If the Trump administration imposes a tariff on imports, it will result in a contraction of trade--both imports and exports. Frequently, it is claimed that this will only occur if foreign governments retaliate to tariffs on their exports to the U.S. with tariffs on U.S. exports to their countries. However, that is not correct. There is a market process that brings this about even without "retaliation."
The most direct process occurs with floating exchange rates, which is at least approximately the U.S. regime. The tariff reduces the amount of dollars paid for imported goods. This reduces the supply of dollars on foreign exchange markets and so requires an increase in the value of the dollar for the market to clear. This partly offsets the tariff by making the dollar prices of imported goods less, but it also makes U.S. exports more expensive for the foreign buyers. This results in a decrease in exports.
The U.S. currently has a trade deficit, and the stronger dollar will tend to reduce it. While an expectation of an increased value of the dollar will encourage foreigners to invest in U.S. financial assets, once the dollar is higher, U.S. real assets will be more expensive for foreigners. So, the result will tend to be a lower trade deficit as well as reduced exports.
The U.S. could use monetary policy to prevent the dollar from rising in value. This is done by increasing the quantity of money. The equilibrium consequence of this policy is higher inflation in the U.S. The higher prices in the U.S. will make foreign imports more attractive, partially offsetting the effect of the tariff, but will also make U.S. exports more expensive for foreigners, causing them to purchase less. Similarly, U.S. assets will be more expensive for foreigners to purchase, so that the trade deficit will be smaller.
Like most market monetarists, I think that prices are sticky, and that includes wages. The inflationary process would not be instantaneous or smooth. The increase in demand though out the economy would likely result in increases in output and employment. Wages are also very sticky, even in an upwards direction, so real wages would be depressed which should help employment. That effect would be especially strong in weak areas of the economy--including import competing manufacturing. Only in "the long run" would higher prices and wages result in a return to equilibrium.
It would be possible for the central bank to keep keep the dollar from rising while avoiding inflation by sterilization. The Federal Reserve would need to sell off dollar assets it holds while purchasing foreign exchange--foreign assets. This can last as long as the Fed has U.S. assets to sell. The result should be a reduction in imports and a reduced trade deficit. Unfortunately, the expansion in demand for the products of U.S. import competing industries will be offset by a reduction in the demand for the products of interest-sensitive industries--construction and capital goods. Once the Fed runs out of U.S. assets, allowing the dollar to rise would result in financial losses to the Fed. Perhaps this would lock in the inflationary equilibrium.
It would also be possible to blame the increase in the value of the dollar on currency manipulations by foreigners. A higher dollar is at the same time a lower pound, euro, yen, and renminbi. How dare they devalue their currencies to offset the effects of the tariffs?
Foreign nations could prevent their currencies from losing value by contracting their quantities of money. In the long run, this would result in them having lower prices and wages, and so U.S. buyers would find their imported goods cheap and they would find U.S. exports expensive. While that process would likely be long and painful, the effect on U.S. exports would be prompt. The recession induced by their monetary contraction would result in reduced U.S. sales in their markets.
Finally, they could keep their currencies from losing value by selling off any U.S. assets they hold and instead accumulating other sorts of foreign exchange or else each their own domestic assets. This would tend to shrink the U.S. trade deficit by reducing the amount of foreign funding of U.S. investment. This could last until they run out of U.S. foreign exchange. Again, any expansion in the demand for import competing industries will be offset by a reduction in demand in interest sensitive industries in the U.S.--construction and capital goods.
Changes in the composition of the Fed's balance sheet or the balance sheets of foreign central banks could shield U.S. exports from the decrease in imports for a time. It is only if such adjustments are made that a contraction in U.S. exports would only occur due to retaliation of increased tariffs.