Worked Tariff Welfare Calculations

Set Up the Tariff Calculation

A tariff calculation links a policy change to several outcomes: the domestic price, domestic production and consumption, imports, and the gains or losses experienced by consumers, producers, and the government. The accounting is most direct in a small-country model, where the country takes the world price as given.

The calculations below are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. Use a specific tariff: a fixed amount charged on each imported unit. Assume competitive domestic markets, unchanged linear demand and supply curves, and no external effects. The tariff does not change the world price. If imports continue after the tariff, the domestic price rises by the tariff amount. If the tariff is sufficiently high to eliminate imports, the domestic price instead reaches the domestic no-trade equilibrium; the formula that adds the full tariff to the world price no longer applies.

Definition: Let $P_w$ be the world price, $t$ the specific tariff per imported unit, $Q_d(P)$ domestic quantity demanded at price $P$, and $Q_s(P)$ domestic quantity supplied. With positive imports both before and after the tariff, the initial domestic price is $P_w$ and the post-tariff domestic price is $P_w+t$. Imports at either price are $M=Q_d-Q_s$.

For the welfare calculations, consumer surplus is the area below the demand curve and above the price; producer surplus is the area above the supply curve and below the price. Tariff revenue equals the tariff per unit multiplied by the number of units imported after the tariff. In this small-country model, changes in these three components are the measured change in national surplus. The model excludes any terms-of-trade gain because the world price is fixed.

Calculate the Price, Quantities, and Imports

Begin by evaluating domestic demand and supply at the world price. Then evaluate both curves at the post-tariff price. Keep the difference between demand and supply explicit: demand is domestic consumption, supply is domestic production, and their difference is imports.

Find the initial outcome.
Set the domestic price equal to $P_w$, then calculate $Q_{d0}=Q_d(P_w)$, $Q_{s0}=Q_s(P_w)$, and $M_0=Q_{d0}-Q_{s0}$.

Apply the tariff.
If imports remain positive, set $P_1=P_w+t$. Calculate $Q_{d1}=Q_d(P_1)$, $Q_{s1}=Q_s(P_1)$, and $M_1=Q_{d1}-Q_{s1}$. Confirm that $M_1>0$.

Calculate welfare changes.
Use the areas under or above the linear curves, and calculate tariff revenue from $tM_1$. Check that the total change equals the negative of the two deadweight-loss triangles.

Worked Example: Calculate All Effects From Inverse Curves

A hypothetical small country faces a world price of $20 per unit and imposes a specific tariff of $5 per unit. Its inverse domestic demand and supply curves are $P=45-Q_d/4$ and $P=Q_s/2-10$. Find the price, quantities, imports, changes in surplus, tariff revenue, and deadweight loss. Use the price-and-area method; in particular, use post-tariff imports for revenue.

Initial outcome. At $P_w=20$, demand satisfies $20=45-Q_{d0}/4$, so $Q_{d0}=4(45-20)=100$. Supply satisfies $20=Q_{s0}/2-10$, so $Q_{s0}=2(20+10)=60$. Initial imports are $M_0=100-60=40$ units.

Post-tariff outcome. The candidate domestic price is $P_1=20+5=25$. At this price, $Q_{d1}=4(45-25)=80$ and $Q_{s1}=2(25+10)=70$. Post-tariff imports are $M_1=80-70=10$, which are positive; therefore, the price formula is applicable.

Consumer surplus change. With linear demand, the loss is the trapezoid between the two prices and the two consumption quantities: $\Delta CS=-5(100+80)/2=-450$ dollars.

Producer surplus change. With linear supply, the gain is $\Delta PS=5(60+70)/2=325$ dollars.

Revenue and net change. Tariff revenue is $TR=5(10)=50$ dollars. The measured total change is $-450+325+50=-75$ dollars.

Deadweight-loss check. Domestic consumption falls by $100-80=20$ units, and domestic production rises by $70-60=10$ units. Thus $DWL=5(20+10)/2=75$ dollars. The net change of $-$75$ matches the deadweight-loss calculation.

Surplus Formulas and Why They Work

For linear curves, changes in surplus are trapezoid areas. When the price rises from $P_w$ to $P_w+t$, consumer surplus falls over the interval between initial and final consumption. Producer surplus rises over the interval between initial and final domestic production. Therefore:

\Delta CS=-\frac{t}{2}(Q_{d0}+Q_{d1}), \qquad \Delta PS=\frac{t}{2}(Q_{s0}+Q_{s1}), \qquad TR=tM_1.

The negative sign for consumers records a loss; the positive sign for producers records a gain. Revenue uses $M_1$, not $M_0$, because only the units still imported after the tariff are taxed.

The deadweight-loss result follows by adding the three welfare changes. Since $M_1=Q_{d1}-Q_{s1}$, substitute that expression for imports in the total:

\begin{aligned} \Delta CS+\Delta PS+TR &=-\frac{t}{2}(Q_{d0}+Q_{d1}) +\frac{t}{2}(Q_{s0}+Q_{s1}) +t(Q_{d1}-Q_{s1})\\ &=-\frac{t}{2}\left[(Q_{d0}-Q_{d1})+(Q_{s1}-Q_{s0})\right]. \end{aligned}

The demand reduction $Q_{d0}-Q_{d1}$ and supply increase $Q_{s1}-Q_{s0}$ are both nonnegative when the tariff raises price along ordinary downward-sloping demand and upward-sloping supply curves. Each is the base of a triangle whose height is $t$. Their combined area is the deadweight loss:

Result: For a small country with linear demand and supply and positive imports before and after a specific tariff, $DWL=\frac{t}{2}[(Q_{d0}-Q_{d1})+(Q_{s1}-Q_{s0})]$. Hence $\Delta CS+\Delta PS+TR=-DWL$. The result does not include any change in the world price or other external effects.

Worked Example: Keep Quantity Units and Money Units Consistent

A hypothetical market reports quantities in thousands of units and prices in dollars per unit. Demand is $Q_d=240-3P$, supply is $Q_s=30+2P$, the world price is $30, and the tariff is $8 per unit. Calculate the market outcomes and welfare effects, reporting money in thousands of dollars. Use the trapezoid-and-triangle method; the common mistake to avoid is treating quantities in thousands as individual units when reporting revenue.

Before the tariff. At $P_w=30$, $Q_{d0}=240-3(30)=150$ thousand units and $Q_{s0}=30+2(30)=90$ thousand units. Thus $M_0=150-90=60$ thousand units.

After the tariff. The domestic price is $P_1=30+8=38$ dollars per unit. Then $Q_{d1}=240-3(38)=126$ thousand units and $Q_{s1}=30+2(38)=106$ thousand units. Imports are $M_1=126-106=20$ thousand units, so the country remains an importer.

Surplus changes. Consumer surplus changes by $-8(150+126)/2=-8(138)=-1{,}104$ thousand dollars. Producer surplus changes by $8(90+106)/2=8(98)=784$ thousand dollars.

Revenue and check. Revenue is $8$ dollars per unit times $20$ thousand units, or $160$ thousand dollars. The total change is $-1{,}104+784+160=-160$ thousand dollars. Quantity demanded falls by $150-126=24$ thousand units and quantity supplied rises by $106-90=16$ thousand units. Thus deadweight loss is $8(24+16)/2=160$ thousand dollars, confirming the result.

Check the Import Condition and the Revenue Base

A calculated price of $P_w+t$ is valid only if the tariff does not eliminate imports. If the resulting demand is less than or equal to supply, the country cannot import the implied negative amount. In that case the domestic price is determined by the intersection of domestic demand and supply, and tariff revenue is zero when imports are zero.

Worked Example: Recover a Tariff From the Observed Price

A hypothetical small country has $Q_d=210-3P$, $Q_s=30+P$, and a world price of $30 per unit. After a tariff, its domestic price is observed to be $36. Find the tariff, imports before and after it, and the welfare effects. Use the observed-price method; it avoids assuming a tariff rate before checking the data.

Tariff and initial quantities. The price increase is $36-30=$6$, so the specific tariff is $t=$6$ per unit. At the world price, $Q_{d0}=210-3(30)=120$ and $Q_{s0}=30+30=60$, giving $M_0=120-60=60$ units.

Post-tariff quantities. At $P_1=36$, $Q_{d1}=210-3(36)=102$ and $Q_{s1}=30+36=66$. Therefore $M_1=102-66=36$ units, confirming that imports remain positive.

Welfare calculations. The consumer loss is $6(120+102)/2=666$ dollars. The producer gain is $6(60+66)/2=378$ dollars. Revenue is $6(36)=216$ dollars. The total change is $-666+378+216=-72$ dollars.

Check. Consumption falls by $120-102=18$ units and production rises by $66-60=6$ units. Deadweight loss is $6(18+6)/2=72$ dollars, equal to the magnitude of the net loss.

Common Calculation Traps

Using initial imports for tariff revenue. In the first worked example, using $5(40)=$200$ wrongly treats all initial imports as still imported. The correct calculation is $5(10)=$50$, based on post-tariff imports.
Using only the final quantity to measure a surplus change. For example, calculating the consumer loss as $tQ_{d1}$ gives a rectangle, not the area under a linear demand curve between the two quantities. Use the trapezoid $t(Q_{d0}+Q_{d1})/2$.
Ignoring units. If quantity is measured in thousands of units, multiplying dollars per unit by that quantity gives thousands of dollars. State that scale in the answer.
Applying the price formula after imports disappear. Always calculate post-tariff imports. A result of zero or less means the country is at or beyond the no-trade outcome, and the usual tariff-revenue and price calculations need reconsideration.

Key takeaway: Calculate both domestic quantities at both prices, verify that post-tariff imports are positive, and base tariff revenue on those post-tariff imports. For linear curves, the consumer loss and producer gain are trapezoids; the remaining net loss must equal the two deadweight-loss triangles.

Check Your Understanding

For each problem, calculate the requested tariff outcomes and welfare quantities. Quantities and money are in the units stated.

Demand is $Q_d=150-2P$, supply is $Q_s=10+P$, the world price is $20, and a $4 tariff is imposed. Find initial and final imports, tariff revenue, consumer and producer surplus changes, and deadweight loss. (Answer: imports fall from 80 to 68 units; revenue $272; $\Delta CS=-$424$, $\Delta PS=+$128$, and $DWL=$24$.)
Demand is $Q_d=300-4P$, supply is $Q_s=20+2P$, the world price is $40, and the tariff is $5. Calculate the two outcomes and the welfare changes. (Answer: imports fall from 40 to 10 units; $\Delta CS=-$650$, $\Delta PS=+$525$, revenue $50, and $DWL=$75$.)
Quantities are in thousands of units. Demand is $Q_d=500-5P$, supply is $Q_s=50+2P$, the world price is $40, and the tariff is $10 per unit. Report welfare amounts in thousands of dollars. (Answer: imports fall from 170 to 100 thousand units; $\Delta CS=-2{,}750$, $\Delta PS=+1{,}400$, revenue $1{,}000$, and $DWL=350$ thousand dollars.)
Demand is $Q_d=180-2P$, supply is $Q_s=20+P$, and the world price is $30. The observed domestic price after the tariff is $38. Calculate the tariff and welfare effects. (Answer: tariff $8; imports fall from 70 to 46 units; $\Delta CS=-$896$, $\Delta PS=+$432$, revenue $368, and $DWL=$96$.)
Demand is $Q_d=260-4P$, supply is $Q_s=20+2P$, and the world price is $30. Find the smallest tariff that eliminates imports, and calculate the welfare changes at that threshold. (Answer: domestic no-trade price $40, so the threshold tariff is $10; imports fall from 60 units to zero; $\Delta CS=-$1{,}200$, $\Delta PS=+$900$, revenue $0, and $DWL=$300$.)

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