It is common for economists to assert that the trade deficit is equal to the gap between national savings and national investment. If the United States invests more than it saves (combining private savings and government savings) then it is running a trade deficit. This is true by definition.
Intro Econ fans may remember that we have the basic accounting identity saying that output is equal to income:
...where C is consumption,
...I is investment,
...G is government spending,
...X-M is net exports (exports minus imports),
and Y is income.
We also can say that Y=S+C+T,
...where S is savings,
...C is consumption,
...and T is taxes.
The basic story is that the government taxes away some of our income and the rest is either saved or consumed (saved means it is not consumed).
This gets us C+I+G+(X-M)=S+C+T
Since we have C on both sides, we can just say I+G+(X-M)=S+T,
...then we subtract I+G from both sides and get (X-M)=(S-I)+(T-G).
This is our basic story where the trade surplus (X-M) is equal to the surplus of private savings (S-I) plus the surplus of public savings (T-G). If we have a trade deficit rather than a trade surplus, then it must mean that the sum of private and public savings is negative.
All of this is 100 percent true, we are in definition land. The problem is when economists try to turn this into a story of causation as we got in Eduardo Porter's NYT piece on trade:
"The American current account deficit — a broad measure of its trade — is the mirror image of the gap between the United States’ national savings and its national investment. Because it invests more than it saves, it draws money from abroad and spends it on foreign goods and services. Until it closes the savings gap, no amount of diplomacy, bullying or cajoling will close the gap in trade. ....
"But even if Mr. Trump’s Nafta gambit worked and bilateral trade came into balance, it wouldn’t necessarily change the balance of American trade over all. ...
"As another Dartmouth economist, Robert Staiger, told me, unless the American savings-investment imbalance corrected itself, too, the former deficit with Mexico would simply pop up somewhere else.
"'Bilateral deficits are going to keep popping up everywhere,' he said. 'Trump is going to be playing Whac-a-Mole.'"
This is a view of the economy that assumes it is at full employment. In other words, an increase in demand cannot lead to an increase in output because the economy lacks the ability to produce more goods and services. In that context, if the trade deficit falls, say because of a rise in exports, it would have to be offset by reduced production in another area such as a fall in investment due to higher interest rates. The economy literally can't produce more since it is at its limits.
However, since the Great Recession, many economists no longer believe that the economy is necessarily close to full employment. The idea of "secular stagnation," which means that the economy is suffering from a shortfall in demand, has been endorsed by numerous prominent economists like Paul Krugman, Larry Summers, and Ben Bernanke.
In this context, the trade deficit is not limited by the balance of domestic savings and investment. If for example, the United States were to suddenly increase exports, then it would lead to more employment and income from the additional wages and profits. While some of this income would be spent, some of it would be saved or taxed away by the government. The identity between net exports and net national savings would still hold, but we would now have a smaller trade deficit and more (less negative) national savings. In this case, national savings is not determining the trade deficit, the trade deficit is determining national savings.
It is important to keep these concepts straight. People can argue that the trade deficit is determined by national savings, but then they believe the economy is typically close to full employment and don't take the concept of secular stagnation seriously. If they do take secular stagnation seriously and still argue that the trade deficit is determined by national savings, then they are confused.