Deriving Tariff Welfare Losses From Supply and Demand

Why a Tariff Creates Two Distortions

A tariff changes the domestic price of an imported good. At the higher price, domestic firms produce more, while domestic buyers consume less. Those two quantity changes are not themselves the full welfare loss: each represents units for which the value of the good differs from its cost to the economy. With linear supply and demand, the losses are triangular areas.

This tutorial derives those areas for a specific tariff in a small importing economy. The key is to separate the extra domestic production from the consumption that is given up, then relate each quantity change to the tariff wedge. The resulting formulas are useful only when their assumptions fit the situation.

Model and assumptions: A small country imports a good at a fixed world price $P_w$, which it cannot affect. A specific tariff $t$ per unit raises the domestic price from $P_w$ to $P_w+t$. Domestic supply and demand are competitive and linear over the quantities considered. There are no externalities, other market distortions, or tariff-collection costs. Tariff revenue is a transfer within the country, not a resource cost. Under these assumptions, the tariff’s net national welfare loss is the sum of its production and consumption distortions.

Let $Q_s$ denote domestic quantity supplied and $Q_d$ domestic quantity demanded. The tariff increases domestic output from $Q_{s0}$ to $Q_{s1}$, and reduces domestic consumption from $Q_{d0}$ to $Q_{d1}$. Define the changes as $\Delta Q_s=Q_{s1}-Q_{s0}$ and $\Delta Q_d=Q_{d0}-Q_{d1}$, both nonnegative for an upward-sloping supply curve and a downward-sloping demand curve. Imports are $Q_d-Q_s$.

Definition: The production distortion is the welfare loss from replacing imports with extra domestic production whose marginal cost exceeds the fixed world price. The consumption distortion is the loss from buyers giving up units whose willingness to pay exceeds the world price. The total deadweight loss is the sum of these two losses.

Deriving the Triangular Areas

For domestic supply, the marginal cost of the last unit is given by the inverse supply curve. Before the tariff, firms produce up to the point at which marginal cost equals $P_w$. After the tariff, they produce up to the point at which marginal cost equals $P_w+t$. The extra units therefore have marginal costs rising from $P_w$ to $P_w+t$. Their total resource cost above the world-price alternative is the area between that rising marginal cost and $P_w$.

For domestic demand, buyers initially consume units with willingness to pay at least $P_w$. After the tariff, they face $P_w+t$ and give up the units whose willingness to pay lies between those two prices. The value of those forgone units above their world-price cost is the area between the demand curve and $P_w$.

When supply and demand are linear, marginal cost and willingness to pay change at constant rates as quantity changes. Each area is therefore a triangle with vertical height $t$. The horizontal base is the corresponding change in quantity.

\text{Production loss}=\frac{1}{2}t\Delta Q_s,\qquad \text{Consumption loss}=\frac{1}{2}t\Delta Q_d

Adding the two areas gives the small-country tariff deadweight loss:

\text{DWL}=\frac{1}{2}t(\Delta Q_s+\Delta Q_d)

The half in each expression follows directly from the area of a triangle. For a linear inverse supply curve, marginal cost rises evenly from $P_w$ to $P_w+t$ across the extra output $\Delta Q_s$. Thus the average excess cost per extra unit is $\bigl(P_w+t-P_w\bigr)/2=t/2$, and multiplying by the number of extra units gives $(t/2)\Delta Q_s$. On the demand side, willingness to pay falls evenly from $P_w+t$ to $P_w$ over the forgone quantity $\Delta Q_d$. Its average value above $P_w$ is also $t/2$, giving $(t/2)\Delta Q_d$. This argument requires linear curves over the affected ranges; with nonlinear curves, the areas are generally not triangles and must be calculated from the curves instead.

Worked Calculations

Worked Example: Separate the Two Losses

A small importing country faces a fixed world price of $40 per unit. Its inverse domestic supply curve is $P=20+2Q_s$, and its inverse domestic demand curve is $P=100-Q_d$. A specific tariff of $12 per unit is imposed. Find the initial and post-tariff quantities, the two welfare losses, and total deadweight loss.

Method. First find the quantities at the two domestic prices, $40 and $52. Then apply the triangle formula separately to supply and demand. This method avoids treating the entire tariff wedge as the loss on every changed unit.

At the world price of $40: Set price equal to inverse supply:

40=20+2Q_{s0} \quad\Rightarrow\quad 2Q_{s0}=20 \quad\Rightarrow\quad Q_{s0}=10.

Set price equal to inverse demand:

40=100-Q_{d0} \quad\Rightarrow\quad Q_{d0}=60.

Initial imports are $60-10=50$ units, so the country is an importer at the world price.

After the tariff: The domestic price is $40+12=$52$. Supply becomes

52=20+2Q_{s1} \quad\Rightarrow\quad Q_{s1}=16,

and demand becomes

52=100-Q_{d1} \quad\Rightarrow\quad Q_{d1}=48.

The domestic production increase is $\Delta Q_s=16-10=6$ units. The consumption reduction is $\Delta Q_d=60-48=12$ units.

Production loss: $\frac{1}{2}(12)(6)=$36.$ Consumption loss: $\frac{1}{2}(12)(12)=$72.$ Therefore total deadweight loss is $36+72=$108.$

Check. Post-tariff imports are $48-16=32$ units, down from 50, a decrease of 18. That equals the production increase plus the consumption reduction: $6+12=18$. The two distortion areas are both nonnegative, and their sum is $108.

Worked Example: Use the Inverse-Curve Slopes

A small importing economy has inverse supply $P=10+Q_s$ and inverse demand $P=90-2Q_d$. The fixed world price is $30 per unit and the specific tariff is $6 per unit. Find the initial and post-tariff imports and the tariff deadweight loss.

Method. Calculate quantities at both prices, then find the triangle areas. As a check, use the linear curve slopes to confirm the size of each quantity change.

At $P=30$, supply is $30=10+Q_{s0}$, so $Q_{s0}=20$. Demand is $30=90-2Q_{d0}$, so $2Q_{d0}=60$ and $Q_{d0}=30$. Initial imports are $30-20=10$ units.

The tariff raises the domestic price to $30+6=$36$. At that price, supply is $36=10+Q_{s1}$, giving $Q_{s1}=26$. Demand is $36=90-2Q_{d1}$, giving $2Q_{d1}=54$ and $Q_{d1}=27$. Post-tariff imports are $27-26=1$ unit.

Thus $\Delta Q_s=26-20=6$, and $\Delta Q_d=30-27=3$. The production loss is $\frac{1}{2}(6)(6)=$18$; the consumption loss is $\frac{1}{2}(6)(3)=$9$. Total deadweight loss is $18+9=$27$.

Slope check. The supply curve’s slope is 1, so a $6 price rise increases supply by $6/1=6$ units. The demand curve’s price coefficient has magnitude 2, so a $6 price rise reduces demand by $6/2=3$ units. These agree with the calculated changes. Imports fall by $10-1=9$ units, also equal to $6+3=9$.

Worked Example: Infer the Tariff From an Output Change

A small importing country has inverse supply $P=12+3Q_s$ and inverse demand $P=96-2Q_d$. The world price is $36 per unit. After a specific tariff is imposed, domestic production is observed to rise by 4 units. Find the tariff, the change in consumption, and total deadweight loss.

Method. Use the supply curve to infer the price increase and therefore the tariff. Then use that same price increase with the demand curve. This avoids assuming that production and consumption change by equal amounts.

At the world price, initial supply satisfies $36=12+3Q_{s0}$, so $3Q_{s0}=24$ and $Q_{s0}=8$. The observed increase of 4 means $Q_{s1}=8+4=12$. The post-tariff price implied by supply is $12+3(12)=48$. Therefore the tariff is $48-36=$12$ per unit.

Initial demand satisfies $36=96-2Q_{d0}$, so $Q_{d0}=30$. At the post-tariff price of $48, demand satisfies $48=96-2Q_{d1}$, so $Q_{d1}=24$. Consumption falls by $\Delta Q_d=30-24=6$ units.

The production loss is $\frac{1}{2}(12)(4)=$24$, and the consumption loss is $\frac{1}{2}(12)(6)=$36$. Total deadweight loss is $24+36=$60$.

Check. Initial imports are $30-8=22$ units; post-tariff imports are $24-12=12$ units. Imports decline by 10, which equals the output increase of 4 plus the consumption reduction of 6. The inferred price rise is $12, consistent with the tariff wedge in the model.

Common Trap: Using the Rectangle Instead of the Triangle

A frequent error is to multiply the full tariff by each quantity change. In the first worked example, that incorrect calculation would be $12(6+12)=$216$. It treats every extra domestic unit as if its cost exceeded the world price by the full $12, and every forgone consumption unit as if its value exceeded the world price by the full $12. Neither is true: on a linear curve, the gap changes gradually from zero to $12 across the affected units.

The correction is to use average gaps of $12/2=$6$ per changed unit. The production loss is $6(6)=$36$, and the consumption loss is $6(12)=$72$, for total deadweight loss of $108. The tariff rectangle associated with government revenue is not a deadweight-loss triangle: revenue is a transfer under the stated assumptions. Do not add it to the two distortion areas as if it were an additional resource loss.

When the Formulas Apply

The triangle formulas are a consequence of the specific model, not a universal shortcut. In particular, they presume that the domestic price rises by the full specific tariff while the world price stays fixed. They also presume competitive domestic markets, linear curves over the changed quantities, and no external costs or benefits that would make private supply or demand differ from social cost or value.

If the country can affect the world price, the domestic price change need not equal the tariff, and the welfare analysis must account for that price change. If the curves are nonlinear, the correct losses are still the areas between marginal cost and the world-price benchmark for extra production, and between willingness to pay and that benchmark for forgone consumption; those areas may not be triangular. If other distortions are present, the simple sum may not capture the tariff’s full welfare effect.

For the small-country, linear case, the most reliable sequence is to establish the domestic price change, calculate both quantity changes, and then evaluate the two triangles separately. Checking that the change in imports equals $\Delta Q_s+\Delta Q_d$ is a useful consistency test, but it does not replace checking that the two distortion areas have been calculated correctly. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, assume a small importing country, a fixed world price, a specific tariff, competitive markets, and linear domestic supply and demand with no other distortions.

Inverse supply is $P=8+2Q_s$, inverse demand is $P=68-Q_d$, the world price is $28, and the tariff is $8. Find initial and post-tariff supply and demand, then calculate both losses and total deadweight loss. (Answer: Initial $Q_s=10, Q_d=40$; post-tariff $Q_s=14, Q_d=32$; production loss $16, consumption loss $32, total deadweight loss $48.)
Inverse supply is $P=5+Q_s$, inverse demand is $P=65-3Q_d$, and the world price is $15. A $2 tariff is imposed. Find imports before and after the tariff and the total deadweight loss. (Answer: Initial imports $50/3-10=20/3$ units; post-tariff imports $16-12=4$ units; production loss $2, consumption loss $2/3$ dollars, total deadweight loss $8/3$ dollars.)
Inverse supply is $P=10+2Q_s$, inverse demand is $P=70-Q_d$, and the world price is $30. A $8 tariff is imposed. Find the production and consumption changes and total deadweight loss, then check the change in imports. (Answer: Production rises by 4 units, consumption falls by 8 units; production loss $16, consumption loss $32, total $48; imports decline by 12 units.)
With linear curves and a fixed world price, a $15 tariff increases domestic production by 5 units and reduces domestic consumption by 10 units. Find the production and consumption losses. If initial imports were 40 units, find post-tariff imports. (Answer: Production loss $37.50, consumption loss $75, total $112.50; post-tariff imports $40-5-10=25$ units.)
Inverse supply is $P=14+2Q_s$, inverse demand is $P=86-Q_d$, and the world price is $34. A tariff raises domestic production from 10 units to 15 units. Infer the tariff, calculate the consumption change, and find total deadweight loss. (Answer: The post-tariff price is $14+2(15)=$44$, so the tariff is $10; demand falls from 52 to 42 units, a reduction of 10; production loss $25, consumption loss $50, total deadweight loss $75.)

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