Comparative statics of a monopoly 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?
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:
$$
\begin{align*}
    &amp;Q + P Q_P - C_Q Q_P - t Q_P = 0,\\
\iff &amp;Q - b P + b C_Q + bt = 0
\end{align*}
$$
then differentiating this with respect to  $t$ gives:
$$
\begin{align*}
&amp;(Q_P - b + b C_{QQ} Q_P)\frac{\partial P}{\partial t} + b = 0,\\
\to &amp;(-2b - b^2 C_{QQ})\frac{\partial P}{\partial t} = -b,\\
\to &amp;\frac{\partial P}{\partial t} = \frac{1}{2 + b C_{QQ}}
\end{align*}
$$
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 $\gamma_\mu &lt; 0$ 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).