That tariff formula

I really want to get back to blogging. And what a time to do so, as the world collapses around us! So I'll start with a take on the (now partially suspended) Trump tariffs and in particular, that formula...

The formula through which Trump set his "reciprocal" tariffs has been roundly ridiculed, and as a determination for a tariff rate that other countries are charging the USA, it rightly deserves to be. But the administration's technical justification does not claim that it is the tariff that other countries are charging the US. Rather, it mislabels "reciprocal tariffs" as the tariff required to be levied by the USA on its trading partners in order to achieve bilateral trade balance. Does it make sense on this basis?

Kind of... They claim the trade elasticity $ϵ=-4$ [reasonable], and the pass-through rate $\varphi =0.25$ [seems very low to me] (conveniently cancelling each other out, but let's take them at their word). So let's consider the case where $x_i=75$ and $m_i=100$ so that $\mathrm{\Delta }{\tau }_i=25\mathrm{\%}$ and consider what this formula would predict happens:

Firstly, the formula assumes that there is no exchange rate adjustment, which is a fairly heroic assumption. And the formula also assumes no impact on exports, which is again questionable (though perhaps follows from no exchange rate adjustment in the absence of retaliation).

Secondly, the low pass-through rate implies that prices rise by 6.25% (25% tariff times 0.25 pass-through), meaning foreign producers accept a price cut of 18.75% (since the consumer price has to cover payment to the foreign producers and the tariff payment).

And thirdly, the 6.25% price rise combined with the trade elasticity of -4 means that imports fall by 25.

So what is the outcome of this? Well there is bilateral trade balance with exports and imports now both 75. But in income flow terms between the countries this is an overreaction, as the foreign producers are only paid 56.25 of this with the 25% tariff raising 18.75 for the US government. And US consumers won't be too happy: a 6.25% price increase, combined with a 25% drop in the nominal value, means they are enjoying 29.4% fewer imported goods.

Overall then, this formula may be an answer to the question: "what tariff should we charge to eliminate the trade deficit?"; but it relies on some heroic assumptions, and even taking it at face value its imposition implies some unpleasant outcomes.