Question

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.

Short answer

In conclusion, the effects of ad valorem and specific taxes on monopolies depend on the underlying demand curve and its elasticity. In the case of a linear demand curve, the price increases due to ad valorem taxes are less than the full extent of the tax, while in the case of a constant elasticity demand curve, the price increase matches the tax. However, in special scenarios with highly elastic demand curves, ad valorem taxes can lead to price increases greater than the tax itself. Lastly, specific taxes tend to reduce output and increase price more than ad valorem taxes that collect the same revenue, because they are applied to the per-unit output and can result in higher marginal costs for the monopolist.

Step-by-step solution

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.

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.

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.

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.

Key concepts

Inverse Elasticity Rule

The Inverse Elasticity Rule is a crucial principle for understanding how monopolies set prices. It suggests that the optimal markup of price over marginal cost is inversely related to the price elasticity of demand. This means that if customers are less sensitive to changes in price—indicating a lower elasticity—a monopoly can charge a higher price. On the other hand, if the demand is highly elastic, a monopoly would need to keep prices lower to maximize profits. This concept is essential because it shows how monopolies strategically adjust their pricing based on consumer responsiveness.
When a monopoly faces a tax, like an ad valorem tax, the Inverse Elasticity Rule helps determine how much of the tax is passed on to consumers through price changes. This decision hinges on the elasticity of the demand curve, guiding how much more consumers have to pay versus how much of the tax eats into the monopolist's profit margin.

Ad Valorem Tax

An ad valorem tax is a percentage-based tax applied to the price of goods. When such a tax is imposed on a monopoly, it effectively reduces the price received by the monopolist by the tax rate proportion, \(1-t\), where \t\ is the tax rate.
This type of tax can influence pricing decisions because the monopolist must consider how much of the tax can be passed onto consumers without significantly reducing demand.

• If the demand curve is linear, the research shows that the price increase due to the ad valorem tax is less than the tax itself. This happens because the quantity demanded decreases along a linear demand curve as price goes up.
• However, in a scenario with constant elasticity demand, the price increase matches the tax completely. This occurs because the demand adjusts proportionally, making it easier for the monopolist to transfer the full cost of the tax to consumers.

Thus, understanding the demand curve's nature is essential in predicting a monopolist's response to ad valorem taxes.

Specific Tax

A specific tax is a fixed amount imposed per unit of a product sold. Unlike an ad valorem tax, which varies with the price, a specific tax adds a fixed cost to each unit produced, designated by \(\tau\).
This type of tax can significantly impact the pricing strategy and output levels of a monopoly. Since the cost is added on a per-unit basis:

• This leads to a greater reduction in output compared to an ad valorem tax of equal total revenue because it effectively raises the marginal cost of production without altering the proportionality to price.
• Consequently, the price of goods is typically increased more under a specific tax to cover these heightened costs, while maintaining monopoly profits.
• Thus, monopolists facing specific taxes must adjust their production and pricing more aggressively than they would under an ad valorem tax.

This emphasizes how different tax structures can influence monopolistic pricing strategies.

Constant Elasticity Demand Curve

A constant elasticity demand curve is when the price elasticity, \(\eta\), remains unchanged across different price levels. This means any percentage change in price results in a proportional percentage change in quantity demanded.
In the context of monopoly taxation:

• With an ad valorem tax, such a demand curve means that a price increase from the tax will fully match the percentage tax rate because the responsiveness to price change is consistently predictable.
• This is different from linear or varying elasticity demand curves where the price adjustment to tax is often incomplete or varies. The reason lies in the consistent "stretchiness" of the constant elasticity demand, making the pass-through of tax to consumers straightforward for pricing decisions.
• This uniformity simplifies the monopolist's strategy in determining how much of a tax should shift onto consumers through price hikes.

Understanding the elasticity shape of demand curves helps in predicting how monopolistic markets react to taxation.