Why Trade-Policy Diagrams Need an Audit
Trade-policy welfare diagrams are easy to misread because several different things happen at once. A policy can change the domestic price, alter domestic production and consumption, and create a government receipt or a quota rent. These effects are not all welfare losses: some are transfers between groups within the country, while deadweight loss is the net loss from distorted production or consumption.
A reliable diagram audit therefore starts with the model and the price wedge, not with shaded areas. This tutorial uses the standard competitive, small-country setting: the country takes the world price as given, and there are no externalities or other market failures. Unless stated otherwise, amounts are domestic national welfare, and policy administration has no cost. In this setting, a transfer between domestic consumers, producers, and government changes who receives surplus but does not, by itself, destroy national welfare.
Start With the Price and Quantity Changes
For an import tariff \(t\) in the small-country model, the domestic price rises from the world price \(P_w\) to \(P_w+t\), provided imports remain positive. Domestic quantity demanded falls, domestic quantity supplied rises, and imports equal quantity demanded minus quantity supplied at the domestic price. Tariff revenue is the tariff per imported unit multiplied by post-tariff imports.
A binding import quota fixes imports at a quantity below free-trade imports. With competitive domestic markets and a fixed \(P_w\), the domestic price rises until domestic excess demand equals the quota. The price gap \(P_d-P_w\), multiplied by quota imports, is quota rent. Who receives that rent depends on license allocation; it is not automatically government revenue.
For an export subsidy \(s\), the standard small-country competitive model gives domestic sellers a price \(P_w+s\). Domestic production rises and domestic consumption falls, so exports increase. The government pays the subsidy on post-policy exports. This differs from the import-tariff revenue rectangle: subsidy expenditure is a fiscal cost.
Write the initial world price and the policy price on the vertical axis. Check the direction and size of the wedge before shading any area.
Read domestic demand and supply separately. For imports, subtract supply from demand; for exports, subtract demand from supply.
Identify tariff revenue, quota rents, or subsidy payments separately from the triangles caused by changed production and consumption.
Add the domestic changes in CS, PS, and government receipts or costs. The result should agree with the net welfare change implied by the distortion areas.
Worked Welfare Checks
Worked Example: Auditing a Tariff Diagram
A small country faces a world price \(P_w=\$10\) and imposes a tariff \(t=\$2\) per unit. Its demand and supply are \(Q_d=100-4P\) and \(Q_s=10+2P\). Find the free-trade and tariff quantities, tariff revenue, and deadweight loss. Assume competitive markets and no other distortions.
Method and price check. Under free trade, the domestic price is \(P=\$10\). After the tariff, it is \(P=\$10+\$2=\$12\). The tariff wedge is \(\$2\), not \(\$12\): the latter is the full domestic price.
Free-trade quantities. At \(P=10\), \(Q_d=100-4(10)=60\), while \(Q_s=10+2(10)=30\). Imports are \(60-30=30\) units.
Post-tariff quantities. At \(P=12\), \(Q_d=100-4(12)=52\), and \(Q_s=10+2(12)=34\). Imports are \(52-34=18\) units. This is the quantity that earns tariff revenue.
Revenue and surplus changes. Tariff revenue is \(2(18)=\$36\). The price increase reduces CS by the price rise times average demand across the two prices: \(2(60+52)/2=\$112\). It increases PS by \(2(30+34)/2=\$64\). The net change is \(-112+64+36=-\$12\).
Area check. The demand-side distortion is \(1/2(2)(60-52)=\$8\). The supply-side distortion is \(1/2(2)(34-30)=\$4\). Thus DWL is \(8+4=\$12\), matching the \(-\$12\) net change. The common mistake this avoids is using the original 30 imports to calculate revenue: the correct rectangle uses the post-tariff 18 imports.
Worked Example: Finding Quota Rents Without Calling Them Revenue
A small country has \(Q_d=120-3P\) and \(Q_s=20+P\). The world price is \(P_w=\$10\), and an import quota is fixed at 30 units. Find the domestic price, quota rent, domestic welfare change, and DWL. Assume the quota binds and domestic firms and buyers are competitive.
Free-trade benchmark. At \(P=10\), demand is \(120-30=90\), supply is \(20+10=30\), and imports are \(90-30=60\). Since the quota permits only 30 imports, it is binding.
Quota price. At the domestic price \(P\), imports demanded are \(Q_d-Q_s=(120-3P)-(20+P)=100-4P\). Set this equal to the quota: \(100-4P=30\), so \(4P=70\) and \(P=\$17.50\). At that price, \(Q_d=120-3(17.5)=67.5\), \(Q_s=20+17.5=37.5\), and imports are \(67.5-37.5=30\).
Rent and surplus changes. The price gap is \(17.50-10=\$7.50\), so quota rent is \(7.50(30)=\$225\). CS falls by \(7.5(90+67.5)/2=\$590.625\), and PS rises by \(7.5(30+37.5)/2=\$253.125\). If domestic residents receive the quota licenses, the rent is a domestic transfer, so the net domestic change is \(-590.625+253.125+225=-\$112.50\).
Area check. Demand falls by \(90-67.5=22.5\) units, giving a consumption-distortion triangle of \(1/2(7.5)(22.5)=\$84.375\). Supply rises by \(37.5-30=7.5\) units, giving a production-distortion triangle of \(1/2(7.5)(7.5)=\$28.125\). Total DWL is \(84.375+28.125=\$112.50\), as in the surplus calculation. The common mistake this avoids is counting the rent as government revenue without evidence: if licenses are allocated to foreign holders, the rent is not received by domestic residents, and the domestic welfare accounting changes.
Worked Example: Checking an Export-Subsidy Diagram
A small exporting country has \(Q_s=20+3P\) and \(Q_d=80-P\). The world price is \(P_w=\$20\), and the government pays an export subsidy \(s=\$4\) per exported unit. Find exports before and after the subsidy, the fiscal cost, and the domestic DWL. Assume competitive markets and that the subsidy applies to every post-policy export.
Free-trade benchmark. At \(P=20\), supply is \(20+3(20)=80\), and demand is \(80-20=60\). Exports are \(80-60=20\) units.
Subsidy price and quantities. The domestic price rises to \(20+4=\$24\). At this price, supply is \(20+3(24)=92\), and demand is \(80-24=56\). Exports are \(92-56=36\), not the original 20 units.
Fiscal and surplus accounting. Government expenditure is \(4(36)=\$144\). CS falls by \(4(60+56)/2=\$232\). PS rises by \(4(80+92)/2=\$344\). The net domestic welfare change is \(-232+344-144=-\$32\).
Area check. Production expands by \(92-80=12\), creating a production distortion of \(1/2(4)(12)=\$24\). Consumption falls by \(60-56=4\), creating a consumption distortion of \(1/2(4)(4)=\$8\). Total DWL is \(\$32\), which matches the net loss. The common mistake this avoids is treating subsidy expenditure as a transfer to domestic taxpayers with no cost: the government’s payment is a cost in national welfare accounting, even though producers receive it.
Transfers, Deadweight Loss, and Signed Arithmetic
The area under a price wedge is not automatically deadweight loss. Under a tariff, the rectangle \(t\) times post-tariff imports is government revenue. Under a quota, the rectangle \((P_d-P_w)\) times quota imports is rent, whose domestic value depends on who holds the licenses. Under an export subsidy, the rectangle \(s\) times post-subsidy exports is government expenditure. The distortion triangles arise from the production and consumption responses, not from simply relabeling these rectangles.
A second audit is arithmetic with signs. Suppose a signed welfare expression is given as \(0.5(4)(3(20)-(150-3(20+4)))+0.5(4)((30+(20+4))-(30+20))\). Evaluate the parentheses before interpreting the result. In the first term, \(3(20)=60\), \(20+4=24\), and \(150-3(24)=78\), so the first bracket is \(60-78=-18\) and its term is \(0.5(4)(-18)=-36\). In the second term, \(30+24=54\), \(30+20=50\), so its term is \(0.5(4)(4)=8\). The signed expression is therefore \(-36+8=-28\), not \(+32\).
That result must not be silently changed to a positive number to make it look like a loss. A signed net-welfare change of \(-28\) represents a loss of 28 in magnitude. If a calculation is explicitly intended to be conventional DWL, which is reported as a nonnegative magnitude, a negative result signals that the signs, labels, or area construction need checking. Preserve the correct signed arithmetic, then interpret it consistently with the definition.
| Policy | Price wedge in the stated model | Rectangle to identify | Common diagram error |
|---|---|---|---|
| Import tariff | Domestic price \(P_w+t\) | Tariff times post-tariff imports | Using pre-tariff imports for revenue |
| Import quota | Domestic price rises until imports equal the quota | Price gap times quota quantity | Assuming quota rent must be government revenue |
| Export subsidy | Domestic price \(P_w+s\) | Subsidy times post-subsidy exports | Using baseline exports for fiscal cost |
Common Errors to Catch
- Using the wrong wedge. The wedge is the difference between the policy price and the world price, not the full domestic price.
- Reading imports or exports from only one curve. Imports are demand minus supply; exports are supply minus demand. Use quantities at the same price.
- Shading a rectangle as DWL. A revenue or rent rectangle is a transfer if received domestically. The usual small-country DWL is the sum of the production and consumption distortion triangles.
- Using the wrong quantity for a rectangle. Tariff revenue and subsidy cost use post-policy trade quantities; quota rent uses the quota quantity.
- Ignoring license ownership. Quota rents count in domestic national welfare only when domestic residents receive them.
- Forcing a signed answer to be positive. Keep the arithmetic sign. A negative net change and a positive loss magnitude are related, but they are not the same signed quantity.
Check Your Understanding
For each problem, state any needed quantities before calculating the requested welfare item. Use competitive markets and the small-country assumptions stated in the problem.
- A country has \(Q_d=90-2P\), \(Q_s=10+P\), \(P_w=\$20\), and a tariff of \(\$5\). Find free-trade imports, post-tariff imports, tariff revenue, and DWL. (Answer: Imports are 20 before and 5 after; revenue is \(\$25\); DWL is \(\$37.50\).)
- Demand is \(Q_d=150-2P\), supply is \(Q_s=30+P\), the world price is \(\$20\), and a binding quota is 30 units. Find the quota price and rent. (Answer: Domestic price is \(\$30\); quota rent is \(\$300\).)
- An exporting country has \(Q_s=5+2P\), \(Q_d=50-P\), and \(P_w=\$20\). A subsidy of \(\$3\) applies to exports. Find exports before and after, fiscal cost, and DWL. (Answer: Exports are 15 before and 24 after; fiscal cost is \(\$72\); DWL is \(\$13.50\).)
- A tariff raises the domestic price by \(\$4\). Demand falls by 12 units and supply rises by 5 units. Under the usual small-country model, calculate the consumption and production distortion areas and total DWL. (Answer: Consumption distortion is \(\$24\); production distortion is \(\$10\); total DWL is \(\$34\).)
- A signed welfare calculation evaluates to \(-\$45\). Explain how to report it if it represents a net welfare change, and distinguish that result from the conventional DWL magnitude. (Answer: Report a net welfare change of \(-\$45\); the corresponding loss magnitude is \(\$45\), if the negative result is correctly identified as a welfare loss.)