A textbook version of a border adjustment has no net impact in the long run. In the very long run, trade between the United States and rest of the world must balance in present discounted value. As a result, imports must equal the present value of all future exports and vice versa. This is because trade represents net borrowing and lending between the US and the rest of the world. What we borrow (import) we have to pay back (export). Economic literature dating to Abba Lerner in 1936 has posited a long-run neutrality between taxing imports and taxing exports, which implies that shifting a system from one treatment to the other would have no impact.
Under a simple long-run model, border adjustments are accounted for in foreign exchange rates, offsetting the tax on imports and the subsidy on exports simultaneously, and leaving the relative prices of imports, exports, and domestic goods all unchanged.
However, in more complex models of the economy, border adjustments can have real economic effects; economists have noted exceptions to the theoretical result, some of which are significant to the US.
There are reasons to believe the dollar would appreciate only partially relative to the theoretical result, especially if the border adjustment fails to capture certain industries or activities in its net. Additionally, a number of transition dynamics—that is, temporary but nonetheless substantial economic effects—would exist if the US were to adopt a border adjustment quickly and unexpectedly. Therefore, the long-run and theoretical impact has significant exceptions.
