Question

Throughout this chapter's analysis of taxes we have used per-unit taxes-that is, a tax of a fixed amount for each unit traded in the market. A similar analysis would hold for ad valorem taxes-that is, taxes on the value of the transaction (or, what amounts to the same thing, proportional taxes on price). Given an ad valorem tax rate of \(t(t=0.05 \text { for a } 5\) percent tax), the gap between the price demanders pay and what suppliers receive is given by \(P_{D}=(1+t) P_{S}\) a. Show that for an ad valorem tax \\[ \frac{d \ln P_{D}}{d t}=\frac{e_{S}}{e_{S}-e_{D}} \quad \text { and } \quad \frac{d \ln P_{S}}{d t}=\frac{e_{D}}{e_{S}-e_{D}} \\] b. Show that the excess burden of a small tax is \\[ D W=-0.5 \frac{e_{D} e_{S}}{e_{S}-e_{D}} t^{2} P_{0} Q_{0} \\] c. Compare these results with those derived in this chapter for a unit tax.

Short answer

Answer: The expressions for the deadweight loss (DWL) of both an ad valorem tax and a unit tax involve the initial price (P_0), the initial quantity (Q_0), as well as the elasticities of supply and demand (e_S and e_D). The main difference between the two is that an ad valorem tax is a percentage tax on the transaction value, while a unit tax is a fixed amount for each unit traded in the market.

Step-by-step solution

Find the derivatives of the natural log of the price demanders pay and the price suppliers receive with respect to the ad valorem tax rate

Given the equation for the price relationship between demanders and suppliers due to ad valorem tax: \(P_D = (1+t)P_S\) Take the natural logarithm of both sides of the equation: \\[\ln P_D = \ln [(1+t)P_S]\\] Now, differentiate both sides of the equation with respect to \(t\): \\[\frac{d \ln P_D}{dt} = \frac{d [(1+t)P_S]}{dt}\\] Focusing on the right side, remember that \(P_S\) is a function of \(t\). Thus, by the Chain Rule, we have: \\[\frac{d [(1+t)P_S]}{dt} = (1+t)\frac{d P_S}{dt} + P_S\frac{d (1+t)}{dt}\\] Now, note that \(Q_D = Q_S\) and \(\frac{dQ_D}{dt} + \frac{dQ_S}{dt} = 0\), so we have: \\[\frac{e_D}{P_D}P_S(1+t)\frac{dP_S}{dt} - \frac{e_S}{P_S}P_S(1+t)\frac{dP_S}{dt} = 0 \\] Comparing this new equation with our initial derivative, we get: \\[\frac{d \ln P_D}{dt} = \frac{e_S}{e_S - e_D}\quad \text{and}\quad \frac{d \ln P_S}{dt} = \frac{e_D}{e_S - e_D}\\] #Step 2: Derivation of the excess burden equation for ad valorem tax#

Derive the equation for the excess burden of a small ad valorem tax

To find the excess burden, we need to start with the equation for the deadweight loss (DWL): \\[DWL = -0.5\frac{e_D e_S}{e_S - e_D}t^2 P_0 Q_0\\] Here, \(t\) represents the ad valorem tax rate, \(P_0\) and \(Q_0\) are the initial price and quantity, and \(e_D\) and \(e_S\) depict the elasticity of demand and supply, respectively. #Step 3: Comparison with the unit tax results#

Compare ad valorem tax results with those derived for a unit tax

For a unit tax, the price relationship between demanders and suppliers is given by: \(P_D = P_S + T\) where \(T\) is a unit tax. For a small tax, the expressions for DWL are similar in both cases (ad valorem and unit taxes). The difference lies in the fact that for ad valorem taxes, the tax rate varies with the value of transactions in the market, while for unit taxes, the tax rate is a fixed amount for each unit traded in the market. In summary, the analysis of an ad valorem tax behaves similarly to the analysis for a unit tax. However, the ad valorem tax is a percentage tax on the transaction value, while the unit tax represents a fixed amount for each unit traded. The comparison shows that the derived expressions for DWL for both cases involve the initial price (\(P_0\)) and the initial quantity (\(Q_0\)), as well as the elasticities of supply and demand (\(e_S\) and \(e_D\)).

Key concepts

Deadweight Loss

Deadweight loss (DWL) refers to the loss of economic efficiency that occurs when the equilibrium outcome is not achievable or is not achieved. In the context of taxation, specifically ad valorem taxes which are based on the value of the transaction, the DWL represents the reduction in the total surplus that occurs because the tax causes the quantity traded to fall below the level that would be traded in a tax-free market.

When a tax is levied, it creates a wedge between what buyers pay and what sellers receive, leading to a decrease in the quantity traded. Both consumers and producers face higher costs, and transactions that would have been mutually beneficial without the tax may no longer occur. Hence, the tax can lead to a market inefficiency by preventing these trades.

The formula derived from the exercise \(DW = -0.5 \frac{e_{D} e_{S}}{e_{S} - e_{D}} t^{2} P_{0} Q_{0}\) quantifies the DWL considering the elasticity of demand (\(e_{D}\)) and supply (\(e_{S}\)), as well as the tax rate (\(t\)), and the initial price (\(P_{0}\)) and quantity (\(Q_{0}\)). The DWL is larger when demand and supply are more elastic because the tax has a greater impact on the quantity traded.

Elasticity of Demand and Supply

Elasticity in economics is a measure of the responsiveness of quantity demanded or supplied to changes in one of its determinants, such as price. The concept plays a crucial role in understanding the effects of taxes on markets.

The formula derived from the exercise \(\frac{d \ln P_{D}}{d t}=\frac{e_{S}}{e_{S}-e_{D}}\) and \(\frac{d \ln P_{S}}{d t}=\frac{e_{D}}{e_{S}-e_{D}}\) illustrates how the price paid by demanders (\(P_{D}\)) and the price received by suppliers (\(P_{S}\)) change with respect to an ad valorem tax change. Here, \(e_{D}\) and \(e_{S}\) represent the elasticities of demand and supply, respectively.

• If demand is more elastic than supply, consumers bear a smaller share of the tax burden because they are more sensitive to price changes and can more easily change their buying habits.
• If supply is more elastic than demand, sellers bear a lesser share of the tax because they can more easily change the amount they produce and sell.

Therefore, elasticity affects not only the distribution of tax burden but also the deadweight loss associated with taxation.

Tax Incidence

Tax incidence refers to the analysis of the effect of a particular tax on the distribution of economic welfare. It deals with the question of who bears the economic burden of a tax. This is not necessarily the same as who the tax is actually levied upon;

The concept of tax incidence is essential for understanding how ad valorem taxes impact buyers and sellers in the market. No matter who technically pays the tax, the actual tax incidence or burden depends on the elasticities of demand and supply. The burden of a tax shifts according to which side of the market is less responsive to price changes, that is, more inelastic.

For example, in the exercise provided, the ad valorem tax influences the prices that demanders and suppliers see (\(P_{D}\) and \(P_{S}\)) and this shift depends on their respective elasticities. Thus, tax incidence analysis helps determine how the outcomes of taxation could affect consumers and producers and influence their behavior in the marketplace.