Monopoly and Taxes (Nicholson Exercise)?

Question

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)$$q_s^m \ imes \ au_s =A$$$$q_a^m \ imes p^m_a \ imes \ au_a=A $$Then I have tried some identities of $q^m$ and $p^m$ but I can't get to the proof. $$q_s^m=\\dfrac{C'(q_s^m)+\ au_s-p^m_s}{P'_s} $$$$q_a^m=\\dfrac{C'(q_a^m)}{(1-\ au_a)P'_a}-\\dfrac{p_a^m}{P'_a}$$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 $\ au$ as $\ au_s=\\dfrac{q_a^m \ imes p_a^m \ imes \ au_a }{q_s^m}$ and then replacing it in $q_s^m$ but I get a quadratic form of no use...

Answer 1

I absolutely love this question. It would probably take a whole tutoring session to run you through it.. I can take you through this but I want to make sure I am not wasting my time.

Let T be a specific tax.

Let P(Q) Be the Pretax demand facing the monopolist.. Let P*(Q) be the Effective demand to the monopolist with a Specific Tax. Show that the specific tax shifts the demand curve straight down (south, whatever without changing the slope. What happens to effective demand elasticity?

Now, Choose some Arbitrary specific tax, T . The monopolist will produce some level Q* such that effective MR = MC. At the level, the difference between the the price the consumer pays, P and the price the Producer gets, P* is T. The average tax rate is T/P*

Draw the Total Revenue functions to the firm with no tax, with a specific tax ans an equivalent Average tax rate. The two after tax total revenue curves will intersect at Q*. Which one is steeper and why?

If you can convince me that you have this, or at least want to have this I will go into more detail.

Send me a message with your answers?