### Lerner's Symmetry Theorem

Export taxes and import tariff have symmetric effects on trade.

International trade depends on relative price, but trade restriction distorts domestic prices and creates a gap between the two relative prices. In the absence of tariffs or export taxes,

Case I: The home country taxes export.

Domestic Relative Price/Foreign Relative Price:

Case II: The home country imposes a tariff.

Domestic relative price/Foreign relative price

Thus, if t1 = t2, then export taxes and import tariffs have the same effect on trade. However, export taxes are not used in the U.S; they are often used by infant governments or LDCs.

While the symmetry theorem is correct in the two-commodity world, countries trade innumerable commodities in the real world. If trade is balanced, the symmetry theorem can still work. A 10% export tax on all exportable goods has the same effect on trade as 10% tariff on all imports. However, this symmetry theorem is not practical, because trade is almost always not balanced.

The symmetry theorem also does not work in the presence of nontraded goods. A 10% export tax and an equal import tariff creates the same gap, , in the relative price of traded goods, but affects differentially the Corden-Neary exchange rate,

In practice, nontraded goods account for a significant share of GDP, the Lerner symmetry theorem does not work. However, any export tax designed to raise the world price of the exportable necessarily raises the relative prices of the importables and has a protective effect on the import sectors.

### Maximal revenue tariff exceeds optimal tariff

Balance of trade:

where is import of good 2, t is an ad valorem tariff and is tariff revenue. Note that  A change (Δ) in tariff implies a change in p2.

For a small country, p*2 is fixed, and a tariff raises domestic price p2 by the same amount. In the case of a large country, p2 rises also but by a smaller amount. (When the tariff is too high, a further increase in tariff may lower p2.)

Observe that

Maximization of utility => MRS =

Maximization of GDP => MRT =

Thus, QΔp2 = . Since Q > 0, this implies that the price of the importable and tariff revenue move in the same direction, i.e., Δp2 and  have the same signs. Moreover, domestic price rises with the tariff (Δp2/Δt2 > 0). Thus, when the optimal tariff is imposed,

That is, when the optimal tariff is imposed, tariff revenue is still increasing in the tariff rate, and has not reached a maximum.

Figure 14. Maximal revenue tariff exceeds optimal tariff.