<p>Most of what you have done was correct. Here are the steps to finish the argument. </p>

<p>Evaluating the profits at the optimum, you get 
\begin{equation}
\pi_G=R(y^*(t))-C(y^*(t))
\quad\text{and}\quad 
\pi_N=R(y^*(t))-C(y^*(t))-ty^*(t).
\end{equation}
Differentiate the net profits with respect to $t$:
\begin{align}
\frac{\mathrm d\pi_N}{\mathrm dt}&amp;=R'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-C'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-t\frac{\mathrm dy^*}{\mathrm dt}-y^*\\&amp;=\underbrace{[R'(y^*)-C'(y^*)-t]}_{=0 \text{ due to FOC}}\frac{\mathrm dy^*}{\mathrm dt}-y^*&lt;0, 
\end{align}
which is true assuming $y^*&gt;0$. 
Differentiate the gross profits with respect to $t$: 
\begin{align}
\frac{\mathrm d\pi_G}{\mathrm dt}&amp;=R'(y^*)\frac{\mathrm dy^*}{\mathrm dt}-C'(y^*)\frac{\mathrm dy^*}{\mathrm dt}=[R'(y^*)-C'(y^*)]\frac{\mathrm dy^*}{\mathrm dt}&lt;0. 
\end{align}
This is true because from the FOC, we know that for any positive tax, $R'(y^*)-C'(y^*)&gt;0$, and since you've correctly derived $\frac{\mathrm dy^*}{\mathrm dt}&lt;0$, their product must be negative. </p>

CC BY-SA 3.0
2018-04-28T23:04:48.687Z
2018-04-28T23:04:48.687Z
Herr K.
<p>I tried to do following question but I am not sure about my solution.  Please tell me your opinions</p>
<p><a href="https://i.sstatic.net/iHUcN.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/iHUcN.jpg" alt="enter image description here" /></a></p>
<p>Since gross profit is strictly concave, I can say that <span class="math-container">$R’’(y)-C’’(y) &lt;0$</span></p>
<p>Now I maximize the net profit and</p>
<p>FOC is <span class="math-container">$R’(y)-C’(y) -t=0$</span></p>
<p>SOC is hold because of strict concavity in y.</p>
<p>That is <span class="math-container">$MR(y)=MC(y)+t$</span> from FOC.</p>
<p>Now show that the implicit relationship btw y and t can be solved for the explicit choice function</p>
<p><span class="math-container">$$y=y^*(t)$$</span></p>
<p>FOC with respect to y combine with above function</p>
<p><span class="math-container">$R’(y^*(t))-C’(y^*(t)) -t=0$</span></p>
<p>Derivative wrt t</p>
<p><span class="math-container">$$R’’(y) (dy^*/dt)-C’’(y) (dy^*/dt)-1=0$$</span></p>
<p><span class="math-container">$$\frac{dy^*}{dt}=\frac{1}{R’’-C’’}&lt;0$$</span></p>
<p>which means that the output will decrease as the tax goes up.</p>
<p>Think again gross profit</p>
<p><span class="math-container">$G\pi =R(y^*(t))-C(y^*(t))$</span></p>
<h2>I am not sure about my solution after this point especially.</h2>
<p>Suppose the firm is price taker then <span class="math-container">$R(y)=py$</span></p>
<p>So
<span class="math-container">$$G\pi =py^*(t)-C(y^*(t))$$</span></p>
<p>That is, since the output will decrease as tax rises, revenue <span class="math-container">$R(y) $</span> will decrease as well. Accordingly gross profit will decrease.</p>
<p>———————</p>
<p>I am not sure about the last part especially. Thank you.</p>

<taxation><self-study><profit-maximization><profits>
The relationship between gross profit and tax
b11bb
# The relationship between gross profit and tax

(Asked by: username_0 on 2018-04-28)

I tried to do following question but I am not sure about my solution. Please tell me your opinions

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

Since gross profit is strictly concave, I can say that $R’’(y)-C’’(y) <0$

Now I maximize the net profit and

FOC is $R’(y)-C’(y) -t=0$

SOC is hold because of strict concavity in y.

That is $MR(y)=MC(y)+t$ from FOC.

Now show that the implicit relationship btw y and t can be solved for the explicit choice function

$$y=y^\*(t)$$

FOC with respect to y combine with above function

$R’(y^\*(t))-C’(y^\*(t)) -t=0$

Derivative wrt t

$$R’’(y) (dy^\*/dt)-C’’(y) (dy^\*/dt)-1=0$$

$$\frac{dy^\*}{dt}=\frac{1}{R’’-C’’}<0$$

which means that the output will decrease as the tax goes up.

Think again gross profit

$G\pi =R(y^\*(t))-C(y^\*(t))$

## I am not sure about my solution after this point especially.

Suppose the firm is price taker then $R(y)=py$

So
$$G\pi =py^\*(t)-C(y^\*(t))$$

That is, since the output will decrease as tax rises, revenue $R(y) $ will decrease as well. Accordingly gross profit will decrease.

———————

I am not sure about the last part especially. Thank you.

**Tags:** taxation, self-study, profit-maximization, profits


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## 1 answers

### Answer by username_1 on 2018-04-28. Score: 2 | Accepted answer

Most of what you have done was correct. Here are the steps to finish the argument.

Evaluating the profits at the optimum, you get
\begin{equation}
\pi\_G=R(y^\*(t))-C(y^\*(t))
\quad\text{and}\quad
\pi\_N=R(y^\*(t))-C(y^\*(t))-ty^\*(t).
\end{equation}
Differentiate the net profits with respect to $t$:
\begin{align}
\frac{\mathrm d\pi\_N}{\mathrm dt}&=R'(y^\*)\frac{\mathrm dy^\*}{\mathrm dt}-C'(y^\*)\frac{\mathrm dy^\*}{\mathrm dt}-t\frac{\mathrm dy^\*}{\mathrm dt}-y^\*\\&=\underbrace{[R'(y^\*)-C'(y^\*)-t]}\_{=0 \text{ due to FOC}}\frac{\mathrm dy^\*}{\mathrm dt}-y^\*<0,
\end{align}
which is true assuming $y^\*>0$.
Differentiate the gross profits with respect to $t$:
\begin{align}
\frac{\mathrm d\pi\_G}{\mathrm dt}&=R'(y^\*)\frac{\mathrm dy^\*}{\mathrm dt}-C'(y^\*)\frac{\mathrm dy^\*}{\mathrm dt}=[R'(y^\*)-C'(y^\*)]\frac{\mathrm dy^\*}{\mathrm dt}<0.
\end{align}
This is true because from the FOC, we know that for any positive tax, $R'(y^\*)-C'(y^\*)>0$, and since you've correctly derived $\frac{\mathrm dy^\*}{\mathrm dt}<0$, their product must be negative.


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