Question

Throughout this chapter our analysis of taxes has assumed that they are imposed on a per-unit basis. Many taxes (such as sales taxes ) are proportional, based on the price of the item. In this problem you are asked to show that, assuming the tax rate is reasonably small, the market consequences of such a \(\operatorname{tax}\) are quite similar to those already analyzed. To do so, we now assume that the price received by suppliers is given by \(P\) and the price paid by demanders is \(P(1+t),\) where \(t\) is the ad valorem tax rate (i.e., with a tax rate of 5 percent, \(t=0.05\), the price paid by demanders is \(1.05 P\) ). In this problem then the supply function is given by \(Q=S(P)\) and the demand function by \(Q=D[(1+t) P]\) a. Show that for such a tax $$\frac{d \ln P}{d t}=\frac{e_{D, P}}{e_{S, P}-e_{D, P}}$$. (Hint: Remember that \(d \ln P / d t=\frac{1}{P} \cdot \frac{d P}{d t}\) and that here we are assuming \(t \otimes 0 .\) b. Show that the excess burden of such a small ad valorem tax is given by: \\[ D W=-0.5 \frac{e_{D . p} e_{S . P}}{e_{S, P}-e_{D, P}} t^{2} P^{*} Q^{*} \\] c. Compare these results to those derived in this chapter for a per-unit tax. Can you make any statements about which tax would be superior in various circumstances?

Short answer

Additionally, provide the expression for the excess burden (deadweight loss) of the ad valorem tax and compare it to the per-unit tax system. Answer: The relationship between the change in price with respect to the ad valorem tax and the price elasticities of demand and supply can be expressed as: $$\frac{d\ln P}{dt} = \frac{e_{D,P}}{e_{S,P} - e_{D,P}}$$ The excess burden of the ad valorem tax is given by: $$DW = -0.5 \cdot \frac{e_{D,P} \cdot e_{S,P}}{e_{S,P}-e_{D,P}} \cdot P^{*} \cdot Q^{*} \cdot t^2$$ Comparing this to the per-unit tax system, where the excess burden is: $$DW = -0.5 \cdot \frac{e_{D,P} \cdot e_{S,P}}{(e_{S,P}+e_{D,P})} \cdot P^{*} \cdot Q^{*} \cdot T^2$$ the superiority of either tax system depends on the price elasticities and tax rates involved. As tax rates decrease, the excess burden of both systems declines, with the ad valorem tax system being more sensitive to the tax rate. The main goal is to minimize excess burden by selecting the tax system that causes the smallest distortion in the market.

Step-by-step solution

Relationship between change in price, tax rate, and price elasticities

First, we are given the supply function \(Q = S(P)\) and the demand function \(Q = D((1+t)P)\). Our task is to show that: $$\frac{d \ln P}{d t}=\frac{e_{D, P}}{e_{S, P}-e_{D, P}}$$ Let's start by differentiating the demand function with respect to \(t\): $$\frac{d Q}{d t} = \frac{d D((1+t)P)}{d t} = D'((1+t)P) \cdot\frac{d(1+t)P}{dt} = D'((1+t)P) \cdot P$$ Now, differentiate the supply function with respect to \(P\): $$\frac{d Q}{d P} = \frac{d S(P)}{d P} = S'(P)$$ Since, \(D((1+t)P) = S(P)\), we can write: $$\frac{d Q}{dt} = \frac{d Q}{d P} \cdot \frac{d P}{dt} = S'(P) \cdot \frac{d P}{dt}$$ And therefore, substituting for \(\frac{d Q}{dt}\) from the step above, we have: $$D'((1+t)P) \cdot P = S'(P) \cdot \frac{d P}{dt}$$ Which gives us: $$\frac{d P}{d t} = \frac{D'((1+t)P)P}{S'(P)}$$ Now, we know that: $$e_{D,P} = \frac{D'((1+t)P)P}{D((1+t)P)}$$ Let's multiply both sides of the above equation by \(\frac{1}{t P}\) and use the hint given which states \(\frac{d \ln P}{dt} = \frac{1}{P} \cdot \frac{d P}{dt}\): $$\frac{d \ln P}{d t} = \frac{D'((1+t)P)P}{D((1+t)P) \cdot t}$$ Similarly, $$e_{S,P} = \frac{S'(P)P}{S(P)}$$ We can rewrite the equation derived earlier for \(\frac{d P}{dt}\) as: $$\frac{d \ln P}{d t} = \frac{e_{D, P} \cdot t}{\frac{S(P)}{P} \cdot S'(P)}$$ Dividing both the numerator and denominator by \(Q\), and substituting the definitions of \(e_{D,P}\) and \(e_{S,P}\): $$\frac{d \ln P}{d t}=\frac{e_{D, P}}{e_{S, P}-e_{D, P}}$$

Excess burden of the ad valorem tax

Now we should find the equation for the excess burden (deadweight loss) due to the small ad valorem tax. The excess burden can be derived from the following formula: $$DW = -0.5 \cdot \text{change in quantity} \cdot \text{change in price per unit}$$ Combining the results from Step 1, the change in price per unit is given by: $$\frac{d P}{dt} = \frac{e_{D,P}}{e_{S,P}-e_{D,P}}P \cdot t$$ And the change in quantity is: $$\frac{d Q}{dt}= \frac{D'((1+t)P)P}{S'(P)} \cdot t$$ Now we can calculate the excess burden: $$DW = -0.5 \cdot \frac{D'((1+t)P)P}{S'(P)} \cdot t \cdot \frac{e_{D,P}}{e_{S,P}-e_{D,P}}P \cdot t$$ After simplifying the expression, we get: $$DW = -0.5 \cdot \frac{e_{D,P} \cdot e_{S,P}}{e_{S,P}-e_{D,P}} \cdot P^{*} \cdot Q^{*} \cdot t^2$$

Compare the results with the per-unit tax

Now let's compare these results to those derived for a per-unit tax system. In a per-unit tax system, the excess burden is given by: $$DW = -0.5 \cdot \frac{e_{D,P} \cdot e_{S,P}}{(e_{S,P}+e_{D,P})} \cdot P^{*} \cdot Q^{*} \cdot T^2$$ Where \(T\) is the per-unit tax rate. In general, the magnitude of the excess burden will depend on the relative price elasticities of supply and demand, and the tax rate. Under certain conditions, either a per-unit tax or an ad valorem tax might be superior depending on the price elasticities and the tax rates involved. Furthermore, as the tax rates grow smaller, the excess burden of both types of taxes will also tend to decrease, with the excess burden of the ad valorem tax being more sensitive to the tax rate (\(t^2\)) compared to the per-unit tax system (\(T^2\)). However, in both cases, the main goal is to minimize the excess burden by choosing the tax system that induces the smallest distortion in the market.

Key concepts

Price Elasticity of Demand

Understanding how the quantity demanded of a good or service responds to changes in its price is crucial in microeconomic theory. This response is measured by the price elasticity of demand. It is defined as the percentage change in quantity demanded resulting from a one percent change in price. Mathematically, it is expressed as:
\[ e_{D,P} = \frac{\text{% change in quantity demanded}}{\text{% change in price}} \]
When the price elasticity of demand is high, consumers are much more sensitive to price changes, often leading to larger shifts in consumption patterns. Conversely, low elasticity indicates that demand is more rigid, and price changes have a lesser impact on the quantity consumers buy. This concept is essential when analyzing the effects of an ad valorem tax because the tax's impact on prices would change the quantity demanded differently depending on the demand's elasticity.

Price Elasticity of Supply

Similarly to demand, the price elasticity of supply measures how much the quantity supplied of a good changes in response to a change in its price. It's given by the following equation:
\[ e_{S,P} = \frac{\text{% change in quantity supplied}}{\text{% change in price}} \]
This elasticity sheds light on a supplier's ability to adjust production in response to price changes. If supply is elastic, producers can adjust quickly and significantly. In contrast, inelastic supply signifies a slower and less flexible response. This property is vital when considering ad valorem taxes because the extent of supply adjustments directly affects market outcomes such as price changes and the tax's excess burden.

Excess Burden of Taxation

The excess burden of taxation, also known as deadweight loss, is a key concept in the study of tax effects in economics. It represents the cost to society that arises from market distortions due to taxation. Taxes can lead to a decrease in the quantity of goods traded below the level of a free market, creating a loss of economic efficiency.
In the context of ad valorem taxes, which are percentage-based, such as sales tax, the excess burden is represented by the formula:
\[ DW = -0.5 \cdot \frac{e_{D,P} \cdot e_{S,P}}{e_{S,P}-e_{D,P}} \cdot P^{*} \cdot Q^{*} \cdot t^2 \]
Where \( P^{*} \) and \( Q^{*} \) are the equilibrium price and quantity. The extent of the excess burden is influenced by the price elasticities of demand and supply, which determine how taxation will alter market activity and the overall efficiency of the market exchange.

Microeconomic Theory

At the heart of these concepts lies microeconomic theory, which aims to understand and explain how individuals and firms make decisions on the allocation of limited resources. This theory investigates factors influencing individuals' and firms' behaviors and how these behaviors determine prices and quantities in specific markets.
In our discussion on ad valorem tax, microeconomic theory helps explain the roles of price elasticities and how the tax affects consumer choices and producer strategies. Taxes can skew market dynamics away from their efficient outcomes, resulting in either an over- or underproduction of goods, and a departure from the ideal market equilibrium. By applying principles of microeconomic theory, one can assess the desirability of different types of taxation and their consequent impact on economic welfare.