The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Optimal Tariffs

  • Nuno Limão
Reference work entry


Optimal tariffs allow a country to exploit its market power in international trade. A country can improve its terms of trade by unilaterally restricting its exports if it faces a downward-sloping demand for them or restricting its imports if it faces an upward-sloping foreign export supply. This argument against unilateral free trade is over 150 years old but it remains central to modern theories that explain trade agreements and their rules. This, along with recent evidence that prior to such agreements countries exploit their market power in trade, shows that optimal tariffs may be an important positive theory of protection.


Corn Laws Cross-elasticities Free trade Imperfect competition Marginal rates of transformation Market power Marketing boards Monopolistic competition Monopsony pricing Optimal tariffs Optimal taxation Smoot Hawley Tariff Act of 1930 (USA) Tariffs Terms of trade Torrens, R. Trade agreements Trade policy, political economy of 

JEL Classifications


A country that faces a downward-sloping demand for its exports has market power and therefore, as a monopolist, can benefit from restricting its export supply. When a country’s exporters are perfectly competitive, the government can coordinate this restriction via an export tax, which increases the world price for its exports and so improves its terms of trade. Analogously, a country facing an upward-sloping export supply has market power in imports and can benefit from restricting them via a tariff. Generally, the optimal tariff is defined as the rate that unilaterally maximizes a country’s welfare and is given by the inverse elasticity of foreign export supply, as determined by optimal monopsony pricing.

The terms-of-trade argument against unilateral free trade is over 150 years old, yet it remains one of the hardest to refute theoretically. The reason is simple. A country’s atomistic consumers impose an externality on each other since, by increasing import demand, they raise the equilibrium price for all. The optimal instrument to correct an externality must target it at the source (Bhagwati and Ramaswami 1963), and the optimal tariff does this by reducing import demand. This quantity reduction entails a cost but, for a sufficiently small tariff, it is more than offset by the improved terms of trade. This is one of the only cases when, in the absence of retaliation, the tariff is a first-best instrument. However, if a tariff improves a country’s terms of trade, it worsens those of its trading partner, who is therefore likely to retaliate. The typical trade war outcome is to leave both worse off relative to free trade, which explains many economists’ opposition to optimal tariffs as a normative theory. The trade war outcome points to the benefits from reciprocal tariff reductions and as such the terms-of-trade argument remains central to modern theories of trade agreements and their rules. This, along with recent evidence that prior to such agreements countries exploit their market power in trade, shows that optimal tariffs may be an important positive theory of protection rather than an irrelevant normative one.

Informal Derivation and Applications

The standard derivation of the optimal tariff focuses on a standard neoclassical economy with no domestic externalities and available lump-sum transfers to address any resulting redistribution issues (see Graaf 1949–50). A Pareto optimum for the closed economy requires the domestic marginal rate of substitution between any two goods i and j to equal their marginal rates of transformation, which in a competitive economy is done via the domestic relative prices, that is, MRSij = pi/pj = MRTij. An open economy can exchange goods at the prevailing world prices, which can be thought of as having access to a new technology or foreign rate of transformation. Now efficient production requires the domestic MRTij to equal the marginal foreign rate of transformation (MFRT). Optimal ad valorem tariffs, ti, imposed on the world prices, πi, ensure that this additional condition for efficiency is met, by introducing a wedge such that the relative price faced by domestic producers is equal to MFRTij, that is, pi/pj = πi(1 + ti)/πj(1 + tj). The final step is to determine the MFRTij, which simply reflects the relative marginal cost of these goods in the world market. The marginal cost for the importer is πi + ai, the price paid for the unit plus the marginal change in price(S) it causes, ai = Σkmkpk/∂mi. Therefore the optimal ad valorem tariff rates are determined by
$$ \frac{\pi_i\left(1+{t}_i\right)}{\pi_j\left(1+{t}_j\right)}=\frac{\pi_i\left(1+{a}_i/{\pi}_i\right)}{\pi_j\left(1+{a}_j/{\pi}_j\right)}\ \mathrm{for}\;\mathrm{all}\;i\;\mathrm{and}\;j, $$
which is satisfied by any tax structure such that
$$ {t}_i=\frac{a_i}{\pi_i} \mathrm{for}\;\mathrm{all}\;i. $$

When all cross-price elasticities are zero, ai/πi is simply the inverse of the foreign export supply for i, 1/εi, and we obtain the standard formula, ti = 1/εi. Otherwise ti also includes the cross-elasticities, as ai captures the weighted sum of marginal world price changes in all goods due to the increase in demand for i.

Since the cross-effects in ai can be negative the optimal tariff may be zero or even a subsidy on any or all goods. However, that can’t be the case with only one import and one export good, i = m and e, as is easily shown if their cross-elasticity is zero. To see this, note that with two goods we can attain the same outcome with either a tariff or an export tax (Lerner 1936), which simultaneously accounts for market power in imports and exports. Solving (1) with te = 0 we have the import tariff rate
$$ {t}_m=\frac{1/{\varepsilon}_m+1/{\varepsilon}_e}{1-1/{\varepsilon}_e}, $$

which is positive given the positively defined elasticities of foreign export supply, εm, and foreign import demand, εe, and 1/εe < 1.

The result extends in several ways under perfect competition settings. If a domestic distortion exists, and is addressed by a first-best instrument, then the rate in (2) is generally still optimal. Graaf (1949–50) shows this for external (dis)economies in production, for example. The equivalence of tariffs and quantity restrictions, that is, quotas, under certainty implies that quotas can be used to the same effect provided that their rents accrue to the country that imposes them (but the welfare and trade volume outcomes of tariff and quota wars differ, as shown by Rodriguez 1974). Kemp (1966) and Jones (1967) derive the optimal tax structure when capital is mobile and a country has market power in goods and factors trade. Similarly to (2) the optimal tariffs on goods take into account their effect on the price of capital and vice versa.

Tariffs can also affect a country’s terms of trade under imperfect competition. However, there are fewer general results in these settings, even in the simpler cases of zero cross-price elasticities. Nonetheless, a few points are worth noting. First, if a country has a monopoly importer or exporter (for example, an agricultural marketing board or a cartel of oil exporters such as OPEC) then there is no first best role for a trade tax – a monopolist would already optimally restrict quantities. A tariff may still be necessary to internalize effects from any cross-elasticities, as Gros (1987) shows for a monopolistic competition model. Second, under imperfect competition a tariff can affect a country’s terms of trade even if it has an infinitesimally small share of the world’s expenditure. For example, if imposing a small ad valorem tariff reduces the import demand elasticity, then it also improves the welfare of a small importer facing a monopoly exporter (Katrak 1977; Brander and Spencer 1984). However, when the country is small the tariff is generally not the first best instrument.

Early Contributions and Current Relevance as a Positive Theory

There have been four important waves in the development and application of the optimal tariffs idea. They were each about 40 to 50 years apart, and at least three appear to be linked to important policy events.

Several early advances in trade theory arose during the debate over the repeal of British import duties imposed by the Corn Laws of 1815. Robert Torrens, a famous classical economist, initially supported the repeal but eventually turned against unilateral free trade as he understood that countries may gain from tariffs through an improvement in their terms of trade. This basic idea and the intuition for it are found in Torrens (1833, 1844) and Mill (1844). However, a country will actually gain only if the terms-of-trade benefit offsets the cost from lower import volume; in a second phase of development, Edgeworth (1894) shows that this is the case unless the foreign country’s offer curve is perfectly elastic, while Bickerdike (1906, 1907) develops the first optimal tariff formula, similar to (3).

Renewed interest in the topic came after the Smoot Hawley Tariff Act of 1930, which raised US average tariffs to about 50 per cent and triggered a cycle of tariff retaliation. The key contributions by Kaldor (1940), Scitovsky (1942) and Johnson (1953–4) focus on the outcomes when countries retaliate. Johnson (1953–4) shows the outcome of a tariff war using tariff reaction curves that summarize a country’s best response. He confirms that two symmetric countries prefer free trade to a trade war; but otherwise one of them may be better off under a trade war.

The latest developments in the topic also came in the wake of important economic events. Mayer (1981) examines the possible tariff outcomes under the tariff cutting formulas used in the 1973–9 multilateral trade negotiations under the General Agreement on Tariffs and Trade (GATT). Since then numerous authors have relied on the tariff war equilibrium as the threat point for the theoretical analysis of multilateral and bilateral trade agreements. Notably, Bagwell and Staiger (1999, 2002) argue that the purpose of the GATT and its successor, the World Trade Organization, is to allow countries to reciprocally lower protection in a way that eliminates the terms-of-trade component of tariffs, and show that such an economic theory of GATT can explain several of its key rules.

Despite the success of the terms-of-trade motive for tariffs in explaining important features of trade agreements, its power as a positive theory of trade protection is often questioned for two reasons. First, governments do not set tariffs to maximize social welfare. Although governments often set tariffs to redistribute income across interest groups, this does not imply that tariffs will not reflect market power. For example, Johnson (1950) derives the revenue-maximizing tariff rate, which does not maximize welfare but is nonetheless increasing in a country’s market power since a given tariff rate yields higher revenue, under a less elastic export supply. Moreover, recent micro-founded political economy models predict that a large country’s unilateral tariff reflects its market power, even if the government places no weight on social welfare (Grossman and Helpman 1995). Thus, even if the primary objective of the government is a political economy one, its tariffs can reflect market power since this allows it to achieve that objective at a lower cost as it captures some income from its trading partners via improved terms of trade.

The second critique is the argument that most countries cannot affect their terms of trade and so it is not an important determinant of protection. Critics concede that certain commodity exporters do have some market power and have at times exerted it (for example, OPEC’s oil restrictions or export taxes by marketing boards). But evidence of market power in exports appears to go beyond these obvious cases since aggregate estimates of εe in (3) are often found to be low, sometimes close to unity. Nonetheless, there are considerable difficulties in estimating and interpreting such aggregate elasticities, which are often estimated only for countries already setting their tariffs cooperatively. Therefore cross-country comparisons of average tariffs and these aggregate elasticities cannot provide much insight into the empirical importance of the terms-of-trade motive.

There is also growing evidence of market power in imports since when countries change their exchange rates or tariffs part of the effect is absorbed by the foreign exporters (cf. Kreinin 1961). Broda et al. (2006) provide compelling evidence that countries have and exploit their market power. They estimate inverse foreign export supply elasticities by good and country, and find that even small countries have some market power, which is increasing in country size and degree of good differentiation. They then examine tariffs for countries that are not setting them cooperatively and find that they are set higher in goods with higher inverse elasticities. They conclude that market power is an economically and statistically important determinant of tariffs.

In sum, optimal tariffs are evolving from a curious normative theory to a positive one. The broad applicability of the terms-of-trade motive for tariffs; its theoretical success in explaining important rules of trade agreements; and the recent evidence that countries exploit their market power, all indicate this will remain a key concept in economics.

See Also


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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Nuno Limão
    • 1
  1. 1.