The taxation of monopoly can sometimes produce results different from those that arise in the competitive case. This problem looks at some of those cases. Most of these can be analyzed by using the inverse elasticity rule (Equation 14.1 ). a. Consider first an ad valorem tax on the price of a monopoly's good. This tax reduces the net price received by the monopoly from \(P\) to \(P(1-t)-\) where \(t\) is the proportional tax rate. Show that, with a linear demand curve and constant marginal cost, the imposition of such a tax causes price to increase by less than the full extent of the tax. b. Suppose that the demand curve in part (a) were a constant elasticity curve. Show that the price would now increase by precisely the full extent of the tax. Explain the difference between these two cases. c. Describe a case where the imposition of an ad valorem tax on a monopoly would cause the price to increase by more than the tax. d. A specific tax is a fixed amount per unit of output. If the tax rate is \(\tau\) per unit, total tax collections are \(\tau Q .\) 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.

Step by step solution

01

a. Ad valorem tax on monopoly with linear demand curve

Let us consider a monopoly with a linear demand curve, \(Q = a - bP\). The marginal cost is constant and equal to \(c\). The monopolist's profit function can be written as: \[ \pi = (P(1-t) - c)Q \] Since the monopolist aims to maximize profit, we need to find the optimal price \(P^*\) that will maximize this expression. To do this, we'll first express the profit function in terms of \(Q\): \[ \pi = ((a/b - Q)(1-t) - c)Q\] Now, we'll differentiate the profit function with respect to \(Q\) and set the result equal to zero, then solve for \(Q\): \[ \frac{d \pi}{d Q} = (a/b - Q)(1-t) - c - Q(1-t) = 0\] \[ Q^* = \frac{a - c/b}{2 - t} \] That gives us the optimal quantity to maximize profits under the given ad valorem tax. Now, we can find the optimal price that corresponds to this optimal quantity using the demand function: \[ P^* = \frac{a - bQ^*}{b} \] \[ P^* = \frac{a - c}{2b - bt} \] Note how the tax \(t\) affects the optimal price. Since \(2b - bt > 2b\) when \(t > 0\), it means that the price increase due to the ad valorem tax is less than the full extent of the tax.

02

b. Ad valorem tax on monopoly with constant elasticity demand curve

Now, let us consider a monopoly with a constant elasticity demand curve, given by \(Q = A P^{\eta}\), where \(\eta\) is the elasticity of demand. Now, the profit function is given by: \[ \pi = (P(1-t) - c)Q \] \[ \pi = (P(1-t) - c)A P^{\eta} \] To maximize profits, we'll differentiate the profit function with respect to \(P\) and set the result equal to zero: \[ \frac{d \pi}{d P} = A (1-t) P^{\eta - 1} (\eta - 1 + \frac{c}{P(1-t)}) = 0\] Now, rearrange the equation to get the expression for the optimal price \(P^*\): \[ P^* = \frac{c}{(1-t)(\eta - 1)} \] This shows that the price increases by the full extent of the tax, since \(P^*\) is directly proportional to \(\frac{1}{1-t}\). As a result, when the demand curve has constant elasticity, the price increase will match the tax. The difference between these two cases lies in the shape of the demand curve. With a linear demand curve, the demand is less sensitive to price changes compared to a constant elasticity demand curve. In the latter case, a price increase due to the tax fully passes through to the consumers.

03

c. Ad valorem tax leading to a price increase larger than the tax

Consider a monopoly facing a demand curve that is very sensitive to price changes (i.e., highly elastic). In this case, the monopolist will try to minimize the impact of the tax on price while maintaining its market share. One potential scenario could be a luxury good, where consumers have many substitutes available and are highly responsive to changes in price. In order to maintain market share, the monopolist might attempt to maintain a low price, absorbing more of the tax itself. In this case, the imposition of an ad valorem tax could lead to price increases greater than the tax, as the monopolist tries to maintain a competitive position in the market.

04

d. Specific tax vs. ad valorem tax on monopoly

Let's analyze the effects of a specific tax and an ad valorem tax on the output and price of a monopoly. Suppose that the demand curve is given by \(Q = a - bP\) and the marginal cost is constant and equal to \(c\). For the specific tax case, the profit function is: \[ \pi_s = (P - c - \tau)Q \] To maximize profits, we can differentiate \(\pi_s\) with respect to \(Q\) and set the result equal to zero: \[ \frac{d \pi_s}{d Q} = (a/b - Q) - c - \tau = 0\] Now, for the ad valorem tax case, we have the following profit function (from part a): \[ \pi_v = (P(1-t) - c)Q \] Differentiating \(\pi_v\) with respect to \(Q\) and setting the result equal to zero: \[ \frac{d \pi_v}{d Q} = (a/b - Q)(1-t) - c = 0\] Comparing these two cases, we can see that the specific tax reduces output more and increases price more than an ad valorem tax that collects the same tax revenue. This is because specific taxes are applied to the per-unit output, leading to potentially higher marginal costs for the monopolist, whereas ad valorem taxes apply proportionally to the price.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!