest value it can take i.e., $y^\*=5$

Now let us consinder the case with the tax.
The tax imposed is per unit so his new cost function is $c(y)=y+1\cdot y=2y$

The new problem of the monopolist is:
$$\begin{aligned}
\max\_{p,y} \quad & py(p)-c(y)\\
\textrm{s.t.} \quad & y(p)=\begin{cases} \frac{100}{p} & \text{if } p \leq 20\\ 0 & \text{if } p>20\end{cases} \\
& c(y)=2y
\end{aligned}$$

using substitution like before gives:
$$\begin{aligned}
\max\_{y\geq 5} \quad & 100-2y\\
\end{aligned}$$
Which again gives $y^\*=5$

Since the equilibrium quantity and price remains unchanged, it must be that the consumer surplus is still the same. Clearly the profit of the monopolist has reduced as compared to before due to paying taxes, but as username_1 pointed out in the comment, the government's revenue increases by the difference in old and new profits of the monopolist. Therefore, the aggregate social surplus stays the same leading to no deadweight loss


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