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Chapter 8 of Taxation
Questions for Review
- The greater the elasticities of demand and supply, the greater the deadweight loss of a tax. Because elasticity measures the response of quantity to a change in price, higher elasticity means the tax induces a greater reduction in quantity, and therefore, a greater distortion to the market.
- Experts disagree about whether labor taxes have small or large deadweight losses because they have different views about the elasticity of labor supply. Some believe that labor supply is inelastic, so a tax on labor has a small deadweight loss. But others think that workers can adjust their hours worked in various ways, so labor supply is elastic, and thus a tax on labor has a large deadweight loss.
- Figure 2 illustrates the deadweight loss and tax revenue from a tax on the sale of a good. Without a tax, the equilibrium quantity would beQ 1 , the equilibrium price would beP 1 , consumer surplus would be A + B + C, and producer surplus would be D + E + F. The imposition of a tax places a wedge between the price buyers pay,P B , and the price sellers receive,P S, whereP B =P S + tax. The quantity sold declines toQ 2. Now consumer surplus is A, producer surplus is F, and government revenue is B + D. The deadweight loss of the tax is C+E, because that area is lost due to the decline in quantity fromQ 1 toQ 2.
Figure 2
APPLICATION: THE COSTS OF
TAXATION
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152 Chapter 8 /Application: The Costs of Taxation
- The deadweight loss of a tax rises more than proportionally as the tax rises. Tax revenue, however, may increase initially as a tax rises, but as the tax rises further, revenue eventually declines.
- When the sale of a good is taxed, both consumer surplus and producer surplus decline. The decline in consumer surplus and producer surplus exceeds the amount of government revenue that is raised, so society's total surplus declines. The tax distorts the incentives of both buyers and sellers, so resources are allocated inefficiently.
Problems and Applications
- a. Figure 7 illustrates the market for socks and the effects of the tax. Without a tax, the equilibrium quantity would beQ 1 , the equilibrium price would beP 1 , total spending by consumers equals total revenue for producers, which isP 1 xQ 1 , which equals area B + C
- D + E + F, and government revenue is zero. The imposition of a tax places a wedge between the price buyers pay,P B , and the price sellers receive,P S, whereP B =P S + tax. The quantity sold declines toQ 2. Now total spending by consumers isP B xQ 2 , which equals area A + B + C + D, total revenue for producers isP S xQ 2 , which is area C + D, and government tax revenue isQ 2 x tax, which is area A + B.
b. Unless supply is perfectly elastic or demand is perfectly inelastic, the price received by producers falls because of the tax. Total receipts for producers fall, because producers lose revenue equal to area B + E + F.
Figure 7
c. The price paid by consumers rises, unless demand is perfectly elastic or supply is perfectly inelastic. Whether total spending by consumers rises or falls depends on the price elasticity of demand. If demand is elastic, the percentage decline in quantity exceeds the percentage increase in price, so total spending declines. If demand is inelastic, the percentage decline in quantity is less than the percentage increase in price, so total spending rises. Whether total consumer spending falls or rises, consumer surplus declines because of the increase in price and reduction in quantity.
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154 Chapter 8 /Application: The Costs of Taxation
Figure 4
b. With very inelastic supply and very elastic demand, the burden of the tax on rubber bands will be borne largely by sellers. As Figure 5 shows, consumer surplus does not decline much, just by area A + B, while producer surplus falls substantially, by area C + D. Compared to part (a), producers bear much more of the burden of the tax, and consumers bear much less.
Figure 5
- Because the demand for food is inelastic, a tax on food is a good way to raise revenue because it does not lead to much of a deadweight loss; thus taxing food is less inefficient than taxing other things. But it is not a good way to raise revenue from an equity point of view, because poorer people spend a higher proportion of their income on food. The tax would hit them harder than it would hit wealthier people.
- a. The statement, "A tax that has no deadweight loss cannot raise any revenue for the government," is incorrect. An example is the case of a tax when either supply or demand
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Chapter 8 /Application: The Costs of Taxation 155 is perfectly inelastic. The tax has neither an effect on quantity nor any deadweight loss, but it does raise revenue.
b. The statement, "A tax that raises no revenue for the government cannot have any deadweight loss," is incorrect. An example is the case of a 100% tax imposed on sellers. With a 100% tax on their sales of the good, sellers will not supply any of the good, so the tax will raise no revenue. Yet the tax has a large deadweight loss, because it reduces the quantity sold to zero.
- a. The deadweight loss from a tax on heating oil is likely to be greater in the fifth year after it is imposed rather than the first year. In the first year, the elasticity of demand is fairly low, as people who own oil heaters are not likely to get rid of them right away. But over time they may switch to other energy sources and people buying new heaters for their homes will more likely choose gas or electric, so the tax will have a greater impact on quantity. Thus, the deadweight loss of the tax will get larger over time.
b. The tax revenue is likely to be higher in the first year after it is imposed than in the fifth year. In the first year, demand is more inelastic, so the quantity does not decline as much and tax revenue is relatively high. As time passes and more people substitute away from oil, the quantity sold declines, as does tax revenue.
- a. Figure 3 illustrates the market for pizza. The equilibrium price isP 1 , the equilibrium quantity isQ 1 , consumer surplus is area A + B + C, and producer surplus is area D + E + F. There is no deadweight loss, as all the potential gains from trade are realized; total surplus is the entire area between the demand and supply curves: A + B + C + D + E + F.
Figure 3
b. With a $1 tax on each pizza sold, the price paid by buyers,P (^) B , is now higher than the price received by sellers,P (^) S , whereP (^) B =P (^) S + $1. The quantity declines toQ 2 , consumer surplus is area A, producer surplus is area F, government revenue is area B + D, and
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Chapter 8 /Application: The Costs of Taxation 157
Figure 8
- a. Figure 9 shows the effect of a $10 tax on hotel rooms. The tax revenue is represented by areas A + B, which are equal to ($10)(900) = $9,000. The deadweight loss from the tax is represented by areas C + D, which are equal to (0.5)($10)(100) = $500.
Figure 9
b. Figure 10 shows the effect of a $20 tax on hotel rooms. The tax revenue is represented by areas A + B, which are equal to ($20)(800) = $16,000. The deadweight loss from the tax is represented by areas C + D, which are equal to (0.5)($20)(200) = $2,000.
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158 Chapter 8 /Application: The Costs of Taxation
Figure 10
When the tax is doubled, the tax revenue rises by less than double, while the deadweight loss rises by more than double.
- a. Setting quantity supplied equal to quantity demanded gives 2P = 300 –P. AddingP to both sides of the equation gives 3P = 300. Dividing both sides by 3 givesP = 100. PluggingP = 100 back into either equation for quantity demanded or supplied givesQ =
b. Now P is the price received by sellers andP +T is the price paid by buyers. Equating quantity demanded to quantity supplied gives2P = 300 - (P+T). AddingP to both sides of the equation gives 3P = 300 –T. Dividing both sides by 3 givesP = 100 –T/3. This is the price received by sellers. The buyers pay a price equal to the price received by sellers plus the tax (P +T = 100 + 2T/3). The quantity sold is nowQ = 2P = 200 – 2T/3.
c. Because tax revenue is equal toT xQ andQ = 200 – 2T/3, tax revenue equals 200 T - 2T 2 /3. Figure 11 shows a graph of this relationship. Tax revenue is zero atT = 0 and atT = 300.
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($15,000 when the tax is $150 versus $13,333 when the tax is $200), and reduce the deadweight loss (7,500 when the tax is $150 compared to 13,333 when the tax is $200).
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