We all know how difficult it is for elite economists at places like Harvard to get information about the economy, so perhaps we should excuse him for this little mess up. Of course, if he had heard of the Great Recession he would not be writing a piece in the New York Times telling us that trade deficits really don't matter:
"Nations run trade deficits when their spending on consumption and investment, both private and public, exceeds the value of goods and services they produce. If you really want to reduce a trade deficit, the way to do it is to bring down spending relative to production, not to demonize trading partners around the world."
If Mankiw had heard about the Great Recession, he would have known that countries often face shortfalls in demand, meaning that we have unemployment because there is not enough demand (i.e. spending on consumption and investment). In this context, if we reduce domestic spending, as Mankiw advocates, it may reduce the trade deficit somewhat, but it will also lead to a further reduction in demand and loss of jobs.
In a context where the economy faces a shortfall of demand, a situation that has become popularly known among non-Mankiw economists as "secular stagnation," a large trade deficit is one of the factors reducing demand in the economy. If we have a trade deficit of 3.0 percent of GDP, it is equivalent to a reduction in domestic consumption equal to 3.0 percentage points of GDP, or the government reducing its spending by 3.0 percentage points of GDP as part of an austerity program.
For this reason, people who see secular stagnation as a problem should be viewing the trade deficit as a problem. That is, at least if they understand basic economics. And, the best remedy is a lower-valued dollar, but we'll get to that another day.
Just to flesh my point a bit. Mankiw is of course right about the accounting identity. The trade deficit is equal to the excess of domestic consumption and investment over domestic savings. The problem is that this accounting identity says nothing about causation.
If an economy is operating below its potential level of output that the additional output stemming from a lower trade deficit can lead to more savings (both private and government) and therefore a lower imbalance between domestic consumption and investment and domestic savings. Without saying it, Mankiw has slipped in the assumption that the economy is already operating at its potential level of output so that a smaller trade defcit cannot boost output. In that context, a smaller trade deficit would lead to higher interest rates, which would crowd out domestic consumption and investment.