<p>you can do this using comparative statics and envelope theorem. The profit function specification was correct given the assumptions, let me just rewrite it as:</p>
<p><span class="math-container">$$\pi (q,t) = p q - C(q) - tq$$</span></p>
<p>However, given the above we also know that the optimal quantity will depend also on a tax rate, so we will have <span class="math-container">$q^* = q(t)$</span>.</p>
<p>Now by envelope theorem:</p>
<p><span class="math-container">$$\frac{\pi^*(t)}{dt} = \pi'_2(q^*(t),t) = - Q^*(t) $$</span></p>
<p>So the optimal level of output given the tax <span class="math-container">$t$</span> will depend inversely  on change in profit when the tax rate changes or  <span class="math-container">$-\pi'_2(q^*(t),t) = Q^*(t)$</span>.</p>

CC BY-SA 4.0
2020-11-24T16:49:43.300Z
2020-11-24T16:49:43.300Z
1muflon1
<p><strong>Intuitive/graphical answer</strong>:</p>
<p>A tax per unit of production basically increases marginal cost of production at each level of production. Effectively, it shifts MC curve up by <span class="math-container">$t$</span> units. Now we know that in a competitive market, firms are price takers (so horizontal demand curve). So they'll produce till the point MC touches <span class="math-container">$P$</span>. Given that MC is increasing<span class="math-container">$^\dagger$</span>, an upward shift in in MC curve means decrease in output. More the shift more the decrease in output.</p>
<p><a href="https://i.sstatic.net/wa5xI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/wa5xI.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$^\dagger:$</span> In a perfect competition, decreasing MC is not possible because there is infinite demand for each firm - so firm will always operate in increasing MC part</p>
<p><strong>Mathematically:</strong>
You have already found the FOC, which states that at tax rate <span class="math-container">$t$</span>, output supplied <span class="math-container">$q^*(t)$</span> will satisfy the following condition:</p>
<p><span class="math-container">\begin{align}
C'(q^*(t))+t-P&amp;=0 \\
\end{align}</span></p>
<p>Differentiating this equation w.r.t <span class="math-container">$t$</span>:
<span class="math-container">\begin{align}
\frac{dC'(q^*(t))}{dt}+1&amp;=0 \\
\frac{\partial C'(q^*)}{\partial q^*} \cdot \frac{dq^*}{dt}&amp;=-1 \\
\frac{dq^*}{dt}&amp;=-1/C''(q^*)&lt;0 
\end{align}</span></p>
<p>The last step uses the SOC:</p>
<p><span class="math-container">$$\frac{d^2\pi}{dq^2}=-C''(q)&lt;0$$</span></p>

CC BY-SA 4.0
2020-11-24T19:55:54.970Z
2020-11-24T19:55:54.970Z
Dayne
<p>Suppose the government imposes a tax on each unit of output
produced of $t. What is the firm’s new profit maximizing level of
output?</p>
<p>The question suggests the use of comparative to determine how a change in tax affects a change in output. The firm would operate under perfect competition conditions.</p>
<p>I started to answer it this way.</p>
<p>Before Tax: <br>
π = revenue - costs <br>
π = pq - C(q) <br>
dπ/dq = p - C'(q) <br></p>
<p>Set dπ/dq for profit-maximisation <br>
0 = p - C'(q) <br>
p = C'(q) <br>
p = MC</p>
<p>After Tax: <br>
π = revenue - costs - tax <br>
π = pq - C(q) -tq<br>
dπ/dq = p - t - C'(q) <br></p>
<p>Set dπ/dq for profit-maximisation <br>
0 = p - t - C'(q) <br>
p = t + C'(q) <br>
p = t + MC <br></p>
<p>Next I would try to find out how quantity changes with tax t by taking derivates? Any ideas?</p>

<microeconomics>
Effects of change in tax on output
pe78600
# Effects of change in tax on output

(Asked by: username_0 on 2020-11-24)

Suppose the government imposes a tax on each unit of output
produced of $t. What is the firm’s new profit maximizing level of
output?

The question suggests the use of comparative to determine how a change in tax affects a change in output. The firm would operate under perfect competition conditions.

I started to answer it this way.

Before Tax:   
π = revenue - costs   
π = pq - C(q)   
dπ/dq = p - C'(q)

Set dπ/dq for profit-maximisation   
0 = p - C'(q)   
p = C'(q)   
p = MC

After Tax:   
π = revenue - costs - tax   
π = pq - C(q) -tq  
dπ/dq = p - t - C'(q)

Set dπ/dq for profit-maximisation   
0 = p - t - C'(q)   
p = t + C'(q)   
p = t + MC

Next I would try to find out how quantity changes with tax t by taking derivates? Any ideas?

**Tags:** microeconomics


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## 2 answers

### Answer by username_1 on 2020-11-24. Score: 0

you can do this using comparative statics and envelope theorem. The profit function specification was correct given the assumptions, let me just rewrite it as:

$$\pi (q,t) = p q - C(q) - tq$$

However, given the above we also know that the optimal quantity will depend also on a tax rate, so we will have $q^\* = q(t)$.

Now by envelope theorem:

$$\frac{\pi^\*(t)}{dt} = \pi'\_2(q^\*(t),t) = - Q^\*(t) $$

So the optimal level of output given the tax $t$ will depend inversely on change in profit when the tax rate changes or $-\pi'\_2(q^\*(t),t) = Q^\*(t)$.


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### Answer by username_2 on 2020-11-24. Score: 0

**Intuitive/graphical answer**:

A tax per unit of production basically increases marginal cost of production at each level of production. Effectively, it shifts MC curve up by $t$ units. Now we know that in a competitive market, firms are price takers (so horizontal demand curve). So they'll produce till the point MC touches $P$. Given that MC is increasing$^\dagger$, an upward shift in in MC curve means decrease in output. More the shift more the decrease in output.

[![enter image description here](https://i.sstatic.net/wa5xI.png)](https://i.sstatic.net/wa5xI.png)

$^\dagger:$ In a perfect competition, decreasing MC is not possible because there is infinite demand for each firm - so firm will always operate in increasing MC part

**Mathematically:**
You have already found the FOC, which states that at tax rate $t$, output supplied $q^\*(t)$ will satisfy the following condition:

\begin{align}
C'(q^\*(t))+t-P&=0 \\
\end{align}

Differentiating this equation w.r.t $t$:
\begin{align}
\frac{dC'(q^\*(t))}{dt}+1&=0 \\
\frac{\partial C'(q^\*)}{\partial q^\*} \cdot \frac{dq^\*}{dt}&=-1 \\
\frac{dq^\*}{dt}&=-1/C''(q^\*)<0
\end{align}

The last step uses the SOC:

$$\frac{d^2\pi}{dq^2}=-C''(q)<0$$


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