International economics · Trade policy

The optimum tariff, retaliation and the Metzler paradox

Q: What is the optimum tariff?

The optimum tariff is the tariff that maximises a country's welfare. It is positive only for a country large enough to affect the world price, where it improves the terms of trade enough to outweigh the deadweight losses. Its size is the reciprocal of the elasticity of foreign export supply: t*/P* = 1/ε. For a small country, whose suppliers are infinitely elastic, the optimum tariff is zero.

Q: Why does a large country gain from a tariff?

A large country's tariff reduces its demand for imports, which lowers the price foreign exporters receive. Part of the tariff is therefore paid by foreigners, and the country gains on its terms of trade. That gain must be set against the two deadweight-loss triangles from distorted production and consumption. For a small tariff the terms-of-trade gain is larger, so welfare rises.

Q: What happens if other countries retaliate?

If the trading partners respond with their own optimum tariffs, each country's terms-of-trade gain is cancelled by the other's, and only the deadweight losses remain. With identical countries, imposing a tariff is each one's best choice whatever the other does, yet when both do it, both end worse off than under free trade. It is a prisoner's dilemma.

Q: What is the Metzler paradox?

The Metzler paradox is the case in which a tariff lowers, instead of raises, the domestic price of the imported good. It happens when the world price falls by more than the tariff adds, which requires that foreign demand for the home country's exports is inelastic. The condition is that the foreign elasticity is less than the home marginal propensity to consume its exportable good.

Q: Does the optimum tariff apply to a small country?

No. A small country cannot influence the world price, so a tariff brings it no terms-of-trade gain and only the two deadweight losses. Free trade is its best policy. JNU asks it more than once: free trade is the optimal trading policy for a small country.

Small countries lose from tariffs. Large ones can gain, up to a point, and the point has a formula. This page derives it, shows why a country that uses it invites retaliation and ends worse off, and then meets the odd case in which a tariff makes the imported good cheaper at home. 10 past-paper questions at the end.

A question to hold on to

A country can increase its welfare by imposing an import tariff if it is small and importing labour-intensive goods; it is small and there is no retaliation; it is large and there is no retaliation; or both of the first two.

JNU 2018Solved at the end ↓

New here?What is the shirts market?The one-good market behind the numbers on this page: demand, supply and a world price of ₹20.

The international economics pages share one invented market, shirts, so a tariff, a quota or a customs union can be compared on the same numbers. Home buyers want 100 − 2P shirts at price P, and home firms supply −20 + 2P. With no trade the price settles at ₹30.

A small country takes the world price of ₹20 as given. At that price home buyers take 60 shirts, home firms make 20, and the other 40 are imported. The numbers are illustrative, not data.

Demand: Qd = 100 − 2P
Home supply: Qs = −20 + 2P
Price with no trade: ₹30, with 40 shirts
World price: ₹20, so 60 bought and 20 made

See it built up →

The idea

Why a large country can gain

The tariffs and quotas page showed that a small country loses from a tariff: the two deadweight triangles are a pure loss, because the world price does not move. A large country is different. Its tariff cuts its demand for imports, so the price exporters can get falls, and the home country pays less for what it still buys. This is the terms-of-trade gain, and it is taken from the foreign suppliers.

Two forces then fight. The terms-of-trade gain is first-order: it is proportional to the size of the tariff. The deadweight loss is second-order: it grows with the square of the tariff. For a small tariff the first wins, so a large country gains. For a big enough one the second wins. That is why CUET 2022 says welfare improves when the terms-of-trade gain exceeds the two losses, and why the best tariff exists.

Pause and predict

A large country raises its tariff, starting from zero. Its welfare

Derivation

The optimum tariff, derived

How big should it be? The derivation uses an analogy: a country with power over the world price behaves like a monopsonist, a single large buyer. It buys until the benefit of one more unit equals its true marginal cost, and the tariff is the gap between the price its buyers pay and the price its suppliers get.

Derive it

Why the optimum tariff is the reciprocal of the supply elasticity

We want to show

A large country maximises its welfare with a tariff of

t^{*} = \frac{P ^{*}}{ε}

, where P* is the world price it faces and ε the elasticity of foreign export supply. Expressed as a rate on the world price,

t^{*} / P^{*} = 1/ ε

A large buyer faces a rising supply curve.
A small country pays the same price per unit whatever it buys. A large country does not: to buy more it must pay foreigners a higher world price, and that higher price applies to every unit it buys, not only the last. In this respect it is like a monopsonist, a single large buyer.
Marginal benefit against marginal cost.
The benefit of one more unit imported is what home buyers will pay for it, net of what home firms would charge: the domestic price p. The cost of one more unit is the marginal expenditure. Total spending is E = P*·M, so
$\frac{d E}{d M} = P^{*} + M \frac{d P ^{*}}{d M} = P^{*} (1 + \frac{1}{ε})$
where ε = (dM/M) ÷ (dP*/P*) is the elasticity of supply. The extra unit costs the price P* plus the rise in price on all the units already bought.
Equate them, and the tariff is the gap.
Free trade buys until p = P*, ignoring the rise in price on earlier units. The best policy buys until p = P*(1 + 1/ε), so the wedge between what buyers pay and what foreigners get is
$t^{*} = p - P^{*} = \frac{P ^{*}}{ε}$
In the shirts market, the optimum is ₹12. There P* = ₹12, home buyers pay p = ₹24 and the elasticity of foreign supply is 1. Marginal expenditure is 12 × (1 + 1) = 24, which is exactly p. A bigger tariff makes buyers face a price above the true marginal cost, and a smaller one below it.
Check the shape and the limits.
In this market the gain from a tariff t is (80 ÷ 3)·t − (10 ÷ 9)·t², which rises, peaks at t = 12 with a gain of 160, and is zero again at t = 24: beyond 24 the country is worse off than under free trade. The more elastic foreign supply is, the smaller t*. For a small country, ε is infinite and t* = 0.

Step 1 of 4

The optimum tariff

\frac{t ^{*}}{P ^{*}} = \frac{1}{ε}

P* is the world price at the optimum and ε the elasticity of foreign export supply at that point. The rate is on the world price.

In the shirts market: the optimum tariff is ₹12, the world price falls from ₹20 to ₹12, home buyers pay ₹24, imports fall from 40 to 24, and the country gains 160: a terms-of-trade gain of 192, less deadweight losses of 32. A small country using the same ₹12 tariff would have lost 200. Hyderabad 2015 asks the name (the optimum tariff), and JNU 2019 two statements about it: the beneficiary is a large country, and any deviation from the optimum lowers welfare. Both are true.

Variant

The same gain through a quota

A quota can reach the same point. If the large country limits imports to the optimum quantity, 24, the price foreign exporters need to supply that amount is ₹12, home buyers pay ₹24, and, provided the licences are sold or kept by the home government, the country gains the same 160. If instead the exporters run the restraint, they keep the rent of 288, and the gain turns into a loss. Who holds the licences decides the sign.

Interactive

Retaliation: the tariff war

The gain from an optimum tariff is a gain at foreign expense, so foreigners have every reason to hit back. This lab uses two identical countries, each exporting one good, each with the same preferences. Slide the tariffs, or use the buttons to go from free trade, to Home alone, to retaliation, to the point where neither wants to change.

Home's tariff on imports of M0%Foreign's tariff on imports of X0%

Home100

Foreign100

Real income, free trade = 100

Real income (Home, Foreign), free trade = 100
	Foreign: free trade	Foreign: tariff 51%
Home: free trade	100, 100	91.2, 104.9
Home: tariff 51%	104.9, 91.2	93.6, 93.6

Whatever the other country does, a tariff pays: 104.9 beats 100, and 93.6 beats 91.2. Both therefore impose one, and both end at 93.6, below the 100 that free trade would have given each.

What it shows

With no tariffs both countries sit at 100.

The table is a prisoner’s dilemma. If Foreign stays at free trade, Home does better with a tariff. If Foreign has a tariff, Home still does better with one. So each imposes a tariff whatever the other does, and both end up in the corner where each is worse off than under free trade. The tariff gains cancel, since each country’s terms-of-trade gain is the other’s loss, and the deadweight losses remain. This is the logic behind trade agreements, in which governments promise not to raise tariffs. JNU 2018’s “no retaliation” condition is the reason the answer to the opening question is qualified.

Derivation and interactive

The Metzler paradox

Everything so far assumed a tariff raises the domestic price of the imported good, by less than the tariff if the country is large. Metzler (1949) showed that this can fail: a tariff can lower the price Home buyers pay. It is possible when the world price falls by more than the tariff adds. The condition is a comparison of two numbers, and the derivation reaches it in four steps.

Derive it

When does a tariff lower the domestic price? The Metzler condition

We want to show

A small tariff lowers the domestic price of the importable if and only if

η^{*} < m_{x}

, where η* is the elasticity of Foreign’s demand for Home’s export, and

m_{x}

is Home’s marginal propensity to consume its export good.

Two forces act on the domestic price.
Home levies a tariff t on its imports. The price Home buyers pay is p = q(1 + t), where q is the world price. The tariff pushes p up directly. But it also cuts Home’s demand for imports, which pushes the world price q down. Everything depends on which is bigger. If q falls by more than the tariff adds, p falls, and that is the paradox.
Write the market-clearing condition and differentiate.
Home’s demand for the import, M, depends on the price p it faces and on its income, which includes the tariff revenue handed back. Foreign supplies M_f(q). Let S < 0 be the substitution effect of a rise in p on Home’s demand, and μ the effect of an extra unit of income. Differentiating M = M_f at t = 0 gives
$(S - M μ - M_{f}^{'}) d q = - S q d t$
and since p = q(1 + t), dp = dq + q·dt. Substituting dq,
$d p = q d t \frac{M μ + M _{f}^{'}}{M μ + M _{f}^{'} - S}$
The denominator is positive because S is negative and the other terms are positive, so the sign of dp is the sign of Mμ + M_f′.
Translate into elasticities.
Multiply by q ÷ M. Then qμ is the share of an extra unit of income that Home spends on the import, which is 1 − m_x, the rest being its marginal propensity to consume its export good. And q·M_f′ ÷ M is the elasticity of Foreign’s supply of M, which equals η* − 1, since what Foreign pays for Home’s exports is q·M_f. So
$sign (d p) = sign [(1 - m_{x}) + (η^{*} - 1)] = sign (η^{*} - m_{x})$
The tariff lowers the domestic price exactly when η* < m_x. Because m_x cannot exceed 1 when there are no inferior goods, η* must be less than 1: Foreign’s demand for Home’s export must be inelastic. That is JNU 2015’s necessary condition, and JNU 2013 states the full condition.
See it on the offer curves.
In the lab’s second case η* = 0.65 and m_x = 0.86, so the condition holds. A tariff of only 10% takes the world price from 0.251 to 0.218, and the price Home buyers pay, q(1 + t), from 0.251 to 0.24: lower than before. In the first case, with η* = 1.25 and m_x = 0.5, it rises.

Step 1 of 4

JNU asks it twice, in two different forms. JNU 2015: the necessary condition for the Metzler paradox, in the absence of inferior goods, is that the foreign import demand is inelastic. JNU 2013: if the home marginal propensity to consume the exportable is greater than the elasticity of the foreigner’s offer curve, a tariff lowers the domestic price of the importable. These are the two parts of the same condition: η* < m_x, and since m_x is at most 1, η* < 1.

What it does to the tariff’s purpose: the tariff was meant to raise the price of the protected industry’s product. If the price falls, the industry is worse off, the factor it uses intensively loses, and the Stolper-Samuelson result is reversed. It is rare in practice, but it is why economists say a tariff does not always protect.

The condition in one line

A tariff lowers the domestic price of the import when η* < m_x: foreign demand for our exports is so inelastic that the world price collapses, and we spend most extra income on our own export good.

Exam focus

The mistakes examiners are fishing for

Watch out for

The mistakes examiners are fishing for

Saying that any country can gain from a tariff. Only a large one, and only if it is not retaliated against.
Believing the optimum tariff is the one that raises most revenue or protects most. It maximises welfare: gains on the terms of trade less deadweight losses.
Forgetting that beyond the optimum the country loses. In the example, any tariff above ₹24 is worse than free trade. A “bigger tariff is better” answer is always wrong.
Saying a deviation from the optimum might improve welfare. By definition it lowers it (JNU 2019).
Ignoring who bears the tariff. In a large country foreigners bear part, as the price they receive falls. In a small country, home buyers bear all of it.
Mixing up the Metzler condition. The paradox needs inelastic foreign demand (η* below m_x, so below 1), not elastic.

Back to the puzzle

The opening question, solved

JNU 2018 · moderate

A country can increase its welfare by imposing an import tariff if…

Test "small"
A small country takes the world price as given, so a tariff gives it only the two deadweight losses. Options A, B and D, which rely on it being small, are out.
Test "large"
A large country gains on its terms of trade. The gain survives only if foreigners do not respond with tariffs of their own.

Answer

It is large and there is no retaliation.

Numbers · the shirts market

Compare a tariff of ₹6 for a small country and for a large country with foreign supply of 2P*.

Small country
It loses the two triangles: 0.5 × 6 × 12 + 0.5 × 6 × 12 = 72.
Large country
The world price falls to ₹16. Terms-of-trade gain 128, deadweight losses 8, net 120.

Answer

A loss of 72 for a small country, a gain of 120 for a large one, from the same tariff.

Revision

The page on one screen

The lesson on one screen

Large country: Its tariff lowers the world price. Foreigners pay part. Gain = terms of trade − two triangles.
First and second order: The terms-of-trade gain grows with t, the deadweight loss with t², so a small tariff gains and a big one loses.
Optimum tariff: t*/P* = 1/ε. Marginal benefit p equals marginal expenditure P*(1 + 1/ε), like a monopsonist.
Small country: ε is infinite, so t* = 0. Free trade is optimal.
Retaliation: Each country gains from a tariff whatever the other does, but both tariffs together leave both worse off: a prisoner’s dilemma.
Quota: The optimum quota reaches the same gain if the home country keeps the licence rent; a VER gives it away.
Metzler paradox: A tariff lowers the domestic price of the import iff η* < m_x. Needs inelastic foreign demand.

Practice

Check yourself

0 of 9 correct so far

Numbers

In the shirts market of the tariffs page, demand is Qd = 100 − 2P and home supply Qs = −20 + 2P. Foreign export supply is 2P*.

The reciprocal rule

At the optimum, the elasticity of foreign supply is 0.5. What is the optimum tariff as a percentage of the world price?

t* as % of P*

A more elastic supplier

Foreign supply has an elasticity of 4 at the optimum. What is the optimum tariff as a percentage of the world price?

t* as % of P*

The world price after a tariff

The country imposes a tariff of ₹6. Using 120 − 4p = 2P* with p = P* + 6, what world price do foreigners now receive? (₹)

World price

The net gain

For the same ₹6 tariff, the net gain is (80 ÷ 3)t − (10 ÷ 9)t². What is it?

Net gain

Concepts

1Free trade is the optimal trade policy for

2A country can raise its welfare by imposing an import tariff if

3Two identical large countries each impose their best tariff. Compared with free trade, each is

4Foreign demand for our export has an elasticity of 0.4, and we spend 70% of any extra income on our export good. A small tariff will

5Hyderabad 2015 asks for the tariff that maximises a country’s welfare. It is called the

Past papers

Past papers, by level

Foundation

Definitions, directions and one-step sums. Most CUET, UGC NET and Hyderabad questions sit here.

3 questionsPractise →

Standard

Classification traps, multi-step numericals and statement questions, up to the hardest JNU, JAM and GATE items.

7 questionsPractise →

All 10 past-paper questions on optimum tariff, the terms-of-trade argument and the metzler paradox, easiest first: practise the full set →

These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

Quick answers

What is the optimum tariff?

The optimum tariff is the tariff that maximises a country's welfare. It is positive only for a country large enough to affect the world price, where it improves the terms of trade enough to outweigh the deadweight losses. Its size is the reciprocal of the elasticity of foreign export supply: t*/P* = 1/ε. For a small country, whose suppliers are infinitely elastic, the optimum tariff is zero.

Why does a large country gain from a tariff?

A large country's tariff reduces its demand for imports, which lowers the price foreign exporters receive. Part of the tariff is therefore paid by foreigners, and the country gains on its terms of trade. That gain must be set against the two deadweight-loss triangles from distorted production and consumption. For a small tariff the terms-of-trade gain is larger, so welfare rises.

What happens if other countries retaliate?

If the trading partners respond with their own optimum tariffs, each country's terms-of-trade gain is cancelled by the other's, and only the deadweight losses remain. With identical countries, imposing a tariff is each one's best choice whatever the other does, yet when both do it, both end worse off than under free trade. It is a prisoner's dilemma.

What is the Metzler paradox?

The Metzler paradox is the case in which a tariff lowers, instead of raises, the domestic price of the imported good. It happens when the world price falls by more than the tariff adds, which requires that foreign demand for the home country's exports is inelastic. The condition is that the foreign elasticity is less than the home marginal propensity to consume its exportable good.

Does the optimum tariff apply to a small country?

No. A small country cannot influence the world price, so a tariff brings it no terms-of-trade gain and only the two deadweight losses. Free trade is its best policy. JNU asks it more than once: free trade is the optimal trading policy for a small country.

The optimum tariff, retaliation and the Metzler paradox

Why a large country can gain

The optimum tariff, derived

The same gain through a quota

Retaliation: the tariff war

The Metzler paradox

The mistakes examiners are fishing for

The mistakes examiners are fishing for

The opening question, solved

The page on one screen

Check yourself

Numbers

Concepts

Past papers, by level

Foundation

Standard

Quick answers

What is the optimum tariff?

Why does a large country gain from a tariff?

What happens if other countries retaliate?

What is the Metzler paradox?

Does the optimum tariff apply to a small country?

Read next

The optimum tariff, retaliation and the Metzler paradox

Why a large country can gain

The optimum tariff, derived

The same gain through a quota

Retaliation: the tariff war

The Metzler paradox

The mistakes examiners are fishing for

The mistakes examiners are fishing for

The opening question, solved

The page on one screen

Check yourself

Numbers

Concepts

Past papers, by level

Foundation

Standard

Quick answers

What is the optimum tariff?

Why does a large country gain from a tariff?

What happens if other countries retaliate?

What is the Metzler paradox?

Does the optimum tariff apply to a small country?

Read next