Four instruments do roughly the same job at very different prices to the country using them. The surplus machinery you already have turns the comparison into arithmetic.
Worked example: a tariff in a small country
Work a small country, meaning one whose purchases do not move the world price. Take a synthetic US market for an industrial input, quantities in millions of units a year and prices in dollars per unit:
Domestic demand: Qd = 1,200 − 20 P
Domestic supply: Qs = 100 + 15 P
World price: Pw = 20Under free trade the domestic price is the world price. Demand is 800, domestic supply is 400, and imports are 400.
A tariff of $5 raises the domestic price to $25 and nothing else changes. Demand falls to 700, domestic supply rises to 475, and imports are squeezed to 225, a fall of 43.75%. Consumers lose $3,750.00m, producers gain $2,187.50m, the government collects $1,125.00m, and the deadweight loss is $437.50m, which Worked Example 3 verifies by two independent routes. Two ratios make it sayable in a sentence: consumers pay $1.714286 for every dollar reaching producers, and the extra 75 million units of domestic output cost consumers $50.00 each against a world price of $20.00. Protection is a transfer with a leak, and the leak is the two triangles.
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Tariff vs quota: who gets the revenue?
A quota set at 225 million units produces an identical price. Solve (1,200 − 20P) − (100 + 15P) = 225 and you get P = 25.000000, the tariff price to six decimals. Consumers lose the same 3,750.00 and producers gain the same 2,187.50. What differs is the 1,125.00. Under a tariff the government collects it. Under a quota it becomes quota rent, and who gets it depends on who holds the licences.
- Licences auctioned by the importing government: the rent is captured domestically and the national loss is 437.50, exactly the tariff.
- Licences handed to importers: the rent goes to domestic firms, a transfer, and the national loss is still 437.50, though the distribution differs.
- Licences held by foreign exporters, which is what a voluntary export restraint does by construction: the rent of 1,125.00 leaves the country and the national loss becomes 1,562.50.
The same protection, delivered the third way, costs 3.571429 times as much as the tariff, an extra $1,125.00 million a year, or 7.031250% of what consumers spent on the good under free trade. A voluntary export restraint is a tariff whose revenue has been handed to the exporting country as an inducement to accept it, which is precisely why exporters prefer one and why it was the instrument of choice in the automotive disputes of the 1980s.
The restriction is identical. The revenue has simply been given away.
What does an export subsidy cost?
An export subsidy works on the other side of the border and loses money in both directions. Take a second synthetic market, Qd = 900 − 10P, Qs = 50 + 20P, world price $40. Free trade gives demand 500, supply 850 and exports 350. A $6 subsidy raises the domestic price to $46, cuts demand to 440, raises supply to 970 and pushes exports to 530. Consumers lose $2,820.00m, producers gain $5,460.00m, the treasury pays $3,180.00m, and the net national change is 5,460.00 − 2,820.00 − 3,180.00 = −$540.00m, which is exactly the two triangles of 180.00 on consumption and 360.00 on production. Every dollar of producer gain cost the treasury $0.582418, and the country lost $3.00 for each extra unit exported. A subsidy that succeeds in raising exports is not thereby a success.
What does the welfare arithmetic leave out?
State the honest counter-case, because the arithmetic is not the whole argument. Nothing above prices national security, and nothing above prices the option value of retaining a capability. A large country, unlike this one, can shift the world price with a tariff and capture a terms-of-trade gain that can, in principle, exceed the triangles, which is the optimal-tariff argument. And infant-industry protection is a claim about dynamics that a static surplus diagram cannot evaluate. What the arithmetic does establish is the bill. Any of those arguments has to be worth more than 437.50, or 1,562.50, or 540.00, and an argument that never names the number it has to beat is not an argument.