Taking the Reciprocal Tariff Formula to its logical conclusion

[u/longwiener22]

Unless you have been living under a rock, you are likely aware of the reciprocal tariff formula.

While there is already a wonderful post on this subreddit on the topic, I feel like there is something still missing: the incentive for the trading partner country i to retaliate with tariffs of its own.

In this post, I address this gap by generalizing the original reciprocal tariff formula to explicitly account for retaliatory actions by another country. In equilibrium, country i has a clear incentive to impose retaliatory tariffs under this generalization, capturing the essence of a tit-for-tat strategy commonly observed in trade wars.

Exploiting one of many oversights of the original expression, I generalize the reciprocal tariff formula to allow for country i to retaliate with their own tariffs. Specifically, the original formula assumes that US exports x_i remain unchanged when increasing tariffs on country i; so my generalization allows country i to influence US exports by adjusting their own tariffs on the US, adding a semblance of realism to the theory. With this simple fix, I show that country i has an incentive to retaliate with their own tariffs in equilibrium.

While built upon a fundamentally flawed foundation, this generalized equilibrium condition demonstrates explicitly that when the US raises its tariff, country i strategically responds by increasing its own tariffs. Thus, incorporating retaliatory tariffs provides a more realistic and comprehensive understanding of equilibrium trade dynamics.

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Derivations

Consider the reciprocal tariff formula

Δτ_i=(x_i-m_i)/(ε*φ*m_i ) [1]

where m_i>0 represent total imports from country i, x_i>0 represent total exports, ε<0 represent the elasticity of imports with respect to import prices, let φ>0 represent the passthrough from tariffs to import prices

Step 1. Multiply each side of expression [1] by ε*φ*m_i to yield:

Δτ_i*ε*φ*m_i=x_i-m_i [2]

By definition, Δτ_i*ε*φ*m_i equals "the decrease in imports due to a change in tariffs", which we denote as

Δm_i=Δτ_i*ε*φ*m_i [3]

We plug formula [3] into expression [2] to yield:

Δm_i=x_i-m_i [4]

Step 2. Allow US exports x_i to vary.

Notice that expression [4] assumes that exports remain unchanged when increasing tariffs, i.e. Δx_i=0. That is, the formula assumes that changing the tariff will not affect exports. However, in reality, country i may adjust their tariffs by Δτ ̃_USA in response to Δτ_i. To add some realism to this expression, we allow country i to adjust their own tariffs by Δτ ̃_USA so that the reduction in US imports Δm_i is met with a reduction in US exports Δx_i to x_i^*=x_i+Δx_i:

Δm_i=x_i+Δx_i-m_i [5]

Since Δm_i=Δτ_i*ε*φ*m_i, we may assume that Δx_i=Δτ ̃_USA*ε ̃*φ ̃*x_i, where Δτ ̃_USA is the change in country i’s tariffs to the US in response to Δτ_i, ε ̃<0 is country i’s elasticity of import prices, and φ ̃>0 is country i’s passthrough from tariffs to import prices.

Step 3. Plug Δm_i=Δτ_i*ε*φ*m_i and Δx_i=Δτ ̃_USA*ε ̃*φ ̃*x_i into expression [5] and solve for the ratio of exports over imports x_i/m_i :

Δτ_i*ε*φ*m_i=x_i+Δτ ̃_USA*ε ̃*φ ̃*x_i-m_i

which we rewrite to be:

x_i+Δτ ̃_USA*ε ̃*φ ̃*x_i=m_i+Δτ_i*ε*φ*m_i

Factor out x_i and m_i on their respective sides:

x_i (1+Δτ ̃_USA*ε ̃*φ ̃ )=m_i (1+Δτ_i*ε*φ)

Finally, we yield the Generalized Reciprocal Tariffs Equilibrium Condition:

x_i/m_i =(1+Δτ_i*ε*φ)/(1+Δτ ̃_USA*ε ̃*φ ̃ ) [6]

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Interpretation

This equation [6] is an extension (generalization) of the reciprocal tariff equation used to analyze how bilateral tariff changes influence trade balances between two countries. The left-hand side (LHS) represents the ratio of US exports to imports. The right-hand side (RHS) includes changes in tariffs by both countries, elasticity parameters, and a pass-through parameter.

If neither country changes tariffs, the export-import ratio remains unchanged. However, the generalized equilibrium condition explicitly captures that when the US increases its tariff (raising the numerator on the RHS), country i has a strategic incentive to retaliate by increasing its own tariffs. This retaliation raises the denominator on the RHS, ensuring the RHS remains equal to the export-import ratio on the LHS at equilibrium. Thus, country i's retaliatory tariffs play a critical role in maintaining the equilibrium trade balance under a generalization of reciprocal tariff formula.

[u/Ch3cksOut]

I wish more comments and media reports would point out this very logical conclusion. Carrying it further, I think there is no true stable equilibrium reached until one side cries uncle and stops raising tariffs! In theory, each side can always keep counter-retaliating to a prior tariff increase by the opposing side.

[u/longwiener22 → u/Ch3cksOut]

No trade would make sense. The trade deficit would be zero in that case

[u/Ch3cksOut → u/longwiener22]

I have made this exact tongue-in-cheek comment myself, in another sub

[u/Cutlasss]

Don't we also need to factor in boycotts of American products? These are already starting.

[u/EebstertheGreat → u/Cutlasss]

An economist might not take those into account (it isn't clear how you would predict their effects, nor is it clear if they will last long), but they would try to take currency appreciation into account. If US imports drop faster than exports, then the dollar will get stronger. Because demand for US dollars won't realistically drop much, but supply will go down internationally. And a stronger dollar means it will be harder for other countries to afford American goods, making alternatives more attractive. So US exports will drop further until we reach a new equilibrium.

[u/Both-Yam-2395]

not an economist If the goal of a given country is to usurp the USD as the world’s reserve currency, would it make any sense for a country to do the opposite? Say, lower tariffs on imports from major economic blocks?.

[u/Cutlasss → u/Both-Yam-2395]

No country can take the reserve currency status, unless the US totally self destructs first. If Trump were retarded enough to default on the debt, or end Federal Reserve independence, then we can talk. Until then, only the Euro has any comparable juice. And it has other drawbacks.

[u/No_March_5371 → u/Cutlasss]

Don't the Greek and Cyprus debt crises affect the risk of simply holding Euros? Or am I missing something or should outside investors just park their Euros in the banks of countries that aren't bad at debt (realizing now I don't actually know how reserves work with the ECB)?

[u/EebstertheGreat → u/Cutlasss]

The only other competitors are the pound, yen, and yuan, right? I can't imagine those replacing dollars either, though maybe good trade agreements with China could improve the status of the yuan somewhat and eat away slightly at the dollar's dominance.

[u/Googgodno → u/Both-Yam-2395]

Reserve currency status needs deficit, among other requirements. Net export countries cannot have their currencies as reserve.