Optimal Tariffs and Retaliation Risk

Why an Optimal Tariff Is a Conditional Result

A large importing country may be able to reduce the price received by foreign exporters by restricting its imports. That improvement in its terms of trade can benefit the country as a whole, even though the tariff also distorts domestic choices and reduces trade. The optimal-tariff argument compares these effects; it does not say that any tariff, or the tariff that maximizes welfare in a one-country calculation, is necessarily desirable once other countries can respond.

The analysis below is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It uses a partial-equilibrium model of one imported good. The model isolates the tariff’s national-welfare effects and treats the rest of the economy as unchanged.

Definition: The optimal tariff is the tariff rate that maximizes the importing country’s national welfare in a specified model. Its potential benefit comes from lowering the foreign price of imports (a terms-of-trade gain); its costs include domestic distortions and, where relevant, the consequences of retaliation. The optimal rate is model-dependent, not a universal recommended policy.

The Terms-of-Trade Gain and the Optimal-Tariff Condition

Let $M$ be the quantity imported, $P_w(M)$ the foreign price received per unit, and $P_d$ the domestic price paid by buyers. With an ad valorem tariff $t$, the price wedge is $P_d=(1+t)P_w$. Let $B(M)$ denote the domestic benefit from the imported good, net of domestic production costs, with marginal benefit $B'(M)=P_d$. In this partial-equilibrium setup, national welfare is domestic benefit less the payment for imports at the foreign price:

$$ W(M)=B(M)-P_w(M)M $$

The tariff is not an additional resource cost in this expression: tariff revenue is paid by domestic buyers and received by the domestic government, so it is a transfer within the country. The foreign price paid on each imported unit is a cost to the country. When imports are reduced, the foreign price may fall too, lowering the amount paid on the remaining imports.

The foreign export-supply elasticity is $\varepsilon_X=(dM/dP_w)(P_w/M)$, evaluated at the relevant equilibrium, and is positive when foreign export supply slopes upward. The reciprocal elasticity measures how strongly the foreign price responds to a change in imports.

Theorem: In the standard one-good, large-importer model with competitive markets and no retaliation or other policy distortions, the interior welfare-maximizing ad valorem tariff satisfies $t^*=1/\varepsilon_X$, where $\varepsilon_X$ is the foreign export-supply elasticity evaluated at the optimum. This condition does not apply to a small country that cannot affect the foreign price, or without modification when other distortions, strategic responses, or policy constraints matter.

The location at which elasticity is evaluated matters. A tariff changes import quantity and the foreign price, so the elasticity at free trade need not equal the elasticity at the optimum. The formula gives a condition at the optimal equilibrium; it is not generally a plug-in rate based on a free-trade elasticity.

Why the Formula Holds

A short derivation makes the trade-off explicit. Since $B'(M)=P_d$, differentiating national welfare with respect to imports gives:

\frac{dW}{dM} =P_d-P_w-M\frac{dP_w}{dM}

At an interior welfare maximum, $dW/dM=0$. The tariff wedge must therefore satisfy $P_d-P_w=M(dP_w/dM)$. Dividing both sides by $P_w$, and using $t=(P_d-P_w)/P_w$, gives:

t=M\frac{1}{P_w}\frac{dP_w}{dM} =\frac{1}{\varepsilon_X}

The condition says that the tariff’s marginal terms-of-trade benefit offsets the marginal domestic distortion at the optimum. It is a first-order condition: it identifies a candidate interior optimum. To establish that it is a maximum, the model must also have the appropriate curvature, and the candidate must be feasible.

Worked Example: Finding the Welfare-Maximizing Tariff

Suppose the inverse import-demand curve is $P_d=50-M/2$, and foreign export supply is $M=2P_w-20$, or equivalently $P_w=10+M/2$. Prices are dollars per unit and quantities are in thousands of units. Find the free-trade equilibrium, the welfare-maximizing tariff, and the national-welfare difference between those outcomes. Assume competitive markets, no retaliation, and no other distortions.

Method. First find free-trade imports where $P_d=P_w$: $50-M/2=10+M/2$, so $M=40$. Substitution gives $P_d=P_w=30$.

The foreign export-supply elasticity at any equilibrium is $\varepsilon_X=(dM/dP_w)(P_w/M)$. Since $M=2P_w-20$, $dM/dP_w=2$. At the optimum, use the welfare condition rather than assuming the free-trade elasticity remains fixed.

Domestic benefit is the area under inverse import demand: $B(M)=\int_0^M(50-m/2)\,dm=50M-M^2/4$. National welfare is therefore $W(M)=50M-M^2/4-(10+M/2)M=40M-3M^2/4$. Differentiating gives $dW/dM=40-3M/2$. Setting this equal to zero gives $M^*=80/3$, or about $26.6667$ thousand units. The second derivative is $-3/2<0$, confirming a maximum.

At this quantity, $P_w=10+(80/3)/2=70/3$, and $P_d=50-(80/3)/2=110/3$. The tariff per unit is $P_d-P_w=40/3$, so the ad valorem rate is $t^*=(40/3)/(70/3)=4/7$, or $57.14\%$ to two decimal places. The elasticity at this optimum is $\varepsilon_X=2(70/3)/(80/3)=7/4$, and $1/\varepsilon_X=4/7$, as required.

At free trade, $W(40)=40(40)-3(40^2)/4=1600-1200=400$ thousand dollars. At the optimum, $W(80/3)=40(80/3)-3(80/3)^2/4=3200/3-1600/3=1600/3$, or $533.3333$ thousand dollars. Thus the model predicts a gain of $1600/3-400=400/3$, or $133.3333$ thousand dollars. The method avoids the common mistake of using the free-trade elasticity as though it were the elasticity at the optimum.

Check: At the calculated optimum, $P_d/P_w=(110/3)/(70/3)=11/7=1+4/7$, so the prices satisfy the stated tariff wedge.

Retaliation Can Reverse the National-Welfare Comparison

A unilateral optimal-tariff calculation holds foreign policy fixed. But a trading partner may retaliate with a tariff on the first country’s exports. That response can reduce export sales and the surplus generated by those sales; it may also bring other costs. These effects are not part of the one-market formula. A terms-of-trade gain in the importing sector is not, by itself, a calculation of the country’s total welfare after retaliation.

Retaliation is a possible response, not an automatic outcome. Its size depends on the partner’s incentives, the products affected, and the resulting market adjustments. A careful comparison states the unilateral gain and the retaliation-related costs separately before combining them.

Worked Example: Separating a Tariff Gain From Retaliation Costs

In a hypothetical two-sector assessment, a unilateral tariff produces an estimated national-welfare gain of 133.333 thousand dollars in the import market, using the assumptions of the previous calculation. A trading partner then retaliates. The resulting decline in domestic exporters’ surplus is estimated at 170 thousand dollars, while an additional domestic adjustment cost is 20 thousand dollars. Treat these amounts as net welfare changes and assume no other effects. What is the overall change relative to the original free-trade benchmark?

Method. Add the separate national-welfare effects, keeping gains positive and losses negative: $133.333-170-20=-56.667$ thousand dollars.

The combined assessment is therefore a loss of about 56.667 thousand dollars relative to the benchmark, or a loss of $\$56,667$ when expressed in dollars to the nearest dollar. The original unilateral gain is real within its stated one-market assumptions, but it is smaller than the stipulated retaliation-related costs.

Check: The retaliation costs total $170+20=190$ thousand dollars, which exceeds the unilateral gain by $190-133.333=56.667$ thousand dollars. A common mistake is to report the unilateral gain as the final national-welfare result while omitting the export-sector loss.

Checking Prices and Quantities Without Rounding Too Early

Tariff calculations often combine a price wedge with demand and supply equations. State what each price means, write the two quantities using those prices, and impose market clearing before rounding. This simple sequence prevents a plausible-looking price from being paired with inconsistent quantities.

Worked Example: Verifying a Price-Wedge Equilibrium

Consider a hypothetical reduced-form market with a producer price $P$, a buyer price $P_b=1.2P$, demand $D=96-2P_b$, and supply $S=2P-16$. Prices are dollars per unit and quantities are in units. Find the market-clearing prices and quantity. The equations are stipulated for this example; they illustrate equilibrium arithmetic for a price wedge rather than a general tariff model.

Method. Substitute the buyer-price relation into demand: $D=96-2(1.2P)=96-2.4P$. Supply is $S=2P-16$. Market clearing requires $D=S$, so $96-2.4P=2P-16$. Adding $2.4P$ and 16 to both sides gives $112=4.4P$, hence $P=112/4.4=280/11$, or approximately $\$25.4545$.

The buyer price is $P_b=1.2(280/11)=336/11$, or approximately $\$30.5455$. Demand is $96-2(336/11)=96-672/11=(1056-672)/11=384/11$, approximately $34.9091$ units. Supply is $2(280/11)-16=560/11-176/11=384/11$, also approximately $34.9091$ units.

The market clears: demand equals supply exactly at $384/11$. Rounding the producer price prematurely to $\$25$ would give demand $96-2(30)=36$ and supply $2(25)-16=34$, which do not match. Keep exact fractions through the equilibrium check; round only the final prices and quantities, here to four decimal places.

Assumptions, Limits, and Common Traps

The optimal-tariff condition is a useful benchmark, but its assumptions matter. It describes a large importer that can influence the foreign price, with competitive markets, a well-defined foreign export-supply curve, and no retaliation or other distortions in the calculation. A small country faces a fixed foreign price, so it has no terms-of-trade gain in this model; its tariff creates a domestic distortion without that offset. The simple condition also does not settle wider questions such as distributional effects, uncertainty, administrative costs, or interactions among several markets.

Keep these distinctions in view when evaluating a proposed tariff:

Do not confuse a national-welfare gain with a gain for every group. Consumers, domestic producers, and the government can experience different effects even when the total for the country rises.
Do not evaluate elasticity at the wrong point. The formula uses the foreign export-supply elasticity at the optimum, which may differ from its free-trade value.
Do not omit the price wedge. For an ad valorem tariff, $P_d=(1+t)P_w$; for a specific tariff $\tau$, $P_d=P_w+\tau$. These are different rate definitions.
Do not treat retaliation as a certainty or as zero by default. State whether the calculation assumes no response, and account for a response separately if the scenario supplies one.

Key takeaway: In the standard large-importer model, the optimal tariff balances a marginal terms-of-trade gain against domestic distortion and satisfies $t^*=1/\varepsilon_X$ at the optimum. That unilateral result is conditional: retaliation can impose costs large enough to eliminate or reverse the predicted national-welfare gain.

Check Your Understanding

Work through each problem using the stated assumptions, and keep exact values until the final step.

At a candidate optimum, foreign export supply is $M=3P_w-30$, with $M=30$ thousand units and $P_w=\$20$ per unit. Find the foreign export-supply elasticity and the optimal ad valorem tariff under the standard no-retaliation model. (Answer: $\varepsilon_X=2$; $t^*=0.50$, or 50%.)
Import demand is $M=90-2P_d$, foreign export supply is $M=2P_w-20$, and an ad valorem tariff is $25\%$. Find the equilibrium foreign price, domestic price, and imports. (Answer: $P_w=\$220/9\approx\$24.4444$, $P_d=\$275/9\approx\$30.5556$, and $M=260/9\approx28.8889$ units.)
A hypothetical one-market calculation gives a unilateral welfare gain of $\$80$ thousand. Retaliation reduces domestic export-sector surplus by $\$65$ thousand and adds $\$25$ thousand of adjustment costs. What is the net change relative to the benchmark? (Answer: $-\$10$ thousand.)
Inverse import demand is $P_d=72-M/3$, and foreign export supply is $P_w=12+M/6$. Under the standard model with no retaliation, find the welfare-maximizing import quantity, the corresponding optimal tariff rate, and welfare at that quantity. Use $W(M)=\int_0^M P_d(m)\,dm-P_w(M)M$. (Answer: $M^*=90$, $P_w=\$27$, $P_d=\$42$, $t^*=5/9\approx55.56\%$, and $W=2700$ thousand dollars.)

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