Ch 8 Worksheet — The Costs of Taxation

Question a

What is the pre-tax equilibrium price and quantity?

Equilibrium Price ($)

Equilibrium Quantity

Hint Set demand equal to supply: 50 − 2Q = 10 + 2Q. Solve for Q, then substitute back.

Explanation Setting 50 − 2Q = 10 + 2Q gives 40 = 4Q, so Q* = 10. Plugging back in: P* = 50 − 2(10) = $30. At this price, Qd = Qs = 10 and there is no tax yet.

Question b

With the $10 tax, what is the price buyers pay (P_b), the price sellers receive (P_s), and the quantity traded (Q_T)?

Buyer Price, P_b ($)

Seller Price, P_s ($)

Quantity, Q_T

Hint The tax wedge means P_b − P_s = $10. Use demand to find P_b = 50 − 2Q_T and supply to find P_s = 10 + 2Q_T. Then set their difference = $10.

Explanation With the tax: P_b − P_s = $10. Using demand and supply: (50 − 2Q) − (10 + 2Q) = 10 ⇒ 40 − 4Q = 10 ⇒ Q_T = 7.5.

Buyer price: P_b = 50 − 2(7.5) = $35.
Seller price: P_s = 35 − 10 = $25 (or: 10 + 2(7.5) = $25). Check: $35 − $25 = $10 ✓

Question c

Calculate the tax revenue collected by the government and the deadweight loss (DWL) from the tax.

Tax Revenue ($)

Deadweight Loss ($)

Hint Tax Revenue = T × Q_T (the rectangle). DWL = ½ × T × ΔQ (the triangle), where ΔQ = Q* − Q_T.

Explanation Tax Revenue = T × Q_T = $10 × 7.5 = $75.

DWL = ½ × T × ΔQ = ½ × $10 × (10 − 7.5) = ½ × $10 × 2.5 = $12.50.

Verify: Before-tax Total Surplus = ½×10×20 + ½×10×20 = $200. After-tax: CS + PS + Rev = $56.25 + $56.25 + $75 = $187.50. DWL = $200 − $187.50 = $12.50 ✓

Question d

With a $20 tax, what is the new quantity traded, tax revenue, and deadweight loss?

Q_T (quantity)

Tax Revenue ($)

DWL ($)

Hint Same process as Q(b) and Q(c) but with T = $20. Notice: DWL should be 4× the DWL at T = $10, because DWL ∝ T².

Explanation With T = $20: (50 − 2Q) − (10 + 2Q) = 20 ⇒ 40 − 4Q = 20 ⇒ Q_T = 5.

P_b = 50 − 10 = $40, P_s = $20.

Tax Revenue = $20 × 5 = $100.
DWL = ½ × $20 × (10 − 5) = ½ × $20 × 5 = $50.

Note: Tax doubled ($10 → $20) but DWL quadrupled ($12.50 → $50). This is the T² relationship.

Question e

In the gasoline market, if the tax were raised from $10 to $30 per unit, how many times larger would the DWL be compared to the $10 tax?

DWL multiplier (e.g., enter 4 if DWL is 4× larger)

Hint DWL ∝ T². If tax triples (from $10 to $30), how does T² change? Compute (30)² / (10)².

Explanation DWL ∝ T². Tripling the tax (10 → 30) means DWL changes by a factor of (30/10)² = 3² = 9.

Verify directly: At T = $30, Q_T = (40−30)/4 = 2.5. DWL = ½ × $30 × (10−2.5) = ½ × $30 × 7.5 = $112.50. Original DWL = $12.50. Ratio: $112.50 / $12.50 = 9×.

Question 1

Refer to Figure 8-1. When the government imposes the tax, consumer surplus equals

A + B + C A + B + D A B + D (the tax revenue)

Hint After the tax, buyers pay P_b. Consumer surplus is the area above P_b and below the demand curve. Which labeled area sits above P_b?

Explanation Consumer surplus after the tax is the triangle above the buyer price P_b and below the demand curve, from Q = 0 to Q_T. That's only area A. Areas B and D are between P_b and P_s (they became tax revenue). Areas C and E are the deadweight loss.

Question 2

Refer to Figure 8-1. The deadweight loss from the tax equals

B + D C + E A + F B + C + D + E

Hint Deadweight loss is the value of trades that would have occurred without the tax but don't occur with it. These units are between Q_T and Q*. Look for triangles on the right side.

Explanation DWL = C + E. These are the two triangles to the right of Q_T and to the left of Q*. Area C is the lost consumer surplus from units Q_T to Q*, and area E is the lost producer surplus from those same units. Together they form the DWL triangle. Areas B and D are tax revenue (transferred to the government, not lost).

Question 3

Refer to Figure 8-1. Tax revenue collected by the government equals

A + B C + E B + D B + C + D + E

Hint Tax revenue = tax per unit × quantity traded. On the diagram, that's the rectangle between P_b and P_s, from Q = 0 to Q_T.

Explanation Tax revenue = T × Q_T = (P_b − P_s) × Q_T = the rectangle between P_b and P_s from 0 to Q_T. That's B + D. Area B was formerly consumer surplus (buyers now pay more), and area D was formerly producer surplus (sellers now receive less). Both flow to the government as tax revenue.

Question 4

If the demand for gasoline is very inelastic (people need it regardless of price), a given tax will generate

more tax revenue and larger DWL compared to elastic demand less tax revenue and smaller DWL compared to elastic demand more tax revenue and smaller DWL compared to elastic demand less tax revenue and larger DWL compared to elastic demand

Hint Inelastic demand means quantity barely changes when price rises. If quantity barely falls, what happens to tax revenue (T×Q) and DWL (½T×ΔQ)?

Explanation With inelastic demand, buyers continue purchasing nearly the same quantity even after the tax. So ΔQ is small, making DWL small (½T×ΔQ is small). But since quantity barely falls, tax revenue (T × Q) remains large. Inelastic goods are attractive tax targets: high revenue, low efficiency cost. This is why governments tax necessities like gasoline, cigarettes, and alcohol.

Question 5

Which of the following correctly describes the Laffer curve insight as it relates to tax revenue?

Tax revenue always increases as the tax rate increases Tax revenue first rises then falls as the tax rate increases, reaching zero at a very high tax rate Tax revenue equals deadweight loss at all tax rates Tax revenue is maximized when the tax rate is as high as possible

Hint What happens to quantity traded as the tax gets very large? If Q → 0, what happens to revenue = T × Q?

Explanation As the tax increases, two opposing forces act on revenue: a higher T raises revenue, but a falling Q lowers it. At very high tax rates, Q falls so far that revenue collapses. In the extreme, a prohibitive tax stops all trade (Q = 0) and collects zero revenue. So revenue follows an inverted-U shape — it first rises then falls, reaching zero at a tax rate where no trades occur. This is the Laffer curve.

Question 6

A tax on sellers and a tax on buyers of the same size will result in

buyers bearing all the burden when taxed on buyers, sellers bearing all the burden when taxed on sellers the same equilibrium price, quantity, and tax burden distribution regardless of who is legally taxed different quantities traded but the same tax revenue more DWL when taxed on sellers than on buyers

Hint Tax incidence is determined by elasticities, not by the legal assignment. A tax creates the same wedge whether placed on buyers or sellers — the market adjusts to reach the same outcome.

Explanation Tax equivalence: It doesn't matter which side of the market is legally required to pay — the final outcome (equilibrium quantity, buyer price, seller price, and how the burden is shared) is identical. A $10 tax on sellers shifts the supply curve up by $10. A $10 tax on buyers shifts the demand curve down by $10. Both create the same wedge and the same result. Burden sharing depends on relative elasticities, not on legal incidence.

Question 7

Deadweight loss from a tax represents money that the government collects but does not return to society.

Hint Think about what deadweight loss actually is. Is it money the government has, or is it something that never exists in the first place?

Explanation False. Deadweight loss is NOT money the government collects — it is gains from trade that never materialize. These are transactions that would have been mutually beneficial but don't happen because the tax makes them unprofitable. Tax revenue (areas B + D in Figure 8-1) does go to the government and can be used for public goods. The DWL (areas C + E) is pure economic waste — it helps nobody.

Question 8

If the supply of a good is perfectly inelastic, a tax on that good creates no deadweight loss.

Hint If supply is perfectly inelastic, quantity supplied is fixed regardless of price. What happens to quantity traded when the tax is imposed? If ΔQ = 0, what is DWL = ½T×ΔQ?

Explanation True. With perfectly inelastic supply, sellers produce the same quantity no matter what price they receive. When a tax is imposed, quantity supplied doesn't change, so quantity traded stays at Q* and ΔQ = 0. Since DWL = ½ × T × ΔQ = ½ × T × 0 = $0. The tax is entirely borne by sellers (the supply curve is vertical, so sellers absorb the full tax), but no trades are destroyed.

Question 9

Doubling a tax rate always doubles the tax revenue collected.

Hint Revenue = T × Q. When T doubles, what happens to Q? Does Q stay constant?

Explanation False. Revenue = T × Q_T. When T doubles, Q_T falls (higher tax reduces quantity traded). So revenue doesn't simply double — it increases by less than double (and eventually decreases). In our gasoline example: at T = $10, Revenue = $75; at T = $20, Revenue = $100 (not $150). At very high tax rates, revenue falls toward zero. This is exactly the Laffer curve insight.

Question 10

A tax placed on sellers has a different economic effect on buyers and sellers than a tax of the same size placed on buyers.

Hint Think about tax equivalence. A tax on sellers shifts supply up; a tax on buyers shifts demand down. Does it matter which curve shifts?

Explanation False. This is the principle of tax equivalence. A tax on sellers and an equal tax on buyers produce identical outcomes: the same wedge between P_b and P_s, the same reduction in quantity, the same DWL, and the same distribution of the burden between buyers and sellers. The legal incidence (who writes the check) doesn't determine the economic incidence (who really pays). That depends only on relative elasticities.

Practice Worksheet

Short Answer — The Gasoline Market

Multiple Choice

True or False