The excess burden of a tax, also known as deadweight loss, occurs when the equilibrium quantity of a good is reduced due to the tax, leading to a loss of economic efficiency. In this scenario, we need to calculate the excess burden of a 2% sales tax on automobiles, given the price elasticity of demand (PED) is 2 and the price elasticity of supply (PES) is 3.
To start, we determine the initial expenditure on automobiles, which is given as $5 billion. When a sales tax is imposed, the price paid by consumers increases while the price received by sellers decreases. The demand and supply curves will shift as a result of the changes in price due to the tax.
First, we need to find the quantity sold (Q) before the tax. If we let P be the original price before tax, the total expenditure (E) is given by:
E = P * Q = $5 billion
After introducing the 2% tax, we will denote the original price as P and the new price consumers pay as P + 0.02P = 1.02P. The price received by the sellers after tax becomes P – 0.02P = 0.98P.
From the elasticities, we find the change in quantity due to the tax. The formula to determine the change in quantity (ΔQ) based on elasticity and tax rate (t) is:
ΔQ = – (PED / (PED + PES)) * E / P * t
With PED = 2 and PES = 3, and substituting those values in:
ΔQ = - (2 / (2 + 3)) * (5 billion / P) * 0.02
This simplifies to:
ΔQ = - (2/5) * (5 billion/P) * 0.02
ΔQ = - (2 * 5 billion * 0.02) / (5P)
ΔQ = - (0.08 billion) / (P)
Next, we need to calculate the loss in consumer and producer surplus due to the change in quantity ΔQ. The excess burden can be approximated by the area of the triangle formed due to the reduction in quantity sold, which is:
Excess Burden = 0.5 * tax rate * ΔQ * ΔP
Where ΔP is the price difference that results from the tax. As this involves further complex calculations depending on the actual prices and quantity involved, we’ll summarize that the excess burden represents the efficiency loss due to the tax.
In summary, while we estimated a formula to analyze it, the exact dollar value of the excess burden would depend on knowing the initial price of automobiles. However, given a price elasticity of demand of 2 and a supply of 3, we can conclude that the excess burden will lead to inefficiencies in the market, causing both consumers and producers to face a loss in surplus that could have been avoided without the tax.