Monopoly and Taxes (Nicholson Exercise) Hello I'm working through Microeconomic Theory : Basic Principles and Extensions of Nicholson and Snyder 10e, for an exam and I fail to get how to answer this question (p.517) :


  A specific tax is a fixed amount per unit of output. If the tax rate
  is t per unit, total tax collections are tQ . Show that the imposition
  of a specific tax on a monopoly will reduce output more (and increase
  price more) than will the imposition of an ad valorem tax that
  collects the same tax revenue.


So I state that both tax collects the same (s for specific and a for ad valorem)
$$q_s^m \times \tau_s =A$$
$$q_a^m \times p^m_a \times \tau_a=A  $$

Then I have tried some identities of $q^m$ and $p^m$ but I can't get to the proof. 

$$q_s^m=\dfrac{C'(q_s^m)+\tau_s-p^m_s}{P'_s} $$
$$q_a^m=\dfrac{C'(q_a^m)}{(1-\tau_a)P'_a}-\dfrac{p_a^m}{P'_a}$$

where $P'_i$ is the derivative of the inverse demand function at the equilibrium point of the monopolist under the tax type $i$

I tried replacing $\tau$ as $\tau_s=\dfrac{q_a^m \times p_a^m \times \tau_a }{q_s^m}$ and then replacing it in $q_s^m$ but I get a quadratic form of no use...
First let's look at the specific tax. The profit is
$$\pi_s=[P(q)-\tau_s]q-C(q).$$
Differentiating to establish the first-order condition:
$$P'(q)q+P(q)-\tau_s-C'(q)=0.$$
If we write $A=\tau_s q$ for the tax revenue then we can rewrite the FOC thus:
$$P'(q)q+P(q)-C'(q)=\frac{A}{q}.$$



Now for ad valorem:
$$\pi_a=(1-\tau_a)P(q)q-C(q)$$
First-order condition:
$$(1-\tau_a)[P'(q)q+P(q)]-C'(q)=0.$$
We have $A=\tau_a q P(q)$, which can can use as follows:
$$[P'(q)q+P(q)]-\tau_aP'(q)q-\tau_aP(q)-C'(q)=0.$$
$$[P'(q)q+P(q)]-\frac{A}{P(q)}P'(q)-\frac{A}{q}-C'(q)=0.$$
$$P'(q)q+P(q)-C'(q)=\frac{A}{q}+\frac{A}{P(q)}P'(q).$$



Note the the left-hand side of the last line in both sections is the same. Moreover, this expression ($P'(q)q+P(q)-C'(q)$) is the derivative of profits assuming zero tax. If profits are concave then this derivative is decreasing in $q$.

We can see that, for any value of $q$, the right-hand side is smaller for the ad valorem tax (because it has an extra term that is negative--remember that $P'(q)&lt;0$). Thus, we have something that looks like this: