That's basically the punch line in a column telling us Thomas Piketty is wrong to worry about rising inequality. After a long digression on motivations for saving among the very rich, Mankiw tells readers:
"When a family saves for future generations, it provides resources to finance capital investments, like the start-up of new businesses and the expansion of old ones. Greater capital, in turn, affects the earnings of both existing capital and workers.
"Because capital is subject to diminishing returns, an increase in its supply causes each unit of capital to earn less. And because increased capital raises labor productivity, workers enjoy higher wages. In other words, by saving rather than spending, those who leave an estate to their heirs induce an unintended redistribution of income from other owners of capital toward workers."
To summarize, the story is that by saving rather than spending their money, rich people will make more capital available to firms to invest, thereby raising productivity and wages.
There are two important problems with this story. First, we are operating well below the economy's potential level of output and are likely to remain below potential for many years into the future according to most projections. This is the story of "secular stagnation" that even folks like Larry Summers have embraced in recent years.
In a context of secular stagnation, more saving is harmful. If people save rather than consume there will be less demand in the economy and less employment. If we think that secular stagnation is likely to be a persistent problem, then the fact the rich save more of their money than everyone is bad news for the economy. It will slow growth and make us all poorer.
The other point is that moderate income and middle income people did actually use to save a larger share of their income. Back in the days when wages were keeping pace with productivity growth, savings rates were considerably higher than they have been in the last two decades when the wealthy got most of the benefits of growth. It tends to be the case that people save a larger share of their income when their income is rising rapidly. This means that we don't need rich people to not spend. Moderate and middle income people will also save a substantial portion of their income during prosperous times.