<p>The profit function is given by:</p>
<p><span class="math-container">$$
\pi = PQ(P) - C(Q(P), \mu) - tQ(P)
$$</span>
Assume demand is linear, s
<span class="math-container">$$
p = \alpha - \beta Q \to Q = b(\alpha - P),
$$</span>
where <span class="math-container">$b = 1/\beta$</span>. So:
<span class="math-container">$$
Q_P = -b,
$$</span>
where I use subscripts to denote the partial derivatives.</p>
<p>The first order condition for profit maximisation gives:
<span class="math-container">$$
\begin{align*}
    &amp;Q + P Q_P - C_Q Q_P - t Q_P = 0,\\
\iff &amp;Q - b P + b C_Q + bt = 0
\end{align*}
$$</span>
then differentiating this with respect to  <span class="math-container">$t$</span> gives:
<span class="math-container">$$
\begin{align*}
&amp;(Q_P - b + b C_{QQ} Q_P)\frac{\partial P}{\partial t} + b = 0,\\
\to &amp;(-2b - b^2 C_{QQ})\frac{\partial P}{\partial t} = -b,\\
\to &amp;\frac{\partial P}{\partial t} = \frac{1}{2 + b C_{QQ}}
\end{align*}
$$</span>
Differentiating the first order condition with respect to <span class="math-container">$\mu$</span> gives:
<span class="math-container">$$
(Q_P - b + b C_{QQ} Q_P)\frac{\partial P}{\partial \mu} + b C_{Q\mu} = 0,\\
\to (-2b - b^2 C_{QQ})\frac{\partial P}{\partial \mu} = - b C_{Q\mu},\\
\to \frac{\partial P}{\partial \mu} = \frac{C_{Q,\mu}}{2 + b C_{QQ}}
$$</span></p>
<p>Then differentiating <span class="math-container">$\frac{\partial P}{\partial t}$</span> once more with respect to <span class="math-container">$\mu$</span> gives:
<span class="math-container">$$
\frac{\partial^2 P}{\partial t \partial \mu} = -\frac{1}{(2 + bC_{QQ})^2}b\left(C_{QQQ}Q_P \frac{\partial P}{\partial \mu}+ C_{QQ\mu}\right),\\
= -\frac{1}{(2 + b C_{QQ})^2}b \left(-b C_{QQQ} \frac{C_{Q,\mu}}{2 + b C_{QQ}} + C_{QQ\mu} \right)
$$</span>
Where the last line uses the expression for <span class="math-container">$\frac{\partial P}{\partial \mu}$</span> from above.</p>
<p>Now assume that costs take the form:
<span class="math-container">$$
C(Q, \mu) = \delta + \eta Q + \frac{\gamma(\mu)\, Q^2}{2}
$$</span>
So the slope of the marginal costs depends on <span class="math-container">$\mu$</span>. Then <span class="math-container">$C_{QQ} = \gamma(\mu)$</span>, <span class="math-container">$C_{QQQ} = 0$</span>  and <span class="math-container">$C_{QQ\mu} = \gamma_\mu$</span> , so:
<span class="math-container">$$
\frac{\partial^2 P}{\partial t \partial \mu} = -\frac{b \gamma_\mu}{(2 + b \gamma)^2}.
$$</span>
We have that <span class="math-container">$\gamma_\mu &lt; 0$</span> by assumption (slope decreases in <span class="math-container">$\mu$</span>) then we see that the sign of <span class="math-container">$\frac{\partial^2 P}{\partial t \partial \mu}$</span> is positive (at least if I didn't make any mistakes).</p>

CC BY-SA 4.0
2021-05-27T17:51:49.497Z
2021-05-27T17:51:49.497Z
tdm
<p>Consider a profit maximising monopolist with linear demand Q(P*) and total production cost C(Q(P*)) who faces a per unit tax t. Suppose the slope of marginal cost is decreasing in some parameter, μ. Let P* denote the monopolist’ profit maximising choice of price. Being careful to explain your method and interpret your result, determine the comparative static:</p>
<p>∂²P*/∂μ∂t</p>
<p>I got to P*= price Q = Q(P*) t=t(Q(P*)) C=C(Q(P*))</p>
<p>πmax = P*(Q(P*))-C(Q(P*))-t(Q(P*))</p>
<p>dπ/dP* = Q'(P*)[-C'(Q(P*))-t'(Q(P*))+P*]+Q(P*)=0 dπ/dP* = μ + Q'(P*)[-t'(Q(P*))+P*]+Q(P*)</p>
<p>Equilibrum: PQ'(P)=t'(Q(P*))Q(P))-μ-Q(P*)</p>
<p>P* = t'(Q(P*))- μ/(Q'(P*)) - Q(P*)/Q'(P*)</p>
<p>This is where I got to, when I solved the comparative static partial differentiation I got an answer of 0, I don't believe this is correct, can anyone help me solve this? Thanks!</p>
<p>If 0 is somehow the correct answer, what does it mean?</p>

CC BY-SA 4.0
2021-05-27T05:49:04.350Z
Hi! Can you please type your calculations into Mathjax and write separate lines separately so that the question is readable?
Giskard
CC BY-SA 4.0
2021-05-27T05:50:14.703Z
Also, what is t'? And do I understand correctly that you just didn't type in your final few lines of calculations?
Giskard
CC BY-SA 4.0
2021-05-27T11:39:21.343Z
So total tax is t*Q  so it comes out as t(Q(P*))  so  t(Q(P*)) differentiates to t'(Q(P*)) * Q'(P*) if I am not mistaken?
Javid
<microeconomics><mathematical-economics><monopoly><comparative-statics>
Comparative statics of a monopoly
Javid
# Comparative statics of a monopoly

(Asked by: username_0 on 2021-05-27)

Consider a profit maximising monopolist with linear demand Q(P\*) and total production cost C(Q(P\*)) who faces a per unit tax t. Suppose the slope of marginal cost is decreasing in some parameter, μ. Let P\* denote the monopolist’ profit maximising choice of price. Being careful to explain your method and interpret your result, determine the comparative static:

∂²P\*/∂μ∂t

I got to P\*= price Q = Q(P\*) t=t(Q(P\*)) C=C(Q(P\*))

πmax = P\*(Q(P\*))-C(Q(P\*))-t(Q(P\*))

dπ/dP\* = Q'(P\*)[-C'(Q(P\*))-t'(Q(P\*))+P\*]+Q(P\*)=0 dπ/dP\* = μ + Q'(P\*)[-t'(Q(P\*))+P\*]+Q(P\*)

Equilibrum: PQ'(P)=t'(Q(P\*))Q(P))-μ-Q(P\*)

P\* = t'(Q(P\*))- μ/(Q'(P\*)) - Q(P\*)/Q'(P\*)

This is where I got to, when I solved the comparative static partial differentiation I got an answer of 0, I don't believe this is correct, can anyone help me solve this? Thanks!

If 0 is somehow the correct answer, what does it mean?

**Tags:** microeconomics, mathematical-economics, monopoly, comparative-statics

> **Comments:**
> **username_1**: Hi! Can you please type your calculations into Mathjax and write separate lines separately so that the question is readable?  
> **username_1**: Also, what is t'? And do I understand correctly that you just didn't type in your final few lines of calculations?  
> **username_0**: So total tax is t*Q  so it comes out as t(Q(P*))  so  t(Q(P*)) differentiates to t'(Q(P*)) * Q'(P*) if I am not mistaken?  

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## 1 answers

### Answer by username_2 on 2021-05-27. Score: 2 | Accepted answer

The profit function is given by:

$$
\pi = PQ(P) - C(Q(P), \mu) - tQ(P)
$$
Assume demand is linear, s
$$
p = \alpha - \beta Q \to Q = b(\alpha - P),
$$
where $b = 1/\beta$. So:
$$
Q\_P = -b,
$$
where I use subscripts to denote the partial derivatives.

The first order condition for profit maximisation gives:
$$
\begin{align\*}
&Q + P Q\_P - C\_Q Q\_P - t Q\_P = 0,\\
\iff &Q - b P + b C\_Q + bt = 0
\end{align\*}
$$
then differentiating this with respect to $t$ gives:
$$
\begin{align\*}
&(Q\_P - b + b C\_{QQ} Q\_P)\frac{\partial P}{\partial t} + b = 0,\\
\to &(-2b - b^2 C\_{QQ})\frac{\partial P}{\partial t} = -b,\\
\to &\frac{\partial P}{\partial t} = \frac{1}{2 + b C\_{QQ}}
\end{align\*}
$$
Differentiating the first order condition with respect to $\mu$ gives:
$$
(Q\_P - b + b C\_{QQ} Q\_P)\frac{\partial P}{\partial \mu} + b C\_{Q\mu} = 0,\\
\to (-2b - b^2 C\_{QQ})\frac{\partial P}{\partial \mu} = - b C\_{Q\mu},\\
\to \frac{\partial P}{\partial \mu} = \frac{C\_{Q,\mu}}{2 + b C\_{QQ}}
$$

Then differentiating $\frac{\partial P}{\partial t}$ once more with respect to $\mu$ gives:
$$
\frac{\partial^2 P}{\partial t \partial \mu} = -\frac{1}{(2 + bC\_{QQ})^2}b\left(C\_{QQQ}Q\_P \frac{\partial P}{\partial \mu}+ C\_{QQ\mu}\right),\\
= -\frac{1}{(2 + b C\_{QQ})^2}b \left(-b C\_{QQQ} \frac{C\_{Q,\mu}}{2 + b C\_{QQ}} + C\_{QQ\mu} \right)
$$
Where the last line uses the expression for $\frac{\partial P}{\partial \mu}$ from above.

Now assume that costs take the form:
$$
C(Q, \mu) = \delta + \eta Q + \frac{\gamma(\mu)\, Q^2}{2}
$$
So the slope of the marginal costs depends on $\mu$. Then $C\_{QQ} = \gamma(\mu)$, $C\_{QQQ} = 0$ and $C\_{QQ\mu} = \gamma\_\mu$ , so:
$$
\frac{\partial^2 P}{\partial t \partial \mu} = -\frac{b \gamma\_\mu}{(2 + b \gamma)^2}.
$$
We have that $\gamma\_\mu < 0$ by assumption (slope decreases in $\mu$) then we see that the sign of $\frac{\partial^2 P}{\partial t \partial \mu}$ is positive (at least if I didn't make any mistakes).


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