Tariff Revenue and Deadweight Loss

From a Tariff Wedge to Revenue and Welfare Loss

An import tariff raises the domestic price above the world price in a small importing economy. That price change affects the government, domestic producers, consumers, and the quantity imported. To assess the tariff’s welfare effect, distinguish two calculations: the government collects revenue on the imports that remain, while the economy loses surplus in two separate places—where domestic production expands and where domestic consumption contracts.

This tutorial considers a specific tariff, a fixed charge per imported unit. The formulas below are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. They use the small-country competitive-market model: the economy takes the world price as given, the tariff is fully reflected in the domestic price, and there are no other distortions or tariff-collection costs.

Definition: Let $P^*$ be the fixed world price and $t$ the specific tariff per imported unit. Under the model assumptions, the domestic price rises from $P^*$ to $P^*+t$. Let $Q_{s0}$ and $Q_{d0}$ be domestic supply and demand at the free-trade price; let $Q_{s1}$ and $Q_{d1}$ be supply and demand after the tariff. Imports are $M_0=Q_{d0}-Q_{s0}$ before the tariff and $M_1=Q_{d1}-Q_{s1}$ after it.

The tariff changes domestic quantities in opposite directions. Producers supply more because the domestic price is higher; consumers demand less for the same reason. The post-tariff import quantity is therefore lower than the free-trade quantity. The government’s revenue depends on this post-tariff quantity, not the imports that would have occurred without the tariff.

$$ R=tM_1 $$

For linear supply and demand, the two deadweight-loss components are triangles. Define the domestic production increase as $\Delta Q_s=Q_{s1}-Q_{s0}$, and the domestic consumption decrease as $\Delta Q_d=Q_{d0}-Q_{d1}$. Then:

DWL_{\text{production}}=\frac{1}{2}t\Delta Q_s \qquad DWL_{\text{consumption}}=\frac{1}{2}t\Delta Q_d

DWL_{\text{total}}=DWL_{\text{production}}+DWL_{\text{consumption}} =\frac{1}{2}t(\Delta Q_s+\Delta Q_d)

The total reduction in imports equals the two domestic quantity responses added together. That identity is a useful check, but it does not mean that each response should be replaced by the total import reduction when calculating a triangle.

M_0-M_1=(Q_{d0}-Q_{s0})-(Q_{d1}-Q_{s1}) =\Delta Q_s+\Delta Q_d

Why the Deadweight-Loss Triangles Appear

At the world price, an additional unit supplied by domestic producers costs the economy the world price: that unit could instead have been imported. After the tariff, domestic producers expand output even though the marginal cost of the additional units rises above the world price. With linear supply, the excess cost rises from zero at the first added unit to $t$ at the last one. The resulting loss is a triangle with height $t$ and base $\Delta Q_s$.

On the consumption side, the tariff causes consumers to give up units whose value to them is above the world price but below the tariff-inclusive domestic price. With linear demand, the value-price gap changes from zero at the first unit given up to $t$ at the last. This creates the second triangle, with height $t$ and base $\Delta Q_d$. Adding the two areas gives the net welfare loss: tariff revenue is a transfer to the government, not an additional resource loss within this model.

The triangle formulas are exact when supply and demand are linear over the affected quantity ranges. With nonlinear curves, welfare loss is still the area between the relevant curves and the price lines, but that area need not be triangular. The formulas should not be used as exact triangle calculations in that case.

Worked Example: Revenue and Both Loss Triangles

A small importing economy has domestic supply $Q_s=2P-20$ and domestic demand $Q_d=100-2P$, where price $P$ is in dollars per unit and quantities are units per period. The world price is $P^*=\$20$. A specific tariff of $t=\$6$ per unit is imposed. Find free-trade imports, post-tariff imports, tariff revenue, and each deadweight-loss component.

Method. Find supply and demand at the world price and at the tariff-inclusive domestic price. Then calculate revenue using post-tariff imports and calculate each triangle from its own quantity change. This avoids applying the tariff to the wrong import quantity.

At the free-trade price of $\$20$, supply is $Q_{s0}=2(20)-20=20$, and demand is $Q_{d0}=100-2(20)=60$. Thus free-trade imports are $M_0=60-20=40$ units.

The domestic price after the tariff is $\$20+\$6=\$26$. At that price, supply is $Q_{s1}=2(26)-20=32$, and demand is $Q_{d1}=100-2(26)=48$. Post-tariff imports are $M_1=48-32=16$ units.

Tariff revenue is $R=(\$6)(16)=\$96$ per period. The production increase is $\Delta Q_s=32-20=12$ units, so its loss triangle is $\frac{1}{2}(\$6)(12)=\$36$. The consumption decrease is $\Delta Q_d=60-48=12$ units, so its loss triangle is $\frac{1}{2}(\$6)(12)=\$36$. Total deadweight loss is $\$36+\$36=\$72$ per period.

Check. Imports fell by $40-16=24$ units. Domestic production rose by 12 and consumption fell by 12, also a combined 24 units. Revenue uses the 16 imports that remain, not the 40 free-trade imports.

Compute the Components Separately

When a problem gives quantity changes directly, the calculation can be shorter, but the interpretation still matters. Revenue needs the post-tariff import quantity. The production triangle needs the increase in domestic output. The consumption triangle needs the reduction in domestic demand. If only the total import reduction is given, it is enough for total deadweight loss under the linear model, but it is not enough to split the loss into its two components.

Worked Example: Starting From Import and Quantity Changes

A tariff of $\$8$ per imported unit leaves 30 units of imports. Before the tariff, imports were 70 units. Domestic production increases by 15 units, while domestic consumption decreases. Find tariff revenue, the consumption decrease, and both loss components.

Method. First use the change in imports to check and determine the missing domestic quantity response. Then apply the revenue and triangle formulas separately.

Tariff revenue is $R=(\$8)(30)=\$240$. Imports declined by $70-30=40$ units. Because $M_0-M_1=\Delta Q_s+\Delta Q_d$, the consumption decrease is $\Delta Q_d=40-15=25$ units.

The production loss is $\frac{1}{2}(\$8)(15)=\$60$. The consumption loss is $\frac{1}{2}(\$8)(25)=\$100$. Total deadweight loss is $\$60+\$100=\$160$.

Check. The quantity responses add to $15+25=40$ units, matching the fall in imports. The government receives $\$240$, while the two triangles total $\$160$; these are different welfare-accounting quantities, not amounts that must be equal.

How a Larger Tariff Changes the Results

A higher tariff increases the price wedge, but it also changes the quantity of imports on which revenue is collected. Consequently, revenue does not necessarily rise whenever the tariff rate rises. Deadweight loss, in contrast, depends on both the height of the wedge and the size of the domestic quantity responses. Comparing two tariff levels requires recalculating quantities at each price rather than multiplying the new tariff by the original imports.

Worked Example: Comparing Two Tariff Levels

Suppose an importing economy has $Q_s=3P-15$, $Q_d=105-3P$, and world price $P^*=\$10$. Compare a tariff of $\$4$ per unit with a tariff of $\$8$ per unit. For each tariff, calculate imports, revenue, and total deadweight loss.

Method. Calculate the free-trade benchmark once, then evaluate supply and demand at each tariff-inclusive price. Use post-tariff imports for revenue and separate quantity changes for the triangles.

At the world price of $\$10$, supply is $3(10)-15=15$, demand is $105-3(10)=75$, and free-trade imports are $75-15=60$ units.

With a $\$4$ tariff, the domestic price is $\$14$. Supply is $3(14)-15=27$, and demand is $105-3(14)=63$, so imports are $63-27=36$. Revenue is $(\$4)(36)=\$144$. Production rises by $27-15=12$ units and consumption falls by $75-63=12$ units. Each triangle is $\frac{1}{2}(\$4)(12)=\$24$, so total deadweight loss is $\$48$.

With an $\$8$ tariff, the domestic price is $\$18$. Supply is $3(18)-15=39$, and demand is $105-3(18)=51$, so imports are $51-39=12$. Revenue is $(\$8)(12)=\$96$. Production rises by $39-15=24$ units and consumption falls by $75-51=24$ units. Each triangle is $\frac{1}{2}(\$8)(24)=\$96$, so total deadweight loss is $\$192$.

Check. At the higher tariff, imports fall from 36 to 12, and revenue falls from $\$144$ to $\$96$ despite the doubled tariff. Total deadweight loss rises from $\$48$ to $\$192$. This comparison shows why revenue and welfare loss must be evaluated from the full quantity response, not from the tariff size alone.

Common Calculation Traps

A frequent error is to calculate revenue as $tM_0$, using free-trade imports. In the first worked example, that wrong calculation would be $(\$6)(40)=\$240$. The correct calculation is $(\$6)(16)=\$96$, because only 16 units are imported after the tariff. Applying the tariff to imports that no longer occur overstates government revenue.

Another error is to use the entire import reduction as the base of each loss triangle. In the first example, imports fell by 24 units, but production rose by 12 and consumption fell by 12. Each triangle therefore has base 12, not 24. Using 24 for both triangles would double-count the quantity response and produce an incorrect total loss.

The units provide a useful check: a tariff measured in dollars per unit multiplied by imports measured in units gives revenue in dollars. A tariff measured in dollars per unit multiplied by a quantity change in units gives a triangle area in dollars. Always state the tariff’s unit and the quantity period so that the revenue and losses refer to the same period.

Key takeaway: Under the small-country model, tariff revenue is the per-unit tariff times post-tariff imports. With linear supply and demand, total deadweight loss is the sum of a production triangle and a consumption triangle, each with height equal to the tariff and base equal to its own domestic quantity response.

Check Your Understanding

For each problem, use the small-country model and the triangle formulas where supply and demand are linear.

A $\$5$ tariff leaves 22 units imported. Domestic production rises by 8 units and domestic consumption falls by 10 units. Find revenue and total deadweight loss. (Answer: Revenue = $\$110$; production loss = $\$20$; consumption loss = $\$25$; total deadweight loss = $\$45$.)
A $\$3$ tariff reduces imports from 50 units to 30 units, while domestic output rises by 6 units. Find the decrease in consumption, revenue, and total deadweight loss. (Answer: Consumption falls by 14 units; revenue = $\$90$; production loss = $\$9$; consumption loss = $\$21$; total deadweight loss = $\$30$.)
Supply is $Q_s=2P-8$, demand is $Q_d=92-2P$, the world price is $\$12$, and the tariff is $\$5$ per unit. Calculate free-trade imports, post-tariff imports, revenue, and total deadweight loss. (Answer: Free-trade imports = 52 units; post-tariff imports = 32 units; revenue = $\$160$; production loss = $\$25$; consumption loss = $\$25$; total deadweight loss = $\$50$.)
Supply is $Q_s=4P-20$, demand is $Q_d=100-2P$, and the world price is $\$15$. Compare tariffs of $\$2$ and $\$4$ per unit. For each, find imports, revenue, and total deadweight loss. (Answer: At $\$2$, imports = 18 units, revenue = $\$36$, and total deadweight loss = $\$12$. At $\$4$, imports = 6 units, revenue = $\$24$, and total deadweight loss = $\$48$.)
A $\$6$ tariff reduces imports from 84 to 54 units, and domestic production rises by 12 units. Find the consumption decrease, revenue, and each deadweight-loss component. (Answer: Consumption falls by 18 units; revenue = $\$324$; production loss = $\$36$; consumption loss = $\$54$; total deadweight loss = $\$90$.)

What You'll Learn